Optimal. Leaf size=26 \[ \text {Int}\left (\frac {(c+d x)^m}{(a+i a \sinh (e+f x))^2},x\right ) \]
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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 = {} \[ \int \frac {(c+d x)^m}{(a+i a \sinh (e+f x))^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {(c+d x)^m}{(a+i a \sinh (e+f x))^2} \, dx &=\int \frac {(c+d x)^m}{(a+i a \sinh (e+f x))^2} \, dx\\ \end {align*}
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Mathematica [A] time = 16.93, size = 0, normalized size = 0.00 \[ \int \frac {(c+d x)^m}{(a+i a \sinh (e+f x))^2} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.45, size = 0, normalized size = 0.00 \[ \frac {{\left (-2 i \, d^{2} f^{2} x^{2} - 4 i \, c d f^{2} x - 2 i \, c^{2} f^{2} + 2 i \, d^{2} m^{2} - 2 i \, d^{2} m + {\left (-2 i \, d^{2} f m x - 2 i \, d^{2} m^{2} + {\left (-2 i \, c d f + 2 i \, d^{2}\right )} m\right )} e^{\left (2 \, f x + 2 \, e\right )} + 2 \, {\left (3 \, d^{2} f^{2} x^{2} + 3 \, c^{2} f^{2} - 2 \, d^{2} m^{2} - {\left (c d f - 2 \, d^{2}\right )} m + {\left (6 \, c d f^{2} - d^{2} f m\right )} x\right )} e^{\left (f x + e\right )}\right )} {\left (d x + c\right )}^{m} + {\left (3 i \, a^{2} d^{2} f^{3} x^{2} + 6 i \, a^{2} c d f^{3} x + 3 i \, a^{2} c^{2} f^{3} + 3 \, {\left (a^{2} d^{2} f^{3} x^{2} + 2 \, a^{2} c d f^{3} x + a^{2} c^{2} f^{3}\right )} e^{\left (3 \, f x + 3 \, e\right )} + {\left (-9 i \, a^{2} d^{2} f^{3} x^{2} - 18 i \, a^{2} c d f^{3} x - 9 i \, a^{2} c^{2} f^{3}\right )} e^{\left (2 \, f x + 2 \, e\right )} - 9 \, {\left (a^{2} d^{2} f^{3} x^{2} + 2 \, a^{2} c d f^{3} x + a^{2} c^{2} f^{3}\right )} e^{\left (f x + e\right )}\right )} {\rm integral}\left (\frac {{\left (-2 i \, d^{3} f^{2} m x^{2} - 4 i \, c d^{2} f^{2} m x + 2 i \, d^{3} m^{3} - 6 i \, d^{3} m^{2} + {\left (-2 i \, c^{2} d f^{2} + 4 i \, d^{3}\right )} m\right )} {\left (d x + c\right )}^{m}}{-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 (f x + e\right )}}, x\right )}{3 i \, a^{2} d^{2} f^{3} x^{2} + 6 i \, a^{2} c d f^{3} x + 3 i \, a^{2} c^{2} f^{3} + 3 \, {\left (a^{2} d^{2} f^{3} x^{2} + 2 \, a^{2} c d f^{3} x + a^{2} c^{2} f^{3}\right )} e^{\left (3 \, f x + 3 \, e\right )} + {\left (-9 i \, a^{2} d^{2} f^{3} x^{2} - 18 i \, a^{2} c d f^{3} x - 9 i \, a^{2} c^{2} f^{3}\right )} e^{\left (2 \, f x + 2 \, e\right )} - 9 \, {\left (a^{2} d^{2} f^{3} x^{2} + 2 \, a^{2} c d f^{3} x + a^{2} c^{2} 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 \[ \int \frac {{\left (d x + c\right )}^{m}}{{\left (i \, a \sinh \left (f x + e\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 0, normalized size = 0.00 \[ \int \frac {\left (d x +c \right )^{m}}{\left (a +i a \sinh \left (f x +e \right )\right )^{2}}\, dx \]
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 \[ \int \frac {{\left (d x + c\right )}^{m}}{{\left (i \, a \sinh \left (f x + e\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {{\left (c+d\,x\right )}^m}{{\left (a+a\,\mathrm {sinh}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^2} \,d x \]
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 \[ - \frac {\int \frac {\left (c + d x\right )^{m}}{\sinh ^{2}{\left (e + f x \right )} - 2 i \sinh {\left (e + f x \right )} - 1}\, dx}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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