Integrand size = 26, antiderivative size = 244 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^3 \left (d+c^2 d x^2\right )^2} \, dx=-\frac {b c (a+b \text {arcsinh}(c x))}{d^2 x \sqrt {1+c^2 x^2}}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d^2 x^2}-\frac {c^2 (a+b \text {arcsinh}(c x))^2}{2 d^2 \left (1+c^2 x^2\right )}+\frac {4 c^2 (a+b \text {arcsinh}(c x))^2 \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right )}{d^2}+\frac {b^2 c^2 \log (x)}{d^2}-\frac {b^2 c^2 \log \left (1+c^2 x^2\right )}{2 d^2}+\frac {2 b c^2 (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{d^2}-\frac {2 b c^2 (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )}{d^2}-\frac {b^2 c^2 \operatorname {PolyLog}\left (3,-e^{2 \text {arcsinh}(c x)}\right )}{d^2}+\frac {b^2 c^2 \operatorname {PolyLog}\left (3,e^{2 \text {arcsinh}(c x)}\right )}{d^2} \] Output:
-b*c*(a+b*arcsinh(c*x))/d^2/x/(c^2*x^2+1)^(1/2)-1/2*(a+b*arcsinh(c*x))^2/d ^2/x^2-1/2*c^2*(a+b*arcsinh(c*x))^2/d^2/(c^2*x^2+1)+4*c^2*(a+b*arcsinh(c*x ))^2*arctanh((c*x+(c^2*x^2+1)^(1/2))^2)/d^2+b^2*c^2*ln(x)/d^2-1/2*b^2*c^2* ln(c^2*x^2+1)/d^2+2*b*c^2*(a+b*arcsinh(c*x))*polylog(2,-(c*x+(c^2*x^2+1)^( 1/2))^2)/d^2-2*b*c^2*(a+b*arcsinh(c*x))*polylog(2,(c*x+(c^2*x^2+1)^(1/2))^ 2)/d^2-b^2*c^2*polylog(3,-(c*x+(c^2*x^2+1)^(1/2))^2)/d^2+b^2*c^2*polylog(3 ,(c*x+(c^2*x^2+1)^(1/2))^2)/d^2
Result contains complex when optimal does not.
Time = 1.45 (sec) , antiderivative size = 506, normalized size of antiderivative = 2.07 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^3 \left (d+c^2 d x^2\right )^2} \, dx=\frac {-\frac {a^2}{x^2}-\frac {a^2 c^2}{1+c^2 x^2}-4 a^2 c^2 \log (x)+2 a^2 c^2 \log \left (1+c^2 x^2\right )+a b \left (-\frac {2 c \sqrt {1+c^2 x^2}}{x}+\frac {c^2 \sqrt {1+c^2 x^2}}{-i+c x}+\frac {c^2 \sqrt {1+c^2 x^2}}{i+c x}-\frac {2 \text {arcsinh}(c x)}{x^2}+\frac {c^2 \text {arcsinh}(c x)}{-1-i c x}-\frac {i c^2 \text {arcsinh}(c x)}{i+c x}+8 c^2 \text {arcsinh}(c x) \log \left (1-i e^{\text {arcsinh}(c x)}\right )+8 c^2 \text {arcsinh}(c x) \log \left (1+i e^{\text {arcsinh}(c x)}\right )-8 c^2 \text {arcsinh}(c x) \log \left (1-e^{2 \text {arcsinh}(c x)}\right )+8 c^2 \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )+8 c^2 \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )-4 c^2 \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )\right )+b^2 c^2 \left (\frac {2 c x \text {arcsinh}(c x)}{\sqrt {1+c^2 x^2}}-\frac {2 \sqrt {1+c^2 x^2} \text {arcsinh}(c x)}{c x}-\frac {\text {arcsinh}(c x)^2}{c^2 x^2}-\frac {\text {arcsinh}(c x)^2}{1+c^2 x^2}-4 \text {arcsinh}(c x)^2 \log \left (1-e^{-2 \text {arcsinh}(c x)}\right )+4 \text {arcsinh}(c x)^2 \log \left (1+e^{-2 \text {arcsinh}(c x)}\right )+2 \log \left (\frac {c x}{\sqrt {1+c^2 x^2}}\right )-4 \text {arcsinh}(c x) \operatorname {PolyLog}\left (2,-e^{-2 \text {arcsinh}(c x)}\right )+4 \text {arcsinh}(c x) \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c x)}\right )-2 \operatorname {PolyLog}\left (3,-e^{-2 \text {arcsinh}(c x)}\right )+2 \operatorname {PolyLog}\left (3,e^{-2 \text {arcsinh}(c x)}\right )\right )}{2 d^2} \] Input:
Integrate[(a + b*ArcSinh[c*x])^2/(x^3*(d + c^2*d*x^2)^2),x]
Output:
