Integrand size = 26, antiderivative size = 381 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^3 \left (d+c^2 d x^2\right )^3} \, dx=\frac {b^2 c^2}{12 d^3 \left (1+c^2 x^2\right )}-\frac {b c (a+b \text {arcsinh}(c x))}{d^3 x \left (1+c^2 x^2\right )^{3/2}}-\frac {5 b c^3 x (a+b \text {arcsinh}(c x))}{6 d^3 \left (1+c^2 x^2\right )^{3/2}}+\frac {4 b c^3 x (a+b \text {arcsinh}(c x))}{3 d^3 \sqrt {1+c^2 x^2}}-\frac {3 c^2 (a+b \text {arcsinh}(c x))^2}{4 d^3 \left (1+c^2 x^2\right )^2}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d^3 x^2 \left (1+c^2 x^2\right )^2}-\frac {3 c^2 (a+b \text {arcsinh}(c x))^2}{2 d^3 \left (1+c^2 x^2\right )}+\frac {6 c^2 (a+b \text {arcsinh}(c x))^2 \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right )}{d^3}+\frac {b^2 c^2 \log (x)}{d^3}-\frac {7 b^2 c^2 \log \left (1+c^2 x^2\right )}{6 d^3}+\frac {3 b c^2 (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{d^3}-\frac {3 b c^2 (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )}{d^3}-\frac {3 b^2 c^2 \operatorname {PolyLog}\left (3,-e^{2 \text {arcsinh}(c x)}\right )}{2 d^3}+\frac {3 b^2 c^2 \operatorname {PolyLog}\left (3,e^{2 \text {arcsinh}(c x)}\right )}{2 d^3} \]
1/12*b^2*c^2/d^3/(c^2*x^2+1)-b*c*(a+b*arcsinh(c*x))/d^3/x/(c^2*x^2+1)^(3/2 )-5/6*b*c^3*x*(a+b*arcsinh(c*x))/d^3/(c^2*x^2+1)^(3/2)-3/4*c^2*(a+b*arcsin h(c*x))^2/d^3/(c^2*x^2+1)^2-1/2*(a+b*arcsinh(c*x))^2/d^3/x^2/(c^2*x^2+1)^2 -3/2*c^2*(a+b*arcsinh(c*x))^2/d^3/(c^2*x^2+1)+6*c^2*(a+b*arcsinh(c*x))^2*a rctanh((c*x+(c^2*x^2+1)^(1/2))^2)/d^3+b^2*c^2*ln(x)/d^3-7/6*b^2*c^2*ln(c^2 *x^2+1)/d^3+3*b*c^2*(a+b*arcsinh(c*x))*polylog(2,-(c*x+(c^2*x^2+1)^(1/2))^ 2)/d^3-3*b*c^2*(a+b*arcsinh(c*x))*polylog(2,(c*x+(c^2*x^2+1)^(1/2))^2)/d^3 -3/2*b^2*c^2*polylog(3,-(c*x+(c^2*x^2+1)^(1/2))^2)/d^3+3/2*b^2*c^2*polylog (3,(c*x+(c^2*x^2+1)^(1/2))^2)/d^3+4/3*b*c^3*x*(a+b*arcsinh(c*x))/d^3/(c^2* x^2+1)^(1/2)
Result contains complex when optimal does not.
Time = 7.93 (sec) , antiderivative size = 701, normalized size of antiderivative = 1.84 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^3 \left (d+c^2 d x^2\right )^3} \, dx=-\frac {\frac {2 a^2}{x^2}+\frac {a^2 c^2}{\left (1+c^2 x^2\right )^2}+\frac {4 a^2 c^2}{1+c^2 x^2}+12 a^2 c^2 \log (x)-6 a^2 c^2 \log \left (1+c^2 x^2\right )-\frac {1}{6} a b \left (\frac {27 c^2 \left (\sqrt {1+c^2 x^2}-i \text {arcsinh}(c x)\right )}{i+c x}+\frac {27 c^2 \left (\sqrt {1+c^2 x^2}+i \text {arcsinh}(c x)\right )}{-i+c x}-\frac {24 \left (c x \sqrt {1+c^2 x^2}+\text {arcsinh}(c x)\right )}{x^2}+\frac {c^2 \left ((-2 i+c x) \sqrt {1+c^2 x^2}+3 \text {arcsinh}(c x)\right )}{(-i+c x)^2}+\frac {c^2 \left ((2 i+c x) \sqrt {1+c^2 x^2}+3 \text {arcsinh}(c x)\right )}{(i+c x)^2}-36 c^2 \left (\text {arcsinh}(c x) \left (\text {arcsinh}(c x)-4 \log \left (1+i e^{\text {arcsinh}(c x)}\right )\right )-4 \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )\right )-36 c^2 \left (\text {arcsinh}(c x) \left (\text {arcsinh}(c x)-4 \log \left (1-i e^{\text {arcsinh}(c x)}\right )\right )-4 \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )\right )+72 c^2 \left (\text {arcsinh}(c x) \left (\text {arcsinh}(c x)-2 \log \left (1-e^{2 \text {arcsinh}(c x)}\right )\right )-\operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )\right )\right )-4 b^2 c^2 \left (-\frac {i \pi ^3}{8}+\frac {1}{12+12 c^2 x^2}+\frac {c x \text {arcsinh}(c x)}{6 \left (1+c^2 x^2\right )^{3/2}}+\frac {7 c x \text {arcsinh}(c x)}{3 \sqrt {1+c^2 x^2}}-\frac {\sqrt {1+c^2 x^2} \text {arcsinh}(c x)}{c x}-\frac {\text {arcsinh}(c x)^2}{2 c^2 x^2}-\frac {\text {arcsinh}(c x)^2}{4 \left (1+c^2 x^2\right )^2}-\frac {\text {arcsinh}(c x)^2}{1+c^2 x^2}+2 \text {arcsinh}(c x)^3+3 \text {arcsinh}(c x)^2 \log \left (1+e^{-2 \text {arcsinh}(c x)}\right )-3 \text {arcsinh}(c x)^2 \log \left (1-e^{2 \text {arcsinh}(c x)}\right )+\log \left (\frac {c x}{\sqrt {1+c^2 x^2}}\right )-\frac {2}{3} \log \left (1+c^2 x^2\right )-3 \text {arcsinh}(c x) \operatorname {PolyLog}\left (2,-e^{-2 \text {arcsinh}(c x)}\right )-3 \text {arcsinh}(c x) \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )-\frac {3}{2} \operatorname {PolyLog}\left (3,-e^{-2 \text {arcsinh}(c x)}\right )+\frac {3}{2} \operatorname {PolyLog}\left (3,e^{2 \text {arcsinh}(c x)}\right )\right )}{4 d^3} \]
-1/4*((2*a^2)/x^2 + (a^2*c^2)/(1 + c^2*x^2)^2 + (4*a^2*c^2)/(1 + c^2*x^2) + 12*a^2*c^2*Log[x] - 6*a^2*c^2*Log[1 + c^2*x^2] - (a*b*((27*c^2*(Sqrt[1 + c^2*x^2] - I*ArcSinh[c*x]))/(I + c*x) + (27*c^2*(Sqrt[1 + c^2*x^2] + I*Ar cSinh[c*x]))/(-I + c*x) - (24*(c*x*Sqrt[1 + c^2*x^2] + ArcSinh[c*x]))/x^2 + (c^2*((-2*I + c*x)*Sqrt[1 + c^2*x^2] + 3*ArcSinh[c*x]))/(-I + c*x)^2 + ( c^2*((2*I + c*x)*Sqrt[1 + c^2*x^2] + 3*ArcSinh[c*x]))/(I + c*x)^2 - 36*c^2 *(ArcSinh[c*x]*(ArcSinh[c*x] - 4*Log[1 + I*E^ArcSinh[c*x]]) - 4*PolyLog[2, (-I)*E^ArcSinh[c*x]]) - 36*c^2*(ArcSinh[c*x]*(ArcSinh[c*x] - 4*Log[1 - I* E^ArcSinh[c*x]]) - 4*PolyLog[2, I*E^ArcSinh[c*x]]) + 72*c^2*(ArcSinh[c*x]* (ArcSinh[c*x] - 2*Log[1 - E^(2*ArcSinh[c*x])]) - PolyLog[2, E^(2*ArcSinh[c *x])])))/6 - 4*b^2*c^2*((-1/8*I)*Pi^3 + (12 + 12*c^2*x^2)^(-1) + (c*x*ArcS inh[c*x])/(6*(1 + c^2*x^2)^(3/2)) + (7*c*x*ArcSinh[c*x])/(3*Sqrt[1 + c^2*x ^2]) - (Sqrt[1 + c^2*x^2]*ArcSinh[c*x])/(c*x) - ArcSinh[c*x]^2/(2*c^2*x^2) - ArcSinh[c*x]^2/(4*(1 + c^2*x^2)^2) - ArcSinh[c*x]^2/(1 + c^2*x^2) + 2*A rcSinh[c*x]^3 + 3*ArcSinh[c*x]^2*Log[1 + E^(-2*ArcSinh[c*x])] - 3*ArcSinh[ c*x]^2*Log[1 - E^(2*ArcSinh[c*x])] + Log[(c*x)/Sqrt[1 + c^2*x^2]] - (2*Log [1 + c^2*x^2])/3 - 3*ArcSinh[c*x]*PolyLog[2, -E^(-2*ArcSinh[c*x])] - 3*Arc Sinh[c*x]*PolyLog[2, E^(2*ArcSinh[c*x])] - (3*PolyLog[3, -E^(-2*ArcSinh[c* x])])/2 + (3*PolyLog[3, E^(2*ArcSinh[c*x])])/2))/d^3
