\(\int \frac {\sin ^3(a+\frac {2}{3} \sqrt {-\frac {1}{n^2}} \log (c x^n))}{x^3} \, dx\) [46]

Optimal result

Integrand size = 28, antiderivative size = 178 \[ \int \frac {\sin ^3\left (a+\frac {2}{3} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^3} \, dx=-\frac {e^{3 a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n \left (c x^n\right )^{-2/n}}{32 x^2}+\frac {9 e^{a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n \left (c x^n\right )^{\left .-\frac {2}{3}\right /n}}{64 x^2}-\frac {9 e^{-a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n \left (c x^n\right )^{\left .\frac {2}{3}\right /n}}{32 x^2}-\frac {e^{-3 a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n \left (c x^n\right )^{2/n} \log (x)}{8 x^2} \] Output:

-1/32*exp(3*a*(-1/n^2)^(1/2)*n)*(-1/n^2)^(1/2)*n/x^2/((c*x^n)^(2/n))+9/64* 
exp(a*(-1/n^2)^(1/2)*n)*(-1/n^2)^(1/2)*n/x^2/((c*x^n)^(2/3/n))-9/32*(-1/n^ 
2)^(1/2)*n*(c*x^n)^(2/3/n)/exp(a*(-1/n^2)^(1/2)*n)/x^2-1/8*(-1/n^2)^(1/2)* 
n*(c*x^n)^(2/n)*ln(x)/exp(3*a*(-1/n^2)^(1/2)*n)/x^2
                                                                                    
                                                                                    
 

Mathematica [F]

\[ \int \frac {\sin ^3\left (a+\frac {2}{3} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^3} \, dx=\int \frac {\sin ^3\left (a+\frac {2}{3} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^3} \, dx \] Input:

Integrate[Sin[a + (2*Sqrt[-n^(-2)]*Log[c*x^n])/3]^3/x^3,x]
 

Output:

Integrate[Sin[a + (2*Sqrt[-n^(-2)]*Log[c*x^n])/3]^3/x^3, x]
 

Rubi [A] (warning: unable to verify)

Time = 0.39 (sec) , antiderivative size = 146, 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.

\(\displaystyle \int \frac {\sin ^3\left (a+\frac {2}{3} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^3} \, dx\)

\(\Big \downarrow \) 4996

\(\displaystyle \frac {\left (c x^n\right )^{2/n} \int \left (c x^n\right )^{-1-\frac {2}{n}} \sin ^3\left (a+\frac {2}{3} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )d\left (c x^n\right )}{n x^2}\)

\(\Big \downarrow \) 4992

\(\displaystyle -\frac {\sqrt {-\frac {1}{n^2}} \left (c x^n\right )^{2/n} \int \left (3 e^{a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{-1-\frac {8}{3 n}}-3 e^{-a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{-1-\frac {4}{3 n}}-e^{3 a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{-\frac {n+4}{n}}+\frac {e^{-3 a \sqrt {-\frac {1}{n^2}} n} x^{-n}}{c}\right )d\left (c x^n\right )}{8 x^2}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\sqrt {-\frac {1}{n^2}} \left (c x^n\right )^{2/n} \left (\frac {1}{4} n e^{3 a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{-4/n}-\frac {9}{8} n e^{a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{\left .-\frac {8}{3}\right /n}+\frac {9}{4} n e^{-a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{\left .-\frac {4}{3}\right /n}+e^{-3 a \sqrt {-\frac {1}{n^2}} n} \log \left (c x^n\right )\right )}{8 x^2}\)

Input:

Int[Sin[a + (2*Sqrt[-n^(-2)]*Log[c*x^n])/3]^3/x^3,x]
 

Output:

-1/8*(Sqrt[-n^(-2)]*(c*x^n)^(2/n)*((E^(3*a*Sqrt[-n^(-2)]*n)*n)/(4*(c*x^n)^ 
(4/n)) - (9*E^(a*Sqrt[-n^(-2)]*n)*n)/(8*(c*x^n)^(8/(3*n))) + (9*n)/(4*E^(a 
*Sqrt[-n^(-2)]*n)*(c*x^n)^(4/(3*n))) + Log[c*x^n]/E^(3*a*Sqrt[-n^(-2)]*n)) 
)/x^2
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 4992

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]
 

rule 4996

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])
 

Maple [A] (verified)

Time = 61.09 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.76

method	result	size
parallelrisch	\(\frac {-47 n \left (n +\frac {40 \ln \left (c \,x^{n}\right )}{47}\right ) \sqrt {-\frac {1}{n^{2}}}\, \cos \left (2 \sqrt {-\frac {1}{n^{2}}}\, \ln \left (c \,x^{n}\right )+3 a \right )+\left (-27 n -40 \ln \left (c \,x^{n}\right )\right ) \sin \left (2 \sqrt {-\frac {1}{n^{2}}}\, \ln \left (c \,x^{n}\right )+3 a \right )-45 n \left (\cos \left (a +\sqrt {-\frac {1}{n^{2}}}\, \ln \left (\left (c \,x^{n}\right )^{\frac {2}{3}}\right )\right ) n \sqrt {-\frac {1}{n^{2}}}+3 \sin \left (a +\sqrt {-\frac {1}{n^{2}}}\, \ln \left (\left (c \,x^{n}\right )^{\frac {2}{3}}\right )\right )\right )}{320 x^{2} n}\)	\(136\)

