\(\int x^2 \sin ^3(a+\sqrt {-\frac {1}{n^2}} \log (c x^n)) \, dx\) [41]

Optimal result

Integrand size = 25, antiderivative size = 172 \[ \int x^2 \sin ^3\left (a+\sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=-\frac {3}{16} e^{a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n x^3 \left (c x^n\right )^{-1/n}+\frac {3}{32} e^{-a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n x^3 \left (c x^n\right )^{\frac {1}{n}}-\frac {1}{48} e^{-3 a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n x^3 \left (c x^n\right )^{3/n}+\frac {1}{8} e^{3 a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n x^3 \left (c x^n\right )^{-3/n} \log (x) \]

[Out]

-3/16*exp(a*n*(-1/n^2)^(1/2))*n*x^3*(-1/n^2)^(1/2)/((c*x^n)^(1/n))+3/32*n*x^3*(c*x^n)^(1/n)*(-1/n^2)^(1/2)/exp
(a*n*(-1/n^2)^(1/2))-1/48*n*x^3*(c*x^n)^(3/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^3*ln(x)*(-1/n^2)^(1/2)/((c*x^n)^(3/n))

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {4581, 4577} \[ \int x^2 \sin ^3\left (a+\sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=-\frac {3}{16} \sqrt {-\frac {1}{n^2}} n x^3 e^{a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{-1/n}-\frac {1}{48} \sqrt {-\frac {1}{n^2}} n x^3 e^{-3 a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{3/n}+\frac {3}{32} \sqrt {-\frac {1}{n^2}} n x^3 e^{-a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{\frac {1}{n}}+\frac {1}{8} \sqrt {-\frac {1}{n^2}} n x^3 e^{3 a \sqrt {-\frac {1}{n^2}} n} \log (x) \left (c x^n\right )^{-3/n} \]

[In]

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

[Out]

(-3*E^(a*Sqrt[-n^(-2)]*n)*Sqrt[-n^(-2)]*n*x^3)/(16*(c*x^n)^n^(-1)) + (3*Sqrt[-n^(-2)]*n*x^3*(c*x^n)^n^(-1))/(3
2*E^(a*Sqrt[-n^(-2)]*n)) - (Sqrt[-n^(-2)]*n*x^3*(c*x^n)^(3/n))/(48*E^(3*a*Sqrt[-n^(-2)]*n)) + (E^(3*a*Sqrt[-n^
(-2)]*n)*Sqrt[-n^(-2)]*n*x^3*Log[x])/(8*(c*x^n)^(3/n))

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]

Rule 4581

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

Rubi steps \begin{align*} \text {integral}& = \frac {\left (x^3 \left (c x^n\right )^{-3/n}\right ) \text {Subst}\left (\int x^{-1+\frac {3}{n}} \sin ^3\left (a+\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^3 \left (c x^n\right )^{-3/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}{n}}+3 e^{-a \sqrt {-\frac {1}{n^2}} n} x^{-1+\frac {4}{n}}-e^{-3 a \sqrt {-\frac {1}{n^2}} n} x^{-1+\frac {6}{n}}\right ) \, dx,x,c x^n\right ) \\ & = -\frac {3}{16} e^{a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n x^3 \left (c x^n\right )^{-1/n}+\frac {3}{32} e^{-a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n x^3 \left (c x^n\right )^{\frac {1}{n}}-\frac {1}{48} e^{-3 a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n x^3 \left (c x^n\right )^{3/n}+\frac {1}{8} e^{3 a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n x^3 \left (c x^n\right )^{-3/n} \log (x) \\ \end{align*}

Mathematica [F]

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

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

int(x^2*sin(a+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 = 82, normalized size of antiderivative = 0.48 \[ \int x^2 \sin ^3\left (a+\sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=\frac {1}{96} \, {\left (-2 i \, x^{6} + 9 i \, x^{4} e^{\left (\frac {2 \, {\left (i \, a n - \log \left (c\right )\right )}}{n}\right )} - 18 i \, x^{2} e^{\left (\frac {4 \, {\left (i \, a n - \log \left (c\right )\right )}}{n}\right )} + 12 i \, e^{\left (\frac {6 \, {\left (i \, a n - \log \left (c\right )\right )}}{n}\right )} \log \left (x\right )\right )} e^{\left (-\frac {3 \, {\left (i \, a n - \log \left (c\right )\right )}}{n}\right )} \]

[In]

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

[Out]

1/96*(-2*I*x^6 + 9*I*x^4*e^(2*(I*a*n - log(c))/n) - 18*I*x^2*e^(4*(I*a*n - log(c))/n) + 12*I*e^(6*(I*a*n - log
(c))/n)*log(x))*e^(-3*(I*a*n - log(c))/n)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.52 \[ \int x^2 \sin ^3\left (a+\sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx=\frac {18 \, c^{\frac {2}{n}} x^{3} \sin \left (a\right ) - 12 \, {\left (x^{n}\right )}^{\left (\frac {1}{n}\right )} \log \left (x\right ) \sin \left (3 \, a\right ) - {\left (2 \, c^{\frac {6}{n}} x^{6} \sin \left (3 \, a\right ) - 9 \, c^{\frac {4}{n}} x^{4} \sin \left (a\right )\right )} {\left (x^{n}\right )}^{\left (\frac {1}{n}\right )}}{96 \, c^{\frac {3}{n}} {\left (x^{n}\right )}^{\left (\frac {1}{n}\right )}} \]

[In]

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

[Out]

1/96*(18*c^(2/n)*x^3*sin(a) - 12*(x^n)^(1/n)*log(x)*sin(3*a) - (2*c^(6/n)*x^6*sin(3*a) - 9*c^(4/n)*x^4*sin(a))
*(x^n)^(1/n))/(c^(3/n)*(x^n)^(1/n))

Giac [F(-2)]

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

[In]

integrate(x^2*sin(a+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^3*exp((-3*i)
*sageVARa)*exp((3*sageVARn*abs(sageVARn)*ln(sageVARx)+3*abs(sageVARn)*ln(sageVARc))/sageVARn^2)+27*i*sageVARn^
4*sageVARx^3*exp((-i)

Mupad [F(-1)]

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

[In]

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

[Out]

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