Optimal. Leaf size=237 \[ -\frac {2 x^3 \text {ArcTan}\left (e^{a+b x}\right )}{b}-\frac {6 x^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b^2}-\frac {x^3 \text {csch}(a+b x)}{b}-\frac {6 x \text {PolyLog}\left (2,-e^{a+b x}\right )}{b^3}+\frac {3 i x^2 \text {PolyLog}\left (2,-i e^{a+b x}\right )}{b^2}-\frac {3 i x^2 \text {PolyLog}\left (2,i e^{a+b x}\right )}{b^2}+\frac {6 x \text {PolyLog}\left (2,e^{a+b x}\right )}{b^3}+\frac {6 \text {PolyLog}\left (3,-e^{a+b x}\right )}{b^4}-\frac {6 i x \text {PolyLog}\left (3,-i e^{a+b x}\right )}{b^3}+\frac {6 i x \text {PolyLog}\left (3,i e^{a+b x}\right )}{b^3}-\frac {6 \text {PolyLog}\left (3,e^{a+b x}\right )}{b^4}+\frac {6 i \text {PolyLog}\left (4,-i e^{a+b x}\right )}{b^4}-\frac {6 i \text {PolyLog}\left (4,i e^{a+b x}\right )}{b^4} \]
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Rubi [A]
time = 0.28, antiderivative size = 237, normalized size of antiderivative = 1.00, number of steps
used = 21, number of rules used = 13, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.722, Rules used = {2701, 327,
213, 5570, 14, 5313, 12, 4265, 2611, 6744, 2320, 6724, 4267} \begin {gather*} -\frac {2 x^3 \text {ArcTan}\left (e^{a+b x}\right )}{b}+\frac {6 \text {Li}_3\left (-e^{a+b x}\right )}{b^4}-\frac {6 \text {Li}_3\left (e^{a+b x}\right )}{b^4}+\frac {6 i \text {Li}_4\left (-i e^{a+b x}\right )}{b^4}-\frac {6 i \text {Li}_4\left (i e^{a+b x}\right )}{b^4}-\frac {6 x \text {Li}_2\left (-e^{a+b x}\right )}{b^3}+\frac {6 x \text {Li}_2\left (e^{a+b x}\right )}{b^3}-\frac {6 i x \text {Li}_3\left (-i e^{a+b x}\right )}{b^3}+\frac {6 i x \text {Li}_3\left (i e^{a+b x}\right )}{b^3}+\frac {3 i x^2 \text {Li}_2\left (-i e^{a+b x}\right )}{b^2}-\frac {3 i x^2 \text {Li}_2\left (i e^{a+b x}\right )}{b^2}-\frac {6 x^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b^2}-\frac {x^3 \text {csch}(a+b x)}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 213
Rule 327
Rule 2320
Rule 2611
Rule 2701
Rule 4265
Rule 4267
Rule 5313
Rule 5570
Rule 6724
Rule 6744
Rubi steps
\begin {align*} \int x^3 \text {csch}^2(a+b x) \text {sech}(a+b x) \, dx &=-\frac {x^3 \tan ^{-1}(\sinh (a+b x))}{b}-\frac {x^3 \text {csch}(a+b x)}{b}-3 \int x^2 \left (-\frac {\tan ^{-1}(\sinh (a+b x))}{b}-\frac {\text {csch}(a+b x)}{b}\right ) \, dx\\ &=-\frac {x^3 \tan ^{-1}(\sinh (a+b x))}{b}-\frac {x^3 \text {csch}(a+b x)}{b}-3 \int \left (-\frac {x^2 \tan ^{-1}(\sinh (a+b x))}{b}-\frac {x^2 \text {csch}(a+b x)}{b}\right ) \, dx\\ &=-\frac {x^3 \tan ^{-1}(\sinh (a+b x))}{b}-\frac {x^3 \text {csch}(a+b x)}{b}+\frac {3 \int x^2 \tan ^{-1}(\sinh (a+b x)) \, dx}{b}+\frac {3 \int x^2 \text {csch}(a+b x) \, dx}{b}\\ &=-\frac {6 x^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b^2}-\frac {x^3 \text {csch}(a+b x)}{b}-\frac {6 \int x \log \left (1-e^{a+b x}\right ) \, dx}{b^2}+\frac {6 \int x \log \left (1+e^{a+b x}\right ) \, dx}{b^2}-\frac {\int b x^3 \text {sech}(a+b x) \, dx}{b}\\ &=-\frac {6 x^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b^2}-\frac {x^3 \text {csch}(a+b x)}{b}-\frac {6 x \text {Li}_2\left (-e^{a+b x}\right )}{b^3}+\frac {6 x \text {Li}_2\left (e^{a+b x}\right )}{b^3}+\frac {6 \int \text {Li}_2\left (-e^{a+b x}\right ) \, dx}{b^3}-\frac {6 \int \text {Li}_2\left (e^{a+b x}\right ) \, dx}{b^3}-\int x^3 \text {sech}(a+b x) \, dx\\ &=-\frac {2 x^3 \tan ^{-1}\left (e^{a+b x}\right )}{b}-\frac {6 x^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b^2}-\frac {x^3 \text {csch}(a+b x)}{b}-\frac {6 x \text {Li}_2\left (-e^{a+b x}\right )}{b^3}+\frac {6 x \text {Li}_2\left (e^{a+b x}\right )}{b^3}+\frac {6 \text {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{a+b x}\right )}{b^4}-\frac {6 \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{a+b x}\right )}{b^4}+\frac {(3 i) \int x^2 \log \left (1-i e^{a+b x}\right ) \, dx}{b}-\frac {(3 i) \int x^2 \log \left (1+i e^{a+b x}\right ) \, dx}{b}\\ &=-\frac {2 x^3 \tan ^{-1}\left (e^{a+b x}\right )}{b}-\frac {6 x^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b^2}-\frac {x^3 \text {csch}(a+b x)}{b}-\frac {6 x \text {Li}_2\left (-e^{a+b x}\right )}{b^3}+\frac {3 i x^2 \text {Li}_2\left (-i e^{a+b x}\right )}{b^2}-\frac {3 i x^2 \text {Li}_2\left (i e^{a+b x}\right )}{b^2}+\frac {6 x \text {Li}_2\left (e^{a+b x}\right )}{b^3}+\frac {6 \text {Li}_3\left (-e^{a+b x}\right )}{b^4}-\frac {6 \text {Li}_3\left (e^{a+b x}\right )}{b^4}-\frac {(6 i) \int x \text {Li}_2\left (-i e^{a+b x}\right ) \, dx}{b^2}+\frac {(6 i) \int x \text {Li}_2\left (i e^{a+b x}\right ) \, dx}{b^2}\\ &=-\frac {2 x^3 \tan ^{-1}\left (e^{a+b x}\right )}{b}-\frac {6 x^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b^2}-\frac {x^3 \text {csch}(a+b x)}{b}-\frac {6 x \text {Li}_2\left (-e^{a+b x}\right )}{b^3}+\frac {3 i x^2 \text {Li}_2\left (-i e^{a+b x}\right )}{b^2}-\frac {3 i x^2 \text {Li}_2\left (i e^{a+b x}\right )}{b^2}+\frac {6 x \text {Li}_2\left (e^{a+b x}\right )}{b^3}+\frac {6 \text {Li}_3\left (-e^{a+b x}\right )}{b^4}-\frac {6 i x \text {Li}_3\left (-i e^{a+b x}\right )}{b^3}+\frac {6 i x \text {Li}_3\left (i e^{a+b x}\right )}{b^3}-\frac {6 \text {Li}_3\left (e^{a+b x}\right )}{b^4}+\frac {(6 i) \int \text {Li}_3\left (-i e^{a+b x}\right ) \, dx}{b^3}-\frac {(6 i) \int \text {Li}_3\left (i e^{a+b x}\right ) \, dx}{b^3}\\ &=-\frac {2 x^3 \tan ^{-1}\left (e^{a+b x}\right )}{b}-\frac {6 x^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b^2}-\frac {x^3 \text {csch}(a+b x)}{b}-\frac {6 x \text {Li}_2\left (-e^{a+b x}\right )}{b^3}+\frac {3 i x^2 \text {Li}_2\left (-i e^{a+b x}\right )}{b^2}-\frac {3 i x^2 \text {Li}_2\left (i e^{a+b x}\right )}{b^2}+\frac {6 x \text {Li}_2\left (e^{a+b x}\right )}{b^3}+\frac {6 \text {Li}_3\left (-e^{a+b x}\right )}{b^4}-\frac {6 i x \text {Li}_3\left (-i e^{a+b x}\right )}{b^3}+\frac {6 i x \text {Li}_3\left (i e^{a+b x}\right )}{b^3}-\frac {6 \text {Li}_3\left (e^{a+b x}\right )}{b^4}+\frac {(6 i) \text {Subst}\left (\int \frac {\text {Li}_3(-i x)}{x} \, dx,x,e^{a+b x}\right )}{b^4}-\frac {(6 i) \text {Subst}\left (\int \frac {\text {Li}_3(i x)}{x} \, dx,x,e^{a+b x}\right )}{b^4}\\ &=-\frac {2 x^3 \tan ^{-1}\left (e^{a+b x}\right )}{b}-\frac {6 x^2 \tanh ^{-1}\left (e^{a+b x}\right )}{b^2}-\frac {x^3 \text {csch}(a+b x)}{b}-\frac {6 x \text {Li}_2\left (-e^{a+b x}\right )}{b^3}+\frac {3 i x^2 \text {Li}_2\left (-i e^{a+b x}\right )}{b^2}-\frac {3 i x^2 \text {Li}_2\left (i e^{a+b x}\right )}{b^2}+\frac {6 x \text {Li}_2\left (e^{a+b x}\right )}{b^3}+\frac {6 \text {Li}_3\left (-e^{a+b x}\right )}{b^4}-\frac {6 i x \text {Li}_3\left (-i e^{a+b x}\right )}{b^3}+\frac {6 i x \text {Li}_3\left (i e^{a+b x}\right )}{b^3}-\frac {6 \text {Li}_3\left (e^{a+b x}\right )}{b^4}+\frac {6 i \text {Li}_4\left (-i e^{a+b x}\right )}{b^4}-\frac {6 i \text {Li}_4\left (i e^{a+b x}\right )}{b^4}\\ \end {align*}
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Mathematica [A]
time = 1.18, size = 333, normalized size = 1.41 \begin {gather*} \frac {-2 b^3 x^3 \text {csch}(a)-12 \left (b^2 x^2 \tanh ^{-1}(\cosh (a+b x)+\sinh (a+b x))+b x \text {PolyLog}(2,-\cosh (a+b x)-\sinh (a+b x))-b x \text {PolyLog}(2,\cosh (a+b x)+\sinh (a+b x))-\text {PolyLog}(3,-\cosh (a+b x)-\sinh (a+b x))+\text {PolyLog}(3,\cosh (a+b x)+\sinh (a+b x))\right )-2 i \left (b^3 x^3 \log \left (1-i e^{a+b x}\right )-b^3 x^3 \log \left (1+i e^{a+b x}\right )-3 b^2 x^2 \text {PolyLog}\left (2,-i e^{a+b x}\right )+3 b^2 x^2 \text {PolyLog}\left (2,i e^{a+b x}\right )+6 b x \text {PolyLog}\left (3,-i e^{a+b x}\right )-6 b x \text {PolyLog}\left (3,i e^{a+b x}\right )-6 \text {PolyLog}\left (4,-i e^{a+b x}\right )+6 \text {PolyLog}\left (4,i e^{a+b x}\right )\right )+b^3 x^3 \text {csch}\left (\frac {a}{2}\right ) \text {csch}\left (\frac {1}{2} (a+b x)\right ) \sinh \left (\frac {b x}{2}\right )+b^3 x^3 \text {sech}\left (\frac {a}{2}\right ) \text {sech}\left (\frac {1}{2} (a+b x)\right ) \sinh \left (\frac {b x}{2}\right )}{2 b^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 1.08, size = 0, normalized size = 0.00 \[\int x^{3} \mathrm {csch}\left (b x +a \right )^{2} \mathrm {sech}\left (b x +a \right )\, dx\]
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] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 1309 vs. \(2 (197) = 394\).
