Optimal. Leaf size=172 \[ \frac{1}{6} a^2 d^3 \sinh (c) \text{Chi}(d x)+\frac{1}{6} a^2 d^3 \cosh (c) \text{Shi}(d x)-\frac{a^2 d^2 \cosh (c+d x)}{6 x}-\frac{a^2 d \sinh (c+d x)}{6 x^2}-\frac{a^2 \cosh (c+d x)}{3 x^3}+a b d^2 \cosh (c) \text{Chi}(d x)+a b d^2 \sinh (c) \text{Shi}(d x)-\frac{a b \cosh (c+d x)}{x^2}-\frac{a b d \sinh (c+d x)}{x}+b^2 d \sinh (c) \text{Chi}(d x)+b^2 d \cosh (c) \text{Shi}(d x)-\frac{b^2 \cosh (c+d x)}{x} \]
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Rubi [A] time = 0.42641, antiderivative size = 172, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294, Rules used = {6742, 3297, 3303, 3298, 3301} \[ \frac{1}{6} a^2 d^3 \sinh (c) \text{Chi}(d x)+\frac{1}{6} a^2 d^3 \cosh (c) \text{Shi}(d x)-\frac{a^2 d^2 \cosh (c+d x)}{6 x}-\frac{a^2 d \sinh (c+d x)}{6 x^2}-\frac{a^2 \cosh (c+d x)}{3 x^3}+a b d^2 \cosh (c) \text{Chi}(d x)+a b d^2 \sinh (c) \text{Shi}(d x)-\frac{a b \cosh (c+d x)}{x^2}-\frac{a b d \sinh (c+d x)}{x}+b^2 d \sinh (c) \text{Chi}(d x)+b^2 d \cosh (c) \text{Shi}(d x)-\frac{b^2 \cosh (c+d x)}{x} \]
Antiderivative was successfully verified.
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Rule 6742
Rule 3297
Rule 3303
Rule 3298
Rule 3301
Rubi steps
\begin{align*} \int \frac{(a+b x)^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^3}+\frac{b^2 \cosh (c+d x)}{x^2}\right ) \, dx\\ &=a^2 \int \frac{\cosh (c+d x)}{x^4} \, dx+(2 a b) \int \frac{\cosh (c+d x)}{x^3} \, dx+b^2 \int \frac{\cosh (c+d x)}{x^2} \, dx\\ &=-\frac{a^2 \cosh (c+d x)}{3 x^3}-\frac{a b \cosh (c+d x)}{x^2}-\frac{b^2 \cosh (c+d x)}{x}+\frac{1}{3} \left (a^2 d\right ) \int \frac{\sinh (c+d x)}{x^3} \, dx+(a b d) \int \frac{\sinh (c+d x)}{x^2} \, dx+\left (b^2 d\right ) \int \frac{\sinh (c+d x)}{x} \, dx\\ &=-\frac{a^2 \cosh (c+d x)}{3 x^3}-\frac{a b \cosh (c+d x)}{x^2}-\frac{b^2 \cosh (c+d x)}{x}-\frac{a^2 d \sinh (c+d x)}{6 x^2}-\frac{a b d \sinh (c+d x)}{x}+\frac{1}{6} \left (a^2 d^2\right ) \int \frac{\cosh (c+d x)}{x^2} \, dx+\left (a b d^2\right ) \int \frac{\cosh (c+d x)}{x} \, dx+\left (b^2 d \cosh (c)\right ) \int \frac{\sinh (d x)}{x} \, dx+\left (b^2 d \sinh (c)\right ) \int \frac{\cosh (d x)}{x} \, dx\\ &=-\frac{a^2 \cosh (c+d x)}{3 x^3}-\frac{a b \cosh (c+d x)}{x^2}-\frac{b^2 \cosh (c+d x)}{x}-\frac{a^2 d^2 \cosh (c+d x)}{6 x}+b^2 d \text{Chi}(d x) \sinh (c)-\frac{a^2 d \sinh (c+d x)}{6 x^2}-\frac{a b d \sinh (c+d x)}{x}+b^2 d \cosh (c) \text{Shi}(d x)+\frac{1}{6} \left (a^2 d^3\right ) \int \frac{\sinh (c+d x)}{x} \, dx+\left (a b d^2 \cosh (c)\right ) \int \frac{\cosh (d x)}{x} \, dx+\left (a b d^2 \sinh (c)\right ) \int \frac{\sinh (d x)}{x} \, dx\\ &=-\frac{a^2 \cosh (c+d x)}{3 x^3}-\frac{a b \cosh (c+d x)}{x^2}-\frac{b^2 \cosh (c+d x)}{x}-\frac{a^2 d^2 \cosh (c+d x)}{6 x}+a b d^2 \cosh (c) \text{Chi}(d x)+b^2 d \text{Chi}(d x) \sinh (c)-\frac{a^2 d \sinh (c+d x)}{6 x^2}-\frac{a b d \sinh (c+d x)}{x}+b^2 d \cosh (c) \text{Shi}(d x)+a b d^2 \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 b \cosh (c+d x)}{x^2}-\frac{b^2 \cosh (c+d x)}{x}-\frac{a^2 d^2 \cosh (c+d x)}{6 x}+a b d^2 \cosh (c) \text{Chi}(d x)+b^2 d \text{Chi}(d x) \sinh (c)+\frac{1}{6} a^2 d^3 \text{Chi}(d x) \sinh (c)-\frac{a^2 d \sinh (c+d x)}{6 x^2}-\frac{a b d \sinh (c+d x)}{x}+b^2 d \cosh (c) \text{Shi}(d x)+\frac{1}{6} a^2 d^3 \cosh (c) \text{Shi}(d x)+a b d^2 \sinh (c) \text{Shi}(d x)\\ \end{align*}
Mathematica [A] time = 0.44756, size = 154, normalized size = 0.9 \[ -\frac{-d x^3 \text{Chi}(d x) \left (\sinh (c) \left (a^2 d^2+6 b^2\right )+6 a b d \cosh (c)\right )-d x^3 \text{Shi}(d x) \left (a^2 d^2 \cosh (c)+6 a b d \sinh (c)+6 b^2 \cosh (c)\right )+a^2 d^2 x^2 \cosh (c+d x)+a^2 d x \sinh (c+d x)+2 a^2 \cosh (c+d x)+6 a b d x^2 \sinh (c+d x)+6 a b x \cosh (c+d x)+6 b^2 x^2 \cosh (c+d x)}{6 x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.06, size = 287, normalized size = 1.7 \begin{align*}{\frac{{d}^{3}{a}^{2}{{\rm e}^{-c}}{\it Ei} \left ( 1,dx \right ) }{12}}-{\frac{{a}^{2}{d}^{2}{{\rm e}^{-dx-c}}}{12\,x}}-{\frac{{a}^{2}{{\rm e}^{-dx-c}}}{6\,{x}^{3}}}+{\frac{d{a}^{2}{{\rm e}^{-dx-c}}}{12\,{x}^{2}}}-{\frac{{b}^{2}{{\rm e}^{-dx-c}}}{2\,x}}+{\frac{bda{{\rm e}^{-dx-c}}}{2\,x}}-{\frac{ab{{\rm e}^{-dx-c}}}{2\,{x}^{2}}}-{\frac{{d}^{2}ab{{\rm e}^{-c}}{\it Ei} \left ( 1,dx \right ) }{2}}+{\frac{d{b}^{2}{{\rm e}^{-c}}{\it Ei} \left ( 1,dx \right ) }{2}}-{\frac{{{\rm e}^{dx+c}}{a}^{2}}{6\,{x}^{3}}}-{\frac{d{a}^{2}{{\rm e}^{dx+c}}}{12\,{x}^{2}}}-{\frac{{a}^{2}{d}^{2}{{\rm e}^{dx+c}}}{12\,x}}-{\frac{{d}^{2}ab{{\rm e}^{c}}{\it Ei} \left ( 1,-dx \right ) }{2}}-{\frac{ab{{\rm e}^{dx+c}}}{2\,{x}^{2}}}-{\frac{bda{{\rm e}^{dx+c}}}{2\,x}}-{\frac{{d}^{3}{a}^{2}{{\rm e}^{c}}{\it Ei} \left ( 1,-dx \right ) }{12}}-{\frac{{{\rm e}^{dx+c}}{b}^{2}}{2\,x}}-{\frac{d{b}^{2}{{\rm e}^{c}}{\it Ei} \left ( 1,-dx \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.43103, size = 158, normalized size = 0.92 \begin{align*} \frac{1}{6} \,{\left (a^{2} d^{2} e^{\left (-c\right )} \Gamma \left (-2, d x\right ) - a^{2} d^{2} e^{c} \Gamma \left (-2, -d x\right ) + 3 \, a b d e^{\left (-c\right )} \Gamma \left (-1, d x\right ) + 3 \, a b d e^{c} \Gamma \left (-1, -d x\right ) - 3 \, b^{2}{\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + 3 \, b^{2}{\rm Ei}\left (d x\right ) e^{c}\right )} d - \frac{{\left (3 \, b^{2} x^{2} + 3 \, a b x + a^{2}\right )} \cosh \left (d x + c\right )}{3 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.01559, size = 431, normalized size = 2.51 \begin{align*} -\frac{2 \,{\left (6 \, a b x +{\left (a^{2} d^{2} + 6 \, b^{2}\right )} x^{2} + 2 \, a^{2}\right )} \cosh \left (d x + c\right ) -{\left ({\left (a^{2} d^{3} + 6 \, a b d^{2} + 6 \, b^{2} d\right )} x^{3}{\rm Ei}\left (d x\right ) -{\left (a^{2} d^{3} - 6 \, a b d^{2} + 6 \, b^{2} d\right )} x^{3}{\rm Ei}\left (-d x\right )\right )} \cosh \left (c\right ) + 2 \,{\left (6 \, a b d x^{2} + a^{2} d x\right )} \sinh \left (d x + c\right ) -{\left ({\left (a^{2} d^{3} + 6 \, a b d^{2} + 6 \, b^{2} d\right )} x^{3}{\rm Ei}\left (d x\right ) +{\left (a^{2} d^{3} - 6 \, a b d^{2} + 6 \, b^{2} d\right )} x^{3}{\rm Ei}\left (-d x\right )\right )} \sinh \left (c\right )}{12 \, 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{\left (a + b x\right )^{2} \cosh{\left (c + d x \right )}}{x^{4}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15162, size = 385, normalized size = 2.24 \begin{align*} -\frac{a^{2} d^{3} x^{3}{\rm Ei}\left (-d x\right ) e^{\left (-c\right )} - a^{2} d^{3} x^{3}{\rm Ei}\left (d x\right ) e^{c} - 6 \, a b d^{2} x^{3}{\rm Ei}\left (-d x\right ) e^{\left (-c\right )} - 6 \, a b d^{2} x^{3}{\rm Ei}\left (d x\right ) e^{c} + 6 \, b^{2} d x^{3}{\rm Ei}\left (-d x\right ) e^{\left (-c\right )} - 6 \, b^{2} d x^{3}{\rm Ei}\left (d x\right ) e^{c} + a^{2} d^{2} x^{2} e^{\left (d x + c\right )} + a^{2} d^{2} x^{2} e^{\left (-d x - c\right )} + 6 \, a b d x^{2} e^{\left (d x + c\right )} - 6 \, a b d x^{2} e^{\left (-d x - c\right )} + a^{2} d x e^{\left (d x + c\right )} + 6 \, b^{2} x^{2} e^{\left (d x + c\right )} - a^{2} d x e^{\left (-d x - c\right )} + 6 \, b^{2} x^{2} e^{\left (-d x - c\right )} + 6 \, a b x e^{\left (d x + c\right )} + 6 \, a b x e^{\left (-d x - c\right )} + 2 \, a^{2} e^{\left (d x + c\right )} + 2 \, a^{2} e^{\left (-d x - c\right )}}{12 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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