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

Optimal. Leaf size=176 \[ -\frac{\sqrt{-\frac{1}{n^2}} n e^{3 a \sqrt{-\frac{1}{n^2}} n} \left (c x^n\right )^{-1/n}}{16 x}+\frac{9 \sqrt{-\frac{1}{n^2}} n e^{a \sqrt{-\frac{1}{n^2}} n} \left (c x^n\right )^{\left .-\frac{1}{3}\right /n}}{32 x}-\frac{9 \sqrt{-\frac{1}{n^2}} n e^{-a \sqrt{-\frac{1}{n^2}} n} \left (c x^n\right )^{\left .\frac{1}{3}\right /n}}{16 x}-\frac{\sqrt{-\frac{1}{n^2}} n e^{-3 a \sqrt{-\frac{1}{n^2}} n} \log (x) \left (c x^n\right )^{\frac{1}{n}}}{8 x} \]

[Out]

-(E^(3*a*Sqrt[-n^(-2)]*n)*Sqrt[-n^(-2)]*n)/(16*x*(c*x^n)^n^(-1)) + (9*E^(a*Sqrt[-n^(-2)]*n)*Sqrt[-n^(-2)]*n)/(
32*x*(c*x^n)^(1/(3*n))) - (9*Sqrt[-n^(-2)]*n*(c*x^n)^(1/(3*n)))/(16*E^(a*Sqrt[-n^(-2)]*n)*x) - (Sqrt[-n^(-2)]*
n*(c*x^n)^n^(-1)*Log[x])/(8*E^(3*a*Sqrt[-n^(-2)]*n)*x)

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Rubi [A]  time = 0.13176, antiderivative size = 176, 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} \[ -\frac{\sqrt{-\frac{1}{n^2}} n e^{3 a \sqrt{-\frac{1}{n^2}} n} \left (c x^n\right )^{-1/n}}{16 x}+\frac{9 \sqrt{-\frac{1}{n^2}} n e^{a \sqrt{-\frac{1}{n^2}} n} \left (c x^n\right )^{\left .-\frac{1}{3}\right /n}}{32 x}-\frac{9 \sqrt{-\frac{1}{n^2}} n e^{-a \sqrt{-\frac{1}{n^2}} n} \left (c x^n\right )^{\left .\frac{1}{3}\right /n}}{16 x}-\frac{\sqrt{-\frac{1}{n^2}} n e^{-3 a \sqrt{-\frac{1}{n^2}} n} \log (x) \left (c x^n\right )^{\frac{1}{n}}}{8 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(E^(3*a*Sqrt[-n^(-2)]*n)*Sqrt[-n^(-2)]*n)/(16*x*(c*x^n)^n^(-1)) + (9*E^(a*Sqrt[-n^(-2)]*n)*Sqrt[-n^(-2)]*n)/(
32*x*(c*x^n)^(1/(3*n))) - (9*Sqrt[-n^(-2)]*n*(c*x^n)^(1/(3*n)))/(16*E^(a*Sqrt[-n^(-2)]*n)*x) - (Sqrt[-n^(-2)]*
n*(c*x^n)^n^(-1)*Log[x])/(8*E^(3*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 ^3\left (a+\frac{1}{3} \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 ^3\left (a+\frac{1}{3} \sqrt{-\frac{1}{n^2}} \log (x)\right ) \, dx,x,c x^n\right )}{n x}\\ &=-\frac{\left (\sqrt{-\frac{1}{n^2}} \left (c x^n\right )^{\frac{1}{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{4}{3 n}}-3 e^{-a \sqrt{-\frac{1}{n^2}} n} x^{-1-\frac{2}{3 n}}-e^{3 a \sqrt{-\frac{1}{n^2}} n} x^{-\frac{2+n}{n}}\right ) \, dx,x,c x^n\right )}{8 x}\\ &=-\frac{e^{3 a \sqrt{-\frac{1}{n^2}} n} \sqrt{-\frac{1}{n^2}} n \left (c x^n\right )^{-1/n}}{16 x}+\frac{9 e^{a \sqrt{-\frac{1}{n^2}} n} \sqrt{-\frac{1}{n^2}} n \left (c x^n\right )^{\left .-\frac{1}{3}\right /n}}{32 x}-\frac{9 e^{-a \sqrt{-\frac{1}{n^2}} n} \sqrt{-\frac{1}{n^2}} n \left (c x^n\right )^{\left .\frac{1}{3}\right /n}}{16 x}-\frac{e^{-3 a \sqrt{-\frac{1}{n^2}} n} \sqrt{-\frac{1}{n^2}} n \left (c x^n\right )^{\frac{1}{n}} \log (x)}{8 x}\\ \end{align*}

