Optimal. Leaf size=67 \[ b^2 \cosh (4 a) \text {Chi}(4 b x)+b^2 \sinh (4 a) \text {Shi}(4 b x)-\frac {\cosh (4 a+4 b x)}{16 x^2}-\frac {b \sinh (4 a+4 b x)}{4 x}+\frac {1}{16 x^2} \]
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Rubi [A] time = 0.14, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5448, 3297, 3303, 3298, 3301} \[ b^2 \cosh (4 a) \text {Chi}(4 b x)+b^2 \sinh (4 a) \text {Shi}(4 b x)-\frac {\cosh (4 a+4 b x)}{16 x^2}-\frac {b \sinh (4 a+4 b x)}{4 x}+\frac {1}{16 x^2} \]
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 ^2(a+b x)}{x^3} \, dx &=\int \left (-\frac {1}{8 x^3}+\frac {\cosh (4 a+4 b x)}{8 x^3}\right ) \, dx\\ &=\frac {1}{16 x^2}+\frac {1}{8} \int \frac {\cosh (4 a+4 b x)}{x^3} \, dx\\ &=\frac {1}{16 x^2}-\frac {\cosh (4 a+4 b x)}{16 x^2}+\frac {1}{4} b \int \frac {\sinh (4 a+4 b x)}{x^2} \, dx\\ &=\frac {1}{16 x^2}-\frac {\cosh (4 a+4 b x)}{16 x^2}-\frac {b \sinh (4 a+4 b x)}{4 x}+b^2 \int \frac {\cosh (4 a+4 b x)}{x} \, dx\\ &=\frac {1}{16 x^2}-\frac {\cosh (4 a+4 b x)}{16 x^2}-\frac {b \sinh (4 a+4 b x)}{4 x}+\left (b^2 \cosh (4 a)\right ) \int \frac {\cosh (4 b x)}{x} \, dx+\left (b^2 \sinh (4 a)\right ) \int \frac {\sinh (4 b x)}{x} \, dx\\ &=\frac {1}{16 x^2}-\frac {\cosh (4 a+4 b x)}{16 x^2}+b^2 \cosh (4 a) \text {Chi}(4 b x)-\frac {b \sinh (4 a+4 b x)}{4 x}+b^2 \sinh (4 a) \text {Shi}(4 b x)\\ \end {align*}
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Mathematica [A] time = 0.11, size = 65, normalized size = 0.97 \[ \frac {16 b^2 x^2 \cosh (4 a) \text {Chi}(4 b x)+16 b^2 x^2 \sinh (4 a) \text {Shi}(4 b x)-4 b x \sinh (4 (a+b x))-\cosh (4 (a+b x))+1}{16 x^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.46, size = 140, normalized size = 2.09 \[ -\frac {16 \, b x \cosh \left (b x + a\right )^{3} \sinh \left (b x + a\right ) + 16 \, b x \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + \cosh \left (b x + a\right )^{4} + 6 \, \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{2} + \sinh \left (b x + a\right )^{4} - 8 \, {\left (b^{2} x^{2} {\rm Ei}\left (4 \, b x\right ) + b^{2} x^{2} {\rm Ei}\left (-4 \, b x\right )\right )} \cosh \left (4 \, a\right ) - 8 \, {\left (b^{2} x^{2} {\rm Ei}\left (4 \, b x\right ) - b^{2} x^{2} {\rm Ei}\left (-4 \, b x\right )\right )} \sinh \left (4 \, a\right ) - 1}{16 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 89, normalized size = 1.33 \[ \frac {16 \, b^{2} x^{2} {\rm Ei}\left (4 \, b x\right ) e^{\left (4 \, a\right )} + 16 \, b^{2} x^{2} {\rm Ei}\left (-4 \, b x\right ) e^{\left (-4 \, a\right )} - 4 \, b x e^{\left (4 \, b x + 4 \, a\right )} + 4 \, b x e^{\left (-4 \, b x - 4 \, a\right )} - e^{\left (4 \, b x + 4 \, a\right )} - e^{\left (-4 \, b x - 4 \, a\right )} + 2}{32 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.35, size = 95, normalized size = 1.42 \[ \frac {1}{16 x^{2}}+\frac {b \,{\mathrm e}^{-4 b x -4 a}}{8 x}-\frac {{\mathrm e}^{-4 b x -4 a}}{32 x^{2}}-\frac {b^{2} {\mathrm e}^{-4 a} \Ei \left (1, 4 b x \right )}{2}-\frac {{\mathrm e}^{4 b x +4 a}}{32 x^{2}}-\frac {b \,{\mathrm e}^{4 b x +4 a}}{8 x}-\frac {b^{2} {\mathrm e}^{4 a} \Ei \left (1, -4 b x \right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 36, normalized size = 0.54 \[ -b^{2} e^{\left (-4 \, a\right )} \Gamma \left (-2, 4 \, b x\right ) - b^{2} e^{\left (4 \, a\right )} \Gamma \left (-2, -4 \, b x\right ) + \frac {1}{16 \, x^{2}} \]
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 \[ \int \frac {{\mathrm {cosh}\left (a+b\,x\right )}^2\,{\mathrm {sinh}\left (a+b\,x\right )}^2}{x^3} \,d x \]
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 \[ \int \frac {\sinh ^{2}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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