Optimal. Leaf size=18 \[ \frac {\log (a+b \cosh (c+d x))}{b d} \]
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Rubi [A]
time = 0.02, 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 = {2747, 31}
\begin {gather*} \frac {\log (a+b \cosh (c+d x))}{b d} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 2747
Rubi steps
\begin {align*} \int \frac {\sinh (c+d x)}{a+b \cosh (c+d x)} \, dx &=\frac {\text {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 = 0.05, size = 18, normalized size = 1.00 \begin {gather*} \frac {\log (a+b \cosh (c+d x))}{b d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.37, size = 19, normalized size = 1.06
method | result | size |
derivativedivides | \(\frac {\ln \left (a +b \cosh \left (d x +c \right )\right )}{b d}\) | \(19\) |
default | \(\frac {\ln \left (a +b \cosh \left (d x +c \right )\right )}{b d}\) | \(19\) |
risch | \(-\frac {x}{b}-\frac {2 c}{b d}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \,{\mathrm e}^{d x +c}}{b}+1\right )}{b d}\) | \(48\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 18, normalized size = 1.00 \begin {gather*} \frac {\log \left (b \cosh \left (d x + c\right ) + a\right )}{b d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 44 vs.
\(2 (18) = 36\).
time = 0.44, size = 44, normalized size = 2.44 \begin {gather*} -\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 41 vs.
\(2 (14) = 28\).
time = 0.43, size = 41, normalized size = 2.28 \begin {gather*} \begin {cases} \frac {x \sinh {\left (c \right )}}{a} & \text {for}\: b = 0 \wedge d = 0 \\\frac {x \sinh {\left (c \right )}}{a + b \cosh {\left (c \right )}} & \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 31, normalized size = 1.72 \begin {gather*} \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} \end {gather*}
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 \begin {gather*} \frac {\ln \left (a+b\,\mathrm {cosh}\left (c+d\,x\right )\right )}{b\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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