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-exp(2*I*a)*x^(4*I))/(1+exp(2*I*a)*x^(4*I)))^p*(1+exp(2*I*a)*x^(4*I))^p*x*AppellF1(-1/4*I,-p,p,1-1/4*I,ex
p(2*I*a)*x^(4*I),-exp(2*I*a)*x^(4*I))/((1-exp(2*I*a)*x^(4*I))^p)

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Rubi [F]  time = 0.02, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, 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*}

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Mathematica [A]  time = 0.51, size = 240, normalized size = 2.00 \[ \frac {(1+4 i) x \left (-\frac {i \left (-1+e^{2 i a} x^{4 i}\right )}{1+e^{2 i a} x^{4 i}}\right )^p F_1\left (-\frac {i}{4};-p,p;1-\frac {i}{4};e^{2 i a} x^{4 i},-e^{2 i a} x^{4 i}\right )}{(1+4 i) F_1\left (-\frac {i}{4};-p,p;1-\frac {i}{4};e^{2 i a} x^{4 i},-e^{2 i a} x^{4 i}\right )-4 i e^{2 i a} p x^{4 i} \left (F_1\left (1-\frac {i}{4};1-p,p;2-\frac {i}{4};e^{2 i a} x^{4 i},-e^{2 i a} x^{4 i}\right )+F_1\left (1-\frac {i}{4};-p,p+1;2-\frac {i}{4};e^{2 i a} x^{4 i},-e^{2 i a} x^{4 i}\right )\right )} \]

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[-1/4*I, -p, p, 1 - I/4,
E^((2*I)*a)*x^(4*I), -(E^((2*I)*a)*x^(4*I))])/((1 + 4*I)*AppellF1[-1/4*I, -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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fricas [F]  time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\tan \left (a + 2 \, \log \relax (x)\right )^{p}, x\right ) \]

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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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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maple [F]  time = 0.32, size = 0, normalized size = 0.00 \[ \int \tan ^{p}\left (a +2 \ln \relax (x )\right )\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int \tan \left (a + 2 \, \log \relax (x)\right )^{p}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int {\mathrm {tan}\left (a+2\,\ln \relax (x)\right )}^p \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(a + 2*log(x))^p,x)

[Out]

int(tan(a + 2*log(x))^p, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \tan ^{p}{\left (a + 2 \log {\relax (x )} \right )}\, dx \]

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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