Optimal. Leaf size=22 \[ \text{Unintegrable}\left (\frac{1}{(c+d x) (a \cosh (e+f x)+a)^2},x\right ) \]
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Rubi [A] time = 0.0543908, 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 = {} \[ \int \frac{1}{(c+d x) (a+a \cosh (e+f x))^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{1}{(c+d x) (a+a \cosh (e+f x))^2} \, dx &=\int \frac{1}{(c+d x) (a+a \cosh (e+f x))^2} \, dx\\ \end{align*}
Mathematica [A] time = 30.1044, size = 0, normalized size = 0. \[ \int \frac{1}{(c+d x) (a+a \cosh (e+f x))^2} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.347, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( dx+c \right ) \left ( a+a\cosh \left ( fx+e \right ) \right ) ^{2}}}\, 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*} -\frac{2 \,{\left (d^{2} f^{2} x^{2} + 2 \, c d f^{2} x + c^{2} f^{2} - 2 \, d^{2} +{\left (d^{2} f x e^{\left (2 \, e\right )} + c d f e^{\left (2 \, e\right )} - 2 \, d^{2} e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )} +{\left (3 \, d^{2} f^{2} x^{2} e^{e} + 3 \, c^{2} f^{2} e^{e} + c d f e^{e} - 4 \, d^{2} e^{e} +{\left (6 \, c d f^{2} e^{e} + d^{2} f e^{e}\right )} x\right )} e^{\left (f x\right )}\right )}}{3 \,{\left (a^{2} d^{3} f^{3} x^{3} + 3 \, a^{2} c d^{2} f^{3} x^{2} + 3 \, a^{2} c^{2} d f^{3} x + a^{2} c^{3} f^{3} +{\left (a^{2} d^{3} f^{3} x^{3} e^{\left (3 \, e\right )} + 3 \, a^{2} c d^{2} f^{3} x^{2} e^{\left (3 \, e\right )} + 3 \, a^{2} c^{2} d f^{3} x e^{\left (3 \, e\right )} + a^{2} c^{3} f^{3} e^{\left (3 \, e\right )}\right )} e^{\left (3 \, f x\right )} + 3 \,{\left (a^{2} d^{3} f^{3} x^{3} e^{\left (2 \, e\right )} + 3 \, a^{2} c d^{2} f^{3} x^{2} e^{\left (2 \, e\right )} + 3 \, a^{2} c^{2} d f^{3} x e^{\left (2 \, e\right )} + a^{2} c^{3} f^{3} e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )} + 3 \,{\left (a^{2} d^{3} f^{3} x^{3} e^{e} + 3 \, a^{2} c d^{2} f^{3} x^{2} e^{e} + 3 \, a^{2} c^{2} d f^{3} x e^{e} + a^{2} c^{3} f^{3} e^{e}\right )} e^{\left (f x\right )}\right )}} - \int \frac{2 \,{\left (d^{3} f^{2} x^{2} + 2 \, c d^{2} f^{2} x + c^{2} d f^{2} - 6 \, d^{3}\right )}}{3 \,{\left (a^{2} d^{4} f^{3} x^{4} + 4 \, a^{2} c d^{3} f^{3} x^{3} + 6 \, a^{2} c^{2} d^{2} f^{3} x^{2} + 4 \, a^{2} c^{3} d f^{3} x + a^{2} c^{4} f^{3} +{\left (a^{2} d^{4} f^{3} x^{4} e^{e} + 4 \, a^{2} c d^{3} f^{3} x^{3} e^{e} + 6 \, a^{2} c^{2} d^{2} f^{3} x^{2} e^{e} + 4 \, a^{2} c^{3} d f^{3} x e^{e} + a^{2} c^{4} f^{3} e^{e}\right )} e^{\left (f 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{1}{a^{2} d x + a^{2} c +{\left (a^{2} d x + a^{2} c\right )} \cosh \left (f x + e\right )^{2} + 2 \,{\left (a^{2} d x + a^{2} c\right )} \cosh \left (f x + e\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*} \frac{\int \frac{1}{c \cosh ^{2}{\left (e + f x \right )} + 2 c \cosh{\left (e + f x \right )} + c + d x \cosh ^{2}{\left (e + f x \right )} + 2 d x \cosh{\left (e + f x \right )} + d x}\, dx}{a^{2}} \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{1}{{\left (d x + c\right )}{\left (a \cosh \left (f x + e\right ) + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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