Optimal. Leaf size=196 \[ -\frac{2 a b x^2 \text{PolyLog}\left (2,-e^{c+d x^2}\right )}{d^2}+\frac{2 a b x^2 \text{PolyLog}\left (2,e^{c+d x^2}\right )}{d^2}+\frac{2 a b \text{PolyLog}\left (3,-e^{c+d x^2}\right )}{d^3}-\frac{2 a b \text{PolyLog}\left (3,e^{c+d x^2}\right )}{d^3}+\frac{b^2 \text{PolyLog}\left (2,e^{2 \left (c+d x^2\right )}\right )}{2 d^3}+\frac{a^2 x^6}{6}-\frac{2 a b x^4 \tanh ^{-1}\left (e^{c+d x^2}\right )}{d}+\frac{b^2 x^2 \log \left (1-e^{2 \left (c+d x^2\right )}\right )}{d^2}-\frac{b^2 x^4 \coth \left (c+d x^2\right )}{2 d}-\frac{b^2 x^4}{2 d} \]
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Rubi [A] time = 0.375086, antiderivative size = 196, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 11, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.611, Rules used = {5437, 4190, 4182, 2531, 2282, 6589, 4184, 3716, 2190, 2279, 2391} \[ -\frac{2 a b x^2 \text{PolyLog}\left (2,-e^{c+d x^2}\right )}{d^2}+\frac{2 a b x^2 \text{PolyLog}\left (2,e^{c+d x^2}\right )}{d^2}+\frac{2 a b \text{PolyLog}\left (3,-e^{c+d x^2}\right )}{d^3}-\frac{2 a b \text{PolyLog}\left (3,e^{c+d x^2}\right )}{d^3}+\frac{b^2 \text{PolyLog}\left (2,e^{2 \left (c+d x^2\right )}\right )}{2 d^3}+\frac{a^2 x^6}{6}-\frac{2 a b x^4 \tanh ^{-1}\left (e^{c+d x^2}\right )}{d}+\frac{b^2 x^2 \log \left (1-e^{2 \left (c+d x^2\right )}\right )}{d^2}-\frac{b^2 x^4 \coth \left (c+d x^2\right )}{2 d}-\frac{b^2 x^4}{2 d} \]
Antiderivative was successfully verified.
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Rule 5437
Rule 4190
Rule 4182
Rule 2531
Rule 2282
Rule 6589
Rule 4184
Rule 3716
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int x^5 \left (a+b \text{csch}\left (c+d x^2\right )\right )^2 \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x^2 (a+b \text{csch}(c+d x))^2 \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (a^2 x^2+2 a b x^2 \text{csch}(c+d x)+b^2 x^2 \text{csch}^2(c+d x)\right ) \, dx,x,x^2\right )\\ &=\frac{a^2 x^6}{6}+(a b) \operatorname{Subst}\left (\int x^2 \text{csch}(c+d x) \, dx,x,x^2\right )+\frac{1}{2} b^2 \operatorname{Subst}\left (\int x^2 \text{csch}^2(c+d x) \, dx,x,x^2\right )\\ &=\frac{a^2 x^6}{6}-\frac{2 a b x^4 \tanh ^{-1}\left (e^{c+d x^2}\right )}{d}-\frac{b^2 x^4 \coth \left (c+d x^2\right )}{2 d}-\frac{(2 a b) \operatorname{Subst}\left (\int x \log \left (1-e^{c+d x}\right ) \, dx,x,x^2\right )}{d}+\frac{(2 a b) \operatorname{Subst}\left (\int x \log \left (1+e^{c+d x}\right ) \, dx,x,x^2\right )}{d}+\frac{b^2 \operatorname{Subst}\left (\int x \coth (c+d x) \, dx,x,x^2\right )}{d}\\ &=-\frac{b^2 x^4}{2 d}+\frac{a^2 x^6}{6}-\frac{2 a b x^4 \tanh ^{-1}\left (e^{c+d x^2}\right )}{d}-\frac{b^2 x^4 \coth \left (c+d x^2\right )}{2 d}-\frac{2 a b x^2 \text{Li}_2\left (-e^{c+d x^2}\right )}{d^2}+\frac{2 