Optimal. Leaf size=104 \[ \frac {a x^6}{6}-\frac {b x^4 \tanh ^{-1}\left (e^{c+d x^2}\right )}{d}-\frac {b x^2 \text {PolyLog}\left (2,-e^{c+d x^2}\right )}{d^2}+\frac {b x^2 \text {PolyLog}\left (2,e^{c+d x^2}\right )}{d^2}+\frac {b \text {PolyLog}\left (3,-e^{c+d x^2}\right )}{d^3}-\frac {b \text {PolyLog}\left (3,e^{c+d x^2}\right )}{d^3} \]
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Rubi [A]
time = 0.10, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 6, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {14, 5545,
4267, 2611, 2320, 6724} \begin {gather*} \frac {a x^6}{6}+\frac {b \text {Li}_3\left (-e^{d x^2+c}\right )}{d^3}-\frac {b \text {Li}_3\left (e^{d x^2+c}\right )}{d^3}-\frac {b x^2 \text {Li}_2\left (-e^{d x^2+c}\right )}{d^2}+\frac {b x^2 \text {Li}_2\left (e^{d x^2+c}\right )}{d^2}-\frac {b x^4 \tanh ^{-1}\left (e^{c+d x^2}\right )}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 2320
Rule 2611
Rule 4267
Rule 5545
Rule 6724
Rubi steps
\begin {align*} \int x^5 \left (a+b \text {csch}\left (c+d x^2\right )\right ) \, dx &=\int \left (a x^5+b x^5 \text {csch}\left (c+d x^2\right )\right ) \, dx\\ &=\frac {a x^6}{6}+b \int x^5 \text {csch}\left (c+d x^2\right ) \, dx\\ &=\frac {a x^6}{6}+\frac {1}{2} b \text {Subst}\left (\int x^2 \text {csch}(c+d x) \, dx,x,x^2\right )\\ &=\frac {a x^6}{6}-\frac {b x^4 \tanh ^{-1}\left (e^{c+d x^2}\right )}{d}-\frac {b \text {Subst}\left (\int x \log \left (1-e^{c+d x}\right ) \, dx,x,x^2\right )}{d}+\frac {b \text {Subst}\left (\int x \log \left (1+e^{c+d x}\right ) \, dx,x,x^2\right )}{d}\\ &=\frac {a x^6}{6}-\frac {b x^4 \tanh ^{-1}\left (e^{c+d x^2}\right )}{d}-\frac {b x^2 \text {Li}_2\left (-e^{c+d x^2}\right )}{d^2}+\frac {b x^2 \text {Li}_2\left (e^{c+d x^2}\right )}{d^2}+\frac {b \text {Subst}\left (\int \text {Li}_2\left (-e^{c+d x}\right ) \, dx,x,x^2\right )}{d^2}-\frac {b \text {Subst}\left (\int \text {Li}_2\left (e^{c+d x}\right ) \, dx,x,x^2\right )}{d^2}\\ &=\frac {a x^6}{6}-\frac {b x^4 \tanh ^{-1}\left (e^{c+d x^2}\right )}{d}-\frac {b x^2 \text {Li}_2\left (-e^{c+d x^2}\right )}{d^2}+\frac {b x^2 \text {Li}_2\left (e^{c+d x^2}\right )}{d^2}+\frac {b \text {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{c+d x^2}\right )}{d^3}-\frac {b \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{c+d x^2}\right )}{d^3}\\ &=\frac {a x^6}{6}-\frac {b x^4 \tanh ^{-1}\left (e^{c+d x^2}\right )}{d}-\frac {b x^2 \text {Li}_2\left (-e^{c+d x^2}\right )}{d^2}+\frac {b x^2 \text {Li}_2\left (e^{c+d x^2}\right )}{d^2}+\frac {b \text {Li}_3\left (-e^{c+d x^2}\right )}{d^3}-\frac {b \text {Li}_3\left (e^{c+d x^2}\right )}{d^3}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 138, normalized size = 1.33 \begin {gather*} \frac {a x^6}{6}-\frac {b \left (d^2 x^4 \tanh ^{-1}\left (\cosh \left (c+d x^2\right )+\sinh \left (c+d x^2\right )\right )+d x^2 \text {PolyLog}\left (2,-\cosh \left (c+d x^2\right )-\sinh \left (c+d x^2\right )\right )-d x^2 \text {PolyLog}\left (2,\cosh \left (c+d x^2\right )+\sinh \left (c+d x^2\right )\right )-\text {PolyLog}\left (3,-\cosh \left (c+d x^2\right )-\sinh \left (c+d x^2\right )\right )+\text {PolyLog}\left (3,\cosh \left (c+d x^2\right )+\sinh \left (c+d x^2\right )\right )\right )}{d^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.92, size = 0, normalized size = 0.00 \[\int x^{5} \left (a +b \,\mathrm {csch}\left (d \,x^{2}+c \right )\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 209 vs.
\(2 (95) = 190\).
time = 0.47, size = 209, normalized size = 2.01 \begin {gather*} \frac {a d^{3} x^{6} - 3 \, b d^{2} x^{4} \log \left (\cosh \left (d x^{2} + c\right ) + \sinh \left (d x^{2} + c\right ) + 1\right ) + 6 \, b d x^{2} {\rm Li}_2\left (\cosh \left (d x^{2} + c\right ) + \sinh \left (d x^{2} + c\right )\right ) - 6 \, b d x^{2} {\rm Li}_2\left (-\cosh \left (d x^{2} + c\right ) - \sinh \left (d x^{2} + c\right )\right ) + 3 \, b c^{2} \log \left (\cosh \left (d x^{2} + c\right ) + \sinh \left (d x^{2} + c\right ) - 1\right ) + 3 \, {\left (b d^{2} x^{4} - b c^{2}\right )} \log \left (-\cosh \left (d x^{2} + c\right ) - \sinh \left (d x^{2} + c\right ) + 1\right ) - 6 \, b {\rm polylog}\left (3, \cosh \left (d x^{2} + c\right ) + \sinh \left (d x^{2} + c\right )\right ) + 6 \, b {\rm polylog}\left (3, -\cosh \left (d x^{2} + c\right ) - \sinh \left (d x^{2} + c\right )\right )}{6 \, d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{5} \left (a + b \operatorname {csch}{\left (c + d x^{2} \right )}\right )\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^5\,\left (a+\frac {b}{\mathrm {sinh}\left (d\,x^2+c\right )}\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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