Optimal. Leaf size=20 \[ \frac {\sinh (x)}{a}-\frac {b \log (a \sinh (x)+b)}{a^2} \]
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Rubi [A] time = 0.08, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {3872, 2833, 12, 43} \[ \frac {\sinh (x)}{a}-\frac {b \log (a \sinh (x)+b)}{a^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 2833
Rule 3872
Rubi steps
\begin {align*} \int \frac {\cosh (x)}{a+b \text {csch}(x)} \, dx &=i \int \frac {\cosh (x) \sinh (x)}{i b+i a \sinh (x)} \, dx\\ &=-\frac {i \operatorname {Subst}\left (\int \frac {x}{a (i b+x)} \, dx,x,i a \sinh (x)\right )}{a}\\ &=-\frac {i \operatorname {Subst}\left (\int \frac {x}{i b+x} \, dx,x,i a \sinh (x)\right )}{a^2}\\ &=-\frac {i \operatorname {Subst}\left (\int \left (1-\frac {b}{b-i x}\right ) \, dx,x,i a \sinh (x)\right )}{a^2}\\ &=-\frac {b \log (b+a \sinh (x))}{a^2}+\frac {\sinh (x)}{a}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 19, normalized size = 0.95 \[ \frac {a \sinh (x)-b \log (a \sinh (x)+b)}{a^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.63, size = 80, normalized size = 4.00 \[ \frac {2 \, b x \cosh \relax (x) + a \cosh \relax (x)^{2} + a \sinh \relax (x)^{2} - 2 \, {\left (b \cosh \relax (x) + b \sinh \relax (x)\right )} \log \left (\frac {2 \, {\left (a \sinh \relax (x) + b\right )}}{\cosh \relax (x) - \sinh \relax (x)}\right ) + 2 \, {\left (b x + a \cosh \relax (x)\right )} \sinh \relax (x) - a}{2 \, {\left (a^{2} \cosh \relax (x) + a^{2} \sinh \relax (x)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 39, normalized size = 1.95 \[ -\frac {e^{\left (-x\right )} - e^{x}}{2 \, a} - \frac {b \log \left ({\left | -a {\left (e^{\left (-x\right )} - e^{x}\right )} + 2 \, b \right |}\right )}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 31, normalized size = 1.55 \[ -\frac {b \ln \left (a +b \,\mathrm {csch}\relax (x )\right )}{a^{2}}+\frac {1}{a \,\mathrm {csch}\relax (x )}+\frac {b \ln \left (\mathrm {csch}\relax (x )\right )}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.31, size = 48, normalized size = 2.40 \[ -\frac {b x}{a^{2}} - \frac {e^{\left (-x\right )}}{2 \, a} + \frac {e^{x}}{2 \, a} - \frac {b \log \left (-2 \, b e^{\left (-x\right )} + a e^{\left (-2 \, x\right )} - a\right )}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 20, normalized size = 1.00 \[ \frac {\mathrm {sinh}\relax (x)}{a}-\frac {b\,\ln \left (b+a\,\mathrm {sinh}\relax (x)\right )}{a^2} \]
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 \[ \int \frac {\cosh {\relax (x )}}{a + b \operatorname {csch}{\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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