Optimal. Leaf size=69 \[ \frac {4 x \text {ArcTan}\left (e^{a+b x}\right )}{b^2}-\frac {2 i \text {PolyLog}\left (2,-i e^{a+b x}\right )}{b^3}+\frac {2 i \text {PolyLog}\left (2,i e^{a+b x}\right )}{b^3}-\frac {x^2 \text {sech}(a+b x)}{b} \]
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Rubi [A]
time = 0.04, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5526, 4265,
2317, 2438} \begin {gather*} \frac {4 x \text {ArcTan}\left (e^{a+b x}\right )}{b^2}-\frac {2 i \text {Li}_2\left (-i e^{a+b x}\right )}{b^3}+\frac {2 i \text {Li}_2\left (i e^{a+b x}\right )}{b^3}-\frac {x^2 \text {sech}(a+b x)}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 2317
Rule 2438
Rule 4265
Rule 5526
Rubi steps
\begin {align*} \int x^2 \text {sech}(a+b x) \tanh (a+b x) \, dx &=-\frac {x^2 \text {sech}(a+b x)}{b}+\frac {2 \int x \text {sech}(a+b x) \, dx}{b}\\ &=\frac {4 x \tan ^{-1}\left (e^{a+b x}\right )}{b^2}-\frac {x^2 \text {sech}(a+b x)}{b}-\frac {(2 i) \int \log \left (1-i e^{a+b x}\right ) \, dx}{b^2}+\frac {(2 i) \int \log \left (1+i e^{a+b x}\right ) \, dx}{b^2}\\ &=\frac {4 x \tan ^{-1}\left (e^{a+b x}\right )}{b^2}-\frac {x^2 \text {sech}(a+b x)}{b}-\frac {(2 i) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{a+b x}\right )}{b^3}+\frac {(2 i) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{a+b x}\right )}{b^3}\\ &=\frac {4 x \tan ^{-1}\left (e^{a+b x}\right )}{b^2}-\frac {2 i \text {Li}_2\left (-i e^{a+b x}\right )}{b^3}+\frac {2 i \text {Li}_2\left (i e^{a+b x}\right )}{b^3}-\frac {x^2 \text {sech}(a+b x)}{b}\\ \end {align*}
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Mathematica [A]
time = 0.30, size = 125, normalized size = 1.81 \begin {gather*} -\frac {(-2 i a+\pi -2 i b x) \left (\log \left (1-i e^{a+b x}\right )-\log \left (1+i e^{a+b x}\right )\right )-(-2 i a+\pi ) \log \left (\cot \left (\frac {1}{4} (2 i a+\pi +2 i b x)\right )\right )+2 i \left (\text {PolyLog}\left (2,-i e^{a+b x}\right )-\text {PolyLog}\left (2,i e^{a+b x}\right )\right )+b^2 x^2 \text {sech}(a+b x)}{b^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 153 vs. \(2 (62 ) = 124\).
time = 1.77, size = 154, normalized size = 2.23
method | result | size |
risch | \(-\frac {2 x^{2} {\mathrm e}^{b x +a}}{b \left ({\mathrm e}^{2 b x +2 a}+1\right )}-\frac {2 i \ln \left (1+i {\mathrm e}^{b x +a}\right ) x}{b^{2}}-\frac {2 i \ln \left (1+i {\mathrm e}^{b x +a}\right ) a}{b^{3}}+\frac {2 i \ln \left (1-i {\mathrm e}^{b x +a}\right ) x}{b^{2}}+\frac {2 i \ln \left (1-i {\mathrm e}^{b x +a}\right ) a}{b^{3}}-\frac {2 i \dilog \left (1+i {\mathrm e}^{b x +a}\right )}{b^{3}}+\frac {2 i \dilog \left (1-i {\mathrm e}^{b x +a}\right )}{b^{3}}-\frac {4 a \arctan \left ({\mathrm e}^{b x +a}\right )}{b^{3}}\) | \(154\) |
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. 468 vs. \(2 (56) = 112\).
time = 0.41, size = 468, normalized size = 6.78 \begin {gather*} -\frac {2 \, {\left (b^{2} x^{2} \cosh \left (b x + a\right ) + b^{2} x^{2} \sinh \left (b x + a\right ) + {\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 Li}_2\left (i \, \cosh \left (b x + a\right ) + i \, \sinh \left (b x + a\right )\right ) + {\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 Li}_2\left (-i \, \cosh \left (b x + a\right ) - i \, \sinh \left (b x + a\right )\right ) + {\left (i \, a \cosh \left (b x + a\right )^{2} + 2 i \, a \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + i \, a \sinh \left (b x + a\right )^{2} + i \, a\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + i\right ) + {\left (-i \, a \cosh \left (b x + a\right )^{2} - 2 i \, a \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) - i \, a \sinh \left (b x + a\right )^{2} - i \, a\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - i\right ) + {\left ({\left (i \, b x + i \, a\right )} \cosh \left (b x + a\right )^{2} + 2 \, {\left (i \, b x + i \, a\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + {\left (i \, b x + i \, a\right )} \sinh \left (b x + a\right )^{2} + i \, b x + i \, a\right )} \log \left (i \, \cosh \left (b x + a\right ) + i \, \sinh \left (b x + a\right ) + 1\right ) + {\left ({\left (-i \, b x - i \, a\right )} \cosh \left (b x + a\right )^{2} + 2 \, {\left (-i \, b x - i \, a\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + {\left (-i \, b x - i \, a\right )} \sinh \left (b x + a\right )^{2} - i \, b x - i \, a\right )} \log \left (-i \, \cosh \left (b x + a\right ) - i \, \sinh \left (b x + a\right ) + 1\right )\right )}}{b^{3} \cosh \left (b x + a\right )^{2} + 2 \, b^{3} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b^{3} \sinh \left (b x + a\right )^{2} + b^{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^{2} \sinh {\left (a + b x \right )} \operatorname {sech}^{2}{\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 = -0.01 \begin {gather*} \int \frac {x^2\,\mathrm {sinh}\left (a+b\,x\right )}{{\mathrm {cosh}\left (a+b\,x\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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