Optimal. Leaf size=150 \[ -\frac{a^3 \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (x d+\frac{a d}{b}\right )}{b^4}-\frac{a^3 \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (x d+\frac{a d}{b}\right )}{b^4}+\frac{a^2 \sinh (c+d x)}{b^3 d}+\frac{a \cosh (c+d x)}{b^2 d^2}-\frac{a x \sinh (c+d x)}{b^2 d}+\frac{2 \sinh (c+d x)}{b d^3}-\frac{2 x \cosh (c+d x)}{b d^2}+\frac{x^2 \sinh (c+d x)}{b d} \]
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Rubi [A] time = 0.328539, antiderivative size = 150, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.412, Rules used = {6742, 2637, 3296, 2638, 3303, 3298, 3301} \[ -\frac{a^3 \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (x d+\frac{a d}{b}\right )}{b^4}-\frac{a^3 \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (x d+\frac{a d}{b}\right )}{b^4}+\frac{a^2 \sinh (c+d x)}{b^3 d}+\frac{a \cosh (c+d x)}{b^2 d^2}-\frac{a x \sinh (c+d x)}{b^2 d}+\frac{2 \sinh (c+d x)}{b d^3}-\frac{2 x \cosh (c+d x)}{b d^2}+\frac{x^2 \sinh (c+d x)}{b d} \]
Antiderivative was successfully verified.
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Rule 6742
Rule 2637
Rule 3296
Rule 2638
Rule 3303
Rule 3298
Rule 3301
Rubi steps
\begin{align*} \int \frac{x^3 \cosh (c+d x)}{a+b x} \, dx &=\int \left (\frac{a^2 \cosh (c+d x)}{b^3}-\frac{a x \cosh (c+d x)}{b^2}+\frac{x^2 \cosh (c+d x)}{b}-\frac{a^3 \cosh (c+d x)}{b^3 (a+b x)}\right ) \, dx\\ &=\frac{a^2 \int \cosh (c+d x) \, dx}{b^3}-\frac{a^3 \int \frac{\cosh (c+d x)}{a+b x} \, dx}{b^3}-\frac{a \int x \cosh (c+d x) \, dx}{b^2}+\frac{\int x^2 \cosh (c+d x) \, dx}{b}\\ &=\frac{a^2 \sinh (c+d x)}{b^3 d}-\frac{a x \sinh (c+d x)}{b^2 d}+\frac{x^2 \sinh (c+d x)}{b d}+\frac{a \int \sinh (c+d x) \, dx}{b^2 d}-\frac{2 \int x \sinh (c+d x) \, dx}{b d}-\frac{\left (a^3 \cosh \left (c-\frac{a d}{b}\right )\right ) \int \frac{\cosh \left (\frac{a d}{b}+d x\right )}{a+b x} \, dx}{b^3}-\frac{\left (a^3 \sinh \left (c-\frac{a d}{b}\right )\right ) \int \frac{\sinh \left (\frac{a d}{b}+d x\right )}{a+b x} \, dx}{b^3}\\ &=\frac{a \cosh (c+d x)}{b^2 d^2}-\frac{2 x \cosh (c+d x)}{b d^2}-\frac{a^3 \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (\frac{a d}{b}+d x\right )}{b^4}+\frac{a^2 \sinh (c+d x)}{b^3 d}-\frac{a x \sinh (c+d x)}{b^2 d}+\frac{x^2 \sinh (c+d x)}{b d}-\frac{a^3 \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (\frac{a d}{b}+d x\right )}{b^4}+\frac{2 \int \cosh (c+d x) \, dx}{b d^2}\\ &=\frac{a \cosh (c+d x)}{b^2 d^2}-\frac{2 x \cosh (c+d x)}{b d^2}-\frac{a^3 \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (\frac{a d}{b}+d x\right )}{b^4}+\frac{2 \sinh (c+d x)}{b d^3}+\frac{a^2 \sinh (c+d x)}{b^3 d}-\frac{a x \sinh (c+d x)}{b^2 d}+\frac{x^2 \sinh (c+d x)}{b d}-\frac{a^3 \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (\frac{a d}{b}+d x\right )}{b^4}\\ \end{align*}