time = 0.39, size = 1309, normalized size = 5.52 \begin {gather*} -\frac {2 \, b^{3} x^{3} \cosh \left (b x + a\right ) + 2 \, b^{3} x^{3} \sinh \left (b x + a\right ) - 6 \, {\left (b x \cosh \left (b x + a\right )^{2} + 2 \, b x \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b x \sinh \left (b x + a\right )^{2} - b x\right )} {\rm Li}_2\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) + 3 \, {\left (i \, b^{2} x^{2} \cosh \left (b x + a\right )^{2} + 2 i \, b^{2} x^{2} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + i \, b^{2} x^{2} \sinh \left (b x + a\right )^{2} - i \, b^{2} x^{2}\right )} {\rm Li}_2\left (i \, \cosh \left (b x + a\right ) + i \, \sinh \left (b x + a\right )\right ) + 3 \, {\left (-i \, b^{2} x^{2} \cosh \left (b x + a\right )^{2} - 2 i \, b^{2} x^{2} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) - i \, b^{2} x^{2} \sinh \left (b x + a\right )^{2} + i \, b^{2} x^{2}\right )} {\rm Li}_2\left (-i \, \cosh \left (b x + a\right ) - i \, \sinh \left (b x + a\right )\right ) + 6 \, {\left (b x \cosh \left (b x + a\right )^{2} + 2 \, b x \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b x \sinh \left (b x + a\right )^{2} - b x\right )} {\rm Li}_2\left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right ) + 3 \, {\left (b^{2} x^{2} \cosh \left (b x + a\right )^{2} + 2 \, b^{2} x^{2} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b^{2} x^{2} \sinh \left (b x + a\right )^{2} - b^{2} x^{2}\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) - {\left (i \, a^{3} \cosh \left (b x + a\right )^{2} + 2 i \, a^{3} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + i \, a^{3} \sinh \left (b x + a\right )^{2} - i \, a^{3}\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + i\right ) - {\left (-i \, a^{3} \cosh \left (b x + a\right )^{2} - 2 i \, a^{3} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) - i \, a^{3} \sinh \left (b x + a\right )^{2} + i \, a^{3}\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - i\right ) - 3 \, {\left (a^{2} \cosh \left (b x + a\right )^{2} + 2 \, a^{2} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + a^{2} \sinh \left (b x + a\right )^{2} - a^{2}\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) - {\left (-i \, b^{3} x^{3} - i \, a^{3} + {\left (i \, b^{3} x^{3} + i \, a^{3}\right )} \cosh \left (b x + a\right )^{2} - 2 \, {\left (-i \, b^{3} x^{3} - i \, a^{3}\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + {\left (i \, b^{3} x^{3} + i \, a^{3}\right )} \sinh \left (b x + a\right )^{2}\right )} \log \left (i \, \cosh \left (b x + a\right ) + i \, \sinh \left (b x + a\right ) + 1\right ) - {\left (i \, b^{3} x^{3} + i \, a^{3} + {\left (-i \, b^{3} x^{3} - i \, a^{3}\right )} \cosh \left (b x + a\right )^{2} - 2 \, {\left (i \, b^{3} x^{3} + i \, a^{3}\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + {\left (-i \, b^{3} x^{3} - i \, a^{3}\right )} \sinh \left (b x + a\right )^{2}\right )} \log \left (-i \, \cosh \left (b x + a\right ) - i \, \sinh \left (b x + a\right ) + 1\right ) + 3 \, {\left (b^{2} x^{2} - {\left (b^{2} x^{2} - a^{2}\right )} \cosh \left (b x + a\right )^{2} - 2 \, {\left (b^{2} x^{2} - a^{2}\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) - {\left (b^{2} x^{2} - a^{2}\right )} \sinh \left (b x + a\right )^{2} - a^{2}\right )} \log \left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right ) + 1\right ) + 6 \, {\left (i \, \cosh \left (b x + a\right )^{2} + 2 i \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + i \, \sinh \left (b x + a\right )^{2} - i\right )} {\rm polylog}\left (4, i \, \cosh \left (b x + a\right ) + i \, \sinh \left (b x + a\right )\right ) + 6 \, {\left (-i \, \cosh \left (b x + a\right )^{2} - 2 i \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) - i \, \sinh \left (b x + a\right )^{2} + i\right )} {\rm polylog}\left (4, -i \, \cosh \left (b x + a\right ) - i \, \sinh \left (b x + a\right )\right ) + 6 \, {\left (\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} - 1\right )} {\rm polylog}\left (3, \cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) + 6 \, {\left (-i \, b x \cosh \left (b x + a\right )^{2} - 2 i \, b x \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) - i \, b x \sinh \left (b x + a\right )^{2} + i \, b x\right )} {\rm polylog}\left (3, i \, \cosh \left (b x + a\right ) + i \, \sinh \left (b x + a\right )\right ) + 6 \, {\left (i \, b x \cosh \left (b x + a\right )^{2} + 2 i \, b x \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + i \, b x \sinh \left (b x + a\right )^{2} - i \, b x\right )} {\rm polylog}\left (3, -i \, \cosh \left (b x + a\right ) - i \, \sinh \left (b x + a\right )\right ) - 6 \, {\left (\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} - 1\right )} {\rm polylog}\left (3, -\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right )}{b^{4} \cosh \left (b x + a\right )^{2} + 2 \, b^{4} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b^{4} \sinh \left (b x + a\right )^{2} - b^{4}} \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^{3} \operatorname {csch}^{2}{\left (a + b x \right )} \operatorname {sech}{\left (a + b x \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 = -0.00 \begin {gather*} \int \frac {x^3}{\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {sinh}\left (a+b\,x\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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