(-(a^2/x^2) - (a^2*c^2)/(1 + c^2*x^2) - 4*a^2*c^2*Log[x] + 2*a^2*c^2*Log[1 + c^2*x^2] + a*b*((-2*c*Sqrt[1 + c^2*x^2])/x + (c^2*Sqrt[1 + c^2*x^2])/(- I + c*x) + (c^2*Sqrt[1 + c^2*x^2])/(I + c*x) - (2*ArcSinh[c*x])/x^2 + (c^2 *ArcSinh[c*x])/(-1 - I*c*x) - (I*c^2*ArcSinh[c*x])/(I + c*x) + 8*c^2*ArcSi nh[c*x]*Log[1 - I*E^ArcSinh[c*x]] + 8*c^2*ArcSinh[c*x]*Log[1 + I*E^ArcSinh [c*x]] - 8*c^2*ArcSinh[c*x]*Log[1 - E^(2*ArcSinh[c*x])] + 8*c^2*PolyLog[2, (-I)*E^ArcSinh[c*x]] + 8*c^2*PolyLog[2, I*E^ArcSinh[c*x]] - 4*c^2*PolyLog [2, E^(2*ArcSinh[c*x])]) + b^2*c^2*((2*c*x*ArcSinh[c*x])/Sqrt[1 + c^2*x^2] - (2*Sqrt[1 + c^2*x^2]*ArcSinh[c*x])/(c*x) - ArcSinh[c*x]^2/(c^2*x^2) - A rcSinh[c*x]^2/(1 + c^2*x^2) - 4*ArcSinh[c*x]^2*Log[1 - E^(-2*ArcSinh[c*x]) ] + 4*ArcSinh[c*x]^2*Log[1 + E^(-2*ArcSinh[c*x])] + 2*Log[(c*x)/Sqrt[1 + c ^2*x^2]] - 4*ArcSinh[c*x]*PolyLog[2, -E^(-2*ArcSinh[c*x])] + 4*ArcSinh[c*x ]*PolyLog[2, E^(-2*ArcSinh[c*x])] - 2*PolyLog[3, -E^(-2*ArcSinh[c*x])] + 2 *PolyLog[3, E^(-2*ArcSinh[c*x])]))/(2*d^2)
Result contains complex when optimal does not.
Time = 1.98 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.27, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.692, Rules used = {6224, 27, 6219, 25, 354, 86, 2009, 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.
| \(\displaystyle \int \frac {(a+b \text {arcsinh}(c x))^2}{x^3 \left (c^2 d x^2+d\right )^2} \, dx\) |
| \(\Big \downarrow \) 6224 |
| \(\displaystyle -2 c^2 \int \frac {(a+b \text {arcsinh}(c x))^2}{d^2 x \left (c^2 x^2+1\right )^2}dx+\frac {b c \int \frac {a+b \text {arcsinh}(c x)}{x^2 \left (c^2 x^2+1\right )^{3/2}}dx}{d^2}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d^2 x^2 \left (c^2 x^2+1\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 c^2 \int \frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 x^2+1\right )^2}dx}{d^2}+\frac {b c \int \frac {a+b \text {arcsinh}(c x)}{x^2 \left (c^2 x^2+1\right )^{3/2}}dx}{d^2}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d^2 x^2 \left (c^2 x^2+1\right )}\) |
| \(\Big \downarrow \) 6219 |
| \(\displaystyle -\frac {2 c^2 \int \frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 x^2+1\right )^2}dx}{d^2}+\frac {b c \left (-b c \int -\frac {2 c^2 x^2+1}{x \left (c^2 x^2+1\right )}dx-\frac {2 c^2 x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}-\frac {a+b \text {arcsinh}(c x)}{x \sqrt {c^2 x^2+1}}\right )}{d^2}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d^2 x^2 \left (c^2 x^2+1\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {2 c^2 \int \frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 x^2+1\right )^2}dx}{d^2}+\frac {b c \left (b c \int \frac {2 c^2 x^2+1}{x \left (c^2 x^2+1\right )}dx-\frac {2 c^2 x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}-\frac {a+b \text {arcsinh}(c x)}{x \sqrt {c^2 x^2+1}}\right )}{d^2}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d^2 x^2 \left (c^2 x^2+1\right )}\) |
| \(\Big \downarrow \) 354 |
| \(\displaystyle -\frac {2 c^2 \int \frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 x^2+1\right )^2}dx}{d^2}+\frac {b c \left (\frac {1}{2} b c \int \frac {2 c^2 x^2+1}{x^2 \left (c^2 x^2+1\right )}dx^2-\frac {2 c^2 x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}-\frac {a+b \text {arcsinh}(c x)}{x \sqrt {c^2 x^2+1}}\right )}{d^2}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d^2 x^2 \left (c^2 x^2+1\right )}\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle -\frac {2 c^2 \int \frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 x^2+1\right )^2}dx}{d^2}+\frac {b c \left (\frac {1}{2} b c \int \left (\frac {c^2}{c^2 x^2+1}+\frac {1}{x^2}\right )dx^2-\frac {2 c^2 x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}-\frac {a+b \text {arcsinh}(c x)}{x \sqrt {c^2 x^2+1}}\right )}{d^2}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d^2 x^2 \left (c^2 x^2+1\right )}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 c^2 \int \frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 x^2+1\right )^2}dx}{d^2}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d^2 x^2 \left (c^2 x^2+1\right )}+\frac {b c \left (-\frac {2 c^2 x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}-\frac {a+b \text {arcsinh}(c x)}{x \sqrt {c^2 x^2+1}}+\frac {1}{2} b c \left (\log \left (c^2 x^2+1\right )+\log \left (x^2\right )\right )\right )}{d^2}\) |
