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

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

[Out]

-1/(4*x^2) + E^(2*a*Sqrt[-n^(-2)]*n)/(16*x^2*(c*x^n)^(2/n)) - ((c*x^n)^(2/n)*Log[x])/(4*E^(2*a*Sqrt[-n^(-2)]*n
)*x^2)

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Rubi [A]  time = 0.06197, antiderivative size = 76, 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} \[ \frac{e^{2 a \sqrt{-\frac{1}{n^2}} n} \left (c x^n\right )^{-2/n}}{16 x^2}-\frac{e^{-2 a \sqrt{-\frac{1}{n^2}} n} \log (x) \left (c x^n\right )^{2/n}}{4 x^2}-\frac{1}{4 x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/(4*x^2) + E^(2*a*Sqrt[-n^(-2)]*n)/(16*x^2*(c*x^n)^(2/n)) - ((c*x^n)^(2/n)*Log[x])/(4*E^(2*a*Sqrt[-n^(-2)]*n
)*x^2)

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+\sqrt{-\frac{1}{n^2}} \log \left (c x^n\right )\right )}{x^3} \, dx &=\frac{\left (c x^n\right )^{2/n} \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 x^2}\\ &=-\frac{\left (c x^n\right )^{2/n} \operatorname{Subst}\left (\int \left (\frac{e^{-2 a \sqrt{-\frac{1}{n^2}} n}}{x}-2 x^{-\frac{2+n}{n}}+e^{2 a \sqrt{-\frac{1}{n^2}} n} x^{-\frac{4+n}{n}}\right ) \, dx,x,c x^n\right )}{4 n x^2}\\ &=-\frac{1}{4 x^2}+\frac{e^{2 a \sqrt{-\frac{1}{n^2}} n} \left (c x^n\right )^{-2/n}}{16 x^2}-\frac{e^{-2 a \sqrt{-\frac{1}{n^2}} n} \left (c x^n\right )^{2/n} \log (x)}{4 x^2}\\ \end{align*}

Mathematica [F]  time = 0.128635, size = 0, normalized size = 0. \[ \int \frac{\sin ^2\left (a+\sqrt{-\frac{1}{n^2}} \log \left (c x^n\right )\right )}{x^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [F]  time = 0.061, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3}} \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(sin(a+ln(c*x^n)*(-1/n^2)^(1/2))^2/x^3,x)

[Out]

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

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Maxima [A]  time = 1.10118, size = 73, normalized size = 0.96 \begin{align*} -\frac{4 \, c^{\frac{4}{n}} x^{6} \cos \left (2 \, a\right ) \log \left (x\right ) + 4 \, c^{\frac{2}{n}} x^{4} - x^{2} \cos \left (2 \, a\right )}{16 \, c^{\frac{2}{n}} x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*(4*c^(4/n)*x^6*cos(2*a)*log(x) + 4*c^(2/n)*x^4 - x^2*cos(2*a))/(c^(2/n)*x^6)

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Fricas [C]  time = 0.469676, size = 151, normalized size = 1.99 \begin{align*} -\frac{{\left (4 \, x^{4} \log \left (x\right ) + 4 \, x^{2} e^{\left (\frac{2 \,{\left (i \, a n - \log \left (c\right )\right )}}{n}\right )} - e^{\left (\frac{4 \,{\left (i \, a n - \log \left (c\right )\right )}}{n}\right )}\right )} e^{\left (-\frac{2 \,{\left (i \, a n - \log \left (c\right )\right )}}{n}\right )}}{16 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [C]  time = 55.4519, size = 464, normalized size = 6.11 \begin{align*} \frac{\log{\left (x \right )} \sin ^{2}{\left (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^{2}} - \frac{\log{\left (x \right )} \cos ^{2}{\left (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^{2}} - \frac{\sin ^{2}{\left (a + i n \sqrt{\frac{1}{n^{2}}} \log{\left (x \right )} + i \sqrt{\frac{1}{n^{2}}} \log{\left (c \right )} \right )}}{2 x^{2}} + \frac{\log{\left (c \right )} \sin ^{2}{\left (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^{2}} - \frac{\log{\left (c \right )} \cos ^{2}{\left (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^{2}} + \frac{i \log{\left (x \right )} \sin{\left (a + i n \sqrt{\frac{1}{n^{2}}} \log{\left (x \right )} + i \sqrt{\frac{1}{n^{2}}} \log{\left (c \right )} \right )} \cos{\left (a + i n \sqrt{\frac{1}{n^{2}}} \log{\left (x \right )} + i \sqrt{\frac{1}{n^{2}}} \log{\left (c \right )} \right )}}{2 n x^{2} \sqrt{\frac{1}{n^{2}}}} - \frac{i \sin{\left (a + i n \sqrt{\frac{1}{n^{2}}} \log{\left (x \right )} + i \sqrt{\frac{1}{n^{2}}} \log{\left (c \right )} \right )} \cos{\left (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^{2} \sqrt{\frac{1}{n^{2}}}} + \frac{i \log{\left (c \right )} \sin{\left (a + i n \sqrt{\frac{1}{n^{2}}} \log{\left (x \right )} + i \sqrt{\frac{1}{n^{2}}} \log{\left (c \right )} \right )} \cos{\left (a + i n \sqrt{\frac{1}{n^{2}}} \log{\left (x \right )} + i \sqrt{\frac{1}{n^{2}}} \log{\left (c \right )} \right )}}{2 n^{2} x^{2} \sqrt{\frac{1}{n^{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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