\(\int x^m \csc ^{\frac {3}{2}}(a+b \log (c x^n)) \, dx\) [324]

Optimal result

Integrand size = 19, antiderivative size = 130 \[ \int x^m \csc ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {2 x^{1+m} \left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{3/2} \csc ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right ) \operatorname {Hypergeometric2F1}\left (\frac {3}{2},-\frac {2 i+2 i m-3 b n}{4 b n},-\frac {2 i+2 i m-7 b n}{4 b n},e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{2+2 m+3 i b n} \]

[Out]

2*x^(1+m)*(1-exp(2*I*a)*(c*x^n)^(2*I*b))^(3/2)*csc(a+b*ln(c*x^n))^(3/2)*hypergeom([3/2, 1/4*(-2*I-2*I*m+3*b*n)
/b/n],[1/4*(-2*I-2*I*m+7*b*n)/b/n],exp(2*I*a)*(c*x^n)^(2*I*b))/(2+2*m+3*I*b*n)

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} \[ \int x^m \csc ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {2 x^{m+1} \left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{3/2} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {1}{4} \left (3-\frac {2 i (m+1)}{b n}\right ),-\frac {2 i m-7 b n+2 i}{4 b n},e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \csc ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{3 i b n+2 m+2} \]

[In]

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

[Out]

(2*x^(1 + m)*(1 - E^((2*I)*a)*(c*x^n)^((2*I)*b))^(3/2)*Csc[a + b*Log[c*x^n]]^(3/2)*Hypergeometric2F1[3/2, (3 -
((2*I)*(1 + m))/(b*n))/4, -1/4*(2*I + (2*I)*m - 7*b*n)/(b*n), E^((2*I)*a)*(c*x^n)^((2*I)*b)])/(2 + 2*m + (3*I
)*b*n)

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 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}} \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 )\right ) \text {Subst}\left (\int \frac {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} \\ & = \frac {2 x^{1+m} \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 ) \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-7 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} \\ \end{align*}

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(466\) vs. \(2(130)=260\).

Time = 7.34 (sec) , antiderivative size = 466, normalized size of antiderivative = 3.58 \[ \int x^m \csc ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {x^{1+m-i b n} \left (\left (4+8 m+4 m^2+b^2 n^2\right ) x^{2 i b n} \sqrt {2-2 e^{2 i a} \left (c x^n\right )^{2 i b}} \sqrt {\frac {i e^{i a} \left (c x^n\right )^{i b}}{-1+e^{2 i a} \left (c x^n\right )^{2 i b}}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-\frac {i \left (1+m+\frac {3 i b n}{2}\right )}{2 b n},-\frac {2 i+2 i m-7 b n}{4 b n},e^{2 i a} \left (c x^n\right )^{2 i b}\right )+(-2 i-2 i m+3 b n) \left ((-2 i-2 i m+b n) \sqrt {2-2 e^{2 i a} \left (c x^n\right )^{2 i b}} \sqrt {\frac {i e^{i a} \left (c x^n\right )^{i b}}{-1+e^{2 i a} \left (c x^n\right )^{2 i b}}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-\frac {2 i+2 i m+b n}{4 b n},-\frac {2 i+2 i m-3 b n}{4 b n},e^{2 i a} \left (c x^n\right )^{2 i b}\right )-2 x^{i b n} \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )} (b n \cos (b n \log (x))-2 (1+m) \sin (b n \log (x)))\right )\right )}{b n (-2 i-2 i m+3 b n) \left (b n \cos \left (a-b n \log (x)+b \log \left (c x^n\right )\right )+2 (1+m) \sin \left (a-b n \log (x)+b \log \left (c x^n\right )\right )\right )} \]

[In]

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

[Out]

(x^(1 + m - I*b*n)*((4 + 8*m + 4*m^2 + b^2*n^2)*x^((2*I)*b*n)*Sqrt[2 - 2*E^((2*I)*a)*(c*x^n)^((2*I)*b)]*Sqrt[(
I*E^(I*a)*(c*x^n)^(I*b))/(-1 + E^((2*I)*a)*(c*x^n)^((2*I)*b))]*Hypergeometric2F1[1/2, ((-1/2*I)*(1 + m + ((3*I
)/2)*b*n))/(b*n), -1/4*(2*I + (2*I)*m - 7*b*n)/(b*n), E^((2*I)*a)*(c*x^n)^((2*I)*b)] + (-2*I - (2*I)*m + 3*b*n
)*((-2*I - (2*I)*m + b*n)*Sqrt[2 - 2*E^((2*I)*a)*(c*x^n)^((2*I)*b)]*Sqrt[(I*E^(I*a)*(c*x^n)^(I*b))/(-1 + E^((2
*I)*a)*(c*x^n)^((2*I)*b))]*Hypergeometric2F1[1/2, -1/4*(2*I + (2*I)*m + b*n)/(b*n), -1/4*(2*I + (2*I)*m - 3*b*
n)/(b*n), E^((2*I)*a)*(c*x^n)^((2*I)*b)] - 2*x^(I*b*n)*Sqrt[Csc[a + b*Log[c*x^n]]]*(b*n*Cos[b*n*Log[x]] - 2*(1
+ m)*Sin[b*n*Log[x]]))))/(b*n*(-2*I - (2*I)*m + 3*b*n)*(b*n*Cos[a - b*n*Log[x] + b*Log[c*x^n]] + 2*(1 + m)*Si
n[a - b*n*Log[x] + b*Log[c*x^n]]))

Maple [F]

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

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

[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 (co
nstant residues)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[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(-1)]

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

[In]

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

[Out]

Timed out

Mupad [F(-1)]

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

[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)