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 = 0.25, antiderivative size = 70, normalized size of antiderivative = 1.00, 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 2637
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^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*}
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Mathematica [A] time = 0.25, 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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fricas [A] time = 0.54, size = 122, normalized size = 1.74 \[ -\frac {2 \, a^{2} d \cosh \left (d x + c\right ) - 2 \, b^{2} x \sinh \left (d x + c\right ) - {\left ({\left (a^{2} d^{2} + 2 \, a b d\right )} x {\rm Ei}\left (d x\right ) - {\left (a^{2} d^{2} - 2 \, a b d\right )} x {\rm Ei}\left (-d x\right )\right )} \cosh \relax (c) - {\left ({\left (a^{2} d^{2} + 2 \, a b d\right )} x {\rm Ei}\left (d x\right ) + {\left (a^{2} d^{2} - 2 \, a b d\right )} x {\rm Ei}\left (-d x\right )\right )} \sinh \relax (c)}{2 \, d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 119, normalized size = 1.70 \[ -\frac {a^{2} d^{2} x {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} - a^{2} d^{2} x {\rm Ei}\left (d x\right ) e^{c} - 2 \, a b d x {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} - 2 \, a b d x {\rm Ei}\left (d x\right ) e^{c} + a^{2} d e^{\left (d x + c\right )} - b^{2} x e^{\left (d x + c\right )} + a^{2} d e^{\left (-d x - c\right )} + b^{2} x e^{\left (-d x - c\right )}}{2 \, d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 118, normalized size = 1.69 \[ -\frac {a^{2} {\mathrm e}^{-d x -c}}{2 x}+\frac {d \,a^{2} {\mathrm e}^{-c} \Ei \left (1, d x \right )}{2}-\frac {b^{2} {\mathrm e}^{-d x -c}}{2 d}-a b \,{\mathrm e}^{-c} \Ei \left (1, d x \right )-\frac {a^{2} {\mathrm e}^{d x +c}}{2 x}-\frac {d \,a^{2} {\mathrm e}^{c} \Ei \left (1, -d x \right )}{2}+\frac {b^{2} {\mathrm e}^{d x +c}}{2 d}-a b \,{\mathrm e}^{c} \Ei \left (1, -d x \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 136, normalized size = 1.94 \[ -\frac {1}{2} \, {\left ({\left ({\rm Ei}\left (-d x\right ) e^{\left (-c\right )} - {\rm Ei}\left (d x\right ) e^{c}\right )} a^{2} + b^{2} {\left (\frac {{\left (d x e^{c} - e^{c}\right )} e^{\left (d x\right )}}{d^{2}} + \frac {{\left (d x + 1\right )} e^{\left (-d x - c\right )}}{d^{2}}\right )} + \frac {4 \, a b \cosh \left (d x + c\right ) \log \relax (x)}{d} - \frac {2 \, {\left ({\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + {\rm Ei}\left (d x\right ) e^{c}\right )} a b}{d}\right )} d + {\left (b^{2} x + 2 \, a b \log \relax (x) - \frac {a^{2}}{x}\right )} \cosh \left (d x + c\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\mathrm {cosh}\left (c+d\,x\right )\,{\left (a+b\,x\right )}^2}{x^2} \,d x \]
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 \[ \int \frac {\left (a + b x\right )^{2} \cosh {\left (c + d x \right )}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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