Optimal. Leaf size=73 \[ \frac {3 \log (x)}{8}-\frac {3 \cosh \left (a+b \log \left (c x^n\right )\right ) \sinh \left (a+b \log \left (c x^n\right )\right )}{8 b n}+\frac {\cosh \left (a+b \log \left (c x^n\right )\right ) \sinh ^3\left (a+b \log \left (c x^n\right )\right )}{4 b n} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.03, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2715, 8}
\begin {gather*} \frac {\sinh ^3\left (a+b \log \left (c x^n\right )\right ) \cosh \left (a+b \log \left (c x^n\right )\right )}{4 b n}-\frac {3 \sinh \left (a+b \log \left (c x^n\right )\right ) \cosh \left (a+b \log \left (c x^n\right )\right )}{8 b n}+\frac {3 \log (x)}{8} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 2715
Rubi steps
\begin {align*} \int \frac {\sinh ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac {\text {Subst}\left (\int \sinh ^4(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac {\cosh \left (a+b \log \left (c x^n\right )\right ) \sinh ^3\left (a+b \log \left (c x^n\right )\right )}{4 b n}-\frac {3 \text {Subst}\left (\int \sinh ^2(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{4 n}\\ &=-\frac {3 \cosh \left (a+b \log \left (c x^n\right )\right ) \sinh \left (a+b \log \left (c x^n\right )\right )}{8 b n}+\frac {\cosh \left (a+b \log \left (c x^n\right )\right ) \sinh ^3\left (a+b \log \left (c x^n\right )\right )}{4 b n}+\frac {3 \text {Subst}\left (\int 1 \, dx,x,\log \left (c x^n\right )\right )}{8 n}\\ &=\frac {3 \log (x)}{8}-\frac {3 \cosh \left (a+b \log \left (c x^n\right )\right ) \sinh \left (a+b \log \left (c x^n\right )\right )}{8 b n}+\frac {\cosh \left (a+b \log \left (c x^n\right )\right ) \sinh ^3\left (a+b \log \left (c x^n\right )\right )}{4 b n}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.04, size = 51, normalized size = 0.70 \begin {gather*} \frac {12 \left (a+b \log \left (c x^n\right )\right )-8 \sinh \left (2 \left (a+b \log \left (c x^n\right )\right )\right )+\sinh \left (4 \left (a+b \log \left (c x^n\right )\right )\right )}{32 b n} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\sinh ^{4}\left (a +b \ln \left (c \,x^{n}\right )\right )}{x}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.26, size = 93, normalized size = 1.27 \begin {gather*} \frac {e^{\left (4 \, b \log \left (c x^{n}\right ) + 4 \, a\right )}}{64 \, b n} - \frac {e^{\left (2 \, b \log \left (c x^{n}\right ) + 2 \, a\right )}}{8 \, b n} + \frac {e^{\left (-2 \, b \log \left (c x^{n}\right ) - 2 \, a\right )}}{8 \, b n} - \frac {e^{\left (-4 \, b \log \left (c x^{n}\right ) - 4 \, a\right )}}{64 \, b n} + \frac {3}{8} \, \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.46, size = 84, normalized size = 1.15 \begin {gather*} \frac {\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} + 3 \, b n \log \left (x\right ) + {\left (\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} - 4 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{8 \, b n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sinh ^{4}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.44, size = 114, normalized size = 1.56 \begin {gather*} \frac {{\left (24 \, b c^{4 \, b} n e^{\left (4 \, a\right )} \log \left (x\right ) + c^{8 \, b} x^{4 \, b n} e^{\left (8 \, a\right )} - 8 \, c^{6 \, b} x^{2 \, b n} e^{\left (6 \, a\right )} - \frac {18 \, c^{4 \, b} x^{4 \, b n} e^{\left (4 \, a\right )} - 8 \, c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} + 1}{x^{4 \, b n}}\right )} e^{\left (-4 \, a\right )}}{64 \, b c^{4 \, b} n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.78, size = 51, normalized size = 0.70 \begin {gather*} \frac {3\,\ln \left (x^n\right )}{8\,n}-\frac {\frac {\mathrm {sinh}\left (2\,a+2\,b\,\ln \left (c\,x^n\right )\right )}{4}-\frac {\mathrm {sinh}\left (4\,a+4\,b\,\ln \left (c\,x^n\right )\right )}{32}}{b\,n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________