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

Optimal. Leaf size=130 \[ \frac{2 x^{m+1} \left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{3/2} \csc ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right ) \text{Hypergeometric2F1}\left (\frac{3}{2},-\frac{-3 b n+2 i m+2 i}{4 b n},-\frac{-7 b n+2 i m+2 i}{4 b n},e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{3 i b n+2 m+2} \]

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

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Rubi [A]  time = 0.09169, antiderivative size = 126, normalized size of antiderivative = 0.97, 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} \[ \frac{2 x^{m+1} \left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{3/2} \, _2F_1\left (\frac{3}{2},\frac{1}{4} \left (3-\frac{2 i (m+1)}{b n}\right );-\frac{2 i m-7 b n+2 i}{4 b n};e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \csc ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right )}{3 i b n+2 m+2} \]

Antiderivative was successfully verified.

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

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])

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 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])

Rubi steps

\begin{align*} \int x^m \csc ^{\frac{3}{2}}\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}} \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 ) \operatorname{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 ) \, _2F_1\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]  time = 9.53125, size = 466, normalized size = 3.58 \[ \frac{x^{-i b n+m+1} \left (\left (b^2 n^2+4 m^2+8 m+4\right ) x^{2 i b n} \sqrt{2-2 e^{2 i a} \left (c x^n\right )^{2 i b}} \sqrt{\frac{i e^{i a} \left (c x^n\right )^{i b}}{-1+e^{2 i a} \left (c x^n\right )^{2 i b}}} \text{Hypergeometric2F1}\left (\frac{1}{2},-\frac{i \left (\frac{3 i b n}{2}+m+1\right )}{2 b n},-\frac{-7 b n+2 i m+2 i}{4 b n},e^{2 i a} \left (c x^n\right )^{2 i b}\right )+(3 b n-2 i m-2 i) \left ((b n-2 i m-2 i) \sqrt{2-2 e^{2 i a} \left (c x^n\right )^{2 i b}} \sqrt{\frac{i e^{i a} \left (c x^n\right )^{i b}}{-1+e^{2 i a} \left (c x^n\right )^{2 i b}}} \text{Hypergeometric2F1}\left (\frac{1}{2},-\frac{b n+2 i m+2 i}{4 b n},-\frac{-3 b n+2 i m+2 i}{4 b n},e^{2 i a} \left (c x^n\right )^{2 i b}\right )-2 x^{i b n} \sqrt{\csc \left (a+b \log \left (c x^n\right )\right )} (b n \cos (b n \log (x))-2 (m+1) \sin (b n \log (x)))\right )\right )}{b n (3 b n-2 i m-2 i) \left (2 (m+1) \sin \left (a+b \log \left (c x^n\right )-b n \log (x)\right )+b n \cos \left (a+b \log \left (c x^n\right )-b n \log (x)\right )\right )} \]

Warning: Unable to verify antiderivative.

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

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Maple [F]  time = 0.298, size = 0, normalized size = 0. \begin{align*} \int{x}^{m} \left ( \csc \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{m} \csc \left (b \log \left (c x^{n}\right ) + a\right )^{\frac{3}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out