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 = 0.289493, antiderivative size = 154, normalized size of antiderivative = 1., 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 5448
Rule 3297
Rule 3303
Rule 3298
Rule 3301
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*}
Mathematica [A] time = 0.349048, size = 138, normalized size = 0.9 \[ -\frac{-b^3 x^3 \cosh (a) \text{Chi}(b x)-27 b^3 x^3 \cosh (3 a) \text{Chi}(3 b x)-b^3 x^3 \sinh (a) \text{Shi}(b x)-27 b^3 x^3 \sinh (3 a) \text{Shi}(3 b x)+b^2 x^2 \sinh (a+b x)+9 b^2 x^2 \sinh (3 (a+b x))+2 \sinh (a+b x)+2 \sinh (3 (a+b x))+b x \cosh (a+b x)+3 b x \cosh (3 (a+b x))}{24 x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.076, size = 234, normalized size = 1.5 \begin{align*}{\frac{3\,{b}^{2}{{\rm e}^{-3\,bx-3\,a}}}{16\,x}}-{\frac{b{{\rm e}^{-3\,bx-3\,a}}}{16\,{x}^{2}}}+{\frac{{{\rm e}^{-3\,bx-3\,a}}}{24\,{x}^{3}}}-{\frac{9\,{b}^{3}{{\rm e}^{-3\,a}}{\it Ei} \left ( 1,3\,bx \right ) }{16}}+{\frac{{b}^{2}{{\rm e}^{-bx-a}}}{48\,x}}-{\frac{b{{\rm e}^{-bx-a}}}{48\,{x}^{2}}}+{\frac{{{\rm e}^{-bx-a}}}{24\,{x}^{3}}}-{\frac{{b}^{3}{{\rm e}^{-a}}{\it Ei} \left ( 1,bx \right ) }{48}}-{\frac{{{\rm e}^{bx+a}}}{24\,{x}^{3}}}-{\frac{b{{\rm e}^{bx+a}}}{48\,{x}^{2}}}-{\frac{{b}^{2}{{\rm e}^{bx+a}}}{48\,x}}-{\frac{{b}^{3}{{\rm e}^{a}}{\it Ei} \left ( 1,-bx \right ) }{48}}-{\frac{{{\rm e}^{3\,bx+3\,a}}}{24\,{x}^{3}}}-{\frac{b{{\rm e}^{3\,bx+3\,a}}}{16\,{x}^{2}}}-{\frac{3\,{b}^{2}{{\rm e}^{3\,bx+3\,a}}}{16\,x}}-{\frac{9\,{b}^{3}{{\rm e}^{3\,a}}{\it Ei} \left ( 1,-3\,bx \right ) }{16}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.26281, size = 78, normalized size = 0.51 \begin{align*} \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.75587, size = 548, normalized size = 3.56 \begin{align*} -\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 \left (a\right ) + 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 \left (a\right )}{48 \, x^{3}} \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{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{x^{4}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16996, size = 301, normalized size = 1.95 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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