Integrand size = 26, antiderivative size = 529 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^4 \left (d+c^2 d x^2\right )^3} \, dx=-\frac {b^2 c^2}{2 d^3 x}+\frac {b^2 c^2}{6 d^3 x \left (1+c^2 x^2\right )}+\frac {b^2 c^4 x}{12 d^3 \left (1+c^2 x^2\right )}-\frac {b c^3 (a+b \text {arcsinh}(c x))}{6 d^3 \left (1+c^2 x^2\right )^{3/2}}-\frac {b c (a+b \text {arcsinh}(c x))}{3 d^3 x^2 \left (1+c^2 x^2\right )^{3/2}}+\frac {29 b c^3 (a+b \text {arcsinh}(c x))}{12 d^3 \sqrt {1+c^2 x^2}}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d^3 x^3 \left (1+c^2 x^2\right )^2}+\frac {7 c^2 (a+b \text {arcsinh}(c x))^2}{3 d^3 x \left (1+c^2 x^2\right )^2}+\frac {35 c^4 x (a+b \text {arcsinh}(c x))^2}{12 d^3 \left (1+c^2 x^2\right )^2}+\frac {35 c^4 x (a+b \text {arcsinh}(c x))^2}{8 d^3 \left (1+c^2 x^2\right )}+\frac {35 c^3 (a+b \text {arcsinh}(c x))^2 \arctan \left (e^{\text {arcsinh}(c x)}\right )}{4 d^3}-\frac {17 b^2 c^3 \arctan (c x)}{6 d^3}+\frac {38 b c^3 (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{3 d^3}+\frac {19 b^2 c^3 \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )}{3 d^3}-\frac {35 i b c^3 (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{4 d^3}+\frac {35 i b c^3 (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{4 d^3}-\frac {19 b^2 c^3 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )}{3 d^3}+\frac {35 i b^2 c^3 \operatorname {PolyLog}\left (3,-i e^{\text {arcsinh}(c x)}\right )}{4 d^3}-\frac {35 i b^2 c^3 \operatorname {PolyLog}\left (3,i e^{\text {arcsinh}(c x)}\right )}{4 d^3} \]
-1/2*b^2*c^2/d^3/x+1/6*b^2*c^2/d^3/x/(c^2*x^2+1)+1/12*b^2*c^4*x/d^3/(c^2*x ^2+1)-1/6*b*c^3*(a+b*arcsinh(c*x))/d^3/(c^2*x^2+1)^(3/2)-1/3*b*c*(a+b*arcs inh(c*x))/d^3/x^2/(c^2*x^2+1)^(3/2)-1/3*(a+b*arcsinh(c*x))^2/d^3/x^3/(c^2* x^2+1)^2+7/3*c^2*(a+b*arcsinh(c*x))^2/d^3/x/(c^2*x^2+1)^2+35/12*c^4*x*(a+b *arcsinh(c*x))^2/d^3/(c^2*x^2+1)^2+35/8*c^4*x*(a+b*arcsinh(c*x))^2/d^3/(c^ 2*x^2+1)+35/4*c^3*(a+b*arcsinh(c*x))^2*arctan(c*x+(c^2*x^2+1)^(1/2))/d^3-1 7/6*b^2*c^3*arctan(c*x)/d^3+38/3*b*c^3*(a+b*arcsinh(c*x))*arctanh(c*x+(c^2 *x^2+1)^(1/2))/d^3+19/3*b^2*c^3*polylog(2,-c*x-(c^2*x^2+1)^(1/2))/d^3-35/4 *I*b*c^3*(a+b*arcsinh(c*x))*polylog(2,-I*(c*x+(c^2*x^2+1)^(1/2)))/d^3+35/4 *I*b*c^3*(a+b*arcsinh(c*x))*polylog(2,I*(c*x+(c^2*x^2+1)^(1/2)))/d^3-19/3* b^2*c^3*polylog(2,c*x+(c^2*x^2+1)^(1/2))/d^3+35/4*I*b^2*c^3*polylog(3,-I*( c*x+(c^2*x^2+1)^(1/2)))/d^3-35/4*I*b^2*c^3*polylog(3,I*(c*x+(c^2*x^2+1)^(1 /2)))/d^3+29/12*b*c^3*(a+b*arcsinh(c*x))/d^3/(c^2*x^2+1)^(1/2)
Time = 9.00 (sec) , antiderivative size = 937, normalized size of antiderivative = 1.77 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^4 \left (d+c^2 d x^2\right )^3} \, dx=-\frac {a^2}{3 d^3 x^3}+\frac {3 a^2 c^2}{d^3 x}+\frac {a^2 c^4 x}{4 d^3 \left (1+c^2 x^2\right )^2}+\frac {11 a^2 c^4 x}{8 d^3 \left (1+c^2 x^2\right )}+\frac {35 a^2 c^3 \arctan (c x)}{8 d^3}+\frac {2 a b \left (-\frac {c \sqrt {1+c^2 x^2}}{6 x^2}+\frac {i c^3 \left ((2 i-c x) \sqrt {1+c^2 x^2}-3 \text {arcsinh}(c x)\right )}{48 (-i+c x)^2}-\frac {11 c^3 \left (\sqrt {1+c^2 x^2}+i \text {arcsinh}(c x)\right )}{16 (-1-i c x)}-\frac {\text {arcsinh}(c x)}{3 x^3}+\frac {11 c^4 \left (i \sqrt {1+c^2 x^2}+\text {arcsinh}(c x)\right )}{16 \left (i c+c^2 x\right )}+\frac {i c^3 \left ((2 i+c x) \sqrt {1+c^2 x^2}+3 \text {arcsinh}(c x)\right )}{48 (i+c x)^2}+\frac {1}{6} c^3 \text {arctanh}\left (\sqrt {1+c^2 x^2}\right )-3 c^2 \left (-\frac {\text {arcsinh}(c x)}{x}-c \text {arctanh}\left (\sqrt {1+c^2 x^2}\right )\right )-\frac {35}{16} i c^4 \left (-\frac {\text {arcsinh}(c x)^2}{2 c}+\frac {2 \text {arcsinh}(c x) \log \left (1+i e^{\text {arcsinh}(c x)}\right )}{c}+\frac {2 \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{c}\right )+\frac {35}{16} i c^4 \left (-\frac {\text {arcsinh}(c x)^2}{2 c}+\frac {2 \text {arcsinh}(c x) \log \left (1-i e^{\text {arcsinh}(c x)}\right )}{c}+\frac {2 \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{c}\right )\right )}{d^3}+\frac {b^2 c^3 \left (-\frac {2 c x}{1+c^2 x^2}+\frac {4 \text {arcsinh}(c x)}{\left (1+c^2 x^2\right )^{3/2}}+\frac {66 \text {arcsinh}(c x)}{\sqrt {1+c^2 x^2}}+\frac {6 c x \text {arcsinh}(c x)^2}{\left (1+c^2 x^2\right )^2}+\frac {33 c x \text {arcsinh}(c x)^2}{1+c^2 x^2}-136 \arctan \left (\tanh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )-4 \coth \left (\frac {1}{2} \text {arcsinh}(c x)\right )+38 \text {arcsinh}(c x)^2 \coth \left (\frac {1}{2} \text {arcsinh}(c x)\right )-2 \text {arcsinh}(c x) \text {csch}^2\left (\frac {1}{2} \text {arcsinh}(c x)\right )-\frac {1}{2} c x \text {arcsinh}(c x)^2 \text {csch}^4\left (\frac {1}{2} \text {arcsinh}(c x)\right )-152 \text {arcsinh}(c x) \log \left (1-e^{-\text {arcsinh}(c x)}\right )-105 i \text {arcsinh}(c x)^2 \log \left (1-i e^{-\text {arcsinh}(c x)}\right )+105 i \text {arcsinh}(c x)^2 \log \left (1+i e^{-\text {arcsinh}(c x)}\right )+152 \text {arcsinh}(c x) \log \left (1+e^{-\text {arcsinh}(c x)}\right )-152 \operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(c x)}\right )-210 i \text {arcsinh}(c x) \operatorname {PolyLog}\left (2,-i e^{-\text {arcsinh}(c x)}\right )+210 i \text {arcsinh}(c x) \operatorname {PolyLog}\left (2,i e^{-\text {arcsinh}(c x)}\right )+152 \operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(c x)}\right )-210 i \operatorname {PolyLog}\left (3,-i e^{-\text {arcsinh}(c x)}\right )+210 i \operatorname {PolyLog}\left (3,i e^{-\text {arcsinh}(c x)}\right )-2 \text {arcsinh}(c x) \text {sech}^2\left (\frac {1}{2} \text {arcsinh}(c x)\right )-\frac {8 \text {arcsinh}(c x)^2 \sinh ^4\left (\frac {1}{2} \text {arcsinh}(c x)\right )}{c^3 x^3}+4 \tanh \left (\frac {1}{2} \text {arcsinh}(c x)\right )-38 \text {arcsinh}(c x)^2 \tanh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )}{24 d^3} \]
