Optimal. Leaf size=70 \[ a^2 d \sinh (c) \text{Chi}(d x)+a^2 d \cosh (c) \text{Shi}(d x)-\frac{a^2 \cosh (c+d x)}{x}+2 a b \cosh (c) \text{Chi}(d x)+2 a b \sinh (c) \text{Shi}(d x)+\frac{b^2 \sinh (c+d x)}{d} \]
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Rubi [A] time = 0.246734, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.353, Rules used = {6742, 2637, 3297, 3303, 3298, 3301} \[ a^2 d \sinh (c) \text{Chi}(d x)+a^2 d \cosh (c) \text{Shi}(d x)-\frac{a^2 \cosh (c+d x)}{x}+2 a b \cosh (c) \text{Chi}(d x)+2 a b \sinh (c) \text{Shi}(d x)+\frac{b^2 \sinh (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 6742
Rule 2637
Rule 3297
Rule 3303
Rule 3298
Rule 3301
Rubi steps
\begin{align*} \int \frac{(a+b x)^2 \cosh (c+d x)}{x^2} \, dx &=\int \left (b^2 \cosh (c+d x)+\frac{a^2 \cosh (c+d x)}{x^2}+\frac{2 a b \cosh (c+d x)}{x}\right ) \, dx\\ &=a^2 \int \frac{\cosh (c+d x)}{x^2} \, dx+(2 a b) \int \frac{\cosh (c+d x)}{x} \, dx+b^2 \int \cosh (c+d x) \, dx\\ &=-\frac{a^2 \cosh (c+d x)}{x}+\frac{b^2 \sinh (c+d x)}{d}+\left (a^2 d\right ) \int \frac{\sinh (c+d x)}{x} \, 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{a^2 \cosh (c+d x)}{x}+2 a b \cosh (c) \text{Chi}(d x)+\frac{b^2 \sinh (c+d x)}{d}+2 a b \sinh (c) \text{Shi}(d x)+\left (a^2 d \cosh (c)\right ) \int \frac{\sinh (d x)}{x} \, dx+\left (a^2 d \sinh (c)\right ) \int \frac{\cosh (d x)}{x} \, dx\\ &=-\frac{a^2 \cosh (c+d x)}{x}+2 a b \cosh (c) \text{Chi}(d x)+a^2 d \text{Chi}(d x) \sinh (c)+\frac{b^2 \sinh (c+d x)}{d}+a^2 d \cosh (c) \text{Shi}(d x)+2 a b \sinh (c) \text{Shi}(d x)\\ \end{align*}
Mathematica [A] time = 0.229813, size = 62, normalized size = 0.89 \[ -\frac{a^2 \cosh (c+d x)}{x}+a \text{Chi}(d x) (a d \sinh (c)+2 b \cosh (c))+a \text{Shi}(d x) (a d \cosh (c)+2 b \sinh (c))+\frac{b^2 \sinh (c+d x)}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 118, normalized size = 1.7 \begin{align*} -{\frac{{a}^{2}{{\rm e}^{-dx-c}}}{2\,x}}+{\frac{d{a}^{2}{{\rm e}^{-c}}{\it Ei} \left ( 1,dx \right ) }{2}}-{\frac{{b}^{2}{{\rm e}^{-dx-c}}}{2\,d}}-ab{{\rm e}^{-c}}{\it Ei} \left ( 1,dx \right ) -{\frac{{{\rm e}^{dx+c}}{a}^{2}}{2\,x}}-{\frac{d{a}^{2}{{\rm e}^{c}}{\it Ei} \left ( 1,-dx \right ) }{2}}+{\frac{{{\rm e}^{dx+c}}{b}^{2}}{2\,d}}-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 = 1.39303, size = 184, normalized size = 2.63 \begin{align*} -\frac{1}{2} \,{\left ({\left ({\rm Ei}\left (-d x\right ) e^{\left (-c\right )} -{\rm Ei}\left (d x\right ) e^{c}\right )} a^{2} + b^{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 )} + \frac{4 \, a b \cosh \left (d x + c\right ) \log \left (x\right )}{d} - \frac{2 \,{\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 +{\left (b^{2} x + 2 \, a b \log \left (x\right ) - \frac{a^{2}}{x}\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 = 1.99353, size = 284, normalized size = 4.06 \begin{align*} -\frac{2 \, a^{2} d \cosh \left (d x + c\right ) - 2 \, b^{2} x \sinh \left (d x + c\right ) -{\left ({\left (a^{2} d^{2} + 2 \, a b d\right )} x{\rm Ei}\left (d x\right ) -{\left (a^{2} d^{2} - 2 \, a b d\right )} x{\rm Ei}\left (-d x\right )\right )} \cosh \left (c\right ) -{\left ({\left (a^{2} d^{2} + 2 \, a b d\right )} x{\rm Ei}\left (d x\right ) +{\left (a^{2} d^{2} - 2 \, a b d\right )} x{\rm Ei}\left (-d x\right )\right )} \sinh \left (c\right )}{2 \, d x} \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\right )^{2} \cosh{\left (c + d x \right )}}{x^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.23853, size = 161, normalized size = 2.3 \begin{align*} -\frac{a^{2} d^{2} x{\rm Ei}\left (-d x\right ) e^{\left (-c\right )} - a^{2} d^{2} x{\rm Ei}\left (d x\right ) e^{c} - 2 \, a b d x{\rm Ei}\left (-d x\right ) e^{\left (-c\right )} - 2 \, a b d x{\rm Ei}\left (d x\right ) e^{c} + a^{2} d e^{\left (d x + c\right )} - b^{2} x e^{\left (d x + c\right )} + a^{2} d e^{\left (-d x - c\right )} + b^{2} x e^{\left (-d x - c\right )}}{2 \, d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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