3.318 \(\int \frac{1}{\csc ^{\frac{5}{2}}(a+b \log (c x^n))} \, dx\)

Optimal. Leaf size=110 \[ \frac{2 x \text{Hypergeometric2F1}\left (-\frac{5}{2},\frac{1}{4} \left (-5-\frac{2 i}{b n}\right ),-\frac{b n+2 i}{4 b n},e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(2-5 i b n) \left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{5/2} \csc ^{\frac{5}{2}}\left (a+b \log \left (c x^n\right )\right )} \]

[Out]

(2*x*Hypergeometric2F1[-5/2, (-5 - (2*I)/(b*n))/4, -(2*I + b*n)/(4*b*n), E^((2*I)*a)*(c*x^n)^((2*I)*b)])/((2 -
(5*I)*b*n)*(1 - E^((2*I)*a)*(c*x^n)^((2*I)*b))^(5/2)*Csc[a + b*Log[c*x^n]]^(5/2))

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Rubi [A]  time = 0.0731525, antiderivative size = 110, 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} \[ \frac{2 x \, _2F_1\left (-\frac{5}{2},\frac{1}{4} \left (-5-\frac{2 i}{b n}\right );-\frac{b n+2 i}{4 b n};e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(2-5 i b n) \left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{5/2} \csc ^{\frac{5}{2}}\left (a+b \log \left (c x^n\right )\right )} \]

Antiderivative was successfully verified.

[In]

Int[Csc[a + b*Log[c*x^n]]^(-5/2),x]

[Out]

(2*x*Hypergeometric2F1[-5/2, (-5 - (2*I)/(b*n))/4, -(2*I + b*n)/(4*b*n), E^((2*I)*a)*(c*x^n)^((2*I)*b)])/((2 -
(5*I)*b*n)*(1 - E^((2*I)*a)*(c*x^n)^((2*I)*b))^(5/2)*Csc[a + b*Log[c*x^n]]^(5/2))

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 \frac{1}{\csc ^{\frac{5}{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 \frac{x^{-1+\frac{1}{n}}}{\csc ^{\frac{5}{2}}(a+b \log (x))} \, dx,x,c x^n\right )}{n}\\ &=\frac{\left (x \left (c x^n\right )^{\frac{5 i b}{2}-\frac{1}{n}}\right ) \operatorname{Subst}\left (\int x^{-1-\frac{5 i b}{2}+\frac{1}{n}} \left (1-e^{2 i a} x^{2 i b}\right )^{5/2} \, dx,x,c x^n\right )}{n \left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{5/2} \csc ^{\frac{5}{2}}\left (a+b \log \left (c x^n\right )\right )}\\ &=\frac{2 x \, _2F_1\left (-\frac{5}{2},\frac{1}{4} \left (-5-\frac{2 i}{b n}\right );-\frac{2 i+b n}{4 b n};e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(2-5 i b n) \left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{5/2} \csc ^{\frac{5}{2}}\left (a+b \log \left (c x^n\right )\right )}\\ \end{align*}

Mathematica [B]  time = 8.75875, size = 876, normalized size = 7.96 \[ \sqrt{\csc \left (a+b n \log (x)+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )} \left (-\frac{x \cos (b n \log (x)) \left (-55 b^2 n^2+65 b^2 \cos \left (2 \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )\right ) n^2+4 b \sin \left (2 \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )\right ) n+12 \cos \left (2 \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )\right )-12\right )}{4 (5 b n-2 i) (5 b n+2 i) \left (b n \cos \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )+2 \sin \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )\right )}+\frac{x \sin (b n \log (x)) \left (65 b^2 \sin \left (2 \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )\right ) n^2+16 b n-4 b \cos \left (2 \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )\right ) n+12 \sin \left (2 \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )\right )\right )}{4 (5 b n-2 i) (5 b n+2 i) \left (b n \cos \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )+2 \sin \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )\right )}+\frac{x \cos (3 b n \log (x)) \left (5 b n \cos \left (3 \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )\right )-2 \sin \left (3 \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )\right )\right )}{2 (5 b n-2 i) (5 b n+2 i)}-\frac{x \sin (3 b n \log (x)) \left (2 \cos \left (3 \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )\right )+5 b n \sin \left (3 \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )\right )\right )}{2 (5 b n-2 i) (5 b n+2 i)}\right )-\frac{30 b^3 e^{i \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )} n^3 x^{1-i b n} \sqrt{2-2 e^{2 i \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )} x^{2 i b n}} \sqrt{\frac{i e^{i \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )} x^{i b n}}{e^{2 i \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )} x^{2 i b n}-1}} \left ((b n+2 i) \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 \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )} x^{2 i b n}\right ) x^{2 i b n}+(3 b n-2 i) \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 \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )} x^{2 i b n}\right )\right )}{(b n+2 i) (3 b n-2 i) (5 b n-2 i) (5 b n+2 i) \left (b n+e^{2 i \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )} (b n-2 i)+2 i\right )} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Csc[a + b*Log[c*x^n]]^(-5/2),x]

