Optimal. Leaf size=26 \[ \text {Int}\left (\frac {1}{(c+d x) (a+i a \sinh (e+f x))^2},x\right ) \]
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Rubi [A] time = 0.06, 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 = {} \[ \int \frac {1}{(c+d x) (a+i a \sinh (e+f x))^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {1}{(c+d x) (a+i a \sinh (e+f x))^2} \, dx &=\int \frac {1}{(c+d x) (a+i a \sinh (e+f x))^2} \, dx\\ \end {align*}
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Mathematica [A] time = 35.25, size = 0, normalized size = 0.00 \[ \int \frac {1}{(c+d x) (a+i a \sinh (e+f x))^2} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.63, size = 0, normalized size = 0.00 \[ \frac {-2 i \, d^{2} f^{2} x^{2} - 4 i \, c d f^{2} x - 2 i \, c^{2} f^{2} + 4 i \, d^{2} + {\left (2 i \, d^{2} f x + 2 i \, c d f - 4 i \, d^{2}\right )} e^{\left (2 \, f x + 2 \, e\right )} + 2 \, {\left (3 \, d^{2} f^{2} x^{2} + 3 \, c^{2} f^{2} + c d f - 4 \, d^{2} + {\left (6 \, c d f^{2} + d^{2} f\right )} x\right )} e^{\left (f x + e\right )} + {\left (3 i \, a^{2} d^{3} f^{3} x^{3} + 9 i \, a^{2} c d^{2} f^{3} x^{2} + 9 i \, a^{2} c^{2} d f^{3} x + 3 i \, a^{2} c^{3} f^{3} + 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}\right )} e^{\left (3 \, f x + 3 \, e\right )} + {\left (-9 i \, a^{2} d^{3} f^{3} x^{3} - 27 i \, a^{2} c d^{2} f^{3} x^{2} - 27 i \, a^{2} c^{2} d f^{3} x - 9 i \, a^{2} c^{3} f^{3}\right )} e^{\left (2 \, f x + 2 \, e\right )} - 9 \, {\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}\right )} e^{\left (f x + e\right )}\right )} {\rm integral}\left (\frac {2 i \, d^{3} f^{2} x^{2} + 4 i \, c d^{2} f^{2} x + 2 i \, c^{2} d f^{2} - 12 i \, d^{3}}{-3 i \, a^{2} d^{4} f^{3} x^{4} - 12 i \, a^{2} c d^{3} f^{3} x^{3} - 18 i \, a^{2} c^{2} d^{2} f^{3} x^{2} - 12 i \, a^{2} c^{3} d f^{3} x - 3 i \, a^{2} c^{4} f^{3} + 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}\right )} e^{\left (f x + e\right )}}, x\right )}{3 i \, a^{2} d^{3} f^{3} x^{3} + 9 i \, a^{2} c d^{2} f^{3} x^{2} + 9 i \, a^{2} c^{2} d f^{3} x + 3 i \, a^{2} c^{3} f^{3} + 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}\right )} e^{\left (3 \, f x + 3 \, e\right )} + {\left (-9 i \, a^{2} d^{3} f^{3} x^{3} - 27 i \, a^{2} c d^{2} f^{3} x^{2} - 27 i \, a^{2} c^{2} d f^{3} x - 9 i \, a^{2} c^{3} f^{3}\right )} e^{\left (2 \, f x + 2 \, e\right )} - 9 \, {\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}\right )} e^{\left (f x + e\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 \[ \int \frac {1}{{\left (d x + c\right )} {\left (i \, a \sinh \left (f x + e\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.92, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (d x +c \right ) \left (a +i a \sinh \left (f x +e \right )\right )^{2}}\, dx \]
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 \[ \frac {-2 i \, d^{2} f^{2} x^{2} - 4 i \, c d f^{2} x - 2 i \, c^{2} f^{2} + 4 i \, d^{2} + {\left (2 i \, d^{2} f x e^{\left (2 \, e\right )} + 2 i \, c d f e^{\left (2 \, e\right )} - 4 i \, d^{2} e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )} + 2 \, {\left (3 \, d^{2} f^{2} x^{2} e^{e} + 3 \, c^{2} f^{2} e^{e} + c d f e^{e} - 4 \, d^{2} e^{e} + {\left (6 \, c d f^{2} e^{e} + d^{2} f e^{e}\right )} x\right )} e^{\left (f x\right )}}{3 i \, a^{2} d^{3} f^{3} x^{3} + 9 i \, a^{2} c d^{2} f^{3} x^{2} + 9 i \, a^{2} c^{2} d f^{3} x + 3 i \, a^{2} c^{3} f^{3} + 3 \, {\left (a^{2} d^{3} f^{3} x^{3} e^{\left (3 \, e\right )} + 3 \, a^{2} c d^{2} f^{3} x^{2} e^{\left (3 \, e\right )} + 3 \, a^{2} c^{2} d f^{3} x e^{\left (3 \, e\right )} + a^{2} c^{3} f^{3} e^{\left (3 \, e\right )}\right )} e^{\left (3 \, f x\right )} + {\left (-9 i \, a^{2} d^{3} f^{3} x^{3} e^{\left (2 \, e\right )} - 27 i \, a^{2} c d^{2} f^{3} x^{2} e^{\left (2 \, e\right )} - 27 i \, a^{2} c^{2} d f^{3} x e^{\left (2 \, e\right )} - 9 i \, a^{2} c^{3} f^{3} e^{\left (2 \, e\right )}\right )} e^{\left (2 \, f x\right )} - 9 \, {\left (a^{2} d^{3} f^{3} x^{3} e^{e} + 3 \, a^{2} c d^{2} f^{3} x^{2} e^{e} + 3 \, a^{2} c^{2} d f^{3} x e^{e} + a^{2} c^{3} f^{3} e^{e}\right )} e^{\left (f x\right )}} - \int \frac {2 \, {\left (d^{3} f^{2} x^{2} + 2 \, c d^{2} f^{2} x + c^{2} d f^{2} - 6 \, d^{3}\right )}}{3 \, a^{2} d^{4} f^{3} x^{4} + 12 \, a^{2} c d^{3} f^{3} x^{3} + 18 \, a^{2} c^{2} d^{2} f^{3} x^{2} + 12 \, a^{2} c^{3} d f^{3} x + 3 \, a^{2} c^{4} f^{3} + {\left (3 i \, a^{2} d^{4} f^{3} x^{4} e^{e} + 12 i \, a^{2} c d^{3} f^{3} x^{3} e^{e} + 18 i \, a^{2} c^{2} d^{2} f^{3} x^{2} e^{e} + 12 i \, a^{2} c^{3} d f^{3} x e^{e} + 3 i \, a^{2} c^{4} f^{3} e^{e}\right )} e^{\left (f x\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {1}{{\left (a+a\,\mathrm {sinh}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^2\,\left (c+d\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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