Optimal. Leaf size=102 \[ -\frac {a b^2 x}{\left (a^2-b^2\right )^2}-\frac {a x}{2 \left (a^2-b^2\right )}+\frac {a^2 b \log (a \cosh (x)+b \sinh (x))}{\left (a^2-b^2\right )^2}+\frac {a \cosh (x) \sinh (x)}{2 \left (a^2-b^2\right )}-\frac {b \sinh ^2(x)}{2 \left (a^2-b^2\right )} \]
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Rubi [A]
time = 0.11, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {3188, 2644, 30,
2715, 8, 3176, 3212} \begin {gather*} -\frac {a x}{2 \left (a^2-b^2\right )}-\frac {a b^2 x}{\left (a^2-b^2\right )^2}-\frac {b \sinh ^2(x)}{2 \left (a^2-b^2\right )}+\frac {a \sinh (x) \cosh (x)}{2 \left (a^2-b^2\right )}+\frac {a^2 b \log (a \cosh (x)+b \sinh (x))}{\left (a^2-b^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 30
Rule 2644
Rule 2715
Rule 3176
Rule 3188
Rule 3212
Rubi steps
\begin {align*} \int \frac {\cosh (x) \sinh ^2(x)}{a \cosh (x)+b \sinh (x)} \, dx &=\frac {a \int \sinh ^2(x) \, dx}{a^2-b^2}-\frac {b \int \cosh (x) \sinh (x) \, dx}{a^2-b^2}+\frac {(a b) \int \frac {\sinh (x)}{a \cosh (x)+b \sinh (x)} \, dx}{a^2-b^2}\\ &=-\frac {a b^2 x}{\left (a^2-b^2\right )^2}+\frac {a \cosh (x) \sinh (x)}{2 \left (a^2-b^2\right )}+\frac {\left (i a^2 b\right ) \int \frac {-i b \cosh (x)-i a \sinh (x)}{a \cosh (x)+b \sinh (x)} \, dx}{\left (a^2-b^2\right )^2}-\frac {a \int 1 \, dx}{2 \left (a^2-b^2\right )}+\frac {b \text {Subst}(\int x \, dx,x,i \sinh (x))}{a^2-b^2}\\ &=-\frac {a b^2 x}{\left (a^2-b^2\right )^2}-\frac {a x}{2 \left (a^2-b^2\right )}+\frac {a^2 b \log (a \cosh (x)+b \sinh (x))}{\left (a^2-b^2\right )^2}+\frac {a \cosh (x) \sinh (x)}{2 \left (a^2-b^2\right )}-\frac {b \sinh ^2(x)}{2 \left (a^2-b^2\right )}\\ \end {align*}
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Mathematica [A]
time = 0.18, size = 73, normalized size = 0.72 \begin {gather*} \frac {\left (-a^2 b+b^3\right ) \cosh (2 x)+a \left (-2 \left (a^2+b^2\right ) x+4 a b \log (a \cosh (x)+b \sinh (x))+\left (a^2-b^2\right ) \sinh (2 x)\right )}{4 (a-b)^2 (a+b)^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.08, size = 145, normalized size = 1.42
method | result | size |
risch | \(-\frac {a x}{2 \left (a +b \right )^{2}}+\frac {{\mathrm e}^{2 x}}{8 a +8 b}-\frac {{\mathrm e}^{-2 x}}{8 \left (a -b \right )}-\frac {2 a^{2} b x}{a^{4}-2 a^{2} b^{2}+b^{4}}+\frac {a^{2} b \ln \left ({\mathrm e}^{2 x}+\frac {a -b}{a +b}\right )}{a^{4}-2 a^{2} b^{2}+b^{4}}\) | \(98\) |
default | \(-\frac {4}{\left (8 a -8 b \right ) \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {8}{\left (16 a -16 b \right ) \left (\tanh \left (\frac {x}{2}\right )+1\right )}-\frac {a \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{2 \left (a -b \right )^{2}}+\frac {4}{\left (8 a +8 b \right ) \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}+\frac {8}{\left (16 a +16 b \right ) \left (\tanh \left (\frac {x}{2}\right )-1\right )}+\frac {a \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{2 \left (a +b \right )^{2}}+\frac {a^{2} b \ln \left (a +2 b \tanh \left (\frac {x}{2}\right )+a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )\right )}{\left (a -b \right )^{2} \left (a +b \right )^{2}}\) | \(145\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 83, normalized size = 0.81 \begin {gather*} \frac {a^{2} b \log \left (-{\left (a - b\right )} e^{\left (-2 \, x\right )} - a - b\right )}{a^{4} - 2 \, a^{2} b^{2} + b^{4}} - \frac {a x}{2 \, {\left (a^{2} + 2 \, a b + b^{2}\right )}} + \frac {e^{\left (2 \, x\right )}}{8 \, {\left (a + b\right )}} - \frac {e^{\left (-2 \, x\right )}}{8 \, {\left (a - b\right )}} \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. 334 vs.
