Optimal. Leaf size=33 \[ \text{Unintegrable}\left (\frac{\text{sech}^2(c+d x)}{(e+f x) (a+i a \sinh (c+d x))},x\right ) \]
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Rubi [A] time = 0.0780557, 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 = {} \[ \int \frac{\text{sech}^2(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx \]
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+i a \sinh (c+d x))} \, dx &=\int \frac{\text{sech}^2(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx\\ \end{align*}
Mathematica [A] time = 122.852, size = 0, normalized size = 0. \[ \int \frac{\text{sech}^2(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.248, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ({\rm sech} \left (dx+c\right ) \right ) ^{2}}{ \left ( fx+e \right ) \left ( a+ia\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 i \, f \int \frac{1}{8 i \, a d f^{2} x^{2} + 16 i \, a d e f x + 8 i \, a d e^{2} + 8 \,{\left (a d f^{2} x^{2} e^{c} + 2 \, a d e f x e^{c} + a d e^{2} e^{c}\right )} e^{\left (d x\right )}}\,{d x} - \frac{4 \,{\left (4 \, d^{2} f^{2} x^{2} + 8 \, d^{2} e f x + 4 \, d^{2} e^{2} - 2 \, f^{2} e^{\left (2 \, d x + 2 \, c\right )} - 2 \, f^{2} +{\left (i \, d f^{2} x e^{\left (3 \, c\right )} +{\left (i \, d e f - 2 i \, f^{2}\right )} e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )} +{\left (8 i \, d^{2} f^{2} x^{2} e^{c} +{\left (16 i \, d^{2} e f + i \, d f^{2}\right )} x e^{c} +{\left (8 i \, d^{2} e^{2} + i \, d e f - 2 i \, f^{2}\right )} e^{c}\right )} e^{\left (d x\right )}\right )}}{12 \, a d^{3} f^{3} x^{3} + 36 \, a d^{3} e f^{2} x^{2} + 36 \, a d^{3} e^{2} f x + 12 \, a d^{3} e^{3} - 12 \,{\left (a d^{3} f^{3} x^{3} e^{\left (4 \, c\right )} + 3 \, a d^{3} e f^{2} x^{2} e^{\left (4 \, c\right )} + 3 \, a d^{3} e^{2} f x e^{\left (4 \, c\right )} + a d^{3} e^{3} e^{\left (4 \, c\right )}\right )} e^{\left (4 \, d x\right )} -{\left (-24 i \, a d^{3} f^{3} x^{3} e^{\left (3 \, c\right )} - 72 i \, a d^{3} e f^{2} x^{2} e^{\left (3 \, c\right )} - 72 i \, a d^{3} e^{2} f x e^{\left (3 \, c\right )} - 24 i \, a d^{3} e^{3} e^{\left (3 \, c\right )}\right )} e^{\left (3 \, d x\right )} -{\left (-24 i \, a d^{3} f^{3} x^{3} e^{c} - 72 i \, a d^{3} e f^{2} x^{2} e^{c} - 72 i \, a d^{3} e^{2} f x e^{c} - 24 i \, a d^{3} e^{3} e^{c}\right )} e^{\left (d x\right )}} - 4 \, \int \frac{5 \, d^{2} f^{3} x^{2} + 10 \, d^{2} e f^{2} x + 5 \, d^{2} e^{2} f - 12 \, f^{3}}{24 \, a d^{3} f^{4} x^{4} + 96 \, a d^{3} e f^{3} x^{3} + 144 \, a d^{3} e^{2} f^{2} x^{2} + 96 \, a d^{3} e^{3} f x + 24 \, a d^{3} e^{4} +{\left (24 i \, a d^{3} f^{4} x^{4} e^{c} + 96 i \, a d^{3} e f^{3} x^{3} e^{c} + 144 i \, a d^{3} e^{2} f^{2} x^{2} e^{c} + 96 i \, a d^{3} e^{3} f x e^{c} + 24 i \, a d^{3} e^{4} e^{c}\right )} e^{\left (d x\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*} \text{result too large to display} \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*} \frac{\int \frac{\operatorname{sech}^{2}{\left (c + d x \right )}}{i e \sinh{\left (c + d x \right )} + e + i f x \sinh{\left (c + d x \right )} + f x}\, dx}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{sech}\left (d x + c\right )^{2}}{{\left (f x + e\right )}{\left (i \, a \sinh \left (d x + c\right ) + a\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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