Optimal. Leaf size=150 \[ \frac {1}{6} a^2 d^3 \sinh (c) \text {Chi}(d x)+\frac {1}{6} a^2 d^3 \cosh (c) \text {Shi}(d x)-\frac {a^2 d^2 \cosh (c+d x)}{6 x}-\frac {a^2 \cosh (c+d x)}{3 x^3}-\frac {a^2 d \sinh (c+d x)}{6 x^2}+2 a b \cosh (c) \text {Chi}(d x)+2 a b \sinh (c) \text {Shi}(d x)+\frac {2 b^2 \sinh (c+d x)}{d^3}-\frac {2 b^2 x \cosh (c+d x)}{d^2}+\frac {b^2 x^2 \sinh (c+d x)}{d} \]
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Rubi [A] time = 0.28, antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 7, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {5287, 3297, 3303, 3298, 3301, 3296, 2637} \[ \frac {1}{6} a^2 d^3 \sinh (c) \text {Chi}(d x)+\frac {1}{6} a^2 d^3 \cosh (c) \text {Shi}(d x)-\frac {a^2 d^2 \cosh (c+d x)}{6 x}-\frac {a^2 d \sinh (c+d x)}{6 x^2}-\frac {a^2 \cosh (c+d x)}{3 x^3}+2 a b \cosh (c) \text {Chi}(d x)+2 a b \sinh (c) \text {Shi}(d x)+\frac {2 b^2 \sinh (c+d x)}{d^3}-\frac {2 b^2 x \cosh (c+d x)}{d^2}+\frac {b^2 x^2 \sinh (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 2637
Rule 3296
Rule 3297
Rule 3298
Rule 3301
Rule 3303
Rule 5287
Rubi steps
\begin {align*} \int \frac {\left (a+b x^3\right )^2 \cosh (c+d x)}{x^4} \, dx &=\int \left (\frac {a^2 \cosh (c+d x)}{x^4}+\frac {2 a b \cosh (c+d x)}{x}+b^2 x^2 \cosh (c+d x)\right ) \, dx\\ &=a^2 \int \frac {\cosh (c+d x)}{x^4} \, dx+(2 a b) \int \frac {\cosh (c+d x)}{x} \, dx+b^2 \int x^2 \cosh (c+d x) \, dx\\ &=-\frac {a^2 \cosh (c+d x)}{3 x^3}+\frac {b^2 x^2 \sinh (c+d x)}{d}-\frac {\left (2 b^2\right ) \int x \sinh (c+d x) \, dx}{d}+\frac {1}{3} \left (a^2 d\right ) \int \frac {\sinh (c+d x)}{x^3} \, dx+(2 a b \cosh (c)) \int \frac {\cosh (d x)}{x} \, dx+(2 a b \sinh (c)) \int \frac {\sinh (d x)}{x} \, dx\\ &=-\frac {a^2 \cosh (c+d x)}{3 x^3}-\frac {2 b^2 x \cosh (c+d x)}{d^2}+2 a b \cosh (c) \text {Chi}(d x)-\frac {a^2 d \sinh (c+d x)}{6 x^2}+\frac {b^2 x^2 \sinh (c+d x)}{d}+2 a b \sinh (c) \text {Shi}(d x)+\frac {\left (2 b^2\right ) \int \cosh (c+d x) \, dx}{d^2}+\frac {1}{6} \left (a^2 d^2\right ) \int \frac {\cosh (c+d x)}{x^2} \, dx\\ &=-\frac {a^2 \cosh (c+d x)}{3 x^3}-\frac {a^2 d^2 \cosh (c+d x)}{6 x}-\frac {2 b^2 x \cosh (c+d x)}{d^2}+2 a b \cosh (c) \text {Chi}(d x)+\frac {2 b^2 \sinh (c+d x)}{d^3}-\frac {a^2 d \sinh (c+d x)}{6 x^2}+\frac {b^2 x^2 \sinh (c+d x)}{d}+2 a b \sinh (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 {a^2 d^2 \cosh (c+d x)}{6 x}-\frac {2 b^2 x \cosh (c+d x)}{d^2}+2 a b \cosh (c) \text {Chi}(d x)+\frac {2 b^2 \sinh (c+d x)}{d^3}-\frac {a^2 d \sinh (c+d x)}{6 x^2}+\frac {b^2 x^2 \sinh (c+d x)}{d}+2 a b \sinh (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 {a^2 d^2 \cosh (c+d x)}{6 x}-\frac {2 b^2 x \cosh (c+d x)}{d^2}+2 a b \cosh (c) \text {Chi}(d x)+\frac {1}{6} a^2 d^3 \text {Chi}(d x) \sinh (c)+\frac {2 b^2 \sinh (c+d x)}{d^3}-\frac {a^2 d \sinh (c+d x)}{6 x^2}+\frac {b^2 x^2 \sinh (c+d x)}{d}+\frac {1}{6} a^2 d^3 \cosh (c) \text {Shi}(d x)+2 a b \sinh (c) \text {Shi}(d x)\\ \end {align*}
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Mathematica [A] time = 0.63, size = 135, normalized size = 0.90 \[ \frac {1}{6} \left (-\frac {a^2 d^2 \cosh (c+d x)}{x}-\frac {2 a^2 \cosh (c+d x)}{x^3}-\frac {a^2 d \sinh (c+d x)}{x^2}+a \text {Chi}(d x) \left (a d^3 \sinh (c)+12 b \cosh (c)\right )+a \text {Shi}(d x) \left (a d^3 \cosh (c)+12 b \sinh (c)\right )+\frac {12 b^2 \sinh (c+d x)}{d^3}-\frac {12 b^2 x \cosh (c+d x)}{d^2}+\frac {6 b^2 x^2 \sinh (c+d x)}{d}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 187, normalized size = 1.25 \[ -\frac {2 \, {\left (a^{2} d^{5} x^{2} + 12 \, b^{2} d x^{4} + 2 \, a^{2} d^{3}\right )} \cosh \left (d x + c\right ) - {\left ({\left (a^{2} d^{6} + 12 \, a b d^{3}\right )} x^{3} {\rm Ei}\left (d x\right ) - {\left (a^{2} d^{6} - 12 \, a b d^{3}\right )} x^{3} {\rm Ei}\left (-d x\right )\right )} \cosh \relax (c) - 2 \, {\left (6 \, b^{2} d^{2} x^{5} - a^{2} d^{4} x + 12 \, b^{2} x^{3}\right )} \sinh \left (d x + c\right ) - {\left ({\left (a^{2} d^{6} + 12 \, a b d^{3}\right )} x^{3} {\rm Ei}\left (d x\right ) + {\left (a^{2} d^{6} - 12 \, a b d^{3}\right )} x^{3} {\rm Ei}\left (-d x\right )\right )} \sinh \relax (c)}{12 \, d^{3} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 279, normalized size = 1.86 \[ -\frac {a^{2} d^{6} x^{3} {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} - a^{2} d^{6} x^{3} {\rm Ei}\left (d x\right ) e^{c} + a^{2} d^{5} x^{2} e^{\left (d x + c\right )} - 6 \, b^{2} d^{2} x^{5} e^{\left (d x + c\right )} + a^{2} d^{5} x^{2} e^{\left (-d x - c\right )} + 6 \, b^{2} d^{2} x^{5} e^{\left (-d x - c\right )} - 12 \, a b d^{3} x^{3} {\rm Ei}\left (-d x\right ) e^{\left (-c\right )} - 12 \, a b d^{3} x^{3} {\rm Ei}\left (d x\right ) e^{c} + a^{2} d^{4} x e^{\left (d x + c\right )} + 12 \, b^{2} d x^{4} e^{\left (d x + c\right )} - a^{2} d^{4} x e^{\left (-d x - c\right )} + 12 \, b^{2} d x^{4} e^{\left (-d x - c\right )} + 2 \, a^{2} d^{3} e^{\left (d x + c\right )} - 12 \, b^{2} x^{3} e^{\left (d x + c\right )} + 2 \, a^{2} d^{3} e^{\left (-d x - c\right )} + 12 \, b^{2} x^{3} e^{\left (-d x - c\right )}}{12 \, d^{3} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.24, size = 261, normalized size = 1.74 \[ -\frac {b^{2} {\mathrm e}^{-d x -c} x^{2}}{2 d}-\frac {b^{2} {\mathrm e}^{-d x -c} x}{d^{2}}+\frac {d^{3} a^{2} {\mathrm e}^{-c} \Ei \left (1, d x \right )}{12}-\frac {b^{2} {\mathrm e}^{-d x -c}}{d^{3}}-\frac {d^{2} a^{2} {\mathrm e}^{-d x -c}}{12 x}+\frac {d \,a^{2} {\mathrm e}^{-d x -c}}{12 x^{2}}-\frac {a^{2} {\mathrm e}^{-d x -c}}{6 x^{3}}-a b \,{\mathrm e}^{-c} \Ei \left (1, d x \right )-\frac {a^{2} {\mathrm e}^{d x +c}}{6 x^{3}}+\frac {b^{2} {\mathrm e}^{d x +c} x^{2}}{2 d}-\frac {b^{2} {\mathrm e}^{d x +c} x}{d^{2}}-\frac {d^{3} a^{2} {\mathrm e}^{c} \Ei \left (1, -d x \right )}{12}+\frac {b^{2} {\mathrm e}^{d x +c}}{d^{3}}-\frac {d \,a^{2} {\mathrm e}^{d x +c}}{12 x^{2}}-\frac {d^{2} a^{2} {\mathrm e}^{d x +c}}{12 x}-a b \,{\mathrm e}^{c} \Ei \left (1, -d x \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 188, normalized size = 1.25 \[ \frac {1}{6} \, {\left ({\left (d^{2} e^{\left (-c\right )} \Gamma \left (-2, d x\right ) - d^{2} e^{c} \Gamma \left (-2, -d x\right )\right )} a^{2} - b^{2} {\left (\frac {{\left (d^{3} x^{3} e^{c} - 3 \, d^{2} x^{2} e^{c} + 6 \, d x e^{c} - 6 \, e^{c}\right )} e^{\left (d x\right )}}{d^{4}} + \frac {{\left (d^{3} x^{3} + 3 \, d^{2} x^{2} + 6 \, d x + 6\right )} e^{\left (-d x - c\right )}}{d^{4}}\right )} - \frac {4 \, a b \cosh \left (d x + c\right ) \log \left (x^{3}\right )}{d} + \frac {6 \, {\left ({\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + {\rm Ei}\left (d x\right ) e^{c}\right )} a b}{d}\right )} d + \frac {1}{3} \, {\left (b^{2} x^{3} + 2 \, a b \log \left (x^{3}\right ) - \frac {a^{2}}{x^{3}}\right )} \cosh \left (d x + c\right ) \]
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 \[ \int \frac {\mathrm {cosh}\left (c+d\,x\right )\,{\left (b\,x^3+a\right )}^2}{x^4} \,d x \]
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 \[ \int \frac {\left (a + b x^{3}\right )^{2} \cosh {\left (c + d x \right )}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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