3.306 \(\int \csc ^p(a+\frac{i \log (c x^n)}{n (-2+p)}) \, dx\)

Optimal. Leaf size=96 \[ -\frac{e^{-2 i a} (2-p) x \left (c x^n\right )^{-\frac{2}{n (2-p)}} \left (1-e^{2 i a} \left (c x^n\right )^{\frac{2}{n (2-p)}}\right ) \csc ^p\left (a-\frac{i \log \left (c x^n\right )}{n (2-p)}\right )}{2 (1-p)} \]

[Out]

-((2 - p)*x*(1 - E^((2*I)*a)*(c*x^n)^(2/(n*(2 - p))))*Csc[a - (I*Log[c*x^n])/(n*(2 - p))]^p)/(2*E^((2*I)*a)*(1
- p)*(c*x^n)^(2/(n*(2 - p))))

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Rubi [A]  time = 0.0883863, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {4504, 4508, 261} \[ -\frac{e^{-2 i a} (2-p) x \left (c x^n\right )^{-\frac{2}{n (2-p)}} \left (1-e^{2 i a} \left (c x^n\right )^{\frac{2}{n (2-p)}}\right ) \csc ^p\left (a-\frac{i \log \left (c x^n\right )}{n (2-p)}\right )}{2 (1-p)} \]

Antiderivative was successfully verified.

[In]

Int[Csc[a + (I*Log[c*x^n])/(n*(-2 + p))]^p,x]

[Out]

-((2 - p)*x*(1 - E^((2*I)*a)*(c*x^n)^(2/(n*(2 - p))))*Csc[a - (I*Log[c*x^n])/(n*(2 - p))]^p)/(2*E^((2*I)*a)*(1
- p)*(c*x^n)^(2/(n*(2 - p))))

Rule 4504

Int[Csc[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_.), x_Symbol] :> Dist[x/(n*(c*x^n)^(1/n)), Subst[Int[x
^(1/n - 1)*Csc[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 4508

Int[Csc[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] :> Dist[(Csc[d*(a + b*Log[x])]^p*(1
- E^(2*I*a*d)*x^(2*I*b*d))^p)/x^(I*b*d*p), Int[((e*x)^m*x^(I*b*d*p))/(1 - E^(2*I*a*d)*x^(2*I*b*d))^p, x], x]
/; FreeQ[{a, b, d, e, m, p}, x] &&  !IntegerQ[p]

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

\begin{align*} \int \csc ^p\left (a+\frac{i \log \left (c x^n\right )}{n (-2+p)}\right ) \, dx &=\frac{\left (x \left (c x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int x^{-1+\frac{1}{n}} \csc ^p\left (a+\frac{i \log (x)}{n (-2+p)}\right ) \, dx,x,c x^n\right )}{n}\\ &=\frac{\left (x \left (c x^n\right )^{-\frac{1}{n}+\frac{p}{n (-2+p)}} \left (1-e^{2 i a} \left (c x^n\right )^{-\frac{2}{n (-2+p)}}\right )^p \csc ^p\left (a+\frac{i \log \left (c x^n\right )}{n (-2+p)}\right )\right ) \operatorname{Subst}\left (\int x^{-1+\frac{1}{n}-\frac{p}{n (-2+p)}} \left (1-e^{2 i a} x^{-\frac{2}{n (-2+p)}}\right )^{-p} \, dx,x,c x^n\right )}{n}\\ &=-\frac{e^{-2 i a} (2-p) x \left (c x^n\right )^{-\frac{2}{n (2-p)}} \left (1-e^{2 i a} \left (c x^n\right )^{\frac{2}{n (2-p)}}\right ) \csc ^p\left (a-\frac{i \log \left (c x^n\right )}{n (2-p)}\right )}{2 (1-p)}\\ \end{align*}

