Optimal. Leaf size=87 \[ -\frac{a}{d (c+d x)}+\frac{b f \text{Chi}\left (x f+\frac{c f}{d}\right ) \sinh \left (e-\frac{c f}{d}\right )}{d^2}+\frac{b f \cosh \left (e-\frac{c f}{d}\right ) \text{Shi}\left (x f+\frac{c f}{d}\right )}{d^2}-\frac{b \cosh (e+f x)}{d (c+d x)} \]
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Rubi [A] time = 0.151882, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 6, 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} \[ -\frac{a}{d (c+d x)}+\frac{b f \text{Chi}\left (x f+\frac{c f}{d}\right ) \sinh \left (e-\frac{c f}{d}\right )}{d^2}+\frac{b f \cosh \left (e-\frac{c f}{d}\right ) \text{Shi}\left (x f+\frac{c f}{d}\right )}{d^2}-\frac{b \cosh (e+f x)}{d (c+d x)} \]
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+b \cosh (e+f x)}{(c+d x)^2} \, dx &=\int \left (\frac{a}{(c+d x)^2}+\frac{b \cosh (e+f x)}{(c+d x)^2}\right ) \, dx\\ &=-\frac{a}{d (c+d x)}+b \int \frac{\cosh (e+f x)}{(c+d x)^2} \, dx\\ &=-\frac{a}{d (c+d x)}-\frac{b \cosh (e+f x)}{d (c+d x)}+\frac{(b f) \int \frac{\sinh (e+f x)}{c+d x} \, dx}{d}\\ &=-\frac{a}{d (c+d x)}-\frac{b \cosh (e+f x)}{d (c+d x)}+\frac{\left (b f \cosh \left (e-\frac{c f}{d}\right )\right ) \int \frac{\sinh \left (\frac{c f}{d}+f x\right )}{c+d x} \, dx}{d}+\frac{\left (b f \sinh \left (e-\frac{c f}{d}\right )\right ) \int \frac{\cosh \left (\frac{c f}{d}+f x\right )}{c+d x} \, dx}{d}\\ &=-\frac{a}{d (c+d x)}-\frac{b \cosh (e+f x)}{d (c+d x)}+\frac{b f \text{Chi}\left (\frac{c f}{d}+f x\right ) \sinh \left (e-\frac{c f}{d}\right )}{d^2}+\frac{b f \cosh \left (e-\frac{c f}{d}\right ) \text{Shi}\left (\frac{c f}{d}+f x\right )}{d^2}\\ \end{align*}
Mathematica [A] time = 0.400004, size = 71, normalized size = 0.82 \[ \frac{-\frac{d (a+b \cosh (e+f x))}{c+d x}+b f \text{Chi}\left (f \left (\frac{c}{d}+x\right )\right ) \sinh \left (e-\frac{c f}{d}\right )+b f \cosh \left (e-\frac{c f}{d}\right ) \text{Shi}\left (f \left (\frac{c}{d}+x\right )\right )}{d^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 180., size = 0, normalized size = 0. \begin{align*} \text{hanged} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.39857, size = 117, normalized size = 1.34 \begin{align*} -\frac{1}{2} \, b{\left (\frac{e^{\left (-e + \frac{c f}{d}\right )} E_{2}\left (\frac{{\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )} d} + \frac{e^{\left (e - \frac{c f}{d}\right )} E_{2}\left (-\frac{{\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )} d}\right )} - \frac{a}{d^{2} x + c d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.35, size = 351, normalized size = 4.03 \begin{align*} -\frac{2 \, b d \cosh \left (f x + e\right ) + 2 \, a d -{\left ({\left (b d f x + b c f\right )}{\rm Ei}\left (\frac{d f x + c f}{d}\right ) -{\left (b d f x + b c f\right )}{\rm Ei}\left (-\frac{d f x + c f}{d}\right )\right )} \cosh \left (-\frac{d e - c f}{d}\right ) +{\left ({\left (b d f x + b c f\right )}{\rm Ei}\left (\frac{d f x + c f}{d}\right ) +{\left (b d f x + b c f\right )}{\rm Ei}\left (-\frac{d f x + c f}{d}\right )\right )} \sinh \left (-\frac{d e - c f}{d}\right )}{2 \,{\left (d^{3} x + c d^{2}\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 [A] time = 1.20195, size = 227, normalized size = 2.61 \begin{align*} -\frac{{\left (d f x{\rm Ei}\left (-\frac{d f x + c f}{d}\right ) e^{\left (\frac{c f}{d} - e\right )} - d f x{\rm Ei}\left (\frac{d f x + c f}{d}\right ) e^{\left (-\frac{c f}{d} + e\right )} + c f{\rm Ei}\left (-\frac{d f x + c f}{d}\right ) e^{\left (\frac{c f}{d} - e\right )} - c f{\rm Ei}\left (\frac{d f x + c f}{d}\right ) e^{\left (-\frac{c f}{d} + e\right )} + d e^{\left (f x + e\right )} + d e^{\left (-f x - e\right )}\right )} b}{2 \,{\left (d^{3} x + c d^{2}\right )}} - \frac{a}{{\left (d x + c\right )} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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