Optimal. Leaf size=133 \[ -\frac {a^2 \cosh (c+d x)}{3 x^3}-\frac {2 a b \cosh (c+d x)}{x}-\frac {a^2 d^2 \cosh (c+d x)}{6 x}+2 a b d \text {Chi}(d x) \sinh (c)+\frac {1}{6} a^2 d^3 \text {Chi}(d x) \sinh (c)+\frac {b^2 \sinh (c+d x)}{d}-\frac {a^2 d \sinh (c+d x)}{6 x^2}+2 a b d \cosh (c) \text {Shi}(d x)+\frac {1}{6} a^2 d^3 \cosh (c) \text {Shi}(d x) \]
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Rubi [A]
time = 0.18, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {5395, 2717,
3378, 3384, 3379, 3382} \begin {gather*} \frac {1}{6} a^2 d^3 \sinh (c) \text {Chi}(d x)+\frac {1}{6} a^2 d^3 \cosh (c) \text {Shi}(d x)-\frac {a^2 d^2 \cosh (c+d x)}{6 x}-\frac {a^2 \cosh (c+d x)}{3 x^3}-\frac {a^2 d \sinh (c+d x)}{6 x^2}+2 a b d \sinh (c) \text {Chi}(d x)+2 a b d \cosh (c) \text {Shi}(d x)-\frac {2 a b \cosh (c+d x)}{x}+\frac {b^2 \sinh (c+d x)}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2717
Rule 3378
Rule 3379
Rule 3382
Rule 3384
Rule 5395
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^2 \cosh (c+d x)}{x^4} \, dx &=\int \left (b^2 \cosh (c+d x)+\frac {a^2 \cosh (c+d x)}{x^4}+\frac {2 a b \cosh (c+d x)}{x^2}\right ) \, dx\\ &=a^2 \int \frac {\cosh (c+d x)}{x^4} \, dx+(2 a b) \int \frac {\cosh (c+d x)}{x^2} \, dx+b^2 \int \cosh (c+d x) \, dx\\ &=-\frac {a^2 \cosh (c+d x)}{3 x^3}-\frac {2 a b \cosh (c+d x)}{x}+\frac {b^2 \sinh (c+d x)}{d}+\frac {1}{3} \left (a^2 d\right ) \int \frac {\sinh (c+d x)}{x^3} \, dx+(2 a b d) \int \frac {\sinh (c+d x)}{x} \, dx\\ &=-\frac {a^2 \cosh (c+d x)}{3 x^3}-\frac {2 a b \cosh (c+d x)}{x}+\frac {b^2 \sinh (c+d x)}{d}-\frac {a^2 d \sinh (c+d x)}{6 x^2}+\frac {1}{6} \left (a^2 d^2\right ) \int \frac {\cosh (c+d x)}{x^2} \, dx+(2 a b d \cosh (c)) \int \frac {\sinh (d x)}{x} \, dx+(2 a b d \sinh (c)) \int \frac {\cosh (d x)}{x} \, dx\\ &=-\frac {a^2 \cosh (c+d x)}{3 x^3}-\frac {2 a b \cosh (c+d x)}{x}-\frac {a^2 d^2 \cosh (c+d x)}{6 x}+2 a b d \text {Chi}(d x) \sinh (c)+\frac {b^2 \sinh (c+d x)}{d}-\frac {a^2 d \sinh (c+d x)}{6 x^2}+2 a b d \cosh (c) \text {Shi}(d x)+\frac {1}{6} \left (a^2 d^3\right ) \int \frac {\sinh (c+d x)}{x} \, dx\\ &=-\frac {a^2 \cosh (c+d x)}{3 x^3}-\frac {2 a b \cosh (c+d x)}{x}-\frac {a^2 d^2 \cosh (c+d x)}{6 x}+2 a b d \text {Chi}(d x) \sinh (c)+\frac {b^2 \sinh (c+d x)}{d}-\frac {a^2 d \sinh (c+d x)}{6 x^2}+2 a b d \cosh (c) \text {Shi}(d x)+\frac {1}{6} \left (a^2 d^3 \cosh (c)\right ) \int \frac {\sinh (d x)}{x} \, dx+\frac {1}{6} \left (a^2 d^3 \sinh (c)\right ) \int \frac {\cosh (d x)}{x} \, dx\\ &=-\frac {a^2 \cosh (c+d x)}{3 x^3}-\frac {2 a b \cosh (c+d x)}{x}-\frac {a^2 d^2 \cosh (c+d x)}{6 x}+2 a b d \text {Chi}(d x) \sinh (c)+\frac {1}{6} a^2 d^3 \text {Chi}(d x) \sinh (c)+\frac {b^2 \sinh (c+d x)}{d}-\frac {a^2 d \sinh (c+d x)}{6 x^2}+2 a b d \cosh (c) \text {Shi}(d x)+\frac {1}{6} a^2 d^3 \cosh (c) \text {Shi}(d x)\\ \end {align*}
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Mathematica [A]
time = 0.28, size = 114, normalized size = 0.86 \begin {gather*} \frac {1}{6} \left (-\frac {2 a^2 \cosh (c+d x)}{x^3}-\frac {12 a b \cosh (c+d x)}{x}-\frac {a^2 d^2 \cosh (c+d x)}{x}+a d \left (12 b+a d^2\right ) \text {Chi}(d x) \sinh (c)+\frac {6 b^2 \sinh (c+d x)}{d}-\frac {a^2 d \sinh (c+d x)}{x^2}+a d \left (12 b+a d^2\right ) \cosh (c) \text {Shi}(d x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.83, size = 222, normalized size = 1.67
method | result | size |
risch | \(-\frac {b^{2} {\mathrm e}^{-d x -c}}{2 d}+\frac {d^{3} a^{2} {\mathrm e}^{-c} \expIntegral \left (1, d x \right )}{12}-\frac {d^{2} a^{2} {\mathrm e}^{-d x -c}}{12 x}+\frac {d \,a^{2} {\mathrm e}^{-d x -c}}{12 x^{2}}-\frac {a^{2} {\mathrm e}^{-d x -c}}{6 x^{3}}-\frac {a b \,{\mathrm e}^{-d x -c}}{x}+d a b \,{\mathrm e}^{-c} \expIntegral \left (1, d x \right )-\frac {d^{3} a^{2} {\mathrm e}^{c} \expIntegral \left (1, -d x \right )}{12}-\frac {a^{2} {\mathrm e}^{d x +c}}{6 x^{3}}+\frac {{\mathrm e}^{d x +c} b^{2}}{2 d}-\frac {a b \,{\mathrm e}^{d x +c}}{x}-d a b \,{\mathrm e}^{c} \expIntegral \left (1, -d x \right )-\frac {d \,a^{2} {\mathrm e}^{d x +c}}{12 x^{2}}-\frac {d^{2} a^{2} {\mathrm e}^{d x +c}}{12 x}\) | \(222\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.33, size = 135, normalized size = 1.02 \begin {gather*} \frac {1}{6} \, {\left (a^{2} d^{2} e^{\left (-c\right )} \Gamma \left (-2, d x\right ) - a^{2} d^{2} e^{c} \Gamma \left (-2, -d x\right ) - 6 \, a b {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + 6 \, a b {\rm Ei}\left (d x\right ) e^{c} - \frac {3 \, {\left (d x e^{c} - e^{c}\right )} b^{2} e^{\left (d x\right )}}{d^{2}} - \frac {3 \, {\left (d x + 1\right )} b^{2} e^{\left (-d x - c\right )}}{d^{2}}\right )} d + \frac {1}{3} \, {\left (3 \, b^{2} x - \frac {6 \, a b x^{2} + a^{2}}{x^{3}}\right )} \cosh \left (d x + c\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 171, normalized size = 1.29 \begin {gather*} -\frac {2 \, {\left (2 \, a^{2} d + {\left (a^{2} d^{3} + 12 \, a b d\right )} x^{2}\right )} \cosh \left (d x + c\right ) - {\left ({\left (a^{2} d^{4} + 12 \, a b d^{2}\right )} x^{3} {\rm Ei}\left (d x\right ) - {\left (a^{2} d^{4} + 12 \, a b d^{2}\right )} x^{3} {\rm Ei}\left (-d x\right )\right )} \cosh \left (c\right ) + 2 \, {\left (a^{2} d^{2} x - 6 \, b^{2} x^{3}\right )} \sinh \left (d x + c\right ) - {\left ({\left (a^{2} d^{4} + 12 \, a b d^{2}\right )} x^{3} {\rm Ei}\left (d x\right ) + {\left (a^{2} d^{4} + 12 \, a b d^{2}\right )} x^{3} {\rm Ei}\left (-d x\right )\right )} \sinh \left (c\right )}{12 \, d x^{3}} \end {gather*}
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 \begin {gather*} \int \frac {\left (a + b x^{2}\right )^{2} \cosh {\left (c + d x \right )}}{x^{4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 236, normalized size = 1.77 \begin {gather*} -\frac {a^{2} d^{4} x^{3} {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} - a^{2} d^{4} x^{3} {\rm Ei}\left (d x\right ) e^{c} + 12 \, a b d^{2} x^{3} {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} - 12 \, a b d^{2} x^{3} {\rm Ei}\left (d x\right ) e^{c} + a^{2} d^{3} x^{2} e^{\left (d x + c\right )} + a^{2} d^{3} x^{2} e^{\left (-d x - c\right )} + a^{2} d^{2} x e^{\left (d x + c\right )} + 12 \, a b d x^{2} e^{\left (d x + c\right )} - 6 \, b^{2} x^{3} e^{\left (d x + c\right )} - a^{2} d^{2} x e^{\left (-d x - c\right )} + 12 \, a b d x^{2} e^{\left (-d x - c\right )} + 6 \, b^{2} x^{3} e^{\left (-d x - c\right )} + 2 \, a^{2} d e^{\left (d x + c\right )} + 2 \, a^{2} d e^{\left (-d x - c\right )}}{12 \, d x^{3}} \end {gather*}
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 \begin {gather*} \int \frac {\mathrm {cosh}\left (c+d\,x\right )\,{\left (b\,x^2+a\right )}^2}{x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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