Input:

int(sin(a+2/3*(-1/n^2)^(1/2)*ln(c*x^n))^3/x^3,x,method=_RETURNVERBOSE)
 

Output:

1/320*(-47*n*(n+40/47*ln(c*x^n))*(-1/n^2)^(1/2)*cos(2*(-1/n^2)^(1/2)*ln(c* 
x^n)+3*a)+(-27*n-40*ln(c*x^n))*sin(2*(-1/n^2)^(1/2)*ln(c*x^n)+3*a)-45*n*(c 
os(a+(-1/n^2)^(1/2)*ln((c*x^n)^(2/3)))*n*(-1/n^2)^(1/2)+3*sin(a+(-1/n^2)^( 
1/2)*ln((c*x^n)^(2/3)))))/x^2/n
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.08 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.49 \[ \int \frac {\sin ^3\left (a+\frac {2}{3} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^3} \, dx=\frac {{\left (-24 i \, x^{4} \log \left (x^{\frac {1}{3}}\right ) - 18 i \, x^{\frac {8}{3}} e^{\left (\frac {2 \, {\left (3 i \, a n - 2 \, \log \left (c\right )\right )}}{3 \, n}\right )} + 9 i \, x^{\frac {4}{3}} e^{\left (\frac {4 \, {\left (3 i \, a n - 2 \, \log \left (c\right )\right )}}{3 \, n}\right )} - 2 i \, e^{\left (\frac {2 \, {\left (3 i \, a n - 2 \, \log \left (c\right )\right )}}{n}\right )}\right )} e^{\left (-\frac {3 i \, a n - 2 \, \log \left (c\right )}{n}\right )}}{64 \, x^{4}} \] Input:

integrate(sin(a+2/3*(-1/n^2)^(1/2)*log(c*x^n))^3/x^3,x, algorithm="fricas" 
)
 

Output:

1/64*(-24*I*x^4*log(x^(1/3)) - 18*I*x^(8/3)*e^(2/3*(3*I*a*n - 2*log(c))/n) 
 + 9*I*x^(4/3)*e^(4/3*(3*I*a*n - 2*log(c))/n) - 2*I*e^(2*(3*I*a*n - 2*log( 
c))/n))*e^(-(3*I*a*n - 2*log(c))/n)/x^4
 

Sympy [A] (verification not implemented)

Time = 64.67 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.03 \[ \int \frac {\sin ^3\left (a+\frac {2}{3} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^3} \, dx=- \frac {9 n \sqrt {- \frac {1}{n^{2}}} \cos {\left (a + \frac {2 \sqrt {- \frac {1}{n^{2}}} \log {\left (c x^{n} \right )}}{3} \right )}}{64 x^{2}} - \frac {\sqrt {- \frac {1}{n^{2}}} \log {\left (c x^{n} \right )} \cos {\left (3 a + 2 \sqrt {- \frac {1}{n^{2}}} \log {\left (c x^{n} \right )} \right )}}{8 x^{2}} - \frac {27 \sin {\left (a + \frac {2 \sqrt {- \frac {1}{n^{2}}} \log {\left (c x^{n} \right )}}{3} \right )}}{64 x^{2}} + \frac {\sin {\left (3 a + 2 \sqrt {- \frac {1}{n^{2}}} \log {\left (c x^{n} \right )} \right )}}{16 x^{2}} - \frac {\log {\left (c x^{n} \right )} \sin {\left (3 a + 2 \sqrt {- \frac {1}{n^{2}}} \log {\left (c x^{n} \right )} \right )}}{8 n x^{2}} \] Input:

integrate(sin(a+2/3*(-1/n**2)**(1/2)*ln(c*x**n))**3/x**3,x)
 

Output:

-9*n*sqrt(-1/n**2)*cos(a + 2*sqrt(-1/n**2)*log(c*x**n)/3)/(64*x**2) - sqrt 
(-1/n**2)*log(c*x**n)*cos(3*a + 2*sqrt(-1/n**2)*log(c*x**n))/(8*x**2) - 27 
*sin(a + 2*sqrt(-1/n**2)*log(c*x**n)/3)/(64*x**2) + sin(3*a + 2*sqrt(-1/n* 
*2)*log(c*x**n))/(16*x**2) - log(c*x**n)*sin(3*a + 2*sqrt(-1/n**2)*log(c*x 
**n))/(8*n*x**2)
 

