Optimal. Leaf size=11 \[ \frac {\log (a+b \sinh (x))}{b} \]
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Rubi [A] time = 0.02, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2668, 31} \[ \frac {\log (a+b \sinh (x))}{b} \]
Antiderivative was successfully verified.
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Rule 31
Rule 2668
Rubi steps
\begin {align*} \int \frac {\cosh (x)}{a+b \sinh (x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{a+x} \, dx,x,b \sinh (x)\right )}{b}\\ &=\frac {\log (a+b \sinh (x))}{b}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 11, normalized size = 1.00 \[ \frac {\log (a+b \sinh (x))}{b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.64, size = 27, normalized size = 2.45 \[ -\frac {x - \log \left (\frac {2 \, {\left (b \sinh \relax (x) + a\right )}}{\cosh \relax (x) - \sinh \relax (x)}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 22, normalized size = 2.00 \[ \frac {\log \left ({\left | -b {\left (e^{\left (-x\right )} - e^{x}\right )} + 2 \, a \right |}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 12, normalized size = 1.09 \[ \frac {\ln \left (a +b \sinh \relax (x )\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 11, normalized size = 1.00 \[ \frac {\log \left (b \sinh \relax (x) + a\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 11, normalized size = 1.00 \[ \frac {\ln \left (a+b\,\mathrm {sinh}\relax (x)\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.31, size = 14, normalized size = 1.27 \[ \begin {cases} \frac {\log {\left (\frac {a}{b} + \sinh {\relax (x )} \right )}}{b} & \text {for}\: b \neq 0 \\\frac {\sinh {\relax (x )}}{a} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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