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

Optimal. Leaf size=172 \[ -\frac{3}{16} \sqrt{-\frac{1}{n^2}} n x^3 e^{a \sqrt{-\frac{1}{n^2}} n} \left (c x^n\right )^{-1/n}-\frac{1}{48} \sqrt{-\frac{1}{n^2}} n x^3 e^{-3 a \sqrt{-\frac{1}{n^2}} n} \left (c x^n\right )^{3/n}+\frac{3}{32} \sqrt{-\frac{1}{n^2}} n x^3 e^{-a \sqrt{-\frac{1}{n^2}} n} \left (c x^n\right )^{\frac{1}{n}}+\frac{1}{8} \sqrt{-\frac{1}{n^2}} n x^3 e^{3 a \sqrt{-\frac{1}{n^2}} n} \log (x) \left (c x^n\right )^{-3/n} \]

[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))

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Rubi [A]  time = 0.160594, antiderivative size = 172, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {4493, 4489} \[ -\frac{3}{16} \sqrt{-\frac{1}{n^2}} n x^3 e^{a \sqrt{-\frac{1}{n^2}} n} \left (c x^n\right )^{-1/n}-\frac{1}{48} \sqrt{-\frac{1}{n^2}} n x^3 e^{-3 a \sqrt{-\frac{1}{n^2}} n} \left (c x^n\right )^{3/n}+\frac{3}{32} \sqrt{-\frac{1}{n^2}} n x^3 e^{-a \sqrt{-\frac{1}{n^2}} n} \left (c x^n\right )^{\frac{1}{n}}+\frac{1}{8} \sqrt{-\frac{1}{n^2}} n x^3 e^{3 a \sqrt{-\frac{1}{n^2}} n} \log (x) \left (c x^n\right )^{-3/n} \]

Antiderivative was successfully verified.

[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 4493

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])

Rule 4489

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*} \int x^2 \sin ^3\left (a+\sqrt{-\frac{1}{n^2}} \log \left (c x^n\right )\right ) \, dx &=\frac{\left (x^3 \left (c x^n\right )^{-3/n}\right ) \operatorname{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 ) \operatorname{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]  time = 0.215729, size = 0, normalized size = 0. \[ \int x^2 \sin ^3\left (a+\sqrt{-\frac{1}{n^2}} \log \left (c x^n\right )\right ) \, dx \]

Verification is Not applicable to the result.

[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]

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Maple [F]  time = 0.072, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \left ( \sin \left ( a+\ln \left ( c{x}^{n} \right ) \sqrt{-{n}^{-2}} \right ) \right ) ^{3}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [A]  time = 1.19871, size = 122, normalized size = 0.71 \begin{align*} \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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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Fricas [C]  time = 0.470377, size = 207, normalized size = 1.2 \begin{align*} \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 )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError