Optimal. Leaf size=30 \[ \text{Unintegrable}\left (\frac{\text{sech}^2(c+d x)}{(e+f x) (a+b \sinh (c+d x))},x\right ) \]
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Rubi [A] time = 0.0769621, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\text{sech}^2(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{\text{sech}^2(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx &=\int \frac{\text{sech}^2(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx\\ \end{align*}
Mathematica [A] time = 68.3938, size = 0, normalized size = 0. \[ \int \frac{\text{sech}^2(c+d x)}{(e+f x) (a+b \sinh (c+d x))} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.567, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ({\rm sech} \left (dx+c\right ) \right ) ^{2}}{ \left ( fx+e \right ) \left ( a+b\sinh \left ( dx+c \right ) \right ) }}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0., size = 0, normalized size = 0. \begin{align*} 4 \, b^{2} \int -\frac{e^{\left (d x + c\right )}}{2 \,{\left (a^{2} b e + b^{3} e +{\left (a^{2} b f + b^{3} f\right )} x -{\left (a^{2} b e e^{\left (2 \, c\right )} + b^{3} e e^{\left (2 \, c\right )} +{\left (a^{2} b f e^{\left (2 \, c\right )} + b^{3} f e^{\left (2 \, c\right )}\right )} x\right )} e^{\left (2 \, d x\right )} - 2 \,{\left (a^{3} e e^{c} + a b^{2} e e^{c} +{\left (a^{3} f e^{c} + a b^{2} f e^{c}\right )} x\right )} e^{\left (d x\right )}\right )}}\,{d x} + \frac{2 \,{\left (b e^{\left (d x + c\right )} - a\right )}}{a^{2} d e + b^{2} d e +{\left (a^{2} d f + b^{2} d f\right )} x +{\left (a^{2} d e e^{\left (2 \, c\right )} + b^{2} d e e^{\left (2 \, c\right )} +{\left (a^{2} d f e^{\left (2 \, c\right )} + b^{2} d f e^{\left (2 \, c\right )}\right )} x\right )} e^{\left (2 \, d x\right )}} + 4 \, \int \frac{b f e^{\left (d x + c\right )} - a f}{2 \,{\left (a^{2} d e^{2} + b^{2} d e^{2} +{\left (a^{2} d f^{2} + b^{2} d f^{2}\right )} x^{2} + 2 \,{\left (a^{2} d e f + b^{2} d e f\right )} x +{\left (a^{2} d e^{2} e^{\left (2 \, c\right )} + b^{2} d e^{2} e^{\left (2 \, c\right )} +{\left (a^{2} d f^{2} e^{\left (2 \, c\right )} + b^{2} d f^{2} e^{\left (2 \, c\right )}\right )} x^{2} + 2 \,{\left (a^{2} d e f e^{\left (2 \, c\right )} + b^{2} d e f e^{\left (2 \, c\right )}\right )} x\right )} e^{\left (2 \, d x\right )}\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\operatorname{sech}\left (d x + c\right )^{2}}{a f x + a e +{\left (b f x + b e\right )} \sinh \left (d x + c\right )}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{sech}^{2}{\left (c + d x \right )}}{\left (a + b \sinh{\left (c + d x \right )}\right ) \left (e + f x\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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