a b x^2 \text{Li}_2\left (e^{c+d x^2}\right )}{d^2}+\frac{(2 a b) \operatorname{Subst}\left (\int \text{Li}_2\left (-e^{c+d x}\right ) \, dx,x,x^2\right )}{d^2}-\frac{(2 a b) \operatorname{Subst}\left (\int \text{Li}_2\left (e^{c+d x}\right ) \, dx,x,x^2\right )}{d^2}-\frac{\left (2 b^2\right ) \operatorname{Subst}\left (\int \frac{e^{2 (c+d x)} x}{1-e^{2 (c+d x)}} \, dx,x,x^2\right )}{d}\\ &=-\frac{b^2 x^4}{2 d}+\frac{a^2 x^6}{6}-\frac{2 a b x^4 \tanh ^{-1}\left (e^{c+d x^2}\right )}{d}-\frac{b^2 x^4 \coth \left (c+d x^2\right )}{2 d}+\frac{b^2 x^2 \log \left (1-e^{2 \left (c+d x^2\right )}\right )}{d^2}-\frac{2 a b x^2 \text{Li}_2\left (-e^{c+d x^2}\right )}{d^2}+\frac{2 a b x^2 \text{Li}_2\left (e^{c+d x^2}\right )}{d^2}+\frac{(2 a b) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{c+d x^2}\right )}{d^3}-\frac{(2 a b) \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{c+d x^2}\right )}{d^3}-\frac{b^2 \operatorname{Subst}\left (\int \log \left (1-e^{2 (c+d x)}\right ) \, dx,x,x^2\right )}{d^2}\\ &=-\frac{b^2 x^4}{2 d}+\frac{a^2 x^6}{6}-\frac{2 a b x^4 \tanh ^{-1}\left (e^{c+d x^2}\right )}{d}-\frac{b^2 x^4 \coth \left (c+d x^2\right )}{2 d}+\frac{b^2 x^2 \log \left (1-e^{2 \left (c+d x^2\right )}\right )}{d^2}-\frac{2 a b x^2 \text{Li}_2\left (-e^{c+d x^2}\right )}{d^2}+\frac{2 a b x^2 \text{Li}_2\left (e^{c+d x^2}\right )}{d^2}+\frac{2 a b \text{Li}_3\left (-e^{c+d x^2}\right )}{d^3}-\frac{2 a b \text{Li}_3\left (e^{c+d x^2}\right )}{d^3}-\frac{b^2 \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 \left (c+d x^2\right )}\right )}{2 d^3}\\ &=-\frac{b^2 x^4}{2 d}+\frac{a^2 x^6}{6}-\frac{2 a b x^4 \tanh ^{-1}\left (e^{c+d x^2}\right )}{d}-\frac{b^2 x^4 \coth \left (c+d x^2\right )}{2 d}+\frac{b^2 x^2 \log \left (1-e^{2 \left (c+d x^2\right )}\right )}{d^2}-\frac{2 a b x^2 \text{Li}_2\left (-e^{c+d x^2}\right )}{d^2}+\frac{2 a b x^2 \text{Li}_2\left (e^{c+d x^2}\right )}{d^2}+\frac{b^2 \text{Li}_2\left (e^{2 \left (c+d x^2\right )}\right )}{2 d^3}+\frac{2 a b \text{Li}_3\left (-e^{c+d x^2}\right )}{d^3}-\frac{2 a b \text{Li}_3\left (e^{c+d x^2}\right )}{d^3}\\ \end{align*}
Mathematica [B] time = 6.18077, size = 595, normalized size = 3.04 \[ -\frac{12 b \left (e^{2 c}-1\right ) \left (b-2 a d x^2\right ) \text{PolyLog}\left (2,-e^{-c-d x^2}\right )+12 b \left (e^{2 c}-1\right ) \left (2 a d x^2+b\right ) \text{PolyLog}\left (2,e^{-c-d x^2}\right )-24 a b e^{2 c} \text{PolyLog}\left (3,-e^{-c-d x^2}\right )+24 a b \text{PolyLog}\left (3,-e^{-c-d x^2}\right )+24 a b e^{2 c} \text{PolyLog}\left (3,e^{-c-d x^2}\right )-24 a b \text{PolyLog}\left (3,e^{-c-d x^2}\right )-2 a^2 e^{2 c} d^3 x^6+2 a^2 d^3 x^6-12 a b e^{2 c} d^2 x^4 \log \left (1-e^{-c-d x^2}\right )+12 a b d^2 x^4 \log \left (1-e^{-c-d x^2}\right )+12 a b e^{2 