Optimal. Leaf size=149 \[ \frac {1}{24} a d^4 \cosh (c) \text {Chi}(d x)+\frac {1}{24} a d^4 \sinh (c) \text {Shi}(d x)-\frac {a d^3 \sinh (c+d x)}{24 x}-\frac {a d^2 \cosh (c+d x)}{24 x^2}-\frac {a \cosh (c+d x)}{4 x^4}-\frac {a d \sinh (c+d x)}{12 x^3}+\frac {1}{2} b d^2 \cosh (c) \text {Chi}(d x)+\frac {1}{2} b d^2 \sinh (c) \text {Shi}(d x)-\frac {b \cosh (c+d x)}{2 x^2}-\frac {b d \sinh (c+d x)}{2 x} \]
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Rubi [A] time = 0.29, antiderivative size = 149, 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 = {5287, 3297, 3303, 3298, 3301} \[ \frac {1}{24} a d^4 \cosh (c) \text {Chi}(d x)+\frac {1}{24} a d^4 \sinh (c) \text {Shi}(d x)-\frac {a d^2 \cosh (c+d x)}{24 x^2}-\frac {a d^3 \sinh (c+d x)}{24 x}-\frac {a d \sinh (c+d x)}{12 x^3}-\frac {a \cosh (c+d x)}{4 x^4}+\frac {1}{2} b d^2 \cosh (c) \text {Chi}(d x)+\frac {1}{2} b d^2 \sinh (c) \text {Shi}(d x)-\frac {b \cosh (c+d x)}{2 x^2}-\frac {b d \sinh (c+d x)}{2 x} \]
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^2\right ) \cosh (c+d x)}{x^5} \, dx &=\int \left (\frac {a \cosh (c+d x)}{x^5}+\frac {b \cosh (c+d x)}{x^3}\right ) \, dx\\ &=a \int \frac {\cosh (c+d x)}{x^5} \, dx+b \int \frac {\cosh (c+d x)}{x^3} \, dx\\ &=-\frac {a \cosh (c+d x)}{4 x^4}-\frac {b \cosh (c+d x)}{2 x^2}+\frac {1}{4} (a d) \int \frac {\sinh (c+d x)}{x^4} \, dx+\frac {1}{2} (b d) \int \frac {\sinh (c+d x)}{x^2} \, dx\\ &=-\frac {a \cosh (c+d x)}{4 x^4}-\frac {b \cosh (c+d x)}{2 x^2}-\frac {a d \sinh (c+d x)}{12 x^3}-\frac {b d \sinh (c+d x)}{2 x}+\frac {1}{12} \left (a d^2\right ) \int \frac {\cosh (c+d x)}{x^3} \, dx+\frac {1}{2} \left (b d^2\right ) \int \frac {\cosh (c+d x)}{x} \, dx\\ &=-\frac {a \cosh (c+d x)}{4 x^4}-\frac {b \cosh (c+d x)}{2 x^2}-\frac {a d^2 \cosh (c+d x)}{24 x^2}-\frac {a d \sinh (c+d x)}{12 x^3}-\frac {b d \sinh (c+d x)}{2 x}+\frac {1}{24} \left (a d^3\right ) \int \frac {\sinh (c+d x)}{x^2} \, dx+\frac {1}{2} \left (b d^2 \cosh (c)\right ) \int \frac {\cosh (d x)}{x} \, dx+\frac {1}{2} \left (b d^2 \sinh (c)\right ) \int \frac {\sinh (d x)}{x} \, dx\\ &=-\frac {a \cosh (c+d x)}{4 x^4}-\frac {b \cosh (c+d x)}{2 x^2}-\frac {a d^2 \cosh (c+d x)}{24 x^2}+\frac {1}{2} b d^2 \cosh (c) \text {Chi}(d x)-\frac {a d \sinh (c+d x)}{12 x^3}-\frac {b d \sinh (c+d x)}{2 x}-\frac {a d^3 \sinh (c+d x)}{24 x}+\frac {1}{2} b d^2 \sinh (c) \text {Shi}(d x)+\frac {1}{24} \left (a d^4\right ) \int \frac {\cosh (c+d x)}{x} \, dx\\ &=-\frac {a \cosh (c+d x)}{4 x^4}-\frac {b \cosh (c+d x)}{2 x^2}-\frac {a d^2 \cosh (c+d x)}{24 x^2}+\frac {1}{2} b d^2 \cosh (c) \text {Chi}(d x)-\frac {a d \sinh (c+d x)}{12 x^3}-\frac {b d \sinh (c+d x)}{2 x}-\frac {a d^3 \sinh (c+d x)}{24 x}+\frac {1}{2} b d^2 \sinh (c) \text {Shi}(d x)+\frac {1}{24} \left (a d^4 \cosh (c)\right ) \int \frac {\cosh (d x)}{x} \, dx+\frac {1}{24} \left (a d^4 \sinh (c)\right ) \int \frac {\sinh (d x)}{x} \, dx\\ &=-\frac {a \cosh (c+d x)}{4 x^4}-\frac {b \cosh (c+d x)}{2 x^2}-\frac {a d^2 \cosh (c+d x)}{24 x^2}+\frac {1}{2} b d^2 \cosh (c) \text {Chi}(d x)+\frac {1}{24} a d^4 \cosh (c) \text {Chi}(d x)-\frac {a d \sinh (c+d x)}{12 x^3}-\frac {b d \sinh (c+d x)}{2 x}-\frac {a d^3 \sinh (c+d x)}{24 x}+\frac {1}{2} b d^2 \sinh (c) \text {Shi}(d x)+\frac {1}{24} a d^4 \sinh (c) \text {Shi}(d x)\\ \end {align*}
