\(\int \tan ^p(a+3 \log (x)) \, dx\) [157]

Optimal result

Integrand size = 9, antiderivative size = 120 \[ \int \tan ^p(a+3 \log (x)) \, dx=\left (1-e^{2 i a} x^{6 i}\right )^{-p} \left (\frac {i \left (1-e^{2 i a} x^{6 i}\right )}{1+e^{2 i a} x^{6 i}}\right )^p \left (1+e^{2 i a} x^{6 i}\right )^p x \operatorname {AppellF1}\left (-\frac {i}{6},-p,p,1-\frac {i}{6},e^{2 i a} x^{6 i},-e^{2 i a} x^{6 i}\right ) \] Output:

(I*(1-exp(2*I*a)*x^(6*I))/(1+exp(2*I*a)*x^(6*I)))^p*(1+exp(2*I*a)*x^(6*I)) 
^p*x*AppellF1(-1/6*I,-p,p,1-1/6*I,exp(2*I*a)*x^(6*I),-exp(2*I*a)*x^(6*I))/ 
((1-exp(2*I*a)*x^(6*I))^p)
 

Mathematica [A] (warning: unable to verify)

Time = 0.40 (sec) , antiderivative size = 240, normalized size of antiderivative = 2.00 \[ \int \tan ^p(a+3 \log (x)) \, dx=\frac {(1+6 i) \left (-\frac {i \left (-1+e^{2 i a} x^{6 i}\right )}{1+e^{2 i a} x^{6 i}}\right )^p x \operatorname {AppellF1}\left (-\frac {i}{6},-p,p,1-\frac {i}{6},e^{2 i a} x^{6 i},-e^{2 i a} x^{6 i}\right )}{(1+6 i) \operatorname {AppellF1}\left (-\frac {i}{6},-p,p,1-\frac {i}{6},e^{2 i a} x^{6 i},-e^{2 i a} x^{6 i}\right )-6 i e^{2 i a} p x^{6 i} \left (\operatorname {AppellF1}\left (1-\frac {i}{6},1-p,p,2-\frac {i}{6},e^{2 i a} x^{6 i},-e^{2 i a} x^{6 i}\right )+\operatorname {AppellF1}\left (1-\frac {i}{6},-p,1+p,2-\frac {i}{6},e^{2 i a} x^{6 i},-e^{2 i a} x^{6 i}\right )\right )} \] Input:

Integrate[Tan[a + 3*Log[x]]^p,x]
 

Output:

((1 + 6*I)*(((-I)*(-1 + E^((2*I)*a)*x^(6*I)))/(1 + E^((2*I)*a)*x^(6*I)))^p 
*x*AppellF1[-1/6*I, -p, p, 1 - I/6, E^((2*I)*a)*x^(6*I), -(E^((2*I)*a)*x^( 
6*I))])/((1 + 6*I)*AppellF1[-1/6*I, -p, p, 1 - I/6, E^((2*I)*a)*x^(6*I), - 
(E^((2*I)*a)*x^(6*I))] - (6*I)*E^((2*I)*a)*p*x^(6*I)*(AppellF1[1 - I/6, 1 
- p, p, 2 - I/6, E^((2*I)*a)*x^(6*I), -(E^((2*I)*a)*x^(6*I))] + AppellF1[1 
 - I/6, -p, 1 + p, 2 - I/6, E^((2*I)*a)*x^(6*I), -(E^((2*I)*a)*x^(6*I))]))
 

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {5002, 2058, 937, 936}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \tan ^p(a+3 \log (x)) \, dx\)

\(\Big \downarrow \) 5002

\(\displaystyle \int \left (\frac {i-i e^{2 i a} x^{6 i}}{1+e^{2 i a} x^{6 i}}\right )^pdx\)

\(\Big \downarrow \) 2058

\(\displaystyle \left (i-i e^{2 i a} x^{6 i}\right )^{-p} \left (\frac {i \left (1-e^{2 i a} x^{6 i}\right )}{1+e^{2 i a} x^{6 i}}\right )^p \left (1+e^{2 i a} x^{6 i}\right )^p \int \left (i-i e^{2 i a} x^{6 i}\right )^p \left (e^{2 i a} x^{6 i}+1\right )^{-p}dx\)

