Optimal. Leaf size=279 \[ \frac{3 i b \text{PolyLog}\left (2,-i e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{c^5 d^2}-\frac{3 i b \text{PolyLog}\left (2,i e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{c^5 d^2}-\frac{3 i b^2 \text{PolyLog}\left (3,-i e^{\sinh ^{-1}(c x)}\right )}{c^5 d^2}+\frac{3 i b^2 \text{PolyLog}\left (3,i e^{\sinh ^{-1}(c x)}\right )}{c^5 d^2}-\frac{x^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^2 d^2 \left (c^2 x^2+1\right )}-\frac{2 b \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{c^5 d^2}+\frac{b \left (a+b \sinh ^{-1}(c x)\right )}{c^5 d^2 \sqrt{c^2 x^2+1}}+\frac{3 x \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^4 d^2}-\frac{3 \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{c^5 d^2}+\frac{2 b^2 x}{c^4 d^2}-\frac{b^2 \tan ^{-1}(c x)}{c^5 d^2} \]
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Rubi [A] time = 0.531413, antiderivative size = 279, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 14, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.538, Rules used = {5751, 5767, 5693, 4180, 2531, 2282, 6589, 5717, 8, 266, 43, 5732, 388, 205} \[ \frac{3 i b \text{PolyLog}\left (2,-i e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{c^5 d^2}-\frac{3 i b \text{PolyLog}\left (2,i e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{c^5 d^2}-\frac{3 i b^2 \text{PolyLog}\left (3,-i e^{\sinh ^{-1}(c x)}\right )}{c^5 d^2}+\frac{3 i b^2 \text{PolyLog}\left (3,i e^{\sinh ^{-1}(c x)}\right )}{c^5 d^2}-\frac{x^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^2 d^2 \left (c^2 x^2+1\right )}-\frac{2 b \sqrt{c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{c^5 d^2}+\frac{b \left (a+b \sinh ^{-1}(c x)\right )}{c^5 d^2 \sqrt{c^2 x^2+1}}+\frac{3 x \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^4 d^2}-\frac{3 \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{c^5 d^2}+\frac{2 b^2 x}{c^4 d^2}-\frac{b^2 \tan ^{-1}(c x)}{c^5 d^2} \]
Antiderivative was successfully verified.
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Rule 5751
Rule 5767
Rule 5693
Rule 4180
Rule 2531
Rule 2282
Rule 6589
Rule 5717
Rule 8
Rule 266
Rule 43
Rule 5732
Rule 388
Rule 205
Rubi steps
\begin{align*} \int \frac{x^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{\left (d+c^2 d x^2\right )^2} \, dx &=-\frac{x^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^2 d^2 \left (1+c^2 x^2\right )}+\frac{b \int \frac{x^3 \left (a+b \sinh ^{-1}(c x)\right )}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{c d^2}+\frac{3 \int \frac{x^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{d+c^2 d x^2} \, dx}{2 c^2 d}\\ &=\frac{b \left (a+b \sinh ^{-1}(c x)\right )}{c^5 d^2 \sqrt{1+c^2 x^2}}+\frac{b \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{c^5 d^2}+\frac{3 x \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^4 d^2}-\frac{x^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^2 d^2 \left (1+c^2 x^2\right )}-\frac{b^2 \int \frac{2+c^2 x^2}{c^4+c^6 x^2} \, dx}{d^2}-\frac{(3 b) \int \frac{x \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{1+c^2 x^2}} \, dx}{c^3 d^2}-\frac{3 \int \frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{d+c^2 d x^2} \, dx}{2 c^4 d}\\ &=-\frac{b^2 x}{c^4 d^2}+\frac{b \left (a+b \sinh ^{-1}(c x)\right )}{c^5 d^2 \sqrt{1+c^2 x^2}}-\frac{2 b \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{c^5 d^2}+\frac{3 x \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^4 d^2}-\frac{x^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^2 d^2 \left (1+c^2 x^2\right )}-\frac{b^2 \int \frac{1}{c^4+c^6 x^2} \, dx}{d^2}-\frac{3 \operatorname{Subst}\left (\int (a+b x)^2 \text{sech}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{2 c^5 d^2}+\frac{\left (3 b^2\right ) \int 1 \, dx}{c^4 d^2}\\ &=\frac{2 b^2 x}{c^4 d^2}+\frac{b \left (a+b \sinh ^{-1}(c x)\right )}{c^5 d^2 \sqrt{1+c^2 x^2}}-\frac{2 b \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{c^5 d^2}+\frac{3 x \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^4 d^2}-\frac{x^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^2 d^2 \left (1+c^2 x^2\right )}-\frac{3 \left (a+b \sinh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{c^5 d^2}-\frac{b^2 \tan ^{-1}(c x)}{c^5 d^2}+\frac{(3 i b) \operatorname{Subst}\left (\int (a+b x) \log \left (1-i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{c^5 d^2}-\frac{(3 i b) \operatorname{Subst}\left (\int (a+b x) \log \left (1+i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{c^5 d^2}\\ &=\frac{2 b^2 x}{c^4 d^2}+\frac{b \left (a+b \sinh ^{-1}(c x)\right )}{c^5 d^2 \sqrt{1+c^2 x^2}}-\frac{2 b \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{c^5 d^2}+\frac{3 x \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^4 d^2}-\frac{x^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^2 d^2 \left (1+c^2 x^2\right )}-\frac{3 \left (a+b \sinh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{c^5 d^2}-\frac{b^2 \tan ^{-1}(c x)}{c^5 d^2}+\frac{3 i b \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (-i e^{\sinh ^{-1}(c x)}\right )}{c^5 d^2}-\frac{3 i b \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (i e^{\sinh ^{-1}(c x)}\right )}{c^5 d^2}-\frac{\left (3 i b^2\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{c^5 d^2}+\frac{\left (3 i b^2\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{c^5 d^2}\\ &=\frac{2 b^2 x}{c^4 d^2}+\frac{b \left (a+b \sinh ^{-1}(c x)\right )}{c^5 d^2 \sqrt{1+c^2 x^2}}-\frac{2 b \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{c^5 d^2}+\frac{3 x \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^4 d^2}-\frac{x^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^2 d^2 \left (1+c^2 x^2\right )}-\frac{3 \left (a+b \sinh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{c^5 d^2}-\frac{b^2 \tan ^{-1}(c x)}{c^5 d^2}+\frac{3 i b \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (-i e^{\sinh ^{-1}(c x)}\right )}{c^5 d^2}-\frac{3 i b \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (i e^{\sinh ^{-1}(c x)}\right )}{c^5 d^2}-\frac{\left (3 i b^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-i x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{c^5 d^2}+\frac{\left (3 i b^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(i x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{c^5 d^2}\\ &=\frac{2 b^2 x}{c^4 d^2}+\frac{b \left (a+b \sinh ^{-1}(c x)\right )}{c^5 d^2 \sqrt{1+c^2 x^2}}-\frac{2 b \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{c^5 d^2}+\frac{3 x \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^4 d^2}-\frac{x^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 c^2 d^2 \left (1+c^2 x^2\right )}-\frac{3 \left (a+b \sinh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{c^5 d^2}-\frac{b^2 \tan ^{-1}(c x)}{c^5 d^2}+\frac{3 i b \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (-i e^{\sinh ^{-1}(c x)}\right )}{c^5 d^2}-\frac{3 i b \left (a+b \sinh ^{-1}(c x)\right ) \text{Li}_2\left (i e^{\sinh ^{-1}(c x)}\right )}{c^5 d^2}-\frac{3 i b^2 \text{Li}_3\left (-i e^{\sinh ^{-1}(c x)}\right )}{c^5 d^2}+\frac{3 i b^2 \text{Li}_3\left (i e^{\sinh ^{-1}(c x)}\right )}{c^5 d^2}\\ \end{align*}
