Integrand size = 15, antiderivative size = 109 \[ \int \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {2 x \sqrt {1-e^{2 i a} \left (c x^n\right )^{2 i b}} \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{4} \left (1-\frac {2 i}{b n}\right ),\frac {1}{4} \left (5-\frac {2 i}{b n}\right ),e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{2+i b n} \] Output:
2*x*(1-exp(2*I*a)*(c*x^n)^(2*I*b))^(1/2)*csc(a+b*ln(c*x^n))^(1/2)*hypergeo m([1/2, 1/4-1/2*I/b/n],[5/4-1/2*I/b/n],exp(2*I*a)*(c*x^n)^(2*I*b))/(2+I*b* n)
Time = 0.31 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.06 \[ \int \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {2 i e^{-2 i a} \left (-1+e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right ) x \left (c x^n\right )^{-2 i b} \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )} \operatorname {Hypergeometric2F1}\left (1,\frac {3}{4}+\frac {i}{2 b n},\frac {5}{4}+\frac {i}{2 b n},e^{-2 i \left (a+b \log \left (c x^n\right )\right )}\right )}{2 i+b n} \] Input:
Integrate[Sqrt[Csc[a + b*Log[c*x^n]]],x]
Output:
((2*I)*(-1 + E^((2*I)*(a + b*Log[c*x^n])))*x*Sqrt[Csc[a + b*Log[c*x^n]]]*H ypergeometric2F1[1, 3/4 + (I/2)/(b*n), 5/4 + (I/2)/(b*n), E^((-2*I)*(a + b *Log[c*x^n]))])/(E^((2*I)*a)*(2*I + b*n)*(c*x^n)^((2*I)*b))
Time = 0.30 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5015, 5019, 888}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )} \, dx\) |
\(\Big \downarrow \) 5015 |
\(\displaystyle \frac {x \left (c x^n\right )^{-1/n} \int \left (c x^n\right )^{\frac {1}{n}-1} \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )}d\left (c x^n\right )}{n}\) |
\(\Big \downarrow \) 5019 |
\(\displaystyle \frac {x \left (c x^n\right )^{-\frac {1}{n}-\frac {i b}{2}} \sqrt {1-e^{2 i a} \left (c x^n\right )^{2 i b}} \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )} \int \frac {\left (c x^n\right )^{\frac {i b}{2}+\frac {1}{n}-1}}{\sqrt {1-e^{2 i a} \left (c x^n\right )^{2 i b}}}d\left (c x^n\right )}{n}\) |
\(\Big \downarrow \) 888 |
\(\displaystyle \frac {2 x \sqrt {1-e^{2 i a} \left (c x^n\right )^{2 i b}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{4} \left (1-\frac {2 i}{b n}\right ),\frac {1}{4} \left (5-\frac {2 i}{b n}\right ),e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )}}{2+i b n}\) |
Input:
Int[Sqrt[Csc[a + b*Log[c*x^n]]],x]
Output:
(2*x*Sqrt[1 - E^((2*I)*a)*(c*x^n)^((2*I)*b)]*Sqrt[Csc[a + b*Log[c*x^n]]]*H ypergeometric2F1[1/2, (1 - (2*I)/(b*n))/4, (5 - (2*I)/(b*n))/4, E^((2*I)*a )*(c*x^n)^((2*I)*b)])/(2 + I*b*n)
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p *((c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/n, (m + 1)/n + 1 , (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] && !IGtQ[p, 0] && (ILt Q[p, 0] || GtQ[a, 0])
Int[Csc[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_.), x_Symbol] :> Si mp[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])
Int[Csc[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] :> Simp[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] /; F reeQ[{a, b, d, e, m, p}, x] && !IntegerQ[p]
\[\int \sqrt {\csc \left (a +b \ln \left (c \,x^{n}\right )\right )}d x\]
Input:
int(csc(a+b*ln(c*x^n))^(1/2),x)
Output:
int(csc(a+b*ln(c*x^n))^(1/2),x)
Exception generated. \[ \int \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(csc(a+b*log(c*x^n))^(1/2),x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )} \, dx=\int \sqrt {\csc {\left (a + b \log {\left (c x^{n} \right )} \right )}}\, dx \] Input:
integrate(csc(a+b*ln(c*x**n))**(1/2),x)
Output:
Integral(sqrt(csc(a + b*log(c*x**n))), x)
\[ \int \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )} \, dx=\int { \sqrt {\csc \left (b \log \left (c x^{n}\right ) + a\right )} \,d x } \] Input:
integrate(csc(a+b*log(c*x^n))^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(csc(b*log(c*x^n) + a)), x)
\[ \int \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )} \, dx=\int { \sqrt {\csc \left (b \log \left (c x^{n}\right ) + a\right )} \,d x } \] Input:
integrate(csc(a+b*log(c*x^n))^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(csc(b*log(c*x^n) + a)), x)
Timed out. \[ \int \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )} \, dx=\int \sqrt {\frac {1}{\sin \left (a+b\,\ln \left (c\,x^n\right )\right )}} \,d x \] Input:
int((1/sin(a + b*log(c*x^n)))^(1/2),x)
Output:
int((1/sin(a + b*log(c*x^n)))^(1/2), x)
\[ \int \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )} \, dx=\int \sqrt {\csc \left (\mathrm {log}\left (x^{n} c \right ) b +a \right )}d x \] Input:
int(csc(a+b*log(c*x^n))^(1/2),x)
Output:
int(sqrt(csc(log(x**n*c)*b + a)),x)