Optimal. Leaf size=108 \[ \frac {a^2 x^4}{4}-\frac {a b \text {Li}_2\left (-e^{d x^2+c}\right )}{d^2}+\frac {a b \text {Li}_2\left (e^{d x^2+c}\right )}{d^2}-\frac {2 a b x^2 \tanh ^{-1}\left (e^{c+d x^2}\right )}{d}+\frac {b^2 \log \left (\sinh \left (c+d x^2\right )\right )}{2 d^2}-\frac {b^2 x^2 \coth \left (c+d x^2\right )}{2 d} \]
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Rubi [A] time = 0.16, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {5437, 4190, 4182, 2279, 2391, 4184, 3475} \[ -\frac {a b \text {PolyLog}\left (2,-e^{c+d x^2}\right )}{d^2}+\frac {a b \text {PolyLog}\left (2,e^{c+d x^2}\right )}{d^2}+\frac {a^2 x^4}{4}-\frac {2 a b x^2 \tanh ^{-1}\left (e^{c+d x^2}\right )}{d}+\frac {b^2 \log \left (\sinh \left (c+d x^2\right )\right )}{2 d^2}-\frac {b^2 x^2 \coth \left (c+d x^2\right )}{2 d} \]
Antiderivative was successfully verified.
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Rule 2279
Rule 2391
Rule 3475
Rule 4182
Rule 4184
Rule 4190
Rule 5437
Rubi steps
\begin {align*} \int x^3 \left (a+b \text {csch}\left (c+d x^2\right )\right )^2 \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int x (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 a b x \text {csch}(c+d x)+b^2 x \text {csch}^2(c+d x)\right ) \, dx,x,x^2\right )\\ &=\frac {a^2 x^4}{4}+(a b) \operatorname {Subst}\left (\int x \text {csch}(c+d x) \, dx,x,x^2\right )+\frac {1}{2} b^2 \operatorname {Subst}\left (\int x \text {csch}^2(c+d x) \, dx,x,x^2\right )\\ &=\frac {a^2 x^4}{4}-\frac {2 a b x^2 \tanh ^{-1}\left (e^{c+d x^2}\right )}{d}-\frac {b^2 x^2 \coth \left (c+d x^2\right )}{2 d}-\frac {(a b) \operatorname {Subst}\left (\int \log \left (1-e^{c+d x}\right ) \, dx,x,x^2\right )}{d}+\frac {(a b) \operatorname {Subst}\left (\int \log \left (1+e^{c+d x}\right ) \, dx,x,x^2\right )}{d}+\frac {b^2 \operatorname {Subst}\left (\int \coth (c+d x) \, dx,x,x^2\right )}{2 d}\\ &=\frac {a^2 x^4}{4}-\frac {2 a b x^2 \tanh ^{-1}\left (e^{c+d x^2}\right )}{d}-\frac {b^2 x^2 \coth \left (c+d x^2\right )}{2 d}+\frac {b^2 \log \left (\sinh \left (c+d x^2\right )\right )}{2 d^2}-\frac {(a b) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{c+d x^2}\right )}{d^2}+\frac {(a b) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{c+d x^2}\right )}{d^2}\\ &=\frac {a^2 x^4}{4}-\frac {2 a b x^2 \tanh ^{-1}\left (e^{c+d x^2}\right )}{d}-\frac {b^2 x^2 \coth \left (c+d x^2\right )}{2 d}+\frac {b^2 \log \left (\sinh \left (c+d x^2\right )\right )}{2 d^2}-\frac {a b \text {Li}_2\left (-e^{c+d x^2}\right )}{d^2}+\frac {a b \text {Li}_2\left (e^{c+d x^2}\right )}{d^2}\\ \end {align*}
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Mathematica [B] time = 4.50, size = 260, normalized size = 2.41 \[ \frac {2 d x^2 \left (a^2 d x^2-2 b^2 \coth (c)\right )+8 a b \left (\frac {\text {sech}(c) \left (\text {Li}_2\left (-e^{-d x^2-\tanh ^{-1}(\tanh (c))}\right )-\text {Li}_2\left (e^{-d x^2-\tanh ^{-1}(\tanh (c))}\right )+\left (\tanh ^{-1}(\tanh (c))+d x^2\right ) \left (\log \left (1-e^{-\tanh ^{-1}(\tanh (c))-d x^2}\right )-\log \left (e^{-\tanh ^{-1}(\tanh (c))-d x^2}+1\right )\right )\right )}{\sqrt {\text {sech}^2(c)}}+2 \tanh ^{-1}(\tanh (c)) \tanh ^{-1}\left (\sinh (c) \tanh \left (\frac {d x^2}{2}\right )+\cosh (c)\right )\right )+4 b^2 d x^2 \coth (c)+2 b^2 d x^2 \text {csch}\left (\frac {c}{2}\right ) \sinh \left (\frac {d x^2}{2}\right ) \text {csch}\left (\frac {1}{2} \left (c+d x^2\right )\right )-2 b^2 d x^2 \text {sech}\left (\frac {c}{2}\right ) \sinh \left (\frac {d x^2}{2}\right ) \text {sech}\left (\frac {1}{2} \left (c+d x^2\right )\right )-4 b^2 \left (d x^2 \coth (c)-\log \left (\sinh \left (c+d x^2\right )\right )\right )}{8 d^2} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.43, size = 