Optimal. Leaf size=71 \[ \frac{b \cosh \left (a-\frac{b c}{d}\right ) \text{Chi}\left (\frac{b c}{d}+b x\right )}{d^2}+\frac{b \sinh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (\frac{b c}{d}+b x\right )}{d^2}-\frac{\sinh (a+b x)}{d (c+d x)} \]
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Rubi [A] time = 0.126858, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {3297, 3303, 3298, 3301} \[ \frac{b \cosh \left (a-\frac{b c}{d}\right ) \text{Chi}\left (\frac{b c}{d}+b x\right )}{d^2}+\frac{b \sinh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (\frac{b c}{d}+b x\right )}{d^2}-\frac{\sinh (a+b x)}{d (c+d x)} \]
Antiderivative was successfully verified.
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Rule 3297
Rule 3303
Rule 3298
Rule 3301
Rubi steps
\begin{align*} \int \frac{\sinh (a+b x)}{(c+d x)^2} \, dx &=-\frac{\sinh (a+b x)}{d (c+d x)}+\frac{b \int \frac{\cosh (a+b x)}{c+d x} \, dx}{d}\\ &=-\frac{\sinh (a+b x)}{d (c+d x)}+\frac{\left (b \cosh \left (a-\frac{b c}{d}\right )\right ) \int \frac{\cosh \left (\frac{b c}{d}+b x\right )}{c+d x} \, dx}{d}+\frac{\left (b \sinh \left (a-\frac{b c}{d}\right )\right ) \int \frac{\sinh \left (\frac{b c}{d}+b x\right )}{c+d x} \, dx}{d}\\ &=\frac{b \cosh \left (a-\frac{b c}{d}\right ) \text{Chi}\left (\frac{b c}{d}+b x\right )}{d^2}-\frac{\sinh (a+b x)}{d (c+d x)}+\frac{b \sinh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (\frac{b c}{d}+b x\right )}{d^2}\\ \end{align*}
Mathematica [A] time = 0.238479, size = 65, normalized size = 0.92 \[ \frac{b \cosh \left (a-\frac{b c}{d}\right ) \text{Chi}\left (b \left (\frac{c}{d}+x\right )\right )+b \sinh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (b \left (\frac{c}{d}+x\right )\right )-\frac{d \sinh (a+b x)}{c+d x}}{d^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.033, size = 133, normalized size = 1.9 \begin{align*}{\frac{b{{\rm e}^{-bx-a}}}{2\,d \left ( bdx+cb \right ) }}-{\frac{b}{2\,{d}^{2}}{{\rm e}^{-{\frac{da-cb}{d}}}}{\it Ei} \left ( 1,bx+a-{\frac{da-cb}{d}} \right ) }-{\frac{b{{\rm e}^{bx+a}}}{2\,{d}^{2}} \left ({\frac{cb}{d}}+bx \right ) ^{-1}}-{\frac{b}{2\,{d}^{2}}{{\rm e}^{{\frac{da-cb}{d}}}}{\it Ei} \left ( 1,-bx-a-{\frac{-da+cb}{d}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.38578, size = 108, normalized size = 1.52 \begin{align*} -\frac{b{\left (\frac{e^{\left (-a + \frac{b c}{d}\right )} E_{1}\left (\frac{{\left (d x + c\right )} b}{d}\right )}{d} + \frac{e^{\left (a - \frac{b c}{d}\right )} E_{1}\left (-\frac{{\left (d x + c\right )} b}{d}\right )}{d}\right )}}{2 \, d} - \frac{\sinh \left (b x + a\right )}{{\left (d x + c\right )} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.67794, size = 315, normalized size = 4.44 \begin{align*} \frac{{\left ({\left (b d x + b c\right )}{\rm Ei}\left (\frac{b d x + b c}{d}\right ) +{\left (b d x + b c\right )}{\rm Ei}\left (-\frac{b d x + b c}{d}\right )\right )} \cosh \left (-\frac{b c - a d}{d}\right ) - 2 \, d \sinh \left (b x + a\right ) +{\left ({\left (b d x + b c\right )}{\rm Ei}\left (\frac{b d x + b c}{d}\right ) -{\left (b d x + b c\right )}{\rm Ei}\left (-\frac{b d x + b c}{d}\right )\right )} \sinh \left (-\frac{b c - a d}{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 [B] time = 1.27503, size = 197, normalized size = 2.77 \begin{align*} \frac{b d x{\rm Ei}\left (\frac{b d x + b c}{d}\right ) e^{\left (a - \frac{b c}{d}\right )} + b d x{\rm Ei}\left (-\frac{b d x + b c}{d}\right ) e^{\left (-a + \frac{b c}{d}\right )} + b c{\rm Ei}\left (\frac{b d x + b c}{d}\right ) e^{\left (a - \frac{b c}{d}\right )} + b c{\rm Ei}\left (-\frac{b d x + b c}{d}\right ) e^{\left (-a + \frac{b c}{d}\right )} - d e^{\left (b x + a\right )} + d e^{\left (-b x - a\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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