Optimal. Leaf size=131 \[ \frac{i a f^2 \text{Chi}\left (x f+\frac{c f}{d}\right ) \sinh \left (e-\frac{c f}{d}\right )}{2 d^3}+\frac{i a f^2 \cosh \left (e-\frac{c f}{d}\right ) \text{Shi}\left (x f+\frac{c f}{d}\right )}{2 d^3}-\frac{i a f \cosh (e+f x)}{2 d^2 (c+d x)}-\frac{i a \sinh (e+f x)}{2 d (c+d x)^2}-\frac{a}{2 d (c+d x)^2} \]
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Rubi [A] time = 0.232228, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {3317, 3297, 3303, 3298, 3301} \[ \frac{i a f^2 \text{Chi}\left (x f+\frac{c f}{d}\right ) \sinh \left (e-\frac{c f}{d}\right )}{2 d^3}+\frac{i a f^2 \cosh \left (e-\frac{c f}{d}\right ) \text{Shi}\left (x f+\frac{c f}{d}\right )}{2 d^3}-\frac{i a f \cosh (e+f x)}{2 d^2 (c+d x)}-\frac{i a \sinh (e+f x)}{2 d (c+d x)^2}-\frac{a}{2 d (c+d x)^2} \]
Antiderivative was successfully verified.
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Rule 3317
Rule 3297
Rule 3303
Rule 3298
Rule 3301
Rubi steps
\begin{align*} \int \frac{a+i a \sinh (e+f x)}{(c+d x)^3} \, dx &=\int \left (\frac{a}{(c+d x)^3}+\frac{i a \sinh (e+f x)}{(c+d x)^3}\right ) \, dx\\ &=-\frac{a}{2 d (c+d x)^2}+(i a) \int \frac{\sinh (e+f x)}{(c+d x)^3} \, dx\\ &=-\frac{a}{2 d (c+d x)^2}-\frac{i a \sinh (e+f x)}{2 d (c+d x)^2}+\frac{(i a f) \int \frac{\cosh (e+f x)}{(c+d x)^2} \, dx}{2 d}\\ &=-\frac{a}{2 d (c+d x)^2}-\frac{i a f \cosh (e+f x)}{2 d^2 (c+d x)}-\frac{i a \sinh (e+f x)}{2 d (c+d x)^2}+\frac{\left (i a f^2\right ) \int \frac{\sinh (e+f x)}{c+d x} \, dx}{2 d^2}\\ &=-\frac{a}{2 d (c+d x)^2}-\frac{i a f \cosh (e+f x)}{2 d^2 (c+d x)}-\frac{i a \sinh (e+f x)}{2 d (c+d x)^2}+\frac{\left (i a f^2 \cosh \left (e-\frac{c f}{d}\right )\right ) \int \frac{\sinh \left (\frac{c f}{d}+f x\right )}{c+d x} \, dx}{2 d^2}+\frac{\left (i a f^2 \sinh \left (e-\frac{c f}{d}\right )\right ) \int \frac{\cosh \left (\frac{c f}{d}+f x\right )}{c+d x} \, dx}{2 d^2}\\ &=-\frac{a}{2 d (c+d x)^2}-\frac{i a f \cosh (e+f x)}{2 d^2 (c+d x)}+\frac{i a f^2 \text{Chi}\left (\frac{c f}{d}+f x\right ) \sinh \left (e-\frac{c f}{d}\right )}{2 d^3}-\frac{i a \sinh (e+f x)}{2 d (c+d x)^2}+\frac{i a f^2 \cosh \left (e-\frac{c f}{d}\right ) \text{Shi}\left (\frac{c f}{d}+f x\right )}{2 d^3}\\ \end{align*}
Mathematica [A] time = 0.669285, size = 109, normalized size = 0.83 \[ \frac{i a \left (f^2 (c+d x)^2 \text{Chi}\left (f \left (\frac{c}{d}+x\right )\right ) \sinh \left (e-\frac{c f}{d}\right )+f^2 (c+d x)^2 \cosh \left (e-\frac{c f}{d}\right ) \text{Shi}\left (f \left (\frac{c}{d}+x\right )\right )-d (f (c+d x) \cosh (e+f x)+d (\sinh (e+f x)-i))\right )}{2 d^3 (c+d x)^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.06, size = 303, normalized size = 2.3 \begin{align*} -{\frac{a}{2\,d \left ( dx+c \right ) ^{2}}}-{\frac{{\frac{i}{4}}a{f}^{3}{{\rm e}^{-fx-e}}x}{d \left ({d}^{2}{f}^{2}{x}^{2}+2\,cd{f}^{2}x+{c}^{2}{f}^{2} \right ) }}-{\frac{{\frac{i}{4}}a{f}^{3}{{\rm e}^{-fx-e}}c}{{d}^{2} \left ({d}^{2}{f}^{2}{x}^{2}+2\,cd{f}^{2}x+{c}^{2}{f}^{2} \right ) }}+{\frac{{\frac{i}{4}}a{f}^{2}{{\rm e}^{-fx-e}}}{d \left ({d}^{2}{f}^{2}{x}^{2}+2\,cd{f}^{2}x+{c}^{2}{f}^{2} \right ) }}+{\frac{{\frac{i}{4}}a{f}^{2}}{{d}^{3}}{{\rm e}^{{\frac{cf-de}{d}}}}{\it Ei} \left ( 1,fx+e+{\frac{cf-de}{d}} \right ) }-{\frac{{\frac{i}{4}}a{f}^{2}{{\rm e}^{fx+e}}}{{d}^{3}} \left ({\frac{cf}{d}}+fx \right ) ^{-2}}-{\frac{{\frac{i}{4}}a{f}^{2}{{\rm e}^{fx+e}}}{{d}^{3}} \left ({\frac{cf}{d}}+fx \right ) ^{-1}}-{\frac{{\frac{i}{4}}a{f}^{2}}{{d}^{3}}{{\rm e}^{-{\frac{cf-de}{d}}}}{\it Ei} \left ( 1,-fx-e-{\frac{cf-de}{d}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.21992, size = 134, normalized size = 1.02 \begin{align*} \frac{1}{2} i \, a{\left (\frac{e^{\left (-e + \frac{c f}{d}\right )} E_{3}\left (\frac{{\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )}^{2} d} - \frac{e^{\left (e - \frac{c f}{d}\right )} E_{3}\left (-\frac{{\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )}^{2} d}\right )} - \frac{a}{2 \,{\left (d^{3} x^{2} + 2 \, c d^{2} x + c^{2} d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.72999, size = 471, normalized size = 3.6 \begin{align*} \frac{{\left (-i \, a d^{2} f x - i \, a c d f + i \, a d^{2} +{\left (-i \, a d^{2} f x - i \, a c d f - i \, a d^{2}\right )} e^{\left (2 \, f x + 2 \, e\right )} -{\left (2 \, a d^{2} -{\left (i \, a d^{2} f^{2} x^{2} + 2 i \, a c d f^{2} x + i \, a c^{2} f^{2}\right )}{\rm Ei}\left (\frac{d f x + c f}{d}\right ) e^{\left (\frac{d e - c f}{d}\right )} -{\left (-i \, a d^{2} f^{2} x^{2} - 2 i \, a c d f^{2} x - i \, a c^{2} f^{2}\right )}{\rm Ei}\left (-\frac{d f x + c f}{d}\right ) e^{\left (-\frac{d e - c f}{d}\right )}\right )} e^{\left (f x + e\right )}\right )} e^{\left (-f x - e\right )}}{4 \,{\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.34846, size = 451, normalized size = 3.44 \begin{align*} \frac{-i \, a d^{2} f^{2} x^{2}{\rm Ei}\left (-\frac{d f x + c f}{d}\right ) e^{\left (\frac{c f}{d} - e\right )} + i \, a d^{2} f^{2} x^{2}{\rm Ei}\left (\frac{d f x + c f}{d}\right ) e^{\left (-\frac{c f}{d} + e\right )} - 2 i \, a c d f^{2} x{\rm Ei}\left (-\frac{d f x + c f}{d}\right ) e^{\left (\frac{c f}{d} - e\right )} + 2 i \, a c d f^{2} x{\rm Ei}\left (\frac{d f x + c f}{d}\right ) e^{\left (-\frac{c f}{d} + e\right )} - i \, a c^{2} f^{2}{\rm Ei}\left (-\frac{d f x + c f}{d}\right ) e^{\left (\frac{c f}{d} - e\right )} + i \, a c^{2} f^{2}{\rm Ei}\left (\frac{d f x + c f}{d}\right ) e^{\left (-\frac{c f}{d} + e\right )} - i \, a d^{2} f x e^{\left (f x + e\right )} - i \, a d^{2} f x e^{\left (-f x - e\right )} - i \, a c d f e^{\left (f x + e\right )} - i \, a c d f e^{\left (-f x - e\right )} - i \, a d^{2} e^{\left (f x + e\right )} + i \, a d^{2} e^{\left (-f x - e\right )} - 2 \, a d^{2}}{4 \,{\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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