3.204 \(\int \cot ^p(a+b \log (x)) \, dx\)

Optimal. Leaf size=142 \[ x \left (1-e^{2 i a} x^{2 i b}\right )^p \left (1+e^{2 i a} x^{2 i b}\right )^{-p} \left (-\frac{i \left (1+e^{2 i a} x^{2 i b}\right )}{1-e^{2 i a} x^{2 i b}}\right )^p F_1\left (-\frac{i}{2 b};p,-p;1-\frac{i}{2 b};e^{2 i a} x^{2 i b},-e^{2 i a} x^{2 i b}\right ) \]

[Out]

(x*(1 - E^((2*I)*a)*x^((2*I)*b))^p*(((-I)*(1 + E^((2*I)*a)*x^((2*I)*b)))/(1 - E^((2*I)*a)*x^((2*I)*b)))^p*Appe
llF1[(-I/2)/b, p, -p, 1 - (I/2)/b, E^((2*I)*a)*x^((2*I)*b), -(E^((2*I)*a)*x^((2*I)*b))])/(1 + E^((2*I)*a)*x^((
2*I)*b))^p

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Rubi [F]  time = 0.0215861, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \cot ^p(a+b \log (x)) \, dx \]

Verification is Not applicable to the result.

[In]

Int[Cot[a + b*Log[x]]^p,x]

[Out]

Defer[Int][Cot[a + b*Log[x]]^p, x]

Rubi steps

\begin{align*} \int \cot ^p(a+b \log (x)) \, dx &=\int \cot ^p(a+b \log (x)) \, dx\\ \end{align*}

Mathematica [B]  time = 0.593524, size = 330, normalized size = 2.32 \[ \frac{(2 b-i) x \left (\frac{i \left (1+e^{2 i a} x^{2 i b}\right )}{-1+e^{2 i a} x^{2 i b}}\right )^p F_1\left (-\frac{i}{2 b};p,-p;1-\frac{i}{2 b};e^{2 i a} x^{2 i b},-e^{2 i a} x^{2 i b}\right )}{2 e^{2 i a} b p x^{2 i b} F_1\left (1-\frac{i}{2 b};p,1-p;2-\frac{i}{2 b};e^{2 i a} x^{2 i b},-e^{2 i a} x^{2 i b}\right )+2 e^{2 i a} b p x^{2 i b} F_1\left (1-\frac{i}{2 b};p+1,-p;2-\frac{i}{2 b};e^{2 i a} x^{2 i b},-e^{2 i a} x^{2 i b}\right )+(2 b-i) F_1\left (-\frac{i}{2 b};p,-p;1-\frac{i}{2 b};e^{2 i a} x^{2 i b},-e^{2 i a} x^{2 i b}\right )} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Cot[a + b*Log[x]]^p,x]

[Out]

((-I + 2*b)*x*((I*(1 + E^((2*I)*a)*x^((2*I)*b)))/(-1 + E^((2*I)*a)*x^((2*I)*b)))^p*AppellF1[(-I/2)/b, p, -p, 1
- (I/2)/b, E^((2*I)*a)*x^((2*I)*b), -(E^((2*I)*a)*x^((2*I)*b))])/(2*b*E^((2*I)*a)*p*x^((2*I)*b)*AppellF1[1 -
(I/2)/b, p, 1 - p, 2 - (I/2)/b, E^((2*I)*a)*x^((2*I)*b), -(E^((2*I)*a)*x^((2*I)*b))] + 2*b*E^((2*I)*a)*p*x^((2
*I)*b)*AppellF1[1 - (I/2)/b, 1 + p, -p, 2 - (I/2)/b, E^((2*I)*a)*x^((2*I)*b), -(E^((2*I)*a)*x^((2*I)*b))] + (-
I + 2*b)*AppellF1[(-I/2)/b, p, -p, 1 - (I/2)/b, E^((2*I)*a)*x^((2*I)*b), -(E^((2*I)*a)*x^((2*I)*b))])

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Maple [F]  time = 0.372, size = 0, normalized size = 0. \begin{align*} \int \left ( \cot \left ( a+b\ln \left ( x \right ) \right ) \right ) ^{p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(a+b*ln(x))^p,x)

[Out]

int(cot(a+b*ln(x))^p,x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(cot(b*log(x) + a)^p, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(cot(b*log(x) + a)^p, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(a+b*ln(x))**p,x)

[Out]

Integral(cot(a + b*log(x))**p, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(cot(b*log(x) + a)^p, x)