Optimal. Leaf size=143 \[ a^2 d \sinh (c) \text{Chi}(d x)+a^2 d \cosh (c) \text{Shi}(d x)-\frac{a^2 \cosh (c+d x)}{x}-\frac{2 a b \cosh (c+d x)}{d^2}+\frac{2 a b x \sinh (c+d x)}{d}+\frac{12 b^2 x^2 \sinh (c+d x)}{d^3}-\frac{4 b^2 x^3 \cosh (c+d x)}{d^2}+\frac{24 b^2 \sinh (c+d x)}{d^5}-\frac{24 b^2 x \cosh (c+d x)}{d^4}+\frac{b^2 x^4 \sinh (c+d x)}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.251996, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421, Rules used = {5287, 3297, 3303, 3298, 3301, 3296, 2638, 2637} \[ a^2 d \sinh (c) \text{Chi}(d x)+a^2 d \cosh (c) \text{Shi}(d x)-\frac{a^2 \cosh (c+d x)}{x}-\frac{2 a b \cosh (c+d x)}{d^2}+\frac{2 a b x \sinh (c+d x)}{d}+\frac{12 b^2 x^2 \sinh (c+d x)}{d^3}-\frac{4 b^2 x^3 \cosh (c+d x)}{d^2}+\frac{24 b^2 \sinh (c+d x)}{d^5}-\frac{24 b^2 x \cosh (c+d x)}{d^4}+\frac{b^2 x^4 \sinh (c+d x)}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5287
Rule 3297
Rule 3303
Rule 3298
Rule 3301
Rule 3296
Rule 2638
Rule 2637
Rubi steps
\begin{align*} \int \frac{\left (a+b x^3\right )^2 \cosh (c+d x)}{x^2} \, dx &=\int \left (\frac{a^2 \cosh (c+d x)}{x^2}+2 a b x \cosh (c+d x)+b^2 x^4 \cosh (c+d x)\right ) \, dx\\ &=a^2 \int \frac{\cosh (c+d x)}{x^2} \, dx+(2 a b) \int x \cosh (c+d x) \, dx+b^2 \int x^4 \cosh (c+d x) \, dx\\ &=-\frac{a^2 \cosh (c+d x)}{x}+\frac{2 a b x \sinh (c+d x)}{d}+\frac{b^2 x^4 \sinh (c+d x)}{d}-\frac{(2 a b) \int \sinh (c+d x) \, dx}{d}-\frac{\left (4 b^2\right ) \int x^3 \sinh (c+d x) \, dx}{d}+\left (a^2 d\right ) \int \frac{\sinh (c+d x)}{x} \, dx\\ &=-\frac{2 a b \cosh (c+d x)}{d^2}-\frac{a^2 \cosh (c+d x)}{x}-\frac{4 b^2 x^3 \cosh (c+d x)}{d^2}+\frac{2 a b x \sinh (c+d x)}{d}+\frac{b^2 x^4 \sinh (c+d x)}{d}+\frac{\left (12 b^2\right ) \int x^2 \cosh (c+d x) \, dx}{d^2}+\left (a^2 d \cosh (c)\right ) \int \frac{\sinh (d x)}{x} \, dx+\left (a^2 d \sinh (c)\right ) \int \frac{\cosh (d x)}{x} \, dx\\ &=-\frac{2 a b \cosh (c+d x)}{d^2}-\frac{a^2 \cosh (c+d x)}{x}-\frac{4 b^2 x^3 \cosh (c+d x)}{d^2}+a^2 d \text{Chi}(d x) \sinh (c)+\frac{2 a b x \sinh (c+d x)}{d}+\frac{12 b^2 x^2 \sinh (c+d x)}{d^3}+\frac{b^2 x^4 \sinh (c+d x)}{d}+a^2 d \cosh (c) \text{Shi}(d x)-\frac{\left (24 b^2\right ) \int x \sinh (c+d x) \, dx}{d^3}\\ &=-\frac{2 a b \cosh (c+d x)}{d^2}-\frac{a^2 \cosh (c+d x)}{x}-\frac{24 b^2 x \cosh (c+d x)}{d^4}-\frac{4 