Optimal. Leaf size=25 \[ \frac {x \sinh (x)}{a+b \cosh (x)}-\frac {\log (a+b \cosh (x))}{b} \]
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Rubi [A] time = 0.06, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {5637, 2668, 31} \[ \frac {x \sinh (x)}{a+b \cosh (x)}-\frac {\log (a+b \cosh (x))}{b} \]
Antiderivative was successfully verified.
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Rule 31
Rule 2668
Rule 5637
Rubi steps
\begin {align*} \int \frac {x (b+a \cosh (x))}{(a+b \cosh (x))^2} \, dx &=\frac {x \sinh (x)}{a+b \cosh (x)}-\int \frac {\sinh (x)}{a+b \cosh (x)} \, dx\\ &=\frac {x \sinh (x)}{a+b \cosh (x)}-\frac {\operatorname {Subst}\left (\int \frac {1}{a+x} \, dx,x,b \cosh (x)\right )}{b}\\ &=-\frac {\log (a+b \cosh (x))}{b}+\frac {x \sinh (x)}{a+b \cosh (x)}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 25, normalized size = 1.00 \[ \frac {x \sinh (x)}{a+b \cosh (x)}-\frac {\log (a+b \cosh (x))}{b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 129, normalized size = 5.16 \[ \frac {2 \, b x \cosh \relax (x)^{2} + 2 \, b x \sinh \relax (x)^{2} + 2 \, a x \cosh \relax (x) - {\left (b \cosh \relax (x)^{2} + b \sinh \relax (x)^{2} + 2 \, a \cosh \relax (x) + 2 \, {\left (b \cosh \relax (x) + a\right )} \sinh \relax (x) + b\right )} \log \left (\frac {2 \, {\left (b \cosh \relax (x) + a\right )}}{\cosh \relax (x) - \sinh \relax (x)}\right ) + 2 \, {\left (2 \, b x \cosh \relax (x) + a x\right )} \sinh \relax (x)}{b^{2} \cosh \relax (x)^{2} + b^{2} \sinh \relax (x)^{2} + 2 \, a b \cosh \relax (x) + b^{2} + 2 \, {\left (b^{2} \cosh \relax (x) + a b\right )} \sinh \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.14, size = 100, normalized size = 4.00 \[ \frac {2 \, b x e^{\left (2 \, x\right )} - b e^{\left (2 \, x\right )} \log \left (-b e^{\left (2 \, x\right )} - 2 \, a e^{x} - b\right ) - 2 \, a e^{x} \log \left (-b e^{\left (2 \, x\right )} - 2 \, a e^{x} - b\right ) - 2 \, b x - b \log \left (-b e^{\left (2 \, x\right )} - 2 \, a e^{x} - b\right )}{b^{2} e^{\left (2 \, x\right )} + 2 \, a b e^{x} + b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.48, size = 55, normalized size = 2.20 \[ \frac {2 x}{b}-\frac {2 x \left (a \,{\mathrm e}^{x}+b \right )}{b \left (b \,{\mathrm e}^{2 x}+2 a \,{\mathrm e}^{x}+b \right )}-\frac {\ln \left ({\mathrm e}^{2 x}+\frac {2 a \,{\mathrm e}^{x}}{b}+1\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.53, size = 105, normalized size = 4.20 \[ \frac {2\,x}{b}+\frac {\frac {2\,\left (b^3\,x-a^2\,b\,x\right )}{a^2\,b-b^3}+\frac {2\,{\mathrm {e}}^x\,\left (a\,b^3\,x-a^3\,b\,x\right )}{b\,\left (a^2\,b-b^3\right )}}{b+2\,a\,{\mathrm {e}}^x+b\,{\mathrm {e}}^{2\,x}}-\frac {\ln \left (b+2\,a\,{\mathrm {e}}^x+b\,{\mathrm {e}}^{2\,x}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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