Optimal. Leaf size=26 \[ \text {Int}\left (\frac {1}{(c+d x) (a+i a \sinh (e+f x))},x\right ) \]
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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 = {} \[ \int \frac {1}{(c+d x) (a+i a \sinh (e+f x))} \, dx \]
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))} \, dx &=\int \frac {1}{(c+d x) (a+i a \sinh (e+f x))} \, dx\\ \end {align*}
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Mathematica [A] time = 20.66, size = 0, normalized size = 0.00 \[ \int \frac {1}{(c+d x) (a+i a \sinh (e+f x))} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.67, size = 0, normalized size = 0.00 \[ \frac {{\left (-i \, a d f x - i \, a c f + {\left (a d f x + a c f\right )} e^{\left (f x + e\right )}\right )} {\rm integral}\left (\frac {2 i \, d}{-i \, a d^{2} f x^{2} - 2 i \, a c d f x - i \, a c^{2} f + {\left (a d^{2} f x^{2} + 2 \, a c d f x + a c^{2} f\right )} e^{\left (f x + e\right )}}, x\right ) + 2 i}{-i \, a d f x - i \, a c f + {\left (a d f x + a c f\right )} e^{\left (f x + e\right )}} \]
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 \[ \int \frac {1}{{\left (d x + c\right )} {\left (i \, a \sinh \left (f x + e\right ) + a\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.13, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (d x +c \right ) \left (a +i a \sinh \left (f x +e \right )\right )}\, dx \]
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 \[ 2 i \, d \int \frac {1}{-i \, a d^{2} f x^{2} - 2 i \, a c d f x - i \, a c^{2} f + {\left (a d^{2} f x^{2} e^{e} + 2 \, a c d f x e^{e} + a c^{2} f e^{e}\right )} e^{\left (f x\right )}}\,{d x} + \frac {2 i}{-i \, a d f x - i \, a c f + {\left (a d f x e^{e} + a c f e^{e}\right )} e^{\left (f x\right )}} \]
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 \[ \int \frac {1}{\left (a+a\,\mathrm {sinh}\left (e+f\,x\right )\,1{}\mathrm {i}\right )\,\left (c+d\,x\right )} \,d x \]
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 \[ \frac {2 e^{e}}{a c f e^{e} + a d f x e^{e} + \left (- i a c f - i a d f x\right ) e^{- f x}} + \frac {2 d e^{e} \int \frac {e^{f x}}{c^{2} e^{e} e^{f x} - i c^{2} + 2 c d x e^{e} e^{f x} - 2 i c d x + d^{2} x^{2} e^{e} e^{f x} - i d^{2} x^{2}}\, dx}{a f} \]
Verification of antiderivative is not currently implemented for this CAS.
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