Integrand size = 23, antiderivative size = 23 \[ \int \frac {1}{(c+d x)^2 (a+i a \sinh (e+f x))} \, dx=\text {Int}\left (\frac {1}{(c+d x)^2 (a+i a \sinh (e+f x))},x\right ) \]
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Not integrable
Time = 0.04 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {1}{(c+d x)^2 (a+i a \sinh (e+f x))} \, dx=\int \frac {1}{(c+d x)^2 (a+i a \sinh (e+f x))} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{(c+d x)^2 (a+i a \sinh (e+f x))} \, dx \\ \end{align*}
Not integrable
Time = 26.72 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {1}{(c+d x)^2 (a+i a \sinh (e+f x))} \, dx=\int \frac {1}{(c+d x)^2 (a+i a \sinh (e+f x))} \, dx \]
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Not integrable
Time = 0.34 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91
\[\int \frac {1}{\left (d x +c \right )^{2} \left (a +i a \sinh \left (f x +e \right )\right )}d x\]
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Not integrable
Time = 0.26 (sec) , antiderivative size = 204, normalized size of antiderivative = 8.87 \[ \int \frac {1}{(c+d x)^2 (a+i a \sinh (e+f x))} \, dx=\int { \frac {1}{{\left (d x + c\right )}^{2} {\left (i \, a \sinh \left (f x + e\right ) + a\right )}} \,d x } \]
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Not integrable
Time = 10.17 (sec) , antiderivative size = 187, normalized size of antiderivative = 8.13 \[ \int \frac {1}{(c+d x)^2 (a+i a \sinh (e+f x))} \, dx=\frac {2 i}{- i a c^{2} f - 2 i a c d f x - i a d^{2} f x^{2} + \left (a c^{2} f e^{e} + 2 a c d f x e^{e} + a d^{2} f x^{2} e^{e}\right ) e^{f x}} + \frac {4 i d \int \frac {1}{c^{3} e^{e} e^{f x} - i c^{3} + 3 c^{2} d x e^{e} e^{f x} - 3 i c^{2} d x + 3 c d^{2} x^{2} e^{e} e^{f x} - 3 i c d^{2} x^{2} + d^{3} x^{3} e^{e} e^{f x} - i d^{3} x^{3}}\, dx}{a f} \]
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Not integrable
Time = 0.33 (sec) , antiderivative size = 158, normalized size of antiderivative = 6.87 \[ \int \frac {1}{(c+d x)^2 (a+i a \sinh (e+f x))} \, dx=\int { \frac {1}{{\left (d x + c\right )}^{2} {\left (i \, a \sinh \left (f x + e\right ) + a\right )}} \,d x } \]
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Not integrable
Time = 0.32 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(c+d x)^2 (a+i a \sinh (e+f x))} \, dx=\int { \frac {1}{{\left (d x + c\right )}^{2} {\left (i \, a \sinh \left (f x + e\right ) + a\right )}} \,d x } \]
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Not integrable
Time = 1.00 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04 \[ \int \frac {1}{(c+d x)^2 (a+i a \sinh (e+f x))} \, dx=\int \frac {1}{\left (a+a\,\mathrm {sinh}\left (e+f\,x\right )\,1{}\mathrm {i}\right )\,{\left (c+d\,x\right )}^2} \,d x \]
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