3.4.6 \(\int \csc ^p(a+\frac {i \log (c x^n)}{n (-2+p)}) \, dx\) [306]

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]

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

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Rubi [A]
time = 0.06, antiderivative size = 96, 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 = {4600, 4604, 267} \begin {gather*} -\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 {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 267

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]

Rule 4600

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 4604

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 ) \text {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 ) \text {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*}

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Mathematica [A]
time = 1.98, size = 155, normalized size = 1.61 \begin {gather*} \frac {2^{-1+p} e^{-\frac {2 i a p}{-2+p}} (-2+p) x \left (e^{\frac {2 i a p}{-2+p}}-e^{\frac {4 i a}{-2+p}} \left (c x^n\right )^{\frac {2}{n (-2+p)}}\right ) \left (-\frac {i e^{\frac {i a (2+p)}{-2+p}} \left (c x^n\right )^{\frac {1}{n (-2+p)}}}{-e^{\frac {2 i a p}{-2+p}}+e^{\frac {4 i a}{-2+p}} \left (c x^n\right )^{\frac {2}{n (-2+p)}}}\right )^p}{-1+p} \end {gather*}

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 = 0.16, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 4.01, size = 150, normalized size = 1.56 \begin {gather*} \frac {{\left ({\left (p - 2\right )} x e^{\left (\frac {2 \, {\left (i \, a n p - 2 i \, a n - n \log \left (x\right ) - \log \left (c\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 - n \log \left (x\right ) - \log \left (c\right )}{n p - 2 \, n}\right )}}{e^{\left (\frac {2 \, {\left (i \, a n p - 2 i \, a n - n \log \left (x\right ) - \log \left (c\right )\right )}}{n p - 2 \, n}\right )} - 1}\right )^{p} e^{\left (-\frac {2 \, {\left (i \, a n p - 2 i \, a n - n \log \left (x\right ) - \log \left (c\right )\right )}}{n p - 2 \, n}\right )}}{2 \, {\left (p - 1\right )}} \end {gather*}

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

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \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 \end {gather*}

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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