Optimal. Leaf size=25 \[ \frac {\log (a+b \sinh (x))}{b}-\frac {x \cosh (x)}{a+b \sinh (x)} \]
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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 = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {5636, 2668, 31} \[ \frac {\log (a+b \sinh (x))}{b}-\frac {x \cosh (x)}{a+b \sinh (x)} \]
Antiderivative was successfully verified.
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Rule 31
Rule 2668
Rule 5636
Rubi steps
\begin {align*} \int \frac {x (b-a \sinh (x))}{(a+b \sinh (x))^2} \, dx &=-\frac {x \cosh (x)}{a+b \sinh (x)}+\int \frac {\cosh (x)}{a+b \sinh (x)} \, dx\\ &=-\frac {x \cosh (x)}{a+b \sinh (x)}+\frac {\operatorname {Subst}\left (\int \frac {1}{a+x} \, dx,x,b \sinh (x)\right )}{b}\\ &=\frac {\log (a+b \sinh (x))}{b}-\frac {x \cosh (x)}{a+b \sinh (x)}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 25, normalized size = 1.00 \[ \frac {\log (a+b \sinh (x))}{b}-\frac {x \cosh (x)}{a+b \sinh (x)} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.42, size = 134, normalized size = 5.36 \[ -\frac {2 \, b x \cosh \relax (x)^{2} + 2 \, b x \sinh \relax (x)^{2} + 2 \, a x \cosh \relax (x) - {\left (b \cosh \relax (x)^{2} + b \sinh \relax (x)^{2} + 2 \, a \cosh \relax (x) + 2 \, {\left (b \cosh \relax (x) + a\right )} \sinh \relax (x) - b\right )} \log \left (\frac {2 \, {\left (b \sinh \relax (x) + a\right )}}{\cosh \relax (x) - \sinh \relax (x)}\right ) + 2 \, {\left (2 \, b x \cosh \relax (x) + a x\right )} \sinh \relax (x)}{b^{2} \cosh \relax (x)^{2} + b^{2} \sinh \relax (x)^{2} + 2 \, a b \cosh \relax (x) - b^{2} + 2 \, {\left (b^{2} \cosh \relax (x) + a b\right )} \sinh \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.16, size = 96, normalized size = 3.84 \[ -\frac {2 \, b x e^{\left (2 \, x\right )} - b e^{\left (2 \, x\right )} \log \left (-b e^{\left (2 \, x\right )} - 2 \, a e^{x} + b\right ) - 2 \, a e^{x} \log \left (-b e^{\left (2 \, x\right )} - 2 \, a e^{x} + b\right ) + 2 \, b x + b \log \left (-b e^{\left (2 \, x\right )} - 2 \, a e^{x} + b\right )}{b^{2} e^{\left (2 \, x\right )} + 2 \, a b e^{x} - b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.91, size = 58, normalized size = 2.32 \[ -\frac {2 x}{b}+\frac {2 x \left (a \,{\mathrm e}^{x}-b \right )}{b \left (b \,{\mathrm e}^{2 x}+2 a \,{\mathrm e}^{x}-b \right )}+\frac {\ln \left ({\mathrm e}^{2 x}+\frac {2 a \,{\mathrm e}^{x}}{b}-1\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.71, size = 62, normalized size = 2.48 \[ -\frac {2 \, {\left (b x e^{\left (2 \, x\right )} + a x e^{x}\right )}}{b^{2} e^{\left (2 \, x\right )} + 2 \, a b e^{x} - b^{2}} + \frac {\log \left (\frac {b e^{\left (2 \, x\right )} + 2 \, a e^{x} - b}{b}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.54, size = 103, normalized size = 4.12 \[ \frac {\ln \left (2\,a\,{\mathrm {e}}^x-b+b\,{\mathrm {e}}^{2\,x}\right )}{b}-\frac {\frac {2\,\left (x\,a^2\,b+x\,b^3\right )}{a^2\,b+b^3}-\frac {2\,{\mathrm {e}}^x\,\left (x\,a^3\,b+x\,a\,b^3\right )}{b\,\left (a^2\,b+b^3\right )}}{2\,a\,{\mathrm {e}}^x-b+b\,{\mathrm {e}}^{2\,x}}-\frac {2\,x}{b} \]
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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