\(\Big \downarrow \) 937

\(\displaystyle \left (1-e^{2 i a} x^{6 i}\right )^{-p} \left (\frac {i \left (1-e^{2 i a} x^{6 i}\right )}{1+e^{2 i a} x^{6 i}}\right )^p \left (1+e^{2 i a} x^{6 i}\right )^p \int \left (1-e^{2 i a} x^{6 i}\right )^p \left (e^{2 i a} x^{6 i}+1\right )^{-p}dx\)

\(\Big \downarrow \) 936

\(\displaystyle x \left (1-e^{2 i a} x^{6 i}\right )^{-p} \left (\frac {i \left (1-e^{2 i a} x^{6 i}\right )}{1+e^{2 i a} x^{6 i}}\right )^p \left (1+e^{2 i a} x^{6 i}\right )^p \operatorname {AppellF1}\left (-\frac {i}{6},-p,p,1-\frac {i}{6},e^{2 i a} x^{6 i},-e^{2 i a} x^{6 i}\right )\)

Input:

Int[Tan[a + 3*Log[x]]^p,x]
 

Output:

(((I*(1 - E^((2*I)*a)*x^(6*I)))/(1 + E^((2*I)*a)*x^(6*I)))^p*(1 + E^((2*I) 
*a)*x^(6*I))^p*x*AppellF1[-1/6*I, -p, p, 1 - I/6, E^((2*I)*a)*x^(6*I), -(E 
^((2*I)*a)*x^(6*I))])/(1 - E^((2*I)*a)*x^(6*I))^p
 

Defintions of rubi rules used

rule 936

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 937

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] 
:> Simp[a^IntPart[p]*((a + b*x^n)^FracPart[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 2058

Int[(u_.)*((e_.)*((a_.) + (b_.)*(x_)^(n_.))^(q_.)*((c_) + (d_.)*(x_)^(n_))^ 
(r_.))^(p_), x_Symbol] :> Simp[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 5002

Int[Tan[((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]
 

Maple [F]

\[\int \tan \left (a +3 \ln \left (x \right )\right )^{p}d x\]

Input:

int(tan(a+3*ln(x))^p,x)
 

Output:

int(tan(a+3*ln(x))^p,x)
 

Fricas [F]

\[ \int \tan ^p(a+3 \log (x)) \, dx=\int { \tan \left (a + 3 \, \log \left (x\right )\right )^{p} \,d x } \] Input:

integrate(tan(a+3*log(x))^p,x, algorithm="fricas")
 

Output:

integral(tan(a + 3*log(x))^p, x)
 

Sympy [F]

\[ \int \tan ^p(a+3 \log (x)) \, dx=\int \tan ^{p}{\left (a + 3 \log {\left (x \right )} \right )}\, dx \] Input:

integrate(tan(a+3*ln(x))**p,x)
 

Output:

Integral(tan(a + 3*log(x))**p, x)
 

Maxima [F]

\[ \int \tan ^p(a+3 \log (x)) \, dx=\int { \tan \left (a + 3 \, \log \left (x\right )\right )^{p} \,d x } \] Input:

integrate(tan(a+3*log(x))^p,x, algorithm="maxima")
 

Output:

integrate(tan(a + 3*log(x))^p, x)
 

Giac [F]

\[ \int \tan ^p(a+3 \log (x)) \, dx=\int { \tan \left (a + 3 \, \log \left (x\right )\right )^{p} \,d x } \] Input:

integrate(tan(a+3*log(x))^p,x, algorithm="giac")
 

Output:

integrate(tan(a + 3*log(x))^p, x)
 

Mupad [F(-1)]

Timed out. \[ \int \tan ^p(a+3 \log (x)) \, dx=\int {\mathrm {tan}\left (a+3\,\ln \left (x\right )\right )}^p \,d x \] Input:

int(tan(a + 3*log(x))^p,x)
 

Output:

int(tan(a + 3*log(x))^p, x)
 

Reduce [F]

\[ \int \tan ^p(a+3 \log (x)) \, dx=\int \tan \left (3 \,\mathrm {log}\left (x \right )+a \right )^{p}d x \] Input:

int(tan(a+3*log(x))^p,x)
 

Output:

int(tan(3*log(x) + a)**p,x)