Optimal. Leaf size=62 \[ a^2 \cosh (c) \text {Chi}(d x)+a^2 \sinh (c) \text {Shi}(d x)+\frac {2 a b \sinh (c+d x)}{d}-\frac {b^2 \cosh (c+d x)}{d^2}+\frac {b^2 x \sinh (c+d x)}{d} \]
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Rubi [A] time = 0.18, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {6742, 2637, 3303, 3298, 3301, 3296, 2638} \[ a^2 \cosh (c) \text {Chi}(d x)+a^2 \sinh (c) \text {Shi}(d x)+\frac {2 a b \sinh (c+d x)}{d}-\frac {b^2 \cosh (c+d x)}{d^2}+\frac {b^2 x \sinh (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 2637
Rule 2638
Rule 3296
Rule 3298
Rule 3301
Rule 3303
Rule 6742
Rubi steps
\begin {align*} \int \frac {(a+b x)^2 \cosh (c+d x)}{x} \, dx &=\int \left (2 a b \cosh (c+d x)+\frac {a^2 \cosh (c+d x)}{x}+b^2 x \cosh (c+d x)\right ) \, dx\\ &=a^2 \int \frac {\cosh (c+d x)}{x} \, dx+(2 a b) \int \cosh (c+d x) \, dx+b^2 \int x \cosh (c+d x) \, dx\\ &=\frac {2 a b \sinh (c+d x)}{d}+\frac {b^2 x \sinh (c+d x)}{d}-\frac {b^2 \int \sinh (c+d x) \, dx}{d}+\left (a^2 \cosh (c)\right ) \int \frac {\cosh (d x)}{x} \, dx+\left (a^2 \sinh (c)\right ) \int \frac {\sinh (d x)}{x} \, dx\\ &=-\frac {b^2 \cosh (c+d x)}{d^2}+a^2 \cosh (c) \text {Chi}(d x)+\frac {2 a b \sinh (c+d x)}{d}+\frac {b^2 x \sinh (c+d x)}{d}+a^2 \sinh (c) \text {Shi}(d x)\\ \end {align*}
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Mathematica [A] time = 0.25, size = 51, normalized size = 0.82 \[ a^2 \cosh (c) \text {Chi}(d x)+a^2 \sinh (c) \text {Shi}(d x)+\frac {b (d (2 a+b x) \sinh (c+d x)-b \cosh (c+d x))}{d^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.62, size = 94, normalized size = 1.52 \[ -\frac {2 \, b^{2} \cosh \left (d x + c\right ) - {\left (a^{2} d^{2} {\rm Ei}\left (d x\right ) + a^{2} d^{2} {\rm Ei}\left (-d x\right )\right )} \cosh \relax (c) - 2 \, {\left (b^{2} d x + 2 \, a b d\right )} \sinh \left (d x + c\right ) - {\left (a^{2} d^{2} {\rm Ei}\left (d x\right ) - a^{2} d^{2} {\rm Ei}\left (-d x\right )\right )} \sinh \relax (c)}{2 \, d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 113, normalized size = 1.82 \[ \frac {a^{2} d^{2} {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + a^{2} d^{2} {\rm Ei}\left (d x\right ) e^{c} + b^{2} d x e^{\left (d x + c\right )} - b^{2} d x e^{\left (-d x - c\right )} + 2 \, a b d e^{\left (d x + c\right )} - 2 \, a b d e^{\left (-d x - c\right )} - b^{2} e^{\left (d x + c\right )} - b^{2} e^{\left (-d x - c\right )}}{2 \, d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 121, normalized size = 1.95 \[ -\frac {a^{2} {\mathrm e}^{-c} \Ei \left (1, d x \right )}{2}-\frac {b^{2} {\mathrm e}^{-d x -c} x}{2 d}-\frac {b^{2} {\mathrm e}^{-d x -c}}{2 d^{2}}-\frac {a b \,{\mathrm e}^{-d x -c}}{d}-\frac {a^{2} {\mathrm e}^{c} \Ei \left (1, -d x \right )}{2}+\frac {b^{2} {\mathrm e}^{d x +c} x}{2 d}-\frac {b^{2} {\mathrm e}^{d x +c}}{2 d^{2}}+\frac {a b \,{\mathrm e}^{d x +c}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.40, size = 175, normalized size = 2.82 \[ -\frac {1}{4} \, {\left (4 \, a b {\left (\frac {{\left (d x e^{c} - e^{c}\right )} e^{\left (d x\right )}}{d^{2}} + \frac {{\left (d x + 1\right )} e^{\left (-d x - c\right )}}{d^{2}}\right )} + b^{2} {\left (\frac {{\left (d^{2} x^{2} e^{c} - 2 \, d x e^{c} + 2 \, e^{c}\right )} e^{\left (d x\right )}}{d^{3}} + \frac {{\left (d^{2} x^{2} + 2 \, d x + 2\right )} e^{\left (-d x - c\right )}}{d^{3}}\right )} + \frac {4 \, a^{2} \cosh \left (d x + c\right ) \log \relax (x)}{d} - \frac {2 \, {\left ({\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + {\rm Ei}\left (d x\right ) e^{c}\right )} a^{2}}{d}\right )} d + \frac {1}{2} \, {\left (b^{2} x^{2} + 4 \, a b x + 2 \, a^{2} \log \relax (x)\right )} \cosh \left (d x + c\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\mathrm {cosh}\left (c+d\,x\right )\,{\left (a+b\,x\right )}^2}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.70, size = 73, normalized size = 1.18 \[ a^{2} \sinh {\relax (c )} \operatorname {Shi}{\left (d x \right )} + a^{2} \cosh {\relax (c )} \operatorname {Chi}\left (d x\right ) + 2 a b \left (\begin {cases} \frac {\sinh {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \cosh {\relax (c )} & \text {otherwise} \end {cases}\right ) + b^{2} \left (\begin {cases} \frac {x \sinh {\left (c + d x \right )}}{d} - \frac {\cosh {\left (c + d x \right )}}{d^{2}} & \text {for}\: d \neq 0 \\\frac {x^{2} \cosh {\relax (c )}}{2} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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