Optimal. Leaf size=114 \[ \frac{1}{2} a^2 d^2 \cosh (c) \text{Chi}(d x)+\frac{1}{2} a^2 d^2 \sinh (c) \text{Shi}(d x)-\frac{a^2 \cosh (c+d x)}{2 x^2}-\frac{a^2 d \sinh (c+d x)}{2 x}+2 a b \cosh (c) \text{Chi}(d x)+2 a b \sinh (c) \text{Shi}(d x)-\frac{b^2 \cosh (c+d x)}{d^2}+\frac{b^2 x \sinh (c+d x)}{d} \]
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Rubi [A] time = 0.225079, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368, Rules used = {5287, 3297, 3303, 3298, 3301, 3296, 2638} \[ \frac{1}{2} a^2 d^2 \cosh (c) \text{Chi}(d x)+\frac{1}{2} a^2 d^2 \sinh (c) \text{Shi}(d x)-\frac{a^2 \cosh (c+d x)}{2 x^2}-\frac{a^2 d \sinh (c+d x)}{2 x}+2 a b \cosh (c) \text{Chi}(d x)+2 a b \sinh (c) \text{Shi}(d x)-\frac{b^2 \cosh (c+d x)}{d^2}+\frac{b^2 x \sinh (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 5287
Rule 3297
Rule 3303
Rule 3298
Rule 3301
Rule 3296
Rule 2638
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^2 \cosh (c+d x)}{x^3} \, dx &=\int \left (\frac{a^2 \cosh (c+d x)}{x^3}+\frac{2 a b \cosh (c+d x)}{x}+b^2 x \cosh (c+d x)\right ) \, dx\\ &=a^2 \int \frac{\cosh (c+d x)}{x^3} \, dx+(2 a b) \int \frac{\cosh (c+d x)}{x} \, dx+b^2 \int x \cosh (c+d x) \, dx\\ &=-\frac{a^2 \cosh (c+d x)}{2 x^2}+\frac{b^2 x \sinh (c+d x)}{d}-\frac{b^2 \int \sinh (c+d x) \, dx}{d}+\frac{1}{2} \left (a^2 d\right ) \int \frac{\sinh (c+d x)}{x^2} \, dx+(2 a b \cosh (c)) \int \frac{\cosh (d x)}{x} \, dx+(2 a b \sinh (c)) \int \frac{\sinh (d x)}{x} \, dx\\ &=-\frac{b^2 \cosh (c+d x)}{d^2}-\frac{a^2 \cosh (c+d x)}{2 x^2}+2 a b \cosh (c) \text{Chi}(d x)-\frac{a^2 d \sinh (c+d x)}{2 x}+\frac{b^2 x \sinh (c+d x)}{d}+2 a b \sinh (c) \text{Shi}(d x)+\frac{1}{2} \left (a^2 d^2\right ) \int \frac{\cosh (c+d x)}{x} \, dx\\ &=-\frac{b^2 \cosh (c+d x)}{d^2}-\frac{a^2 \cosh (c+d x)}{2 x^2}+2 a b \cosh (c) \text{Chi}(d x)-\frac{a^2 d \sinh (c+d x)}{2 x}+\frac{b^2 x \sinh (c+d x)}{d}+2 a b \sinh (c) \text{Shi}(d x)+\frac{1}{2} \left (a^2 d^2 \cosh (c)\right ) \int \frac{\cosh (d x)}{x} \, dx+\frac{1}{2} \left (a^2 d^2 \sinh (c)\right ) \int \frac{\sinh (d x)}{x} \, dx\\ &=-\frac{b^2 \cosh (c+d x)}{d^2}-\frac{a^2 \cosh (c+d x)}{2 x^2}+2 a b \cosh (c) \text{Chi}(d x)+\frac{1}{2} a^2 d^2 \cosh (c) \text{Chi}(d x)-\frac{a^2 d \sinh (c+d x)}{2 x}+\frac{b^2 x \sinh (c+d x)}{d}+2 a b \sinh (c) \text{Shi}(d x)+\frac{1}{2} a^2 d^2 \sinh (c) \text{Shi}(d x)\\ \end{align*}
Mathematica [A] time = 0.348361, size = 97, normalized size = 0.85 \[ \frac{1}{2} \left (-\frac{a^2 \cosh (c+d x)}{x^2}-\frac{a^2 d \sinh (c+d x)}{x}+a \cosh (c) \left (a d^2+4 b\right ) \text{Chi}(d x)+a \sinh (c) \left (a d^2+4 b\right ) \text{Shi}(d x)-\frac{2 b^2 \cosh (c+d x)}{d^2}+\frac{2 b^2 x \sinh (c+d x)}{d}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.084, size = 188, normalized size = 1.7 \begin{align*} -{\frac{{d}^{2}{a}^{2}{{\rm