\(\int \csc ^p(a+b \log (c x^n)) \, dx\) [330]

Optimal result

Integrand size = 13, antiderivative size = 107 \[ \int \csc ^p\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {x \left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^p \csc ^p\left (a+b \log \left (c x^n\right )\right ) \operatorname {Hypergeometric2F1}\left (p,-\frac {i-b n p}{2 b n},\frac {1}{2} \left (2-\frac {i}{b n}+p\right ),e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{1+i b n p} \]

[Out]

x*(1-exp(2*I*a)*(c*x^n)^(2*I*b))^p*csc(a+b*ln(c*x^n))^p*hypergeom([p, 1/2*(-I+b*n*p)/b/n],[1-1/2*I/b/n+1/2*p],
exp(2*I*a)*(c*x^n)^(2*I*b))/(1+I*b*n*p)

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {4600, 4604, 371} \[ \int \csc ^p\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {x \left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^p \operatorname {Hypergeometric2F1}\left (p,-\frac {i-b n p}{2 b n},\frac {1}{2} \left (p-\frac {i}{b n}+2\right ),e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \csc ^p\left (a+b \log \left (c x^n\right )\right )}{1+i b n p} \]

[In]

Int[Csc[a + b*Log[c*x^n]]^p,x]

[Out]

(x*(1 - E^((2*I)*a)*(c*x^n)^((2*I)*b))^p*Csc[a + b*Log[c*x^n]]^p*Hypergeometric2F1[p, -1/2*(I - b*n*p)/(b*n),
(2 - I/(b*n) + p)/2, E^((2*I)*a)*(c*x^n)^((2*I)*b)])/(1 + I*b*n*p)

Rule 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

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*} \text {integral}& = \frac {\left (x \left (c x^n\right )^{-1/n}\right ) \text {Subst}\left (\int x^{-1+\frac {1}{n}} \csc ^p(a+b \log (x)) \, dx,x,c x^n\right )}{n} \\ & = \frac {\left (x \left (c x^n\right )^{-\frac {1}{n}-i b p} \left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^p \csc ^p\left (a+b \log \left (c x^n\right )\right )\right ) \text {Subst}\left (\int x^{-1+\frac {1}{n}+i b p} \left (1-e^{2 i a} x^{2 i b}\right )^{-p} \, dx,x,c x^n\right )}{n} \\ & = \frac {x \left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^p \csc ^p\left (a+b \log \left (c x^n\right )\right ) \operatorname {Hypergeometric2F1}\left (p,-\frac {i-b n p}{2 b n},\frac {1}{2} \left (2-\frac {i}{b n}+p\right ),e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{1+i b n p} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.76 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.33 \[ \int \csc ^p\left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {i x \left (2-2 e^{2 i a} \left (c x^n\right )^{2 i b}\right )^p \left (\frac {i e^{i a} \left (c x^n\right )^{i b}}{-1+e^{2 i a} \left (c x^n\right )^{2 i b}}\right )^p \operatorname {Hypergeometric2F1}\left (p,\frac {-i+b n p}{2 b n},\frac {1}{2} \left (2-\frac {i}{b n}+p\right ),e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{-i+b n p} \]

[In]

Integrate[Csc[a + b*Log[c*x^n]]^p,x]

[Out]

((-I)*x*(2 - 2*E^((2*I)*a)*(c*x^n)^((2*I)*b))^p*((I*E^(I*a)*(c*x^n)^(I*b))/(-1 + E^((2*I)*a)*(c*x^n)^((2*I)*b)
))^p*Hypergeometric2F1[p, (-I + b*n*p)/(2*b*n), (2 - I/(b*n) + p)/2, E^((2*I)*a)*(c*x^n)^((2*I)*b)])/(-I + b*n
*p)

Maple [F]

\[\int {\csc \left (a +b \ln \left (c \,x^{n}\right )\right )}^{p}d x\]

[In]

int(csc(a+b*ln(c*x^n))^p,x)

[Out]

int(csc(a+b*ln(c*x^n))^p,x)

Fricas [F]

\[ \int \csc ^p\left (a+b \log \left (c x^n\right )\right ) \, dx=\int { \csc \left (b \log \left (c x^{n}\right ) + a\right )^{p} \,d x } \]

[In]

integrate(csc(a+b*log(c*x^n))^p,x, algorithm="fricas")

[Out]

integral(csc(b*log(c*x^n) + a)^p, x)

Sympy [F]

\[ \int \csc ^p\left (a+b \log \left (c x^n\right )\right ) \, dx=\int \csc ^{p}{\left (a + b \log {\left (c x^{n} \right )} \right )}\, dx \]

[In]

integrate(csc(a+b*ln(c*x**n))**p,x)

[Out]

Integral(csc(a + b*log(c*x**n))**p, x)

Maxima [F]

\[ \int \csc ^p\left (a+b \log \left (c x^n\right )\right ) \, dx=\int { \csc \left (b \log \left (c x^{n}\right ) + a\right )^{p} \,d x } \]

[In]

integrate(csc(a+b*log(c*x^n))^p,x, algorithm="maxima")

[Out]

integrate(csc(b*log(c*x^n) + a)^p, x)

Giac [F]

\[ \int \csc ^p\left (a+b \log \left (c x^n\right )\right ) \, dx=\int { \csc \left (b \log \left (c x^{n}\right ) + a\right )^{p} \,d x } \]

[In]

integrate(csc(a+b*log(c*x^n))^p,x, algorithm="giac")

[Out]

integrate(csc(b*log(c*x^n) + a)^p, x)

Mupad [F(-1)]

Timed out. \[ \int \csc ^p\left (a+b \log \left (c x^n\right )\right ) \, dx=\int {\left (\frac {1}{\sin \left (a+b\,\ln \left (c\,x^n\right )\right )}\right )}^p \,d x \]

[In]

int((1/sin(a + b*log(c*x^n)))^p,x)

[Out]

int((1/sin(a + b*log(c*x^n)))^p, x)