\(\int \sin ^3(a+\frac {1}{3} \sqrt {-\frac {1}{n^2}} \log (c x^n)) \, dx\) [43]

Optimal result

Integrand size = 24, antiderivative size = 168 \[ \int \sin ^3\left (a+\frac {1}{3} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=-\frac {9}{16} e^{a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n x \left (c x^n\right )^{\left .-\frac {1}{3}\right /n}+\frac {9}{32} e^{-a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n x \left (c x^n\right )^{\left .\frac {1}{3}\right /n}-\frac {1}{16} e^{-3 a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n x \left (c x^n\right )^{\frac {1}{n}}+\frac {1}{8} e^{3 a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n x \left (c x^n\right )^{-1/n} \log (x) \]

[Out]

-9/16*exp(a*n*(-1/n^2)^(1/2))*n*x*(-1/n^2)^(1/2)/((c*x^n)^(1/3/n))+9/32*n*x*(c*x^n)^(1/3/n)*(-1/n^2)^(1/2)/exp
(a*n*(-1/n^2)^(1/2))-1/16*n*x*(c*x^n)^(1/n)*(-1/n^2)^(1/2)/exp(3*a*n*(-1/n^2)^(1/2))+1/8*exp(3*a*n*(-1/n^2)^(1
/2))*n*x*ln(x)*(-1/n^2)^(1/2)/((c*x^n)^(1/n))

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {4571, 4577} \[ \int \sin ^3\left (a+\frac {1}{3} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=-\frac {9}{16} \sqrt {-\frac {1}{n^2}} n x e^{a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{\left .-\frac {1}{3}\right /n}+\frac {9}{32} \sqrt {-\frac {1}{n^2}} n x e^{-a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{\left .\frac {1}{3}\right /n}-\frac {1}{16} \sqrt {-\frac {1}{n^2}} n x e^{-3 a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{\frac {1}{n}}+\frac {1}{8} \sqrt {-\frac {1}{n^2}} n x e^{3 a \sqrt {-\frac {1}{n^2}} n} \log (x) \left (c x^n\right )^{-1/n} \]

[In]

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

[Out]

(-9*E^(a*Sqrt[-n^(-2)]*n)*Sqrt[-n^(-2)]*n*x)/(16*(c*x^n)^(1/(3*n))) + (9*Sqrt[-n^(-2)]*n*x*(c*x^n)^(1/(3*n)))/
(32*E^(a*Sqrt[-n^(-2)]*n)) - (Sqrt[-n^(-2)]*n*x*(c*x^n)^n^(-1))/(16*E^(3*a*Sqrt[-n^(-2)]*n)) + (E^(3*a*Sqrt[-n
^(-2)]*n)*Sqrt[-n^(-2)]*n*x*Log[x])/(8*(c*x^n)^n^(-1))

Rule 4571

Int[Sin[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_.), x_Symbol] :> Dist[x/(n*(c*x^n)^(1/n)), Subst[Int[x
^(1/n - 1)*Sin[d*(a + b*Log[x])]^p, x], x, c*x^n], x] /; FreeQ[{a, b, c, d, n, p}, x] && (NeQ[c, 1] || NeQ[n,
1])

Rule 4577

Int[((e_.)*(x_))^(m_.)*Sin[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_.), x_Symbol] :> Dist[(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]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (x \left (c x^n\right )^{-1/n}\right ) \text {Subst}\left (\int x^{-1+\frac {1}{n}} \sin ^3\left (a+\frac {1}{3} \sqrt {-\frac {1}{n^2}} \log (x)\right ) \, dx,x,c x^n\right )}{n} \\ & = \frac {1}{8} \left (\sqrt {-\frac {1}{n^2}} x \left (c x^n\right )^{-1/n}\right ) \text {Subst}\left (\int \left (\frac {e^{3 a \sqrt {-\frac {1}{n^2}} n}}{x}-3 e^{a \sqrt {-\frac {1}{n^2}} n} x^{-1+\frac {2}{3 n}}+3 e^{-a \sqrt {-\frac {1}{n^2}} n} x^{-1+\frac {4}{3 n}}-e^{-3 a \sqrt {-\frac {1}{n^2}} n} x^{-1+\frac {2}{n}}\right ) \, dx,x,c x^n\right ) \\ & = -\frac {9}{16} e^{a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n x \left (c x^n\right )^{\left .-\frac {1}{3}\right /n}+\frac {9}{32} e^{-a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n x \left (c x^n\right )^{\left .\frac {1}{3}\right /n}-\frac {1}{16} e^{-3 a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n x \left (c x^n\right )^{\frac {1}{n}}+\frac {1}{8} e^{3 a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n x \left (c x^n\right )^{-1/n} \log (x) \\ \end{align*}

Mathematica [F]