| \(\Big \downarrow \) 6226 |
| \(\displaystyle -\frac {2 c^2 \left (-b c \int \frac {a+b \text {arcsinh}(c x)}{\left (c^2 x^2+1\right )^{3/2}}dx+\int \frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 x^2+1\right )}dx+\frac {(a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}\right )}{d^2}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d^2 x^2 \left (c^2 x^2+1\right )}+\frac {b c \left (-\frac {2 c^2 x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}-\frac {a+b \text {arcsinh}(c x)}{x \sqrt {c^2 x^2+1}}+\frac {1}{2} b c \left (\log \left (c^2 x^2+1\right )+\log \left (x^2\right )\right )\right )}{d^2}\) |
| \(\Big \downarrow \) 6202 |
| \(\displaystyle -\frac {2 c^2 \left (-b c \left (\frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}-b c \int \frac {x}{c^2 x^2+1}dx\right )+\int \frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 x^2+1\right )}dx+\frac {(a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}\right )}{d^2}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d^2 x^2 \left (c^2 x^2+1\right )}+\frac {b c \left (-\frac {2 c^2 x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}-\frac {a+b \text {arcsinh}(c x)}{x \sqrt {c^2 x^2+1}}+\frac {1}{2} b c \left (\log \left (c^2 x^2+1\right )+\log \left (x^2\right )\right )\right )}{d^2}\) |
| \(\Big \downarrow \) 240 |
| \(\displaystyle -\frac {2 c^2 \left (\int \frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 x^2+1\right )}dx+\frac {(a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}-b c \left (\frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}-\frac {b \log \left (c^2 x^2+1\right )}{2 c}\right )\right )}{d^2}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d^2 x^2 \left (c^2 x^2+1\right )}+\frac {b c \left (-\frac {2 c^2 x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}-\frac {a+b \text {arcsinh}(c x)}{x \sqrt {c^2 x^2+1}}+\frac {1}{2} b c \left (\log \left (c^2 x^2+1\right )+\log \left (x^2\right )\right )\right )}{d^2}\) |
| \(\Big \downarrow \) 6214 |
| \(\displaystyle -\frac {2 c^2 \left (\int \frac {(a+b \text {arcsinh}(c x))^2}{c x \sqrt {c^2 x^2+1}}d\text {arcsinh}(c x)+\frac {(a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}-b c \left (\frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}-\frac {b \log \left (c^2 x^2+1\right )}{2 c}\right )\right )}{d^2}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d^2 x^2 \left (c^2 x^2+1\right )}+\frac {b c \left (-\frac {2 c^2 x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}-\frac {a+b \text {arcsinh}(c x)}{x \sqrt {c^2 x^2+1}}+\frac {1}{2} b c \left (\log \left (c^2 x^2+1\right )+\log \left (x^2\right )\right )\right )}{d^2}\) |
| \(\Big \downarrow \) 5984 |
