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

Optimal. Leaf size=109 \[ \frac{2 x \left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{3/2} \text{Hypergeometric2F1}\left (\frac{3}{2},\frac{1}{4} \left (3-\frac{2 i}{b n}\right ),\frac{1}{4} \left (7-\frac{2 i}{b n}\right ),e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \csc ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right )}{2+3 i b n} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0701248, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {4504, 4508, 364} \[ \frac{2 x \left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{3/2} \, _2F_1\left (\frac{3}{2},\frac{1}{4} \left (3-\frac{2 i}{b n}\right );\frac{1}{4} \left (7-\frac{2 i}{b n}\right );e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \csc ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right )}{2+3 i b n} \]

Antiderivative was successfully verified.

[In]

Int[Csc[a + b*Log[c*x^n]]^(3/2),x]

[Out]

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

Rule 4504

Int[Csc[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_.), x_Symbol] :> Dist[x/(n*(c*x^n)^(1/n)), Subst[Int[x
^(1/n - 1)*Csc[d*(a + b*Log[x])]^p, x], x, c*x^n], x] /; FreeQ[{a, b, c, d, 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 \csc ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac{\left (x \left (c x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int x^{-1+\frac{1}{n}} \csc ^{\frac{3}{2}}(a+b \log (x)) \, dx,x,c x^n\right )}{n}\\ &=\frac{\left (x \left (c x^n\right )^{-\frac{3 i b}{2}-\frac{1}{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}{n}}}{\left (1-e^{2 i a} x^{2 i b}\right )^{3/2}} \, dx,x,c x^n\right )}{n}\\ &=\frac{2 x \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}{b n}\right );\frac{1}{4} \left (7-\frac{2 i}{b n}\right );e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{2+3 i b n}\\ \end{align*}

Mathematica [B]  time = 6.05775, size = 411, normalized size = 3.77 \[ \frac{x \left (\left (b^2 n^2+4\right ) x^{i b n} \sqrt{2-2 e^{2 i a} \left (c x^n\right )^{2 i b}} \sqrt{\frac{i e^{i a} \left (c x^n\right )^{i b}}{-1+e^{2 i a} \left (c x^n\right )^{2 i b}}} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{3}{4}-\frac{i}{2 b n},\frac{7}{4}-\frac{i}{2 b n},e^{2 i a} \left (c x^n\right )^{2 i b}\right )-(3 b n-2 i) x^{-i b n} \left ((-b n+2 i) \sqrt{2-2 e^{2 i a} \left (c x^n\right )^{2 i b}} \sqrt{\frac{i e^{i a} \left (c x^n\right )^{i b}}{-1+e^{2 i a} \left (c x^n\right )^{2 i b}}} \text{Hypergeometric2F1}\left (\frac{1}{2},-\frac{b n+2 i}{4 b n},\frac{3}{4}-\frac{i}{2 b n},e^{2 i a} \left (c x^n\right )^{2 i b}\right )+2 x^{i b n} (b n \cos (b n \log (x))-2 \sin (b n \log (x))) \sqrt{\csc \left (a+b \log \left (c x^n\right )\right )}\right )\right )}{b n (3 b n-2 i) \left (2 \sin \left (a+b \log \left (c x^n\right )-b n \log (x)\right )+b n \cos \left (a+b \log \left (c x^n\right )-b n \log (x)\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 0.295, size = 0, normalized size = 0. \begin{align*} \int \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(csc(a+b*ln(c*x^n))^(3/2),x)

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \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(csc(a+b*log(c*x^n))^(3/2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

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

[Out]

Exception raised: UnboundLocalError

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

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

[Out]

Timed out