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

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.25 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.50 \[ \int \sin ^3\left (a+\frac {1}{3} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=\frac {1}{32} \, {\left (9 i \, x^{\frac {4}{3}} e^{\left (\frac {2 \, {\left (3 i \, a n - \log \left (c\right )\right )}}{3 \, n}\right )} - 2 i \, x^{2} + 12 i \, e^{\left (\frac {2 \, {\left (3 i \, a n - \log \left (c\right )\right )}}{n}\right )} \log \left (x^{\frac {1}{3}}\right ) - 18 i \, x^{\frac {2}{3}} e^{\left (\frac {4 \, {\left (3 i \, a n - \log \left (c\right )\right )}}{3 \, n}\right )}\right )} e^{\left (-\frac {3 i \, a n - \log \left (c\right )}{n}\right )} \]

[In]

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

[Out]

1/32*(9*I*x^(4/3)*e^(2/3*(3*I*a*n - log(c))/n) - 2*I*x^2 + 12*I*e^(2*(3*I*a*n - log(c))/n)*log(x^(1/3)) - 18*I
*x^(2/3)*e^(4/3*(3*I*a*n - log(c))/n))*e^(-(3*I*a*n - log(c))/n)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.63 \[ \int \sin ^3\left (a+\frac {1}{3} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=-\frac {4 \, c^{\frac {1}{3 \, n}} {\left (x^{n}\right )}^{\frac {1}{3 \, n}} \log \left (x\right ) \sin \left (3 \, a\right ) - 9 \, c^{\frac {5}{3 \, n}} x {\left (x^{n}\right )}^{\frac {2}{3 \, n}} \sin \left (a\right ) + 2 \, c^{\frac {7}{3 \, n}} e^{\left (\frac {\log \left (x^{n}\right )}{3 \, n} + 2 \, \log \left (x\right )\right )} \sin \left (3 \, a\right ) - 18 \, c^{\left (\frac {1}{n}\right )} x \sin \left (a\right )}{32 \, c^{\frac {4}{3 \, n}} {\left (x^{n}\right )}^{\frac {1}{3 \, n}}} \]

[In]

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

[Out]

-1/32*(4*c^(1/3/n)*(x^n)^(1/3/n)*log(x)*sin(3*a) - 9*c^(5/3/n)*x*(x^n)^(2/3/n)*sin(a) + 2*c^(7/3/n)*e^(1/3*log
(x^n)/n + 2*log(x))*sin(3*a) - 18*c^(1/n)*x*sin(a))/(c^(4/3/n)*(x^n)^(1/3/n))

Giac [F(-2)]

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

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 27.26 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.92 \[ \int \sin ^3\left (a+\frac {1}{3} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=-x\,{\mathrm {e}}^{-a\,1{}\mathrm {i}}\,\frac {1}{{\left (c\,x^n\right )}^{\frac {\sqrt {-\frac {1}{n^2}}\,1{}\mathrm {i}}{3}}}\,\left (\frac {9\,n\,\sqrt {-\frac {1}{n^2}}}{64}-\frac {27}{64}{}\mathrm {i}\right )-x\,{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\left (c\,x^n\right )}^{\frac {\sqrt {-\frac {1}{n^2}}\,1{}\mathrm {i}}{3}}\,\left (\frac {9\,n\,\sqrt {-\frac {1}{n^2}}}{64}+\frac {27}{64}{}\mathrm {i}\right )+\frac {x\,{\mathrm {e}}^{-a\,3{}\mathrm {i}}\,\frac {1}{{\left (c\,x^n\right )}^{\sqrt {-\frac {1}{n^2}}\,1{}\mathrm {i}}}}{8\,n\,\sqrt {-\frac {1}{n^2}}+8{}\mathrm {i}}+\frac {x\,{\mathrm {e}}^{a\,3{}\mathrm {i}}\,{\left (c\,x^n\right )}^{\sqrt {-\frac {1}{n^2}}\,1{}\mathrm {i}}}{8\,n\,\sqrt {-\frac {1}{n^2}}-8{}\mathrm {i}} \]

[In]

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

[Out]

(x*exp(-a*3i)/(c*x^n)^((-1/n^2)^(1/2)*1i))/(8*n*(-1/n^2)^(1/2) + 8i) - x*exp(a*1i)*(c*x^n)^(((-1/n^2)^(1/2)*1i
)/3)*((9*n*(-1/n^2)^(1/2))/64 + 27i/64) - x*exp(-a*1i)/(c*x^n)^(((-1/n^2)^(1/2)*1i)/3)*((9*n*(-1/n^2)^(1/2))/6
4 - 27i/64) + (x*exp(a*3i)*(c*x^n)^((-1/n^2)^(1/2)*1i))/(8*n*(-1/n^2)^(1/2) - 8i)