\(2 (96) = 192\).
time = 0.39, size = 334, normalized size = 3.27 \begin {gather*} \frac {{\left (a^{3} - a^{2} b - a b^{2} + b^{3}\right )} \cosh \left (x\right )^{4} + 4 \, {\left (a^{3} - a^{2} b - a b^{2} + b^{3}\right )} \cosh \left (x\right ) \sinh \left (x\right )^{3} + {\left (a^{3} - a^{2} b - a b^{2} + b^{3}\right )} \sinh \left (x\right )^{4} - 4 \, {\left (a^{3} + 2 \, a^{2} b + a b^{2}\right )} x \cosh \left (x\right )^{2} - a^{3} - a^{2} b + a b^{2} + b^{3} + 2 \, {\left (3 \, {\left (a^{3} - a^{2} b - a b^{2} + b^{3}\right )} \cosh \left (x\right )^{2} - 2 \, {\left (a^{3} + 2 \, a^{2} b + a b^{2}\right )} x\right )} \sinh \left (x\right )^{2} + 8 \, {\left (a^{2} b \cosh \left (x\right )^{2} + 2 \, a^{2} b \cosh \left (x\right ) \sinh \left (x\right ) + a^{2} b \sinh \left (x\right )^{2}\right )} \log \left (\frac {2 \, {\left (a \cosh \left (x\right ) + b \sinh \left (x\right )\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + 4 \, {\left ({\left (a^{3} - a^{2} b - a b^{2} + b^{3}\right )} \cosh \left (x\right )^{3} - 2 \, {\left (a^{3} + 2 \, a^{2} b + a b^{2}\right )} x \cosh \left (x\right )\right )} \sinh \left (x\right )}{8 \, {\left ({\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \cosh \left (x\right ) \sinh \left (x\right ) + {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \sinh \left (x\right )^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 101, normalized size = 0.99 \begin {gather*} \frac {a^{2} b \log \left ({\left | a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} + a - b \right |}\right )}{a^{4} - 2 \, a^{2} b^{2} + b^{4}} - \frac {a x}{2 \, {\left (a^{2} - 2 \, a b + b^{2}\right )}} + \frac {{\left (2 \, a e^{\left (2 \, x\right )} - a + b\right )} e^{\left (-2 \, x\right )}}{8 \, {\left (a^{2} - 2 \, a b + b^{2}\right )}} + \frac {e^{\left (2 \, x\right )}}{8 \, {\left (a + b\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.68, size = 81, normalized size = 0.79 \begin {gather*} \frac {{\mathrm {e}}^{2\,x}}{8\,a+8\,b}-\frac {{\mathrm {e}}^{-2\,x}}{8\,a-8\,b}-\frac {a\,x}{2\,{\left (a-b\right )}^2}+\frac {a^2\,b\,\ln \left (a-b+a\,{\mathrm {e}}^{2\,x}+b\,{\mathrm {e}}^{2\,x}\right )}{a^4-2\,a^2\,b^2+b^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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