Optimal. Leaf size=123 \[ \frac{a f^2 \text{Chi}\left (x f+\frac{c f}{d}\right ) \cosh \left (e-\frac{c f}{d}\right )}{2 d^3}+\frac{a f^2 \sinh \left (e-\frac{c f}{d}\right ) \text{Shi}\left (x f+\frac{c f}{d}\right )}{2 d^3}-\frac{a f \sinh (e+f x)}{2 d^2 (c+d x)}-\frac{a \cosh (e+f x)}{2 d (c+d x)^2}-\frac{a}{2 d (c+d x)^2} \]
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Rubi [A] time = 0.215713, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278, Rules used = {3317, 3297, 3303, 3298, 3301} \[ \frac{a f^2 \text{Chi}\left (x f+\frac{c f}{d}\right ) \cosh \left (e-\frac{c f}{d}\right )}{2 d^3}+\frac{a f^2 \sinh \left (e-\frac{c f}{d}\right ) \text{Shi}\left (x f+\frac{c f}{d}\right )}{2 d^3}-\frac{a f \sinh (e+f x)}{2 d^2 (c+d x)}-\frac{a \cosh (e+f x)}{2 d (c+d x)^2}-\frac{a}{2 d (c+d x)^2} \]
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+a \cosh (e+f x)}{(c+d x)^3} \, dx &=\int \left (\frac{a}{(c+d x)^3}+\frac{a \cosh (e+f x)}{(c+d x)^3}\right ) \, dx\\ &=-\frac{a}{2 d (c+d x)^2}+a \int \frac{\cosh (e+f x)}{(c+d x)^3} \, dx\\ &=-\frac{a}{2 d (c+d x)^2}-\frac{a \cosh (e+f x)}{2 d (c+d x)^2}+\frac{(a f) \int \frac{\sinh (e+f x)}{(c+d x)^2} \, dx}{2 d}\\ &=-\frac{a}{2 d (c+d x)^2}-\frac{a \cosh (e+f x)}{2 d (c+d x)^2}-\frac{a f \sinh (e+f x)}{2 d^2 (c+d x)}+\frac{\left (a f^2\right ) \int \frac{\cosh (e+f x)}{c+d x} \, dx}{2 d^2}\\ &=-\frac{a}{2 d (c+d x)^2}-\frac{a \cosh (e+f x)}{2 d (c+d x)^2}-\frac{a f \sinh (e+f x)}{2 d^2 (c+d x)}+\frac{\left (a f^2 \cosh \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{\left (a f^2 \sinh \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{a}{2 d (c+d x)^2}-\frac{a \cosh (e+f x)}{2 d (c+d x)^2}+\frac{a f^2 \cosh \left (e-\frac{c f}{d}\right ) \text{Chi}\left (\frac{c f}{d}+f x\right )}{2 d^3}-\frac{a f \sinh (e+f x)}{2 d^2 (c+d x)}+\frac{a f^2 \sinh \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 = 0.455808, size = 90, normalized size = 0.73 \[ \frac{a \left (f^2 \text{Chi}\left (f \left (\frac{c}{d}+x\right )\right ) \cosh \left (e-\frac{c f}{d}\right )+f^2 \sinh \left (e-\frac{c f}{d}\right ) \text{Shi}\left (f \left (\frac{c}{d}+x\right )\right )-\frac{d (f (c+d x) \sinh (e+f x)+d \cosh (e+f x)+d)}{(c+d x)^2}\right )}{2 d^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.053, size = 296, normalized size = 2.4 \begin{align*} -{\frac{a}{2\,d \left ( dx+c \right ) ^{2}}}+{\frac{{f}^{3}a{{\rm e}^{-fx-e}}x}{4\,d \left ({d}^{2}{f}^{2}{x}^{2}+2\,cd{f}^{2}x+{c}^{2}{f}^{2} \right ) }}+{\frac{{f}^{3}a{{\rm e}^{-fx-e}}c}{4\,{d}^{2} \left ({d}^{2}{f}^{2}{x}^{2}+2\,cd{f}^{2}x+{c}^{2}{f}^{2} \right ) }}-{\frac{{f}^{2}a{{\rm e}^{-fx-e}}}{4\,d \left ({d}^{2}{f}^{2}{x}^{2}+2\,cd{f}^{2}x+{c}^{2}{f}^{2} \right ) }}-{\frac{{f}^{2}a}{4\,{d}^{3}}{{\rm e}^{{\frac{cf-de}{d}}}}{\it Ei} \left ( 1,fx+e+{\frac{cf-de}{d}} \right ) }-{\frac{{f}^{2}a{{\rm e}^{fx+e}}}{4\,{d}^{3}} \left ({\frac{cf}{d}}+fx \right ) ^{-2}}-{\frac{{f}^{2}a{{\rm e}^{fx+e}}}{4\,{d}^{3}} \left ({\frac{cf}{d}}+fx \right ) ^{-1}}-{\frac{{f}^{2}a}{4\,{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 = 1.22622, size = 132, normalized size = 1.07 \begin{align*} -\frac{1}{2} \, 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 [B] time = 2.07163, size = 572, normalized size = 4.65 \begin{align*} -\frac{2 \, a d^{2} \cosh \left (f x + e\right ) + 2 \, a d^{2} -{\left ({\left (a d^{2} f^{2} x^{2} + 2 \, a c d f^{2} x + a c^{2} f^{2}\right )}{\rm Ei}\left (\frac{d f x + c f}{d}\right ) +{\left (a d^{2} f^{2} x^{2} + 2 \, a c d f^{2} x + a c^{2} f^{2}\right )}{\rm Ei}\left (-\frac{d f x + c f}{d}\right )\right )} \cosh \left (-\frac{d e - c f}{d}\right ) + 2 \,{\left (a d^{2} f x + a c d f\right )} \sinh \left (f x + e\right ) +{\left ({\left (a d^{2} f^{2} x^{2} + 2 \, a c d f^{2} x + a c^{2} f^{2}\right )}{\rm Ei}\left (\frac{d f x + c f}{d}\right ) -{\left (a d^{2} f^{2} x^{2} + 2 \, a c d f^{2} x + a c^{2} f^{2}\right )}{\rm Ei}\left (-\frac{d f x + c f}{d}\right )\right )} \sinh \left (-\frac{d e - c f}{d}\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 = 1.19746, size = 443, normalized size = 3.6 \begin{align*} \frac{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 )} + 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 \, 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 \, 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 )} + a c^{2} f^{2}{\rm Ei}\left (-\frac{d f x + c f}{d}\right ) e^{\left (\frac{c f}{d} - e\right )} + a c^{2} f^{2}{\rm Ei}\left (\frac{d f x + c f}{d}\right ) e^{\left (-\frac{c f}{d} + e\right )} - a d^{2} f x e^{\left (f x + e\right )} + a d^{2} f x e^{\left (-f x - e\right )} - a c d f e^{\left (f x + e\right )} + a c d f e^{\left (-f x - e\right )} - a d^{2} e^{\left (f x + e\right )} - 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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