683, normalized size = 6.32 \[ -\frac {a^{2} d^{2} x^{4} - 4 \, b^{2} c - {\left (a^{2} d^{2} x^{4} - 4 \, b^{2} d x^{2} - 4 \, b^{2} c\right )} \cosh \left (d x^{2} + c\right )^{2} - 2 \, {\left (a^{2} d^{2} x^{4} - 4 \, b^{2} d x^{2} - 4 \, b^{2} c\right )} \cosh \left (d x^{2} + c\right ) \sinh \left (d x^{2} + c\right ) - {\left (a^{2} d^{2} x^{4} - 4 \, b^{2} d x^{2} - 4 \, b^{2} c\right )} \sinh \left (d x^{2} + c\right )^{2} - 4 \, {\left (a b \cosh \left (d x^{2} + c\right )^{2} + 2 \, a b \cosh \left (d x^{2} + c\right ) \sinh \left (d x^{2} + c\right ) + a b \sinh \left (d x^{2} + c\right )^{2} - a b\right )} {\rm Li}_2\left (\cosh \left (d x^{2} + c\right ) + \sinh \left (d x^{2} + c\right )\right ) + 4 \, {\left (a b \cosh \left (d x^{2} + c\right )^{2} + 2 \, a b \cosh \left (d x^{2} + c\right ) \sinh \left (d x^{2} + c\right ) + a b \sinh \left (d x^{2} + c\right )^{2} - a b\right )} {\rm Li}_2\left (-\cosh \left (d x^{2} + c\right ) - \sinh \left (d x^{2} + c\right )\right ) - 2 \, {\left (2 \, a b d x^{2} - {\left (2 \, a b d x^{2} - b^{2}\right )} \cosh \left (d x^{2} + c\right )^{2} - 2 \, {\left (2 \, a b d x^{2} - b^{2}\right )} \cosh \left (d x^{2} + c\right ) \sinh \left (d x^{2} + c\right ) - {\left (2 \, a b d x^{2} - b^{2}\right )} \sinh \left (d x^{2} + c\right )^{2} - b^{2}\right )} \log \left (\cosh \left (d x^{2} + c\right ) + \sinh \left (d x^{2} + c\right ) + 1\right ) - 2 \, {\left (2 \, a b c - {\left (2 \, a b c - b^{2}\right )} \cosh \left (d x^{2} + c\right )^{2} - 2 \, {\left (2 \, a b c - b^{2}\right )} \cosh \left (d x^{2} + c\right ) \sinh \left (d x^{2} + c\right ) - {\left (2 \, a b c - b^{2}\right )} \sinh \left (d x^{2} + c\right )^{2} - b^{2}\right )} \log \left (\cosh \left (d x^{2} + c\right ) + \sinh \left (d x^{2} + c\right ) - 1\right ) + 4 \, {\left (a b d x^{2} + a b c - {\left (a b d x^{2} + a b c\right )} \cosh \left (d x^{2} + c\right )^{2} - 2 \, {\left (a b d x^{2} + a b c\right )} \cosh \left (d x^{2} + c\right ) \sinh \left (d x^{2} + c\right ) - {\left (a b d x^{2} + a b c\right )} \sinh \left (d x^{2} + c\right )^{2}\right )} \log \left (-\cosh \left (d x^{2} + c\right ) - \sinh \left (d x^{2} + c\right ) + 1\right )}{4 \, {\left (d^{2} \cosh \left (d x^{2} + c\right )^{2} + 2 \, d^{2} \cosh \left (d x^{2} + c\right ) \sinh \left (d x^{2} + c\right ) + d^{2} \sinh \left (d x^{2} + c\right )^{2} - d^{2}\right )}} \]
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 \[ \int {\left (b \operatorname {csch}\left (d x^{2} + c\right ) + a\right )}^{2} x^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.46, size = 0, normalized size = 0.00 \[ \int x^{3} \left (a +b \,\mathrm {csch}\left (d \,x^{2}+c \right )\right )^{2}\, dx \]
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 \[ \frac {1}{4} \, a^{2} x^{4} - \frac {1}{2} \, {\left (\frac {2 \, x^{2} e^{\left (2 \, d x^{2} + 2 \, c\right )}}{d e^{\left (2 \, d x^{2} + 2 \, c\right )} - d} - \frac {\log \left ({\left (e^{\left (d x^{2} + c\right )} + 1\right )} e^{\left (-c\right )}\right )}{d^{2}} - \frac {\log \left ({\left (e^{\left (d x^{2} + c\right )} - 1\right )} e^{\left (-c\right )}\right )}{d^{2}}\right )} b^{2} + 4 \, a b {\left (\int \frac {x^{3}}{2 \, {\left (e^{\left (d x^{2} + c\right )} + 1\right )}}\,{d x} + \int \frac {x^{3}}{2 \, {\left (e^{\left (d x^{2} + c\right )} - 1\right )}}\,{d x}\right )} \]
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 \[ \int x^3\,{\left (a+\frac {b}{\mathrm {sinh}\left (d\,x^2+c\right )}\right )}^2 \,d x \]
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 \[ \int x^{3} \left (a + b \operatorname {csch}{\left (c + d x^{2} \right )}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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