Integrand size = 28, antiderivative size = 176 \[ \int \frac {\sin ^3\left (a+\frac {1}{3} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^2} \, dx=-\frac {e^{3 a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n \left (c x^n\right )^{-1/n}}{16 x}+\frac {9 e^{a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n \left (c x^n\right )^{\left .-\frac {1}{3}\right /n}}{32 x}-\frac {9 e^{-a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n \left (c x^n\right )^{\left .\frac {1}{3}\right /n}}{16 x}-\frac {e^{-3 a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n \left (c x^n\right )^{\frac {1}{n}} \log (x)}{8 x} \] Output:
-1/16*exp(3*a*(-1/n^2)^(1/2)*n)*(-1/n^2)^(1/2)*n/x/((c*x^n)^(1/n))+9/32*ex p(a*(-1/n^2)^(1/2)*n)*(-1/n^2)^(1/2)*n/x/((c*x^n)^(1/3/n))-9/16*(-1/n^2)^( 1/2)*n*(c*x^n)^(1/3/n)/exp(a*(-1/n^2)^(1/2)*n)/x-1/8*(-1/n^2)^(1/2)*n*(c*x ^n)^(1/n)*ln(x)/exp(3*a*(-1/n^2)^(1/2)*n)/x
\[ \int \frac {\sin ^3\left (a+\frac {1}{3} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^2} \, dx=\int \frac {\sin ^3\left (a+\frac {1}{3} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^2} \, dx \] Input:
Integrate[Sin[a + (Sqrt[-n^(-2)]*Log[c*x^n])/3]^3/x^2,x]
Output:
Integrate[Sin[a + (Sqrt[-n^(-2)]*Log[c*x^n])/3]^3/x^2, x]
Time = 0.39 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.82, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {4996, 4992, 2009}
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.
| \(\displaystyle \int \frac {\sin ^3\left (a+\frac {1}{3} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^2} \, dx\) |
| \(\Big \downarrow \) 4996 |
| \(\displaystyle \frac {\left (c x^n\right )^{\frac {1}{n}} \int \left (c x^n\right )^{-1-\frac {1}{n}} \sin ^3\left (a+\frac {1}{3} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )d\left (c x^n\right )}{n x}\) |
| \(\Big \downarrow \) 4992 |
| \(\displaystyle -\frac {\sqrt {-\frac {1}{n^2}} \left (c x^n\right )^{\frac {1}{n}} \int \left (3 e^{a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{-1-\frac {4}{3 n}}-3 e^{-a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{-1-\frac {2}{3 n}}-e^{3 a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{-\frac {n+2}{n}}+\frac {e^{-3 a \sqrt {-\frac {1}{n^2}} n} x^{-n}}{c}\right )d\left (c x^n\right )}{8 x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\sqrt {-\frac {1}{n^2}} \left (c x^n\right )^{\frac {1}{n}} \left (\frac {1}{2} n e^{3 a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{-2/n}-\frac {9}{4} n e^{a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{\left .-\frac {4}{3}\right /n}+\frac {9}{2} n e^{-a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{\left .-\frac {2}{3}\right /n}+e^{-3 a \sqrt {-\frac {1}{n^2}} n} \log \left (c x^n\right )\right )}{8 x}\) |
Input:
Int[Sin[a + (Sqrt[-n^(-2)]*Log[c*x^n])/3]^3/x^2,x]
Output:
-1/8*(Sqrt[-n^(-2)]*(c*x^n)^n^(-1)*((E^(3*a*Sqrt[-n^(-2)]*n)*n)/(2*(c*x^n) ^(2/n)) - (9*E^(a*Sqrt[-n^(-2)]*n)*n)/(4*(c*x^n)^(4/(3*n))) + (9*n)/(2*E^( a*Sqrt[-n^(-2)]*n)*(c*x^n)^(2/(3*n))) + Log[c*x^n]/E^(3*a*Sqrt[-n^(-2)]*n) ))/x
Int[((e_.)*(x_))^(m_.)*Sin[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_.), x_Symbol] :> Simp[(m + 1)^p/(2^p*b^p*d^p*p^p) Int[ExpandIntegrand[(e*x)^m*(E^(a*b*d ^2*(p/(m + 1)))/x^((m + 1)/p) - x^((m + 1)/p)/E^(a*b*d^2*(p/(m + 1))))^p, x ], x], x] /; FreeQ[{a, b, d, e, m}, x] && IGtQ[p, 0] && EqQ[b^2*d^2*p^2 + ( m + 1)^2, 0]
Int[((e_.)*(x_))^(m_.)*Sin[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_ .), x_Symbol] :> Simp[(e*x)^(m + 1)/(e*n*(c*x^n)^((m + 1)/n)) Subst[Int[x ^((m + 1)/n - 1)*Sin[d*(a + b*Log[x])]^p, x], x, c*x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && (NeQ[c, 1] || NeQ[n, 1])
Time = 32.05 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.61
| method | result | size |
| parallelrisch | \(\frac {12 \left (n +\frac {5 \ln \left (c \,x^{n}\right )}{12}\right ) n \sqrt {-\frac {1}{n^{2}}}\, {\tan \left (\frac {a}{2}+\sqrt {-\frac {1}{n^{2}}}\, \ln \left (\left (c \,x^{n}\right )^{\frac {1}{6}}\right )\right )}^{6}+\left (-30 \ln \left (c \,x^{n}\right )-42 n \right ) {\tan \left (\frac {a}{2}+\sqrt {-\frac {1}{n^{2}}}\, \ln \left (\left (c \,x^{n}\right )^{\frac {1}{6}}\right )\right )}^{5}-75 \sqrt {-\frac {1}{n^{2}}}\, {\tan \left (\frac {a}{2}+\sqrt {-\frac {1}{n^{2}}}\, \ln \left (\left (c \,x^{n}\right )^{\frac {1}{6}}\right )\right )}^{4} \ln \left (c \,x^{n}\right ) n +\left (100 \ln \left (c \,x^{n}\right )-220 n \right ) {\tan \left (\frac {a}{2}+\sqrt {-\frac {1}{n^{2}}}\, \ln \left (\left (c \,x^{n}\right )^{\frac {1}{6}}\right )\right )}^{3}+75 \sqrt {-\frac {1}{n^{2}}}\, {\tan \left (\frac {a}{2}+\sqrt {-\frac {1}{n^{2}}}\, \ln \left (\left (c \,x^{n}\right )^{\frac {1}{6}}\right )\right )}^{2} \ln \left (c \,x^{n}\right ) n +\left (-30 \ln \left (c \,x^{n}\right )-42 n \right ) \tan \left (\frac {a}{2}+\sqrt {-\frac {1}{n^{2}}}\, \ln \left (\left (c \,x^{n}\right )^{\frac {1}{6}}\right )\right )-12 \left (n +\frac {5 \ln \left (c \,x^{n}\right )}{12}\right ) n \sqrt {-\frac {1}{n^{2}}}}{40 x n {\left (1+{\tan \left (\frac {a}{2}+\sqrt {-\frac {1}{n^{2}}}\, \ln \left (\left (c \,x^{n}\right )^{\frac {1}{6}}\right )\right )}^{2}\right )}^{3}}\) | \(284\) |
Input:
int(sin(a+1/3*(-1/n^2)^(1/2)*ln(c*x^n))^3/x^2,x,method=_RETURNVERBOSE)
Output:
1/40*(12*(n+5/12*ln(c*x^n))*n*(-1/n^2)^(1/2)*tan(1/2*a+(-1/n^2)^(1/2)*ln(( c*x^n)^(1/6)))^6+(-30*ln(c*x^n)-42*n)*tan(1/2*a+(-1/n^2)^(1/2)*ln((c*x^n)^ (1/6)))^5-75*(-1/n^2)^(1/2)*tan(1/2*a+(-1/n^2)^(1/2)*ln((c*x^n)^(1/6)))^4* ln(c*x^n)*n+(100*ln(c*x^n)-220*n)*tan(1/2*a+(-1/n^2)^(1/2)*ln((c*x^n)^(1/6 )))^3+75*(-1/n^2)^(1/2)*tan(1/2*a+(-1/n^2)^(1/2)*ln((c*x^n)^(1/6)))^2*ln(c *x^n)*n+(-30*ln(c*x^n)-42*n)*tan(1/2*a+(-1/n^2)^(1/2)*ln((c*x^n)^(1/6)))-1 2*(n+5/12*ln(c*x^n))*n*(-1/n^2)^(1/2))/x/n/(1+tan(1/2*a+(-1/n^2)^(1/2)*ln( (c*x^n)^(1/6)))^2)^3
Result contains complex when optimal does not.
Time = 0.07 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.49 \[ \int \frac {\sin ^3\left (a+\frac {1}{3} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^2} \, dx=\frac {{\left (-12 i \, x^{2} \log \left (x^{\frac {1}{3}}\right ) - 18 i \, x^{\frac {4}{3}} e^{\left (\frac {2 \, {\left (3 i \, a n - \log \left (c\right )\right )}}{3 \, n}\right )} + 9 i \, x^{\frac {2}{3}} e^{\left (\frac {4 \, {\left (3 i \, a n - \log \left (c\right )\right )}}{3 \, n}\right )} - 2 i \, e^{\left (\frac {2 \, {\left (3 i \, a n - \log \left (c\right )\right )}}{n}\right )}\right )} e^{\left (-\frac {3 i \, a n - \log \left (c\right )}{n}\right )}}{32 \, x^{2}} \] Input:
integrate(sin(a+1/3*(-1/n^2)^(1/2)*log(c*x^n))^3/x^2,x, algorithm="fricas" )
Output:
1/32*(-12*I*x^2*log(x^(1/3)) - 18*I*x^(4/3)*e^(2/3*(3*I*a*n - log(c))/n) + 9*I*x^(2/3)*e^(4/3*(3*I*a*n - log(c))/n) - 2*I*e^(2*(3*I*a*n - log(c))/n) )*e^(-(3*I*a*n - log(c))/n)/x^2
Time = 53.37 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.02 \[ \int \frac {\sin ^3\left (a+\frac {1}{3} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^2} \, dx=- \frac {9 n \sqrt {- \frac {1}{n^{2}}} \cos {\left (a + \frac {\sqrt {- \frac {1}{n^{2}}} \log {\left (c x^{n} \right )}}{3} \right )}}{32 x} - \frac {n \sqrt {- \frac {1}{n^{2}}} \cos {\left (3 a + \sqrt {- \frac {1}{n^{2}}} \log {\left (c x^{n} \right )} \right )}}{8 x} - \frac {\sqrt {- \frac {1}{n^{2}}} \log {\left (c x^{n} \right )} \cos {\left (3 a + \sqrt {- \frac {1}{n^{2}}} \log {\left (c x^{n} \right )} \right )}}{8 x} - \frac {27 \sin {\left (a + \frac {\sqrt {- \frac {1}{n^{2}}} \log {\left (c x^{n} \right )}}{3} \right )}}{32 x} - \frac {\log {\left (c x^{n} \right )} \sin {\left (3 a + \sqrt {- \frac {1}{n^{2}}} \log {\left (c x^{n} \right )} \right )}}{8 n x} \] Input:
integrate(sin(a+1/3*(-1/n**2)**(1/2)*ln(c*x**n))**3/x**2,x)
Output:
-9*n*sqrt(-1/n**2)*cos(a + sqrt(-1/n**2)*log(c*x**n)/3)/(32*x) - n*sqrt(-1 /n**2)*cos(3*a + sqrt(-1/n**2)*log(c*x**n))/(8*x) - sqrt(-1/n**2)*log(c*x* *n)*cos(3*a + sqrt(-1/n**2)*log(c*x**n))/(8*x) - 27*sin(a + sqrt(-1/n**2)* log(c*x**n)/3)/(32*x) - log(c*x**n)*sin(3*a + sqrt(-1/n**2)*log(c*x**n))/( 8*n*x)
Time = 0.07 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.69 \[ \int \frac {\sin ^3\left (a+\frac {1}{3} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^2} \, dx=-\frac {{\left (4 \, c^{\frac {7}{3 \, n}} x e^{\left (\frac {\log \left (x^{n}\right )}{3 \, n} + 2 \, \log \left (x\right )\right )} \log \left (x\right ) \sin \left (3 \, a\right ) - 2 \, c^{\frac {1}{3 \, n}} x {\left (x^{n}\right )}^{\frac {1}{3 \, n}} \sin \left (3 \, a\right ) + 9 \, c^{\left (\frac {1}{n}\right )} x^{2} \sin \left (a\right ) + 18 \, c^{\frac {5}{3 \, n}} e^{\left (\frac {2 \, \log \left (x^{n}\right )}{3 \, n} + 2 \, \log \left (x\right )\right )} \sin \left (a\right )\right )} e^{\left (-\frac {\log \left (x^{n}\right )}{3 \, n} - 2 \, \log \left (x\right )\right )}}{32 \, c^{\frac {4}{3 \, n}} x} \] Input:
integrate(sin(a+1/3*(-1/n^2)^(1/2)*log(c*x^n))^3/x^2,x, algorithm="maxima" )
Output:
-1/32*(4*c^(7/3/n)*x*e^(1/3*log(x^n)/n + 2*log(x))*log(x)*sin(3*a) - 2*c^( 1/3/n)*x*(x^n)^(1/3/n)*sin(3*a) + 9*c^(1/n)*x^2*sin(a) + 18*c^(5/3/n)*e^(2 /3*log(x^n)/n + 2*log(x))*sin(a))*e^(-1/3*log(x^n)/n - 2*log(x))/(c^(4/3/n )*x)
\[ \int \frac {\sin ^3\left (a+\frac {1}{3} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^2} \, dx=\int { \frac {\sin \left (\frac {1}{3} \, \sqrt {-\frac {1}{n^{2}}} \log \left (c x^{n}\right ) + a\right )^{3}}{x^{2}} \,d x } \] Input:
integrate(sin(a+1/3*(-1/n^2)^(1/2)*log(c*x^n))^3/x^2,x, algorithm="giac")
Output:
integrate(sin(1/3*sqrt(-1/n^2)*log(c*x^n) + a)^3/x^2, x)
Timed out. \[ \int \frac {\sin ^3\left (a+\frac {1}{3} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^2} \, dx=\int \frac {{\sin \left (a+\frac {\ln \left (c\,x^n\right )\,\sqrt {-\frac {1}{n^2}}}{3}\right )}^3}{x^2} \,d x \] Input:
int(sin(a + (log(c*x^n)*(-1/n^2)^(1/2))/3)^3/x^2,x)
Output:
int(sin(a + (log(c*x^n)*(-1/n^2)^(1/2))/3)^3/x^2, x)
Time = 0.17 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.32 \[ \int \frac {\sin ^3\left (a+\frac {1}{3} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^2} \, dx=\frac {4 \cos \left (\frac {\mathrm {log}\left (x^{n} c \right ) i +3 a n}{3 n}\right ) \mathrm {log}\left (x^{n} c \right ) {\sin \left (\frac {\mathrm {log}\left (x^{n} c \right ) i +3 a n}{3 n}\right )}^{2} i -\cos \left (\frac {\mathrm {log}\left (x^{n} c \right ) i +3 a n}{3 n}\right ) \mathrm {log}\left (x^{n} c \right ) i -5 \cos \left (\frac {\mathrm {log}\left (x^{n} c \right ) i +3 a n}{3 n}\right ) {\sin \left (\frac {\mathrm {log}\left (x^{n} c \right ) i +3 a n}{3 n}\right )}^{2} i n -\cos \left (\frac {\mathrm {log}\left (x^{n} c \right ) i +3 a n}{3 n}\right ) i n +4 \,\mathrm {log}\left (x^{n} c \right ) {\sin \left (\frac {\mathrm {log}\left (x^{n} c \right ) i +3 a n}{3 n}\right )}^{3}-3 \,\mathrm {log}\left (x^{n} c \right ) \sin \left (\frac {\mathrm {log}\left (x^{n} c \right ) i +3 a n}{3 n}\right )-9 {\sin \left (\frac {\mathrm {log}\left (x^{n} c \right ) i +3 a n}{3 n}\right )}^{3} n}{8 n x} \] Input:
int(sin(a+1/3*(-1/n^2)^(1/2)*log(c*x^n))^3/x^2,x)
Output:
(4*cos((log(x**n*c)*i + 3*a*n)/(3*n))*log(x**n*c)*sin((log(x**n*c)*i + 3*a *n)/(3*n))**2*i - cos((log(x**n*c)*i + 3*a*n)/(3*n))*log(x**n*c)*i - 5*cos ((log(x**n*c)*i + 3*a*n)/(3*n))*sin((log(x**n*c)*i + 3*a*n)/(3*n))**2*i*n - cos((log(x**n*c)*i + 3*a*n)/(3*n))*i*n + 4*log(x**n*c)*sin((log(x**n*c)* i + 3*a*n)/(3*n))**3 - 3*log(x**n*c)*sin((log(x**n*c)*i + 3*a*n)/(3*n)) - 9*sin((log(x**n*c)*i + 3*a*n)/(3*n))**3*n)/(8*n*x)