Optimal. Leaf size=42 \[ a d \sinh (c) \text{Chi}(d x)+a d \cosh (c) \text{Shi}(d x)-\frac{a \cosh (c+d x)}{x}+\frac{b \sinh (c+d x)}{d} \]
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Rubi [A] time = 0.106847, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.353, Rules used = {5287, 2637, 3297, 3303, 3298, 3301} \[ a d \sinh (c) \text{Chi}(d x)+a d \cosh (c) \text{Shi}(d x)-\frac{a \cosh (c+d x)}{x}+\frac{b \sinh (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 5287
Rule 2637
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^2} \, dx &=\int \left (b \cosh (c+d x)+\frac{a \cosh (c+d x)}{x^2}\right ) \, dx\\ &=a \int \frac{\cosh (c+d x)}{x^2} \, dx+b \int \cosh (c+d x) \, dx\\ &=-\frac{a \cosh (c+d x)}{x}+\frac{b \sinh (c+d x)}{d}+(a d) \int \frac{\sinh (c+d x)}{x} \, dx\\ &=-\frac{a \cosh (c+d x)}{x}+\frac{b \sinh (c+d x)}{d}+(a d \cosh (c)) \int \frac{\sinh (d x)}{x} \, dx+(a d \sinh (c)) \int \frac{\cosh (d x)}{x} \, dx\\ &=-\frac{a \cosh (c+d x)}{x}+a d \text{Chi}(d x) \sinh (c)+\frac{b \sinh (c+d x)}{d}+a d \cosh (c) \text{Shi}(d x)\\ \end{align*}
Mathematica [A] time = 0.0881973, size = 42, normalized size = 1. \[ a d \sinh (c) \text{Chi}(d x)+a d \cosh (c) \text{Shi}(d x)-\frac{a \cosh (c+d x)}{x}+\frac{b \sinh (c+d x)}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 81, normalized size = 1.9 \begin{align*} -{\frac{a{{\rm e}^{-dx-c}}}{2\,x}}+{\frac{da{{\rm e}^{-c}}{\it Ei} \left ( 1,dx \right ) }{2}}-{\frac{b{{\rm e}^{-dx-c}}}{2\,d}}-{\frac{a{{\rm e}^{dx+c}}}{2\,x}}-{\frac{da{{\rm e}^{c}}{\it Ei} \left ( 1,-dx \right ) }{2}}+{\frac{b{{\rm e}^{dx+c}}}{2\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.18785, size = 108, normalized size = 2.57 \begin{align*} -\frac{1}{2} \,{\left (a{\rm Ei}\left (-d x\right ) e^{\left (-c\right )} - a{\rm Ei}\left (d x\right ) e^{c} + \frac{{\left (d x e^{c} - e^{c}\right )} b e^{\left (d x\right )}}{d^{2}} + \frac{{\left (d x + 1\right )} b e^{\left (-d x - c\right )}}{d^{2}}\right )} d +{\left (b x - \frac{a}{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 = 2.084, size = 203, normalized size = 4.83 \begin{align*} -\frac{2 \, a d \cosh \left (d x + c\right ) - 2 \, b x \sinh \left (d x + c\right ) -{\left (a d^{2} x{\rm Ei}\left (d x\right ) - a d^{2} x{\rm Ei}\left (-d x\right )\right )} \cosh \left (c\right ) -{\left (a d^{2} x{\rm Ei}\left (d x\right ) + a d^{2} 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^{2}\right ) \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 = 1.22212, size = 108, normalized size = 2.57 \begin{align*} -\frac{a d^{2} x{\rm Ei}\left (-d x\right ) e^{\left (-c\right )} - a d^{2} x{\rm Ei}\left (d x\right ) e^{c} + a d e^{\left (d x + c\right )} - b x e^{\left (d x + c\right )} + a d e^{\left (-d x - c\right )} + b 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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