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

Optimal. Leaf size=88 \[ \frac{1}{12} \sqrt{-\frac{1}{n^2}} n x^3 e^{-a \sqrt{-\frac{1}{n^2}} n} \left (c x^n\right )^{3/n}-\frac{1}{2} \sqrt{-\frac{1}{n^2}} n x^3 e^{a \sqrt{-\frac{1}{n^2}} n} \log (x) \left (c x^n\right )^{-3/n} \]

[Out]

(Sqrt[-n^(-2)]*n*x^3*(c*x^n)^(3/n))/(12*E^(a*Sqrt[-n^(-2)]*n)) - (E^(a*Sqrt[-n^(-2)]*n)*Sqrt[-n^(-2)]*n*x^3*Lo
g[x])/(2*(c*x^n)^(3/n))

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Rubi [A]  time = 0.0988647, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {4493, 4489} \[ \frac{1}{12} \sqrt{-\frac{1}{n^2}} n x^3 e^{-a \sqrt{-\frac{1}{n^2}} n} \left (c x^n\right )^{3/n}-\frac{1}{2} \sqrt{-\frac{1}{n^2}} n x^3 e^{a \sqrt{-\frac{1}{n^2}} n} \log (x) \left (c x^n\right )^{-3/n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[-n^(-2)]*n*x^3*(c*x^n)^(3/n))/(12*E^(a*Sqrt[-n^(-2)]*n)) - (E^(a*Sqrt[-n^(-2)]*n)*Sqrt[-n^(-2)]*n*x^3*Lo
g[x])/(2*(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 \left (a+3 \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 \left (a+3 \sqrt{-\frac{1}{n^2}} \log (x)\right ) \, dx,x,c x^n\right )}{n}\\ &=-\left (\frac{1}{2} \left (\sqrt{-\frac{1}{n^2}} x^3 \left (c x^n\right )^{-3/n}\right ) \operatorname{Subst}\left (\int \left (\frac{e^{a \sqrt{-\frac{1}{n^2}} n}}{x}-e^{-a \sqrt{-\frac{1}{n^2}} n} x^{-1+\frac{6}{n}}\right ) \, dx,x,c x^n\right )\right )\\ &=\frac{1}{12} e^{-a \sqrt{-\frac{1}{n^2}} n} \sqrt{-\frac{1}{n^2}} n x^3 \left (c x^n\right )^{3/n}-\frac{1}{2} e^{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 = 0.145668, size = 0, normalized size = 0. \[ \int x^2 \sin \left (a+3 \sqrt{-\frac{1}{n^2}} \log \left (c x^n\right )\right ) \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [F]  time = 0.037, size = 0, normalized size = 0. \begin{align*} \int{x}^{2}\sin \left ( a+3\,\ln \left ( c{x}^{n} \right ) \sqrt{-{n}^{-2}} \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.09538, size = 42, normalized size = 0.48 \begin{align*} \frac{c^{\frac{6}{n}} x^{6} \sin \left (a\right ) + 6 \, \log \left (x\right ) \sin \left (a\right )}{12 \, c^{\frac{3}{n}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(c^(6/n)*x^6*sin(a) + 6*log(x)*sin(a))/c^(3/n)

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Fricas [C]  time = 0.468447, size = 108, normalized size = 1.23 \begin{align*} \frac{1}{12} \,{\left (i \, x^{6} - 6 i \, e^{\left (\frac{2 \,{\left (i \, a n - 3 \, \log \left (c\right )\right )}}{n}\right )} \log \left (x\right )\right )} e^{\left (-\frac{i \, a n - 3 \, \log \left (c\right )}{n}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(I*x^6 - 6*I*e^(2*(I*a*n - 3*log(c))/n)*log(x))*e^(-(I*a*n - 3*log(c))/n)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \sin{\left (a + 3 \sqrt{- \frac{1}{n^{2}}} \log{\left (c x^{n} \right )} \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.46388, size = 1, normalized size = 0.01 \begin{align*} +\infty \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

+Infinity