-1/3*a^2/(d^3*x^3) + (3*a^2*c^2)/(d^3*x) + (a^2*c^4*x)/(4*d^3*(1 + c^2*x^2 )^2) + (11*a^2*c^4*x)/(8*d^3*(1 + c^2*x^2)) + (35*a^2*c^3*ArcTan[c*x])/(8* d^3) + (2*a*b*(-1/6*(c*Sqrt[1 + c^2*x^2])/x^2 + ((I/48)*c^3*((2*I - c*x)*S qrt[1 + c^2*x^2] - 3*ArcSinh[c*x]))/(-I + c*x)^2 - (11*c^3*(Sqrt[1 + c^2*x ^2] + I*ArcSinh[c*x]))/(16*(-1 - I*c*x)) - ArcSinh[c*x]/(3*x^3) + (11*c^4* (I*Sqrt[1 + c^2*x^2] + ArcSinh[c*x]))/(16*(I*c + c^2*x)) + ((I/48)*c^3*((2 *I + c*x)*Sqrt[1 + c^2*x^2] + 3*ArcSinh[c*x]))/(I + c*x)^2 + (c^3*ArcTanh[ Sqrt[1 + c^2*x^2]])/6 - 3*c^2*(-(ArcSinh[c*x]/x) - c*ArcTanh[Sqrt[1 + c^2* x^2]]) - ((35*I)/16)*c^4*(-1/2*ArcSinh[c*x]^2/c + (2*ArcSinh[c*x]*Log[1 + I*E^ArcSinh[c*x]])/c + (2*PolyLog[2, (-I)*E^ArcSinh[c*x]])/c) + ((35*I)/16 )*c^4*(-1/2*ArcSinh[c*x]^2/c + (2*ArcSinh[c*x]*Log[1 - I*E^ArcSinh[c*x]])/ c + (2*PolyLog[2, I*E^ArcSinh[c*x]])/c)))/d^3 + (b^2*c^3*((-2*c*x)/(1 + c^ 2*x^2) + (4*ArcSinh[c*x])/(1 + c^2*x^2)^(3/2) + (66*ArcSinh[c*x])/Sqrt[1 + c^2*x^2] + (6*c*x*ArcSinh[c*x]^2)/(1 + c^2*x^2)^2 + (33*c*x*ArcSinh[c*x]^ 2)/(1 + c^2*x^2) - 136*ArcTan[Tanh[ArcSinh[c*x]/2]] - 4*Coth[ArcSinh[c*x]/ 2] + 38*ArcSinh[c*x]^2*Coth[ArcSinh[c*x]/2] - 2*ArcSinh[c*x]*Csch[ArcSinh[ c*x]/2]^2 - (c*x*ArcSinh[c*x]^2*Csch[ArcSinh[c*x]/2]^4)/2 - 152*ArcSinh[c* x]*Log[1 - E^(-ArcSinh[c*x])] - (105*I)*ArcSinh[c*x]^2*Log[1 - I/E^ArcSinh [c*x]] + (105*I)*ArcSinh[c*x]^2*Log[1 + I/E^ArcSinh[c*x]] + 152*ArcSinh[c* x]*Log[1 + E^(-ArcSinh[c*x])] - 152*PolyLog[2, -E^(-ArcSinh[c*x])] - (2...
Time = 6.15 (sec) , antiderivative size = 727, normalized size of antiderivative = 1.37, number of steps used = 29, number of rules used = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.077, Rules used = {6224, 27, 6224, 253, 264, 216, 6203, 6203, 6204, 3042, 4668, 3011, 2720, 6213, 215, 216, 6226, 215, 216, 6226, 216, 6231, 3042, 26, 4670, 2715, 2838, 7143}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {(a+b \text {arcsinh}(c x))^2}{x^4 \left (c^2 d x^2+d\right )^3} \, dx\) |
\(\Big \downarrow \) 6224 |
\(\displaystyle -\frac {7}{3} c^2 \int \frac {(a+b \text {arcsinh}(c x))^2}{d^3 x^2 \left (c^2 x^2+1\right )^3}dx+\frac {2 b c \int \frac {a+b \text {arcsinh}(c x)}{x^3 \left (c^2 x^2+1\right )^{5/2}}dx}{3 d^3}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d^3 x^3 \left (c^2 x^2+1\right )^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {7 c^2 \int \frac {(a+b \text {arcsinh}(c x))^2}{x^2 \left (c^2 x^2+1\right )^3}dx}{3 d^3}+\frac {2 b c \int \frac {a+b \text {arcsinh}(c x)}{x^3 \left (c^2 x^2+1\right )^{5/2}}dx}{3 d^3}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d^3 x^3 \left (c^2 x^2+1\right )^2}\) |
\(\Big \downarrow \) 6224 |
\(\displaystyle -\frac {7 c^2 \left (-5 c^2 \int \frac {(a+b \text {arcsinh}(c x))^2}{\left (c^2 x^2+1\right )^3}dx+2 b c \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{5/2}}dx-\frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 x^2+1\right )^2}\right )}{3 d^3}+\frac {2 b c \left (-\frac {5}{2} c^2 \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{5/2}}dx+\frac {1}{2} b c \int \frac {1}{x^2 \left (c^2 x^2+1\right )^2}dx-\frac {a+b \text {arcsinh}(c x)}{2 x^2 \left (c^2 x^2+1\right )^{3/2}}\right )}{3 d^3}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d^3 x^3 \left (c^2 x^2+1\right )^2}\) |
\(\Big \downarrow \) 253 |
\(\displaystyle -\frac {7 c^2 \left (-5 c^2 \int \frac {(a+b \text {arcsinh}(c x))^2}{\left (c^2 x^2+1\right )^3}dx+2 b c \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{5/2}}dx-\frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 x^2+1\right )^2}\right )}{3 d^3}+\frac {2 b c \left (-\frac {5}{2} c^2 \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{5/2}}dx+\frac {1}{2} b c \left (\frac {3}{2} \int \frac {1}{x^2 \left (c^2 x^2+1\right )}dx+\frac {1}{2 x \left (c^2 x^2+1\right )}\right )-\frac {a+b \text {arcsinh}(c x)}{2 x^2 \left (c^2 x^2+1\right )^{3/2}}\right )}{3 d^3}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d^3 x^3 \left (c^2 x^2+1\right )^2}\) |
\(\Big \downarrow \) 264 |
\(\displaystyle -\frac {7 c^2 \left (-5 c^2 \int \frac {(a+b \text {arcsinh}(c x))^2}{\left (c^2 x^2+1\right )^3}dx+2 b c \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{5/2}}dx-\frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 x^2+1\right )^2}\right )}{3 d^3}+\frac {2 b c \left (-\frac {5}{2} c^2 \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{5/2}}dx+\frac {1}{2} b c \left (\frac {3}{2} \left (c^2 \left (-\int \frac {1}{c^2 x^2+1}dx\right )-\frac {1}{x}\right )+\frac {1}{2 x \left (c^2 x^2+1\right )}\right )-\frac {a+b \text {arcsinh}(c x)}{2 x^2 \left (c^2 x^2+1\right )^{3/2}}\right )}{3 d^3}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d^3 x^3 \left (c^2 x^2+1\right )^2}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {2 b c \left (-\frac {5}{2} c^2 \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{5/2}}dx-\frac {a+b \text {arcsinh}(c x)}{2 x^2 \left (c^2 x^2+1\right )^{3/2}}+\frac {1}{2} b c \left (\frac {3}{2} \left (-c \arctan (c x)-\frac {1}{x}\right )+\frac {1}{2 x \left (c^2 x^2+1\right )}\right )\right )}{3 d^3}-\frac {7 c^2 \left (-5 c^2 \int \frac {(a+b \text {arcsinh}(c x))^2}{\left (c^2 x^2+1\right )^3}dx+2 b c \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{5/2}}dx-\frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 x^2+1\right )^2}\right )}{3 d^3}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d^3 x^3 \left (c^2 x^2+1\right )^2}\) |
\(\Big \downarrow \) 6203 |
\(\displaystyle \frac {2 b c \left (-\frac {5}{2} c^2 \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{5/2}}dx-\frac {a+b \text {arcsinh}(c x)}{2 x^2 \left (c^2 x^2+1\right )^{3/2}}+\frac {1}{2} b c \left (\frac {3}{2} \left (-c \arctan (c x)-\frac {1}{x}\right )+\frac {1}{2 x \left (c^2 x^2+1\right )}\right )\right )}{3 d^3}-\frac {7 c^2 \left (-5 c^2 \left (-\frac {1}{2} b c \int \frac {x (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^{5/2}}dx+\frac {3}{4} \int \frac {(a+b \text {arcsinh}(c x))^2}{\left (c^2 x^2+1\right )^2}dx+\frac {x (a+b \text {arcsinh}(c x))^2}{4 \left (c^2 x^2+1\right )^2}\right )+2 b c \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{5/2}}dx-\frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 x^2+1\right )^2}\right )}{3 d^3}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d^3 x^3 \left (c^2 x^2+1\right )^2}\) |
\(\Big \downarrow \) 6203 |
\(\displaystyle \frac {2 b c \left (-\frac {5}{2} c^2 \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{5/2}}dx-\frac {a+b \text {arcsinh}(c x)}{2 x^2 \left (c^2 x^2+1\right )^{3/2}}+\frac {1}{2} b c \left (\frac {3}{2} \left (-c \arctan (c x)-\frac {1}{x}\right )+\frac {1}{2 x \left (c^2 x^2+1\right )}\right )\right )}{3 d^3}-\frac {7 c^2 \left (-5 c^2 \left (-\frac {1}{2} b c \int \frac {x (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^{5/2}}dx+\frac {3}{4} \left (-b c \int \frac {x (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^{3/2}}dx+\frac {1}{2} \int \frac {(a+b \text {arcsinh}(c x))^2}{c^2 x^2+1}dx+\frac {x (a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}\right )+\frac {x (a+b \text {arcsinh}(c x))^2}{4 \left (c^2 x^2+1\right )^2}\right )+2 b c \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{5/2}}dx-\frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 x^2+1\right )^2}\right )}{3 d^3}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d^3 x^3 \left (c^2 x^2+1\right )^2}\) |