Result contains complex when optimal does not.
Time = 3.45 (sec) , antiderivative size = 482, normalized size of antiderivative = 1.27, number of steps used = 24, number of rules used = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.885, Rules used = {6224, 27, 6219, 27, 1578, 1195, 2009, 6226, 6203, 241, 6202, 240, 6226, 6202, 240, 6214, 5984, 3042, 26, 4670, 3011, 2720, 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.
| \(\displaystyle \int \frac {(a+b \text {arcsinh}(c x))^2}{x^3 \left (c^2 d x^2+d\right )^3} \, dx\) |
| \(\Big \downarrow \) 6224 |
| \(\displaystyle -3 c^2 \int \frac {(a+b \text {arcsinh}(c x))^2}{d^3 x \left (c^2 x^2+1\right )^3}dx+\frac {b c \int \frac {a+b \text {arcsinh}(c x)}{x^2 \left (c^2 x^2+1\right )^{5/2}}dx}{d^3}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d^3 x^2 \left (c^2 x^2+1\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {3 c^2 \int \frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 x^2+1\right )^3}dx}{d^3}+\frac {b c \int \frac {a+b \text {arcsinh}(c x)}{x^2 \left (c^2 x^2+1\right )^{5/2}}dx}{d^3}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d^3 x^2 \left (c^2 x^2+1\right )^2}\) |
| \(\Big \downarrow \) 6219 |
| \(\displaystyle -\frac {3 c^2 \int \frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 x^2+1\right )^3}dx}{d^3}+\frac {b c \left (-b c \int -\frac {8 c^4 x^4+12 c^2 x^2+3}{3 x \left (c^2 x^2+1\right )^2}dx-\frac {8 c^2 x (a+b \text {arcsinh}(c x))}{3 \sqrt {c^2 x^2+1}}-\frac {4 c^2 x (a+b \text {arcsinh}(c x))}{3 \left (c^2 x^2+1\right )^{3/2}}-\frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{3/2}}\right )}{d^3}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d^3 x^2 \left (c^2 x^2+1\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {3 c^2 \int \frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 x^2+1\right )^3}dx}{d^3}+\frac {b c \left (\frac {1}{3} b c \int \frac {8 c^4 x^4+12 c^2 x^2+3}{x \left (c^2 x^2+1\right )^2}dx-\frac {8 c^2 x (a+b \text {arcsinh}(c x))}{3 \sqrt {c^2 x^2+1}}-\frac {4 c^2 x (a+b \text {arcsinh}(c x))}{3 \left (c^2 x^2+1\right )^{3/2}}-\frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{3/2}}\right )}{d^3}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d^3 x^2 \left (c^2 x^2+1\right )^2}\) |
| \(\Big \downarrow \) 1578 |
| \(\displaystyle -\frac {3 c^2 \int \frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 x^2+1\right )^3}dx}{d^3}+\frac {b c \left (\frac {1}{6} b c \int \frac {8 c^4 x^4+12 c^2 x^2+3}{x^2 \left (c^2 x^2+1\right )^2}dx^2-\frac {8 c^2 x (a+b \text {arcsinh}(c x))}{3 \sqrt {c^2 x^2+1}}-\frac {4 c^2 x (a+b \text {arcsinh}(c x))}{3 \left (c^2 x^2+1\right )^{3/2}}-\frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{3/2}}\right )}{d^3}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d^3 x^2 \left (c^2 x^2+1\right )^2}\) |
| \(\Big \downarrow \) 1195 |
| \(\displaystyle -\frac {3 c^2 \int \frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 x^2+1\right )^3}dx}{d^3}+\frac {b c \left (\frac {1}{6} b c \int \left (\frac {5 c^2}{c^2 x^2+1}+\frac {c^2}{\left (c^2 x^2+1\right )^2}+\frac {3}{x^2}\right )dx^2-\frac {8 c^2 x (a+b \text {arcsinh}(c x))}{3 \sqrt {c^2 x^2+1}}-\frac {4 c^2 x (a+b \text {arcsinh}(c x))}{3 \left (c^2 x^2+1\right )^{3/2}}-\frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{3/2}}\right )}{d^3}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d^3 x^2 \left (c^2 x^2+1\right )^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 c^2 \int \frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 x^2+1\right )^3}dx}{d^3}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d^3 x^2 \left (c^2 x^2+1\right )^2}+\frac {b c \left (-\frac {8 c^2 x (a+b \text {arcsinh}(c x))}{3 \sqrt {c^2 x^2+1}}-\frac {4 c^2 x (a+b \text {arcsinh}(c x))}{3 \left (c^2 x^2+1\right )^{3/2}}-\frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{3/2}}+\frac {1}{6} b c \left (-\frac {1}{c^2 x^2+1}+5 \log \left (c^2 x^2+1\right )+3 \log \left (x^2\right )\right )\right )}{d^3}\) |
| \(\Big \downarrow \) 6226 |
| \(\displaystyle -\frac {3 c^2 \left (-\frac {1}{2} b c \int \frac {a+b \text {arcsinh}(c x)}{\left (c^2 x^2+1\right )^{5/2}}dx+\int \frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 x^2+1\right )^2}dx+\frac {(a+b \text {arcsinh}(c x))^2}{4 \left (c^2 x^2+1\right )^2}\right )}{d^3}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d^3 x^2 \left (c^2 x^2+1\right )^2}+\frac {b c \left (-\frac {8 c^2 x (a+b \text {arcsinh}(c x))}{3 \sqrt {c^2 x^2+1}}-\frac {4 c^2 x (a+b \text {arcsinh}(c x))}{3 \left (c^2 x^2+1\right )^{3/2}}-\frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{3/2}}+\frac {1}{6} b c \left (-\frac {1}{c^2 x^2+1}+5 \log \left (c^2 x^2+1\right )+3 \log \left (x^2\right )\right )\right )}{d^3}\) |
