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

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

[Out]

(x*((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*(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 = 0.0148668, 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 = {} \[ \int \tan ^p\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx \]

Verification is Not applicable to the result.

[In]

Int[Tan[d*(a + b*Log[c*x^n])]^p,x]

[Out]

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

Rubi steps

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

Mathematica [B]  time = 1.41376, size = 458, normalized size = 2.41 \[ \frac{x (2 b d n-i) \left (-\frac{i \left (-1+e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{1+e^{2 i a d} \left (c x^n\right )^{2 i b d}}\right )^p F_1\left (-\frac{i}{2 b d n};-p,p;1-\frac{i}{2 b d n};e^{2 i a d} \left (c x^n\right )^{2 i b d},-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )}{-2 b d n p e^{2 i a d} \left (c x^n\right )^{2 i b d} F_1\left (1-\frac{i}{2 b d n};1-p,p;2-\frac{i}{2 b d n};e^{2 i a d} \left (c x^n\right )^{2 i b d},-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )-2 b d n p e^{2 i a d} \left (c x^n\right )^{2 i b d} F_1\left (1-\frac{i}{2 b d n};-p,p+1;2-\frac{i}{2 b d n};e^{2 i a d} \left (c x^n\right )^{2 i b d},-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )+(2 b d n-i) F_1\left (-\frac{i}{2 b d n};-p,p;1-\frac{i}{2 b d n};e^{2 i a d} \left (c x^n\right )^{2 i b d},-e^{2 i a d} \left (c x^n\right )^{2 i b d}\right )} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Tan[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*
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))])/(-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))] - 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))] + (-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 = 0.177, size = 0, normalized size = 0. \begin{align*} \int \left ( \tan \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(tan(d*(a+b*ln(c*x^n)))^p,x)

[Out]

int(tan(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 \tan \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(tan(d*(a+b*log(c*x^n)))^p,x, algorithm="maxima")

[Out]

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

[Out]

integral(tan(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 \tan ^{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(tan(d*(a+b*ln(c*x**n)))**p,x)

[Out]

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

[Out]

Timed out