Optimal. Leaf size=67 \[ -\frac {3 \sinh (a) \text {Chi}\left (b x^n\right )}{4 n}+\frac {\sinh (3 a) \text {Chi}\left (3 b x^n\right )}{4 n}-\frac {3 \cosh (a) \text {Shi}\left (b x^n\right )}{4 n}+\frac {\cosh (3 a) \text {Shi}\left (3 b x^n\right )}{4 n} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.10, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {5362, 5318, 5317, 5316} \[ -\frac {3 \sinh (a) \text {Chi}\left (b x^n\right )}{4 n}+\frac {\sinh (3 a) \text {Chi}\left (3 b x^n\right )}{4 n}-\frac {3 \cosh (a) \text {Shi}\left (b x^n\right )}{4 n}+\frac {\cosh (3 a) \text {Shi}\left (3 b x^n\right )}{4 n} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5316
Rule 5317
Rule 5318
Rule 5362
Rubi steps
\begin {align*} \int \frac {\sinh ^3\left (a+b x^n\right )}{x} \, dx &=\int \left (-\frac {3 \sinh \left (a+b x^n\right )}{4 x}+\frac {\sinh \left (3 a+3 b x^n\right )}{4 x}\right ) \, dx\\ &=\frac {1}{4} \int \frac {\sinh \left (3 a+3 b x^n\right )}{x} \, dx-\frac {3}{4} \int \frac {\sinh \left (a+b x^n\right )}{x} \, dx\\ &=-\left (\frac {1}{4} (3 \cosh (a)) \int \frac {\sinh \left (b x^n\right )}{x} \, dx\right )+\frac {1}{4} \cosh (3 a) \int \frac {\sinh \left (3 b x^n\right )}{x} \, dx-\frac {1}{4} (3 \sinh (a)) \int \frac {\cosh \left (b x^n\right )}{x} \, dx+\frac {1}{4} \sinh (3 a) \int \frac {\cosh \left (3 b x^n\right )}{x} \, dx\\ &=-\frac {3 \text {Chi}\left (b x^n\right ) \sinh (a)}{4 n}+\frac {\text {Chi}\left (3 b x^n\right ) \sinh (3 a)}{4 n}-\frac {3 \cosh (a) \text {Shi}\left (b x^n\right )}{4 n}+\frac {\cosh (3 a) \text {Shi}\left (3 b x^n\right )}{4 n}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.06, size = 52, normalized size = 0.78 \[ \frac {-3 \sinh (a) \text {Chi}\left (b x^n\right )+\sinh (3 a) \text {Chi}\left (3 b x^n\right )-3 \cosh (a) \text {Shi}\left (b x^n\right )+\cosh (3 a) \text {Shi}\left (3 b x^n\right )}{4 n} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.60, size = 115, normalized size = 1.72 \[ \frac {{\left (\cosh \left (3 \, a\right ) + \sinh \left (3 \, a\right )\right )} {\rm Ei}\left (3 \, b \cosh \left (n \log \relax (x)\right ) + 3 \, b \sinh \left (n \log \relax (x)\right )\right ) - 3 \, {\left (\cosh \relax (a) + \sinh \relax (a)\right )} {\rm Ei}\left (b \cosh \left (n \log \relax (x)\right ) + b \sinh \left (n \log \relax (x)\right )\right ) + 3 \, {\left (\cosh \relax (a) - \sinh \relax (a)\right )} {\rm Ei}\left (-b \cosh \left (n \log \relax (x)\right ) - b \sinh \left (n \log \relax (x)\right )\right ) - {\left (\cosh \left (3 \, a\right ) - \sinh \left (3 \, a\right )\right )} {\rm Ei}\left (-3 \, b \cosh \left (n \log \relax (x)\right ) - 3 \, b \sinh \left (n \log \relax (x)\right )\right )}{8 \, n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sinh \left (b x^{n} + a\right )^{3}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.19, size = 67, normalized size = 1.00 \[ \frac {{\mathrm e}^{-3 a} \Ei \left (1, 3 b \,x^{n}\right )}{8 n}-\frac {3 \,{\mathrm e}^{-a} \Ei \left (1, b \,x^{n}\right )}{8 n}-\frac {{\mathrm e}^{3 a} \Ei \left (1, -3 b \,x^{n}\right )}{8 n}+\frac {3 \,{\mathrm e}^{a} \Ei \left (1, -b \,x^{n}\right )}{8 n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.43, size = 62, normalized size = 0.93 \[ \frac {{\rm Ei}\left (3 \, b x^{n}\right ) e^{\left (3 \, a\right )}}{8 \, n} + \frac {3 \, {\rm Ei}\left (-b x^{n}\right ) e^{\left (-a\right )}}{8 \, n} - \frac {{\rm Ei}\left (-3 \, b x^{n}\right ) e^{\left (-3 \, a\right )}}{8 \, n} - \frac {3 \, {\rm Ei}\left (b x^{n}\right ) e^{a}}{8 \, n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {sinh}\left (a+b\,x^n\right )}^3}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sinh ^{3}{\left (a + b x^{n} \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________