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 = 0.135611, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, 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 5448
Rule 3297
Rule 3303
Rule 3298
Rule 3301
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*}
Mathematica [A] time = 0.10449, 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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Maple [A] time = 0.055, size = 95, normalized size = 1.4 \begin{align*}{\frac{1}{16\,{x}^{2}}}+{\frac{b{{\rm e}^{-4\,bx-4\,a}}}{8\,x}}-{\frac{{{\rm e}^{-4\,bx-4\,a}}}{32\,{x}^{2}}}-{\frac{{b}^{2}{{\rm e}^{-4\,a}}{\it Ei} \left ( 1,4\,bx \right ) }{2}}-{\frac{{{\rm e}^{4\,bx+4\,a}}}{32\,{x}^{2}}}-{\frac{b{{\rm e}^{4\,bx+4\,a}}}{8\,x}}-{\frac{{b}^{2}{{\rm e}^{4\,a}}{\it Ei} \left ( 1,-4\,bx \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.20587, size = 49, normalized size = 0.73 \begin{align*} -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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.76163, size = 371, normalized size = 5.54 \begin{align*} -\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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sinh ^{2}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{x^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18005, size = 120, normalized size = 1.79 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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