Mathematica [F]  time = 0.170141, size = 0, normalized size = 0. \[ \int \frac{\sin ^3\left (a+\frac{1}{3} \sqrt{-\frac{1}{n^2}} \log \left (c x^n\right )\right )}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [F]  time = 0.069, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}} \left ( \sin \left ( a+{\frac{\ln \left ( c{x}^{n} \right ) }{3}\sqrt{-{n}^{-2}}} \right ) \right ) ^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.14978, size = 165, normalized size = 0.94 \begin{align*} -\frac{{\left (4 \, c^{\frac{7}{3 \, n}} x e^{\left (\frac{\log \left (x^{n}\right )}{3 \, n} + 2 \, \log \left (x\right )\right )} \log \left (x\right ) \sin \left (3 \, a\right ) - 2 \, c^{\frac{1}{3 \, n}} x{\left (x^{n}\right )}^{\frac{1}{3 \, n}} \sin \left (3 \, a\right ) + 9 \, c^{\left (\frac{1}{n}\right )} x^{2} \sin \left (a\right ) + 18 \, c^{\frac{5}{3 \, n}} e^{\left (\frac{2 \, \log \left (x^{n}\right )}{3 \, n} + 2 \, \log \left (x\right )\right )} \sin \left (a\right )\right )} e^{\left (-\frac{\log \left (x^{n}\right )}{3 \, n} - 2 \, \log \left (x\right )\right )}}{32 \, c^{\frac{4}{3 \, n}} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/32*(4*c^(7/3/n)*x*e^(1/3*log(x^n)/n + 2*log(x))*log(x)*sin(3*a) - 2*c^(1/3/n)*x*(x^n)^(1/3/n)*sin(3*a) + 9*
c^(1/n)*x^2*sin(a) + 18*c^(5/3/n)*e^(2/3*log(x^n)/n + 2*log(x))*sin(a))*e^(-1/3*log(x^n)/n - 2*log(x))/(c^(4/3
/n)*x)

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Fricas [C]  time = 0.478729, size = 244, normalized size = 1.39 \begin{align*} \frac{{\left (-12 i \, x^{2} \log \left (x^{\frac{1}{3}}\right ) - 18 i \, x^{\frac{4}{3}} e^{\left (\frac{2 \,{\left (3 i \, a n - \log \left (c\right )\right )}}{3 \, n}\right )} + 9 i \, x^{\frac{2}{3}} e^{\left (\frac{4 \,{\left (3 i \, a n - \log \left (c\right )\right )}}{3 \, n}\right )} - 2 i \, e^{\left (\frac{2 \,{\left (3 i \, a n - \log \left (c\right )\right )}}{n}\right )}\right )} e^{\left (-\frac{3 i \, a n - \log \left (c\right )}{n}\right )}}{32 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/32*(-12*I*x^2*log(x^(1/3)) - 18*I*x^(4/3)*e^(2/3*(3*I*a*n - log(c))/n) + 9*I*x^(2/3)*e^(4/3*(3*I*a*n - log(c
))/n) - 2*I*e^(2*(3*I*a*n - log(c))/n))*e^(-(3*I*a*n - log(c))/n)/x^2

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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