Integrand size = 26, antiderivative size = 389 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^2 \left (d+c^2 d x^2\right )^3} \, dx=\frac {b^2 c^2 x}{12 d^3 \left (1+c^2 x^2\right )}-\frac {b c (a+b \text {arcsinh}(c x))}{6 d^3 \left (1+c^2 x^2\right )^{3/2}}-\frac {7 b c (a+b \text {arcsinh}(c x))}{4 d^3 \sqrt {1+c^2 x^2}}-\frac {(a+b \text {arcsinh}(c x))^2}{d^3 x \left (1+c^2 x^2\right )^2}-\frac {5 c^2 x (a+b \text {arcsinh}(c x))^2}{4 d^3 \left (1+c^2 x^2\right )^2}-\frac {15 c^2 x (a+b \text {arcsinh}(c x))^2}{8 d^3 \left (1+c^2 x^2\right )}-\frac {15 c (a+b \text {arcsinh}(c x))^2 \arctan \left (e^{\text {arcsinh}(c x)}\right )}{4 d^3}+\frac {11 b^2 c \arctan (c x)}{6 d^3}-\frac {4 b c (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{d^3}-\frac {2 b^2 c \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )}{d^3}+\frac {15 i b c (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{4 d^3}-\frac {15 i b c (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{4 d^3}+\frac {2 b^2 c \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )}{d^3}-\frac {15 i b^2 c \operatorname {PolyLog}\left (3,-i e^{\text {arcsinh}(c x)}\right )}{4 d^3}+\frac {15 i b^2 c \operatorname {PolyLog}\left (3,i e^{\text {arcsinh}(c x)}\right )}{4 d^3} \] Output:
1/12*b^2*c^2*x/d^3/(c^2*x^2+1)-1/6*b*c*(a+b*arcsinh(c*x))/d^3/(c^2*x^2+1)^ (3/2)-7/4*b*c*(a+b*arcsinh(c*x))/d^3/(c^2*x^2+1)^(1/2)-(a+b*arcsinh(c*x))^ 2/d^3/x/(c^2*x^2+1)^2-5/4*c^2*x*(a+b*arcsinh(c*x))^2/d^3/(c^2*x^2+1)^2-15/ 8*c^2*x*(a+b*arcsinh(c*x))^2/d^3/(c^2*x^2+1)-15/4*c*(a+b*arcsinh(c*x))^2*a rctan(c*x+(c^2*x^2+1)^(1/2))/d^3+11/6*b^2*c*arctan(c*x)/d^3-4*b*c*(a+b*arc sinh(c*x))*arctanh(c*x+(c^2*x^2+1)^(1/2))/d^3-2*b^2*c*polylog(2,-c*x-(c^2* x^2+1)^(1/2))/d^3+15/4*I*b*c*(a+b*arcsinh(c*x))*polylog(2,-I*(c*x+(c^2*x^2 +1)^(1/2)))/d^3-15/4*I*b*c*(a+b*arcsinh(c*x))*polylog(2,I*(c*x+(c^2*x^2+1) ^(1/2)))/d^3+2*b^2*c*polylog(2,c*x+(c^2*x^2+1)^(1/2))/d^3-15/4*I*b^2*c*pol ylog(3,-I*(c*x+(c^2*x^2+1)^(1/2)))/d^3+15/4*I*b^2*c*polylog(3,I*(c*x+(c^2* x^2+1)^(1/2)))/d^3
Time = 7.33 (sec) , antiderivative size = 716, normalized size of antiderivative = 1.84 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^2 \left (d+c^2 d x^2\right )^3} \, dx =\text {Too large to display} \] Input:
Integrate[(a + b*ArcSinh[c*x])^2/(x^2*(d + c^2*d*x^2)^3),x]
Output:
-(a^2/(d^3*x)) - (a^2*c^2*x)/(4*d^3*(1 + c^2*x^2)^2) - (7*a^2*c^2*x)/(8*d^ 3*(1 + c^2*x^2)) - (15*a^2*c*ArcTan[c*x])/(8*d^3) + (2*a*b*c*((7*(Sqrt[1 + c^2*x^2] + I*ArcSinh[c*x]))/(16*(-1 - I*c*x)) - ArcSinh[c*x]/(c*x) - (7*( I*Sqrt[1 + c^2*x^2] + ArcSinh[c*x]))/(16*(I + c*x)) + ((I/48)*((-2*I + c*x )*Sqrt[1 + c^2*x^2] + 3*ArcSinh[c*x]))/(-I + c*x)^2 - ((I/48)*((2*I + c*x) *Sqrt[1 + c^2*x^2] + 3*ArcSinh[c*x]))/(I + c*x)^2 - ArcTanh[Sqrt[1 + c^2*x ^2]] + ((15*I)/16)*(-1/2*ArcSinh[c*x]^2 + 2*ArcSinh[c*x]*Log[1 + I*E^ArcSi nh[c*x]] + 2*PolyLog[2, (-I)*E^ArcSinh[c*x]]) - ((15*I)/16)*(-1/2*ArcSinh[ c*x]^2 + 2*ArcSinh[c*x]*Log[1 - I*E^ArcSinh[c*x]] + 2*PolyLog[2, I*E^ArcSi nh[c*x]])))/d^3 + (b^2*c*((2*c*x)/(1 + c^2*x^2) - (4*ArcSinh[c*x])/(1 + c^ 2*x^2)^(3/2) - (42*ArcSinh[c*x])/Sqrt[1 + c^2*x^2] - (6*c*x*ArcSinh[c*x]^2 )/(1 + c^2*x^2)^2 - (21*c*x*ArcSinh[c*x]^2)/(1 + c^2*x^2) + 88*ArcTan[Tanh [ArcSinh[c*x]/2]] - 12*ArcSinh[c*x]^2*Coth[ArcSinh[c*x]/2] + 48*ArcSinh[c* x]*Log[1 - E^(-ArcSinh[c*x])] + (45*I)*ArcSinh[c*x]^2*Log[1 - I/E^ArcSinh[ c*x]] - (45*I)*ArcSinh[c*x]^2*Log[1 + I/E^ArcSinh[c*x]] - 48*ArcSinh[c*x]* Log[1 + E^(-ArcSinh[c*x])] + 48*PolyLog[2, -E^(-ArcSinh[c*x])] + (90*I)*Ar cSinh[c*x]*PolyLog[2, (-I)/E^ArcSinh[c*x]] - (90*I)*ArcSinh[c*x]*PolyLog[2 , I/E^ArcSinh[c*x]] - 48*PolyLog[2, E^(-ArcSinh[c*x])] + (90*I)*PolyLog[3, (-I)/E^ArcSinh[c*x]] - (90*I)*PolyLog[3, I/E^ArcSinh[c*x]] + 12*ArcSinh[c *x]^2*Tanh[ArcSinh[c*x]/2]))/(24*d^3)
Time = 4.87 (sec) , antiderivative size = 466, normalized size of antiderivative = 1.20, number of steps used = 25, number of rules used = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.923, Rules used = {6224, 27, 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.
| \(\displaystyle \int \frac {(a+b \text {arcsinh}(c x))^2}{x^2 \left (c^2 d x^2+d\right )^3} \, dx\) |
| \(\Big \downarrow \) 6224 |
| \(\displaystyle -5 c^2 \int \frac {(a+b \text {arcsinh}(c x))^2}{d^3 \left (c^2 x^2+1\right )^3}dx+\frac {2 b c \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{5/2}}dx}{d^3}-\frac {(a+b \text {arcsinh}(c x))^2}{d^3 x \left (c^2 x^2+1\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5 c^2 \int \frac {(a+b \text {arcsinh}(c x))^2}{\left (c^2 x^2+1\right )^3}dx}{d^3}+\frac {2 b c \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{5/2}}dx}{d^3}-\frac {(a+b \text {arcsinh}(c x))^2}{d^3 x \left (c^2 x^2+1\right )^2}\) |
| \(\Big \downarrow \) 6203 |
| \(\displaystyle -\frac {5 c^2 \left (-\frac {1}{2} b c \int \frac {x (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^{5/2}}dx+\frac {3}{4} \int \frac {(a+b \text {arcsinh}(c x))^2}{\left (c^2 x^2+1\right )^2}dx+\frac {x (a+b \text {arcsinh}(c x))^2}{4 \left (c^2 x^2+1\right )^2}\right )}{d^3}+\frac {2 b c \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{5/2}}dx}{d^3}-\frac {(a+b \text {arcsinh}(c x))^2}{d^3 x \left (c^2 x^2+1\right )^2}\) |
| \(\Big \downarrow \) 6203 |
| \(\displaystyle -\frac {5 c^2 \left (-\frac {1}{2} b c \int \frac {x (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^{5/2}}dx+\frac {3}{4} \left (-b c \int \frac {x (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^{3/2}}dx+\frac {1}{2} \int \frac {(a+b \text {arcsinh}(c x))^2}{c^2 x^2+1}dx+\frac {x (a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}\right )+\frac {x (a+b \text {arcsinh}(c x))^2}{4 \left (c^2 x^2+1\right )^2}\right )}{d^3}+\frac {2 b c \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{5/2}}dx}{d^3}-\frac {(a+b \text {arcsinh}(c x))^2}{d^3 x \left (c^2 x^2+1\right )^2}\) |
| \(\Big \downarrow \) 6204 |
| \(\displaystyle -\frac {5 c^2 \left (-\frac {1}{2} b c \int \frac {x (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^{5/2}}dx+\frac {3}{4} \left (-b c \int \frac {x (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^{3/2}}dx+\frac {\int \frac {(a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}d\text {arcsinh}(c x)}{2 c}+\frac {x (a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}\right )+\frac {x (a+b \text {arcsinh}(c x))^2}{4 \left (c^2 x^2+1\right )^2}\right )}{d^3}+\frac {2 b c \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{5/2}}dx}{d^3}-\frac {(a+b \text {arcsinh}(c x))^2}{d^3 x \left (c^2 x^2+1\right )^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 b c \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{5/2}}dx}{d^3}-\frac {5 c^2 \left (-\frac {1}{2} b c \int \frac {x (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^{5/2}}dx+\frac {3}{4} \left (-b c \int \frac {x (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^{3/2}}dx+\frac {\int (a+b \text {arcsinh}(c x))^2 \csc \left (i \text {arcsinh}(c x)+\frac {\pi }{2}\right )d\text {arcsinh}(c x)}{2 c}+\frac {x (a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}\right )+\frac {x (a+b \text {arcsinh}(c x))^2}{4 \left (c^2 x^2+1\right )^2}\right )}{d^3}-\frac {(a+b \text {arcsinh}(c x))^2}{d^3 x \left (c^2 x^2+1\right )^2}\) |
| \(\Big \downarrow \) 4668 |
| \(\displaystyle -\frac {5 c^2 \left (\frac {3}{4} \left (\frac {-2 i b \int (a+b \text {arcsinh}(c x)) \log \left (1-i e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)+2 i b \int (a+b \text {arcsinh}(c x)) \log \left (1+i e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2}{2 c}-b c \int \frac {x (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^{3/2}}dx+\frac {x (a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}\right )-\frac {1}{2} b c \int \frac {x (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^{5/2}}dx+\frac {x (a+b \text {arcsinh}(c x))^2}{4 \left (c^2 x^2+1\right )^2}\right )}{d^3}+\frac {2 b c \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{5/2}}dx}{d^3}-\frac {(a+b \text {arcsinh}(c x))^2}{d^3 x \left (c^2 x^2+1\right )^2}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle -\frac {5 c^2 \left (\frac {3}{4} \left (\frac {2 i b \left (b \int \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)-\operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-2 i b \left (b \int \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)-\operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2}{2 c}-b c \int \frac {x (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^{3/2}}dx+\frac {x (a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}\right )-\frac {1}{2} b c \int \frac {x (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^{5/2}}dx+\frac {x (a+b \text {arcsinh}(c x))^2}{4 \left (c^2 x^2+1\right )^2}\right )}{d^3}+\frac {2 b c \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{5/2}}dx}{d^3}-\frac {(a+b \text {arcsinh}(c x))^2}{d^3 x \left (c^2 x^2+1\right )^2}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle -\frac {5 c^2 \left (\frac {3}{4} \left (\frac {2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2}{2 c}-b c \int \frac {x (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^{3/2}}dx+\frac {x (a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}\right )-\frac {1}{2} b c \int \frac {x (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^{5/2}}dx+\frac {x (a+b \text {arcsinh}(c x))^2}{4 \left (c^2 x^2+1\right )^2}\right )}{d^3}+\frac {2 b c \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{5/2}}dx}{d^3}-\frac {(a+b \text {arcsinh}(c x))^2}{d^3 x \left (c^2 x^2+1\right )^2}\) |
| \(\Big \downarrow \) 6213 |
| \(\displaystyle -\frac {5 c^2 \left (\frac {3}{4} \left (\frac {2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2}{2 c}-b c \left (\frac {b \int \frac {1}{c^2 x^2+1}dx}{c}-\frac {a+b \text {arcsinh}(c x)}{c^2 \sqrt {c^2 x^2+1}}\right )+\frac {x (a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}\right )-\frac {1}{2} b c \left (\frac {b \int \frac {1}{\left (c^2 x^2+1\right )^2}dx}{3 c}-\frac {a+b \text {arcsinh}(c x)}{3 c^2 \left (c^2 x^2+1\right )^{3/2}}\right )+\frac {x (a+b \text {arcsinh}(c x))^2}{4 \left (c^2 x^2+1\right )^2}\right )}{d^3}+\frac {2 b c \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{5/2}}dx}{d^3}-\frac {(a+b \text {arcsinh}(c x))^2}{d^3 x \left (c^2 x^2+1\right )^2}\) |
| \(\Big \downarrow \) 215 |
| \(\displaystyle -\frac {5 c^2 \left (\frac {3}{4} \left (\frac {2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2}{2 c}-b c \left (\frac {b \int \frac {1}{c^2 x^2+1}dx}{c}-\frac {a+b \text {arcsinh}(c x)}{c^2 \sqrt {c^2 x^2+1}}\right )+\frac {x (a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}\right )-\frac {1}{2} b c \left (\frac {b \left (\frac {1}{2} \int \frac {1}{c^2 x^2+1}dx+\frac {x}{2 \left (c^2 x^2+1\right )}\right )}{3 c}-\frac {a+b \text {arcsinh}(c x)}{3 c^2 \left (c^2 x^2+1\right )^{3/2}}\right )+\frac {x (a+b \text {arcsinh}(c x))^2}{4 \left (c^2 x^2+1\right )^2}\right )}{d^3}+\frac {2 b c \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{5/2}}dx}{d^3}-\frac {(a+b \text {arcsinh}(c x))^2}{d^3 x \left (c^2 x^2+1\right )^2}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle -\frac {5 c^2 \left (\frac {3}{4} \left (\frac {2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2}{2 c}-b c \left (\frac {b \arctan (c x)}{c^2}-\frac {a+b \text {arcsinh}(c x)}{c^2 \sqrt {c^2 x^2+1}}\right )+\frac {x (a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}\right )-\frac {1}{2} b c \left (\frac {b \left (\frac {\arctan (c x)}{2 c}+\frac {x}{2 \left (c^2 x^2+1\right )}\right )}{3 c}-\frac {a+b \text {arcsinh}(c x)}{3 c^2 \left (c^2 x^2+1\right )^{3/2}}\right )+\frac {x (a+b \text {arcsinh}(c x))^2}{4 \left (c^2 x^2+1\right )^2}\right )}{d^3}+\frac {2 b c \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{5/2}}dx}{d^3}-\frac {(a+b \text {arcsinh}(c x))^2}{d^3 x \left (c^2 x^2+1\right )^2}\) |
| \(\Big \downarrow \) 6226 |
| \(\displaystyle -\frac {5 c^2 \left (\frac {3}{4} \left (\frac {2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2}{2 c}-b c \left (\frac {b \arctan (c x)}{c^2}-\frac {a+b \text {arcsinh}(c x)}{c^2 \sqrt {c^2 x^2+1}}\right )+\frac {x (a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}\right )-\frac {1}{2} b c \left (\frac {b \left (\frac {\arctan (c x)}{2 c}+\frac {x}{2 \left (c^2 x^2+1\right )}\right )}{3 c}-\frac {a+b \text {arcsinh}(c x)}{3 c^2 \left (c^2 x^2+1\right )^{3/2}}\right )+\frac {x (a+b \text {arcsinh}(c x))^2}{4 \left (c^2 x^2+1\right )^2}\right )}{d^3}+\frac {2 b c \left (\int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{3/2}}dx-\frac {1}{3} b c \int \frac {1}{\left (c^2 x^2+1\right )^2}dx+\frac {a+b \text {arcsinh}(c x)}{3 \left (c^2 x^2+1\right )^{3/2}}\right )}{d^3}-\frac {(a+b \text {arcsinh}(c x))^2}{d^3 x \left (c^2 x^2+1\right )^2}\) |
| \(\Big \downarrow \) 215 |
| \(\displaystyle -\frac {5 c^2 \left (\frac {3}{4} \left (\frac {2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2}{2 c}-b c \left (\frac {b \arctan (c x)}{c^2}-\frac {a+b \text {arcsinh}(c x)}{c^2 \sqrt {c^2 x^2+1}}\right )+\frac {x (a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}\right )-\frac {1}{2} b c \left (\frac {b \left (\frac {\arctan (c x)}{2 c}+\frac {x}{2 \left (c^2 x^2+1\right )}\right )}{3 c}-\frac {a+b \text {arcsinh}(c x)}{3 c^2 \left (c^2 x^2+1\right )^{3/2}}\right )+\frac {x (a+b \text {arcsinh}(c x))^2}{4 \left (c^2 x^2+1\right )^2}\right )}{d^3}+\frac {2 b c \left (\int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{3/2}}dx-\frac {1}{3} b c \left (\frac {1}{2} \int \frac {1}{c^2 x^2+1}dx+\frac {x}{2 \left (c^2 x^2+1\right )}\right )+\frac {a+b \text {arcsinh}(c x)}{3 \left (c^2 x^2+1\right )^{3/2}}\right )}{d^3}-\frac {(a+b \text {arcsinh}(c x))^2}{d^3 x \left (c^2 x^2+1\right )^2}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle -\frac {5 c^2 \left (\frac {3}{4} \left (\frac {2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2}{2 c}-b c \left (\frac {b \arctan (c x)}{c^2}-\frac {a+b \text {arcsinh}(c x)}{c^2 \sqrt {c^2 x^2+1}}\right )+\frac {x (a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}\right )-\frac {1}{2} b c \left (\frac {b \left (\frac {\arctan (c x)}{2 c}+\frac {x}{2 \left (c^2 x^2+1\right )}\right )}{3 c}-\frac {a+b \text {arcsinh}(c x)}{3 c^2 \left (c^2 x^2+1\right )^{3/2}}\right )+\frac {x (a+b \text {arcsinh}(c x))^2}{4 \left (c^2 x^2+1\right )^2}\right )}{d^3}+\frac {2 b c \left (\int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{3/2}}dx+\frac {a+b \text {arcsinh}(c x)}{3 \left (c^2 x^2+1\right )^{3/2}}-\frac {1}{3} b c \left (\frac {\arctan (c x)}{2 c}+\frac {x}{2 \left (c^2 x^2+1\right )}\right )\right )}{d^3}-\frac {(a+b \text {arcsinh}(c x))^2}{d^3 x \left (c^2 x^2+1\right )^2}\) |
| \(\Big \downarrow \) 6226 |
| \(\displaystyle -\frac {5 c^2 \left (\frac {3}{4} \left (\frac {2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2}{2 c}-b c \left (\frac {b \arctan (c x)}{c^2}-\frac {a+b \text {arcsinh}(c x)}{c^2 \sqrt {c^2 x^2+1}}\right )+\frac {x (a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}\right )-\frac {1}{2} b c \left (\frac {b \left (\frac {\arctan (c x)}{2 c}+\frac {x}{2 \left (c^2 x^2+1\right )}\right )}{3 c}-\frac {a+b \text {arcsinh}(c x)}{3 c^2 \left (c^2 x^2+1\right )^{3/2}}\right )+\frac {x (a+b \text {arcsinh}(c x))^2}{4 \left (c^2 x^2+1\right )^2}\right )}{d^3}+\frac {2 b c \left (\int \frac {a+b \text {arcsinh}(c x)}{x \sqrt {c^2 x^2+1}}dx-b c \int \frac {1}{c^2 x^2+1}dx+\frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}+\frac {a+b \text {arcsinh}(c x)}{3 \left (c^2 x^2+1\right )^{3/2}}-\frac {1}{3} b c \left (\frac {\arctan (c x)}{2 c}+\frac {x}{2 \left (c^2 x^2+1\right )}\right )\right )}{d^3}-\frac {(a+b \text {arcsinh}(c x))^2}{d^3 x \left (c^2 x^2+1\right )^2}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle -\frac {5 c^2 \left (\frac {3}{4} \left (\frac {2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2}{2 c}-b c \left (\frac {b \arctan (c x)}{c^2}-\frac {a+b \text {arcsinh}(c x)}{c^2 \sqrt {c^2 x^2+1}}\right )+\frac {x (a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}\right )-\frac {1}{2} b c \left (\frac {b \left (\frac {\arctan (c x)}{2 c}+\frac {x}{2 \left (c^2 x^2+1\right )}\right )}{3 c}-\frac {a+b \text {arcsinh}(c x)}{3 c^2 \left (c^2 x^2+1\right )^{3/2}}\right )+\frac {x (a+b \text {arcsinh}(c x))^2}{4 \left (c^2 x^2+1\right )^2}\right )}{d^3}+\frac {2 b c \left (\int \frac {a+b \text {arcsinh}(c x)}{x \sqrt {c^2 x^2+1}}dx+\frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}+\frac {a+b \text {arcsinh}(c x)}{3 \left (c^2 x^2+1\right )^{3/2}}-\frac {1}{3} b c \left (\frac {\arctan (c x)}{2 c}+\frac {x}{2 \left (c^2 x^2+1\right )}\right )-b \arctan (c x)\right )}{d^3}-\frac {(a+b \text {arcsinh}(c x))^2}{d^3 x \left (c^2 x^2+1\right )^2}\) |
| \(\Big \downarrow \) 6231 |
| \(\displaystyle -\frac {5 c^2 \left (\frac {3}{4} \left (\frac {2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2}{2 c}-b c \left (\frac {b \arctan (c x)}{c^2}-\frac {a+b \text {arcsinh}(c x)}{c^2 \sqrt {c^2 x^2+1}}\right )+\frac {x (a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}\right )-\frac {1}{2} b c \left (\frac {b \left (\frac {\arctan (c x)}{2 c}+\frac {x}{2 \left (c^2 x^2+1\right )}\right )}{3 c}-\frac {a+b \text {arcsinh}(c x)}{3 c^2 \left (c^2 x^2+1\right )^{3/2}}\right )+\frac {x (a+b \text {arcsinh}(c x))^2}{4 \left (c^2 x^2+1\right )^2}\right )}{d^3}+\frac {2 b c \left (\int \frac {a+b \text {arcsinh}(c x)}{c x}d\text {arcsinh}(c x)+\frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}+\frac {a+b \text {arcsinh}(c x)}{3 \left (c^2 x^2+1\right )^{3/2}}-\frac {1}{3} b c \left (\frac {\arctan (c x)}{2 c}+\frac {x}{2 \left (c^2 x^2+1\right )}\right )-b \arctan (c x)\right )}{d^3}-\frac {(a+b \text {arcsinh}(c x))^2}{d^3 x \left (c^2 x^2+1\right )^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {5 c^2 \left (\frac {3}{4} \left (\frac {2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2}{2 c}-b c \left (\frac {b \arctan (c x)}{c^2}-\frac {a+b \text {arcsinh}(c x)}{c^2 \sqrt {c^2 x^2+1}}\right )+\frac {x (a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}\right )-\frac {1}{2} b c \left (\frac {b \left (\frac {\arctan (c x)}{2 c}+\frac {x}{2 \left (c^2 x^2+1\right )}\right )}{3 c}-\frac {a+b \text {arcsinh}(c x)}{3 c^2 \left (c^2 x^2+1\right )^{3/2}}\right )+\frac {x (a+b \text {arcsinh}(c x))^2}{4 \left (c^2 x^2+1\right )^2}\right )}{d^3}+\frac {2 b c \left (\int i (a+b \text {arcsinh}(c x)) \csc (i \text {arcsinh}(c x))d\text {arcsinh}(c x)+\frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}+\frac {a+b \text {arcsinh}(c x)}{3 \left (c^2 x^2+1\right )^{3/2}}-\frac {1}{3} b c \left (\frac {\arctan (c x)}{2 c}+\frac {x}{2 \left (c^2 x^2+1\right )}\right )-b \arctan (c x)\right )}{d^3}-\frac {(a+b \text {arcsinh}(c x))^2}{d^3 x \left (c^2 x^2+1\right )^2}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {5 c^2 \left (\frac {3}{4} \left (\frac {2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2}{2 c}-b c \left (\frac {b \arctan (c x)}{c^2}-\frac {a+b \text {arcsinh}(c x)}{c^2 \sqrt {c^2 x^2+1}}\right )+\frac {x (a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}\right )-\frac {1}{2} b c \left (\frac {b \left (\frac {\arctan (c x)}{2 c}+\frac {x}{2 \left (c^2 x^2+1\right )}\right )}{3 c}-\frac {a+b \text {arcsinh}(c x)}{3 c^2 \left (c^2 x^2+1\right )^{3/2}}\right )+\frac {x (a+b \text {arcsinh}(c x))^2}{4 \left (c^2 x^2+1\right )^2}\right )}{d^3}+\frac {2 b c \left (i \int (a+b \text {arcsinh}(c x)) \csc (i \text {arcsinh}(c x))d\text {arcsinh}(c x)+\frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}+\frac {a+b \text {arcsinh}(c x)}{3 \left (c^2 x^2+1\right )^{3/2}}-\frac {1}{3} b c \left (\frac {\arctan (c x)}{2 c}+\frac {x}{2 \left (c^2 x^2+1\right )}\right )-b \arctan (c x)\right )}{d^3}-\frac {(a+b \text {arcsinh}(c x))^2}{d^3 x \left (c^2 x^2+1\right )^2}\) |
| \(\Big \downarrow \) 4670 |
| \(\displaystyle \frac {2 b c \left (i \left (i b \int \log \left (1-e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)-i b \int \log \left (1+e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)+2 i \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+\frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}+\frac {a+b \text {arcsinh}(c x)}{3 \left (c^2 x^2+1\right )^{3/2}}-\frac {1}{3} b c \left (\frac {\arctan (c x)}{2 c}+\frac {x}{2 \left (c^2 x^2+1\right )}\right )-b \arctan (c x)\right )}{d^3}-\frac {5 c^2 \left (\frac {3}{4} \left (\frac {2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2}{2 c}-b c \left (\frac {b \arctan (c x)}{c^2}-\frac {a+b \text {arcsinh}(c x)}{c^2 \sqrt {c^2 x^2+1}}\right )+\frac {x (a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}\right )-\frac {1}{2} b c \left (\frac {b \left (\frac {\arctan (c x)}{2 c}+\frac {x}{2 \left (c^2 x^2+1\right )}\right )}{3 c}-\frac {a+b \text {arcsinh}(c x)}{3 c^2 \left (c^2 x^2+1\right )^{3/2}}\right )+\frac {x (a+b \text {arcsinh}(c x))^2}{4 \left (c^2 x^2+1\right )^2}\right )}{d^3}-\frac {(a+b \text {arcsinh}(c x))^2}{d^3 x \left (c^2 x^2+1\right )^2}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {2 b c \left (i \left (i b \int e^{-\text {arcsinh}(c x)} \log \left (1-e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-i b \int e^{-\text {arcsinh}(c x)} \log \left (1+e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}+2 i \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+\frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}+\frac {a+b \text {arcsinh}(c x)}{3 \left (c^2 x^2+1\right )^{3/2}}-\frac {1}{3} b c \left (\frac {\arctan (c x)}{2 c}+\frac {x}{2 \left (c^2 x^2+1\right )}\right )-b \arctan (c x)\right )}{d^3}-\frac {5 c^2 \left (\frac {3}{4} \left (\frac {2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2}{2 c}-b c \left (\frac {b \arctan (c x)}{c^2}-\frac {a+b \text {arcsinh}(c x)}{c^2 \sqrt {c^2 x^2+1}}\right )+\frac {x (a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}\right )-\frac {1}{2} b c \left (\frac {b \left (\frac {\arctan (c x)}{2 c}+\frac {x}{2 \left (c^2 x^2+1\right )}\right )}{3 c}-\frac {a+b \text {arcsinh}(c x)}{3 c^2 \left (c^2 x^2+1\right )^{3/2}}\right )+\frac {x (a+b \text {arcsinh}(c x))^2}{4 \left (c^2 x^2+1\right )^2}\right )}{d^3}-\frac {(a+b \text {arcsinh}(c x))^2}{d^3 x \left (c^2 x^2+1\right )^2}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -\frac {5 c^2 \left (\frac {3}{4} \left (\frac {2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2}{2 c}-b c \left (\frac {b \arctan (c x)}{c^2}-\frac {a+b \text {arcsinh}(c x)}{c^2 \sqrt {c^2 x^2+1}}\right )+\frac {x (a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}\right )-\frac {1}{2} b c \left (\frac {b \left (\frac {\arctan (c x)}{2 c}+\frac {x}{2 \left (c^2 x^2+1\right )}\right )}{3 c}-\frac {a+b \text {arcsinh}(c x)}{3 c^2 \left (c^2 x^2+1\right )^{3/2}}\right )+\frac {x (a+b \text {arcsinh}(c x))^2}{4 \left (c^2 x^2+1\right )^2}\right )}{d^3}+\frac {2 b c \left (i \left (2 i \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )\right )+\frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}+\frac {a+b \text {arcsinh}(c x)}{3 \left (c^2 x^2+1\right )^{3/2}}-\frac {1}{3} b c \left (\frac {\arctan (c x)}{2 c}+\frac {x}{2 \left (c^2 x^2+1\right )}\right )-b \arctan (c x)\right )}{d^3}-\frac {(a+b \text {arcsinh}(c x))^2}{d^3 x \left (c^2 x^2+1\right )^2}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle \frac {2 b c \left (i \left (2 i \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )\right )+\frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}+\frac {a+b \text {arcsinh}(c x)}{3 \left (c^2 x^2+1\right )^{3/2}}-\frac {1}{3} b c \left (\frac {\arctan (c x)}{2 c}+\frac {x}{2 \left (c^2 x^2+1\right )}\right )-b \arctan (c x)\right )}{d^3}-\frac {5 c^2 \left (\frac {3}{4} \left (-b c \left (\frac {b \arctan (c x)}{c^2}-\frac {a+b \text {arcsinh}(c x)}{c^2 \sqrt {c^2 x^2+1}}\right )+\frac {2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2+2 i b \left (b \operatorname {PolyLog}\left (3,-i e^{\text {arcsinh}(c x)}\right )-\operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-2 i b \left (b \operatorname {PolyLog}\left (3,i e^{\text {arcsinh}(c x)}\right )-\operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )}{2 c}+\frac {x (a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}\right )-\frac {1}{2} b c \left (\frac {b \left (\frac {\arctan (c x)}{2 c}+\frac {x}{2 \left (c^2 x^2+1\right )}\right )}{3 c}-\frac {a+b \text {arcsinh}(c x)}{3 c^2 \left (c^2 x^2+1\right )^{3/2}}\right )+\frac {x (a+b \text {arcsinh}(c x))^2}{4 \left (c^2 x^2+1\right )^2}\right )}{d^3}-\frac {(a+b \text {arcsinh}(c x))^2}{d^3 x \left (c^2 x^2+1\right )^2}\) |
Input:
Int[(a + b*ArcSinh[c*x])^2/(x^2*(d + c^2*d*x^2)^3),x]
Output:
-((a + b*ArcSinh[c*x])^2/(d^3*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])*ArcTanh[E^ArcSinh[c*x]] + I*b*PolyLog[2, -E^ArcSin h[c*x]] - I*b*PolyLog[2, E^ArcSinh[c*x]])))/d^3 - (5*c^2*((x*(a + b*ArcSin h[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*ArcSin h[c*x])/(c^2*Sqrt[1 + c^2*x^2])) + (b*ArcTan[c*x])/c^2) + (2*(a + b*ArcSin h[c*x])^2*ArcTan[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*I)*b*( -((a + b*ArcSinh[c*x])*PolyLog[2, I*E^ArcSinh[c*x]]) + b*PolyLog[3, I*E^Ar cSinh[c*x]]))/(2*c)))/4))/d^3
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[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]
\[\int \frac {\left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2}}{x^{2} \left (c^{2} d \,x^{2}+d \right )^{3}}d x\]
Input:
int((a+b*arcsinh(x*c))^2/x^2/(c^2*d*x^2+d)^3,x)
Output:
int((a+b*arcsinh(x*c))^2/x^2/(c^2*d*x^2+d)^3,x)
\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^2 \left (d+c^2 d x^2\right )^3} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} + d\right )}^{3} x^{2}} \,d x } \] Input:
integrate((a+b*arcsinh(c*x))^2/x^2/(c^2*d*x^2+d)^3,x, algorithm="fricas")
Output:
integral((b^2*arcsinh(c*x)^2 + 2*a*b*arcsinh(c*x) + a^2)/(c^6*d^3*x^8 + 3* c^4*d^3*x^6 + 3*c^2*d^3*x^4 + d^3*x^2), x)
\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^2 \left (d+c^2 d x^2\right )^3} \, dx=\frac {\int \frac {a^{2}}{c^{6} x^{8} + 3 c^{4} x^{6} + 3 c^{2} x^{4} + x^{2}}\, dx + \int \frac {b^{2} \operatorname {asinh}^{2}{\left (c x \right )}}{c^{6} x^{8} + 3 c^{4} x^{6} + 3 c^{2} x^{4} + x^{2}}\, dx + \int \frac {2 a b \operatorname {asinh}{\left (c x \right )}}{c^{6} x^{8} + 3 c^{4} x^{6} + 3 c^{2} x^{4} + x^{2}}\, dx}{d^{3}} \] Input:
integrate((a+b*asinh(c*x))**2/x**2/(c**2*d*x**2+d)**3,x)
Output:
(Integral(a**2/(c**6*x**8 + 3*c**4*x**6 + 3*c**2*x**4 + x**2), x) + Integr al(b**2*asinh(c*x)**2/(c**6*x**8 + 3*c**4*x**6 + 3*c**2*x**4 + x**2), x) + Integral(2*a*b*asinh(c*x)/(c**6*x**8 + 3*c**4*x**6 + 3*c**2*x**4 + x**2), x))/d**3
\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^2 \left (d+c^2 d x^2\right )^3} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} + d\right )}^{3} x^{2}} \,d x } \] Input:
integrate((a+b*arcsinh(c*x))^2/x^2/(c^2*d*x^2+d)^3,x, algorithm="maxima")
Output:
-1/8*a^2*((15*c^4*x^4 + 25*c^2*x^2 + 8)/(c^4*d^3*x^5 + 2*c^2*d^3*x^3 + d^3 *x) + 15*c*arctan(c*x)/d^3) + integrate(b^2*log(c*x + sqrt(c^2*x^2 + 1))^2 /(c^6*d^3*x^8 + 3*c^4*d^3*x^6 + 3*c^2*d^3*x^4 + d^3*x^2) + 2*a*b*log(c*x + sqrt(c^2*x^2 + 1))/(c^6*d^3*x^8 + 3*c^4*d^3*x^6 + 3*c^2*d^3*x^4 + d^3*x^2 ), x)
\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^2 \left (d+c^2 d x^2\right )^3} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} + d\right )}^{3} x^{2}} \,d x } \] Input:
integrate((a+b*arcsinh(c*x))^2/x^2/(c^2*d*x^2+d)^3,x, algorithm="giac")
Output:
integrate((b*arcsinh(c*x) + a)^2/((c^2*d*x^2 + d)^3*x^2), x)
Timed out. \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^2 \left (d+c^2 d x^2\right )^3} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{x^2\,{\left (d\,c^2\,x^2+d\right )}^3} \,d x \] Input:
int((a + b*asinh(c*x))^2/(x^2*(d + c^2*d*x^2)^3),x)
Output:
int((a + b*asinh(c*x))^2/(x^2*(d + c^2*d*x^2)^3), x)
\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^2 \left (d+c^2 d x^2\right )^3} \, dx=\frac {-15 \mathit {atan} \left (c x \right ) a^{2} c^{5} x^{5}-30 \mathit {atan} \left (c x \right ) a^{2} c^{3} x^{3}-15 \mathit {atan} \left (c x \right ) a^{2} c x +16 \left (\int \frac {\mathit {asinh} \left (c x \right )}{c^{6} x^{8}+3 c^{4} x^{6}+3 c^{2} x^{4}+x^{2}}d x \right ) a b \,c^{4} x^{5}+32 \left (\int \frac {\mathit {asinh} \left (c x \right )}{c^{6} x^{8}+3 c^{4} x^{6}+3 c^{2} x^{4}+x^{2}}d x \right ) a b \,c^{2} x^{3}+16 \left (\int \frac {\mathit {asinh} \left (c x \right )}{c^{6} x^{8}+3 c^{4} x^{6}+3 c^{2} x^{4}+x^{2}}d x \right ) a b x +8 \left (\int \frac {\mathit {asinh} \left (c x \right )^{2}}{c^{6} x^{8}+3 c^{4} x^{6}+3 c^{2} x^{4}+x^{2}}d x \right ) b^{2} c^{4} x^{5}+16 \left (\int \frac {\mathit {asinh} \left (c x \right )^{2}}{c^{6} x^{8}+3 c^{4} x^{6}+3 c^{2} x^{4}+x^{2}}d x \right ) b^{2} c^{2} x^{3}+8 \left (\int \frac {\mathit {asinh} \left (c x \right )^{2}}{c^{6} x^{8}+3 c^{4} x^{6}+3 c^{2} x^{4}+x^{2}}d x \right ) b^{2} x -15 a^{2} c^{4} x^{4}-25 a^{2} c^{2} x^{2}-8 a^{2}}{8 d^{3} x \left (c^{4} x^{4}+2 c^{2} x^{2}+1\right )} \] Input:
int((a+b*asinh(c*x))^2/x^2/(c^2*d*x^2+d)^3,x)
Output:
( - 15*atan(c*x)*a**2*c**5*x**5 - 30*atan(c*x)*a**2*c**3*x**3 - 15*atan(c* x)*a**2*c*x + 16*int(asinh(c*x)/(c**6*x**8 + 3*c**4*x**6 + 3*c**2*x**4 + x **2),x)*a*b*c**4*x**5 + 32*int(asinh(c*x)/(c**6*x**8 + 3*c**4*x**6 + 3*c** 2*x**4 + x**2),x)*a*b*c**2*x**3 + 16*int(asinh(c*x)/(c**6*x**8 + 3*c**4*x* *6 + 3*c**2*x**4 + x**2),x)*a*b*x + 8*int(asinh(c*x)**2/(c**6*x**8 + 3*c** 4*x**6 + 3*c**2*x**4 + x**2),x)*b**2*c**4*x**5 + 16*int(asinh(c*x)**2/(c** 6*x**8 + 3*c**4*x**6 + 3*c**2*x**4 + x**2),x)*b**2*c**2*x**3 + 8*int(asinh (c*x)**2/(c**6*x**8 + 3*c**4*x**6 + 3*c**2*x**4 + x**2),x)*b**2*x - 15*a** 2*c**4*x**4 - 25*a**2*c**2*x**2 - 8*a**2)/(8*d**3*x*(c**4*x**4 + 2*c**2*x* *2 + 1))