[Out]

(-30*b^3*E^(I*(a + b*(-(n*Log[x]) + Log[c*x^n])))*n^3*x^(1 - I*b*n)*Sqrt[2 - 2*E^((2*I)*(a + b*(-(n*Log[x]) +
Log[c*x^n])))*x^((2*I)*b*n)]*Sqrt[(I*E^(I*(a + b*(-(n*Log[x]) + Log[c*x^n])))*x^(I*b*n))/(-1 + E^((2*I)*(a + b
*(-(n*Log[x]) + Log[c*x^n])))*x^((2*I)*b*n))]*((2*I + b*n)*x^((2*I)*b*n)*Hypergeometric2F1[1/2, 3/4 - (I/2)/(b
*n), 7/4 - (I/2)/(b*n), E^((2*I)*(a + b*(-(n*Log[x]) + Log[c*x^n])))*x^((2*I)*b*n)] + (-2*I + 3*b*n)*Hypergeom
etric2F1[1/2, -(2*I + b*n)/(4*b*n), 3/4 - (I/2)/(b*n), E^((2*I)*(a + b*(-(n*Log[x]) + Log[c*x^n])))*x^((2*I)*b
*n)]))/((2*I + b*n)*(-2*I + 3*b*n)*(-2*I + 5*b*n)*(2*I + 5*b*n)*(2*I + b*n + E^((2*I)*(a + b*(-(n*Log[x]) + Lo
g[c*x^n])))*(-2*I + b*n))) + Sqrt[Csc[a + b*n*Log[x] + b*(-(n*Log[x]) + Log[c*x^n])]]*(-(x*Cos[b*n*Log[x]]*(-1
2 - 55*b^2*n^2 + 12*Cos[2*(a + b*(-(n*Log[x]) + Log[c*x^n]))] + 65*b^2*n^2*Cos[2*(a + b*(-(n*Log[x]) + Log[c*x
^n]))] + 4*b*n*Sin[2*(a + b*(-(n*Log[x]) + Log[c*x^n]))]))/(4*(-2*I + 5*b*n)*(2*I + 5*b*n)*(b*n*Cos[a + b*(-(n
*Log[x]) + Log[c*x^n])] + 2*Sin[a + b*(-(n*Log[x]) + Log[c*x^n])])) + (x*Sin[b*n*Log[x]]*(16*b*n - 4*b*n*Cos[2
*(a + b*(-(n*Log[x]) + Log[c*x^n]))] + 12*Sin[2*(a + b*(-(n*Log[x]) + Log[c*x^n]))] + 65*b^2*n^2*Sin[2*(a + b*
(-(n*Log[x]) + Log[c*x^n]))]))/(4*(-2*I + 5*b*n)*(2*I + 5*b*n)*(b*n*Cos[a + b*(-(n*Log[x]) + Log[c*x^n])] + 2*
Sin[a + b*(-(n*Log[x]) + Log[c*x^n])])) + (x*Cos[3*b*n*Log[x]]*(5*b*n*Cos[3*(a + b*(-(n*Log[x]) + Log[c*x^n]))
] - 2*Sin[3*(a + b*(-(n*Log[x]) + Log[c*x^n]))]))/(2*(-2*I + 5*b*n)*(2*I + 5*b*n)) - (x*Sin[3*b*n*Log[x]]*(2*C
os[3*(a + b*(-(n*Log[x]) + Log[c*x^n]))] + 5*b*n*Sin[3*(a + b*(-(n*Log[x]) + Log[c*x^n]))]))/(2*(-2*I + 5*b*n)
*(2*I + 5*b*n)))

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Maple [F]  time = 0.3, size = 0, normalized size = 0. \begin{align*} \int \left ( \csc \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{-{\frac{5}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/csc(a+b*ln(c*x^n))^(5/2),x)

[Out]

int(1/csc(a+b*ln(c*x^n))^(5/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\csc \left (b \log \left (c x^{n}\right ) + a\right )^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/csc(a+b*log(c*x^n))^(5/2),x, algorithm="maxima")

[Out]

integrate(csc(b*log(c*x^n) + a)^(-5/2), 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(1/csc(a+b*log(c*x^n))^(5/2),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(1/csc(a+b*ln(c*x**n))**(5/2),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(1/csc(a+b*log(c*x^n))^(5/2),x, algorithm="giac")

[Out]

Timed out