| \(\Big \downarrow \) 6203 |
| \(\displaystyle -\frac {3 c^2 \left (-\frac {1}{2} b c \left (\frac {2}{3} \int \frac {a+b \text {arcsinh}(c x)}{\left (c^2 x^2+1\right )^{3/2}}dx-\frac {1}{3} b c \int \frac {x}{\left (c^2 x^2+1\right )^2}dx+\frac {x (a+b \text {arcsinh}(c x))}{3 \left (c^2 x^2+1\right )^{3/2}}\right )+\int \frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 x^2+1\right )^2}dx+\frac {(a+b \text {arcsinh}(c x))^2}{4 \left (c^2 x^2+1\right )^2}\right )}{d^3}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d^3 x^2 \left (c^2 x^2+1\right )^2}+\frac {b c \left (-\frac {8 c^2 x (a+b \text {arcsinh}(c x))}{3 \sqrt {c^2 x^2+1}}-\frac {4 c^2 x (a+b \text {arcsinh}(c x))}{3 \left (c^2 x^2+1\right )^{3/2}}-\frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{3/2}}+\frac {1}{6} b c \left (-\frac {1}{c^2 x^2+1}+5 \log \left (c^2 x^2+1\right )+3 \log \left (x^2\right )\right )\right )}{d^3}\) |
| \(\Big \downarrow \) 241 |
| \(\displaystyle -\frac {3 c^2 \left (-\frac {1}{2} b c \left (\frac {2}{3} \int \frac {a+b \text {arcsinh}(c x)}{\left (c^2 x^2+1\right )^{3/2}}dx+\frac {x (a+b \text {arcsinh}(c x))}{3 \left (c^2 x^2+1\right )^{3/2}}+\frac {b}{6 c \left (c^2 x^2+1\right )}\right )+\int \frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 x^2+1\right )^2}dx+\frac {(a+b \text {arcsinh}(c x))^2}{4 \left (c^2 x^2+1\right )^2}\right )}{d^3}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d^3 x^2 \left (c^2 x^2+1\right )^2}+\frac {b c \left (-\frac {8 c^2 x (a+b \text {arcsinh}(c x))}{3 \sqrt {c^2 x^2+1}}-\frac {4 c^2 x (a+b \text {arcsinh}(c x))}{3 \left (c^2 x^2+1\right )^{3/2}}-\frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{3/2}}+\frac {1}{6} b c \left (-\frac {1}{c^2 x^2+1}+5 \log \left (c^2 x^2+1\right )+3 \log \left (x^2\right )\right )\right )}{d^3}\) |
| \(\Big \downarrow \) 6202 |
| \(\displaystyle -\frac {3 c^2 \left (-\frac {1}{2} b c \left (\frac {2}{3} \left (\frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}-b c \int \frac {x}{c^2 x^2+1}dx\right )+\frac {x (a+b \text {arcsinh}(c x))}{3 \left (c^2 x^2+1\right )^{3/2}}+\frac {b}{6 c \left (c^2 x^2+1\right )}\right )+\int \frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 x^2+1\right )^2}dx+\frac {(a+b \text {arcsinh}(c x))^2}{4 \left (c^2 x^2+1\right )^2}\right )}{d^3}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d^3 x^2 \left (c^2 x^2+1\right )^2}+\frac {b c \left (-\frac {8 c^2 x (a+b \text {arcsinh}(c x))}{3 \sqrt {c^2 x^2+1}}-\frac {4 c^2 x (a+b \text {arcsinh}(c x))}{3 \left (c^2 x^2+1\right )^{3/2}}-\frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{3/2}}+\frac {1}{6} b c \left (-\frac {1}{c^2 x^2+1}+5 \log \left (c^2 x^2+1\right )+3 \log \left (x^2\right )\right )\right )}{d^3}\) |
| \(\Big \downarrow \) 240 |
| \(\displaystyle -\frac {3 c^2 \left (\int \frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 x^2+1\right )^2}dx+\frac {(a+b \text {arcsinh}(c x))^2}{4 \left (c^2 x^2+1\right )^2}-\frac {1}{2} b c \left (\frac {x (a+b \text {arcsinh}(c x))}{3 \left (c^2 x^2+1\right )^{3/2}}+\frac {2}{3} \left (\frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}-\frac {b \log \left (c^2 x^2+1\right )}{2 c}\right )+\frac {b}{6 c \left (c^2 x^2+1\right )}\right )\right )}{d^3}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d^3 x^2 \left (c^2 x^2+1\right )^2}+\frac {b c \left (-\frac {8 c^2 x (a+b \text {arcsinh}(c x))}{3 \sqrt {c^2 x^2+1}}-\frac {4 c^2 x (a+b \text {arcsinh}(c x))}{3 \left (c^2 x^2+1\right )^{3/2}}-\frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{3/2}}+\frac {1}{6} b c \left (-\frac {1}{c^2 x^2+1}+5 \log \left (c^2 x^2+1\right )+3 \log \left (x^2\right )\right )\right )}{d^3}\) |
| \(\Big \downarrow \) 6226 |
| \(\displaystyle -\frac {3 c^2 \left (-b c \int \frac {a+b \text {arcsinh}(c x)}{\left (c^2 x^2+1\right )^{3/2}}dx+\int \frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 x^2+1\right )}dx+\frac {(a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}+\frac {(a+b \text {arcsinh}(c x))^2}{4 \left (c^2 x^2+1\right )^2}-\frac {1}{2} b c \left (\frac {x (a+b \text {arcsinh}(c x))}{3 \left (c^2 x^2+1\right )^{3/2}}+\frac {2}{3} \left (\frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}-\frac {b \log \left (c^2 x^2+1\right )}{2 c}\right )+\frac {b}{6 c \left (c^2 x^2+1\right )}\right )\right )}{d^3}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d^3 x^2 \left (c^2 x^2+1\right )^2}+\frac {b c \left (-\frac {8 c^2 x (a+b \text {arcsinh}(c x))}{3 \sqrt {c^2 x^2+1}}-\frac {4 c^2 x (a+b \text {arcsinh}(c x))}{3 \left (c^2 x^2+1\right )^{3/2}}-\frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{3/2}}+\frac {1}{6} b c \left (-\frac {1}{c^2 x^2+1}+5 \log \left (c^2 x^2+1\right )+3 \log \left (x^2\right )\right )\right )}{d^3}\) |
| \(\Big \downarrow \) 6202 |
