Optimal. Leaf size=35 \[ \text {Int}\left (\frac {\sinh (c+d x) \cosh (c+d x)}{(e+f x) (a \sinh (c+d x)+b)},x\right ) \]
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Rubi [A] time = 0.11, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\cosh (c+d x)}{(e+f x) (a+b \text {csch}(c+d x))} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\cosh (c+d x)}{(e+f x) (a+b \text {csch}(c+d x))} \, dx &=\int \frac {\cosh (c+d x) \sinh (c+d x)}{(e+f x) (b+a \sinh (c+d x))} \, dx\\ \end {align*}
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Mathematica [A] time = 95.58, size = 0, normalized size = 0.00 \[ \int \frac {\cosh (c+d x)}{(e+f x) (a+b \text {csch}(c+d x))} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\cosh \left (d x + c\right )}{a f x + a e + {\left (b f x + b e\right )} \operatorname {csch}\left (d x + c\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cosh \left (d x + c\right )}{{\left (f x + e\right )} {\left (b \operatorname {csch}\left (d x + c\right ) + a\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.99, size = 0, normalized size = 0.00 \[ \int \frac {\cosh \left (d x +c \right )}{\left (f x +e \right ) \left (a +b \,\mathrm {csch}\left (d x +c \right )\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {e^{\left (-c + \frac {d e}{f}\right )} E_{1}\left (\frac {{\left (f x + e\right )} d}{f}\right )}{2 \, a f} - \frac {e^{\left (c - \frac {d e}{f}\right )} E_{1}\left (-\frac {{\left (f x + e\right )} d}{f}\right )}{2 \, a f} - \frac {b \log \left (f x + e\right )}{a^{2} f} + \frac {1}{2} \, \int -\frac {4 \, {\left (b^{2} e^{\left (d x + c\right )} - a b\right )}}{a^{3} f x + a^{3} e - {\left (a^{3} f x e^{\left (2 \, c\right )} + a^{3} e e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )} - 2 \, {\left (a^{2} b f x e^{c} + a^{2} b e e^{c}\right )} e^{\left (d x\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {\mathrm {cosh}\left (c+d\,x\right )}{\left (e+f\,x\right )\,\left (a+\frac {b}{\mathrm {sinh}\left (c+d\,x\right )}\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cosh {\left (c + d x \right )}}{\left (a + b \operatorname {csch}{\left (c + d x \right )}\right ) \left (e + f x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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