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

Optimal. Leaf size=76 \[ -\frac{1}{16} x^2 e^{-2 a \sqrt{-\frac{1}{n^2}} n} \left (c x^n\right )^{2/n}-\frac{1}{4} x^2 e^{2 a \sqrt{-\frac{1}{n^2}} n} \log (x) \left (c x^n\right )^{-2/n}+\frac{x^2}{4} \]

[Out]

x^2/4 - (x^2*(c*x^n)^(2/n))/(16*E^(2*a*Sqrt[-n^(-2)]*n)) - (E^(2*a*Sqrt[-n^(-2)]*n)*x^2*Log[x])/(4*(c*x^n)^(2/
n))

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Rubi [A]  time = 0.05761, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {4493, 4489} \[ -\frac{1}{16} x^2 e^{-2 a \sqrt{-\frac{1}{n^2}} n} \left (c x^n\right )^{2/n}-\frac{1}{4} x^2 e^{2 a \sqrt{-\frac{1}{n^2}} n} \log (x) \left (c x^n\right )^{-2/n}+\frac{x^2}{4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x^2/4 - (x^2*(c*x^n)^(2/n))/(16*E^(2*a*Sqrt[-n^(-2)]*n)) - (E^(2*a*Sqrt[-n^(-2)]*n)*x^2*Log[x])/(4*(c*x^n)^(2/
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 \sin ^2\left (a+\sqrt{-\frac{1}{n^2}} \log \left (c x^n\right )\right ) \, dx &=\frac{\left (x^2 \left (c x^n\right )^{-2/n}\right ) \operatorname{Subst}\left (\int x^{-1+\frac{2}{n}} \sin ^2\left (a+\sqrt{-\frac{1}{n^2}} \log (x)\right ) \, dx,x,c x^n\right )}{n}\\ &=-\frac{\left (x^2 \left (c x^n\right )^{-2/n}\right ) \operatorname{Subst}\left (\int \left (\frac{e^{2 a \sqrt{-\frac{1}{n^2}} n}}{x}-2 x^{-1+\frac{2}{n}}+e^{-2 a \sqrt{-\frac{1}{n^2}} n} x^{-1+\frac{4}{n}}\right ) \, dx,x,c x^n\right )}{4 n}\\ &=\frac{x^2}{4}-\frac{1}{16} e^{-2 a \sqrt{-\frac{1}{n^2}} n} x^2 \left (c x^n\right )^{2/n}-\frac{1}{4} e^{2 a \sqrt{-\frac{1}{n^2}} n} x^2 \left (c x^n\right )^{-2/n} \log (x)\\ \end{align*}

Mathematica [F]  time = 0.151923, size = 0, normalized size = 0. \[ \int x \sin ^2\left (a+\sqrt{-\frac{1}{n^2}} \log \left (c x^n\right )\right ) \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [F]  time = 0.076, size = 0, normalized size = 0. \begin{align*} \int x \left ( \sin \left ( a+\ln \left ( c{x}^{n} \right ) \sqrt{-{n}^{-2}} \right ) \right ) ^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.11589, size = 63, normalized size = 0.83 \begin{align*} -\frac{c^{\frac{4}{n}} x^{4} \cos \left (2 \, a\right ) - 4 \, c^{\frac{2}{n}} x^{2} + 4 \, \cos \left (2 \, a\right ) \log \left (x\right )}{16 \, c^{\frac{2}{n}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*(c^(4/n)*x^4*cos(2*a) - 4*c^(2/n)*x^2 + 4*cos(2*a)*log(x))/c^(2/n)

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Fricas [C]  time = 0.475053, size = 146, normalized size = 1.92 \begin{align*} -\frac{1}{16} \,{\left (x^{4} - 4 \, x^{2} e^{\left (\frac{2 \,{\left (i \, a n - \log \left (c\right )\right )}}{n}\right )} + 4 \, e^{\left (\frac{4 \,{\left (i \, a n - \log \left (c\right )\right )}}{n}\right )} \log \left (x\right )\right )} e^{\left (-\frac{2 \,{\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*sin(a+log(c*x^n)*(-1/n^2)^(1/2))^2,x, algorithm="fricas")

[Out]

-1/16*(x^4 - 4*x^2*e^(2*(I*a*n - log(c))/n) + 4*e^(4*(I*a*n - log(c))/n)*log(x))*e^(-2*(I*a*n - log(c))/n)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x \sin ^{2}{\left (a + \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*sin(a+ln(c*x**n)*(-1/n**2)**(1/2))**2,x)

[Out]

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

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Giac [A]  time = 1.82429, size = 1, normalized size = 0.01 \begin{align*} +\infty \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

+Infinity