Optimal. Leaf size=34 \[ \frac {\sinh (c+d x)}{a d}-\frac {b \log (a \sinh (c+d x)+b)}{a^2 d} \]
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Rubi [A] time = 0.09, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {3872, 2833, 12, 43} \[ \frac {\sinh (c+d x)}{a d}-\frac {b \log (a \sinh (c+d x)+b)}{a^2 d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 2833
Rule 3872
Rubi steps
\begin {align*} \int \frac {\cosh (c+d x)}{a+b \text {csch}(c+d x)} \, dx &=i \int \frac {\cosh (c+d x) \sinh (c+d x)}{i b+i a \sinh (c+d x)} \, dx\\ &=-\frac {i \operatorname {Subst}\left (\int \frac {x}{a (i b+x)} \, dx,x,i a \sinh (c+d x)\right )}{a d}\\ &=-\frac {i \operatorname {Subst}\left (\int \frac {x}{i b+x} \, dx,x,i a \sinh (c+d x)\right )}{a^2 d}\\ &=-\frac {i \operatorname {Subst}\left (\int \left (1-\frac {b}{b-i x}\right ) \, dx,x,i a \sinh (c+d x)\right )}{a^2 d}\\ &=-\frac {b \log (b+a \sinh (c+d x))}{a^2 d}+\frac {\sinh (c+d x)}{a d}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 30, normalized size = 0.88 \[ \frac {a \sinh (c+d x)-b \log (a \sinh (c+d x)+b)}{a^2 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 132, normalized size = 3.88 \[ \frac {2 \, b d x \cosh \left (d x + c\right ) + a \cosh \left (d x + c\right )^{2} + a \sinh \left (d x + c\right )^{2} - 2 \, {\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )\right )} \log \left (\frac {2 \, {\left (a \sinh \left (d x + c\right ) + b\right )}}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right ) + 2 \, {\left (b d x + a \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) - a}{2 \, {\left (a^{2} d \cosh \left (d x + c\right ) + a^{2} d \sinh \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 60, normalized size = 1.76 \[ \frac {\frac {e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}}{a} - \frac {2 \, b \log \left ({\left | a {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 2 \, b \right |}\right )}{a^{2}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.13, size = 52, normalized size = 1.53 \[ -\frac {b \ln \left (a +b \,\mathrm {csch}\left (d x +c \right )\right )}{d \,a^{2}}+\frac {1}{d a \,\mathrm {csch}\left (d x +c \right )}+\frac {b \ln \left (\mathrm {csch}\left (d x +c \right )\right )}{d \,a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.68, size = 83, normalized size = 2.44 \[ -\frac {{\left (d x + c\right )} b}{a^{2} d} + \frac {e^{\left (d x + c\right )}}{2 \, a d} - \frac {e^{\left (-d x - c\right )}}{2 \, a d} - \frac {b \log \left (-2 \, b e^{\left (-d x - c\right )} + a e^{\left (-2 \, d x - 2 \, c\right )} - a\right )}{a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.34, size = 31, normalized size = 0.91 \[ -\frac {b\,\ln \left (b+a\,\mathrm {sinh}\left (c+d\,x\right )\right )-a\,\mathrm {sinh}\left (c+d\,x\right )}{a^2\,d} \]
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 \[ \int \frac {\cosh {\left (c + d x \right )}}{a + b \operatorname {csch}{\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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