3.325 \(\int x^m \sqrt {\csc (a+b \log (c x^n))} \, dx\)

Optimal. Leaf size=130 \[ \frac {2 x^{m+1} \sqrt {1-e^{2 i a} \left (c x^n\right )^{2 i b}} \, _2F_1\left (\frac {1}{2},-\frac {2 i m-b n+2 i}{4 b n};-\frac {2 i m-5 b n+2 i}{4 b n};e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )}}{i b n+2 m+2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.09, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {4510, 4508, 364} \[ \frac {2 x^{m+1} \sqrt {1-e^{2 i a} \left (c x^n\right )^{2 i b}} \, _2F_1\left (\frac {1}{2},-\frac {2 i m-b n+2 i}{4 b n};-\frac {2 i m-5 b n+2 i}{4 b n};e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )}}{i b n+2 m+2} \]

Antiderivative was successfully verified.

[In]

Int[x^m*Sqrt[Csc[a + b*Log[c*x^n]]],x]

[Out]

(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, -(2*I
+ (2*I)*m - b*n)/(4*b*n), -(2*I + (2*I)*m - 5*b*n)/(4*b*n), E^((2*I)*a)*(c*x^n)^((2*I)*b)])/(2 + 2*m + I*b*n)

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

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]

Rule 4510

Int[Csc[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] :> Dist[(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])

Rubi steps

\begin {align*} \int x^m \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )} \, dx &=\frac {\left (x^{1+m} \left (c x^n\right )^{-\frac {1+m}{n}}\right ) \operatorname {Subst}\left (\int x^{-1+\frac {1+m}{n}} \sqrt {\csc (a+b \log (x))} \, dx,x,c x^n\right )}{n}\\ &=\frac {\left (x^{1+m} \left (c x^n\right )^{-\frac {i b}{2}-\frac {1+m}{n}} \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 )}\right ) \operatorname {Subst}\left (\int \frac {x^{-1+\frac {i b}{2}+\frac {1+m}{n}}}{\sqrt {1-e^{2 i a} x^{2 i b}}} \, dx,x,c x^n\right )}{n}\\ &=\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 )} \, _2F_1\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}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.93, size = 138, normalized size = 1.06 \[ \frac {2 e^{-2 i a} x^{m+1} \left (c x^n\right )^{-2 i b} \left (-1+e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )} \, _2F_1\left (1,\frac {2 i m+3 b n+2 i}{4 b n};\frac {2 i m+5 b n+2 i}{4 b n};e^{-2 i \left (a+b \log \left (c x^n\right )\right )}\right )}{-i b n+2 m+2} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^m*Sqrt[Csc[a + b*Log[c*x^n]]],x]

[Out]

(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))

________________________________________________________________________________________

fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*csc(a+b*log(c*x^n))^(1/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (ha
s polynomial part)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{m} \sqrt {\csc \left (b \log \left (c x^{n}\right ) + a\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*csc(a+b*log(c*x^n))^(1/2),x, algorithm="giac")

[Out]

integrate(x^m*sqrt(csc(b*log(c*x^n) + a)), x)

________________________________________________________________________________________

maple [F]  time = 0.17, size = 0, normalized size = 0.00 \[ \int x^{m} \left (\sqrt {\csc }\left (a +b \ln \left (c \,x^{n}\right )\right )\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*csc(a+b*ln(c*x^n))^(1/2),x)

[Out]

int(x^m*csc(a+b*ln(c*x^n))^(1/2),x)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{m} \sqrt {\csc \left (b \log \left (c x^{n}\right ) + a\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*csc(a+b*log(c*x^n))^(1/2),x, algorithm="maxima")

[Out]

integrate(x^m*sqrt(csc(b*log(c*x^n) + a)), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int x^m\,\sqrt {\frac {1}{\sin \left (a+b\,\ln \left (c\,x^n\right )\right )}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*(1/sin(a + b*log(c*x^n)))^(1/2),x)

[Out]

int(x^m*(1/sin(a + b*log(c*x^n)))^(1/2), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{m} \sqrt {\csc {\left (a + b \log {\left (c x^{n} \right )} \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**m*csc(a+b*ln(c*x**n))**(1/2),x)

[Out]

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

________________________________________________________________________________________