| \(\displaystyle -\frac {2 c^2 \left (2 \int (a+b \text {arcsinh}(c x))^2 \text {csch}(2 \text {arcsinh}(c x))d\text {arcsinh}(c x)+\frac {(a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}-b c \left (\frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}-\frac {b \log \left (c^2 x^2+1\right )}{2 c}\right )\right )}{d^2}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d^2 x^2 \left (c^2 x^2+1\right )}+\frac {b c \left (-\frac {2 c^2 x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}-\frac {a+b \text {arcsinh}(c x)}{x \sqrt {c^2 x^2+1}}+\frac {1}{2} b c \left (\log \left (c^2 x^2+1\right )+\log \left (x^2\right )\right )\right )}{d^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {2 c^2 \left (2 \int i (a+b \text {arcsinh}(c x))^2 \csc (2 i \text {arcsinh}(c x))d\text {arcsinh}(c x)+\frac {(a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}-b c \left (\frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}-\frac {b \log \left (c^2 x^2+1\right )}{2 c}\right )\right )}{d^2}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d^2 x^2 \left (c^2 x^2+1\right )}+\frac {b c \left (-\frac {2 c^2 x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}-\frac {a+b \text {arcsinh}(c x)}{x \sqrt {c^2 x^2+1}}+\frac {1}{2} b c \left (\log \left (c^2 x^2+1\right )+\log \left (x^2\right )\right )\right )}{d^2}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {2 c^2 \left (2 i \int (a+b \text {arcsinh}(c x))^2 \csc (2 i \text {arcsinh}(c x))d\text {arcsinh}(c x)+\frac {(a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}-b c \left (\frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}-\frac {b \log \left (c^2 x^2+1\right )}{2 c}\right )\right )}{d^2}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d^2 x^2 \left (c^2 x^2+1\right )}+\frac {b c \left (-\frac {2 c^2 x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}-\frac {a+b \text {arcsinh}(c x)}{x \sqrt {c^2 x^2+1}}+\frac {1}{2} b c \left (\log \left (c^2 x^2+1\right )+\log \left (x^2\right )\right )\right )}{d^2}\) |
| \(\Big \downarrow \) 4670 |
| \(\displaystyle -\frac {2 c^2 \left (2 i \left (i b \int (a+b \text {arcsinh}(c x)) \log \left (1-e^{2 \text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)-i b \int (a+b \text {arcsinh}(c x)) \log \left (1+e^{2 \text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)+i \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2\right )+\frac {(a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}-b c \left (\frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}-\frac {b \log \left (c^2 x^2+1\right )}{2 c}\right )\right )}{d^2}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d^2 x^2 \left (c^2 x^2+1\right )}+\frac {b c \left (-\frac {2 c^2 x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}-\frac {a+b \text {arcsinh}(c x)}{x \sqrt {c^2 x^2+1}}+\frac {1}{2} b c \left (\log \left (c^2 x^2+1\right )+\log \left (x^2\right )\right )\right )}{d^2}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle -\frac {2 c^2 \left (2 i \left (-i b \left (\frac {1}{2} b \int \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)-\frac {1}{2} \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+i b \left (\frac {1}{2} b \int \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)-\frac {1}{2} \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+i \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2\right )+\frac {(a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}-b c \left (\frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}-\frac {b \log \left (c^2 x^2+1\right )}{2 c}\right )\right )}{d^2}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d^2 x^2 \left (c^2 x^2+1\right )}+\frac {b c \left (-\frac {2 c^2 x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}-\frac {a+b \text {arcsinh}(c x)}{x \sqrt {c^2 x^2+1}}+\frac {1}{2} b c \left (\log \left (c^2 x^2+1\right )+\log \left (x^2\right )\right )\right )}{d^2}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle -\frac {2 c^2 \left (2 i \left (-i b \left (\frac {1}{4} b \int e^{-2 \text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )de^{2 \text {arcsinh}(c x)}-\frac {1}{2} \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+i b \left (\frac {1}{4} b \int e^{-2 \text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )de^{2 \text {arcsinh}(c x)}-\frac {1}{2} \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+i \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2\right )+\frac {(a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}-b c \left (\frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}-\frac {b \log \left (c^2 x^2+1\right )}{2 c}\right )\right )}{d^2}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d^2 x^2 \left (c^2 x^2+1\right )}+\frac {b c \left (-\frac {2 c^2 x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}-\frac {a+b \text {arcsinh}(c x)}{x \sqrt {c^2 x^2+1}}+\frac {1}{2} b c \left (\log \left (c^2 x^2+1\right )+\log \left (x^2\right )\right )\right )}{d^2}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle -\frac {2 c^2 \left (2 i \left (i \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2-i b \left (\frac {1}{4} b \operatorname {PolyLog}\left (3,-e^{2 \text {arcsinh}(c x)}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+i b \left (\frac {1}{4} b \operatorname {PolyLog}\left (3,e^{2 \text {arcsinh}(c x)}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )\right )+\frac {(a+b \text {arcsinh}(c x))^2}{2 \left (c^2 x^2+1\right )}-b c \left (\frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}-\frac {b \log \left (c^2 x^2+1\right )}{2 c}\right )\right )}{d^2}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d^2 x^2 \left (c^2 x^2+1\right )}+\frac {b c \left (-\frac {2 c^2 x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}-\frac {a+b \text {arcsinh}(c x)}{x \sqrt {c^2 x^2+1}}+\frac {1}{2} b c \left (\log \left (c^2 x^2+1\right )+\log \left (x^2\right )\right )\right )}{d^2}\) |
Input:
Int[(a + b*ArcSinh[c*x])^2/(x^3*(d + c^2*d*x^2)^2),x]
Output:
-1/2*(a + b*ArcSinh[c*x])^2/(d^2*x^2*(1 + c^2*x^2)) + (b*c*(-((a + b*ArcSi nh[c*x])/(x*Sqrt[1 + c^2*x^2])) - (2*c^2*x*(a + b*ArcSinh[c*x]))/Sqrt[1 + c^2*x^2] + (b*c*(Log[x^2] + Log[1 + c^2*x^2]))/2))/d^2 - (2*c^2*((a + b*Ar cSinh[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)) + (2*I)*(I*(a + b*ArcSinh[c*x])^2*Ar cTanh[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*A rcSinh[c*x])])/4))))/d^2
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_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ .), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 ] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
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_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S ymbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x , x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ [(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_.)/((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. \(585\) vs. \(2(281)=562\).
Time = 1.61 (sec) , antiderivative size = 586, normalized size of antiderivative = 2.40
| method | result | size |
