3.156 \(\int \tan ^p(a+2 \log (x)) \, dx\)

Optimal. Leaf size=120 \[ x \left (1-e^{2 i a} x^{4 i}\right )^{-p} \left (\frac{i \left (1-e^{2 i a} x^{4 i}\right )}{1+e^{2 i a} x^{4 i}}\right )^p \left (1+e^{2 i a} x^{4 i}\right )^p F_1\left (-\frac{i}{4};-p,p;1-\frac{i}{4};e^{2 i a} x^{4 i},-e^{2 i a} x^{4 i}\right ) \]

[Out]

(((I*(1 - E^((2*I)*a)*x^(4*I)))/(1 + E^((2*I)*a)*x^(4*I)))^p*(1 + E^((2*I)*a)*x^(4*I))^p*x*AppellF1[-I/4, -p,
p, 1 - I/4, E^((2*I)*a)*x^(4*I), -(E^((2*I)*a)*x^(4*I))])/(1 - E^((2*I)*a)*x^(4*I))^p

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Rubi [F]  time = 0.0196184, 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 = {} \[ \int \tan ^p(a+2 \log (x)) \, dx \]

Verification is Not applicable to the result.

[In]

Int[Tan[a + 2*Log[x]]^p,x]

[Out]

Defer[Int][Tan[a + 2*Log[x]]^p, x]

Rubi steps

\begin{align*} \int \tan ^p(a+2 \log (x)) \, dx &=\int \tan ^p(a+2 \log (x)) \, dx\\ \end{align*}

Mathematica [A]  time = 0.495122, size = 240, normalized size = 2. \[ \frac{(1+4 i) x \left (-\frac{i \left (-1+e^{2 i a} x^{4 i}\right )}{1+e^{2 i a} x^{4 i}}\right )^p F_1\left (-\frac{i}{4};-p,p;1-\frac{i}{4};e^{2 i a} x^{4 i},-e^{2 i a} x^{4 i}\right )}{(1+4 i) F_1\left (-\frac{i}{4};-p,p;1-\frac{i}{4};e^{2 i a} x^{4 i},-e^{2 i a} x^{4 i}\right )-4 i e^{2 i a} p x^{4 i} \left (F_1\left (1-\frac{i}{4};1-p,p;2-\frac{i}{4};e^{2 i a} x^{4 i},-e^{2 i a} x^{4 i}\right )+F_1\left (1-\frac{i}{4};-p,p+1;2-\frac{i}{4};e^{2 i a} x^{4 i},-e^{2 i a} x^{4 i}\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Tan[a + 2*Log[x]]^p,x]

[Out]

((1 + 4*I)*(((-I)*(-1 + E^((2*I)*a)*x^(4*I)))/(1 + E^((2*I)*a)*x^(4*I)))^p*x*AppellF1[-I/4, -p, p, 1 - I/4, E^
((2*I)*a)*x^(4*I), -(E^((2*I)*a)*x^(4*I))])/((1 + 4*I)*AppellF1[-I/4, -p, p, 1 - I/4, E^((2*I)*a)*x^(4*I), -(E
^((2*I)*a)*x^(4*I))] - (4*I)*E^((2*I)*a)*p*x^(4*I)*(AppellF1[1 - I/4, 1 - p, p, 2 - I/4, E^((2*I)*a)*x^(4*I),
-(E^((2*I)*a)*x^(4*I))] + AppellF1[1 - I/4, -p, 1 + p, 2 - I/4, E^((2*I)*a)*x^(4*I), -(E^((2*I)*a)*x^(4*I))]))

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Maple [F]  time = 0.322, size = 0, normalized size = 0. \begin{align*} \int \left ( \tan \left ( a+2\,\ln \left ( x \right ) \right ) \right ) ^{p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(a+2*ln(x))^p,x)

[Out]

int(tan(a+2*ln(x))^p,x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(a+2*log(x))^p,x, algorithm="maxima")

[Out]

integrate(tan(a + 2*log(x))^p, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(a+2*log(x))^p,x, algorithm="fricas")

[Out]

integral(tan(a + 2*log(x))^p, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(a+2*ln(x))**p,x)

[Out]

Integral(tan(a + 2*log(x))**p, x)

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(a+2*log(x))^p,x, algorithm="giac")

[Out]

Timed out