| \(\displaystyle -\frac {3 c^2 \left (-b c \left (\frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}-b c \int \frac {x}{c^2 x^2+1}dx\right )+\int \frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 x^2+1\right )}dx+\frac {(a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}+\frac {(a+b \text {arcsinh}(c x))^2}{4 \left (c^2 x^2+1\right )^2}-\frac {1}{2} b c \left (\frac {x (a+b \text {arcsinh}(c x))}{3 \left (c^2 x^2+1\right )^{3/2}}+\frac {2}{3} \left (\frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}-\frac {b \log \left (c^2 x^2+1\right )}{2 c}\right )+\frac {b}{6 c \left (c^2 x^2+1\right )}\right )\right )}{d^3}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d^3 x^2 \left (c^2 x^2+1\right )^2}+\frac {b c \left (-\frac {8 c^2 x (a+b \text {arcsinh}(c x))}{3 \sqrt {c^2 x^2+1}}-\frac {4 c^2 x (a+b \text {arcsinh}(c x))}{3 \left (c^2 x^2+1\right )^{3/2}}-\frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{3/2}}+\frac {1}{6} b c \left (-\frac {1}{c^2 x^2+1}+5 \log \left (c^2 x^2+1\right )+3 \log \left (x^2\right )\right )\right )}{d^3}\) |
| \(\Big \downarrow \) 240 |
| \(\displaystyle -\frac {3 c^2 \left (\int \frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 x^2+1\right )}dx+\frac {(a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}+\frac {(a+b \text {arcsinh}(c x))^2}{4 \left (c^2 x^2+1\right )^2}-b c \left (\frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}-\frac {b \log \left (c^2 x^2+1\right )}{2 c}\right )-\frac {1}{2} b c \left (\frac {x (a+b \text {arcsinh}(c x))}{3 \left (c^2 x^2+1\right )^{3/2}}+\frac {2}{3} \left (\frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}-\frac {b \log \left (c^2 x^2+1\right )}{2 c}\right )+\frac {b}{6 c \left (c^2 x^2+1\right )}\right )\right )}{d^3}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d^3 x^2 \left (c^2 x^2+1\right )^2}+\frac {b c \left (-\frac {8 c^2 x (a+b \text {arcsinh}(c x))}{3 \sqrt {c^2 x^2+1}}-\frac {4 c^2 x (a+b \text {arcsinh}(c x))}{3 \left (c^2 x^2+1\right )^{3/2}}-\frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{3/2}}+\frac {1}{6} b c \left (-\frac {1}{c^2 x^2+1}+5 \log \left (c^2 x^2+1\right )+3 \log \left (x^2\right )\right )\right )}{d^3}\) |
| \(\Big \downarrow \) 6214 |
| \(\displaystyle -\frac {3 c^2 \left (\int \frac {(a+b \text {arcsinh}(c x))^2}{c x \sqrt {c^2 x^2+1}}d\text {arcsinh}(c x)+\frac {(a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}+\frac {(a+b \text {arcsinh}(c x))^2}{4 \left (c^2 x^2+1\right )^2}-b c \left (\frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}-\frac {b \log \left (c^2 x^2+1\right )}{2 c}\right )-\frac {1}{2} b c \left (\frac {x (a+b \text {arcsinh}(c x))}{3 \left (c^2 x^2+1\right )^{3/2}}+\frac {2}{3} \left (\frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}-\frac {b \log \left (c^2 x^2+1\right )}{2 c}\right )+\frac {b}{6 c \left (c^2 x^2+1\right )}\right )\right )}{d^3}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d^3 x^2 \left (c^2 x^2+1\right )^2}+\frac {b c \left (-\frac {8 c^2 x (a+b \text {arcsinh}(c x))}{3 \sqrt {c^2 x^2+1}}-\frac {4 c^2 x (a+b \text {arcsinh}(c x))}{3 \left (c^2 x^2+1\right )^{3/2}}-\frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{3/2}}+\frac {1}{6} b c \left (-\frac {1}{c^2 x^2+1}+5 \log \left (c^2 x^2+1\right )+3 \log \left (x^2\right )\right )\right )}{d^3}\) |
| \(\Big \downarrow \) 5984 |
| \(\displaystyle -\frac {3 c^2 \left (2 \int (a+b \text {arcsinh}(c x))^2 \text {csch}(2 \text {arcsinh}(c x))d\text {arcsinh}(c x)+\frac {(a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}+\frac {(a+b \text {arcsinh}(c x))^2}{4 \left (c^2 x^2+1\right )^2}-b c \left (\frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}-\frac {b \log \left (c^2 x^2+1\right )}{2 c}\right )-\frac {1}{2} b c \left (\frac {x (a+b \text {arcsinh}(c x))}{3 \left (c^2 x^2+1\right )^{3/2}}+\frac {2}{3} \left (\frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}-\frac {b \log \left (c^2 x^2+1\right )}{2 c}\right )+\frac {b}{6 c \left (c^2 x^2+1\right )}\right )\right )}{d^3}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d^3 x^2 \left (c^2 x^2+1\right )^2}+\frac {b c \left (-\frac {8 c^2 x (a+b \text {arcsinh}(c x))}{3 \sqrt {c^2 x^2+1}}-\frac {4 c^2 x (a+b \text {arcsinh}(c x))}{3 \left (c^2 x^2+1\right )^{3/2}}-\frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{3/2}}+\frac {1}{6} b c \left (-\frac {1}{c^2 x^2+1}+5 \log \left (c^2 x^2+1\right )+3 \log \left (x^2\right )\right )\right )}{d^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {3 c^2 \left (2 \int i (a+b \text {arcsinh}(c x))^2 \csc (2 i \text {arcsinh}(c x))d\text {arcsinh}(c x)+\frac {(a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}+\frac {(a+b \text {arcsinh}(c x))^2}{4 \left (c^2 x^2+1\right )^2}-b c \left (\frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}-\frac {b \log \left (c^2 x^2+1\right )}{2 c}\right )-\frac {1}{2} b c \left (\frac {x (a+b \text {arcsinh}(c x))}{3 \left (c^2 x^2+1\right )^{3/2}}+\frac {2}{3} \left (\frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}-\frac {b \log \left (c^2 x^2+1\right )}{2 c}\right )+\frac {b}{6 c \left (c^2 x^2+1\right )}\right )\right )}{d^3}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d^3 