Mathematica [A]  time = 2.06899, size = 155, normalized size = 1.61 \[ \frac{2^{p-1} (p-2) x e^{-\frac{2 i a p}{p-2}} \left (e^{\frac{2 i a p}{p-2}}-e^{\frac{4 i a}{p-2}} \left (c x^n\right )^{\frac{2}{n (p-2)}}\right ) \left (-\frac{i e^{\frac{i a (p+2)}{p-2}} \left (c x^n\right )^{\frac{1}{n (p-2)}}}{e^{\frac{4 i a}{p-2}} \left (c x^n\right )^{\frac{2}{n (p-2)}}-e^{\frac{2 i a p}{p-2}}}\right )^p}{p-1} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Csc[a + (I*Log[c*x^n])/(n*(-2 + p))]^p,x]

[Out]

(2^(-1 + p)*(-2 + p)*x*(E^(((2*I)*a*p)/(-2 + p)) - E^(((4*I)*a)/(-2 + p))*(c*x^n)^(2/(n*(-2 + p))))*(((-I)*E^(
(I*a*(2 + p))/(-2 + p))*(c*x^n)^(1/(n*(-2 + p))))/(-E^(((2*I)*a*p)/(-2 + p)) + E^(((4*I)*a)/(-2 + p))*(c*x^n)^
(2/(n*(-2 + p)))))^p)/(E^(((2*I)*a*p)/(-2 + p))*(-1 + p))

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Maple [F]  time = 0.278, size = 0, normalized size = 0. \begin{align*} \int \left ( \csc \left ( a+{\frac{i\ln \left ( c{x}^{n} \right ) }{n \left ( p-2 \right ) }} \right ) \right ) ^{p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(a+I*ln(c*x^n)/n/(p-2))^p,x)

[Out]

int(csc(a+I*ln(c*x^n)/n/(p-2))^p,x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(a+I*log(c*x^n)/n/(-2+p))^p,x, algorithm="maxima")

[Out]

integrate(csc(a + I*log(c*x^n)/(n*(p - 2)))^p, x)

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Fricas [A]  time = 0.510156, size = 343, normalized size = 3.57 \begin{align*} \frac{{\left ({\left (p - 2\right )} x e^{\left (\frac{2 \,{\left (i \, a n p - 2 i \, a n - \log \left (c x^{n}\right )\right )}}{n p - 2 \, n}\right )} -{\left (p - 2\right )} x\right )} \left (\frac{2 i \, e^{\left (\frac{i \, a n p - 2 i \, a n - \log \left (c x^{n}\right )}{n p - 2 \, n}\right )}}{e^{\left (\frac{2 \,{\left (i \, a n p - 2 i \, a n - \log \left (c x^{n}\right )\right )}}{n p - 2 \, n}\right )} - 1}\right )^{p} e^{\left (-\frac{2 \,{\left (i \, a n p - 2 i \, a n - \log \left (c x^{n}\right )\right )}}{n p - 2 \, n}\right )}}{2 \,{\left (p - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(a+I*log(c*x^n)/n/(-2+p))^p,x, algorithm="fricas")

[Out]

1/2*((p - 2)*x*e^(2*(I*a*n*p - 2*I*a*n - log(c*x^n))/(n*p - 2*n)) - (p - 2)*x)*(2*I*e^((I*a*n*p - 2*I*a*n - lo
g(c*x^n))/(n*p - 2*n))/(e^(2*(I*a*n*p - 2*I*a*n - log(c*x^n))/(n*p - 2*n)) - 1))^p*e^(-2*(I*a*n*p - 2*I*a*n -
log(c*x^n))/(n*p - 2*n))/(p - 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \csc ^{p}{\left (a + \frac{i \log{\left (c x^{n} \right )}}{n \left (p - 2\right )} \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(a+I*ln(c*x**n)/n/(-2+p))**p,x)

[Out]

Integral(csc(a + I*log(c*x**n)/(n*(p - 2)))**p, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(a+I*log(c*x^n)/n/(-2+p))^p,x, algorithm="giac")

[Out]

integrate(csc(a + I*log(c*x^n)/(n*(p - 2)))^p, x)