Optimal. Leaf size=149 \[ \frac{1}{24} a d^4 \cosh (c) \text{Chi}(d x)+\frac{1}{24} a d^4 \sinh (c) \text{Shi}(d x)-\frac{a d^2 \cosh (c+d x)}{24 x^2}-\frac{a d^3 \sinh (c+d x)}{24 x}-\frac{a d \sinh (c+d x)}{12 x^3}-\frac{a \cosh (c+d x)}{4 x^4}+\frac{1}{2} b d^2 \cosh (c) \text{Chi}(d x)+\frac{1}{2} b d^2 \sinh (c) \text{Shi}(d x)-\frac{b \cosh (c+d x)}{2 x^2}-\frac{b d \sinh (c+d x)}{2 x} \]
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Rubi [A] time = 0.290435, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294, Rules used = {5287, 3297, 3303, 3298, 3301} \[ \frac{1}{24} a d^4 \cosh (c) \text{Chi}(d x)+\frac{1}{24} a d^4 \sinh (c) \text{Shi}(d x)-\frac{a d^2 \cosh (c+d x)}{24 x^2}-\frac{a d^3 \sinh (c+d x)}{24 x}-\frac{a d \sinh (c+d x)}{12 x^3}-\frac{a \cosh (c+d x)}{4 x^4}+\frac{1}{2} b d^2 \cosh (c) \text{Chi}(d x)+\frac{1}{2} b d^2 \sinh (c) \text{Shi}(d x)-\frac{b \cosh (c+d x)}{2 x^2}-\frac{b d \sinh (c+d x)}{2 x} \]
Antiderivative was successfully verified.
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Rule 5287
Rule 3297
Rule 3303
Rule 3298
Rule 3301
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right ) \cosh (c+d x)}{x^5} \, dx &=\int \left (\frac{a \cosh (c+d x)}{x^5}+\frac{b \cosh (c+d x)}{x^3}\right ) \, dx\\ &=a \int \frac{\cosh (c+d x)}{x^5} \, dx+b \int \frac{\cosh (c+d x)}{x^3} \, dx\\ &=-\frac{a \cosh (c+d x)}{4 x^4}-\frac{b \cosh (c+d x)}{2 x^2}+\frac{1}{4} (a d) \int \frac{\sinh (c+d x)}{x^4} \, dx+\frac{1}{2} (b d) \int \frac{\sinh (c+d x)}{x^2} \, dx\\ &=-\frac{a \cosh (c+d x)}{4 x^4}-\frac{b \cosh (c+d x)}{2 x^2}-\frac{a d \sinh (c+d x)}{12 x^3}-\frac{b d \sinh (c+d x)}{2 x}+\frac{1}{12} \left (a d^2\right ) \int \frac{\cosh (c+d x)}{x^3} \, dx+\frac{1}{2} \left (b d^2\right ) \int \frac{\cosh (c+d x)}{x} \, dx\\ &=-\frac{a \cosh (c+d x)}{4 x^4}-\frac{b \cosh (c+d x)}{2 x^2}-\frac{a d^2 \cosh (c+d x)}{24 x^2}-\frac{a d \sinh (c+d x)}{12 x^3}-\frac{b d \sinh (c+d x)}{2 x}+\frac{1}{24} \left (a d^3\right ) \int \frac{\sinh (c+d x)}{x^2} \, dx+\frac{1}{2} \left (b d^2 \cosh (c)\right ) \int \frac{\cosh (d x)}{x} \, dx+\frac{1}{2} \left (b d^2 \sinh (c)\right ) \int \frac{\sinh (d x)}{x} \, dx\\ &=-\frac{a \cosh (c+d x)}{4 x^4}-\frac{b \cosh (c+d x)}{2 x^2}-\frac{a d^2 \cosh (c+d x)}{24 x^2}+\frac{1}{2} b d^2 \cosh (c) \text{Chi}(d x)-\frac{a d \sinh (c+d x)}{12 x^3}-\frac{b d \sinh (c+d x)}{2 x}-\frac{a d^3 \sinh (c+d x)}{24 x}+\frac{1}{2} b d^2 \sinh (c) \text{Shi}(d x)+\frac{1}{24} \left (a d^4\right ) \int \frac{\cosh (c+d x)}{x} \, dx\\ &=-\frac{a \cosh (c+d x)}{4 x^4}-\frac{b \cosh (c+d x)}{2 x^2}-\frac{a d^2 \cosh (c+d x)}{24 x^2}+\frac{1}{2} b d^2 \cosh (c) \text{Chi}(d x)-\frac{a d \sinh (c+d x)}{12 x^3}-\frac{b d \sinh (c+d x)}{2 x}-\frac{a d^3 \sinh (c+d x)}{24 x}+\frac{1}{2} b d^2 \sinh (c) \text{Shi}(d x)+\frac{1}{24} \left (a d^4 \cosh (c)\right ) \int \frac{\cosh (d x)}{x} \, dx+\frac{1}{24} \left (a d^4 \sinh (c)\right ) \int \frac{\sinh (d x)}{x} \, dx\\ &=-\frac{a \cosh (c+d x)}{4 x^4}-\frac{b \cosh (c+d x)}{2 x^2}-\frac{a d^2 \cosh (c+d x)}{24 x^2}+\frac{1}{2} b d^2 \cosh (c) \text{Chi}(d x)+\frac{1}{24} a d^4 \cosh (c) \text{Chi}(d x)-\frac{a d \sinh (c+d x)}{12 x^3}-\frac{b d \sinh (c+d x)}{2 x}-\frac{a d^3 \sinh (c+d x)}{24 x}+\frac{1}{2} b d^2 \sinh (c) \text{Shi}(d x)+\frac{1}{24} a d^4 \sinh (c) \text{Shi}(d x)\\ \end{align*}
