\(\int x \sin ^3(a+\frac {2}{3} \sqrt {-\frac {1}{n^2}} \log (c x^n)) \, dx\) [42]

Optimal result

Integrand size = 26, antiderivative size = 178 \[ \int x \sin ^3\left (a+\frac {2}{3} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=-\frac {9}{32} e^{a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n x^2 \left (c x^n\right )^{\left .-\frac {2}{3}\right /n}+\frac {9}{64} e^{-a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n x^2 \left (c x^n\right )^{\left .\frac {2}{3}\right /n}-\frac {1}{32} e^{-3 a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n x^2 \left (c x^n\right )^{2/n}+\frac {1}{8} e^{3 a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n x^2 \left (c x^n\right )^{-2/n} \log (x) \] Output:

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

Mathematica [F]

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

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

Output:

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

Rubi [A] (warning: unable to verify)

Time = 0.37 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.82, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, 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.

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

\(\Big \downarrow \) 4996

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

\(\Big \downarrow \) 4992

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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 [F]

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

Input:

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

Output:

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.07 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.47 \[ \int x \sin ^3\left (a+\frac {2}{3} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=\frac {1}{64} \, {\left (-2 i \, x^{4} + 9 i \, x^{\frac {8}{3}} e^{\left (\frac {2 \, {\left (3 i \, a n - 2 \, \log \left (c\right )\right )}}{3 \, n}\right )} - 18 i \, x^{\frac {4}{3}} e^{\left (\frac {4 \, {\left (3 i \, a n - 2 \, \log \left (c\right )\right )}}{3 \, n}\right )} + 24 i \, e^{\left (\frac {2 \, {\left (3 i \, a n - 2 \, \log \left (c\right )\right )}}{n}\right )} \log \left (x^{\frac {1}{3}}\right )\right )} e^{\left (-\frac {3 i \, a n - 2 \, \log \left (c\right )}{n}\right )} \] Input:

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
                                                                                    
                                                                                    
 

Maxima [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.63 \[ \int x \sin ^3\left (a+\frac {2}{3} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=\frac {9 \, c^{\frac {10}{3 \, n}} x^{2} {\left (x^{n}\right )}^{\frac {4}{3 \, n}} \sin \left (a\right ) - 8 \, c^{\frac {2}{3 \, n}} {\left (x^{n}\right )}^{\frac {2}{3 \, n}} \log \left (x\right ) \sin \left (3 \, a\right ) + 18 \, c^{\frac {2}{n}} x^{2} \sin \left (a\right ) - 2 \, c^{\frac {14}{3 \, n}} e^{\left (\frac {2 \, \log \left (x^{n}\right )}{3 \, n} + 4 \, \log \left (x\right )\right )} \sin \left (3 \, a\right )}{64 \, c^{\frac {8}{3 \, n}} {\left (x^{n}\right )}^{\frac {2}{3 \, n}}} \] Input:

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

Output:

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

Giac [F(-2)]

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

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

Output:

Exception raised: NotImplementedError >> unable to parse Giac output: ((-9 
*i)*sageVARn^4*sageVARx^2*exp((-3*i)*sageVARa)*exp((2*sageVARn*abs(sageVAR 
n)*ln(sageVARx)+2*abs(sageVARn)*ln(sageVARc))/sageVARn^2)+27*i*sageVARn^4* 
sageVARx^2*exp((-i)
 

Mupad [B] (verification not implemented)

Time = 20.52 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.92 \[ \int x \sin ^3\left (a+\frac {2}{3} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=-x^2\,{\mathrm {e}}^{-a\,1{}\mathrm {i}}\,\frac {1}{{\left (c\,x^n\right )}^{\frac {\sqrt {-\frac {1}{n^2}}\,2{}\mathrm {i}}{3}}}\,\left (\frac {9\,n\,\sqrt {-\frac {1}{n^2}}}{128}-\frac {27}{128}{}\mathrm {i}\right )-x^2\,{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\left (c\,x^n\right )}^{\frac {\sqrt {-\frac {1}{n^2}}\,2{}\mathrm {i}}{3}}\,\left (\frac {9\,n\,\sqrt {-\frac {1}{n^2}}}{128}+\frac {27}{128}{}\mathrm {i}\right )+\frac {x^2\,{\mathrm {e}}^{-a\,3{}\mathrm {i}}\,\frac {1}{{\left (c\,x^n\right )}^{\sqrt {-\frac {1}{n^2}}\,2{}\mathrm {i}}}}{16\,n\,\sqrt {-\frac {1}{n^2}}+16{}\mathrm {i}}+\frac {x^2\,{\mathrm {e}}^{a\,3{}\mathrm {i}}\,{\left (c\,x^n\right )}^{\sqrt {-\frac {1}{n^2}}\,2{}\mathrm {i}}}{16\,n\,\sqrt {-\frac {1}{n^2}}-16{}\mathrm {i}} \] Input:

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

Output:

(x^2*exp(-a*3i)/(c*x^n)^((-1/n^2)^(1/2)*2i))/(16*n*(-1/n^2)^(1/2) + 16i) - 
 x^2*exp(a*1i)*(c*x^n)^(((-1/n^2)^(1/2)*2i)/3)*((9*n*(-1/n^2)^(1/2))/128 + 
 27i/128) - x^2*exp(-a*1i)/(c*x^n)^(((-1/n^2)^(1/2)*2i)/3)*((9*n*(-1/n^2)^ 
(1/2))/128 - 27i/128) + (x^2*exp(a*3i)*(c*x^n)^((-1/n^2)^(1/2)*2i))/(16*n* 
(-1/n^2)^(1/2) - 16i)
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.36 \[ \int x \sin ^3\left (a+\frac {2}{3} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=\frac {x^{2} \left (-8 \cos \left (\frac {2 \,\mathrm {log}\left (x^{n} c \right ) i +3 a n}{3 n}\right ) \mathrm {log}\left (x^{n} c \right ) {\sin \left (\frac {2 \,\mathrm {log}\left (x^{n} c \right ) i +3 a n}{3 n}\right )}^{2} i +2 \cos \left (\frac {2 \,\mathrm {log}\left (x^{n} c \right ) i +3 a n}{3 n}\right ) \mathrm {log}\left (x^{n} c \right ) i -5 \cos \left (\frac {2 \,\mathrm {log}\left (x^{n} c \right ) i +3 a n}{3 n}\right ) {\sin \left (\frac {2 \,\mathrm {log}\left (x^{n} c \right ) i +3 a n}{3 n}\right )}^{2} i n -\cos \left (\frac {2 \,\mathrm {log}\left (x^{n} c \right ) i +3 a n}{3 n}\right ) i n +8 \,\mathrm {log}\left (x^{n} c \right ) {\sin \left (\frac {2 \,\mathrm {log}\left (x^{n} c \right ) i +3 a n}{3 n}\right )}^{3}-6 \,\mathrm {log}\left (x^{n} c \right ) \sin \left (\frac {2 \,\mathrm {log}\left (x^{n} c \right ) i +3 a n}{3 n}\right )+9 {\sin \left (\frac {2 \,\mathrm {log}\left (x^{n} c \right ) i +3 a n}{3 n}\right )}^{3} n \right )}{16 n} \] Input:

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

Output:

(x**2*( - 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)