Optimal. Leaf size=121 \[ \frac {1}{2} a^2 d^2 \cosh (c) \text {Chi}(d x)+\frac {1}{2} a^2 d^2 \sinh (c) \text {Shi}(d x)-\frac {a^2 \cosh (c+d x)}{2 x^2}-\frac {a^2 d \sinh (c+d x)}{2 x}+2 a b d \sinh (c) \text {Chi}(d x)+2 a b d \cosh (c) \text {Shi}(d x)-\frac {2 a b \cosh (c+d x)}{x}+b^2 \cosh (c) \text {Chi}(d x)+b^2 \sinh (c) \text {Shi}(d x) \]
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Rubi [A] time = 0.34, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 5, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {6742, 3297, 3303, 3298, 3301} \[ \frac {1}{2} a^2 d^2 \cosh (c) \text {Chi}(d x)+\frac {1}{2} a^2 d^2 \sinh (c) \text {Shi}(d x)-\frac {a^2 \cosh (c+d x)}{2 x^2}-\frac {a^2 d \sinh (c+d x)}{2 x}+2 a b d \sinh (c) \text {Chi}(d x)+2 a b d \cosh (c) \text {Shi}(d x)-\frac {2 a b \cosh (c+d x)}{x}+b^2 \cosh (c) \text {Chi}(d x)+b^2 \sinh (c) \text {Shi}(d x) \]
Antiderivative was successfully verified.
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Rule 3297
Rule 3298
Rule 3301
Rule 3303
Rule 6742
Rubi steps
\begin {align*} \int \frac {(a+b x)^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^2}+\frac {b^2 \cosh (c+d x)}{x}\right ) \, dx\\ &=a^2 \int \frac {\cosh (c+d x)}{x^3} \, dx+(2 a b) \int \frac {\cosh (c+d x)}{x^2} \, dx+b^2 \int \frac {\cosh (c+d x)}{x} \, dx\\ &=-\frac {a^2 \cosh (c+d x)}{2 x^2}-\frac {2 a b \cosh (c+d x)}{x}+\frac {1}{2} \left (a^2 d\right ) \int \frac {\sinh (c+d x)}{x^2} \, dx+(2 a b d) \int \frac {\sinh (c+d x)}{x} \, dx+\left (b^2 \cosh (c)\right ) \int \frac {\cosh (d x)}{x} \, dx+\left (b^2 \sinh (c)\right ) \int \frac {\sinh (d x)}{x} \, dx\\ &=-\frac {a^2 \cosh (c+d x)}{2 x^2}-\frac {2 a b \cosh (c+d x)}{x}+b^2 \cosh (c) \text {Chi}(d x)-\frac {a^2 d \sinh (c+d x)}{2 x}+b^2 \sinh (c) \text {Shi}(d x)+\frac {1}{2} \left (a^2 d^2\right ) \int \frac {\cosh (c+d x)}{x} \, dx+(2 a b d \cosh (c)) \int \frac {\sinh (d x)}{x} \, dx+(2 a b d \sinh (c)) \int \frac {\cosh (d x)}{x} \, dx\\ &=-\frac {a^2 \cosh (c+d x)}{2 x^2}-\frac {2 a b \cosh (c+d x)}{x}+b^2 \cosh (c) \text {Chi}(d x)+2 a b d \text {Chi}(d x) \sinh (c)-\frac {a^2 d \sinh (c+d x)}{2 x}+2 a b d \cosh (c) \text {Shi}(d x)+b^2 \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 {a^2 \cosh (c+d x)}{2 x^2}-\frac {2 a b \cosh (c+d x)}{x}+b^2 \cosh (c) \text {Chi}(d x)+\frac {1}{2} a^2 d^2 \cosh (c) \text {Chi}(d x)+2 a b d \text {Chi}(d x) \sinh (c)-\frac {a^2 d \sinh (c+d x)}{2 x}+2 a b d \cosh (c) \text {Shi}(d x)+b^2 \sinh (c) \text {Shi}(d x)+\frac {1}{2} a^2 d^2 \sinh (c) \text {Shi}(d x)\\ \end {align*}
