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

Optimal. Leaf size=85 \[ -\frac{2 e^{i a} \left (c x^n\right )^{i b} \text{Hypergeometric2F1}\left (1,\frac{1}{2} \left (1+\frac{i}{b n}\right ),\frac{1}{2} \left (3+\frac{i}{b n}\right ),e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{x (b n+i)} \]

[Out]

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

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Rubi [A]  time = 0.059023, antiderivative size = 85, 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 = {4510, 4506, 364} \[ -\frac{2 e^{i a} \left (c x^n\right )^{i b} \, _2F_1\left (1,\frac{1}{2} \left (1+\frac{i}{b n}\right );\frac{1}{2} \left (3+\frac{i}{b n}\right );e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{x (b n+i)} \]

Antiderivative was successfully verified.

[In]

Int[Csc[a + b*Log[c*x^n]]/x^2,x]

[Out]

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

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 4506

Int[Csc[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] :> Dist[(-2*I)^p*E^(I*a*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}, 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{\csc \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx &=\frac{\left (c x^n\right )^{\frac{1}{n}} \operatorname{Subst}\left (\int x^{-1-\frac{1}{n}} \csc (a+b \log (x)) \, dx,x,c x^n\right )}{n x}\\ &=-\frac{\left (2 i e^{i a} \left (c x^n\right )^{\frac{1}{n}}\right ) \operatorname{Subst}\left (\int \frac{x^{-1+i b-\frac{1}{n}}}{1-e^{2 i a} x^{2 i b}} \, dx,x,c x^n\right )}{n x}\\ &=-\frac{2 e^{i a} \left (c x^n\right )^{i b} \, _2F_1\left (1,\frac{1}{2} \left (1+\frac{i}{b n}\right );\frac{1}{2} \left (3+\frac{i}{b n}\right );e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(i+b n) x}\\ \end{align*}

Mathematica [A]  time = 1.07642, size = 82, normalized size = 0.96 \[ -\frac{2 e^{i a} \left (c x^n\right )^{i b} \text{Hypergeometric2F1}\left (1,\frac{1}{2}+\frac{i}{2 b n},\frac{3}{2}+\frac{i}{2 b n},e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )}{x (b n+i)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.377, size = 0, normalized size = 0. \begin{align*} \int{\frac{\csc \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) }{{x}^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(csc(a + b*log(c*x**n))/x**2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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