Optimal. Leaf size=71 \[ \frac{d \sinh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (x d+\frac{a d}{b}\right )}{b^2}+\frac{d \cosh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (x d+\frac{a d}{b}\right )}{b^2}-\frac{\cosh (c+d x)}{b (a+b x)} \]
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Rubi [A] time = 0.109993, 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} \[ \frac{d \sinh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (x d+\frac{a d}{b}\right )}{b^2}+\frac{d \cosh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (x d+\frac{a d}{b}\right )}{b^2}-\frac{\cosh (c+d x)}{b (a+b x)} \]
Antiderivative was successfully verified.
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Rule 3297
Rule 3303
Rule 3298
Rule 3301
Rubi steps
\begin{align*} \int \frac{\cosh (c+d x)}{(a+b x)^2} \, dx &=-\frac{\cosh (c+d x)}{b (a+b x)}+\frac{d \int \frac{\sinh (c+d x)}{a+b x} \, dx}{b}\\ &=-\frac{\cosh (c+d x)}{b (a+b x)}+\frac{\left (d \cosh \left (c-\frac{a d}{b}\right )\right ) \int \frac{\sinh \left (\frac{a d}{b}+d x\right )}{a+b x} \, dx}{b}+\frac{\left (d \sinh \left (c-\frac{a d}{b}\right )\right ) \int \frac{\cosh \left (\frac{a d}{b}+d x\right )}{a+b x} \, dx}{b}\\ &=-\frac{\cosh (c+d x)}{b (a+b x)}+\frac{d \text{Chi}\left (\frac{a d}{b}+d x\right ) \sinh \left (c-\frac{a d}{b}\right )}{b^2}+\frac{d \cosh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (\frac{a d}{b}+d x\right )}{b^2}\\ \end{align*}
Mathematica [A] time = 0.201944, size = 65, normalized size = 0.92 \[ \frac{d \sinh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (d \left (\frac{a}{b}+x\right )\right )+d \cosh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (d \left (\frac{a}{b}+x\right )\right )-\frac{b \cosh (c+d x)}{a+b x}}{b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.025, size = 132, normalized size = 1.9 \begin{align*} -{\frac{d{{\rm e}^{-dx-c}}}{2\,b \left ( bdx+da \right ) }}+{\frac{d}{2\,{b}^{2}}{{\rm e}^{{\frac{da-cb}{b}}}}{\it Ei} \left ( 1,dx+c+{\frac{da-cb}{b}} \right ) }-{\frac{d{{\rm e}^{dx+c}}}{2\,{b}^{2}} \left ({\frac{da}{b}}+dx \right ) ^{-1}}-{\frac{d}{2\,{b}^{2}}{{\rm e}^{-{\frac{da-cb}{b}}}}{\it Ei} \left ( 1,-dx-c-{\frac{da-cb}{b}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.20947, size = 109, normalized size = 1.54 \begin{align*} \frac{d{\left (\frac{e^{\left (-c + \frac{a d}{b}\right )} E_{1}\left (\frac{{\left (b x + a\right )} d}{b}\right )}{b} - \frac{e^{\left (c - \frac{a d}{b}\right )} E_{1}\left (-\frac{{\left (b x + a\right )} d}{b}\right )}{b}\right )}}{2 \, b} - \frac{\cosh \left (d x + c\right )}{{\left (b x + a\right )} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.03666, size = 316, normalized size = 4.45 \begin{align*} -\frac{2 \, b \cosh \left (d x + c\right ) -{\left ({\left (b d x + a d\right )}{\rm Ei}\left (\frac{b d x + a d}{b}\right ) -{\left (b d x + a d\right )}{\rm Ei}\left (-\frac{b d x + a d}{b}\right )\right )} \cosh \left (-\frac{b c - a d}{b}\right ) +{\left ({\left (b d x + a d\right )}{\rm Ei}\left (\frac{b d x + a d}{b}\right ) +{\left (b d x + a d\right )}{\rm Ei}\left (-\frac{b d x + a d}{b}\right )\right )} \sinh \left (-\frac{b c - a d}{b}\right )}{2 \,{\left (b^{3} x + a b^{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 = 1.21064, size = 201, normalized size = 2.83 \begin{align*} \frac{b d x{\rm Ei}\left (\frac{b d x + a d}{b}\right ) e^{\left (c - \frac{a d}{b}\right )} - b d x{\rm Ei}\left (-\frac{b d x + a d}{b}\right ) e^{\left (-c + \frac{a d}{b}\right )} + a d{\rm Ei}\left (\frac{b d x + a d}{b}\right ) e^{\left (c - \frac{a d}{b}\right )} - a d{\rm Ei}\left (-\frac{b d x + a d}{b}\right ) e^{\left (-c + \frac{a d}{b}\right )} - b e^{\left (d x + c\right )} - b e^{\left (-d x - c\right )}}{2 \,{\left (b^{3} x + a b^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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