Optimal. Leaf size=219 \[ \frac{a^4 \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (x d+\frac{a d}{b}\right )}{b^5}-\frac{a^2 \cosh (c+d x)}{b^3 d^2}+\frac{a^4 \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (x d+\frac{a d}{b}\right )}{b^5}-\frac{a^3 \sinh (c+d x)}{b^4 d}+\frac{a^2 x \sinh (c+d x)}{b^3 d}-\frac{2 a \sinh (c+d x)}{b^2 d^3}+\frac{2 a x \cosh (c+d x)}{b^2 d^2}-\frac{a x^2 \sinh (c+d x)}{b^2 d}-\frac{3 x^2 \cosh (c+d x)}{b d^2}+\frac{6 x \sinh (c+d x)}{b d^3}-\frac{6 \cosh (c+d x)}{b d^4}+\frac{x^3 \sinh (c+d x)}{b d} \]
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Rubi [A] time = 0.484027, antiderivative size = 219, normalized size of antiderivative = 1., number of steps used = 15, 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} \[ \frac{a^4 \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (x d+\frac{a d}{b}\right )}{b^5}-\frac{a^2 \cosh (c+d x)}{b^3 d^2}+\frac{a^4 \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (x d+\frac{a d}{b}\right )}{b^5}-\frac{a^3 \sinh (c+d x)}{b^4 d}+\frac{a^2 x \sinh (c+d x)}{b^3 d}-\frac{2 a \sinh (c+d x)}{b^2 d^3}+\frac{2 a x \cosh (c+d x)}{b^2 d^2}-\frac{a x^2 \sinh (c+d x)}{b^2 d}-\frac{3 x^2 \cosh (c+d x)}{b d^2}+\frac{6 x \sinh (c+d x)}{b d^3}-\frac{6 \cosh (c+d x)}{b d^4}+\frac{x^3 \sinh (c+d x)}{b d} \]
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^4 \cosh (c+d x)}{a+b x} \, dx &=\int \left (-\frac{a^3 \cosh (c+d x)}{b^4}+\frac{a^2 x \cosh (c+d x)}{b^3}-\frac{a x^2 \cosh (c+d x)}{b^2}+\frac{x^3 \cosh (c+d x)}{b}+\frac{a^4 \cosh (c+d x)}{b^4 (a+b x)}\right ) \, dx\\ &=-\frac{a^3 \int \cosh (c+d x) \, dx}{b^4}+\frac{a^4 \int \frac{\cosh (c+d x)}{a+b x} \, dx}{b^4}+\frac{a^2 \int x \cosh (c+d x) \, dx}{b^3}-\frac{a \int x^2 \cosh (c+d x) \, dx}{b^2}+\frac{\int x^3 \cosh (c+d x) \, dx}{b}\\ &=-\frac{a^3 \sinh (c+d x)}{b^4 d}+\frac{a^2 x \sinh (c+d x)}{b^3 d}-\frac{a x^2 \sinh (c+d x)}{b^2 d}+\frac{x^3 \sinh (c+d x)}{b d}-\frac{a^2 \int \sinh (c+d x) \, dx}{b^3 d}+\frac{(2 a) \int x \sinh (c+d x) \, dx}{b^2 d}-\frac{3 \int x^2 \sinh (c+d x) \, dx}{b d}+\frac{\left (a^4 \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^4}+\frac{\left (a^4 \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^4}\\ &=-\frac{a^2 \cosh (c+d x)}{b^3 d^2}+\frac{2 a x \cosh (c+d x)}{b^2 d^2}-\frac{3 x^2 \cosh (c+d x)}{b d^2}+\frac{a^4 \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (\frac{a d}{b}+d x\right )}{b^5}-\frac{a^3 \sinh (c+d x)}{b^4 d}+\frac{a^2 x \sinh (c+d x)}{b^3 d}-\frac{a x^2 \sinh (c+d x)}{b^2 d}+\frac{x^3 \sinh (c+d x)}{b d}+\frac{a^4 \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (\frac{a d}{b}+d x\right )}{b^5}-\frac{(2 a) \int \cosh (c+d x) \, dx}{b^2 d^2}+\frac{6 \int x \cosh (c+d x) \, dx}{b d^2}\\ &=-\frac{a^2 \cosh (c+d x)}{b^3 d^2}+\frac{2 a x \cosh (c+d x)}{b^2 d^2}-\frac{3 x^2 \cosh (c+d x)}{b d^2}+\frac{a^4 \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (\frac{a d}{b}+d x\right )}{b^5}-\frac{2 a \sinh (c+d x)}{b^2 d^3}-\frac{a^3 \sinh (c+d x)}{b^4 d}+\frac{6 x \sinh (c+d x)}{b d^3}+\frac{a^2 x \sinh (c+d x)}{b^3 d}-\frac{a x^2 \sinh (c+d x)}{b^2 d}+\frac{x^3 \sinh (c+d x)}{b d}+\frac{a^4 \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (\frac{a d}{b}+d x\right )}{b^5}-\frac{6 \int \sinh (c+d x) \, dx}{b d^3}\\ &=-\frac{6 \cosh (c+d x)}{b d^4}-\frac{a^2 \cosh (c+d x)}{b^3 d^2}+\frac{2 a x \cosh (c+d x)}{b^2 d^2}-\frac{3 x^2 \cosh (c+d x)}{b d^2}+\frac{a^4 \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (\frac{a d}{b}+d x\right )}{b^5}-\frac{2 a \sinh (c+d x)}{b^2 d^3}-\frac{a^3 \sinh (c+d x)}{b^4 d}+\frac{6 x \sinh (c+d x)}{b d^3}+\frac{a^2 x \sinh (c+d x)}{b^3 d}-\frac{a x^2 \sinh (c+d x)}{b^2 d}+\frac{x^3 \sinh (c+d x)}{b d}+\frac{a^4 \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (\frac{a d}{b}+d x\right )}{b^5}\\ \end{align*}
Mathematica [A] time = 0.631353, size = 159, normalized size = 0.73 \[ \frac{-b \left (d \left (-a^2 b d^2 x+a^3 d^2+a b^2 \left (d^2 x^2+2\right )-b^3 x \left (d^2 x^2+6\right )\right ) \sinh (c+d x)+b \left (a^2 d^2-2 a b d^2 x+3 b^2 \left (d^2 x^2+2\right )\right ) \cosh (c+d x)\right )+a^4 d^4 \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (d \left (\frac{a}{b}+x\right )\right )+a^4 d^4 \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (d \left (\frac{a}{b}+x\right )\right )}{b^5 d^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.133, size = 442, normalized size = 2. \begin{align*} -{\frac{{a}^{4}}{2\,{b}^{5}}{{\rm e}^{{\frac{da-cb}{b}}}}{\it Ei} \left ( 1,dx+c+{\frac{da-cb}{b}} \right ) }-{\frac{3\,{{\rm e}^{-dx-c}}{x}^{2}}{2\,b{d}^{2}}}-{\frac{{{\rm e}^{-dx-c}}{a}^{2}}{2\,{d}^{2}{b}^{3}}}-3\,{\frac{{{\rm e}^{-dx-c}}}{b{d}^{4}}}+{\frac{{{\rm e}^{-dx-c}}ax}{{d}^{2}{b}^{2}}}-{\frac{{{\rm e}^{-dx-c}}{x}^{3}}{2\,bd}}+{\frac{{{\rm e}^{-dx-c}}{a}^{3}}{2\,d{b}^{4}}}-3\,{\frac{{{\rm e}^{-dx-c}}x}{{d}^{3}b}}+{\frac{{{\rm e}^{-dx-c}}a}{{d}^{3}{b}^{2}}}+{\frac{{{\rm e}^{-dx-c}}a{x}^{2}}{2\,{b}^{2}d}}-{\frac{{{\rm e}^{-dx-c}}{a}^{2}x}{2\,{b}^{3}d}}-3\,{\frac{{{\rm e}^{dx+c}}}{b{d}^{4}}}-{\frac{a{{\rm e}^{dx+c}}{x}^{2}}{2\,{b}^{2}d}}+{\frac{a{{\rm e}^{dx+c}}x}{{d}^{2}{b}^{2}}}-{\frac{{a}^{4}}{2\,{b}^{5}}{{\rm e}^{-{\frac{da-cb}{b}}}}{\it Ei} \left ( 1,-dx-c-{\frac{da-cb}{b}} \right ) }+{\frac{{{\rm e}^{dx+c}}{x}^{3}}{2\,bd}}-{\frac{3\,{{\rm e}^{dx+c}}{x}^{2}}{2\,b{d}^{2}}}+3\,{\frac{{{\rm e}^{dx+c}}x}{{d}^{3}b}}-{\frac{{{\rm e}^{dx+c}}{a}^{3}}{2\,d{b}^{4}}}-{\frac{a{{\rm e}^{dx+c}}}{{d}^{3}{b}^{2}}}-{\frac{{{\rm e}^{dx+c}}{a}^{2}}{2\,{d}^{2}{b}^{3}}}+{\frac{{{\rm e}^{dx+c}}{a}^{2}x}{2\,{b}^{3}d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.44775, size = 590, normalized size = 2.69 \begin{align*} -\frac{1}{24} \, d{\left (\frac{12 \, a^{4}{\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^{4} d} - \frac{12 \, a^{3}{\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^{4}} + \frac{6 \, a^{2}{\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^{3}} - \frac{4 \, a{\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^{2}} + \frac{3 \,{\left (\frac{{\left (d^{4} x^{4} e^{c} - 4 \, d^{3} x^{3} e^{c} + 12 \, d^{2} x^{2} e^{c} - 24 \, d x e^{c} + 24 \, e^{c}\right )} e^{\left (d x\right )}}{d^{5}} + \frac{{\left (d^{4} x^{4} + 4 \, d^{3} x^{3} + 12 \, d^{2} x^{2} + 24 \, d x + 24\right )} e^{\left (-d x - c\right )}}{d^{5}}\right )}}{b} + \frac{24 \, a^{4} \cosh \left (d x + c\right ) \log \left (b x + a\right )}{b^{5} d}\right )} + \frac{1}{12} \,{\left (\frac{12 \, a^{4} \log \left (b x + a\right )}{b^{5}} + \frac{3 \, b^{3} x^{4} - 4 \, a b^{2} x^{3} + 6 \, a^{2} b x^{2} - 12 \, a^{3} x}{b^{4}}\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 = 2.09321, size = 481, normalized size = 2.2 \begin{align*} -\frac{2 \,{\left (3 \, b^{4} d^{2} x^{2} - 2 \, a b^{3} d^{2} x + a^{2} b^{2} d^{2} + 6 \, b^{4}\right )} \cosh \left (d x + c\right ) -{\left (a^{4} d^{4}{\rm Ei}\left (\frac{b d x + a d}{b}\right ) + a^{4} d^{4}{\rm Ei}\left (-\frac{b d x + a d}{b}\right )\right )} \cosh \left (-\frac{b c - a d}{b}\right ) - 2 \,{\left (b^{4} d^{3} x^{3} - a b^{3} d^{3} x^{2} - a^{3} b d^{3} - 2 \, a b^{3} d +{\left (a^{2} b^{2} d^{3} + 6 \, b^{4} d\right )} x\right )} \sinh \left (d x + c\right ) +{\left (a^{4} d^{4}{\rm Ei}\left (\frac{b d x + a d}{b}\right ) - a^{4} d^{4}{\rm Ei}\left (-\frac{b d x + a d}{b}\right )\right )} \sinh \left (-\frac{b c - a d}{b}\right )}{2 \, b^{5} d^{4}} \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^{4} \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 = 1.14898, size = 84, normalized size = 0.38 \begin{align*} \frac{a^{4}{\rm Ei}\left (\frac{b d x + a d}{b}\right ) e^{\left (c - \frac{a d}{b}\right )} + a^{4}{\rm Ei}\left (-\frac{b d x + a d}{b}\right ) e^{\left (-c + \frac{a d}{b}\right )}}{2 \, b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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