Optimal. Leaf size=231 \[ \frac{a^4 d \sinh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (x d+\frac{a d}{b}\right )}{b^6}-\frac{4 a^3 \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (x d+\frac{a d}{b}\right )}{b^5}-\frac{4 a^3 \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (x d+\frac{a d}{b}\right )}{b^5}+\frac{a^4 d \cosh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (x d+\frac{a d}{b}\right )}{b^6}+\frac{3 a^2 \sinh (c+d x)}{b^4 d}-\frac{a^4 \cosh (c+d x)}{b^5 (a+b x)}+\frac{2 a \cosh (c+d x)}{b^3 d^2}-\frac{2 a x \sinh (c+d x)}{b^3 d}+\frac{2 \sinh (c+d x)}{b^2 d^3}-\frac{2 x \cosh (c+d x)}{b^2 d^2}+\frac{x^2 \sinh (c+d x)}{b^2 d} \]
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Rubi [A] time = 0.536144, antiderivative size = 231, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 8, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.471, Rules used = {6742, 2637, 3296, 2638, 3297, 3303, 3298, 3301} \[ \frac{a^4 d \sinh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (x d+\frac{a d}{b}\right )}{b^6}-\frac{4 a^3 \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (x d+\frac{a d}{b}\right )}{b^5}-\frac{4 a^3 \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (x d+\frac{a d}{b}\right )}{b^5}+\frac{a^4 d \cosh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (x d+\frac{a d}{b}\right )}{b^6}+\frac{3 a^2 \sinh (c+d x)}{b^4 d}-\frac{a^4 \cosh (c+d x)}{b^5 (a+b x)}+\frac{2 a \cosh (c+d x)}{b^3 d^2}-\frac{2 a x \sinh (c+d x)}{b^3 d}+\frac{2 \sinh (c+d x)}{b^2 d^3}-\frac{2 x \cosh (c+d x)}{b^2 d^2}+\frac{x^2 \sinh (c+d x)}{b^2 d} \]
Antiderivative was successfully verified.
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Rule 6742
Rule 2637
Rule 3296
Rule 2638
Rule 3297
Rule 3303
Rule 3298
Rule 3301
Rubi steps
\begin{align*} \int \frac{x^4 \cosh (c+d x)}{(a+b x)^2} \, dx &=\int \left (\frac{3 a^2 \cosh (c+d x)}{b^4}-\frac{2 a x \cosh (c+d x)}{b^3}+\frac{x^2 \cosh (c+d x)}{b^2}+\frac{a^4 \cosh (c+d x)}{b^4 (a+b x)^2}-\frac{4 a^3 \cosh (c+d x)}{b^4 (a+b x)}\right ) \, dx\\ &=\frac{\left (3 a^2\right ) \int \cosh (c+d x) \, dx}{b^4}-\frac{\left (4 a^3\right ) \int \frac{\cosh (c+d x)}{a+b x} \, dx}{b^4}+\frac{a^4 \int \frac{\cosh (c+d x)}{(a+b x)^2} \, dx}{b^4}-\frac{(2 a) \int x \cosh (c+d x) \, dx}{b^3}+\frac{\int x^2 \cosh (c+d x) \, dx}{b^2}\\ &=-\frac{a^4 \cosh (c+d x)}{b^5 (a+b x)}+\frac{3 a^2 \sinh (c+d x)}{b^4 d}-\frac{2 a x \sinh (c+d x)}{b^3 d}+\frac{x^2 \sinh (c+d x)}{b^2 d}+\frac{(2 a) \int \sinh (c+d x) \, dx}{b^3 d}-\frac{2 \int x \sinh (c+d x) \, dx}{b^2 d}+\frac{\left (a^4 d\right ) \int \frac{\sinh (c+d x)}{a+b x} \, dx}{b^5}-\frac{\left (4 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^4}-\frac{\left (4 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^4}\\ &=\frac{2 a \cosh (c+d x)}{b^3 d^2}-\frac{2 x \cosh (c+d x)}{b^2 d^2}-\frac{a^4 \cosh (c+d x)}{b^5 (a+b x)}-\frac{4 a^3 \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (\frac{a