Integrand size = 17, antiderivative size = 73 \[ \int \frac {\sinh ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {3 \log (x)}{8}-\frac {3 \cosh \left (a+b \log \left (c x^n\right )\right ) \sinh \left (a+b \log \left (c x^n\right )\right )}{8 b n}+\frac {\cosh \left (a+b \log \left (c x^n\right )\right ) \sinh ^3\left (a+b \log \left (c x^n\right )\right )}{4 b n} \]
3/8*ln(x)-3/8*cosh(a+b*ln(c*x^n))*sinh(a+b*ln(c*x^n))/b/n+1/4*cosh(a+b*ln( c*x^n))*sinh(a+b*ln(c*x^n))^3/b/n
Time = 0.03 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.70 \[ \int \frac {\sinh ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {12 \left (a+b \log \left (c x^n\right )\right )-8 \sinh \left (2 \left (a+b \log \left (c x^n\right )\right )\right )+\sinh \left (4 \left (a+b \log \left (c x^n\right )\right )\right )}{32 b n} \]
(12*(a + b*Log[c*x^n]) - 8*Sinh[2*(a + b*Log[c*x^n])] + Sinh[4*(a + b*Log[ c*x^n])])/(32*b*n)
Time = 0.29 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.10, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.471, Rules used = {3039, 3042, 3115, 25, 3042, 25, 3115, 24}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\sinh ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx\) |
\(\Big \downarrow \) 3039 |
\(\displaystyle \frac {\int \sinh ^4\left (a+b \log \left (c x^n\right )\right )d\log \left (c x^n\right )}{n}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \sin \left (i a+i b \log \left (c x^n\right )\right )^4d\log \left (c x^n\right )}{n}\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle \frac {\frac {3}{4} \int -\sinh ^2\left (a+b \log \left (c x^n\right )\right )d\log \left (c x^n\right )+\frac {\sinh ^3\left (a+b \log \left (c x^n\right )\right ) \cosh \left (a+b \log \left (c x^n\right )\right )}{4 b}}{n}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\frac {\sinh ^3\left (a+b \log \left (c x^n\right )\right ) \cosh \left (a+b \log \left (c x^n\right )\right )}{4 b}-\frac {3}{4} \int \sinh ^2\left (a+b \log \left (c x^n\right )\right )d\log \left (c x^n\right )}{n}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {\sinh ^3\left (a+b \log \left (c x^n\right )\right ) \cosh \left (a+b \log \left (c x^n\right )\right )}{4 b}-\frac {3}{4} \int -\sin \left (i a+i b \log \left (c x^n\right )\right )^2d\log \left (c x^n\right )}{n}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\frac {\sinh ^3\left (a+b \log \left (c x^n\right )\right ) \cosh \left (a+b \log \left (c x^n\right )\right )}{4 b}+\frac {3}{4} \int \sin \left (i a+i b \log \left (c x^n\right )\right )^2d\log \left (c x^n\right )}{n}\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle \frac {\frac {3}{4} \left (\frac {1}{2} \int 1d\log \left (c x^n\right )-\frac {\sinh \left (a+b \log \left (c x^n\right )\right ) \cosh \left (a+b \log \left (c x^n\right )\right )}{2 b}\right )+\frac {\sinh ^3\left (a+b \log \left (c x^n\right )\right ) \cosh \left (a+b \log \left (c x^n\right )\right )}{4 b}}{n}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle \frac {\frac {\sinh ^3\left (a+b \log \left (c x^n\right )\right ) \cosh \left (a+b \log \left (c x^n\right )\right )}{4 b}+\frac {3}{4} \left (\frac {1}{2} \log \left (c x^n\right )-\frac {\sinh \left (a+b \log \left (c x^n\right )\right ) \cosh \left (a+b \log \left (c x^n\right )\right )}{2 b}\right )}{n}\) |
((Cosh[a + b*Log[c*x^n]]*Sinh[a + b*Log[c*x^n]]^3)/(4*b) + (3*(Log[c*x^n]/ 2 - (Cosh[a + b*Log[c*x^n]]*Sinh[a + b*Log[c*x^n]])/(2*b)))/4)/n
3.3.77.3.1 Defintions of rubi rules used
