Optimal. Leaf size=41 \[ a \cosh (c) \text{Chi}(d x)+a \sinh (c) \text{Shi}(d x)-\frac{b \cosh (c+d x)}{d^2}+\frac{b x \sinh (c+d x)}{d} \]
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Rubi [A] time = 0.0980776, antiderivative size = 41, 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, 3303, 3298, 3301, 3296, 2638} \[ a \cosh (c) \text{Chi}(d x)+a \sinh (c) \text{Shi}(d x)-\frac{b \cosh (c+d x)}{d^2}+\frac{b x \sinh (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 5287
Rule 3303
Rule 3298
Rule 3301
Rule 3296
Rule 2638
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right ) \cosh (c+d x)}{x} \, dx &=\int \left (\frac{a \cosh (c+d x)}{x}+b x \cosh (c+d x)\right ) \, dx\\ &=a \int \frac{\cosh (c+d x)}{x} \, dx+b \int x \cosh (c+d x) \, dx\\ &=\frac{b x \sinh (c+d x)}{d}-\frac{b \int \sinh (c+d x) \, dx}{d}+(a \cosh (c)) \int \frac{\cosh (d x)}{x} \, dx+(a \sinh (c)) \int \frac{\sinh (d x)}{x} \, dx\\ &=-\frac{b \cosh (c+d x)}{d^2}+a \cosh (c) \text{Chi}(d x)+\frac{b x \sinh (c+d x)}{d}+a \sinh (c) \text{Shi}(d x)\\ \end{align*}
Mathematica [A] time = 0.109761, size = 55, normalized size = 1.34 \[ a \cosh (c) \text{Chi}(d x)+a \sinh (c) \text{Shi}(d x)+\frac{b \cosh (d x) (d x \sinh (c)-\cosh (c))}{d^2}+\frac{b \sinh (d x) (d x \cosh (c)-\sinh (c))}{d^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.035, size = 81, normalized size = 2. \begin{align*} -{\frac{a{{\rm e}^{-c}}{\it Ei} \left ( 1,dx \right ) }{2}}-{\frac{b{{\rm e}^{-dx-c}}x}{2\,d}}-{\frac{b{{\rm e}^{-dx-c}}}{2\,{d}^{2}}}-{\frac{a{{\rm e}^{c}}{\it Ei} \left ( 1,-dx \right ) }{2}}+{\frac{b{{\rm e}^{dx+c}}x}{2\,d}}-{\frac{b{{\rm e}^{dx+c}}}{2\,{d}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.16856, size = 165, normalized size = 4.02 \begin{align*} -\frac{1}{4} \,{\left (b{\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{2 \, a \cosh \left (d x + c\right ) \log \left (x^{2}\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}{d}\right )} d + \frac{1}{2} \,{\left (b x^{2} + a \log \left (x^{2}\right )\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 = 2.06274, size = 188, normalized size = 4.59 \begin{align*} \frac{2 \, b d x \sinh \left (d x + c\right ) - 2 \, b \cosh \left (d x + c\right ) +{\left (a d^{2}{\rm Ei}\left (d x\right ) + a d^{2}{\rm Ei}\left (-d x\right )\right )} \cosh \left (c\right ) +{\left (a d^{2}{\rm Ei}\left (d x\right ) - a d^{2}{\rm Ei}\left (-d x\right )\right )} \sinh \left (c\right )}{2 \, d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.56531, size = 49, normalized size = 1.2 \begin{align*} a \sinh{\left (c \right )} \operatorname{Shi}{\left (d x \right )} + a \cosh{\left (c \right )} \operatorname{Chi}\left (d x\right ) + b \left (\begin{cases} \frac{x \sinh{\left (c + d x \right )}}{d} - \frac{\cosh{\left (c + d x \right )}}{d^{2}} & \text{for}\: d \neq 0 \\\frac{x^{2} \cosh{\left (c \right )}}{2} & \text{otherwise} \end{cases}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17627, size = 103, normalized size = 2.51 \begin{align*} \frac{a d^{2}{\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + a d^{2}{\rm Ei}\left (d x\right ) e^{c} + b d x e^{\left (d x + c\right )} - b d x e^{\left (-d x - c\right )} - b e^{\left (d x + c\right )} - b e^{\left (-d x - c\right )}}{2 \, d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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