x^2 \left (c^2 x^2+1\right )^2}+\frac {b c \left (-\frac {8 c^2 x (a+b \text {arcsinh}(c x))}{3 \sqrt {c^2 x^2+1}}-\frac {4 c^2 x (a+b \text {arcsinh}(c x))}{3 \left (c^2 x^2+1\right )^{3/2}}-\frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{3/2}}+\frac {1}{6} b c \left (-\frac {1}{c^2 x^2+1}+5 \log \left (c^2 x^2+1\right )+3 \log \left (x^2\right )\right )\right )}{d^3}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {3 c^2 \left (2 i \int (a+b \text {arcsinh}(c x))^2 \csc (2 i \text {arcsinh}(c x))d\text {arcsinh}(c x)+\frac {(a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}+\frac {(a+b \text {arcsinh}(c x))^2}{4 \left (c^2 x^2+1\right )^2}-b c \left (\frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}-\frac {b \log \left (c^2 x^2+1\right )}{2 c}\right )-\frac {1}{2} b c \left (\frac {x (a+b \text {arcsinh}(c x))}{3 \left (c^2 x^2+1\right )^{3/2}}+\frac {2}{3} \left (\frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}-\frac {b \log \left (c^2 x^2+1\right )}{2 c}\right )+\frac {b}{6 c \left (c^2 x^2+1\right )}\right )\right )}{d^3}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d^3 x^2 \left (c^2 x^2+1\right )^2}+\frac {b c \left (-\frac {8 c^2 x (a+b \text {arcsinh}(c x))}{3 \sqrt {c^2 x^2+1}}-\frac {4 c^2 x (a+b \text {arcsinh}(c x))}{3 \left (c^2 x^2+1\right )^{3/2}}-\frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{3/2}}+\frac {1}{6} b c \left (-\frac {1}{c^2 x^2+1}+5 \log \left (c^2 x^2+1\right )+3 \log \left (x^2\right )\right )\right )}{d^3}\) |
| \(\Big \downarrow \) 4670 |
| \(\displaystyle -\frac {3 c^2 \left (2 i \left (i b \int (a+b \text {arcsinh}(c x)) \log \left (1-e^{2 \text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)-i b \int (a+b \text {arcsinh}(c x)) \log \left (1+e^{2 \text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)+i \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2\right )+\frac {(a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}+\frac {(a+b \text {arcsinh}(c x))^2}{4 \left (c^2 x^2+1\right )^2}-b c \left (\frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}-\frac {b \log \left (c^2 x^2+1\right )}{2 c}\right )-\frac {1}{2} b c \left (\frac {x (a+b \text {arcsinh}(c x))}{3 \left (c^2 x^2+1\right )^{3/2}}+\frac {2}{3} \left (\frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}-\frac {b \log \left (c^2 x^2+1\right )}{2 c}\right )+\frac {b}{6 c \left (c^2 x^2+1\right )}\right )\right )}{d^3}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d^3 x^2 \left (c^2 x^2+1\right )^2}+\frac {b c \left (-\frac {8 c^2 x (a+b \text {arcsinh}(c x))}{3 \sqrt {c^2 x^2+1}}-\frac {4 c^2 x (a+b \text {arcsinh}(c x))}{3 \left (c^2 x^2+1\right )^{3/2}}-\frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{3/2}}+\frac {1}{6} b c \left (-\frac {1}{c^2 x^2+1}+5 \log \left (c^2 x^2+1\right )+3 \log \left (x^2\right )\right )\right )}{d^3}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle -\frac {3 c^2 \left (2 i \left (-i b \left (\frac {1}{2} b \int \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)-\frac {1}{2} \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+i b \left (\frac {1}{2} b \int \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)-\frac {1}{2} \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+i \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2\right )+\frac {(a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}+\frac {(a+b \text {arcsinh}(c x))^2}{4 \left (c^2 x^2+1\right )^2}-b c \left (\frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}-\frac {b \log \left (c^2 x^2+1\right )}{2 c}\right )-\frac {1}{2} b c \left (\frac {x (a+b \text {arcsinh}(c x))}{3 \left (c^2 x^2+1\right )^{3/2}}+\frac {2}{3} \left (\frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}-\frac {b \log \left (c^2 x^2+1\right )}{2 c}\right )+\frac {b}{6 c \left (c^2 x^2+1\right )}\right )\right )}{d^3}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d^3 x^2 \left (c^2 x^2+1\right )^2}+\frac {b c \left (-\frac {8 c^2 x (a+b \text {arcsinh}(c x))}{3 \sqrt {c^2 x^2+1}}-\frac {4 c^2 x (a+b \text {arcsinh}(c x))}{3 \left (c^2 x^2+1\right )^{3/2}}-\frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{3/2}}+\frac {1}{6} b c \left (-\frac {1}{c^2 x^2+1}+5 \log \left (c^2 x^2+1\right )+3 \log \left (x^2\right )\right )\right )}{d^3}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle -\frac {3 c^2 \left (2 i \left (-i b \left (\frac {1}{4} b \int e^{-2 \text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )de^{2 \text {arcsinh}(c x)}-\frac {1}{2} \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+i b \left (\frac {1}{4} b \int e^{-2 \text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )de^{2 \text {arcsinh}(c x)}-\frac {1}{2} \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+i \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2\right )+\frac {(a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}+\frac {(a+b \text {arcsinh}(c x))^2}{4 \left (c^2 x^2+1\right )^2}-b c \left (\frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}-\frac {b \log \left (c^2 x^2+1\right )}{2 