e}^{-c}}{\it Ei} \left ( 1,dx \right ) }{4}}+{\frac{d{a}^{2}{{\rm e}^{-dx-c}}}{4\,x}}-{\frac{{a}^{2}{{\rm e}^{-dx-c}}}{4\,{x}^{2}}}-{\frac{{b}^{2}{{\rm e}^{-dx-c}}}{2\,{d}^{2}}}-ab{{\rm e}^{-c}}{\it Ei} \left ( 1,dx \right ) -{\frac{{b}^{2}{{\rm e}^{-dx-c}}x}{2\,d}}-{\frac{{{\rm e}^{dx+c}}{b}^{2}}{2\,{d}^{2}}}-{\frac{{d}^{2}{a}^{2}{{\rm e}^{c}}{\it Ei} \left ( 1,-dx \right ) }{4}}+{\frac{{{\rm e}^{dx+c}}{b}^{2}x}{2\,d}}-{\frac{{{\rm e}^{dx+c}}{a}^{2}}{4\,{x}^{2}}}-{\frac{d{a}^{2}{{\rm e}^{dx+c}}}{4\,x}}-ab{{\rm e}^{c}}{\it Ei} \left ( 1,-dx \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.3475, size = 223, normalized size = 1.96 \begin{align*} \frac{1}{4} \,{\left ({\left (d e^{\left (-c\right )} \Gamma \left (-1, d x\right ) + d e^{c} \Gamma \left (-1, -d x\right )\right )} a^{2} - b^{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 )} - \frac{4 \, a b \cosh \left (d x + c\right ) \log \left (x^{2}\right )}{d} + \frac{4 \,{\left ({\rm Ei}\left (-d x\right ) e^{\left (-c\right )} +{\rm Ei}\left (d x\right ) e^{c}\right )} a b}{d}\right )} d + \frac{1}{2} \,{\left (b^{2} x^{2} + 2 \, a b \log \left (x^{2}\right ) - \frac{a^{2}}{x^{2}}\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 = 1.99347, size = 359, normalized size = 3.15 \begin{align*} -\frac{2 \,{\left (a^{2} d^{2} + 2 \, b^{2} x^{2}\right )} \cosh \left (d x + c\right ) -{\left ({\left (a^{2} d^{4} + 4 \, a b d^{2}\right )} x^{2}{\rm Ei}\left (d x\right ) +{\left (a^{2} d^{4} + 4 \, a b d^{2}\right )} x^{2}{\rm Ei}\left (-d x\right )\right )} \cosh \left (c\right ) + 2 \,{\left (a^{2} d^{3} x - 2 \, b^{2} d x^{3}\right )} \sinh \left (d x + c\right ) -{\left ({\left (a^{2} d^{4} + 4 \, a b d^{2}\right )} x^{2}{\rm Ei}\left (d x\right ) -{\left (a^{2} d^{4} + 4 \, a b d^{2}\right )} x^{2}{\rm Ei}\left (-d x\right )\right )} \sinh \left (c\right )}{4 \, d^{2} x^{2}} \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{\left (a + b x^{2}\right )^{2} \cosh{\left (c + d x \right )}}{x^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21957, size = 278, normalized size = 2.44 \begin{align*} \frac{a^{2} d^{4} x^{2}{\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + a^{2} d^{4} x^{2}{\rm Ei}\left (d x\right ) e^{c} + 4 \, a b d^{2} x^{2}{\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + 4 \, a b d^{2} x^{2}{\rm Ei}\left (d x\right ) e^{c} - a^{2} d^{3} x e^{\left (d x + c\right )} + 2 \, b^{2} d x^{3} e^{\left (d x + c\right )} + a^{2} d^{3} x e^{\left (-d x - c\right )} - 2 \, b^{2} d x^{3} e^{\left (-d x - c\right )} - a^{2} d^{2} e^{\left (d x + c\right )} - 2 \, b^{2} x^{2} e^{\left (d x + c\right )} - a^{2} d^{2} e^{\left (-d x - c\right )} - 2 \, b^{2} x^{2} e^{\left (-d x - c\right )}}{4 \, d^{2} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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