Integrand size = 19, antiderivative size = 130 \[ \int x^m \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {2 x^{1+m} \sqrt {1-e^{2 i a} \left (c x^n\right )^{2 i b}} \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-\frac {2 i+2 i m-b n}{4 b n},-\frac {2 i+2 i m-5 b n}{4 b n},e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{2+2 m+i b n} \] Output:
2*x^(1+m)*(1-exp(2*I*a)*(c*x^n)^(2*I*b))^(1/2)*csc(a+b*ln(c*x^n))^(1/2)*hy pergeom([1/2, -1/4*(2*I+2*I*m-b*n)/b/n],[-1/4*(2*I+2*I*m-5*b*n)/b/n],exp(2 *I*a)*(c*x^n)^(2*I*b))/(2+2*m+I*b*n)
Time = 0.67 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.06 \[ \int x^m \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {2 e^{-2 i a} x^{1+m} \left (c x^n\right )^{-2 i b} \left (-1+e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )} \operatorname {Hypergeometric2F1}\left (1,\frac {2 i+2 i m+3 b n}{4 b n},\frac {2 i+2 i m+5 b n}{4 b n},e^{-2 i \left (a+b \log \left (c x^n\right )\right )}\right )}{2+2 m-i b n} \] Input:
Integrate[x^m*Sqrt[Csc[a + b*Log[c*x^n]]],x]
Output:
(2*x^(1 + m)*(-1 + E^((2*I)*a)*(c*x^n)^((2*I)*b))*Sqrt[Csc[a + b*Log[c*x^n ]]]*Hypergeometric2F1[1, (2*I + (2*I)*m + 3*b*n)/(4*b*n), (2*I + (2*I)*m + 5*b*n)/(4*b*n), E^((-2*I)*(a + b*Log[c*x^n]))])/(E^((2*I)*a)*(2 + 2*m - I *b*n)*(c*x^n)^((2*I)*b))
Time = 0.32 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {5021, 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.
\(\displaystyle \int x^m \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )} \, dx\) |
\(\Big \downarrow \) 5021 |
\(\displaystyle \frac {x^{m+1} \left (c x^n\right )^{-\frac {m+1}{n}} \int \left (c x^n\right )^{\frac {m+1}{n}-1} \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )}d\left (c x^n\right )}{n}\) |
\(\Big \downarrow \) 5019 |
\(\displaystyle \frac {x^{m+1} \sqrt {1-e^{2 i a} \left (c x^n\right )^{2 i b}} \left (c x^n\right )^{-\frac {m+1}{n}-\frac {i b}{2}} \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )} \int \frac {\left (c x^n\right )^{\frac {i b}{2}+\frac {m+1}{n}-1}}{\sqrt {1-e^{2 i a} \left (c x^n\right )^{2 i b}}}d\left (c x^n\right )}{n}\) |
\(\Big \downarrow \) 888 |
\(\displaystyle \frac {2 x^{m+1} \sqrt {1-e^{2 i a} \left (c x^n\right )^{2 i b}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-\frac {2 i m-b n+2 i}{4 b n},-\frac {2 i m-5 b n+2 i}{4 b n},e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )}}{i b n+2 m+2}\) |
Input:
Int[x^m*Sqrt[Csc[a + b*Log[c*x^n]]],x]
Output:
(2*x^(1 + m)*Sqrt[1 - E^((2*I)*a)*(c*x^n)^((2*I)*b)]*Sqrt[Csc[a + b*Log[c* x^n]]]*Hypergeometric2F1[1/2, -1/4*(2*I + (2*I)*m - b*n)/(b*n), -1/4*(2*I + (2*I)*m - 5*b*n)/(b*n), E^((2*I)*a)*(c*x^n)^((2*I)*b)])/(2 + 2*m + 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[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[Csc[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_.)*((e_.)*(x_))^(m_ .), x_Symbol] :> Simp[(e*x)^(m + 1)/(e*n*(c*x^n)^((m + 1)/n)) Subst[Int[x ^((m + 1)/n - 1)*Csc[d*(a + b*Log[x])]^p, x], x, c*x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && (NeQ[c, 1] || NeQ[n, 1])
\[\int x^{m} \sqrt {\csc \left (a +b \ln \left (c \,x^{n}\right )\right )}d x\]
Input:
int(x^m*csc(a+b*ln(c*x^n))^(1/2),x)
Output:
int(x^m*csc(a+b*ln(c*x^n))^(1/2),x)
Exception generated. \[ \int x^m \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x^m*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 (has polynomial part)
\[ \int x^m \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )} \, dx=\int x^{m} \sqrt {\csc {\left (a + b \log {\left (c x^{n} \right )} \right )}}\, dx \] Input:
integrate(x**m*csc(a+b*ln(c*x**n))**(1/2),x)
Output:
Integral(x**m*sqrt(csc(a + b*log(c*x**n))), x)
\[ \int x^m \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )} \, dx=\int { x^{m} \sqrt {\csc \left (b \log \left (c x^{n}\right ) + a\right )} \,d x } \] Input:
integrate(x^m*csc(a+b*log(c*x^n))^(1/2),x, algorithm="maxima")
Output:
integrate(x^m*sqrt(csc(b*log(c*x^n) + a)), x)
\[ \int x^m \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )} \, dx=\int { x^{m} \sqrt {\csc \left (b \log \left (c x^{n}\right ) + a\right )} \,d x } \] Input:
integrate(x^m*csc(a+b*log(c*x^n))^(1/2),x, algorithm="giac")
Output:
integrate(x^m*sqrt(csc(b*log(c*x^n) + a)), x)
Timed out. \[ \int x^m \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )} \, dx=\int x^m\,\sqrt {\frac {1}{\sin \left (a+b\,\ln \left (c\,x^n\right )\right )}} \,d x \] Input:
int(x^m*(1/sin(a + b*log(c*x^n)))^(1/2),x)
Output:
int(x^m*(1/sin(a + b*log(c*x^n)))^(1/2), x)
\[ \int x^m \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )} \, dx=\int x^{m} \sqrt {\csc \left (\mathrm {log}\left (x^{n} c \right ) b +a \right )}d x \] Input:
int(x^m*csc(a+b*log(c*x^n))^(1/2),x)
Output:
int(x**m*sqrt(csc(log(x**n*c)*b + a)),x)