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Mathematica [A] time = 0.40, size = 93, normalized size = 0.77 \[ \frac {1}{2} \left (\text {Chi}(d x) \left (\cosh (c) \left (a^2 d^2+2 b^2\right )+4 a b d \sinh (c)\right )+\text {Shi}(d x) \left (\sinh (c) \left (a^2 d^2+2 b^2\right )+4 a b d \cosh (c)\right )-\frac {a ((a+4 b x) \cosh (c+d x)+a d x \sinh (c+d x))}{x^2}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 153, normalized size = 1.26 \[ -\frac {2 \, a^{2} d x \sinh \left (d x + c\right ) + 2 \, {\left (4 \, a b x + a^{2}\right )} \cosh \left (d x + c\right ) - {\left ({\left (a^{2} d^{2} + 4 \, a b d + 2 \, b^{2}\right )} x^{2} {\rm Ei}\left (d x\right ) + {\left (a^{2} d^{2} - 4 \, a b d + 2 \, b^{2}\right )} x^{2} {\rm Ei}\left (-d x\right )\right )} \cosh \relax (c) - {\left ({\left (a^{2} d^{2} + 4 \, a b d + 2 \, b^{2}\right )} x^{2} {\rm Ei}\left (d x\right ) - {\left (a^{2} d^{2} - 4 \, a b d + 2 \, b^{2}\right )} x^{2} {\rm Ei}\left (-d x\right )\right )} \sinh \relax (c)}{4 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 181, normalized size = 1.50 \[ \frac {a^{2} d^{2} x^{2} {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + a^{2} d^{2} x^{2} {\rm Ei}\left (d x\right ) e^{c} - 4 \, a b d x^{2} {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + 4 \, a b d x^{2} {\rm Ei}\left (d x\right ) e^{c} + 2 \, b^{2} x^{2} {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + 2 \, b^{2} x^{2} {\rm Ei}\left (d x\right ) e^{c} - a^{2} d x e^{\left (d x + c\right )} + a^{2} d x e^{\left (-d x - c\right )} - 4 \, a b x e^{\left (d x + c\right )} - 4 \, a b x e^{\left (-d x - c\right )} - a^{2} e^{\left (d x + c\right )} - a^{2} e^{\left (-d x - c\right )}}{4 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 181, normalized size = 1.50 \[ -\frac {b^{2} {\mathrm e}^{-c} \Ei \left (1, d x \right )}{2}-\frac {a^{2} {\mathrm e}^{-d x -c}}{4 x^{2}}+\frac {d \,a^{2} {\mathrm e}^{-d x -c}}{4 x}-\frac {a b \,{\mathrm e}^{-d x -c}}{x}+d a b \,{\mathrm e}^{-c} \Ei \left (1, d x \right )-\frac {d^{2} a^{2} {\mathrm e}^{-c} \Ei \left (1, d x \right )}{4}-\frac {b^{2} {\mathrm e}^{c} \Ei \left (1, -d x \right )}{2}-\frac {a^{2} {\mathrm e}^{d x +c}}{4 x^{2}}-\frac {d \,a^{2} {\mathrm e}^{d x +c}}{4 x}-\frac {a b \,{\mathrm e}^{d x +c}}{x}-d a b \,{\mathrm e}^{c} \Ei \left (1, -d x \right )-\frac {d^{2} a^{2} {\mathrm e}^{c} \Ei \left (1, -d x \right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 126, normalized size = 1.04 \[ \frac {1}{4} \, {\left ({\left (d e^{\left (-c\right )} \Gamma \left (-1, d x\right ) + d e^{c} \Gamma \left (-1, -d x\right )\right )} a^{2} - 4 \, {\left ({\rm Ei}\left (-d x\right ) e^{\left (-c\right )} - {\rm Ei}\left (d x\right ) e^{c}\right )} a b - \frac {4 \, b^{2} \cosh \left (d x + c\right ) \log \relax (x)}{d} + \frac {2 \, {\left ({\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + {\rm Ei}\left (d x\right ) e^{c}\right )} b^{2}}{d}\right )} d + \frac {1}{2} \, {\left (2 \, b^{2} \log \relax (x) - \frac {4 \, a b x + a^{2}}{x^{2}}\right )} \cosh \left (d x + c\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\mathrm {cosh}\left (c+d\,x\right )\,{\left (a+b\,x\right )}^2}{x^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b x\right )^{2} \cosh {\left (c + d x \right )}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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