Mathematica [A] time = 0.274468, size = 127, normalized size = 0.85 \[ -\frac{-d^2 x^4 \cosh (c) \left (a d^2+12 b\right ) \text{Chi}(d x)-d^2 x^4 \sinh (c) \left (a d^2+12 b\right ) \text{Shi}(d x)+a d^3 x^3 \sinh (c+d x)+a d^2 x^2 \cosh (c+d x)+2 a d x \sinh (c+d x)+6 a \cosh (c+d x)+12 b d x^3 \sinh (c+d x)+12 b x^2 \cosh (c+d x)}{24 x^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.066, size = 238, normalized size = 1.6 \begin{align*}{\frac{a{d}^{3}{{\rm e}^{-dx-c}}}{48\,x}}-{\frac{a{d}^{2}{{\rm e}^{-dx-c}}}{48\,{x}^{2}}}+{\frac{da{{\rm e}^{-dx-c}}}{24\,{x}^{3}}}-{\frac{a{{\rm e}^{-dx-c}}}{8\,{x}^{4}}}+{\frac{bd{{\rm e}^{-dx-c}}}{4\,x}}-{\frac{b{{\rm e}^{-dx-c}}}{4\,{x}^{2}}}-{\frac{{d}^{2}b{{\rm e}^{-c}}{\it Ei} \left ( 1,dx \right ) }{4}}-{\frac{{d}^{4}a{{\rm e}^{-c}}{\it Ei} \left ( 1,dx \right ) }{48}}-{\frac{{d}^{4}a{{\rm e}^{c}}{\it Ei} \left ( 1,-dx \right ) }{48}}-{\frac{a{{\rm e}^{dx+c}}}{8\,{x}^{4}}}-{\frac{da{{\rm e}^{dx+c}}}{24\,{x}^{3}}}-{\frac{a{d}^{2}{{\rm e}^{dx+c}}}{48\,{x}^{2}}}-{\frac{a{d}^{3}{{\rm e}^{dx+c}}}{48\,x}}-{\frac{b{{\rm e}^{dx+c}}}{4\,{x}^{2}}}-{\frac{bd{{\rm e}^{dx+c}}}{4\,x}}-{\frac{{d}^{2}b{{\rm e}^{c}}{\it Ei} \left ( 1,-dx \right ) }{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.21251, size = 103, normalized size = 0.69 \begin{align*} \frac{1}{8} \,{\left (a d^{3} e^{\left (-c\right )} \Gamma \left (-3, d x\right ) + a d^{3} e^{c} \Gamma \left (-3, -d x\right ) + 2 \, b d e^{\left (-c\right )} \Gamma \left (-1, d x\right ) + 2 \, b d e^{c} \Gamma \left (-1, -d x\right )\right )} d - \frac{{\left (2 \, b x^{2} + a\right )} \cosh \left (d x + c\right )}{4 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.00653, size = 352, normalized size = 2.36 \begin{align*} -\frac{2 \,{\left ({\left (a d^{2} + 12 \, b\right )} x^{2} + 6 \, a\right )} \cosh \left (d x + c\right ) -{\left ({\left (a d^{4} + 12 \, b d^{2}\right )} x^{4}{\rm Ei}\left (d x\right ) +{\left (a d^{4} + 12 \, b d^{2}\right )} x^{4}{\rm Ei}\left (-d x\right )\right )} \cosh \left (c\right ) + 2 \,{\left ({\left (a d^{3} + 12 \, b d\right )} x^{3} + 2 \, a d x\right )} \sinh \left (d x + c\right ) -{\left ({\left (a d^{4} + 12 \, b d^{2}\right )} x^{4}{\rm Ei}\left (d x\right ) -{\left (a d^{4} + 12 \, b d^{2}\right )} x^{4}{\rm Ei}\left (-d x\right )\right )} \sinh \left (c\right )}{48 \, x^{4}} \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 [A] time = 1.19852, size = 320, normalized size = 2.15 \begin{align*} \frac{a d^{4} x^{4}{\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + a d^{4} x^{4}{\rm Ei}\left (d x\right ) e^{c} + 12 \, b d^{2} x^{4}{\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + 12 \, b d^{2} x^{4}{\rm Ei}\left (d x\right ) e^{c} - a d^{3} x^{3} e^{\left (d x + c\right )} + a d^{3} x^{3} e^{\left (-d x - c\right )} - a d^{2} x^{2} e^{\left (d x + c\right )} - 12 \, b d x^{3} e^{\left (d x + c\right )} - a d^{2} x^{2} e^{\left (-d x - c\right )} + 12 \, b d x^{3} e^{\left (-d x - c\right )} - 2 \, a d x e^{\left (d x + c\right )} - 12 \, b x^{2} e^{\left (d x + c\right )} + 2 \, a d x e^{\left (-d x - c\right )} - 12 \, b x^{2} e^{\left (-d x - c\right )} - 6 \, a e^{\left (d x + c\right )} - 6 \, a e^{\left (-d x - c\right )}}{48 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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