Optimal. Leaf size=68 \[ \frac {a x^4}{4}-\frac {b \text {Li}_2\left (-e^{d x^2+c}\right )}{2 d^2}+\frac {b \text {Li}_2\left (e^{d x^2+c}\right )}{2 d^2}-\frac {b x^2 \tanh ^{-1}\left (e^{c+d x^2}\right )}{d} \]
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Rubi [A] time = 0.08, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {14, 5437, 4182, 2279, 2391} \[ -\frac {b \text {PolyLog}\left (2,-e^{c+d x^2}\right )}{2 d^2}+\frac {b \text {PolyLog}\left (2,e^{c+d x^2}\right )}{2 d^2}+\frac {a x^4}{4}-\frac {b x^2 \tanh ^{-1}\left (e^{c+d x^2}\right )}{d} \]
Antiderivative was successfully verified.
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Rule 14
Rule 2279
Rule 2391
Rule 4182
Rule 5437
Rubi steps
\begin {align*} \int x^3 \left (a+b \text {csch}\left (c+d x^2\right )\right ) \, dx &=\int \left (a x^3+b x^3 \text {csch}\left (c+d x^2\right )\right ) \, dx\\ &=\frac {a x^4}{4}+b \int x^3 \text {csch}\left (c+d x^2\right ) \, dx\\ &=\frac {a x^4}{4}+\frac {1}{2} b \operatorname {Subst}\left (\int x \text {csch}(c+d x) \, dx,x,x^2\right )\\ &=\frac {a x^4}{4}-\frac {b x^2 \tanh ^{-1}\left (e^{c+d x^2}\right )}{d}-\frac {b \operatorname {Subst}\left (\int \log \left (1-e^{c+d x}\right ) \, dx,x,x^2\right )}{2 d}+\frac {b \operatorname {Subst}\left (\int \log \left (1+e^{c+d x}\right ) \, dx,x,x^2\right )}{2 d}\\ &=\frac {a x^4}{4}-\frac {b x^2 \tanh ^{-1}\left (e^{c+d x^2}\right )}{d}-\frac {b \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{c+d x^2}\right )}{2 d^2}+\frac {b \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{c+d x^2}\right )}{2 d^2}\\ &=\frac {a x^4}{4}-\frac {b x^2 \tanh ^{-1}\left (e^{c+d x^2}\right )}{d}-\frac {b \text {Li}_2\left (-e^{c+d x^2}\right )}{2 d^2}+\frac {b \text {Li}_2\left (e^{c+d x^2}\right )}{2 d^2}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 108, normalized size = 1.59 \[ \frac {1}{4} \left (a x^4+\frac {2 b \left (\text {Li}_2\left (-e^{-d x^2-c}\right )-\text {Li}_2\left (e^{-d x^2-c}\right )+\left (c+d x^2\right ) \left (\log \left (1-e^{-c-d x^2}\right )-\log \left (e^{-c-d x^2}+1\right )\right )-c \log \left (\tanh \left (\frac {1}{2} \left (c+d x^2\right )\right )\right )\right )}{d^2}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 144, normalized size = 2.12 \[ \frac {a d^{2} x^{4} - 2 \, b d x^{2} \log \left (\cosh \left (d x^{2} + c\right ) + \sinh \left (d x^{2} + c\right ) + 1\right ) - 2 \, b c \log \left (\cosh \left (d x^{2} + c\right ) + \sinh \left (d x^{2} + c\right ) - 1\right ) + 2 \, b {\rm Li}_2\left (\cosh \left (d x^{2} + c\right ) + \sinh \left (d x^{2} + c\right )\right ) - 2 \, b {\rm Li}_2\left (-\cosh \left (d x^{2} + c\right ) - \sinh \left (d x^{2} + c\right )\right ) + 2 \, {\left (b d x^{2} + b c\right )} \log \left (-\cosh \left (d x^{2} + c\right ) - \sinh \left (d x^{2} + c\right ) + 1\right )}{4 \, d^{2}} \]
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 \[ \int {\left (b \operatorname {csch}\left (d x^{2} + c\right ) + a\right )} x^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.33, size = 0, normalized size = 0.00 \[ \int x^{3} \left (a +b \,\mathrm {csch}\left (d \,x^{2}+c \right )\right )\, dx \]
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 \[ \frac {1}{4} \, a x^{4} + 2 \, b \int \frac {x^{3}}{e^{\left (d x^{2} + c\right )} - e^{\left (-d x^{2} - c\right )}}\,{d x} \]
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 \[ \int x^3\,\left (a+\frac {b}{\mathrm {sinh}\left (d\,x^2+c\right )}\right ) \,d x \]
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 \[ \int x^{3} \left (a + b \operatorname {csch}{\left (c + d x^{2} \right )}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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