Optimal. Leaf size=175 \[ \frac {1}{24} a^2 d^4 \cosh (c) \text {Chi}(d x)+\frac {1}{24} a^2 d^4 \sinh (c) \text {Shi}(d x)-\frac {a^2 d^3 \sinh (c+d x)}{24 x}-\frac {a^2 d^2 \cosh (c+d x)}{24 x^2}-\frac {a^2 \cosh (c+d x)}{4 x^4}-\frac {a^2 d \sinh (c+d x)}{12 x^3}+a b d^2 \cosh (c) \text {Chi}(d x)+a b d^2 \sinh (c) \text {Shi}(d x)-\frac {a b \cosh (c+d x)}{x^2}-\frac {a b d \sinh (c+d x)}{x}+b^2 \cosh (c) \text {Chi}(d x)+b^2 \sinh (c) \text {Shi}(d x) \]
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Rubi [A] time = 0.36, antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {5287, 3297, 3303, 3298, 3301} \[ \frac {1}{24} a^2 d^4 \cosh (c) \text {Chi}(d x)+\frac {1}{24} a^2 d^4 \sinh (c) \text {Shi}(d x)-\frac {a^2 d^2 \cosh (c+d x)}{24 x^2}-\frac {a^2 d^3 \sinh (c+d x)}{24 x}-\frac {a^2 d \sinh (c+d x)}{12 x^3}-\frac {a^2 \cosh (c+d x)}{4 x^4}+a b d^2 \cosh (c) \text {Chi}(d x)+a b d^2 \sinh (c) \text {Shi}(d x)-\frac {a b \cosh (c+d x)}{x^2}-\frac {a b d \sinh (c+d x)}{x}+b^2 \cosh (c) \text {Chi}(d x)+b^2 \sinh (c) \text {Shi}(d x) \]
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^2\right )^2 \cosh (c+d x)}{x^5} \, dx &=\int \left (\frac {a^2 \cosh (c+d x)}{x^5}+\frac {2 a b \cosh (c+d x)}{x^3}+\frac {b^2 \cosh (c+d x)}{x}\right ) \, dx\\ &=a^2 \int \frac {\cosh (c+d x)}{x^5} \, dx+(2 a b) \int \frac {\cosh (c+d x)}{x^3} \, dx+b^2 \int \frac {\cosh (c+d x)}{x} \, dx\\ &=-\frac {a^2 \cosh (c+d x)}{4 x^4}-\frac {a b \cosh (c+d x)}{x^2}+\frac {1}{4} \left (a^2 d\right ) \int \frac {\sinh (c+d x)}{x^4} \, dx+(a b d) \int \frac {\sinh (c+d x)}{x^2} \, dx+\left (b^2 \cosh (c)\right ) \int \frac {\cosh (d x)}{x} \, dx+\left (b^2 \sinh (c)\right ) \int \frac {\sinh (d x)}{x} \, dx\\ &=-\frac {a^2 \cosh (c+d x)}{4 x^4}-\frac {a b \cosh (c+d x)}{x^2}+b^2 \cosh (c) \text {Chi}(d x)-\frac {a^2 d \sinh (c+d x)}{12 x^3}-\frac {a b d \sinh (c+d x)}{x}+b^2 \sinh (c) \text {Shi}(d x)+\frac {1}{12} \left (a^2 d^2\right ) \int \frac {\cosh (c+d x)}{x^3} \, dx+\left (a b d^2\right ) \int \frac {\cosh (c+d x)}{x} \, dx\\ &=-\frac {a^2 \cosh (c+d x)}{4 x^4}-\frac {a b \cosh (c+d x)}{x^2}-\frac {a^2 d^2 \cosh (c+d x)}{24 x^2}+b^2 \cosh (c) \text {Chi}(d x)-\frac {a^2 d \sinh (c+d x)}{12 x^3}-\frac {a b d \sinh (c+d x)}{x}+b^2 \sinh (c) \text {Shi}(d x)+\frac {1}{24} \left (a^2 d^3\right ) \int \frac {\sinh (c+d x)}{x^2} \, dx+\left (a b d^2 \cosh (c)\right ) \int \frac {\cosh (d x)}{x} \, dx+\left (a b d^2 \sinh (c)\right ) \int \frac {\sinh (d x)}{x} \, dx\\ &=-\frac {a^2 \cosh (c+d x)}{4 x^4}-\frac {a b \cosh (c+d x)}{x^2}-\frac {a^2 d^2 \cosh (c+d x)}{24 x^2}+b^2 \cosh (c) \text {Chi}(d x)+a b d^2 \cosh (c) \text {Chi}(d x)-\frac {a^2 d \sinh (c+d x)}{12 x^3}-\frac {a b d \sinh (c+d x)}{x}-\frac {a^2 d^3 \sinh (c+d x)}{24 x}+b^2 \sinh (c) \text {Shi}(d x)+a b d^2 \sinh (c) \text {Shi}(d x)+\frac {1}{24} \left (a^2 d^4\right ) \int \frac {\cosh (c+d x)}{x} \, dx\\ &=-\frac {a^2 \cosh (c+d x)}{4 x^4}-\frac {a b \cosh (c+d x)}{x^2}-\frac {a^2 d^2 \cosh (c+d x)}{24 x^2}+b^2 \cosh (c) \text {Chi}(d x)+a b d^2 \cosh (c) \text {Chi}(d x)-\frac {a^2 d \sinh (c+d x)}{12 x^3}-\frac {a b d \sinh (c+d x)}{x}-\frac {a^2 d^3 \sinh (c+d x)}{24 x}+b^2 \sinh (c) \text {Shi}(d x)+a b d^2 \sinh (c) \text {Shi}(d x)+\frac {1}{24} \left (a^2 d^4 \cosh (c)\right ) \int \frac {\cosh (d x)}{x} \, dx+\frac {1}{24} \left (a^2 d^4 \sinh (c)\right ) \int \frac {\sinh (d x)}{x} \, dx\\ &=-\frac {a^2 \cosh (c+d x)}{4 x^4}-\frac {a b \cosh (c+d x)}{x^2}-\frac {a^2 d^2 \cosh (c+d x)}{24 x^2}+b^2 \cosh (c) \text {Chi}(d x)+a b d^2 \cosh (c) \text {Chi}(d x)+\frac {1}{24} a^2 d^4 \cosh (c) \text {Chi}(d x)-\frac {a^2 d \sinh (c+d x)}{12 x^3}-\frac {a b d \sinh (c+d x)}{x}-\frac {a^2 d^3 \sinh (c+d x)}{24 x}+b^2 \sinh (c) \text {Shi}(d x)+a b d^2 \sinh (c) \text {Shi}(d x)+\frac {1}{24} a^2 d^4 \sinh (c) \text {Shi}(d x)\\ \end {align*}
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Mathematica [A] time = 0.49, size = 124, normalized size = 0.71 \[ \frac {x^4 \cosh (c) \left (a^2 d^4+24 a b d^2+24 b^2\right ) \text {Chi}(d x)+x^4 \sinh (c) \left (a^2 d^4+24 a b d^2+24 b^2\right ) \text {Shi}(d x)-a \left (d x \left (a d^2 x^2+2 a+24 b x^2\right ) \sinh (c+d x)+\left (a d^2 x^2+6 a+24 b x^2\right ) \cosh (c+d x)\right )}{24 x^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 194, normalized size = 1.11 \[ -\frac {2 \, {\left ({\left (a^{2} d^{2} + 24 \, a b\right )} x^{2} + 6 \, a^{2}\right )} \cosh \left (d x + c\right ) - {\left ({\left (a^{2} d^{4} + 24 \, a b d^{2} + 24 \, b^{2}\right )} x^{4} {\rm Ei}\left (d x\right ) + {\left (a^{2} d^{4} + 24 \, a b d^{2} + 24 \, b^{2}\right )} x^{4} {\rm Ei}\left (-d x\right )\right )} \cosh \relax (c) + 2 \, {\left (2 \, a^{2} d x + {\left (a^{2} d^{3} + 24 \, a b d\right )} x^{3}\right )} \sinh \left (d x + c\right ) - {\left ({\left (a^{2} d^{4} + 24 \, a b d^{2} + 24 \, b^{2}\right )} x^{4} {\rm Ei}\left (d x\right ) - {\left (a^{2} d^{4} + 24 \, a b d^{2} + 24 \, b^{2}\right )} x^{4} {\rm Ei}\left (-d x\right )\right )} \sinh \relax (c)}{48 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 294, normalized size = 1.68 \[ \frac {a^{2} d^{4} x^{4} {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + a^{2} d^{4} x^{4} {\rm Ei}\left (d x\right ) e^{c} + 24 \, a b d^{2} x^{4} {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + 24 \, a b d^{2} x^{4} {\rm Ei}\left (d x\right ) e^{c} - a^{2} d^{3} x^{3} e^{\left (d x + c\right )} + a^{2} d^{3} x^{3} e^{\left (-d x - c\right )} + 24 \, b^{2} x^{4} {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + 24 \, b^{2} x^{4} {\rm Ei}\left (d x\right ) e^{c} - a^{2} d^{2} x^{2} e^{\left (d x + c\right )} - 24 \, a b d x^{3} e^{\left (d x + c\right )} - a^{2} d^{2} x^{2} e^{\left (-d x - c\right )} + 24 \, a b d x^{3} e^{\left (-d x - c\right )} - 2 \, a^{2} d x e^{\left (d x + c\right )} - 24 \, a b x^{2} e^{\left (d x + c\right )} + 2 \, a^{2} d x e^{\left (-d x - c\right )} - 24 \, a b x^{2} e^{\left (-d x - c\right )} - 6 \, a^{2} e^{\left (d x + c\right )} - 6 \, a^{2} e^{\left (-d x - c\right )}}{48 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.20, size = 291, normalized size = 1.66 \[ -\frac {b^{2} {\mathrm e}^{-c} \Ei \left (1, d x \right )}{2}+\frac {d^{3} a^{2} {\mathrm e}^{-d x -c}}{48 x}-\frac {d^{2} a^{2} {\mathrm e}^{-d x -c}}{48 x^{2}}+\frac {d \,a^{2} {\mathrm e}^{-d x -c}}{24 x^{3}}-\frac {a^{2} {\mathrm e}^{-d x -c}}{8 x^{4}}+\frac {d a b \,{\mathrm e}^{-d x -c}}{2 x}-\frac {a b \,{\mathrm e}^{-d x -c}}{2 x^{2}}-\frac {d^{2} a b \,{\mathrm e}^{-c} \Ei \left (1, d x \right )}{2}-\frac {d^{4} a^{2} {\mathrm e}^{-c} \Ei \left (1, d x \right )}{48}-\frac {a b \,{\mathrm e}^{d x +c}}{2 x^{2}}-\frac {d a b \,{\mathrm e}^{d x +c}}{2 x}-\frac {d^{2} a b \,{\mathrm e}^{c} \Ei \left (1, -d x \right )}{2}-\frac {a^{2} {\mathrm e}^{d x +c}}{8 x^{4}}-\frac {d \,a^{2} {\mathrm e}^{d x +c}}{24 x^{3}}-\frac {d^{2} a^{2} {\mathrm e}^{d x +c}}{48 x^{2}}-\frac {d^{3} a^{2} {\mathrm e}^{d x +c}}{48 x}-\frac {b^{2} {\mathrm e}^{c} \Ei \left (1, -d x \right )}{2}-\frac {d^{4} a^{2} {\mathrm e}^{c} \Ei \left (1, -d x \right )}{48} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 139, normalized size = 0.79 \[ \frac {1}{8} \, {\left ({\left (d^{3} e^{\left (-c\right )} \Gamma \left (-3, d x\right ) + d^{3} e^{c} \Gamma \left (-3, -d x\right )\right )} a^{2} + 4 \, {\left (d e^{\left (-c\right )} \Gamma \left (-1, d x\right ) + d e^{c} \Gamma \left (-1, -d x\right )\right )} a b - \frac {4 \, b^{2} \cosh \left (d x + c\right ) \log \left (x^{2}\right )}{d} + \frac {4 \, {\left ({\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + {\rm Ei}\left (d x\right ) e^{c}\right )} b^{2}}{d}\right )} d + \frac {1}{4} \, {\left (2 \, b^{2} \log \left (x^{2}\right ) - \frac {4 \, a b x^{2} + a^{2}}{x^{4}}\right )} \cosh \left (d x + c\right ) \]
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 \[ \int \frac {\mathrm {cosh}\left (c+d\,x\right )\,{\left (b\,x^2+a\right )}^2}{x^5} \,d x \]
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 \[ \int \frac {\left (a + b x^{2}\right )^{2} \cosh {\left (c + d x \right )}}{x^{5}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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