Optimal. Leaf size=154 \[ \frac {1}{24} b^3 \cosh (a) \text {Chi}(b x)+\frac {9}{8} b^3 \cosh (3 a) \text {Chi}(3 b x)+\frac {1}{24} b^3 \sinh (a) \text {Shi}(b x)+\frac {9}{8} b^3 \sinh (3 a) \text {Shi}(3 b x)-\frac {b^2 \sinh (a+b x)}{24 x}-\frac {3 b^2 \sinh (3 a+3 b x)}{8 x}-\frac {\sinh (a+b x)}{12 x^3}-\frac {\sinh (3 a+3 b x)}{12 x^3}-\frac {b \cosh (a+b x)}{24 x^2}-\frac {b \cosh (3 a+3 b x)}{8 x^2} \]
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Rubi [A] time = 0.29, antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 5, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {5448, 3297, 3303, 3298, 3301} \[ \frac {1}{24} b^3 \cosh (a) \text {Chi}(b x)+\frac {9}{8} b^3 \cosh (3 a) \text {Chi}(3 b x)+\frac {1}{24} b^3 \sinh (a) \text {Shi}(b x)+\frac {9}{8} b^3 \sinh (3 a) \text {Shi}(3 b x)-\frac {b^2 \sinh (a+b x)}{24 x}-\frac {3 b^2 \sinh (3 a+3 b x)}{8 x}-\frac {\sinh (a+b x)}{12 x^3}-\frac {\sinh (3 a+3 b x)}{12 x^3}-\frac {b \cosh (a+b x)}{24 x^2}-\frac {b \cosh (3 a+3 b x)}{8 x^2} \]
Antiderivative was successfully verified.
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Rule 3297
Rule 3298
Rule 3301
Rule 3303
Rule 5448
Rubi steps
\begin {align*} \int \frac {\cosh ^2(a+b x) \sinh (a+b x)}{x^4} \, dx &=\int \left (\frac {\sinh (a+b x)}{4 x^4}+\frac {\sinh (3 a+3 b x)}{4 x^4}\right ) \, dx\\ &=\frac {1}{4} \int \frac {\sinh (a+b x)}{x^4} \, dx+\frac {1}{4} \int \frac {\sinh (3 a+3 b x)}{x^4} \, dx\\ &=-\frac {\sinh (a+b x)}{12 x^3}-\frac {\sinh (3 a+3 b x)}{12 x^3}+\frac {1}{12} b \int \frac {\cosh (a+b x)}{x^3} \, dx+\frac {1}{4} b \int \frac {\cosh (3 a+3 b x)}{x^3} \, dx\\ &=-\frac {b \cosh (a+b x)}{24 x^2}-\frac {b \cosh (3 a+3 b x)}{8 x^2}-\frac {\sinh (a+b x)}{12 x^3}-\frac {\sinh (3 a+3 b x)}{12 x^3}+\frac {1}{24} b^2 \int \frac {\sinh (a+b x)}{x^2} \, dx+\frac {1}{8} \left (3 b^2\right ) \int \frac {\sinh (3 a+3 b x)}{x^2} \, dx\\ &=-\frac {b \cosh (a+b x)}{24 x^2}-\frac {b \cosh (3 a+3 b x)}{8 x^2}-\frac {\sinh (a+b x)}{12 x^3}-\frac {b^2 \sinh (a+b x)}{24 x}-\frac {\sinh (3 a+3 b x)}{12 x^3}-\frac {3 b^2 \sinh (3 a+3 b x)}{8 x}+\frac {1}{24} b^3 \int \frac {\cosh (a+b x)}{x} \, dx+\frac {1}{8} \left (9 b^3\right ) \int \frac {\cosh (3 a+3 b x)}{x} \, dx\\ &=-\frac {b \cosh (a+b x)}{24 x^2}-\frac {b \cosh (3 a+3 b x)}{8 x^2}-\frac {\sinh (a+b x)}{12 x^3}-\frac {b^2 \sinh (a+b x)}{24 x}-\frac {\sinh (3 a+3 b x)}{12 x^3}-\frac {3 b^2 \sinh (3 a+3 b x)}{8 x}+\frac {1}{24} \left (b^3 \cosh (a)\right ) \int \frac {\cosh (b x)}{x} \, dx+\frac {1}{8} \left (9 b^3 \cosh (3 a)\right ) \int \frac {\cosh (3 b x)}{x} \, dx+\frac {1}{24} \left (b^3 \sinh (a)\right ) \int \frac {\sinh (b x)}{x} \, dx+\frac {1}{8} \left (9 b^3 \sinh (3 a)\right ) \int \frac {\sinh (3 b x)}{x} \, dx\\ &=-\frac {b \cosh (a+b x)}{24 x^2}-\frac {b \cosh (3 a+3 b x)}{8 x^2}+\frac {1}{24} b^3 \cosh (a) \text {Chi}(b x)+\frac {9}{8} b^3 \cosh (3 a) \text {Chi}(3 b x)-\frac {\sinh (a+b x)}{12 x^3}-\frac {b^2 \sinh (a+b x)}{24 x}-\frac {\sinh (3 a+3 b x)}{12 x^3}-\frac {3 b^2 \sinh (3 a+3 b x)}{8 x}+\frac {1}{24} b^3 \sinh (a) \text {Shi}(b x)+\frac {9}{8} b^3 \sinh (3 a) \text {Shi}(3 b x)\\ \end {align*}