c}\right )-\frac {1}{2} b c \left (\frac {x (a+b \text {arcsinh}(c x))}{3 \left (c^2 x^2+1\right )^{3/2}}+\frac {2}{3} \left (\frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}-\frac {b \log \left (c^2 x^2+1\right )}{2 c}\right )+\frac {b}{6 c \left (c^2 x^2+1\right )}\right )\right )}{d^3}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d^3 x^2 \left (c^2 x^2+1\right )^2}+\frac {b c \left (-\frac {8 c^2 x (a+b \text {arcsinh}(c x))}{3 \sqrt {c^2 x^2+1}}-\frac {4 c^2 x (a+b \text {arcsinh}(c x))}{3 \left (c^2 x^2+1\right )^{3/2}}-\frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{3/2}}+\frac {1}{6} b c \left (-\frac {1}{c^2 x^2+1}+5 \log \left (c^2 x^2+1\right )+3 \log \left (x^2\right )\right )\right )}{d^3}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle -\frac {3 c^2 \left (2 i \left (i \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2-i b \left (\frac {1}{4} b \operatorname {PolyLog}\left (3,-e^{2 \text {arcsinh}(c x)}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+i b \left (\frac {1}{4} b \operatorname {PolyLog}\left (3,e^{2 \text {arcsinh}(c x)}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )\right )+\frac {(a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}+\frac {(a+b \text {arcsinh}(c x))^2}{4 \left (c^2 x^2+1\right )^2}-b c \left (\frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}-\frac {b \log \left (c^2 x^2+1\right )}{2 c}\right )-\frac {1}{2} b c \left (\frac {x (a+b \text {arcsinh}(c x))}{3 \left (c^2 x^2+1\right )^{3/2}}+\frac {2}{3} \left (\frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}-\frac {b \log \left (c^2 x^2+1\right )}{2 c}\right )+\frac {b}{6 c \left (c^2 x^2+1\right )}\right )\right )}{d^3}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d^3 x^2 \left (c^2 x^2+1\right )^2}+\frac {b c \left (-\frac {8 c^2 x (a+b \text {arcsinh}(c x))}{3 \sqrt {c^2 x^2+1}}-\frac {4 c^2 x (a+b \text {arcsinh}(c x))}{3 \left (c^2 x^2+1\right )^{3/2}}-\frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^{3/2}}+\frac {1}{6} b c \left (-\frac {1}{c^2 x^2+1}+5 \log \left (c^2 x^2+1\right )+3 \log \left (x^2\right )\right )\right )}{d^3}\) |
-1/2*(a + b*ArcSinh[c*x])^2/(d^3*x^2*(1 + c^2*x^2)^2) + (b*c*(-((a + b*Arc Sinh[c*x])/(x*(1 + c^2*x^2)^(3/2))) - (4*c^2*x*(a + b*ArcSinh[c*x]))/(3*(1 + c^2*x^2)^(3/2)) - (8*c^2*x*(a + b*ArcSinh[c*x]))/(3*Sqrt[1 + c^2*x^2]) + (b*c*(-(1 + c^2*x^2)^(-1) + 3*Log[x^2] + 5*Log[1 + c^2*x^2]))/6))/d^3 - (3*c^2*((a + b*ArcSinh[c*x])^2/(4*(1 + c^2*x^2)^2) + (a + b*ArcSinh[c*x])^ 2/(2*(1 + c^2*x^2)) - b*c*((x*(a + b*ArcSinh[c*x]))/Sqrt[1 + c^2*x^2] - (b *Log[1 + c^2*x^2])/(2*c)) - (b*c*(b/(6*c*(1 + c^2*x^2)) + (x*(a + b*ArcSin h[c*x]))/(3*(1 + c^2*x^2)^(3/2)) + (2*((x*(a + b*ArcSinh[c*x]))/Sqrt[1 + c ^2*x^2] - (b*Log[1 + c^2*x^2])/(2*c)))/3))/2 + (2*I)*(I*(a + b*ArcSinh[c*x ])^2*ArcTanh[E^(2*ArcSinh[c*x])] - I*b*(-1/2*((a + b*ArcSinh[c*x])*PolyLog [2, -E^(2*ArcSinh[c*x])]) + (b*PolyLog[3, -E^(2*ArcSinh[c*x])])/4) + I*b*( -1/2*((a + b*ArcSinh[c*x])*PolyLog[2, E^(2*ArcSinh[c*x])]) + (b*PolyLog[3, E^(2*ArcSinh[c*x])])/4))))/d^3
3.3.50.3.1 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[(x_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[Log[RemoveContent[a + b*x ^2, x]]/(2*b), x] /; FreeQ[{a, b}, x]
Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x^2)^(p + 1)/ (2*b*(p + 1)), x] /; FreeQ[{a, b, p}, x] && NeQ[p, -1]
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x _) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x ] && IGtQ[p, 0]
Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_ )^4)^(p_.), x_Symbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && Int egerQ[(m - 1)/2]
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[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_.) + (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[Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Simp[2^n Int[(c + d*x)^m*Csch[2*a + 2*b*x ]^n, x], x] /; FreeQ[{a, b, c, d}, x] && RationalQ[m] && IntegerQ[n]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[x*((a + b*ArcSinh[c*x])^n/(d*Sqrt[d + e*x^2])), x] - Simp [b*c*(n/d)*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]] Int[x*((a + b*ArcSinh[ c*x])^(n - 1)/(1 + c^2*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[n, 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_.)/((x_)*((d_) + (e_.)*(x_)^2)), x_Symbol] :> Simp[1/d Subst[Int[(a + b*x)^n/(Cosh[x]*Sinh[x]), x], x, Ar cSinh[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[n, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(p_ ), x_Symbol] :> With[{u = IntHide[x^m*(d + e*x^2)^p, x]}, Simp[(a + b*ArcSi nh[c*x]) u, x] - Simp[b*c*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]] Int[S implifyIntegrand[u/Sqrt[d + e*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e}, x ] && EqQ[e, c^2*d] && IntegerQ[p - 1/2] && NeQ[p, -2^(-1)] && (IGtQ[(m + 1) /2, 0] || ILtQ[(m + 2*p + 3)/2, 0])
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[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]
Leaf count of result is larger than twice the leaf count of optimal. \(831\) vs. \(2(402)=804\).