d}{b}+d x\right )}{b^5}+\frac{3 a^2 \sinh (c+d x)}{b^4 d}-\frac{2 a x \sinh (c+d x)}{b^3 d}+\frac{x^2 \sinh (c+d x)}{b^2 d}-\frac{4 a^3 \sinh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (\frac{a d}{b}+d x\right )}{b^5}+\frac{2 \int \cosh (c+d x) \, dx}{b^2 d^2}+\frac{\left (a^4 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^5}+\frac{\left (a^4 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^5}\\ &=\frac{2 a \cosh (c+d x)}{b^3 d^2}-\frac{2 x \cosh (c+d x)}{b^2 d^2}-\frac{a^4 \cosh (c+d x)}{b^5 (a+b x)}-\frac{4 a^3 \cosh \left (c-\frac{a d}{b}\right ) \text{Chi}\left (\frac{a d}{b}+d x\right )}{b^5}+\frac{a^4 d \text{Chi}\left (\frac{a d}{b}+d x\right ) \sinh \left (c-\frac{a d}{b}\right )}{b^6}+\frac{2 \sinh (c+d x)}{b^2 d^3}+\frac{3 a^2 \sinh (c+d x)}{b^4 d}-\frac{2 a x \sinh (c+d x)}{b^3 d}+\frac{x^2 \sinh (c+d x)}{b^2 d}+\frac{a^4 d \cosh \left (c-\frac{a d}{b}\right ) \text{Shi}\left (\frac{a d}{b}+d x\right )}{b^6}-\frac{4 a^3 \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 = 1.20078, size = 173, normalized size = 0.75 \[ \frac{\frac{b^2 \left (3 a^2 d^2-2 a b d^2 x+b^2 \left (d^2 x^2+2\right )\right ) \sinh (c+d x)}{d^3}-\frac{b \left (-2 a^2 b^2+a^4 d^2+2 b^4 x^2\right ) \cosh (c+d x)}{d^2 (a+b x)}+a^3 \text{Chi}\left (d \left (\frac{a}{b}+x\right )\right ) \left (a d \sinh \left (c-\frac{a d}{b}\right )-4 b \cosh \left (c-\frac{a d}{b}\right )\right )+a^3 \text{Shi}\left (d \left (\frac{a}{b}+x\right )\right ) \left (a d \cosh \left (c-\frac{a d}{b}\right )-4 b \sinh \left (c-\frac{a d}{b}\right )\right )}{b^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.122, size = 431, normalized size = 1.9 \begin{align*} -{\frac{{{\rm e}^{-dx-c}}}{{d}^{3}{b}^{2}}}+{\frac{d{a}^{4}}{2\,{b}^{6}}{{\rm e}^{{\frac{da-cb}{b}}}}{\it Ei} \left ( 1,dx+c+{\frac{da-cb}{b}} \right ) }+2\,{\frac{{a}^{3}}{{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}{{b}^{2}{d}^{2}}}+{\frac{{{\rm e}^{-dx-c}}a}{{d}^{2}{b}^{3}}}-{\frac{{{\rm e}^{-dx-c}}{x}^{2}}{2\,d{b}^{2}}}-{\frac{3\,{{\rm e}^{-dx-c}}{a}^{2}}{2\,d{b}^{4}}}-{\frac{d{{\rm e}^{-dx-c}}{a}^{4}}{2\,{b}^{5} \left ( bdx+da \right ) }}+{\frac{{{\rm e}^{-dx-c}}ax}{d{b}^{3}}}-{\frac{a{{\rm e}^{dx+c}}x}{d{b}^{3}}}+{\frac{3\,{{\rm e}^{dx+c}}{a}^{2}}{2\,d{b}^{4}}}+{\frac{a{{\rm e}^{dx+c}}}{{d}^{2}{b}^{3}}}+{\frac{{{\rm e}^{dx+c}}}{{d}^{3}{b}^{2}}}-{\frac{d{{\rm e}^{dx+c}}{a}^{4}}{2\,{b}^{6}} \left ({\frac{da}{b}}+dx \right ) ^{-1}}-{\frac{d{a}^{4}}{2\,{b}^{6}}{{\rm e}^{-{\frac{da-cb}{b}}}}{\it Ei} \left ( 1,-dx-c-{\frac{da-cb}{b}} \right ) }+2\,{\frac{{a}^{3}}{{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}^{2}}{2\,d{b}^{2}}}-{\frac{{{\rm e}^{dx+c}}x}{{b}^{2}{d}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.51568, size = 548, normalized size = 2.37 \begin{align*} \frac{1}{6} \,{\left (3 \, 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^{6}} - \frac{e^{\left (c - \frac{a d}{b}\right )} E_{1}\left (-\frac{{\left (b x + a\right )} d}{b}\right )}{b^{6}}\right )} + \frac{12 \, 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^{4} d} - \frac{9 \, 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^{4}} + \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^{3}} - \frac{\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}}}{b^{2}} + \frac{24 \, a^{3} \cosh \left (d x + c\right ) \log \left (b x + a\right )}{b^{5} d}\right )} d - \frac{1}{3} \,{\left (\frac{3 \, a^{4}}{b^{6} x + a b^{5}} + \frac{12 \, a^{3} \log \left (b x + a\right )}{b^{5}} - \frac{b^{2} x^{3} - 3 \, a b x^{2} + 9 \, a^{2} 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 = 2.07989, size = 740, normalized size = 3.2 \begin{align*} -\frac{2 \,{\left (a^{4} b d^{3} + 2 \, b^{5} d x^{2} - 2 \, a^{2} b^{3} d\right )} \cosh \left (d x + c\right ) -{\left ({\left (a^{5} d^{4} - 4 \, a^{4} b d^{3} +{\left (a^{4} b d^{4} - 4 \, a^{3} b^{2} d^{3}\right )} x\right )}{\rm Ei}\left (\frac{b d x + a d}{b}\right ) -{\left (a^{5} d^{4} + 4 \, a^{4} b d^{3} +{\left (a^{4} b d^{4} + 4 \, a^{3} b^{2} d^{3}\right )} x\right )}{\rm Ei}\left (-\frac{b d x + a d}{b}\right )\right )} \cosh \left (-\frac{b c - a d}{b}\right ) - 2 \,{\left (b^{5} d^{2} x^{3} - a b^{4} d^{2} x^{2} + 3 \, a^{3} b^{2} d^{2} + 2 \, a b^{4} +{\left (a^{2} b^{3} d^{2} + 2 \, b^{5}\right )} x\right )} \sinh \left (d x + c\right ) +{\left ({\left (a^{5} d^{4} - 4 \, a^{4} b d^{3} +{\left (a^{4} b d^{4} - 4 \, a^{3} b^{2} d^{3}\right )} x\right )}{\rm Ei}\left (\frac{b d x + a d}{b}\right ) +{\left (a^{5} d^{4} + 4 \, a^{4} b d^{3} +{\left (a^{4} b d^{4} + 4 \, a^{3} b^{2} d^{3}\right )} x\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^{7} d^{3} x + a b^{6} d^{3}\right )}} \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 )}}{\left (a + b x\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18744, size = 393, normalized size = 1.7 \begin{align*} \frac{a^{4} b d x{\rm Ei}\left (\frac{b d x + a d}{b}\right ) e^{\left (c - \frac{a d}{b}\right )} - a^{4} b d x{\rm Ei}\left (-\frac{b d x + a d}{b}\right ) e^{\left (-c + \frac{a d}{b}\right )} + a^{5} d{\rm Ei}\left (\frac{b d x + a d}{b}\right ) e^{\left (c - \frac{a d}{b}\right )} - 4 \, a^{3} b^{2} x{\rm Ei}\left (\frac{b d x + a d}{b}\right ) e^{\left (c - \frac{a d}{b}\right )} - a^{5} d{\rm Ei}\left (-\frac{b d x + a d}{b}\right ) e^{\left (-c + \frac{a d}{b}\right )} - 4 \, a^{3} b^{2} x{\rm Ei}\left (-\frac{b d x + a d}{b}\right ) e^{\left (-c + \frac{a d}{b}\right )} - 4 \, a^{4} b{\rm Ei}\left (\frac{b d x + a d}{b}\right ) e^{\left (c - \frac{a d}{b}\right )} - 4 \, a^{4} b{\rm Ei}\left (-\frac{b d x + a d}{b}\right ) e^{\left (-c + \frac{a d}{b}\right )} - a^{4} b e^{\left (d x + c\right )} - a^{4} b e^{\left (-d x - c\right )}}{2 \,{\left (b^{7} x + a b^{6}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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