Optimal. Leaf size=53 \[ \frac {1}{4} \sinh (2 a) \text {Chi}(2 b x)+\frac {1}{8} \sinh (4 a) \text {Chi}(4 b x)+\frac {1}{4} \cosh (2 a) \text {Shi}(2 b x)+\frac {1}{8} \cosh (4 a) \text {Shi}(4 b x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.14, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {5448, 3303, 3298, 3301} \[ \frac {1}{4} \sinh (2 a) \text {Chi}(2 b x)+\frac {1}{8} \sinh (4 a) \text {Chi}(4 b x)+\frac {1}{4} \cosh (2 a) \text {Shi}(2 b x)+\frac {1}{8} \cosh (4 a) \text {Shi}(4 b x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3298
Rule 3301
Rule 3303
Rule 5448
Rubi steps
\begin {align*} \int \frac {\cosh ^3(a+b x) \sinh (a+b x)}{x} \, dx &=\int \left (\frac {\sinh (2 a+2 b x)}{4 x}+\frac {\sinh (4 a+4 b x)}{8 x}\right ) \, dx\\ &=\frac {1}{8} \int \frac {\sinh (4 a+4 b x)}{x} \, dx+\frac {1}{4} \int \frac {\sinh (2 a+2 b x)}{x} \, dx\\ &=\frac {1}{4} \cosh (2 a) \int \frac {\sinh (2 b x)}{x} \, dx+\frac {1}{8} \cosh (4 a) \int \frac {\sinh (4 b x)}{x} \, dx+\frac {1}{4} \sinh (2 a) \int \frac {\cosh (2 b x)}{x} \, dx+\frac {1}{8} \sinh (4 a) \int \frac {\cosh (4 b x)}{x} \, dx\\ &=\frac {1}{4} \text {Chi}(2 b x) \sinh (2 a)+\frac {1}{8} \text {Chi}(4 b x) \sinh (4 a)+\frac {1}{4} \cosh (2 a) \text {Shi}(2 b x)+\frac {1}{8} \cosh (4 a) \text {Shi}(4 b x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.08, size = 47, normalized size = 0.89 \[ \frac {1}{8} (2 \sinh (2 a) \text {Chi}(2 b x)+\sinh (4 a) \text {Chi}(4 b x)+2 \cosh (2 a) \text {Shi}(2 b x)+\cosh (4 a) \text {Shi}(4 b x)) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.13, size = 73, normalized size = 1.38 \[ \frac {1}{16} \, {\left ({\rm Ei}\left (4 \, b x\right ) - {\rm Ei}\left (-4 \, b x\right )\right )} \cosh \left (4 \, a\right ) + \frac {1}{8} \, {\left ({\rm Ei}\left (2 \, b x\right ) - {\rm Ei}\left (-2 \, b x\right )\right )} \cosh \left (2 \, a\right ) + \frac {1}{16} \, {\left ({\rm Ei}\left (4 \, b x\right ) + {\rm Ei}\left (-4 \, b x\right )\right )} \sinh \left (4 \, a\right ) + \frac {1}{8} \, {\left ({\rm Ei}\left (2 \, b x\right ) + {\rm Ei}\left (-2 \, b x\right )\right )} \sinh \left (2 \, a\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.12, size = 45, normalized size = 0.85 \[ \frac {1}{16} \, {\rm Ei}\left (4 \, b x\right ) e^{\left (4 \, a\right )} + \frac {1}{8} \, {\rm Ei}\left (2 \, b x\right ) e^{\left (2 \, a\right )} - \frac {1}{8} \, {\rm Ei}\left (-2 \, b x\right ) e^{\left (-2 \, a\right )} - \frac {1}{16} \, {\rm Ei}\left (-4 \, b x\right ) e^{\left (-4 \, a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.47, size = 50, normalized size = 0.94 \[ \frac {{\mathrm e}^{-4 a} \Ei \left (1, 4 b x \right )}{16}+\frac {{\mathrm e}^{-2 a} \Ei \left (1, 2 b x \right )}{8}-\frac {{\mathrm e}^{2 a} \Ei \left (1, -2 b x \right )}{8}-\frac {{\mathrm e}^{4 a} \Ei \left (1, -4 b x \right )}{16} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.42, size = 45, normalized size = 0.85 \[ \frac {1}{16} \, {\rm Ei}\left (4 \, b x\right ) e^{\left (4 \, a\right )} + \frac {1}{8} \, {\rm Ei}\left (2 \, b x\right ) e^{\left (2 \, a\right )} - \frac {1}{8} \, {\rm Ei}\left (-2 \, b x\right ) e^{\left (-2 \, a\right )} - \frac {1}{16} \, {\rm Ei}\left (-4 \, b x\right ) e^{\left (-4 \, a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\mathrm {cosh}\left (a+b\,x\right )}^3\,\mathrm {sinh}\left (a+b\,x\right )}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sinh {\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________