3.330 \(\int \csc ^p(a+b \log (c x^n)) \, dx\)

Optimal. Leaf size=107 \[ \frac{x \left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^p \text{Hypergeometric2F1}\left (p,-\frac{-b n p+i}{2 b n},\frac{1}{2} \left (-\frac{i}{b n}+p+2\right ),e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \csc ^p\left (a+b \log \left (c x^n\right )\right )}{1+i b n p} \]

[Out]

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

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Rubi [A]  time = 0.0664191, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {4504, 4508, 364} \[ \frac{x \left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^p \, _2F_1\left (p,-\frac{i-b n p}{2 b n};\frac{1}{2} \left (p-\frac{i}{b n}+2\right );e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \csc ^p\left (a+b \log \left (c x^n\right )\right )}{1+i b n p} \]

Antiderivative was successfully verified.

[In]

Int[Csc[a + b*Log[c*x^n]]^p,x]

[Out]

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

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

Mathematica [A]  time = 0.869853, size = 142, normalized size = 1.33 \[ -\frac{i x \left (2-2 e^{2 i a} \left (c x^n\right )^{2 i b}\right )^p \left (\frac{i e^{i a} \left (c x^n\right )^{i b}}{-1+e^{2 i a} \left (c x^n\right )^{2 i b}}\right )^p \text{Hypergeometric2F1}\left (p,\frac{b n p-i}{2 b n},\frac{1}{2} \left (-\frac{i}{b n}+p+2\right ),e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{b n p-i} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Csc[a + b*Log[c*x^n]]^p,x]

[Out]

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

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Maple [F]  time = 0.208, size = 0, normalized size = 0. \begin{align*} \int \left ( \csc \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(a+b*ln(c*x^n))^p,x)

[Out]

int(csc(a+b*ln(c*x^n))^p,x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(a+b*log(c*x^n))^p,x, algorithm="maxima")

[Out]

integrate(csc(b*log(c*x^n) + a)^p, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\csc \left (b \log \left (c x^{n}\right ) + a\right )^{p}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(a+b*log(c*x^n))^p,x, algorithm="fricas")

[Out]

integral(csc(b*log(c*x^n) + a)^p, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \csc ^{p}{\left (a + b \log{\left (c x^{n} \right )} \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(a+b*ln(c*x**n))**p,x)

[Out]

Integral(csc(a + b*log(c*x**n))**p, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(a+b*log(c*x^n))^p,x, algorithm="giac")

[Out]

integrate(csc(b*log(c*x^n) + a)^p, x)