Mathematica [A] time = 0.430323, size = 118, normalized size = 0.79 \[ \frac{b \left (\left (a^2 d^2-a b d^2 x+b^2 \left (d^2 x^2+2\right )\right ) \sinh (c+d x)+b d (a-2 b x) \cosh (c+d x)\right )-a^3 d^3 \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (d \left (\frac{a}{b}+x\right )\right )-a^3 d^3 \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (d \left (\frac{a}{b}+x\right )\right )}{b^4 d^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.034, size = 292, normalized size = 2. \begin{align*} -{\frac{{{\rm e}^{-dx-c}}{x}^{2}}{2\,bd}}+{\frac{{{\rm e}^{-dx-c}}ax}{2\,d{b}^{2}}}-{\frac{{{\rm e}^{-dx-c}}{a}^{2}}{2\,d{b}^{3}}}-{\frac{{{\rm e}^{-dx-c}}x}{b{d}^{2}}}+{\frac{{{\rm e}^{-dx-c}}a}{2\,{d}^{2}{b}^{2}}}-{\frac{{{\rm e}^{-dx-c}}}{{d}^{3}b}}+{\frac{{a}^{3}}{2\,{b}^{4}}{{\rm e}^{{\frac{da-cb}{b}}}}{\it Ei} \left ( 1,dx+c+{\frac{da-cb}{b}} \right ) }-{\frac{{{\rm e}^{dx+c}}x}{b{d}^{2}}}+{\frac{{{\rm e}^{dx+c}}{x}^{2}}{2\,bd}}+{\frac{{{\rm e}^{dx+c}}}{{d}^{3}b}}+{\frac{{a}^{3}}{2\,{b}^{4}}{{\rm e}^{-{\frac{da-cb}{b}}}}{\it Ei} \left ( 1,-dx-c-{\frac{da-cb}{b}} \right ) }-{\frac{a{{\rm e}^{dx+c}}x}{2\,d{b}^{2}}}+{\frac{a{{\rm e}^{dx+c}}}{2\,{d}^{2}{b}^{2}}}+{\frac{{{\rm e}^{dx+c}}{a}^{2}}{2\,d{b}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.27553, size = 443, normalized size = 2.95 \begin{align*} \frac{1}{12} \, d{\left (\frac{6 \, a^{3}{\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 )}}{b^{3} d} - \frac{6 \, a^{2}{\left (\frac{{\left (d x e^{c} - e^{c}\right )} e^{\left (d x\right )}}{d^{2}} + \frac{{\left (d x + 1\right )} e^{\left (-d x - c\right )}}{d^{2}}\right )}}{b^{3}} + \frac{3 \, a{\left (\frac{{\left (d^{2} x^{2} e^{c} - 2 \, d x e^{c} + 2 \, e^{c}\right )} e^{\left (d x\right )}}{d^{3}} + \frac{{\left (d^{2} x^{2} + 2 \, d x + 2\right )} e^{\left (-d x - c\right )}}{d^{3}}\right )}}{b^{2}} - \frac{2 \,{\left (\frac{{\left (d^{3} x^{3} e^{c} - 3 \, d^{2} x^{2} e^{c} + 6 \, d x e^{c} - 6 \, e^{c}\right )} e^{\left (d x\right )}}{d^{4}} + \frac{{\left (d^{3} x^{3} + 3 \, d^{2} x^{2} + 6 \, d x + 6\right )} e^{\left (-d x - c\right )}}{d^{4}}\right )}}{b} + \frac{12 \, a^{3} \cosh \left (d x + c\right ) \log \left (b x + a\right )}{b^{4} d}\right )} - \frac{1}{6} \,{\left (\frac{6 \, a^{3} \log \left (b x + a\right )}{b^{4}} - \frac{2 \, b^{2} x^{3} - 3 \, a b x^{2} + 6 \, a^{2} x}{b^{3}}\right )} \cosh \left (d x + c\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.09382, size = 392, normalized size = 2.61 \begin{align*} -\frac{2 \,{\left (2 \, b^{3} d x - a b^{2} d\right )} \cosh \left (d x + c\right ) +{\left (a^{3} d^{3}{\rm Ei}\left (\frac{b d x + a d}{b}\right ) + a^{3} d^{3}{\rm Ei}\left (-\frac{b d x + a d}{b}\right )\right )} \cosh \left (-\frac{b c - a d}{b}\right ) - 2 \,{\left (b^{3} d^{2} x^{2} - a b^{2} d^{2} x + a^{2} b d^{2} + 2 \, b^{3}\right )} \sinh \left (d x + c\right ) -{\left (a^{3} d^{3}{\rm Ei}\left (\frac{b d x + a d}{b}\right ) - a^{3} d^{3}{\rm Ei}\left (-\frac{b d x + a d}{b}\right )\right )} \sinh \left (-\frac{b c - a d}{b}\right )}{2 \, b^{4} d^{3}} \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{x^{3} \cosh{\left (c + d x \right )}}{a + b x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1756, size = 84, normalized size = 0.56 \begin{align*} -\frac{a^{3}{\rm Ei}\left (\frac{b d x + a d}{b}\right ) e^{\left (c - \frac{a d}{b}\right )} + a^{3}{\rm Ei}\left (-\frac{b d x + a d}{b}\right ) e^{\left (-c + \frac{a d}{b}\right )}}{2 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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