b^2 x^3 \cosh (c+d x)}{d^2}+a^2 d \text{Chi}(d x) \sinh (c)+\frac{2 a b x \sinh (c+d x)}{d}+\frac{12 b^2 x^2 \sinh (c+d x)}{d^3}+\frac{b^2 x^4 \sinh (c+d x)}{d}+a^2 d \cosh (c) \text{Shi}(d x)+\frac{\left (24 b^2\right ) \int \cosh (c+d x) \, dx}{d^4}\\ &=-\frac{2 a b \cosh (c+d x)}{d^2}-\frac{a^2 \cosh (c+d x)}{x}-\frac{24 b^2 x \cosh (c+d x)}{d^4}-\frac{4 b^2 x^3 \cosh (c+d x)}{d^2}+a^2 d \text{Chi}(d x) \sinh (c)+\frac{24 b^2 \sinh (c+d x)}{d^5}+\frac{2 a b x \sinh (c+d x)}{d}+\frac{12 b^2 x^2 \sinh (c+d x)}{d^3}+\frac{b^2 x^4 \sinh (c+d x)}{d}+a^2 d \cosh (c) \text{Shi}(d x)\\ \end{align*}
Mathematica [A] time = 0.347878, size = 143, normalized size = 1. \[ a^2 d \sinh (c) \text{Chi}(d x)+a^2 d \cosh (c) \text{Shi}(d x)-\frac{a^2 \cosh (c+d x)}{x}-\frac{2 a b \cosh (c+d x)}{d^2}+\frac{2 a b x \sinh (c+d x)}{d}+\frac{12 b^2 x^2 \sinh (c+d x)}{d^3}-\frac{4 b^2 x^3 \cosh (c+d x)}{d^2}+\frac{24 b^2 \sinh (c+d x)}{d^5}-\frac{24 b^2 x \cosh (c+d x)}{d^4}+\frac{b^2 x^4 \sinh (c+d x)}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.102, size = 296, normalized size = 2.1 \begin{align*} -{\frac{{b}^{2}{{\rm e}^{-dx-c}}{x}^{4}}{2\,d}}-2\,{\frac{{b}^{2}{{\rm e}^{-dx-c}}{x}^{3}}{{d}^{2}}}-6\,{\frac{{b}^{2}{{\rm e}^{-dx-c}}{x}^{2}}{{d}^{3}}}-12\,{\frac{{b}^{2}{{\rm e}^{-dx-c}}x}{{d}^{4}}}+{\frac{d{a}^{2}{{\rm e}^{-c}}{\it Ei} \left ( 1,dx \right ) }{2}}-{\frac{ab{{\rm e}^{-dx-c}}}{{d}^{2}}}-{\frac{{a}^{2}{{\rm e}^{-dx-c}}}{2\,x}}-12\,{\frac{{b}^{2}{{\rm e}^{-dx-c}}}{{d}^{5}}}-{\frac{ab{{\rm e}^{-dx-c}}x}{d}}-{\frac{{{\rm e}^{dx+c}}{a}^{2}}{2\,x}}-{\frac{d{a}^{2}{{\rm e}^{c}}{\it Ei} \left ( 1,-dx \right ) }{2}}+12\,{\frac{{{\rm e}^{dx+c}}{b}^{2}}{{d}^{5}}}+{\frac{ab{{\rm e}^{dx+c}}x}{d}}-{\frac{ab{{\rm e}^{dx+c}}}{{d}^{2}}}+6\,{\frac{{{\rm e}^{dx+c}}{b}^{2}{x}^{2}}{{d}^{3}}}-12\,{\frac{{{\rm e}^{dx+c}}{b}^{2}x}{{d}^{4}}}+{\frac{{{\rm e}^{dx+c}}{b}^{2}{x}^{4}}{2\,d}}-2\,{\frac{{{\rm e}^{dx+c}}{b}^{2}{x}^{3}}{{d}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.19747, size = 317, normalized size = 2.22 \begin{align*} -\frac{1}{10} \,{\left (5 \, a^{2}{\rm Ei}\left (-d x\right ) e^{\left (-c\right )} - 5 \, a^{2}{\rm Ei}\left (d x\right ) e^{c} + \frac{5 \,{\left (d^{2} x^{2} e^{c} - 2 \, d x e^{c} + 2 \, e^{c}\right )} a b e^{\left (d x\right )}}{d^{3}} + \frac{5 \,{\left (d^{2} x^{2} + 2 \, d x + 2\right )} a b e^{\left (-d x - c\right )}}{d^{3}} + \frac{{\left (d^{5} x^{5} e^{c} - 5 \, d^{4} x^{4} e^{c} + 20 \, d^{3} x^{3} e^{c} - 60 \, d^{2} x^{2} e^{c} + 120 \, d x e^{c} - 120 \, e^{c}\right )} b^{2} e^{\left (d x\right )}}{d^{6}} + \frac{{\left (d^{5} x^{5} + 5 \, d^{4} x^{4} + 20 \, d^{3} x^{3} + 60 \, d^{2} x^{2} + 120 \, d x + 120\right )} b^{2} e^{\left (-d x - c\right )}}{d^{6}}\right )} d + \frac{1}{5} \,{\left (b^{2} x^{5} + 5 \, a b x^{2} - \frac{5 \, a^{2}}{x}\right )} \cosh \left (d x + c\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.77474, size = 358, normalized size = 2.5 \begin{align*} -\frac{2 \,{\left (4 \, b^{2} d^{3} x^{4} + a^{2} d^{5} + 2 \, a b d^{3} x + 24 \, b^{2} d x^{2}\right )} \cosh \left (d x + c\right ) -{\left (a^{2} d^{6} x{\rm Ei}\left (d x\right ) - a^{2} d^{6} x{\rm Ei}\left (-d x\right )\right )} \cosh \left (c\right ) - 2 \,{\left (b^{2} d^{4} x^{5} + 2 \, a b d^{4} x^{2} + 12 \, b^{2} d^{2} x^{3} + 24 \, b^{2} x\right )} \sinh \left (d x + c\right ) -{\left (a^{2} d^{6} x{\rm Ei}\left (d x\right ) + a^{2} d^{6} x{\rm Ei}\left (-d x\right )\right )} \sinh \left (c\right )}{2 \, d^{5} x} \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^{3}\right )^{2} \cosh{\left (c + d x \right )}}{x^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.26092, size = 416, normalized size = 2.91 \begin{align*} \frac{b^{2} d^{4} x^{5} e^{\left (d x + c\right )} - b^{2} d^{4} x^{5} e^{\left (-d x - c\right )} - a^{2} d^{6} x{\rm Ei}\left (-d x\right ) e^{\left (-c\right )} + a^{2} d^{6} x{\rm Ei}\left (d x\right ) e^{c} - 4 \, b^{2} d^{3} x^{4} e^{\left (d x + c\right )} - 4 \, b^{2} d^{3} x^{4} e^{\left (-d x - c\right )} + 2 \, a b d^{4} x^{2} e^{\left (d x + c\right )} - 2 \, a b d^{4} x^{2} e^{\left (-d x - c\right )} - a^{2} d^{5} e^{\left (d x + c\right )} + 12 \, b^{2} d^{2} x^{3} e^{\left (d x + c\right )} - a^{2} d^{5} e^{\left (-d x - c\right )} - 12 \, b^{2} d^{2} x^{3} e^{\left (-d x - c\right )} - 2 \, a b d^{3} x e^{\left (d x + c\right )} - 2 \, a b d^{3} x e^{\left (-d x - c\right )} - 24 \, b^{2} d x^{2} e^{\left (d x + c\right )} - 24 \, b^{2} d x^{2} e^{\left (-d x - c\right )} + 24 \, b^{2} x e^{\left (d x + c\right )} - 24 \, b^{2} x e^{\left (-d x - c\right )}}{2 \, d^{5} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]