Optimal. Leaf size=18 \[ \frac {\log (a+b \cosh (c+d x))}{b d} \]
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Rubi [A] time = 0.03, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2668, 31} \[ \frac {\log (a+b \cosh (c+d x))}{b d} \]
Antiderivative was successfully verified.
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Rule 31
Rule 2668
Rubi steps
\begin {align*} \int \frac {\sinh (c+d x)}{a+b \cosh (c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{a+x} \, dx,x,b \cosh (c+d x)\right )}{b d}\\ &=\frac {\log (a+b \cosh (c+d x))}{b d}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 18, normalized size = 1.00 \[ \frac {\log (a+b \cosh (c+d x))}{b d} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.00, size = 44, normalized size = 2.44 \[ -\frac {d x - \log \left (\frac {2 \, {\left (b \cosh \left (d x + c\right ) + a\right )}}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right )}{b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 31, normalized size = 1.72 \[ \frac {\log \left ({\left | b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} + 2 \, a \right |}\right )}{b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 19, normalized size = 1.06 \[ \frac {\ln \left (a +b \cosh \left (d x +c \right )\right )}{b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 18, normalized size = 1.00 \[ \frac {\log \left (b \cosh \left (d x + c\right ) + a\right )}{b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 18, normalized size = 1.00 \[ \frac {\ln \left (a+b\,\mathrm {cosh}\left (c+d\,x\right )\right )}{b\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.97, size = 41, normalized size = 2.28 \[ \begin {cases} \frac {x \sinh {\relax (c )}}{a} & \text {for}\: b = 0 \wedge d = 0 \\\frac {x \sinh {\relax (c )}}{a + b \cosh {\relax (c )}} & \text {for}\: d = 0 \\\frac {\cosh {\left (c + d x \right )}}{a d} & \text {for}\: b = 0 \\\frac {\log {\left (\frac {a}{b} + \cosh {\left (c + d x \right )} \right )}}{b d} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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