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

Optimal. Leaf size=82 \[ \frac{1}{4} \sqrt{-\frac{1}{n^2}} n x e^{-a \sqrt{-\frac{1}{n^2}} n} \left (c x^n\right )^{\frac{1}{n}}-\frac{1}{2} \sqrt{-\frac{1}{n^2}} n x e^{a \sqrt{-\frac{1}{n^2}} n} \log (x) \left (c x^n\right )^{-1/n} \]

[Out]

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

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Rubi [A]  time = 0.0517531, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {4483, 4489} \[ \frac{1}{4} \sqrt{-\frac{1}{n^2}} n x e^{-a \sqrt{-\frac{1}{n^2}} n} \left (c x^n\right )^{\frac{1}{n}}-\frac{1}{2} \sqrt{-\frac{1}{n^2}} n x e^{a \sqrt{-\frac{1}{n^2}} n} \log (x) \left (c x^n\right )^{-1/n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 4483

Int[Sin[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_.), x_Symbol] :> Dist[x/(n*(c*x^n)^(1/n)), Subst[Int[x
^(1/n - 1)*Sin[d*(a + b*Log[x])]^p, x], x, c*x^n], x] /; FreeQ[{a, b, c, d, 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 \sin \left (a+\sqrt{-\frac{1}{n^2}} \log \left (c x^n\right )\right ) \, dx &=\frac{\left (x \left (c x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int x^{-1+\frac{1}{n}} \sin \left (a+\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 \left (c x^n\right )^{-1/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{2}{n}}\right ) \, dx,x,c x^n\right )\right )\\ &=\frac{1}{4} e^{-a \sqrt{-\frac{1}{n^2}} n} \sqrt{-\frac{1}{n^2}} n x \left (c x^n\right )^{\frac{1}{n}}-\frac{1}{2} e^{a \sqrt{-\frac{1}{n^2}} n} \sqrt{-\frac{1}{n^2}} n x \left (c x^n\right )^{-1/n} \log (x)\\ \end{align*}

Mathematica [F]  time = 0.0858786, size = 0, normalized size = 0. \[ \int \sin \left (a+\sqrt{-\frac{1}{n^2}} \log \left (c x^n\right )\right ) \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [F]  time = 0.034, size = 0, normalized size = 0. \begin{align*} \int \sin \left ( a+\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(sin(a+ln(c*x^n)*(-1/n^2)^(1/2)),x)

[Out]

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

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Maxima [A]  time = 1.17032, size = 39, normalized size = 0.48 \begin{align*} \frac{c^{\frac{2}{n}} x^{2} \sin \left (a\right ) + 2 \, \log \left (x\right ) \sin \left (a\right )}{4 \, c^{\left (\frac{1}{n}\right )}} \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)),x, algorithm="maxima")

[Out]

1/4*(c^(2/n)*x^2*sin(a) + 2*log(x)*sin(a))/c^(1/n)

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Fricas [C]  time = 0.466307, size = 101, normalized size = 1.23 \begin{align*} \frac{1}{4} \,{\left (i \, x^{2} - 2 i \, e^{\left (\frac{2 \,{\left (i \, a n - \log \left (c\right )\right )}}{n}\right )} \log \left (x\right )\right )} e^{\left (-\frac{i \, a n - \log \left (c\right )}{n}\right )} \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)),x, algorithm="fricas")

[Out]

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

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

[Out]

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

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Giac [A]  time = 1.27703, size = 1, normalized size = 0.01 \begin{align*} +\infty \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)),x, algorithm="giac")

[Out]

+Infinity