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Mathematica [A] time = 0.29, size = 127, normalized size = 0.85 \[ -\frac {-d^2 x^4 \cosh (c) \left (a d^2+12 b\right ) \text {Chi}(d x)-d^2 x^4 \sinh (c) \left (a d^2+12 b\right ) \text {Shi}(d x)+a d^3 x^3 \sinh (c+d x)+a d^2 x^2 \cosh (c+d x)+2 a d x \sinh (c+d x)+6 a \cosh (c+d x)+12 b d x^3 \sinh (c+d x)+12 b x^2 \cosh (c+d x)}{24 x^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 152, normalized size = 1.02 \[ -\frac {2 \, {\left ({\left (a d^{2} + 12 \, b\right )} x^{2} + 6 \, a\right )} \cosh \left (d x + c\right ) - {\left ({\left (a d^{4} + 12 \, b d^{2}\right )} x^{4} {\rm Ei}\left (d x\right ) + {\left (a d^{4} + 12 \, b d^{2}\right )} x^{4} {\rm Ei}\left (-d x\right )\right )} \cosh \relax (c) + 2 \, {\left ({\left (a d^{3} + 12 \, b d\right )} x^{3} + 2 \, a d x\right )} \sinh \left (d x + c\right ) - {\left ({\left (a d^{4} + 12 \, b d^{2}\right )} x^{4} {\rm Ei}\left (d x\right ) - {\left (a d^{4} + 12 \, b d^{2}\right )} x^{4} {\rm Ei}\left (-d x\right )\right )} \sinh \relax (c)}{48 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 237, normalized size = 1.59 \[ \frac {a d^{4} x^{4} {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + a d^{4} x^{4} {\rm Ei}\left (d x\right ) e^{c} + 12 \, b d^{2} x^{4} {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + 12 \, b d^{2} x^{4} {\rm Ei}\left (d x\right ) e^{c} - a d^{3} x^{3} e^{\left (d x + c\right )} + a d^{3} x^{3} e^{\left (-d x - c\right )} - a d^{2} x^{2} e^{\left (d x + c\right )} - 12 \, b d x^{3} e^{\left (d x + c\right )} - a d^{2} x^{2} e^{\left (-d x - c\right )} + 12 \, b d x^{3} e^{\left (-d x - c\right )} - 2 \, a d x e^{\left (d x + c\right )} - 12 \, b x^{2} e^{\left (d x + c\right )} + 2 \, a d x e^{\left (-d x - c\right )} - 12 \, b x^{2} e^{\left (-d x - c\right )} - 6 \, a e^{\left (d x + c\right )} - 6 \, a e^{\left (-d x - c\right )}}{48 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 238, normalized size = 1.60 \[ \frac {d b \,{\mathrm e}^{-d x -c}}{4 x}-\frac {b \,{\mathrm e}^{-d x -c}}{4 x^{2}}-\frac {d^{2} b \,{\mathrm e}^{-c} \Ei \left (1, d x \right )}{4}+\frac {d^{3} a \,{\mathrm e}^{-d x -c}}{48 x}-\frac {d^{2} a \,{\mathrm e}^{-d x -c}}{48 x^{2}}+\frac {d a \,{\mathrm e}^{-d x -c}}{24 x^{3}}-\frac {a \,{\mathrm e}^{-d x -c}}{8 x^{4}}-\frac {d^{4} a \,{\mathrm e}^{-c} \Ei \left (1, d x \right )}{48}-\frac {b \,{\mathrm e}^{d x +c}}{4 x^{2}}-\frac {d b \,{\mathrm e}^{d x +c}}{4 x}-\frac {d^{2} b \,{\mathrm e}^{c} \Ei \left (1, -d x \right )}{4}-\frac {a \,{\mathrm e}^{d x +c}}{8 x^{4}}-\frac {d a \,{\mathrm e}^{d x +c}}{24 x^{3}}-\frac {d^{2} a \,{\mathrm e}^{d x +c}}{48 x^{2}}-\frac {d^{3} a \,{\mathrm e}^{d x +c}}{48 x}-\frac {d^{4} a \,{\mathrm e}^{c} \Ei \left (1, -d x \right )}{48} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 76, normalized size = 0.51 \[ \frac {1}{8} \, {\left (a d^{3} e^{\left (-c\right )} \Gamma \left (-3, d x\right ) + a d^{3} e^{c} \Gamma \left (-3, -d x\right ) + 2 \, b d e^{\left (-c\right )} \Gamma \left (-1, d x\right ) + 2 \, b d e^{c} \Gamma \left (-1, -d x\right )\right )} d - \frac {{\left (2 \, b x^{2} + a\right )} \cosh \left (d x + c\right )}{4 \, x^{4}} \]
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 \[ \int \frac {\mathrm {cosh}\left (c+d\,x\right )\,\left (b\,x^2+a\right )}{x^5} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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