| derivativedivides | \(c^{2} \left (\frac {a^{2} \left (\ln \left (c^{2} x^{2}+1\right )-\frac {1}{2 \left (c^{2} x^{2}+1\right )}-\frac {1}{2 c^{2} x^{2}}-2 \ln \left (x c \right )\right )}{d^{2}}+\frac {b^{2} \left (-\frac {\left (2 \,\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}+2 \sqrt {c^{2} x^{2}+1}\, x c +\operatorname {arcsinh}\left (x c \right )\right ) \operatorname {arcsinh}\left (x c \right )}{2 \left (c^{2} x^{2}+1\right ) x^{2} c^{2}}-\ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+\ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )+\ln \left (x c +\sqrt {c^{2} x^{2}+1}-1\right )-2 \operatorname {arcsinh}\left (x c \right )^{2} \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )-4 \,\operatorname {arcsinh}\left (x c \right ) \operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right )+4 \operatorname {polylog}\left (3, -x c -\sqrt {c^{2} x^{2}+1}\right )-2 \operatorname {arcsinh}\left (x c \right )^{2} \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right )-4 \,\operatorname {arcsinh}\left (x c \right ) \operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right )+4 \operatorname {polylog}\left (3, x c +\sqrt {c^{2} x^{2}+1}\right )+2 \operatorname {arcsinh}\left (x c \right )^{2} \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+2 \,\operatorname {arcsinh}\left (x c \right ) \operatorname {polylog}\left (2, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )-\operatorname {polylog}\left (3, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )\right )}{d^{2}}+\frac {2 a b \left (-\frac {2 \,\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}\, x c +\operatorname {arcsinh}\left (x c \right )}{2 \left (c^{2} x^{2}+1\right ) x^{2} c^{2}}+2 \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+\operatorname {polylog}\left (2, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )-2 \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right )-2 \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right )\right )}{d^{2}}\right )\) | \(586\) |
| default | \(c^{2} \left (\frac {a^{2} \left (\ln \left (c^{2} x^{2}+1\right )-\frac {1}{2 \left (c^{2} x^{2}+1\right )}-\frac {1}{2 c^{2} x^{2}}-2 \ln \left (x c \right )\right )}{d^{2}}+\frac {b^{2} \left (-\frac {\left (2 \,\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}+2 \sqrt {c^{2} x^{2}+1}\, x c +\operatorname {arcsinh}\left (x c \right )\right ) \operatorname {arcsinh}\left (x c \right )}{2 \left (c^{2} x^{2}+1\right ) x^{2} c^{2}}-\ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+\ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )+\ln \left (x c +\sqrt {c^{2} x^{2}+1}-1\right )-2 \operatorname {arcsinh}\left (x c \right )^{2} \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )-4 \,\operatorname {arcsinh}\left (x c \right ) \operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right )+4 \operatorname {polylog}\left (3, -x c -\sqrt {c^{2} x^{2}+1}\right )-2 \operatorname {arcsinh}\left (x c \right )^{2} \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right )-4 \,\operatorname {arcsinh}\left (x c \right ) \operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right )+4 \operatorname {polylog}\left (3, x c +\sqrt {c^{2} x^{2}+1}\right )+2 \operatorname {arcsinh}\left (x c \right )^{2} \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+2 \,\operatorname {arcsinh}\left (x c \right ) \operatorname {polylog}\left (2, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )-\operatorname {polylog}\left (3, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )\right )}{d^{2}}+\frac {2 a b \left (-\frac {2 \,\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}\, x c +\operatorname {arcsinh}\left (x c \right )}{2 \left (c^{2} x^{2}+1\right ) x^{2} c^{2}}+2 \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+\operatorname {polylog}\left (2, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )-2 \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right )-2 \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right )\right )}{d^{2}}\right )\) | \(586\) |