c} d^2 x^4 \log \left (e^{-c-d x^2}+1\right )-12 a b d^2 x^4 \log \left (e^{-c-d x^2}+1\right )-3 b^2 e^{2 c} d^2 x^4 \text{csch}\left (\frac{c}{2}\right ) \sinh \left (\frac{d x^2}{2}\right ) \text{csch}\left (\frac{1}{2} \left (c+d x^2\right )\right )+3 b^2 d^2 x^4 \text{csch}\left (\frac{c}{2}\right ) \sinh \left (\frac{d x^2}{2}\right ) \text{csch}\left (\frac{1}{2} \left (c+d x^2\right )\right )+3 b^2 e^{2 c} d^2 x^4 \text{sech}\left (\frac{c}{2}\right ) \sinh \left (\frac{d x^2}{2}\right ) \text{sech}\left (\frac{1}{2} \left (c+d x^2\right )\right )-3 b^2 d^2 x^4 \text{sech}\left (\frac{c}{2}\right ) \sinh \left (\frac{d x^2}{2}\right ) \text{sech}\left (\frac{1}{2} \left (c+d x^2\right )\right )-12 b^2 e^{2 c} d x^2 \log \left (1-e^{-c-d x^2}\right )+12 b^2 d x^2 \log \left (1-e^{-c-d x^2}\right )-12 b^2 e^{2 c} d x^2 \log \left (e^{-c-d x^2}+1\right )+12 b^2 d x^2 \log \left (e^{-c-d x^2}+1\right )+12 b^2 d^2 x^4}{12 \left (e^{2 c}-1\right ) d^3} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.12, size = 0, normalized size = 0. \begin{align*} \int{x}^{5} \left ( a+b{\rm csch} \left (d{x}^{2}+c\right ) \right ) ^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.0402, size = 366, normalized size = 1.87 \begin{align*} \frac{1}{6} \, a^{2} x^{6} - \frac{b^{2} x^{4}}{d e^{\left (2 \, d x^{2} + 2 \, c\right )} - d} - \frac{{\left (d^{2} x^{4} \log \left (e^{\left (d x^{2} + c\right )} + 1\right ) + 2 \, d x^{2}{\rm Li}_2\left (-e^{\left (d x^{2} + c\right )}\right ) - 2 \,{\rm Li}_{3}(-e^{\left (d x^{2} + c\right )})\right )} a b}{d^{3}} + \frac{{\left (d^{2} x^{4} \log \left (-e^{\left (d x^{2} + c\right )} + 1\right ) + 2 \, d x^{2}{\rm Li}_2\left (e^{\left (d x^{2} + c\right )}\right ) - 2 \,{\rm Li}_{3}(e^{\left (d x^{2} + c\right )})\right )} a b}{d^{3}} + \frac{{\left (d x^{2} \log \left (e^{\left (d x^{2} + c\right )} + 1\right ) +{\rm Li}_2\left (-e^{\left (d x^{2} + c\right )}\right )\right )} b^{2}}{d^{3}} + \frac{{\left (d x^{2} \log \left (-e^{\left (d x^{2} + c\right )} + 1\right ) +{\rm Li}_2\left (e^{\left (d x^{2} + c\right )}\right )\right )} b^{2}}{d^{3}} - \frac{2 \, a b d^{3} x^{6} + 3 \, b^{2} d^{2} x^{4}}{6 \, d^{3}} + \frac{2 \, a b d^{3} x^{6} - 3 \, b^{2} d^{2} x^{4}}{6 \, d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 1.71733, size = 2354, normalized size = 12.01 \begin{align*} \text{result too large to display} \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*} \int x^{5} \left (a + b \operatorname{csch}{\left (c + d x^{2} \right )}\right )^{2}\, dx \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{\left (b \operatorname{csch}\left (d x^{2} + c\right ) + a\right )}^{2} x^{5}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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