Time = 0.36 (sec) , antiderivative size = 832, normalized size of antiderivative = 2.18
| method | result | size |
| derivativedivides | \(c^{2} \left (\frac {a^{2} \left (-\frac {1}{2 c^{2} x^{2}}-3 \ln \left (c x \right )-\frac {1}{4 \left (c^{2} x^{2}+1\right )^{2}}-\frac {1}{c^{2} x^{2}+1}+\frac {3 \ln \left (c^{2} x^{2}+1\right )}{2}\right )}{d^{3}}+\frac {b^{2} \left (-\frac {-16 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{5} c^{5}+16 \,\operatorname {arcsinh}\left (c x \right ) c^{6} x^{6}+18 \operatorname {arcsinh}\left (c x \right )^{2} x^{4} c^{4}-6 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}+32 \,\operatorname {arcsinh}\left (c x \right ) c^{4} x^{4}+27 \operatorname {arcsinh}\left (c x \right )^{2} x^{2} c^{2}-c^{4} x^{4}+12 \,\operatorname {arcsinh}\left (c x \right ) c x \sqrt {c^{2} x^{2}+1}+16 \,\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}+6 \operatorname {arcsinh}\left (c x \right )^{2}-c^{2} x^{2}}{12 \left (c^{4} x^{4}+2 c^{2} x^{2}+1\right ) c^{2} x^{2}}+\ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+\frac {8 \ln \left (c x +\sqrt {c^{2} x^{2}+1}\right )}{3}-\frac {7 \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{3}+\ln \left (c x +\sqrt {c^{2} x^{2}+1}-1\right )-3 \operatorname {arcsinh}\left (c x \right )^{2} \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )-6 \,\operatorname {arcsinh}\left (c x \right ) \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+6 \operatorname {polylog}\left (3, -c x -\sqrt {c^{2} x^{2}+1}\right )+3 \operatorname {arcsinh}\left (c x \right )^{2} \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+3 \,\operatorname {arcsinh}\left (c x \right ) \operatorname {polylog}\left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )-\frac {3 \operatorname {polylog}\left (3, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2}-3 \operatorname {arcsinh}\left (c x \right )^{2} \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )-6 \,\operatorname {arcsinh}\left (c x \right ) \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )+6 \operatorname {polylog}\left (3, c x +\sqrt {c^{2} x^{2}+1}\right )\right )}{d^{3}}+\frac {2 a b \left (-\frac {-8 c^{5} x^{5} \sqrt {c^{2} x^{2}+1}+8 c^{6} x^{6}+18 \,\operatorname {arcsinh}\left (c x \right ) c^{4} x^{4}-3 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+16 c^{4} x^{4}+27 \,\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}+6 c x \sqrt {c^{2} x^{2}+1}+8 c^{2} x^{2}+6 \,\operatorname {arcsinh}\left (c x \right )}{12 \left (c^{4} x^{4}+2 c^{2} x^{2}+1\right ) c^{2} x^{2}}-3 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )-3 \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+3 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+\frac {3 \operatorname {polylog}\left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2}-3 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )-3 \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )\right )}{d^{3}}\right )\) | \(832\) |
| default | \(c^{2} \left (\frac {a^{2} \left (-\frac {1}{2 c^{2} x^{2}}-3 \ln \left (c x \right )-\frac {1}{4 \left (c^{2} x^{2}+1\right )^{2}}-\frac {1}{c^{2} x^{2}+1}+\frac {3 \ln \left (c^{2} x^{2}+1\right )}{2}\right )}{d^{3}}+\frac {b^{2} \left (-\frac {-16 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{5} c^{5}+16 \,\operatorname {arcsinh}\left (c x \right ) c^{6} x^{6}+18 \operatorname {arcsinh}\left (c x \right )^{2} x^{4} c^{4}-6 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}+32 \,\operatorname {arcsinh}\left (c x \right ) c^{4} x^{4}+27 \operatorname {arcsinh}\left (c x \right )^{2} x^{2} c^{2}-c^{4} x^{4}+12 \,\operatorname {arcsinh}\left (c x \right ) c x \sqrt {c^{2} x^{2}+1}+16 \,\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}+6 \operatorname {arcsinh}\left (c x \right )^{2}-c^{2} x^{2}}{12 \left (c^{4} x^{4}+2 c^{2} x^{2}+1\right ) c^{2} x^{2}}+\ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+\frac {8 \ln \left (c x +\sqrt {c^{2} x^{2}+1}\right )}{3}-\frac {7 \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{3}+\ln \left (c x +\sqrt {c^{2} x^{2}+1}-1\right )-3 \operatorname {arcsinh}\left (c x \right )^{2} \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )-6 \,\operatorname {arcsinh}\left (c x \right ) \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+6 \operatorname {polylog}\left (3, -c x -\sqrt {c^{2} x^{2}+1}\right )+3 \operatorname {arcsinh}\left (c x \right )^{2} \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+3 \,\operatorname {arcsinh}\left (c x \right ) \operatorname {polylog}\left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )-\frac {3 \operatorname {polylog}\left (3, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2}-3 \operatorname {arcsinh}\left (c x \right )^{2} \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )-6 \,\operatorname {arcsinh}\left (c x \right ) \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )+6 \operatorname {polylog}\left (3, c x +\sqrt {c^{2} x^{2}+1}\right )\right )}{d^{3}}+\frac {2 a b \left (-\frac {-8 c^{5} x^{5} \sqrt {c^{2} x^{2}+1}+8 c^{6} x^{6}+18 \,\operatorname {arcsinh}\left (c x \right ) c^{4} x^{4}-3 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+16 c^{4} x^{4}+27 \,\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}+6 c x \sqrt {c^{2} x^{2}+1}+8 c^{2} x^{2}+6 \,\operatorname {arcsinh}\left (c x \right )}{12 \left (c^{4} x^{4}+2 c^{2} x^{2}+1\right ) c^{2} x^{2}}-3 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )-3 \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+3 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+\frac {3 \operatorname {polylog}\left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2}-3 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )-3 \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )\right )}{d^{3}}\right )\) | \(832\) |
| parts | \(\frac {a^{2} \left (\frac {c^{4} \left (-\frac {2}{c^{2} \left (c^{2} x^{2}+1\right )}-\frac {1}{2 c^{2} \left (c^{2} x^{2}+1\right )^{2}}+\frac {3 \ln \left (c^{2} x^{2}+1\right )}{c^{2}}\right )}{2}-\frac {1}{2 x^{2}}-3 c^{2} \ln \left (x \right )\right )}{d^{3}}+\frac {b^{2} c^{2} \left (-\frac {-16 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{5} c^{5}+16 \,\operatorname {arcsinh}\left (c x \right ) c^{6} x^{6}+18 \operatorname {arcsinh}\left (c x \right )^{2} x^{4} c^{4}-6 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}+32 \,\operatorname {arcsinh}\left (c x \right ) c^{4} x^{4}+27 \operatorname {arcsinh}\left (c x \right )^{2} x^{2} c^{2}-c^{4} x^{4}+12 \,\operatorname {arcsinh}\left (c x \right ) c x \sqrt {c^{2} x^{2}+1}+16 \,\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}+6 \operatorname {arcsinh}\left (c x \right )^{2}-c^{2} x^{2}}{12 \left (c^{4} x^{4}+2 c^{2} x^{2}+1\right ) c^{2} x^{2}}+\ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+\frac {8 \ln \left (c x +\sqrt {c^{2} x^{2}+1}\right )}{3}-\frac {7 \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{3}+\ln \left (c x +\sqrt {c^{2} x^{2}+1}-1\right )-3 \operatorname {arcsinh}\left (c x \right )^{2} \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )-6 \,\operatorname {arcsinh}\left (c x \right ) \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+6 \operatorname {polylog}\left (3, -c x -\sqrt {c^{2} x^{2}+1}\right )+3 \operatorname {arcsinh}\left (c x \right )^{2} \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+3 \,\operatorname {arcsinh}\left (c x \right ) \operatorname {polylog}\left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )-\frac {3 \operatorname {polylog}\left (3, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2}-3 \operatorname {arcsinh}\left (c x \right )^{2} \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )-6 \,\operatorname {arcsinh}\left (c x \right ) \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )+6 \operatorname {polylog}\left (3, c x +\sqrt {c^{2} x^{2}+1}\right )\right )}{d^{3}}+\frac {2 a b \,c^{2} \left (-\frac {-8 c^{5} x^{5} \sqrt {c^{2} x^{2}+1}+8 c^{6} x^{6}+18 \,\operatorname {arcsinh}\left (c x \right ) c^{4} x^{4}-3 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+16 c^{4} x^{4}+27 \,\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}+6 c x \sqrt {c^{2} x^{2}+1}+8 c^{2} x^{2}+6 \,\operatorname {arcsinh}\left (c x \right )}{12 \left (c^{4} x^{4}+2 c^{2} x^{2}+1\right ) c^{2} x^{2}}-3 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )-3 \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+3 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+\frac {3 \operatorname {polylog}\left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2}-3 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )-3 \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )\right )}{d^{3}}\) | \(847\) |
c^2*(a^2/d^3*(-1/2/c^2/x^2-3*ln(c*x)-1/4/(c^2*x^2+1)^2-1/(c^2*x^2+1)+3/2*l n(c^2*x^2+1))+b^2/d^3*(-1/12/(c^4*x^4+2*c^2*x^2+1)/c^2/x^2*(-16*arcsinh(c* x)*(c^2*x^2+1)^(1/2)*x^5*c^5+16*arcsinh(c*x)*c^6*x^6+18*arcsinh(c*x)^2*x^4 *c^4-6*arcsinh(c*x)*(c^2*x^2+1)^(1/2)*x^3*c^3+32*arcsinh(c*x)*c^4*x^4+27*a rcsinh(c*x)^2*x^2*c^2-c^4*x^4+12*arcsinh(c*x)*c*x*(c^2*x^2+1)^(1/2)+16*arc sinh(c*x)*c^2*x^2+6*arcsinh(c*x)^2-c^2*x^2)+ln(1+c*x+(c^2*x^2+1)^(1/2))+8/ 3*ln(c*x+(c^2*x^2+1)^(1/2))-7/3*ln(1+(c*x+(c^2*x^2+1)^(1/2))^2)+ln(c*x+(c^ 2*x^2+1)^(1/2)-1)-3*arcsinh(c*x)^2*ln(1+c*x+(c^2*x^2+1)^(1/2))-6*arcsinh(c *x)*polylog(2,-c*x-(c^2*x^2+1)^(1/2))+6*polylog(3,-c*x-(c^2*x^2+1)^(1/2))+ 3*arcsinh(c*x)^2*ln(1+(c*x+(c^2*x^2+1)^(1/2))^2)+3*arcsinh(c*x)*polylog(2, -(c*x+(c^2*x^2+1)^(1/2))^2)-3/2*polylog(3,-(c*x+(c^2*x^2+1)^(1/2))^2)-3*ar csinh(c*x)^2*ln(1-c*x-(c^2*x^2+1)^(1/2))-6*arcsinh(c*x)*polylog(2,c*x+(c^2 *x^2+1)^(1/2))+6*polylog(3,c*x+(c^2*x^2+1)^(1/2)))+2*a*b/d^3*(-1/12/(c^4*x ^4+2*c^2*x^2+1)/c^2/x^2*(-8*c^5*x^5*(c^2*x^2+1)^(1/2)+8*c^6*x^6+18*arcsinh (c*x)*c^4*x^4-3*c^3*x^3*(c^2*x^2+1)^(1/2)+16*c^4*x^4+27*arcsinh(c*x)*c^2*x ^2+6*c*x*(c^2*x^2+1)^(1/2)+8*c^2*x^2+6*arcsinh(c*x))-3*arcsinh(c*x)*ln(1+c *x+(c^2*x^2+1)^(1/2))-3*polylog(2,-c*x-(c^2*x^2+1)^(1/2))+3*arcsinh(c*x)*l n(1+(c*x+(c^2*x^2+1)^(1/2))^2)+3/2*polylog(2,-(c*x+(c^2*x^2+1)^(1/2))^2)-3 *arcsinh(c*x)*ln(1-c*x-(c^2*x^2+1)^(1/2))-3*polylog(2,c*x+(c^2*x^2+1)^(1/2 ))))
\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^3 \left (d+c^2 d x^2\right )^3} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} + d\right )}^{3} x^{3}} \,d x } \]
integral((b^2*arcsinh(c*x)^2 + 2*a*b*arcsinh(c*x) + a^2)/(c^6*d^3*x^9 + 3* c^4*d^3*x^7 + 3*c^2*d^3*x^5 + d^3*x^3), x)
\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^3 \left (d+c^2 d x^2\right )^3} \, dx=\frac {\int \frac {a^{2}}{c^{6} x^{9} + 3 c^{4} x^{7} + 3 c^{2} x^{5} + x^{3}}\, dx + \int \frac {b^{2} \operatorname {asinh}^{2}{\left (c x \right )}}{c^{6} x^{9} + 3 c^{4} x^{7} + 3 c^{2} x^{5} + x^{3}}\, dx + \int \frac {2 a b \operatorname {asinh}{\left (c x \right )}}{c^{6} x^{9} + 3 c^{4} x^{7} + 3 c^{2} x^{5} + x^{3}}\, dx}{d^{3}} \]
(Integral(a**2/(c**6*x**9 + 3*c**4*x**7 + 3*c**2*x**5 + x**3), x) + Integr al(b**2*asinh(c*x)**2/(c**6*x**9 + 3*c**4*x**7 + 3*c**2*x**5 + x**3), x) + Integral(2*a*b*asinh(c*x)/(c**6*x**9 + 3*c**4*x**7 + 3*c**2*x**5 + x**3), x))/d**3
\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^3 \left (d+c^2 d x^2\right )^3} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} + d\right )}^{3} x^{3}} \,d x } \]
-1/4*a^2*((6*c^4*x^4 + 9*c^2*x^2 + 2)/(c^4*d^3*x^6 + 2*c^2*d^3*x^4 + d^3*x ^2) - 6*c^2*log(c^2*x^2 + 1)/d^3 + 12*c^2*log(x)/d^3) + integrate(b^2*log( c*x + sqrt(c^2*x^2 + 1))^2/(c^6*d^3*x^9 + 3*c^4*d^3*x^7 + 3*c^2*d^3*x^5 + d^3*x^3) + 2*a*b*log(c*x + sqrt(c^2*x^2 + 1))/(c^6*d^3*x^9 + 3*c^4*d^3*x^7 + 3*c^2*d^3*x^5 + d^3*x^3), x)
\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^3 \left (d+c^2 d x^2\right )^3} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} + d\right )}^{3} x^{3}} \,d x } \]
Timed out. \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^3 \left (d+c^2 d x^2\right )^3} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{x^3\,{\left (d\,c^2\,x^2+d\right )}^3} \,d x \]