Optimal. Leaf size=55 \[ a d \sinh (c) \text {Chi}(d x)+a d \cosh (c) \text {Shi}(d x)-\frac {a \cosh (c+d x)}{x}-\frac {b \cosh (c+d x)}{d^2}+\frac {b x \sinh (c+d x)}{d} \]
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Rubi [A] time = 0.13, antiderivative size = 55, 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 = {5287, 3297, 3303, 3298, 3301, 3296, 2638} \[ a d \sinh (c) \text {Chi}(d x)+a d \cosh (c) \text {Shi}(d x)-\frac {a \cosh (c+d x)}{x}-\frac {b \cosh (c+d x)}{d^2}+\frac {b x \sinh (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 2638
Rule 3296
Rule 3297
Rule 3298
Rule 3301
Rule 3303
Rule 5287
Rubi steps
\begin {align*} \int \frac {\left (a+b x^3\right ) \cosh (c+d x)}{x^2} \, dx &=\int \left (\frac {a \cosh (c+d x)}{x^2}+b x \cosh (c+d x)\right ) \, dx\\ &=a \int \frac {\cosh (c+d x)}{x^2} \, dx+b \int x \cosh (c+d x) \, dx\\ &=-\frac {a \cosh (c+d x)}{x}+\frac {b x \sinh (c+d x)}{d}-\frac {b \int \sinh (c+d x) \, dx}{d}+(a d) \int \frac {\sinh (c+d x)}{x} \, dx\\ &=-\frac {b \cosh (c+d x)}{d^2}-\frac {a \cosh (c+d x)}{x}+\frac {b x \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 {b \cosh (c+d x)}{d^2}-\frac {a \cosh (c+d x)}{x}+a d \text {Chi}(d x) \sinh (c)+\frac {b x \sinh (c+d x)}{d}+a d \cosh (c) \text {Shi}(d x)\\ \end {align*}
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Mathematica [A] time = 0.14, size = 55, normalized size = 1.00 \[ a d \sinh (c) \text {Chi}(d x)+a d \cosh (c) \text {Shi}(d x)-\frac {a \cosh (c+d x)}{x}-\frac {b \cosh (c+d x)}{d^2}+\frac {b x \sinh (c+d x)}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 90, normalized size = 1.64 \[ \frac {2 \, b d x^{2} \sinh \left (d x + c\right ) - 2 \, {\left (a d^{2} + b x\right )} \cosh \left (d x + c\right ) + {\left (a d^{3} x {\rm Ei}\left (d x\right ) - a d^{3} x {\rm Ei}\left (-d x\right )\right )} \cosh \relax (c) + {\left (a d^{3} x {\rm Ei}\left (d x\right ) + a d^{3} x {\rm Ei}\left (-d x\right )\right )} \sinh \relax (c)}{2 \, d^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.14, size = 111, normalized size = 2.02 \[ -\frac {a d^{3} x {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} - a d^{3} x {\rm Ei}\left (d x\right ) e^{c} - b d x^{2} e^{\left (d x + c\right )} + b d x^{2} e^{\left (-d x - c\right )} + a d^{2} e^{\left (d x + c\right )} + a d^{2} e^{\left (-d x - c\right )} + b x e^{\left (d x + c\right )} + b x e^{\left (-d x - c\right )}}{2 \, d^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 110, normalized size = 2.00 \[ -\frac {a \,{\mathrm e}^{-d x -c}}{2 x}+\frac {d a \,{\mathrm e}^{-c} \Ei \left (1, d x \right )}{2}-\frac {b \,{\mathrm e}^{-d x -c} x}{2 d}-\frac {b \,{\mathrm e}^{-d x -c}}{2 d^{2}}-\frac {a \,{\mathrm e}^{d x +c}}{2 x}-\frac {d a \,{\mathrm e}^{c} \Ei \left (1, -d x \right )}{2}+\frac {b \,{\mathrm e}^{d x +c} x}{2 d}-\frac {b \,{\mathrm e}^{d x +c}}{2 d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 102, normalized size = 1.85 \[ -\frac {1}{4} \, {\left (2 \, a {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} - 2 \, a {\rm Ei}\left (d x\right ) e^{c} + \frac {{\left (d^{2} x^{2} e^{c} - 2 \, d x e^{c} + 2 \, e^{c}\right )} b e^{\left (d x\right )}}{d^{3}} + \frac {{\left (d^{2} x^{2} + 2 \, d x + 2\right )} b e^{\left (-d x - c\right )}}{d^{3}}\right )} d + \frac {1}{2} \, {\left (b x^{2} - \frac {2 \, a}{x}\right )} \cosh \left (d x + c\right ) \]
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 \[ \int \frac {\mathrm {cosh}\left (c+d\,x\right )\,\left (b\,x^3+a\right )}{x^2} \,d x \]
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 \[ \int \frac {\left (a + b x^{3}\right ) \cosh {\left (c + d x \right )}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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