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Mathematica [A] time = 0.36, size = 138, normalized size = 0.90 \[ -\frac {-b^3 x^3 \cosh (a) \text {Chi}(b x)-27 b^3 x^3 \cosh (3 a) \text {Chi}(3 b x)-b^3 x^3 \sinh (a) \text {Shi}(b x)-27 b^3 x^3 \sinh (3 a) \text {Shi}(3 b x)+b^2 x^2 \sinh (a+b x)+9 b^2 x^2 \sinh (3 (a+b x))+2 \sinh (a+b x)+2 \sinh (3 (a+b x))+b x \cosh (a+b x)+3 b x \cosh (3 (a+b x))}{24 x^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 223, normalized size = 1.45 \[ -\frac {6 \, b x \cosh \left (b x + a\right )^{3} + 18 \, b x \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + 2 \, {\left (9 \, b^{2} x^{2} + 2\right )} \sinh \left (b x + a\right )^{3} + 2 \, b x \cosh \left (b x + a\right ) - 27 \, {\left (b^{3} x^{3} {\rm Ei}\left (3 \, b x\right ) + b^{3} x^{3} {\rm Ei}\left (-3 \, b x\right )\right )} \cosh \left (3 \, a\right ) - {\left (b^{3} x^{3} {\rm Ei}\left (b x\right ) + b^{3} x^{3} {\rm Ei}\left (-b x\right )\right )} \cosh \relax (a) + 2 \, {\left (b^{2} x^{2} + 3 \, {\left (9 \, b^{2} x^{2} + 2\right )} \cosh \left (b x + a\right )^{2} + 2\right )} \sinh \left (b x + a\right ) - 27 \, {\left (b^{3} x^{3} {\rm Ei}\left (3 \, b x\right ) - b^{3} x^{3} {\rm Ei}\left (-3 \, b x\right )\right )} \sinh \left (3 \, a\right ) - {\left (b^{3} x^{3} {\rm Ei}\left (b x\right ) - b^{3} x^{3} {\rm Ei}\left (-b x\right )\right )} \sinh \relax (a)}{48 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 223, normalized size = 1.45 \[ \frac {27 \, b^{3} x^{3} {\rm Ei}\left (3 \, b x\right ) e^{\left (3 \, a\right )} + b^{3} x^{3} {\rm Ei}\left (-b x\right ) e^{\left (-a\right )} + 27 \, b^{3} x^{3} {\rm Ei}\left (-3 \, b x\right ) e^{\left (-3 \, a\right )} + b^{3} x^{3} {\rm Ei}\left (b x\right ) e^{a} - 9 \, b^{2} x^{2} e^{\left (3 \, b x + 3 \, a\right )} - b^{2} x^{2} e^{\left (b x + a\right )} + b^{2} x^{2} e^{\left (-b x - a\right )} + 9 \, b^{2} x^{2} e^{\left (-3 \, b x - 3 \, a\right )} - 3 \, b x e^{\left (3 \, b x + 3 \, a\right )} - b x e^{\left (b x + a\right )} - b x e^{\left (-b x - a\right )} - 3 \, b x e^{\left (-3 \, b x - 3 \, a\right )} - 2 \, e^{\left (3 \, b x + 3 \, a\right )} - 2 \, e^{\left (b x + a\right )} + 2 \, e^{\left (-b x - a\right )} + 2 \, e^{\left (-3 \, b x - 3 \, a\right )}}{48 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.37, size = 234, normalized size = 1.52 \[ \frac {3 b^{2} {\mathrm e}^{-3 b x -3 a}}{16 x}-\frac {b \,{\mathrm e}^{-3 b x -3 a}}{16 x^{2}}+\frac {{\mathrm e}^{-3 b x -3 a}}{24 x^{3}}-\frac {9 b^{3} {\mathrm e}^{-3 a} \Ei \left (1, 3 b x \right )}{16}+\frac {b^{2} {\mathrm e}^{-b x -a}}{48 x}-\frac {b \,{\mathrm e}^{-b x -a}}{48 x^{2}}+\frac {{\mathrm e}^{-b x -a}}{24 x^{3}}-\frac {b^{3} {\mathrm e}^{-a} \Ei \left (1, b x \right )}{48}-\frac {{\mathrm e}^{b x +a}}{24 x^{3}}-\frac {b \,{\mathrm e}^{b x +a}}{48 x^{2}}-\frac {b^{2} {\mathrm e}^{b x +a}}{48 x}-\frac {b^{3} {\mathrm e}^{a} \Ei \left (1, -b x \right )}{48}-\frac {{\mathrm e}^{3 b x +3 a}}{24 x^{3}}-\frac {b \,{\mathrm e}^{3 b x +3 a}}{16 x^{2}}-\frac {3 b^{2} {\mathrm e}^{3 b x +3 a}}{16 x}-\frac {9 b^{3} {\mathrm e}^{3 a} \Ei \left (1, -3 b x \right )}{16} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 58, normalized size = 0.38 \[ \frac {27}{8} \, b^{3} e^{\left (-3 \, a\right )} \Gamma \left (-3, 3 \, b x\right ) + \frac {1}{8} \, b^{3} e^{\left (-a\right )} \Gamma \left (-3, b x\right ) + \frac {1}{8} \, b^{3} e^{a} \Gamma \left (-3, -b x\right ) + \frac {27}{8} \, b^{3} e^{\left (3 \, a\right )} \Gamma \left (-3, -3 \, b x\right ) \]
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 \[ \int \frac {{\mathrm {cosh}\left (a+b\,x\right )}^2\,\mathrm {sinh}\left (a+b\,x\right )}{x^4} \,d x \]
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 \[ \int \frac {\sinh {\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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