Optimal. Leaf size=51 \[ \frac{\sinh \left (a-\frac{b c}{d}\right ) \text{Chi}\left (\frac{b c}{d}+b x\right )}{d}+\frac{\cosh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (\frac{b c}{d}+b x\right )}{d} \]
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Rubi [A] time = 0.109284, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {3303, 3298, 3301} \[ \frac{\sinh \left (a-\frac{b c}{d}\right ) \text{Chi}\left (\frac{b c}{d}+b x\right )}{d}+\frac{\cosh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (\frac{b c}{d}+b x\right )}{d} \]
Antiderivative was successfully verified.
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Rule 3303
Rule 3298
Rule 3301
Rubi steps
\begin{align*} \int \frac{\sinh (a+b x)}{c+d x} \, dx &=\cosh \left (a-\frac{b c}{d}\right ) \int \frac{\sinh \left (\frac{b c}{d}+b x\right )}{c+d x} \, dx+\sinh \left (a-\frac{b c}{d}\right ) \int \frac{\cosh \left (\frac{b c}{d}+b x\right )}{c+d x} \, dx\\ &=\frac{\text{Chi}\left (\frac{b c}{d}+b x\right ) \sinh \left (a-\frac{b c}{d}\right )}{d}+\frac{\cosh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (\frac{b c}{d}+b x\right )}{d}\\ \end{align*}
Mathematica [A] time = 0.0848559, size = 49, normalized size = 0.96 \[ \frac{\sinh \left (a-\frac{b c}{d}\right ) \text{Chi}\left (\frac{b c}{d}+b x\right )+\cosh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (\frac{b c}{d}+b x\right )}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.059, size = 82, normalized size = 1.6 \begin{align*}{\frac{1}{2\,d}{{\rm e}^{-{\frac{da-cb}{d}}}}{\it Ei} \left ( 1,bx+a-{\frac{da-cb}{d}} \right ) }-{\frac{1}{2\,d}{{\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 = 1.2888, size = 77, normalized size = 1.51 \begin{align*} \frac{e^{\left (-a + \frac{b c}{d}\right )} E_{1}\left (\frac{{\left (d x + c\right )} b}{d}\right )}{2 \, d} - \frac{e^{\left (a - \frac{b c}{d}\right )} E_{1}\left (-\frac{{\left (d x + c\right )} b}{d}\right )}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.58023, size = 193, normalized size = 3.78 \begin{align*} \frac{{\left ({\rm Ei}\left (\frac{b d x + b c}{d}\right ) -{\rm Ei}\left (-\frac{b d x + b c}{d}\right )\right )} \cosh \left (-\frac{b c - a d}{d}\right ) +{\left ({\rm Ei}\left (\frac{b d x + b c}{d}\right ) +{\rm Ei}\left (-\frac{b d x + b c}{d}\right )\right )} \sinh \left (-\frac{b c - a d}{d}\right )}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sinh{\left (a + b x \right )}}{c + d x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15718, size = 77, normalized size = 1.51 \begin{align*} \frac{{\rm Ei}\left (\frac{b d x + b c}{d}\right ) e^{\left (a - \frac{b c}{d}\right )} -{\rm Ei}\left (-\frac{b d x + b c}{d}\right ) e^{\left (-a + \frac{b c}{d}\right )}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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