3.3.6 \(\int \cot ^p(a+\log (x)) \, dx\) [206]

Optimal. Leaf size=120 \[ \left (1-e^{2 i a} x^{2 i}\right )^p \left (1+e^{2 i a} x^{2 i}\right )^{-p} \left (-\frac {i \left (1+e^{2 i a} x^{2 i}\right )}{1-e^{2 i a} x^{2 i}}\right )^p x F_1\left (-\frac {i}{2};p,-p;1-\frac {i}{2};e^{2 i a} x^{2 i},-e^{2 i a} x^{2 i}\right ) \]

[Out]

(1-exp(2*I*a)*x^(2*I))^p*(-I*(1+exp(2*I*a)*x^(2*I))/(1-exp(2*I*a)*x^(2*I)))^p*x*AppellF1(-1/2*I,p,-p,1-1/2*I,e
xp(2*I*a)*x^(2*I),-exp(2*I*a)*x^(2*I))/((1+exp(2*I*a)*x^(2*I))^p)

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Rubi [A]
time = 0.04, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {4588, 1986, 441, 440} \begin {gather*} x \left (1-e^{2 i a} x^{2 i}\right )^p \left (1+e^{2 i a} x^{2 i}\right )^{-p} \left (-\frac {i \left (1+e^{2 i a} x^{2 i}\right )}{1-e^{2 i a} x^{2 i}}\right )^p F_1\left (-\frac {i}{2};p,-p;1-\frac {i}{2};e^{2 i a} x^{2 i},-e^{2 i a} x^{2 i}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cot[a + Log[x]]^p,x]

[Out]

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

Rule 440

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*x*AppellF1[1/n, -p,
-q, 1 + 1/n, (-b)*(x^n/a), (-d)*(x^n/c)], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n
, -1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rule 441

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[a^IntPart[p]*((a + b*x^n)^F
racPart[p]/(1 + b*(x^n/a))^FracPart[p]), Int[(1 + b*(x^n/a))^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n,
p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n, -1] &&  !(IntegerQ[p] || GtQ[a, 0])

Rule 1986

Int[(u_.)*((e_.)*((a_.) + (b_.)*(x_)^(n_.))^(q_.)*((c_) + (d_.)*(x_)^(n_))^(r_.))^(p_), x_Symbol] :> Dist[Simp
[(e*(a + b*x^n)^q*(c + d*x^n)^r)^p/((a + b*x^n)^(p*q)*(c + d*x^n)^(p*r))], Int[u*(a + b*x^n)^(p*q)*(c + d*x^n)
^(p*r), x], x] /; FreeQ[{a, b, c, d, e, n, p, q, r}, x]

Rule 4588

Int[Cot[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_.), x_Symbol] :> Int[((-I - I*E^(2*I*a*d)*x^(2*I*b*d))/(1 - E^(2*I*a
*d)*x^(2*I*b*d)))^p, x] /; FreeQ[{a, b, d, p}, x]

Rubi steps

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

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Mathematica [A]
time = 0.56, size = 238, normalized size = 1.98 \begin {gather*} \frac {(2-i) \left (\frac {i \left (1+e^{2 i a} x^{2 i}\right )}{-1+e^{2 i a} x^{2 i}}\right )^p x F_1\left (-\frac {i}{2};p,-p;1-\frac {i}{2};e^{2 i a} x^{2 i},-e^{2 i a} x^{2 i}\right )}{(2-i) F_1\left (-\frac {i}{2};p,-p;1-\frac {i}{2};e^{2 i a} x^{2 i},-e^{2 i a} x^{2 i}\right )+2 e^{2 i a} p x^{2 i} \left (F_1\left (1-\frac {i}{2};p,1-p;2-\frac {i}{2};e^{2 i a} x^{2 i},-e^{2 i a} x^{2 i}\right )+F_1\left (1-\frac {i}{2};1+p,-p;2-\frac {i}{2};e^{2 i a} x^{2 i},-e^{2 i a} x^{2 i}\right )\right )} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[Cot[a + Log[x]]^p,x]

[Out]

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

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Maple [F]
time = 0.07, size = 0, normalized size = 0.00 \[\int \cot ^{p}\left (a +\ln \left (x \right )\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(cot(a + log(x))^p, x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(cot(a + log(x))^p, x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \cot ^{p}{\left (a + \log {\left (x \right )} \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(cot(a + log(x))**p, x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(cot(a + log(x))^p, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\mathrm {cot}\left (a+\ln \left (x\right )\right )}^p \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(a + log(x))^p,x)

[Out]

int(cot(a + log(x))^p, x)

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