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

Optimal. Leaf size=74 \[ \frac{e^{2 a \sqrt{-\frac{1}{n^2}} n} \left (c x^n\right )^{-1/n}}{8 x}-\frac{e^{-2 a \sqrt{-\frac{1}{n^2}} n} \log (x) \left (c x^n\right )^{\frac{1}{n}}}{4 x}-\frac{1}{2 x} \]

[Out]

-1/(2*x) + E^(2*a*Sqrt[-n^(-2)]*n)/(8*x*(c*x^n)^n^(-1)) - ((c*x^n)^n^(-1)*Log[x])/(4*E^(2*a*Sqrt[-n^(-2)]*n)*x
)

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Rubi [A]  time = 0.068497, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {4493, 4489} \[ \frac{e^{2 a \sqrt{-\frac{1}{n^2}} n} \left (c x^n\right )^{-1/n}}{8 x}-\frac{e^{-2 a \sqrt{-\frac{1}{n^2}} n} \log (x) \left (c x^n\right )^{\frac{1}{n}}}{4 x}-\frac{1}{2 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/(2*x) + E^(2*a*Sqrt[-n^(-2)]*n)/(8*x*(c*x^n)^n^(-1)) - ((c*x^n)^n^(-1)*Log[x])/(4*E^(2*a*Sqrt[-n^(-2)]*n)*x
)

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 \frac{\sin ^2\left (a+\frac{1}{2} \sqrt{-\frac{1}{n^2}} \log \left (c x^n\right )\right )}{x^2} \, dx &=\frac{\left (c x^n\right )^{\frac{1}{n}} \operatorname{Subst}\left (\int x^{-1-\frac{1}{n}} \sin ^2\left (a+\frac{1}{2} \sqrt{-\frac{1}{n^2}} \log (x)\right ) \, dx,x,c x^n\right )}{n x}\\ &=-\frac{\left (c x^n\right )^{\frac{1}{n}} \operatorname{Subst}\left (\int \left (\frac{e^{-2 a \sqrt{-\frac{1}{n^2}} n}}{x}-2 x^{-\frac{1+n}{n}}+e^{2 a \sqrt{-\frac{1}{n^2}} n} x^{-\frac{2+n}{n}}\right ) \, dx,x,c x^n\right )}{4 n x}\\ &=-\frac{1}{2 x}+\frac{e^{2 a \sqrt{-\frac{1}{n^2}} n} \left (c x^n\right )^{-1/n}}{8 x}-\frac{e^{-2 a \sqrt{-\frac{1}{n^2}} n} \left (c x^n\right )^{\frac{1}{n}} \log (x)}{4 x}\\ \end{align*}

Mathematica [F]  time = 0.149265, size = 0, normalized size = 0. \[ \int \frac{\sin ^2\left (a+\frac{1}{2} \sqrt{-\frac{1}{n^2}} \log \left (c x^n\right )\right )}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [F]  time = 0.065, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}} \left ( \sin \left ( a+{\frac{\ln \left ( c{x}^{n} \right ) }{2}\sqrt{-{n}^{-2}}} \right ) \right ) ^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.1636, size = 65, normalized size = 0.88 \begin{align*} -\frac{2 \, c^{\frac{2}{n}} x^{3} \cos \left (2 \, a\right ) \log \left (x\right ) + 4 \, c^{\left (\frac{1}{n}\right )} x^{2} - x \cos \left (2 \, a\right )}{8 \, c^{\left (\frac{1}{n}\right )} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(2*c^(2/n)*x^3*cos(2*a)*log(x) + 4*c^(1/n)*x^2 - x*cos(2*a))/(c^(1/n)*x^3)

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Fricas [C]  time = 0.471367, size = 150, normalized size = 2.03 \begin{align*} -\frac{{\left (2 \, x^{2} \log \left (x\right ) + 4 \, x e^{\left (\frac{2 i \, a n - \log \left (c\right )}{n}\right )} - e^{\left (\frac{2 \,{\left (2 i \, a n - \log \left (c\right )\right )}}{n}\right )}\right )} e^{\left (-\frac{2 i \, a n - \log \left (c\right )}{n}\right )}}{8 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(2*x^2*log(x) + 4*x*e^((2*I*a*n - log(c))/n) - e^(2*(2*I*a*n - log(c))/n))*e^(-(2*I*a*n - log(c))/n)/x^2

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Sympy [C]  time = 75.7666, size = 240, normalized size = 3.24 \begin{align*} \frac{i n \sqrt{\frac{1}{n^{2}}} \log{\left (x \right )} \sin{\left (2 a + i n \sqrt{\frac{1}{n^{2}}} \log{\left (x \right )} + i \sqrt{\frac{1}{n^{2}}} \log{\left (c \right )} \right )}}{4 x} + \frac{i n \sqrt{\frac{1}{n^{2}}} \sin{\left (2 a + i n \sqrt{\frac{1}{n^{2}}} \log{\left (x \right )} + i \sqrt{\frac{1}{n^{2}}} \log{\left (c \right )} \right )}}{4 x} + \frac{i \sqrt{\frac{1}{n^{2}}} \log{\left (c \right )} \sin{\left (2 a + i n \sqrt{\frac{1}{n^{2}}} \log{\left (x \right )} + i \sqrt{\frac{1}{n^{2}}} \log{\left (c \right )} \right )}}{4 x} - \frac{\log{\left (x \right )} \cos{\left (2 a + i n \sqrt{\frac{1}{n^{2}}} \log{\left (x \right )} + i \sqrt{\frac{1}{n^{2}}} \log{\left (c \right )} \right )}}{4 x} - \frac{1}{2 x} - \frac{\log{\left (c \right )} \cos{\left (2 a + i n \sqrt{\frac{1}{n^{2}}} \log{\left (x \right )} + i \sqrt{\frac{1}{n^{2}}} \log{\left (c \right )} \right )}}{4 n x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

I*n*sqrt(n**(-2))*log(x)*sin(2*a + I*n*sqrt(n**(-2))*log(x) + I*sqrt(n**(-2))*log(c))/(4*x) + I*n*sqrt(n**(-2)
)*sin(2*a + I*n*sqrt(n**(-2))*log(x) + I*sqrt(n**(-2))*log(c))/(4*x) + I*sqrt(n**(-2))*log(c)*sin(2*a + I*n*sq
rt(n**(-2))*log(x) + I*sqrt(n**(-2))*log(c))/(4*x) - log(x)*cos(2*a + I*n*sqrt(n**(-2))*log(x) + I*sqrt(n**(-2
))*log(c))/(4*x) - 1/(2*x) - log(c)*cos(2*a + I*n*sqrt(n**(-2))*log(x) + I*sqrt(n**(-2))*log(c))/(4*n*x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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