Optimal. Leaf size=121 \[ \frac{3 \cosh \left (a-\frac{b c}{d}\right ) \text{Chi}\left (\frac{b c}{d}+b x\right )}{4 d}+\frac{\cosh \left (3 a-\frac{3 b c}{d}\right ) \text{Chi}\left (\frac{3 b c}{d}+3 b x\right )}{4 d}+\frac{3 \sinh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (\frac{b c}{d}+b x\right )}{4 d}+\frac{\sinh \left (3 a-\frac{3 b c}{d}\right ) \text{Shi}\left (\frac{3 b c}{d}+3 b x\right )}{4 d} \]
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Rubi [A] time = 0.23984, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3312, 3303, 3298, 3301} \[ \frac{3 \cosh \left (a-\frac{b c}{d}\right ) \text{Chi}\left (\frac{b c}{d}+b x\right )}{4 d}+\frac{\cosh \left (3 a-\frac{3 b c}{d}\right ) \text{Chi}\left (\frac{3 b c}{d}+3 b x\right )}{4 d}+\frac{3 \sinh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (\frac{b c}{d}+b x\right )}{4 d}+\frac{\sinh \left (3 a-\frac{3 b c}{d}\right ) \text{Shi}\left (\frac{3 b c}{d}+3 b x\right )}{4 d} \]
Antiderivative was successfully verified.
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Rule 3312
Rule 3303
Rule 3298
Rule 3301
Rubi steps
\begin{align*} \int \frac{\cosh ^3(a+b x)}{c+d x} \, dx &=\int \left (\frac{3 \cosh (a+b x)}{4 (c+d x)}+\frac{\cosh (3 a+3 b x)}{4 (c+d x)}\right ) \, dx\\ &=\frac{1}{4} \int \frac{\cosh (3 a+3 b x)}{c+d x} \, dx+\frac{3}{4} \int \frac{\cosh (a+b x)}{c+d x} \, dx\\ &=\frac{1}{4} \cosh \left (3 a-\frac{3 b c}{d}\right ) \int \frac{\cosh \left (\frac{3 b c}{d}+3 b x\right )}{c+d x} \, dx+\frac{1}{4} \left (3 \cosh \left (a-\frac{b c}{d}\right )\right ) \int \frac{\cosh \left (\frac{b c}{d}+b x\right )}{c+d x} \, dx+\frac{1}{4} \sinh \left (3 a-\frac{3 b c}{d}\right ) \int \frac{\sinh \left (\frac{3 b c}{d}+3 b x\right )}{c+d x} \, dx+\frac{1}{4} \left (3 \sinh \left (a-\frac{b c}{d}\right )\right ) \int \frac{\sinh \left (\frac{b c}{d}+b x\right )}{c+d x} \, dx\\ &=\frac{3 \cosh \left (a-\frac{b c}{d}\right ) \text{Chi}\left (\frac{b c}{d}+b x\right )}{4 d}+\frac{\cosh \left (3 a-\frac{3 b c}{d}\right ) \text{Chi}\left (\frac{3 b c}{d}+3 b x\right )}{4 d}+\frac{3 \sinh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (\frac{b c}{d}+b x\right )}{4 d}+\frac{\sinh \left (3 a-\frac{3 b c}{d}\right ) \text{Shi}\left (\frac{3 b c}{d}+3 b x\right )}{4 d}\\ \end{align*}
Mathematica [A] time = 0.227433, size = 102, normalized size = 0.84 \[ \frac{3 \cosh \left (a-\frac{b c}{d}\right ) \text{Chi}\left (b \left (\frac{c}{d}+x\right )\right )+\cosh \left (3 a-\frac{3 b c}{d}\right ) \text{Chi}\left (\frac{3 b (c+d x)}{d}\right )+3 \sinh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (b \left (\frac{c}{d}+x\right )\right )+\sinh \left (3 a-\frac{3 b c}{d}\right ) \text{Shi}\left (\frac{3 b (c+d x)}{d}\right )}{4 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.088, size = 166, normalized size = 1.4 \begin{align*} -{\frac{1}{8\,d}{{\rm e}^{-3\,{\frac{da-cb}{d}}}}{\it Ei} \left ( 1,3\,bx+3\,a-3\,{\frac{da-cb}{d}} \right ) }-{\frac{3}{8\,d}{{\rm e}^{-{\frac{da-cb}{d}}}}{\it Ei} \left ( 1,bx+a-{\frac{da-cb}{d}} \right ) }-{\frac{3}{8\,d}{{\rm e}^{{\frac{da-cb}{d}}}}{\it Ei} \left ( 1,-bx-a-{\frac{-da+cb}{d}} \right ) }-{\frac{1}{8\,d}{{\rm e}^{3\,{\frac{da-cb}{d}}}}{\it Ei} \left ( 1,-3\,bx-3\,a-3\,{\frac{-da+cb}{d}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.27836, size = 158, normalized size = 1.31 \begin{align*} -\frac{e^{\left (-3 \, a + \frac{3 \, b c}{d}\right )} E_{1}\left (\frac{3 \,{\left (d x + c\right )} b}{d}\right )}{8 \, d} - \frac{3 \, e^{\left (-a + \frac{b c}{d}\right )} E_{1}\left (\frac{{\left (d x + c\right )} b}{d}\right )}{8 \, d} - \frac{3 \, e^{\left (a - \frac{b c}{d}\right )} E_{1}\left (-\frac{{\left (d x + c\right )} b}{d}\right )}{8 \, d} - \frac{e^{\left (3 \, a - \frac{3 \, b c}{d}\right )} E_{1}\left (-\frac{3 \,{\left (d x + c\right )} b}{d}\right )}{8 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.98404, size = 398, normalized size = 3.29 \begin{align*} \frac{3 \,{\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{3 \,{\left (b d x + b c\right )}}{d}\right ) +{\rm Ei}\left (-\frac{3 \,{\left (b d x + b c\right )}}{d}\right )\right )} \cosh \left (-\frac{3 \,{\left (b c - a d\right )}}{d}\right ) + 3 \,{\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 ) +{\left ({\rm Ei}\left (\frac{3 \,{\left (b d x + b c\right )}}{d}\right ) -{\rm Ei}\left (-\frac{3 \,{\left (b d x + b c\right )}}{d}\right )\right )} \sinh \left (-\frac{3 \,{\left (b c - a d\right )}}{d}\right )}{8 \, 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{\cosh ^{3}{\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 = 1.33214, size = 151, normalized size = 1.25 \begin{align*} \frac{{\rm Ei}\left (\frac{3 \,{\left (b d x + b c\right )}}{d}\right ) e^{\left (3 \, a - \frac{3 \, b c}{d}\right )} + 3 \,{\rm Ei}\left (\frac{b d x + b c}{d}\right ) e^{\left (a - \frac{b c}{d}\right )} + 3 \,{\rm Ei}\left (-\frac{b d x + b c}{d}\right ) e^{\left (-a + \frac{b c}{d}\right )} +{\rm Ei}\left (-\frac{3 \,{\left (b d x + b c\right )}}{d}\right ) e^{\left (-3 \, a + \frac{3 \, b c}{d}\right )}}{8 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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