3.229 \(\int \cot ^p(d (a+b \log (c x^n))) \, dx\)

Optimal. Leaf size=190 \[ x \left (1-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^p \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )^{-p} \left (-\frac{i \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{1-e^{2 i a d} \left (c x^n\right )^{2 i b d}}\right )^p F_1\left (-\frac{i}{2 b d n};p,-p;1-\frac{i}{2 b d n};e^{2 i a d} \left (c x^n\right )^{2 i b d},-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right ) \]

[Out]

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

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Rubi [F]  time = 0.0150207, 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 = {} \[ \int \cot ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx \]

Verification is Not applicable to the result.

[In]

Int[Cot[d*(a + b*Log[c*x^n])]^p,x]

[Out]

Defer[Int][Cot[d*(a + b*Log[c*x^n])]^p, x]

Rubi steps

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

Mathematica [B]  time = 1.30346, size = 458, normalized size = 2.41 \[ \frac{x (2 b d n-i) \left (\frac{i \left (1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{-1+e^{2 i a d} \left (c x^n\right )^{2 i b d}}\right )^p F_1\left (-\frac{i}{2 b d n};p,-p;1-\frac{i}{2 b d n};e^{2 i a d} \left (c x^n\right )^{2 i b d},-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{2 b d n p e^{2 i a d} \left (c x^n\right )^{2 i b d} F_1\left (1-\frac{i}{2 b d n};p,1-p;2-\frac{i}{2 b d n};e^{2 i a d} \left (c x^n\right )^{2 i b d},-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )+2 b d n p e^{2 i a d} \left (c x^n\right )^{2 i b d} F_1\left (1-\frac{i}{2 b d n};p+1,-p;2-\frac{i}{2 b d n};e^{2 i a d} \left (c x^n\right )^{2 i b d},-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )+(2 b d n-i) F_1\left (-\frac{i}{2 b d n};p,-p;1-\frac{i}{2 b d n};e^{2 i a d} \left (c x^n\right )^{2 i b d},-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Cot[d*(a + b*Log[c*x^n])]^p,x]

[Out]

((-I + 2*b*d*n)*x*((I*(1 + E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d)))/(-1 + E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d)))^p*App
ellF1[(-I/2)/(b*d*n), p, -p, 1 - (I/2)/(b*d*n), E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d), -(E^((2*I)*a*d)*(c*x^n)^((2
*I)*b*d))])/(2*b*d*E^((2*I)*a*d)*n*p*(c*x^n)^((2*I)*b*d)*AppellF1[1 - (I/2)/(b*d*n), p, 1 - p, 2 - (I/2)/(b*d*
n), E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d), -(E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d))] + 2*b*d*E^((2*I)*a*d)*n*p*(c*x^n)
^((2*I)*b*d)*AppellF1[1 - (I/2)/(b*d*n), 1 + p, -p, 2 - (I/2)/(b*d*n), E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d), -(E^
((2*I)*a*d)*(c*x^n)^((2*I)*b*d))] + (-I + 2*b*d*n)*AppellF1[(-I/2)/(b*d*n), p, -p, 1 - (I/2)/(b*d*n), E^((2*I)
*a*d)*(c*x^n)^((2*I)*b*d), -(E^((2*I)*a*d)*(c*x^n)^((2*I)*b*d))])

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Maple [F]  time = 0.233, size = 0, normalized size = 0. \begin{align*} \int \left ( \cot \left ( d \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) \right ) ^{p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(cot((b*log(c*x^n) + a)*d)^p, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(cot(b*d*log(c*x^n) + a*d)^p, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError