Optimal. Leaf size=91 \[ \frac {1}{6} a d^3 \sinh (c) \text {Chi}(d x)+\frac {1}{6} a d^3 \cosh (c) \text {Shi}(d x)-\frac {a d^2 \cosh (c+d x)}{6 x}-\frac {a \cosh (c+d x)}{3 x^3}-\frac {a d \sinh (c+d x)}{6 x^2}+b \cosh (c) \text {Chi}(d x)+b \sinh (c) \text {Shi}(d x) \]
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Rubi [A] time = 0.22, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 5, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {5287, 3297, 3303, 3298, 3301} \[ \frac {1}{6} a d^3 \sinh (c) \text {Chi}(d x)+\frac {1}{6} a d^3 \cosh (c) \text {Shi}(d x)-\frac {a d^2 \cosh (c+d x)}{6 x}-\frac {a d \sinh (c+d x)}{6 x^2}-\frac {a \cosh (c+d x)}{3 x^3}+b \cosh (c) \text {Chi}(d x)+b \sinh (c) \text {Shi}(d x) \]
Antiderivative was successfully verified.
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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^4} \, dx &=\int \left (\frac {a \cosh (c+d x)}{x^4}+\frac {b \cosh (c+d x)}{x}\right ) \, dx\\ &=a \int \frac {\cosh (c+d x)}{x^4} \, dx+b \int \frac {\cosh (c+d x)}{x} \, dx\\ &=-\frac {a \cosh (c+d x)}{3 x^3}+\frac {1}{3} (a d) \int \frac {\sinh (c+d x)}{x^3} \, dx+(b \cosh (c)) \int \frac {\cosh (d x)}{x} \, dx+(b \sinh (c)) \int \frac {\sinh (d x)}{x} \, dx\\ &=-\frac {a \cosh (c+d x)}{3 x^3}+b \cosh (c) \text {Chi}(d x)-\frac {a d \sinh (c+d x)}{6 x^2}+b \sinh (c) \text {Shi}(d x)+\frac {1}{6} \left (a d^2\right ) \int \frac {\cosh (c+d x)}{x^2} \, dx\\ &=-\frac {a \cosh (c+d x)}{3 x^3}-\frac {a d^2 \cosh (c+d x)}{6 x}+b \cosh (c) \text {Chi}(d x)-\frac {a d \sinh (c+d x)}{6 x^2}+b \sinh (c) \text {Shi}(d x)+\frac {1}{6} \left (a d^3\right ) \int \frac {\sinh (c+d x)}{x} \, dx\\ &=-\frac {a \cosh (c+d x)}{3 x^3}-\frac {a d^2 \cosh (c+d x)}{6 x}+b \cosh (c) \text {Chi}(d x)-\frac {a d \sinh (c+d x)}{6 x^2}+b \sinh (c) \text {Shi}(d x)+\frac {1}{6} \left (a d^3 \cosh (c)\right ) \int \frac {\sinh (d x)}{x} \, dx+\frac {1}{6} \left (a d^3 \sinh (c)\right ) \int \frac {\cosh (d x)}{x} \, dx\\ &=-\frac {a \cosh (c+d x)}{3 x^3}-\frac {a d^2 \cosh (c+d x)}{6 x}+b \cosh (c) \text {Chi}(d x)+\frac {1}{6} a d^3 \text {Chi}(d x) \sinh (c)-\frac {a d \sinh (c+d x)}{6 x^2}+\frac {1}{6} a d^3 \cosh (c) \text {Shi}(d x)+b \sinh (c) \text {Shi}(d x)\\ \end {align*}
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Mathematica [A] time = 0.26, size = 73, normalized size = 0.80 \[ \frac {1}{6} \left (\text {Chi}(d x) \left (a d^3 \sinh (c)+6 b \cosh (c)\right )+\text {Shi}(d x) \left (a d^3 \cosh (c)+6 b \sinh (c)\right )-\frac {a \left (\left (d^2 x^2+2\right ) \cosh (c+d x)+d x \sinh (c+d x)\right )}{x^3}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 118, normalized size = 1.30 \[ -\frac {2 \, a d x \sinh \left (d x + c\right ) + 2 \, {\left (a d^{2} x^{2} + 2 \, a\right )} \cosh \left (d x + c\right ) - {\left ({\left (a d^{3} + 6 \, b\right )} x^{3} {\rm Ei}\left (d x\right ) - {\left (a d^{3} - 6 \, b\right )} x^{3} {\rm Ei}\left (-d x\right )\right )} \cosh \relax (c) - {\left ({\left (a d^{3} + 6 \, b\right )} x^{3} {\rm Ei}\left (d x\right ) + {\left (a d^{3} - 6 \, b\right )} x^{3} {\rm Ei}\left (-d x\right )\right )} \sinh \relax (c)}{12 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 141, normalized size = 1.55 \[ -\frac {a d^{3} x^{3} {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} - a d^{3} x^{3} {\rm Ei}\left (d x\right ) e^{c} + a d^{2} x^{2} e^{\left (d x + c\right )} + a d^{2} x^{2} e^{\left (-d x - c\right )} - 6 \, b x^{3} {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} - 6 \, b x^{3} {\rm Ei}\left (d x\right ) e^{c} + a d x e^{\left (d x + c\right )} - a d x e^{\left (-d x - c\right )} + 2 \, a e^{\left (d x + c\right )} + 2 \, a e^{\left (-d x - c\right )}}{12 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.13, size = 143, normalized size = 1.57 \[ -\frac {d^{2} a \,{\mathrm e}^{-d x -c}}{12 x}+\frac {d a \,{\mathrm e}^{-d x -c}}{12 x^{2}}-\frac {a \,{\mathrm e}^{-d x -c}}{6 x^{3}}+\frac {d^{3} a \,{\mathrm e}^{-c} \Ei \left (1, d x \right )}{12}-\frac {b \,{\mathrm e}^{-c} \Ei \left (1, d x \right )}{2}-\frac {b \,{\mathrm e}^{c} \Ei \left (1, -d x \right )}{2}-\frac {a \,{\mathrm e}^{d x +c}}{6 x^{3}}-\frac {d a \,{\mathrm e}^{d x +c}}{12 x^{2}}-\frac {d^{2} a \,{\mathrm e}^{d x +c}}{12 x}-\frac {d^{3} a \,{\mathrm e}^{c} \Ei \left (1, -d x \right )}{12} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 95, normalized size = 1.04 \[ \frac {1}{6} \, {\left ({\left (d^{2} e^{\left (-c\right )} \Gamma \left (-2, d x\right ) - d^{2} e^{c} \Gamma \left (-2, -d x\right )\right )} a - \frac {2 \, b \cosh \left (d x + c\right ) \log \left (x^{3}\right )}{d} + \frac {3 \, {\left ({\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + {\rm Ei}\left (d x\right ) e^{c}\right )} b}{d}\right )} d + \frac {1}{3} \, {\left (b \log \left (x^{3}\right ) - \frac {a}{x^{3}}\right )} \cosh \left (d x + c\right ) \]
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 \[ \int \frac {\mathrm {cosh}\left (c+d\,x\right )\,\left (b\,x^3+a\right )}{x^4} \,d x \]
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 \[ \int \frac {\left (a + b x^{3}\right ) \cosh {\left (c + d x \right )}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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