Optimal. Leaf size=69 \[ \frac {1}{2} a d^2 \cosh (c) \text {Chi}(d x)+\frac {1}{2} a d^2 \sinh (c) \text {Shi}(d x)-\frac {a \cosh (c+d x)}{2 x^2}-\frac {a d \sinh (c+d x)}{2 x}+\frac {b \sinh (c+d x)}{d} \]
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Rubi [A] time = 0.14, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {5287, 2637, 3297, 3303, 3298, 3301} \[ \frac {1}{2} a d^2 \cosh (c) \text {Chi}(d x)+\frac {1}{2} a d^2 \sinh (c) \text {Shi}(d x)-\frac {a \cosh (c+d x)}{2 x^2}-\frac {a d \sinh (c+d x)}{2 x}+\frac {b \sinh (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 2637
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^3} \, dx &=\int \left (b \cosh (c+d x)+\frac {a \cosh (c+d x)}{x^3}\right ) \, dx\\ &=a \int \frac {\cosh (c+d x)}{x^3} \, dx+b \int \cosh (c+d x) \, dx\\ &=-\frac {a \cosh (c+d x)}{2 x^2}+\frac {b \sinh (c+d x)}{d}+\frac {1}{2} (a d) \int \frac {\sinh (c+d x)}{x^2} \, dx\\ &=-\frac {a \cosh (c+d x)}{2 x^2}+\frac {b \sinh (c+d x)}{d}-\frac {a d \sinh (c+d x)}{2 x}+\frac {1}{2} \left (a d^2\right ) \int \frac {\cosh (c+d x)}{x} \, dx\\ &=-\frac {a \cosh (c+d x)}{2 x^2}+\frac {b \sinh (c+d x)}{d}-\frac {a d \sinh (c+d x)}{2 x}+\frac {1}{2} \left (a d^2 \cosh (c)\right ) \int \frac {\cosh (d x)}{x} \, dx+\frac {1}{2} \left (a d^2 \sinh (c)\right ) \int \frac {\sinh (d x)}{x} \, dx\\ &=-\frac {a \cosh (c+d x)}{2 x^2}+\frac {1}{2} a d^2 \cosh (c) \text {Chi}(d x)+\frac {b \sinh (c+d x)}{d}-\frac {a d \sinh (c+d x)}{2 x}+\frac {1}{2} a d^2 \sinh (c) \text {Shi}(d x)\\ \end {align*}
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Mathematica [A] time = 0.15, size = 86, normalized size = 1.25 \[ \frac {1}{2} a d^2 (\cosh (c) \text {Chi}(d x)+\sinh (c) \text {Shi}(d x))-\frac {a \cosh (d x) (d x \sinh (c)+\cosh (c))}{2 x^2}-\frac {a \sinh (d x) (d x \cosh (c)+\sinh (c))}{2 x^2}+\frac {b \sinh (c) \cosh (d x)}{d}+\frac {b \cosh (c) \sinh (d x)}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 101, normalized size = 1.46 \[ -\frac {2 \, a d \cosh \left (d x + c\right ) - {\left (a d^{3} x^{2} {\rm Ei}\left (d x\right ) + a d^{3} x^{2} {\rm Ei}\left (-d x\right )\right )} \cosh \relax (c) + 2 \, {\left (a d^{2} x - 2 \, b x^{2}\right )} \sinh \left (d x + c\right ) - {\left (a d^{3} x^{2} {\rm Ei}\left (d x\right ) - a d^{3} x^{2} {\rm Ei}\left (-d x\right )\right )} \sinh \relax (c)}{4 \, d x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 118, normalized size = 1.71 \[ \frac {a d^{3} x^{2} {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + a d^{3} x^{2} {\rm Ei}\left (d x\right ) e^{c} - a d^{2} x e^{\left (d x + c\right )} + a d^{2} x e^{\left (-d x - c\right )} + 2 \, b x^{2} e^{\left (d x + c\right )} - 2 \, b x^{2} e^{\left (-d x - c\right )} - a d e^{\left (d x + c\right )} - a d e^{\left (-d x - c\right )}}{4 \, d x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.13, size = 114, normalized size = 1.65 \[ \frac {d a \,{\mathrm e}^{-d x -c}}{4 x}-\frac {a \,{\mathrm e}^{-d x -c}}{4 x^{2}}-\frac {d^{2} a \,{\mathrm e}^{-c} \Ei \left (1, d x \right )}{4}-\frac {b \,{\mathrm e}^{-d x -c}}{2 d}-\frac {a \,{\mathrm e}^{d x +c}}{4 x^{2}}-\frac {d a \,{\mathrm e}^{d x +c}}{4 x}-\frac {d^{2} a \,{\mathrm e}^{c} \Ei \left (1, -d x \right )}{4}+\frac {b \,{\mathrm e}^{d x +c}}{2 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 87, normalized size = 1.26 \[ \frac {1}{4} \, {\left (a d e^{\left (-c\right )} \Gamma \left (-1, d x\right ) + a d e^{c} \Gamma \left (-1, -d x\right ) - \frac {2 \, {\left (d x e^{c} - e^{c}\right )} b e^{\left (d x\right )}}{d^{2}} - \frac {2 \, {\left (d x + 1\right )} b e^{\left (-d x - c\right )}}{d^{2}}\right )} d + \frac {1}{2} \, {\left (2 \, b x - \frac {a}{x^{2}}\right )} \cosh \left (d x + c\right ) \]
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 \[ \int \frac {\mathrm {cosh}\left (c+d\,x\right )\,\left (b\,x^3+a\right )}{x^3} \,d x \]
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 \[ \int \frac {\left (a + b x^{3}\right ) \cosh {\left (c + d x \right )}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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