\(\Big \downarrow \) 6204 |
\(\displaystyle \frac {2 b c \left (-\frac {5}{2} c^2 \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{5/2}}dx-\frac {a+b \text {arcsinh}(c x)}{2 x^2 \left (c^2 x^2+1\right )^{3/2}}+\frac {1}{2} b c \left (\frac {3}{2} \left (-c \arctan (c x)-\frac {1}{x}\right )+\frac {1}{2 x \left (c^2 x^2+1\right )}\right )\right )}{3 d^3}-\frac {7 c^2 \left (-5 c^2 \left (-\frac {1}{2} b c \int \frac {x (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^{5/2}}dx+\frac {3}{4} \left (-b c \int \frac {x (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^{3/2}}dx+\frac {\int \frac {(a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}d\text {arcsinh}(c x)}{2 c}+\frac {x (a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}\right )+\frac {x (a+b \text {arcsinh}(c x))^2}{4 \left (c^2 x^2+1\right )^2}\right )+2 b c \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{5/2}}dx-\frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 x^2+1\right )^2}\right )}{3 d^3}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d^3 x^3 \left (c^2 x^2+1\right )^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 b c \left (-\frac {5}{2} c^2 \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{5/2}}dx-\frac {a+b \text {arcsinh}(c x)}{2 x^2 \left (c^2 x^2+1\right )^{3/2}}+\frac {1}{2} b c \left (\frac {3}{2} \left (-c \arctan (c x)-\frac {1}{x}\right )+\frac {1}{2 x \left (c^2 x^2+1\right )}\right )\right )}{3 d^3}-\frac {7 c^2 \left (2 b c \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{5/2}}dx-5 c^2 \left (-\frac {1}{2} b c \int \frac {x (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^{5/2}}dx+\frac {3}{4} \left (-b c \int \frac {x (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^{3/2}}dx+\frac {\int (a+b \text {arcsinh}(c x))^2 \csc \left (i \text {arcsinh}(c x)+\frac {\pi }{2}\right )d\text {arcsinh}(c x)}{2 c}+\frac {x (a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}\right )+\frac {x (a+b \text {arcsinh}(c x))^2}{4 \left (c^2 x^2+1\right )^2}\right )-\frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 x^2+1\right )^2}\right )}{3 d^3}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d^3 x^3 \left (c^2 x^2+1\right )^2}\) |
\(\Big \downarrow \) 4668 |
\(\displaystyle \frac {2 b c \left (-\frac {5}{2} c^2 \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{5/2}}dx-\frac {a+b \text {arcsinh}(c x)}{2 x^2 \left (c^2 x^2+1\right )^{3/2}}+\frac {1}{2} b c \left (\frac {3}{2} \left (-c \arctan (c x)-\frac {1}{x}\right )+\frac {1}{2 x \left (c^2 x^2+1\right )}\right )\right )}{3 d^3}-\frac {7 c^2 \left (-5 c^2 \left (\frac {3}{4} \left (\frac {-2 i b \int (a+b \text {arcsinh}(c x)) \log \left (1-i e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)+2 i b \int (a+b \text {arcsinh}(c x)) \log \left (1+i e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2}{2 c}-b c \int \frac {x (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^{3/2}}dx+\frac {x (a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}\right )-\frac {1}{2} b c \int \frac {x (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^{5/2}}dx+\frac {x (a+b \text {arcsinh}(c x))^2}{4 \left (c^2 x^2+1\right )^2}\right )+2 b c \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{5/2}}dx-\frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 x^2+1\right )^2}\right )}{3 d^3}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d^3 x^3 \left (c^2 x^2+1\right )^2}\) |
\(\Big \downarrow \) 3011 |
\(\displaystyle -\frac {7 c^2 \left (-5 c^2 \left (\frac {3}{4} \left (\frac {2 i b \left (b \int \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)-\operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-2 i b \left (b \int \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)-\operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2}{2 c}-b c \int \frac {x (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^{3/2}}dx+\frac {x (a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}\right )-\frac {1}{2} b c \int \frac {x (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^{5/2}}dx+\frac {x (a+b \text {arcsinh}(c x))^2}{4 \left (c^2 x^2+1\right )^2}\right )+2 b c \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{5/2}}dx-\frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 x^2+1\right )^2}\right )}{3 d^3}+\frac {2 b c \left (-\frac {5}{2} c^2 \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{5/2}}dx-\frac {a+b \text {arcsinh}(c x)}{2 x^2 \left (c^2 x^2+1\right )^{3/2}}+\frac {1}{2} b c \left (\frac {3}{2} \left (-c \arctan (c x)-\frac {1}{x}\right )+\frac {1}{2 x \left (c^2 x^2+1\right )}\right )\right )}{3 d^3}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d^3 x^3 \left (c^2 x^2+1\right )^2}\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle -\frac {7 c^2 \left (-5 c^2 \left (\frac {3}{4} \left (\frac {2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2}{2 c}-b c \int \frac {x (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^{3/2}}dx+\frac {x (a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}\right )-\frac {1}{2} b c \int \frac {x (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^{5/2}}dx+\frac {x (a+b \text {arcsinh}(c x))^2}{4 \left (c^2 x^2+1\right )^2}\right )+2 b c \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{5/2}}dx-\frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 x^2+1\right )^2}\right )}{3 d^3}+\frac {2 b c \left (-\frac {5}{2} c^2 \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{5/2}}dx-\frac {a+b \text {arcsinh}(c x)}{2 x^2 \left (c^2 x^2+1\right )^{3/2}}+\frac {1}{2} b c \left (\frac {3}{2} \left (-c \arctan (c x)-\frac {1}{x}\right )+\frac {1}{2 x \left (c^2 x^2+1\right )}\right )\right )}{3 d^3}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d^3 x^3 \left (c^2 x^2+1\right )^2}\) |
\(\Big \downarrow \) 6213 |
\(\displaystyle -\frac {7 c^2 \left (-5 c^2 \left (\frac {3}{4} \left (\frac {2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2}{2 c}-b c \left (\frac {b \int \frac {1}{c^2 x^2+1}dx}{c}-\frac {a+b \text {arcsinh}(c x)}{c^2 \sqrt {c^2 x^2+1}}\right )+\frac {x (a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}\right )-\frac {1}{2} b c \left (\frac {b \int \frac {1}{\left (c^2 x^2+1\right )^2}dx}{3 c}-\frac {a+b \text {arcsinh}(c x)}{3 c^2 \left (c^2 x^2+1\right )^{3/2}}\right )+\frac {x (a+b \text {arcsinh}(c x))^2}{4 \left (c^2 x^2+1\right )^2}\right )+2 b c \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{5/2}}dx-\frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 x^2+1\right )^2}\right )}{3 d^3}+\frac {2 b c \left (-\frac {5}{2} c^2 \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{5/2}}dx-\frac {a+b \text {arcsinh}(c x)}{2 x^2 \left (c^2 x^2+1\right )^{3/2}}+\frac {1}{2} b c \left (\frac {3}{2} \left (-c \arctan (c x)-\frac {1}{x}\right )+\frac {1}{2 x \left (c^2 x^2+1\right )}\right )\right )}{3 d^3}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d^3 x^3 \left (c^2 x^2+1\right )^2}\) |
\(\Big \downarrow \) 215 |
\(\displaystyle -\frac {7 c^2 \left (-5 c^2 \left (\frac {3}{4} \left (\frac {2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2}{2 c}-b c \left (\frac {b \int \frac {1}{c^2 x^2+1}dx}{c}-\frac {a+b \text {arcsinh}(c x)}{c^2 \sqrt {c^2 x^2+1}}\right )+\frac {x (a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}\right )-\frac {1}{2} b c \left (\frac {b \left (\frac {1}{2} \int \frac {1}{c^2 x^2+1}dx+\frac {x}{2 \left (c^2 x^2+1\right )}\right )}{3 c}-\frac {a+b \text {arcsinh}(c x)}{3 c^2 \left (c^2 x^2+1\right )^{3/2}}\right )+\frac {x (a+b \text {arcsinh}(c x))^2}{4 \left (c^2 x^2+1\right )^2}\right )+2 b c \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{5/2}}dx-\frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 x^2+1\right )^2}\right )}{3 d^3}+\frac {2 b c \left (-\frac {5}{2} c^2 \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{5/2}}dx-\frac {a+b \text {arcsinh}(c x)}{2 x^2 \left (c^2 x^2+1\right )^{3/2}}+\frac {1}{2} b c \left (\frac {3}{2} \left (-c \arctan (c x)-\frac {1}{x}\right )+\frac {1}{2 x \left (c^2 x^2+1\right )}\right )\right )}{3 d^3}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d^3 x^3 \left (c^2 x^2+1\right )^2}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle -\frac {7 c^2 \left (-5 c^2 \left (\frac {3}{4} \left (\frac {2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2}{2 c}-b c \left (\frac {b \arctan (c x)}{c^2}-\frac {a+b \text {arcsinh}(c x)}{c^2 \sqrt {c^2 x^2+1}}\right )+\frac {x (a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}\right )-\frac {1}{2} b c \left (\frac {b \left (\frac {\arctan (c x)}{2 c}+\frac {x}{2 \left (c^2 x^2+1\right )}\right )}{3 c}-\frac {a+b \text {arcsinh}(c x)}{3 c^2 \left (c^2 x^2+1\right )^{3/2}}\right )+\frac {x (a+b \text {arcsinh}(c x))^2}{4 \left (c^2 x^2+1\right )^2}\right )+2 b c \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{5/2}}dx-\frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 x^2+1\right )^2}\right )}{3 d^3}+\frac {2 b c \left (-\frac {5}{2} c^2 \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{5/2}}dx-\frac {a+b \text {arcsinh}(c x)}{2 x^2 \left (c^2 x^2+1\right )^{3/2}}+\frac {1}{2} b c \left (\frac {3}{2} \left (-c \arctan (c x)-\frac {1}{x}\right )+\frac {1}{2 x \left (c^2 x^2+1\right )}\right )\right )}{3 d^3}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d^3 x^3 \left (c^2 x^2+1\right )^2}\) |
\(\Big \downarrow \) 6226 |
\(\displaystyle -\frac {7 c^2 \left (-5 c^2 \left (\frac {3}{4} \left (\frac {2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2}{2 c}-b c \left (\frac {b \arctan (c x)}{c^2}-\frac {a+b \text {arcsinh}(c x)}{c^2 \sqrt {c^2 x^2+1}}\right )+\frac {x (a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}\right )-\frac {1}{2} b c \left (\frac {b \left (\frac {\arctan (c x)}{2 c}+\frac {x}{2 \left (c^2 x^2+1\right )}\right )}{3 c}-\frac {a+b \text {arcsinh}(c x)}{3 c^2 \left (c^2 x^2+1\right )^{3/2}}\right )+\frac {x (a+b \text {arcsinh}(c x))^2}{4 \left (c^2 x^2+1\right )^2}\right )+2 b c \left (\int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{3/2}}dx-\frac {1}{3} b c \int \frac {1}{\left (c^2 x^2+1\right )^2}dx+\frac {a+b \text {arcsinh}(c x)}{3 \left (c^2 x^2+1\right )^{3/2}}\right )-\frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 x^2+1\right )^2}\right )}{3 d^3}+\frac {2 b c \left (-\frac {5}{2} c^2 \left (\int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{3/2}}dx-\frac {1}{3} b c \int \frac {1}{\left (c^2 x^2+1\right )^2}dx+\frac {a+b \text {arcsinh}(c x)}{3 \left (c^2 x^2+1\right )^{3/2}}\right )-\frac {a+b \text {arcsinh}(c x)}{2 x^2 \left (c^2 x^2+1\right )^{3/2}}+\frac {1}{2} b c \left (\frac {3}{2} \left (-c \arctan (c x)-\frac {1}{x}\right )+\frac {1}{2 x \left (c^2 x^2+1\right )}\right )\right )}{3 d^3}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d^3 x^3 \left (c^2 x^2+1\right )^2}\) |
\(\Big \downarrow \) 215 |
\(\displaystyle -\frac {7 c^2 \left (-5 c^2 \left (\frac {3}{4} \left (\frac {2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2}{2 c}-b c \left (\frac {b \arctan (c x)}{c^2}-\frac {a+b \text {arcsinh}(c x)}{c^2 \sqrt {c^2 x^2+1}}\right )+\frac {x (a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}\right )-\frac {1}{2} b c \left (\frac {b \left (\frac {\arctan (c x)}{2 c}+\frac {x}{2 \left (c^2 x^2+1\right )}\right )}{3 c}-\frac {a+b \text {arcsinh}(c x)}{3 c^2 \left (c^2 x^2+1\right )^{3/2}}\right )+\frac {x (a+b \text {arcsinh}(c x))^2}{4 \left (c^2 x^2+1\right )^2}\right )+2 b c \left (\int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{3/2}}dx-\frac {1}{3} b c \left (\frac {1}{2} \int \frac {1}{c^2 x^2+1}dx+\frac {x}{2 \left (c^2 x^2+1\right )}\right )+\frac {a+b \text {arcsinh}(c x)}{3 \left (c^2 x^2+1\right )^{3/2}}\right )-\frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 x^2+1\right )^2}\right )}{3 d^3}+\frac {2 b c \left (-\frac {5}{2} c^2 \left (\int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{3/2}}dx-\frac {1}{3} b c \left (\frac {1}{2} \int \frac {1}{c^2 x^2+1}dx+\frac {x}{2 \left (c^2 x^2+1\right )}\right )+\frac {a+b \text {arcsinh}(c x)}{3 \left (c^2 x^2+1\right )^{3/2}}\right )-\frac {a+b \text {arcsinh}(c x)}{2 x^2 \left (c^2 x^2+1\right )^{3/2}}+\frac {1}{2} b c \left (\frac {3}{2} \left (-c \arctan (c x)-\frac {1}{x}\right )+\frac {1}{2 x \left (c^2 x^2+1\right )}\right )\right )}{3 d^3}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d^3 x^3 \left (c^2 x^2+1\right )^2}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle -\frac {7 c^2 \left (-5 c^2 \left (\frac {3}{4} \left (\frac {2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2}{2 c}-b c \left (\frac {b \arctan (c x)}{c^2}-\frac {a+b \text {arcsinh}(c x)}{c^2 \sqrt {c^2 x^2+1}}\right )+\frac {x (a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}\right )-\frac {1}{2} b c \left (\frac {b \left (\frac {\arctan (c x)}{2 c}+\frac {x}{2 \left (c^2 x^2+1\right )}\right )}{3 c}-\frac {a+b \text {arcsinh}(c x)}{3 c^2 \left (c^2 x^2+1\right )^{3/2}}\right )+\frac {x (a+b \text {arcsinh}(c x))^2}{4 \left (c^2 x^2+1\right )^2}\right )+2 b c \left (\int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{3/2}}dx+\frac {a+b \text {arcsinh}(c x)}{3 \left (c^2 x^2+1\right )^{3/2}}-\frac {1}{3} b c \left (\frac {\arctan (c x)}{2 c}+\frac {x}{2 \left (c^2 x^2+1\right )}\right )\right )-\frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 x^2+1\right )^2}\right )}{3 d^3}+\frac {2 b c \left (-\frac {5}{2} c^2 \left (\int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{3/2}}dx+\frac {a+b \text {arcsinh}(c x)}{3 \left (c^2 x^2+1\right )^{3/2}}-\frac {1}{3} b c \left (\frac {\arctan (c x)}{2 c}+\frac {x}{2 \left (c^2 x^2+1\right )}\right )\right )-\frac {a+b \text {arcsinh}(c x)}{2 x^2 \left (c^2 x^2+1\right )^{3/2}}+\frac {1}{2} b c \left (\frac {3}{2} \left (-c \arctan (c x)-\frac {1}{x}\right )+\frac {1}{2 x \left (c^2 x^2+1\right )}\right )\right )}{3 d^3}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d^3 x^3 \left (c^2 x^2+1\right )^2}\) |
\(\Big \downarrow \) 6226 |
\(\displaystyle -\frac {7 c^2 \left (-5 c^2 \left (\frac {3}{4} \left (\frac {2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2}{2 c}-b c \left (\frac {b \arctan (c x)}{c^2}-\frac {a+b \text {arcsinh}(c x)}{c^2 \sqrt {c^2 x^2+1}}\right )+\frac {x (a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}\right )-\frac {1}{2} b c \left (\frac {b \left (\frac {\arctan (c x)}{2 c}+\frac {x}{2 \left (c^2 x^2+1\right )}\right )}{3 c}-\frac {a+b \text {arcsinh}(c x)}{3 c^2 \left (c^2 x^2+1\right )^{3/2}}\right )+\frac {x (a+b \text {arcsinh}(c x))^2}{4 \left (c^2 x^2+1\right )^2}\right )+2 b c \left (\int \frac {a+b \text {arcsinh}(c x)}{x \sqrt {c^2 x^2+1}}dx-b c \int \frac {1}{c^2 x^2+1}dx+\frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}+\frac {a+b \text {arcsinh}(c x)}{3 \left (c^2 x^2+1\right )^{3/2}}-\frac {1}{3} b c \left (\frac {\arctan (c x)}{2 c}+\frac {x}{2 \left (c^2 x^2+1\right )}\right )\right )-\frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 x^2+1\right )^2}\right )}{3 d^3}+\frac {2 b c \left (-\frac {5}{2} c^2 \left (\int \frac {a+b \text {arcsinh}(c x)}{x \sqrt {c^2 x^2+1}}dx-b c \int \frac {1}{c^2 x^2+1}dx+\frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}+\frac {a+b \text {arcsinh}(c x)}{3 \left (c^2 x^2+1\right )^{3/2}}-\frac {1}{3} b c \left (\frac {\arctan (c x)}{2 c}+\frac {x}{2 \left (c^2 x^2+1\right )}\right )\right )-\frac {a+b \text {arcsinh}(c x)}{2 x^2 \left (c^2 x^2+1\right )^{3/2}}+\frac {1}{2} b c \left (\frac {3}{2} \left (-c \arctan (c x)-\frac {1}{x}\right )+\frac {1}{2 x \left (c^2 x^2+1\right )}\right )\right )}{3 d^3}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d^3 x^3 \left (c^2 x^2+1\right )^2}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle -\frac {7 c^2 \left (-5 c^2 \left (\frac {3}{4} \left (\frac {2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2}{2 c}-b c \left (\frac {b \arctan (c x)}{c^2}-\frac {a+b \text {arcsinh}(c x)}{c^2 \sqrt {c^2 x^2+1}}\right )+\frac {x (a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}\right )-\frac {1}{2} b c \left (\frac {b \left (\frac {\arctan (c x)}{2 c}+\frac {x}{2 \left (c^2 x^2+1\right )}\right )}{3 c}-\frac {a+b \text {arcsinh}(c x)}{3 c^2 \left (c^2 x^2+1\right )^{3/2}}\right )+\frac {x (a+b \text {arcsinh}(c x))^2}{4 \left (c^2 x^2+1\right )^2}\right )+2 b c \left (\int \frac {a+b \text {arcsinh}(c x)}{x \sqrt {c^2 x^2+1}}dx+\frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}+\frac {a+b \text {arcsinh}(c x)}{3 \left (c^2 x^2+1\right )^{3/2}}-\frac {1}{3} b c \left (\frac {\arctan (c x)}{2 c}+\frac {x}{2 \left (c^2 x^2+1\right )}\right )-b \arctan (c x)\right )-\frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 x^2+1\right )^2}\right )}{3 d^3}+\frac {2 b c \left (-\frac {5}{2} c^2 \left (\int \frac {a+b \text {arcsinh}(c x)}{x \sqrt {c^2 x^2+1}}dx+\frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}+\frac {a+b \text {arcsinh}(c x)}{3 \left (c^2 x^2+1\right )^{3/2}}-\frac {1}{3} b c \left (\frac {\arctan (c x)}{2 c}+\frac {x}{2 \left (c^2 x^2+1\right )}\right )-b \arctan (c x)\right )-\frac {a+b \text {arcsinh}(c x)}{2 x^2 \left (c^2 x^2+1\right )^{3/2}}+\frac {1}{2} b c \left (\frac {3}{2} \left (-c \arctan (c x)-\frac {1}{x}\right )+\frac {1}{2 x \left (c^2 x^2+1\right )}\right )\right )}{3 d^3}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d^3 x^3 \left (c^2 x^2+1\right )^2}\) |
\(\Big \downarrow \) 6231 |
\(\displaystyle -\frac {7 c^2 \left (-5 c^2 \left (\frac {3}{4} \left (\frac {2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2}{2 c}-b c \left (\frac {b \arctan (c x)}{c^2}-\frac {a+b \text {arcsinh}(c x)}{c^2 \sqrt {c^2 x^2+1}}\right )+\frac {x (a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}\right )-\frac {1}{2} b c \left (\frac {b \left (\frac {\arctan (c x)}{2 c}+\frac {x}{2 \left (c^2 x^2+1\right )}\right )}{3 c}-\frac {a+b \text {arcsinh}(c x)}{3 c^2 \left (c^2 x^2+1\right )^{3/2}}\right )+\frac {x (a+b \text {arcsinh}(c x))^2}{4 \left (c^2 x^2+1\right )^2}\right )+2 b c \left (\int \frac {a+b \text {arcsinh}(c x)}{c x}d\text {arcsinh}(c x)+\frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}+\frac {a+b \text {arcsinh}(c x)}{3 \left (c^2 x^2+1\right )^{3/2}}-\frac {1}{3} b c \left (\frac {\arctan (c x)}{2 c}+\frac {x}{2 \left (c^2 x^2+1\right )}\right )-b \arctan (c x)\right )-\frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 x^2+1\right )^2}\right )}{3 d^3}+\frac {2 b c \left (-\frac {5}{2} c^2 \left (\int \frac {a+b \text {arcsinh}(c x)}{c x}d\text {arcsinh}(c x)+\frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}+\frac {a+b \text {arcsinh}(c x)}{3 \left (c^2 x^2+1\right )^{3/2}}-\frac {1}{3} b c \left (\frac {\arctan (c x)}{2 c}+\frac {x}{2 \left (c^2 x^2+1\right )}\right )-b \arctan (c x)\right )-\frac {a+b \text {arcsinh}(c x)}{2 x^2 \left (c^2 x^2+1\right )^{3/2}}+\frac {1}{2} b c \left (\frac {3}{2} \left (-c \arctan (c x)-\frac {1}{x}\right )+\frac {1}{2 x \left (c^2 x^2+1\right )}\right )\right )}{3 d^3}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d^3 x^3 \left (c^2 x^2+1\right )^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {7 c^2 \left (-5 c^2 \left (\frac {3}{4} \left (\frac {2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2}{2 c}-b c \left (\frac {b \arctan (c x)}{c^2}-\frac {a+b \text {arcsinh}(c x)}{c^2 \sqrt {c^2 x^2+1}}\right )+\frac {x (a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}\right )-\frac {1}{2} b c \left (\frac {b \left (\frac {\arctan (c x)}{2 c}+\frac {x}{2 \left (c^2 x^2+1\right )}\right )}{3 c}-\frac {a+b \text {arcsinh}(c x)}{3 c^2 \left (c^2 x^2+1\right )^{3/2}}\right )+\frac {x (a+b \text {arcsinh}(c x))^2}{4 \left (c^2 x^2+1\right )^2}\right )+2 b c \left (\int i (a+b \text {arcsinh}(c x)) \csc (i \text {arcsinh}(c x))d\text {arcsinh}(c x)+\frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}+\frac {a+b \text {arcsinh}(c x)}{3 \left (c^2 x^2+1\right )^{3/2}}-\frac {1}{3} b c \left (\frac {\arctan (c x)}{2 c}+\frac {x}{2 \left (c^2 x^2+1\right )}\right )-b \arctan (c x)\right )-\frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 x^2+1\right )^2}\right )}{3 d^3}+\frac {2 b c \left (-\frac {5}{2} c^2 \left (\int i (a+b \text {arcsinh}(c x)) \csc (i \text {arcsinh}(c x))d\text {arcsinh}(c x)+\frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}+\frac {a+b \text {arcsinh}(c x)}{3 \left (c^2 x^2+1\right )^{3/2}}-\frac {1}{3} b c \left (\frac {\arctan (c x)}{2 c}+\frac {x}{2 \left (c^2 x^2+1\right )}\right )-b \arctan (c x)\right )-\frac {a+b \text {arcsinh}(c x)}{2 x^2 \left (c^2 x^2+1\right )^{3/2}}+\frac {1}{2} b c \left (\frac {3}{2} \left (-c \arctan (c x)-\frac {1}{x}\right )+\frac {1}{2 x \left (c^2 x^2+1\right )}\right )\right )}{3 d^3}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d^3 x^3 \left (c^2 x^2+1\right )^2}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -\frac {7 c^2 \left (-5 c^2 \left (\frac {3}{4} \left (\frac {2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2}{2 c}-b c \left (\frac {b \arctan (c x)}{c^2}-\frac {a+b \text {arcsinh}(c x)}{c^2 \sqrt {c^2 x^2+1}}\right )+\frac {x (a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}\right )-\frac {1}{2} b c \left (\frac {b \left (\frac {\arctan (c x)}{2 c}+\frac {x}{2 \left (c^2 x^2+1\right )}\right )}{3 c}-\frac {a+b \text {arcsinh}(c x)}{3 c^2 \left (c^2 x^2+1\right )^{3/2}}\right )+\frac {x (a+b \text {arcsinh}(c x))^2}{4 \left (c^2 x^2+1\right )^2}\right )+2 b c \left (i \int (a+b \text {arcsinh}(c x)) \csc (i \text {arcsinh}(c x))d\text {arcsinh}(c x)+\frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}+\frac {a+b \text {arcsinh}(c x)}{3 \left (c^2 x^2+1\right )^{3/2}}-\frac {1}{3} b c \left (\frac {\arctan (c x)}{2 c}+\frac {x}{2 \left (c^2 x^2+1\right )}\right )-b \arctan (c x)\right )-\frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 x^2+1\right )^2}\right )}{3 d^3}+\frac {2 b c \left (-\frac {5}{2} c^2 \left (i \int (a+b \text {arcsinh}(c x)) \csc (i \text {arcsinh}(c x))d\text {arcsinh}(c x)+\frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}+\frac {a+b \text {arcsinh}(c x)}{3 \left (c^2 x^2+1\right )^{3/2}}-\frac {1}{3} b c \left (\frac {\arctan (c x)}{2 c}+\frac {x}{2 \left (c^2 x^2+1\right )}\right )-b \arctan (c x)\right )-\frac {a+b \text {arcsinh}(c x)}{2 x^2 \left (c^2 x^2+1\right )^{3/2}}+\frac {1}{2} b c \left (\frac {3}{2} \left (-c \arctan (c x)-\frac {1}{x}\right )+\frac {1}{2 x \left (c^2 x^2+1\right )}\right )\right )}{3 d^3}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d^3 x^3 \left (c^2 x^2+1\right )^2}\) |
\(\Big \downarrow \) 4670 |
\(\displaystyle -\frac {7 c^2 \left (2 b c \left (i \left (i b \int \log \left (1-e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)-i b \int \log \left (1+e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)+2 i \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+\frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}+\frac {a+b \text {arcsinh}(c x)}{3 \left (c^2 x^2+1\right )^{3/2}}-\frac {1}{3} b c \left (\frac {\arctan (c x)}{2 c}+\frac {x}{2 \left (c^2 x^2+1\right )}\right )-b \arctan (c x)\right )-5 c^2 \left (\frac {3}{4} \left (\frac {2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2}{2 c}-b c \left (\frac {b \arctan (c x)}{c^2}-\frac {a+b \text {arcsinh}(c x)}{c^2 \sqrt {c^2 x^2+1}}\right )+\frac {x (a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}\right )-\frac {1}{2} b c \left (\frac {b \left (\frac {\arctan (c x)}{2 c}+\frac {x}{2 \left (c^2 x^2+1\right )}\right )}{3 c}-\frac {a+b \text {arcsinh}(c x)}{3 c^2 \left (c^2 x^2+1\right )^{3/2}}\right )+\frac {x (a+b \text {arcsinh}(c x))^2}{4 \left (c^2 x^2+1\right )^2}\right )-\frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 x^2+1\right )^2}\right )}{3 d^3}+\frac {2 b c \left (-\frac {5}{2} c^2 \left (i \left (i b \int \log \left (1-e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)-i b \int \log \left (1+e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)+2 i \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+\frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}+\frac {a+b \text {arcsinh}(c x)}{3 \left (c^2 x^2+1\right )^{3/2}}-\frac {1}{3} b c \left (\frac {\arctan (c x)}{2 c}+\frac {x}{2 \left (c^2 x^2+1\right )}\right )-b \arctan (c x)\right )-\frac {a+b \text {arcsinh}(c x)}{2 x^2 \left (c^2 x^2+1\right )^{3/2}}+\frac {1}{2} b c \left (\frac {3}{2} \left (-c \arctan (c x)-\frac {1}{x}\right )+\frac {1}{2 x \left (c^2 x^2+1\right )}\right )\right )}{3 d^3}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d^3 x^3 \left (c^2 x^2+1\right )^2}\) |
\(\Big \downarrow \) 2715 |
\(\displaystyle -\frac {7 \left (-5 \left (\frac {x (a+b \text {arcsinh}(c x))^2}{4 \left (c^2 x^2+1\right )^2}-\frac {1}{2} b c \left (\frac {b \left (\frac {x}{2 \left (c^2 x^2+1\right )}+\frac {\arctan (c x)}{2 c}\right )}{3 c}-\frac {a+b \text {arcsinh}(c x)}{3 c^2 \left (c^2 x^2+1\right )^{3/2}}\right )+\frac {3}{4} \left (\frac {x (a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}-b c \left (\frac {b \arctan (c x)}{c^2}-\frac {a+b \text {arcsinh}(c x)}{c^2 \sqrt {c^2 x^2+1}}\right )+\frac {2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2+2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-(a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )\right )-2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-(a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )\right )}{2 c}\right )\right ) c^2+2 b \left (\frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}+\frac {a+b \text {arcsinh}(c x)}{3 \left (c^2 x^2+1\right )^{3/2}}-b \arctan (c x)-\frac {1}{3} b c \left (\frac {x}{2 \left (c^2 x^2+1\right )}+\frac {\arctan (c x)}{2 c}\right )+i \left (2 i (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )+i b \int e^{-\text {arcsinh}(c x)} \log \left (1-e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-i b \int e^{-\text {arcsinh}(c x)} \log \left (1+e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}\right )\right ) c-\frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 x^2+1\right )^2}\right ) c^2}{3 d^3}+\frac {2 b \left (-\frac {5}{2} \left (\frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}+\frac {a+b \text {arcsinh}(c x)}{3 \left (c^2 x^2+1\right )^{3/2}}-b \arctan (c x)-\frac {1}{3} b c \left (\frac {x}{2 \left (c^2 x^2+1\right )}+\frac {\arctan (c x)}{2 c}\right )+i \left (2 i (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )+i b \int e^{-\text {arcsinh}(c x)} \log \left (1-e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-i b \int e^{-\text {arcsinh}(c x)} \log \left (1+e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}\right )\right ) c^2+\frac {1}{2} b \left (\frac {3}{2} \left (-c \arctan (c x)-\frac {1}{x}\right )+\frac {1}{2 x \left (c^2 x^2+1\right )}\right ) c-\frac {a+b \text {arcsinh}(c x)}{2 x^2 \left (c^2 x^2+1\right )^{3/2}}\right ) c}{3 d^3}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d^3 x^3 \left (c^2 x^2+1\right )^2}\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle -\frac {7 c^2 \left (-5 c^2 \left (\frac {3}{4} \left (\frac {2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2}{2 c}-b c \left (\frac {b \arctan (c x)}{c^2}-\frac {a+b \text {arcsinh}(c x)}{c^2 \sqrt {c^2 x^2+1}}\right )+\frac {x (a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}\right )-\frac {1}{2} b c \left (\frac {b \left (\frac {\arctan (c x)}{2 c}+\frac {x}{2 \left (c^2 x^2+1\right )}\right )}{3 c}-\frac {a+b \text {arcsinh}(c x)}{3 c^2 \left (c^2 x^2+1\right )^{3/2}}\right )+\frac {x (a+b \text {arcsinh}(c x))^2}{4 \left (c^2 x^2+1\right )^2}\right )+2 b c \left (i \left (2 i \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )\right )+\frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}+\frac {a+b \text {arcsinh}(c x)}{3 \left (c^2 x^2+1\right )^{3/2}}-\frac {1}{3} b c \left (\frac {\arctan (c x)}{2 c}+\frac {x}{2 \left (c^2 x^2+1\right )}\right )-b \arctan (c x)\right )-\frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 x^2+1\right )^2}\right )}{3 d^3}+\frac {2 b c \left (-\frac {5}{2} c^2 \left (i \left (2 i \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )\right )+\frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}+\frac {a+b \text {arcsinh}(c x)}{3 \left (c^2 x^2+1\right )^{3/2}}-\frac {1}{3} b c \left (\frac {\arctan (c x)}{2 c}+\frac {x}{2 \left (c^2 x^2+1\right )}\right )-b \arctan (c x)\right )-\frac {a+b \text {arcsinh}(c x)}{2 x^2 \left (c^2 x^2+1\right )^{3/2}}+\frac {1}{2} b c \left (\frac {3}{2} \left (-c \arctan (c x)-\frac {1}{x}\right )+\frac {1}{2 x \left (c^2 x^2+1\right )}\right )\right )}{3 d^3}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d^3 x^3 \left (c^2 x^2+1\right )^2}\) |
\(\Big \downarrow \) 7143 |
\(\displaystyle -\frac {7 c^2 \left (2 b c \left (i \left (2 i \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )\right )+\frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}+\frac {a+b \text {arcsinh}(c x)}{3 \left (c^2 x^2+1\right )^{3/2}}-\frac {1}{3} b c \left (\frac {\arctan (c x)}{2 c}+\frac {x}{2 \left (c^2 x^2+1\right )}\right )-b \arctan (c x)\right )-5 c^2 \left (\frac {3}{4} \left (-b c \left (\frac {b \arctan (c x)}{c^2}-\frac {a+b \text {arcsinh}(c x)}{c^2 \sqrt {c^2 x^2+1}}\right )+\frac {2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2+2 i b \left (b \operatorname {PolyLog}\left (3,-i e^{\text {arcsinh}(c x)}\right )-\operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-2 i b \left (b \operatorname {PolyLog}\left (3,i e^{\text {arcsinh}(c x)}\right )-\operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )}{2 c}+\frac {x (a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}\right )-\frac {1}{2} b c \left (\frac {b \left (\frac {\arctan (c x)}{2 c}+\frac {x}{2 \left (c^2 x^2+1\right )}\right )}{3 c}-\frac {a+b \text {arcsinh}(c x)}{3 c^2 \left (c^2 x^2+1\right )^{3/2}}\right )+\frac {x (a+b \text {arcsinh}(c x))^2}{4 \left (c^2 x^2+1\right )^2}\right )-\frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 x^2+1\right )^2}\right )}{3 d^3}+\frac {2 b c \left (-\frac {5}{2} c^2 \left (i \left (2 i \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )\right )+\frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}+\frac {a+b \text {arcsinh}(c x)}{3 \left (c^2 x^2+1\right )^{3/2}}-\frac {1}{3} b c \left (\frac {\arctan (c x)}{2 c}+\frac {x}{2 \left (c^2 x^2+1\right )}\right )-b \arctan (c x)\right )-\frac {a+b \text {arcsinh}(c x)}{2 x^2 \left (c^2 x^2+1\right )^{3/2}}+\frac {1}{2} b c \left (\frac {3}{2} \left (-c \arctan (c x)-\frac {1}{x}\right )+\frac {1}{2 x \left (c^2 x^2+1\right )}\right )\right )}{3 d^3}-\frac {(a+b \text {arcsinh}(c x))^2}{3 d^3 x^3 \left (c^2 x^2+1\right )^2}\) |
-1/3*(a + b*ArcSinh[c*x])^2/(d^3*x^3*(1 + c^2*x^2)^2) + (2*b*c*(-1/2*(a + b*ArcSinh[c*x])/(x^2*(1 + c^2*x^2)^(3/2)) + (b*c*(1/(2*x*(1 + c^2*x^2)) + (3*(-x^(-1) - c*ArcTan[c*x]))/2))/2 - (5*c^2*((a + b*ArcSinh[c*x])/(3*(1 + c^2*x^2)^(3/2)) + (a + b*ArcSinh[c*x])/Sqrt[1 + c^2*x^2] - b*ArcTan[c*x] - (b*c*(x/(2*(1 + c^2*x^2)) + ArcTan[c*x]/(2*c)))/3 + I*((2*I)*(a + b*ArcS inh[c*x])*ArcTanh[E^ArcSinh[c*x]] + I*b*PolyLog[2, -E^ArcSinh[c*x]] - I*b* PolyLog[2, E^ArcSinh[c*x]])))/2))/(3*d^3) - (7*c^2*(-((a + b*ArcSinh[c*x]) ^2/(x*(1 + c^2*x^2)^2)) + 2*b*c*((a + b*ArcSinh[c*x])/(3*(1 + c^2*x^2)^(3/ 2)) + (a + b*ArcSinh[c*x])/Sqrt[1 + c^2*x^2] - b*ArcTan[c*x] - (b*c*(x/(2* (1 + c^2*x^2)) + ArcTan[c*x]/(2*c)))/3 + I*((2*I)*(a + b*ArcSinh[c*x])*Arc Tanh[E^ArcSinh[c*x]] + I*b*PolyLog[2, -E^ArcSinh[c*x]] - I*b*PolyLog[2, E^ ArcSinh[c*x]])) - 5*c^2*((x*(a + b*ArcSinh[c*x])^2)/(4*(1 + c^2*x^2)^2) - (b*c*(-1/3*(a + b*ArcSinh[c*x])/(c^2*(1 + c^2*x^2)^(3/2)) + (b*(x/(2*(1 + c^2*x^2)) + ArcTan[c*x]/(2*c)))/(3*c)))/2 + (3*((x*(a + b*ArcSinh[c*x])^2) /(2*(1 + c^2*x^2)) - b*c*(-((a + b*ArcSinh[c*x])/(c^2*Sqrt[1 + c^2*x^2])) + (b*ArcTan[c*x])/c^2) + (2*(a + b*ArcSinh[c*x])^2*ArcTan[E^ArcSinh[c*x]] + (2*I)*b*(-((a + b*ArcSinh[c*x])*PolyLog[2, (-I)*E^ArcSinh[c*x]]) + b*Pol yLog[3, (-I)*E^ArcSinh[c*x]]) - (2*I)*b*(-((a + b*ArcSinh[c*x])*PolyLog[2, I*E^ArcSinh[c*x]]) + b*PolyLog[3, I*E^ArcSinh[c*x]]))/(2*c)))/4)))/(3*d^3 )
3.3.51.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) /(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1)) Int[(a + b*x^2)^(p + 1 ), x], x] /; FreeQ[{a, b}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[6 *p])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(c*x )^(m + 1))*((a + b*x^2)^(p + 1)/(2*a*c*(p + 1))), x] + Simp[(m + 2*p + 3)/( 2*a*(p + 1)) Int[(c*x)^m*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, m }, x] && LtQ[p, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c ^2*(m + 1))) Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p }, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[1/(d*e*n*Log[F]) Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) ))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) *(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F])) Int[(f + g*x)^( m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e , f, g, n}, x] && GtQ[m, 0]
Int[csc[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_ ))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)/E^( I*k*Pi)]/(f*fz*I)), x] + (-Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[ 1 - E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x] + Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x]) /; FreeQ[{c , d, e, f, fz}, x] && IntegerQ[2*k] && IGtQ[m, 0]
Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x _Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] + (-Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*fz*x )], x], x] + Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_), x _Symbol] :> Simp[(-x)*(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(2*d*(p + 1))), x] + (Simp[(2*p + 3)/(2*d*(p + 1)) Int[(d + e*x^2)^(p + 1)*(a + b* ArcSinh[c*x])^n, x], x] + Simp[b*c*(n/(2*(p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p] Int[x*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x ], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && LtQ[p, -1] && NeQ[p, -3/2]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2), x_Symb ol] :> Simp[1/(c*d) Subst[Int[(a + b*x)^n*Sech[x], x], x, ArcSinh[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[n, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(2*e*(p + 1))), x] - Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p] Int[(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x] /; FreeQ[ {a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && NeQ[p, -1]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ .)*(x_)^2)^(p_), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(d*f*(m + 1))), x] + (-Simp[c^2*((m + 2*p + 3)/(f^2*(m + 1))) Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n, x], x] - Sim p[b*c*(n/(f*(m + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p] Int[(f*x)^(m + 1)*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{ a, b, c, d, e, f, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && ILtQ[m, -1]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ .)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(f*x)^(m + 1))*(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(2*d*f*(p + 1))), x] + (Simp[(m + 2*p + 3)/(2*d*(p + 1 )) Int[(f*x)^m*(d + e*x^2)^(p + 1)*(a + b*ArcSinh[c*x])^n, x], x] + Simp[ b*c*(n/(2*f*(p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p] Int[(f*x)^(m + 1)*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{ a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && LtQ[p, -1] && !G tQ[m, 1] && (IntegerQ[m] || IntegerQ[p] || EqQ[n, 1])
Int[(((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/Sqrt[(d_) + (e_.) *(x_)^2], x_Symbol] :> Simp[(1/c^(m + 1))*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e *x^2]] Subst[Int[(a + b*x)^n*Sinh[x]^m, x], x, ArcSinh[c*x]], x] /; FreeQ [{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[n, 0] && IntegerQ[m]
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d , e, n, p}, x] && EqQ[b*d, a*e]
\[\int \frac {\left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )^{2}}{x^{4} \left (c^{2} d \,x^{2}+d \right )^{3}}d x\]
\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^4 \left (d+c^2 d x^2\right )^3} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} + d\right )}^{3} x^{4}} \,d x } \]
integral((b^2*arcsinh(c*x)^2 + 2*a*b*arcsinh(c*x) + a^2)/(c^6*d^3*x^10 + 3 *c^4*d^3*x^8 + 3*c^2*d^3*x^6 + d^3*x^4), x)
\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^4 \left (d+c^2 d x^2\right )^3} \, dx=\frac {\int \frac {a^{2}}{c^{6} x^{10} + 3 c^{4} x^{8} + 3 c^{2} x^{6} + x^{4}}\, dx + \int \frac {b^{2} \operatorname {asinh}^{2}{\left (c x \right )}}{c^{6} x^{10} + 3 c^{4} x^{8} + 3 c^{2} x^{6} + x^{4}}\, dx + \int \frac {2 a b \operatorname {asinh}{\left (c x \right )}}{c^{6} x^{10} + 3 c^{4} x^{8} + 3 c^{2} x^{6} + x^{4}}\, dx}{d^{3}} \]
(Integral(a**2/(c**6*x**10 + 3*c**4*x**8 + 3*c**2*x**6 + x**4), x) + Integ ral(b**2*asinh(c*x)**2/(c**6*x**10 + 3*c**4*x**8 + 3*c**2*x**6 + x**4), x) + Integral(2*a*b*asinh(c*x)/(c**6*x**10 + 3*c**4*x**8 + 3*c**2*x**6 + x** 4), x))/d**3
\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^4 \left (d+c^2 d x^2\right )^3} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} + d\right )}^{3} x^{4}} \,d x } \]
1/24*a^2*(105*c^3*arctan(c*x)/d^3 + (105*c^6*x^6 + 175*c^4*x^4 + 56*c^2*x^ 2 - 8)/(c^4*d^3*x^7 + 2*c^2*d^3*x^5 + d^3*x^3)) + integrate(b^2*log(c*x + sqrt(c^2*x^2 + 1))^2/(c^6*d^3*x^10 + 3*c^4*d^3*x^8 + 3*c^2*d^3*x^6 + d^3*x ^4) + 2*a*b*log(c*x + sqrt(c^2*x^2 + 1))/(c^6*d^3*x^10 + 3*c^4*d^3*x^8 + 3 *c^2*d^3*x^6 + d^3*x^4), x)
\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^4 \left (d+c^2 d x^2\right )^3} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} + d\right )}^{3} x^{4}} \,d x } \]
Timed out. \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^4 \left (d+c^2 d x^2\right )^3} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{x^4\,{\left (d\,c^2\,x^2+d\right )}^3} \,d x \]