\(\int \frac {x^m}{\csc ^{\frac {3}{2}}(a+b \log (c x^n))} \, dx\) [327]

Optimal result

Integrand size = 19, antiderivative size = 130 \[ \int \frac {x^m}{\csc ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {2 x^{1+m} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-\frac {2 i+2 i m+3 b n}{4 b n},-\frac {2 i+2 i m-b n}{4 b n},e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(2+2 m-3 i b n) \left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{3/2} \csc ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \]

[Out]

2*x^(1+m)*hypergeom([-3/2, 1/4*(-2*I-2*I*m-3*b*n)/b/n],[1/4*(-2*I-2*I*m+b*n)/b/n],exp(2*I*a)*(c*x^n)^(2*I*b))/
(2+2*m-3*I*b*n)/(1-exp(2*I*a)*(c*x^n)^(2*I*b))^(3/2)/csc(a+b*ln(c*x^n))^(3/2)

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.97, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {4606, 4604, 371} \[ \int \frac {x^m}{\csc ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {2 x^{m+1} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1}{4} \left (-\frac {2 i (m+1)}{b n}-3\right ),-\frac {2 i m-b n+2 i}{4 b n},e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(-3 i b n+2 m+2) \left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{3/2} \csc ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \]

[In]

Int[x^m/Csc[a + b*Log[c*x^n]]^(3/2),x]

[Out]

(2*x^(1 + m)*Hypergeometric2F1[-3/2, (-3 - ((2*I)*(1 + m))/(b*n))/4, -1/4*(2*I + (2*I)*m - b*n)/(b*n), E^((2*I
)*a)*(c*x^n)^((2*I)*b)])/((2 + 2*m - (3*I)*b*n)*(1 - E^((2*I)*a)*(c*x^n)^((2*I)*b))^(3/2)*Csc[a + b*Log[c*x^n]
]^(3/2))

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 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]

Rule 4606

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*} \text {integral}& = \frac {\left (x^{1+m} \left (c x^n\right )^{-\frac {1+m}{n}}\right ) \text {Subst}\left (\int \frac {x^{-1+\frac {1+m}{n}}}{\csc ^{\frac {3}{2}}(a+b \log (x))} \, dx,x,c x^n\right )}{n} \\ & = \frac {\left (x^{1+m} \left (c x^n\right )^{\frac {3 i b}{2}-\frac {1+m}{n}}\right ) \text {Subst}\left (\int x^{-1-\frac {3 i b}{2}+\frac {1+m}{n}} \left (1-e^{2 i a} x^{2 i b}\right )^{3/2} \, dx,x,c x^n\right )}{n \left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{3/2} \csc ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \\ & = \frac {2 x^{1+m} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1}{4} \left (-3-\frac {2 i (1+m)}{b n}\right ),-\frac {2 i+2 i m-b n}{4 b n},e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(2+2 m-3 i b n) \left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{3/2} \csc ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.72 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.68 \[ \int \frac {x^m}{\csc ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {2 x^{1+m} \left ((2+2 m-i b n) \left (2+2 m-3 b n \cot \left (a+b \log \left (c x^n\right )\right )\right )+3 b^2 e^{-2 i a} n^2 \left (c x^n\right )^{-2 i b} \left (-1+e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \csc ^2\left (a+b \log \left (c x^n\right )\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {2 i+2 i m+3 b n}{4 b n},\frac {2 i+2 i m+5 b n}{4 b n},e^{-2 i \left (a+b \log \left (c x^n\right )\right )}\right )\right )}{(2+2 m-i b n) (2+2 m-3 i b n) (2+2 m+3 i b n) \csc ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \]

[In]

Integrate[x^m/Csc[a + b*Log[c*x^n]]^(3/2),x]

[Out]

(2*x^(1 + m)*((2 + 2*m - I*b*n)*(2 + 2*m - 3*b*n*Cot[a + b*Log[c*x^n]]) + (3*b^2*n^2*(-1 + E^((2*I)*a)*(c*x^n)
^((2*I)*b))*Csc[a + b*Log[c*x^n]]^2*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)*(c*x^n)^((2*I)*b))))/((2 + 2*m - I*b*n)*(2 + 2*m - (
3*I)*b*n)*(2 + 2*m + (3*I)*b*n)*Csc[a + b*Log[c*x^n]]^(3/2))

Maple [F]

\[\int \frac {x^{m}}{{\csc \left (a +b \ln \left (c \,x^{n}\right )\right )}^{\frac {3}{2}}}d x\]

[In]

int(x^m/csc(a+b*ln(c*x^n))^(3/2),x)

[Out]

int(x^m/csc(a+b*ln(c*x^n))^(3/2),x)

Fricas [F(-2)]

Exception generated. \[ \int \frac {x^m}{\csc ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(x^m/csc(a+b*log(c*x^n))^(3/2),x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {x^m}{\csc ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\int \frac {x^{m}}{\csc ^{\frac {3}{2}}{\left (a + b \log {\left (c x^{n} \right )} \right )}}\, dx \]

[In]

integrate(x**m/csc(a+b*ln(c*x**n))**(3/2),x)

[Out]

Integral(x**m/csc(a + b*log(c*x**n))**(3/2), x)

Maxima [F]

\[ \int \frac {x^m}{\csc ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\int { \frac {x^{m}}{\csc \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {3}{2}}} \,d x } \]

[In]

integrate(x^m/csc(a+b*log(c*x^n))^(3/2),x, algorithm="maxima")

[Out]

integrate(x^m/csc(b*log(c*x^n) + a)^(3/2), x)

Giac [F]

\[ \int \frac {x^m}{\csc ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\int { \frac {x^{m}}{\csc \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {3}{2}}} \,d x } \]

[In]

integrate(x^m/csc(a+b*log(c*x^n))^(3/2),x, algorithm="giac")

[Out]

integrate(x^m/csc(b*log(c*x^n) + a)^(3/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^m}{\csc ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\int \frac {x^m}{{\left (\frac {1}{\sin \left (a+b\,\ln \left (c\,x^n\right )\right )}\right )}^{3/2}} \,d x \]

[In]

int(x^m/(1/sin(a + b*log(c*x^n)))^(3/2),x)

[Out]

int(x^m/(1/sin(a + b*log(c*x^n)))^(3/2), x)