| parts | \(\frac {a^{2} \left (-\frac {1}{2 x^{2}}-2 c^{2} \ln \left (x \right )+\frac {c^{4} \left (-\frac {1}{c^{2} \left (c^{2} x^{2}+1\right )}+\frac {2 \ln \left (c^{2} x^{2}+1\right )}{c^{2}}\right )}{2}\right )}{d^{2}}+\frac {b^{2} c^{2} \left (-\frac {\left (2 \,\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}+2 \sqrt {c^{2} x^{2}+1}\, x c +\operatorname {arcsinh}\left (x c \right )\right ) \operatorname {arcsinh}\left (x c \right )}{2 \left (c^{2} x^{2}+1\right ) x^{2} c^{2}}-\ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+\ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )+\ln \left (x c +\sqrt {c^{2} x^{2}+1}-1\right )-2 \operatorname {arcsinh}\left (x c \right )^{2} \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )-4 \,\operatorname {arcsinh}\left (x c \right ) \operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right )+4 \operatorname {polylog}\left (3, -x c -\sqrt {c^{2} x^{2}+1}\right )-2 \operatorname {arcsinh}\left (x c \right )^{2} \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right )-4 \,\operatorname {arcsinh}\left (x c \right ) \operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right )+4 \operatorname {polylog}\left (3, x c +\sqrt {c^{2} x^{2}+1}\right )+2 \operatorname {arcsinh}\left (x c \right )^{2} \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+2 \,\operatorname {arcsinh}\left (x c \right ) \operatorname {polylog}\left (2, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )-\operatorname {polylog}\left (3, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )\right )}{d^{2}}+\frac {2 a b \,c^{2} \left (-\frac {2 \,\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}\, x c +\operatorname {arcsinh}\left (x c \right )}{2 \left (c^{2} x^{2}+1\right ) x^{2} c^{2}}+2 \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+\operatorname {polylog}\left (2, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )-2 \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right )-2 \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right )\right )}{d^{2}}\) | \(600\) |
Input:
int((a+b*arcsinh(x*c))^2/x^3/(c^2*d*x^2+d)^2,x,method=_RETURNVERBOSE)
Output:
c^2*(a^2/d^2*(ln(c^2*x^2+1)-1/2/(c^2*x^2+1)-1/2/c^2/x^2-2*ln(x*c))+b^2/d^2 *(-1/2*(2*arcsinh(x*c)*c^2*x^2+2*(c^2*x^2+1)^(1/2)*x*c+arcsinh(x*c))*arcsi nh(x*c)/(c^2*x^2+1)/x^2/c^2-ln(1+(x*c+(c^2*x^2+1)^(1/2))^2)+ln(1+x*c+(c^2* x^2+1)^(1/2))+ln(x*c+(c^2*x^2+1)^(1/2)-1)-2*arcsinh(x*c)^2*ln(1+x*c+(c^2*x ^2+1)^(1/2))-4*arcsinh(x*c)*polylog(2,-x*c-(c^2*x^2+1)^(1/2))+4*polylog(3, -x*c-(c^2*x^2+1)^(1/2))-2*arcsinh(x*c)^2*ln(1-x*c-(c^2*x^2+1)^(1/2))-4*arc sinh(x*c)*polylog(2,x*c+(c^2*x^2+1)^(1/2))+4*polylog(3,x*c+(c^2*x^2+1)^(1/ 2))+2*arcsinh(x*c)^2*ln(1+(x*c+(c^2*x^2+1)^(1/2))^2)+2*arcsinh(x*c)*polylo g(2,-(x*c+(c^2*x^2+1)^(1/2))^2)-polylog(3,-(x*c+(c^2*x^2+1)^(1/2))^2))+2*a *b/d^2*(-1/2*(2*arcsinh(x*c)*c^2*x^2+(c^2*x^2+1)^(1/2)*x*c+arcsinh(x*c))/( c^2*x^2+1)/x^2/c^2+2*arcsinh(x*c)*ln(1+(x*c+(c^2*x^2+1)^(1/2))^2)+polylog( 2,-(x*c+(c^2*x^2+1)^(1/2))^2)-2*arcsinh(x*c)*ln(1-x*c-(c^2*x^2+1)^(1/2))-2 *polylog(2,x*c+(c^2*x^2+1)^(1/2))-2*arcsinh(x*c)*ln(1+x*c+(c^2*x^2+1)^(1/2 ))-2*polylog(2,-x*c-(c^2*x^2+1)^(1/2))))
