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

Optimal. Leaf size=71 \[ \frac {(2-p) x \left (1-e^{2 i a} \left (c x^n\right )^{-\frac {2}{n (2-p)}}\right ) \csc ^p\left (a+\frac {i \log \left (c x^n\right )}{n (2-p)}\right )}{2 (1-p)} \]

[Out]

1/2*(2-p)*x*(1-exp(2*I*a)/((c*x^n)^(2/n/(2-p))))*csc(a+I*ln(c*x^n)/n/(2-p))^p/(1-p)

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Rubi [A]  time = 0.08, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {4504, 4508, 264} \[ \frac {(2-p) x \left (1-e^{2 i a} \left (c x^n\right )^{-\frac {2}{n (2-p)}}\right ) \csc ^p\left (a+\frac {i \log \left (c x^n\right )}{n (2-p)}\right )}{2 (1-p)} \]

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*(1 - p))

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

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]

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 {(2-p) x \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*}

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Mathematica [A]  time = 3.10, size = 128, normalized size = 1.80 \[ \frac {2^{p-1} (p-2) x \left (\frac {i e^{i a} \left (c x^n\right )^{\frac {1}{n (p-2)}}}{-1+e^{2 i a} \left (c x^n\right )^{\frac {2}{n (p-2)}}}\right )^p \left (1+e^{2 i a} \left (c x^n\right )^{\frac {2}{n (p-2)}} \left (-1+\left (1-e^{-2 i a} \left (c x^n\right )^{-\frac {2}{n (p-2)}}\right )^p\right )\right )}{p-1} \]

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

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fricas [B]  time = 0.83, size = 150, normalized size = 2.11 \[ \frac {{\left ({\left (p - 2\right )} x e^{\left (\frac {2 \, {\left (-i \, a n p + 2 i \, a n - n \log \relax (x) - \log \relax (c)\right )}}{n p - 2 \, n}\right )} - {\left (p - 2\right )} x\right )} \left (-\frac {2 i \, e^{\left (\frac {-i \, a n p + 2 i \, a n - n \log \relax (x) - \log \relax (c)}{n p - 2 \, n}\right )}}{e^{\left (\frac {2 \, {\left (-i \, a n p + 2 i \, a n - n \log \relax (x) - \log \relax (c)\right )}}{n p - 2 \, n}\right )} - 1}\right )^{p} e^{\left (-\frac {2 \, {\left (-i \, a n p + 2 i \, a n - n \log \relax (x) - \log \relax (c)\right )}}{n p - 2 \, n}\right )}}{2 \, {\left (p - 1\right )}} \]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \csc \left (a - \frac {i \, \log \left (c x^{n}\right )}{n {\left (p - 2\right )}}\right )^{p}\,{d x} \]

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)

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maple [F]  time = 0.24, size = 0, normalized size = 0.00 \[ \int \csc ^{p}\left (a -\frac {i \ln \left (c \,x^{n}\right )}{n \left (-2+p \right )}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (-\csc \left (-a + \frac {i \, \log \left (c x^{n}\right )}{n {\left (p - 2\right )}}\right )\right )^{p}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (\frac {1}{\sin \left (a-\frac {\ln \left (c\,x^n\right )\,1{}\mathrm {i}}{n\,\left (p-2\right )}\right )}\right )}^p \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1/sin(a - (log(c*x^n)*1i)/(n*(p - 2))))^p,x)

[Out]

int((1/sin(a - (log(c*x^n)*1i)/(n*(p - 2))))^p, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \csc ^{p}{\left (a - \frac {i \log {\left (c x^{n} \right )}}{n \left (p - 2\right )} \right )}\, dx \]

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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