Optimal. Leaf size=104 \[ \frac{b^2 \cosh \left (a-\frac{b c}{d}\right ) \text{Chi}\left (\frac{b c}{d}+b x\right )}{2 d^3}+\frac{b^2 \sinh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (\frac{b c}{d}+b x\right )}{2 d^3}-\frac{b \sinh (a+b x)}{2 d^2 (c+d x)}-\frac{\cosh (a+b x)}{2 d (c+d x)^2} \]
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Rubi [A] time = 0.162841, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {3297, 3303, 3298, 3301} \[ \frac{b^2 \cosh \left (a-\frac{b c}{d}\right ) \text{Chi}\left (\frac{b c}{d}+b x\right )}{2 d^3}+\frac{b^2 \sinh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (\frac{b c}{d}+b x\right )}{2 d^3}-\frac{b \sinh (a+b x)}{2 d^2 (c+d x)}-\frac{\cosh (a+b x)}{2 d (c+d x)^2} \]
Antiderivative was successfully verified.
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Rule 3297
Rule 3303
Rule 3298
Rule 3301
Rubi steps
\begin{align*} \int \frac{\cosh (a+b x)}{(c+d x)^3} \, dx &=-\frac{\cosh (a+b x)}{2 d (c+d x)^2}+\frac{b \int \frac{\sinh (a+b x)}{(c+d x)^2} \, dx}{2 d}\\ &=-\frac{\cosh (a+b x)}{2 d (c+d x)^2}-\frac{b \sinh (a+b x)}{2 d^2 (c+d x)}+\frac{b^2 \int \frac{\cosh (a+b x)}{c+d x} \, dx}{2 d^2}\\ &=-\frac{\cosh (a+b x)}{2 d (c+d x)^2}-\frac{b \sinh (a+b x)}{2 d^2 (c+d x)}+\frac{\left (b^2 \cosh \left (a-\frac{b c}{d}\right )\right ) \int \frac{\cosh \left (\frac{b c}{d}+b x\right )}{c+d x} \, dx}{2 d^2}+\frac{\left (b^2 \sinh \left (a-\frac{b c}{d}\right )\right ) \int \frac{\sinh \left (\frac{b c}{d}+b x\right )}{c+d x} \, dx}{2 d^2}\\ &=-\frac{\cosh (a+b x)}{2 d (c+d x)^2}+\frac{b^2 \cosh \left (a-\frac{b c}{d}\right ) \text{Chi}\left (\frac{b c}{d}+b x\right )}{2 d^3}-\frac{b \sinh (a+b x)}{2 d^2 (c+d x)}+\frac{b^2 \sinh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (\frac{b c}{d}+b x\right )}{2 d^3}\\ \end{align*}
Mathematica [A] time = 0.525075, size = 88, normalized size = 0.85 \[ \frac{b^2 \cosh \left (a-\frac{b c}{d}\right ) \text{Chi}\left (b \left (\frac{c}{d}+x\right )\right )+b^2 \sinh \left (a-\frac{b c}{d}\right ) \text{Shi}\left (b \left (\frac{c}{d}+x\right )\right )-\frac{d (b (c+d x) \sinh (a+b x)+d \cosh (a+b x))}{(c+d x)^2}}{2 d^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.039, size = 277, normalized size = 2.7 \begin{align*}{\frac{{b}^{3}{{\rm e}^{-bx-a}}x}{4\,d \left ({b}^{2}{d}^{2}{x}^{2}+2\,{b}^{2}cdx+{c}^{2}{b}^{2} \right ) }}+{\frac{{b}^{3}{{\rm e}^{-bx-a}}c}{4\,{d}^{2} \left ({b}^{2}{d}^{2}{x}^{2}+2\,{b}^{2}cdx+{c}^{2}{b}^{2} \right ) }}-{\frac{{b}^{2}{{\rm e}^{-bx-a}}}{4\,d \left ({b}^{2}{d}^{2}{x}^{2}+2\,{b}^{2}cdx+{c}^{2}{b}^{2} \right ) }}-{\frac{{b}^{2}}{4\,{d}^{3}}{{\rm e}^{-{\frac{da-cb}{d}}}}{\it Ei} \left ( 1,bx+a-{\frac{da-cb}{d}} \right ) }-{\frac{{b}^{2}{{\rm e}^{bx+a}}}{4\,{d}^{3}} \left ({\frac{cb}{d}}+bx \right ) ^{-2}}-{\frac{{b}^{2}{{\rm e}^{bx+a}}}{4\,{d}^{3}} \left ({\frac{cb}{d}}+bx \right ) ^{-1}}-{\frac{{b}^{2}}{4\,{d}^{3}}{{\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 = 1.21465, size = 128, normalized size = 1.23 \begin{align*} \frac{b{\left (\frac{e^{\left (-a + \frac{b c}{d}\right )} E_{2}\left (\frac{{\left (d x + c\right )} b}{d}\right )}{{\left (d x + c\right )} d} - \frac{e^{\left (a - \frac{b c}{d}\right )} E_{2}\left (-\frac{{\left (d x + c\right )} b}{d}\right )}{{\left (d x + c\right )} d}\right )}}{4 \, d} - \frac{\cosh \left (b x + a\right )}{2 \,{\left (d x + c\right )}^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.84861, size = 518, normalized size = 4.98 \begin{align*} -\frac{2 \, d^{2} \cosh \left (b x + a\right ) -{\left ({\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )}{\rm Ei}\left (\frac{b d x + b c}{d}\right ) +{\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )}{\rm Ei}\left (-\frac{b d x + b c}{d}\right )\right )} \cosh \left (-\frac{b c - a d}{d}\right ) + 2 \,{\left (b d^{2} x + b c d\right )} \sinh \left (b x + a\right ) -{\left ({\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )}{\rm Ei}\left (\frac{b d x + b c}{d}\right ) -{\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )}{\rm Ei}\left (-\frac{b d x + b c}{d}\right )\right )} \sinh \left (-\frac{b c - a d}{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 = 1.25575, size = 402, normalized size = 3.87 \begin{align*} \frac{b^{2} d^{2} x^{2}{\rm Ei}\left (\frac{b d x + b c}{d}\right ) e^{\left (a - \frac{b c}{d}\right )} + b^{2} d^{2} x^{2}{\rm Ei}\left (-\frac{b d x + b c}{d}\right ) e^{\left (-a + \frac{b c}{d}\right )} + 2 \, b^{2} c d x{\rm Ei}\left (\frac{b d x + b c}{d}\right ) e^{\left (a - \frac{b c}{d}\right )} + 2 \, b^{2} c d x{\rm Ei}\left (-\frac{b d x + b c}{d}\right ) e^{\left (-a + \frac{b c}{d}\right )} + b^{2} c^{2}{\rm Ei}\left (\frac{b d x + b c}{d}\right ) e^{\left (a - \frac{b c}{d}\right )} + b^{2} c^{2}{\rm Ei}\left (-\frac{b d x + b c}{d}\right ) e^{\left (-a + \frac{b c}{d}\right )} - b d^{2} x e^{\left (b x + a\right )} + b d^{2} x e^{\left (-b x - a\right )} - b c d e^{\left (b x + a\right )} + b c d e^{\left (-b x - a\right )} - d^{2} e^{\left (b x + a\right )} - d^{2} e^{\left (-b x - a\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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