\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^3 \left (d+c^2 d x^2\right )^2} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} + d\right )}^{2} x^{3}} \,d x } \] Input:
integrate((a+b*arcsinh(c*x))^2/x^3/(c^2*d*x^2+d)^2,x, algorithm="fricas")
Output:
integral((b^2*arcsinh(c*x)^2 + 2*a*b*arcsinh(c*x) + a^2)/(c^4*d^2*x^7 + 2* c^2*d^2*x^5 + d^2*x^3), x)
\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^3 \left (d+c^2 d x^2\right )^2} \, dx=\frac {\int \frac {a^{2}}{c^{4} x^{7} + 2 c^{2} x^{5} + x^{3}}\, dx + \int \frac {b^{2} \operatorname {asinh}^{2}{\left (c x \right )}}{c^{4} x^{7} + 2 c^{2} x^{5} + x^{3}}\, dx + \int \frac {2 a b \operatorname {asinh}{\left (c x \right )}}{c^{4} x^{7} + 2 c^{2} x^{5} + x^{3}}\, dx}{d^{2}} \] Input:
integrate((a+b*asinh(c*x))**2/x**3/(c**2*d*x**2+d)**2,x)
Output:
(Integral(a**2/(c**4*x**7 + 2*c**2*x**5 + x**3), x) + Integral(b**2*asinh( c*x)**2/(c**4*x**7 + 2*c**2*x**5 + x**3), x) + Integral(2*a*b*asinh(c*x)/( c**4*x**7 + 2*c**2*x**5 + x**3), x))/d**2
\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^3 \left (d+c^2 d x^2\right )^2} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} + d\right )}^{2} x^{3}} \,d x } \] Input:
integrate((a+b*arcsinh(c*x))^2/x^3/(c^2*d*x^2+d)^2,x, algorithm="maxima")
Output:
1/2*a^2*(2*c^2*log(c^2*x^2 + 1)/d^2 - 4*c^2*log(x)/d^2 - (2*c^2*x^2 + 1)/( c^2*d^2*x^4 + d^2*x^2)) + integrate(b^2*log(c*x + sqrt(c^2*x^2 + 1))^2/(c^ 4*d^2*x^7 + 2*c^2*d^2*x^5 + d^2*x^3) + 2*a*b*log(c*x + sqrt(c^2*x^2 + 1))/ (c^4*d^2*x^7 + 2*c^2*d^2*x^5 + d^2*x^3), x)
\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^3 \left (d+c^2 d x^2\right )^2} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} + d\right )}^{2} x^{3}} \,d x } \] Input:
integrate((a+b*arcsinh(c*x))^2/x^3/(c^2*d*x^2+d)^2,x, algorithm="giac")
Output:
integrate((b*arcsinh(c*x) + a)^2/((c^2*d*x^2 + d)^2*x^3), x)
Timed out. \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^3 \left (d+c^2 d x^2\right )^2} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{x^3\,{\left (d\,c^2\,x^2+d\right )}^2} \,d x \] Input:
int((a + b*asinh(c*x))^2/(x^3*(d + c^2*d*x^2)^2),x)
Output:
int((a + b*asinh(c*x))^2/(x^3*(d + c^2*d*x^2)^2), x)
\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^3 \left (d+c^2 d x^2\right )^2} \, dx=\frac {4 \left (\int \frac {\mathit {asinh} \left (c x \right )}{c^{4} x^{7}+2 c^{2} x^{5}+x^{3}}d x \right ) a b \,c^{2} x^{4}+4 \left (\int \frac {\mathit {asinh} \left (c x \right )}{c^{4} x^{7}+2 c^{2} x^{5}+x^{3}}d x \right ) a b \,x^{2}+2 \left (\int \frac {\mathit {asinh} \left (c x \right )^{2}}{c^{4} x^{7}+2 c^{2} x^{5}+x^{3}}d x \right ) b^{2} c^{2} x^{4}+2 \left (\int \frac {\mathit {asinh} \left (c x \right )^{2}}{c^{4} x^{7}+2 c^{2} x^{5}+x^{3}}d x \right ) b^{2} x^{2}+2 \,\mathrm {log}\left (c^{2} x^{2}+1\right ) a^{2} c^{4} x^{4}+2 \,\mathrm {log}\left (c^{2} x^{2}+1\right ) a^{2} c^{2} x^{2}-4 \,\mathrm {log}\left (x \right ) a^{2} c^{4} x^{4}-4 \,\mathrm {log}\left (x \right ) a^{2} c^{2} x^{2}+2 a^{2} c^{4} x^{4}-a^{2}}{2 d^{2} x^{2} \left (c^{2} x^{2}+1\right )} \] Input:
int((a+b*asinh(c*x))^2/x^3/(c^2*d*x^2+d)^2,x)
Output:
(4*int(asinh(c*x)/(c**4*x**7 + 2*c**2*x**5 + x**3),x)*a*b*c**2*x**4 + 4*in t(asinh(c*x)/(c**4*x**7 + 2*c**2*x**5 + x**3),x)*a*b*x**2 + 2*int(asinh(c* x)**2/(c**4*x**7 + 2*c**2*x**5 + x**3),x)*b**2*c**2*x**4 + 2*int(asinh(c*x )**2/(c**4*x**7 + 2*c**2*x**5 + x**3),x)*b**2*x**2 + 2*log(c**2*x**2 + 1)* a**2*c**4*x**4 + 2*log(c**2*x**2 + 1)*a**2*c**2*x**2 - 4*log(x)*a**2*c**4* x**4 - 4*log(x)*a**2*c**2*x**2 + 2*a**2*c**4*x**4 - a**2)/(2*d**2*x**2*(c* *2*x**2 + 1))