Int[u_, x_Symbol] :> With[{lst = FunctionOfLog[Cancel[x*u], x]}, Simp[1/lst [[3]] Subst[Int[lst[[1]], x], x, Log[lst[[2]]]], x] /; !FalseQ[lst]] /; NonsumQ[u]
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
Time = 21.06 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.63
method | result | size |
parallelrisch | \(\frac {12 \ln \left (x \right ) b n -8 \sinh \left (2 b \ln \left (c \,x^{n}\right )+2 a \right )+\sinh \left (4 b \ln \left (c \,x^{n}\right )+4 a \right )}{32 b n}\) | \(46\) |
derivativedivides | \(\frac {\left (\frac {{\sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}^{3}}{4}-\frac {3 \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}{8}\right ) \cosh \left (a +b \ln \left (c \,x^{n}\right )\right )+\frac {3 b \ln \left (c \,x^{n}\right )}{8}+\frac {3 a}{8}}{n b}\) | \(62\) |
default | \(\frac {\left (\frac {{\sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}^{3}}{4}-\frac {3 \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}{8}\right ) \cosh \left (a +b \ln \left (c \,x^{n}\right )\right )+\frac {3 b \ln \left (c \,x^{n}\right )}{8}+\frac {3 a}{8}}{n b}\) | \(62\) |
Time = 0.28 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.15 \[ \int \frac {\sinh ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} + 3 \, b n \log \left (x\right ) + {\left (\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} - 4 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{8 \, b n} \]
1/8*(cosh(b*n*log(x) + b*log(c) + a)*sinh(b*n*log(x) + b*log(c) + a)^3 + 3 *b*n*log(x) + (cosh(b*n*log(x) + b*log(c) + a)^3 - 4*cosh(b*n*log(x) + b*l og(c) + a))*sinh(b*n*log(x) + b*log(c) + a))/(b*n)
\[ \int \frac {\sinh ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int \frac {\sinh ^{4}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{x}\, dx \]
Time = 0.20 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.27 \[ \int \frac {\sinh ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {e^{\left (4 \, b \log \left (c x^{n}\right ) + 4 \, a\right )}}{64 \, b n} - \frac {e^{\left (2 \, b \log \left (c x^{n}\right ) + 2 \, a\right )}}{8 \, b n} + \frac {e^{\left (-2 \, b \log \left (c x^{n}\right ) - 2 \, a\right )}}{8 \, b n} - \frac {e^{\left (-4 \, b \log \left (c x^{n}\right ) - 4 \, a\right )}}{64 \, b n} + \frac {3}{8} \, \log \left (x\right ) \]
1/64*e^(4*b*log(c*x^n) + 4*a)/(b*n) - 1/8*e^(2*b*log(c*x^n) + 2*a)/(b*n) + 1/8*e^(-2*b*log(c*x^n) - 2*a)/(b*n) - 1/64*e^(-4*b*log(c*x^n) - 4*a)/(b*n ) + 3/8*log(x)
Time = 0.28 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.56 \[ \int \frac {\sinh ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {{\left (24 \, b c^{4 \, b} n e^{\left (4 \, a\right )} \log \left (x\right ) + c^{8 \, b} x^{4 \, b n} e^{\left (8 \, a\right )} - 8 \, c^{6 \, b} x^{2 \, b n} e^{\left (6 \, a\right )} - \frac {18 \, c^{4 \, b} x^{4 \, b n} e^{\left (4 \, a\right )} - 8 \, c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} + 1}{x^{4 \, b n}}\right )} e^{\left (-4 \, a\right )}}{64 \, b c^{4 \, b} n} \]
1/64*(24*b*c^(4*b)*n*e^(4*a)*log(x) + c^(8*b)*x^(4*b*n)*e^(8*a) - 8*c^(6*b )*x^(2*b*n)*e^(6*a) - (18*c^(4*b)*x^(4*b*n)*e^(4*a) - 8*c^(2*b)*x^(2*b*n)* e^(2*a) + 1)/x^(4*b*n))*e^(-4*a)/(b*c^(4*b)*n)
Time = 1.55 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.70 \[ \int \frac {\sinh ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {3\,\ln \left (x^n\right )}{8\,n}-\frac {\frac {\mathrm {sinh}\left (2\,a+2\,b\,\ln \left (c\,x^n\right )\right )}{4}-\frac {\mathrm {sinh}\left (4\,a+4\,b\,\ln \left (c\,x^n\right )\right )}{32}}{b\,n} \]