Mathematica [A] time = 2.19116, size = 482, normalized size = 1.73 \[ \frac{-\frac{2 a b \left (-3 i \left (c^2 x^2+1\right ) \text{PolyLog}\left (2,-i e^{\sinh ^{-1}(c x)}\right )+3 i \left (c^2 x^2+1\right ) \text{PolyLog}\left (2,i e^{\sinh ^{-1}(c x)}\right )+2 c^2 x^2 \sqrt{c^2 x^2+1}+\sqrt{c^2 x^2+1}-2 c^3 x^3 \sinh ^{-1}(c x)+3 i c^2 x^2 \sinh ^{-1}(c x) \log \left (1-i e^{\sinh ^{-1}(c x)}\right )-3 i c^2 x^2 \sinh ^{-1}(c x) \log \left (1+i e^{\sinh ^{-1}(c x)}\right )-3 c x \sinh ^{-1}(c x)+3 i \sinh ^{-1}(c x) \log \left (1-i e^{\sinh ^{-1}(c x)}\right )-3 i \sinh ^{-1}(c x) \log \left (1+i e^{\sinh ^{-1}(c x)}\right )\right )}{c^7 x^2+c^5}+\frac{2 b^2 \left (\frac{1}{2} i \left (6 \sinh ^{-1}(c x) \text{PolyLog}\left (2,-i e^{-\sinh ^{-1}(c x)}\right )-6 \sinh ^{-1}(c x) \text{PolyLog}\left (2,i e^{-\sinh ^{-1}(c x)}\right )+6 \text{PolyLog}\left (3,-i e^{-\sinh ^{-1}(c x)}\right )-6 \text{PolyLog}\left (3,i e^{-\sinh ^{-1}(c x)}\right )+3 \sinh ^{-1}(c x)^2 \log \left (1-i e^{-\sinh ^{-1}(c x)}\right )-3 \sinh ^{-1}(c x)^2 \log \left (1+i e^{-\sinh ^{-1}(c x)}\right )+4 i \tan ^{-1}\left (\tanh \left (\frac{1}{2} \sinh ^{-1}(c x)\right )\right )\right )+\frac{c x \sinh ^{-1}(c x)^2}{2 c^2 x^2+2}-2 \sqrt{c^2 x^2+1} \sinh ^{-1}(c x)+\frac{\sinh ^{-1}(c x)}{\sqrt{c^2 x^2+1}}+c x \left (\sinh ^{-1}(c x)^2+2\right )\right )}{c^5}+\frac{a^2 x}{c^6 x^2+c^4}+\frac{2 a^2 x}{c^4}-\frac{3 a^2 \tan ^{-1}(c x)}{c^5}}{2 d^2} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.26, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{4} \left ( a+b{\it Arcsinh} \left ( cx \right ) \right ) ^{2}}{ \left ({c}^{2}d{x}^{2}+d \right ) ^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{2} \, a^{2}{\left (\frac{x}{c^{6} d^{2} x^{2} + c^{4} d^{2}} + \frac{2 \, x}{c^{4} d^{2}} - \frac{3 \, \arctan \left (c x\right )}{c^{5} d^{2}}\right )} + \int \frac{b^{2} x^{4} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )^{2}}{c^{4} d^{2} x^{4} + 2 \, c^{2} d^{2} x^{2} + d^{2}} + \frac{2 \, a b x^{4} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )}{c^{4} d^{2} x^{4} + 2 \, c^{2} d^{2} x^{2} + d^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b^{2} x^{4} \operatorname{arsinh}\left (c x\right )^{2} + 2 \, a b x^{4} \operatorname{arsinh}\left (c x\right ) + a^{2} x^{4}}{c^{4} d^{2} x^{4} + 2 \, c^{2} d^{2} x^{2} + d^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{a^{2} x^{4}}{c^{4} x^{4} + 2 c^{2} x^{2} + 1}\, dx + \int \frac{b^{2} x^{4} \operatorname{asinh}^{2}{\left (c x \right )}}{c^{4} x^{4} + 2 c^{2} x^{2} + 1}\, dx + \int \frac{2 a b x^{4} \operatorname{asinh}{\left (c x \right )}}{c^{4} x^{4} + 2 c^{2} x^{2} + 1}\, dx}{d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}^{2} x^{4}}{{\left (c^{2} d x^{2} + d\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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