Maxima [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.72 \[ \int \frac {\sin ^3\left (a+\frac {2}{3} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^3} \, dx=-\frac {{\left (8 \, c^{\frac {14}{3 \, n}} x^{2} e^{\left (\frac {2 \, \log \left (x^{n}\right )}{3 \, n} + 4 \, \log \left (x\right )\right )} \log \left (x\right ) \sin \left (3 \, a\right ) + 9 \, c^{\frac {2}{n}} x^{4} \sin \left (a\right ) - 2 \, c^{\frac {2}{3 \, n}} x^{2} {\left (x^{n}\right )}^{\frac {2}{3 \, n}} \sin \left (3 \, a\right ) + 18 \, c^{\frac {10}{3 \, n}} e^{\left (\frac {4 \, \log \left (x^{n}\right )}{3 \, n} + 4 \, \log \left (x\right )\right )} \sin \left (a\right )\right )} e^{\left (-\frac {2 \, \log \left (x^{n}\right )}{3 \, n} - 4 \, \log \left (x\right )\right )}}{64 \, c^{\frac {8}{3 \, n}} x^{2}} \] Input:

integrate(sin(a+2/3*(-1/n^2)^(1/2)*log(c*x^n))^3/x^3,x, algorithm="maxima" 
)
 

Output:

-1/64*(8*c^(14/3/n)*x^2*e^(2/3*log(x^n)/n + 4*log(x))*log(x)*sin(3*a) + 9* 
c^(2/n)*x^4*sin(a) - 2*c^(2/3/n)*x^2*(x^n)^(2/3/n)*sin(3*a) + 18*c^(10/3/n 
)*e^(4/3*log(x^n)/n + 4*log(x))*sin(a))*e^(-2/3*log(x^n)/n - 4*log(x))/(c^ 
(8/3/n)*x^2)
 

Giac [F]

\[ \int \frac {\sin ^3\left (a+\frac {2}{3} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^3} \, dx=\int { \frac {\sin \left (\frac {2}{3} \, \sqrt {-\frac {1}{n^{2}}} \log \left (c x^{n}\right ) + a\right )^{3}}{x^{3}} \,d x } \] Input:

integrate(sin(a+2/3*(-1/n^2)^(1/2)*log(c*x^n))^3/x^3,x, algorithm="giac")
 

Output:

integrate(sin(2/3*sqrt(-1/n^2)*log(c*x^n) + a)^3/x^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sin ^3\left (a+\frac {2}{3} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^3} \, dx=\int \frac {{\sin \left (a+\frac {2\,\ln \left (c\,x^n\right )\,\sqrt {-\frac {1}{n^2}}}{3}\right )}^3}{x^3} \,d x \] Input:

int(sin(a + (2*log(c*x^n)*(-1/n^2)^(1/2))/3)^3/x^3,x)
 

Output:

int(sin(a + (2*log(c*x^n)*(-1/n^2)^(1/2))/3)^3/x^3, x)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.36 \[ \int \frac {\sin ^3\left (a+\frac {2}{3} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^3} \, dx=\frac {8 \cos \left (\frac {2 \,\mathrm {log}\left (x^{n} c \right ) i +3 a n}{3 n}\right ) \mathrm {log}\left (x^{n} c \right ) {\sin \left (\frac {2 \,\mathrm {log}\left (x^{n} c \right ) i +3 a n}{3 n}\right )}^{2} i -2 \cos \left (\frac {2 \,\mathrm {log}\left (x^{n} c \right ) i +3 a n}{3 n}\right ) \mathrm {log}\left (x^{n} c \right ) i -5 \cos \left (\frac {2 \,\mathrm {log}\left (x^{n} c \right ) i +3 a n}{3 n}\right ) {\sin \left (\frac {2 \,\mathrm {log}\left (x^{n} c \right ) i +3 a n}{3 n}\right )}^{2} i n -\cos \left (\frac {2 \,\mathrm {log}\left (x^{n} c \right ) i +3 a n}{3 n}\right ) i n +8 \,\mathrm {log}\left (x^{n} c \right ) {\sin \left (\frac {2 \,\mathrm {log}\left (x^{n} c \right ) i +3 a n}{3 n}\right )}^{3}-6 \,\mathrm {log}\left (x^{n} c \right ) \sin \left (\frac {2 \,\mathrm {log}\left (x^{n} c \right ) i +3 a n}{3 n}\right )-9 {\sin \left (\frac {2 \,\mathrm {log}\left (x^{n} c \right ) i +3 a n}{3 n}\right )}^{3} n}{16 n \,x^{2}} \] Input:

int(sin(a+2/3*(-1/n^2)^(1/2)*log(c*x^n))^3/x^3,x)
 

Output:

(8*cos((2*log(x**n*c)*i + 3*a*n)/(3*n))*log(x**n*c)*sin((2*log(x**n*c)*i + 
 3*a*n)/(3*n))**2*i - 2*cos((2*log(x**n*c)*i + 3*a*n)/(3*n))*log(x**n*c)*i 
 - 5*cos((2*log(x**n*c)*i + 3*a*n)/(3*n))*sin((2*log(x**n*c)*i + 3*a*n)/(3 
*n))**2*i*n - cos((2*log(x**n*c)*i + 3*a*n)/(3*n))*i*n + 8*log(x**n*c)*sin 
((2*log(x**n*c)*i + 3*a*n)/(3*n))**3 - 6*log(x**n*c)*sin((2*log(x**n*c)*i 
+ 3*a*n)/(3*n)) - 9*sin((2*log(x**n*c)*i + 3*a*n)/(3*n))**3*n)/(16*n*x**2)