Optimal. Leaf size=121 \[ \frac{1}{2} a^2 d^2 \cosh (c) \text{Chi}(d x)+\frac{1}{2} a^2 d^2 \sinh (c) \text{Shi}(d x)-\frac{a^2 \cosh (c+d x)}{2 x^2}-\frac{a^2 d \sinh (c+d x)}{2 x}+2 a b d \sinh (c) \text{Chi}(d x)+2 a b d \cosh (c) \text{Shi}(d x)-\frac{2 a b \cosh (c+d x)}{x}+b^2 \cosh (c) \text{Chi}(d x)+b^2 \sinh (c) \text{Shi}(d x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.338688, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 14, 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} \[ \frac{1}{2} a^2 d^2 \cosh (c) \text{Chi}(d x)+\frac{1}{2} a^2 d^2 \sinh (c) \text{Shi}(d x)-\frac{a^2 \cosh (c+d x)}{2 x^2}-\frac{a^2 d \sinh (c+d x)}{2 x}+2 a b d \sinh (c) \text{Chi}(d x)+2 a b d \cosh (c) \text{Shi}(d x)-\frac{2 a b \cosh (c+d x)}{x}+b^2 \cosh (c) \text{Chi}(d x)+b^2 \sinh (c) \text{Shi}(d x) \]
Antiderivative was successfully verified.
[In]
[Out]
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^3} \, dx &=\int \left (\frac{a^2 \cosh (c+d x)}{x^3}+\frac{2 a b \cosh (c+d x)}{x^2}+\frac{b^2 \cosh (c+d x)}{x}\right ) \, dx\\ &=a^2 \int \frac{\cosh (c+d x)}{x^3} \, dx+(2 a b) \int \frac{\cosh (c+d x)}{x^2} \, dx+b^2 \int \frac{\cosh (c+d x)}{x} \, dx\\ &=-\frac{a^2 \cosh (c+d x)}{2 x^2}-\frac{2 a b \cosh (c+d x)}{x}+\frac{1}{2} \left (a^2 d\right ) \int \frac{\sinh (c+d x)}{x^2} \, dx+(2 a b d) \int \frac{\sinh (c+d x)}{x} \, dx+\left (b^2 \cosh (c)\right ) \int \frac{\cosh (d x)}{x} \, dx+\left (b^2 \sinh (c)\right ) \int \frac{\sinh (d x)}{x} \, dx\\ &=-\frac{a^2 \cosh (c+d x)}{2 x^2}-\frac{2 a b \cosh (c+d x)}{x}+b^2 \cosh (c) \text{Chi}(d x)-\frac{a^2 d \sinh (c+d x)}{2 x}+b^2 \sinh (c) \text{Shi}(d x)+\frac{1}{2} \left (a^2 d^2\right ) \int \frac{\cosh (c+d x)}{x} \, dx+(2 a b d \cosh (c)) \int \frac{\sinh (d x)}{x} \, dx+(2 a b d \sinh (c)) \int \frac{\cosh (d x)}{x} \, dx\\ &=-\frac{a^2 \cosh (c+d x)}{2 x^2}-\frac{2 a b \cosh (c+d x)}{x}+b^2 \cosh (c) \text{Chi}(d x)+2 a b d \text{Chi}(d x) \sinh (c)-\frac{a^2 d \sinh (c+d x)}{2 x}+2 a b d \cosh (c) \text{Shi}(d x)+b^2 \sinh (c) \text{Shi}(d x)+\frac{1}{2} \left (a^2 d^2 \cosh (c)\right ) \int \frac{\cosh (d x)}{x} \, dx+\frac{1}{2} \left (a^2 d^2 \sinh (c)\right ) \int \frac{\sinh (d x)}{x} \, dx\\ &=-\frac{a^2 \cosh (c+d x)}{2 x^2}-\frac{2 a b \cosh (c+d x)}{x}+b^2 \cosh (c) \text{Chi}(d x)+\frac{1}{2} a^2 d^2 \cosh (c) \text{Chi}(d x)+2 a b d \text{Chi}(d x) \sinh (c)-\frac{a^2 d \sinh (c+d x)}{2 x}+2 a b d \cosh (c) \text{Shi}(d x)+b^2 \sinh (c) \text{Shi}(d x)+\frac{1}{2} a^2 d^2 \sinh (c) \text{Shi}(d x)\\ \end{align*}
Mathematica [A] time = 0.373391, size = 93, normalized size = 0.77 \[ \frac{1}{2} \left (\text{Chi}(d x) \left (\cosh (c) \left (a^2 d^2+2 b^2\right )+4 a b d \sinh (c)\right )+\text{Shi}(d x) \left (\sinh (c) \left (a^2 d^2+2 b^2\right )+4 a b d \cosh (c)\right )-\frac{a ((a+4 b x) \cosh (c+d x)+a d x \sinh (c+d x))}{x^2}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.052, size = 181, normalized size = 1.5 \begin{align*} -{\frac{{d}^{2}{a}^{2}{{\rm e}^{-c}}{\it Ei} \left ( 1,dx \right ) }{4}}-{\frac{ab{{\rm e}^{-dx-c}}}{x}}+dab{{\rm e}^{-c}}{\it Ei} \left ( 1,dx \right ) -{\frac{{b}^{2}{{\rm e}^{-c}}{\it Ei} \left ( 1,dx \right ) }{2}}+{\frac{d{a}^{2}{{\rm e}^{-dx-c}}}{4\,x}}-{\frac{{a}^{2}{{\rm e}^{-dx-c}}}{4\,{x}^{2}}}-{\frac{ab{{\rm e}^{dx+c}}}{x}}-dab{{\rm e}^{c}}{\it Ei} \left ( 1,-dx \right ) -{\frac{{b}^{2}{{\rm e}^{c}}{\it Ei} \left ( 1,-dx \right ) }{2}}-{\frac{{d}^{2}{a}^{2}{{\rm e}^{c}}{\it Ei} \left ( 1,-dx \right ) }{4}}-{\frac{{{\rm e}^{dx+c}}{a}^{2}}{4\,{x}^{2}}}-{\frac{d{a}^{2}{{\rm e}^{dx+c}}}{4\,x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.37234, size = 170, normalized size = 1.4 \begin{align*} \frac{1}{4} \,{\left ({\left (d e^{\left (-c\right )} \Gamma \left (-1, d x\right ) + d e^{c} \Gamma \left (-1, -d x\right )\right )} a^{2} - 4 \,{\left ({\rm Ei}\left (-d x\right ) e^{\left (-c\right )} -{\rm Ei}\left (d x\right ) e^{c}\right )} a b - \frac{4 \, b^{2} \cosh \left (d x + c\right ) \log \left (x\right )}{d} + \frac{2 \,{\left ({\rm Ei}\left (-d x\right ) e^{\left (-c\right )} +{\rm Ei}\left (d x\right ) e^{c}\right )} b^{2}}{d}\right )} d + \frac{1}{2} \,{\left (2 \, b^{2} \log \left (x\right ) - \frac{4 \, a b x + a^{2}}{x^{2}}\right )} \cosh \left (d x + c\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.94994, size = 351, normalized size = 2.9 \begin{align*} -\frac{2 \, a^{2} d x \sinh \left (d x + c\right ) + 2 \,{\left (4 \, a b x + a^{2}\right )} \cosh \left (d x + c\right ) -{\left ({\left (a^{2} d^{2} + 4 \, a b d + 2 \, b^{2}\right )} x^{2}{\rm Ei}\left (d x\right ) +{\left (a^{2} d^{2} - 4 \, a b d + 2 \, b^{2}\right )} x^{2}{\rm Ei}\left (-d x\right )\right )} \cosh \left (c\right ) -{\left ({\left (a^{2} d^{2} + 4 \, a b d + 2 \, b^{2}\right )} x^{2}{\rm Ei}\left (d x\right ) -{\left (a^{2} d^{2} - 4 \, a b d + 2 \, b^{2}\right )} x^{2}{\rm Ei}\left (-d x\right )\right )} \sinh \left (c\right )}{4 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.17773, size = 244, normalized size = 2.02 \begin{align*} \frac{a^{2} d^{2} x^{2}{\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + a^{2} d^{2} x^{2}{\rm Ei}\left (d x\right ) e^{c} - 4 \, a b d x^{2}{\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + 4 \, a b d x^{2}{\rm Ei}\left (d x\right ) e^{c} + 2 \, b^{2} x^{2}{\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + 2 \, b^{2} x^{2}{\rm Ei}\left (d x\right ) e^{c} - a^{2} d x e^{\left (d x + c\right )} + a^{2} d x e^{\left (-d x - c\right )} - 4 \, a b x e^{\left (d x + c\right )} - 4 \, a b x e^{\left (-d x - c\right )} - a^{2} e^{\left (d x + c\right )} - a^{2} e^{\left (-d x - c\right )}}{4 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]