Integrand size = 9, antiderivative size = 120 \[ \int \tan ^p(a+2 \log (x)) \, dx=\left (1-e^{2 i a} x^{4 i}\right )^{-p} \left (\frac {i \left (1-e^{2 i a} x^{4 i}\right )}{1+e^{2 i a} x^{4 i}}\right )^p \left (1+e^{2 i a} x^{4 i}\right )^p x \operatorname {AppellF1}\left (-\frac {i}{4},-p,p,1-\frac {i}{4},e^{2 i a} x^{4 i},-e^{2 i a} x^{4 i}\right ) \]
(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,exp(2*I*a)*x^(4*I),-exp(2*I*a)*x^(4*I))/ ((1-exp(2*I*a)*x^(4*I))^p)
Time = 0.39 (sec) , antiderivative size = 240, normalized size of antiderivative = 2.00 \[ \int \tan ^p(a+2 \log (x)) \, dx=\frac {(1+4 i) \left (-\frac {i \left (-1+e^{2 i a} x^{4 i}\right )}{1+e^{2 i a} x^{4 i}}\right )^p x \operatorname {AppellF1}\left (-\frac {i}{4},-p,p,1-\frac {i}{4},e^{2 i a} x^{4 i},-e^{2 i a} x^{4 i}\right )}{(1+4 i) \operatorname {AppellF1}\left (-\frac {i}{4},-p,p,1-\frac {i}{4},e^{2 i a} x^{4 i},-e^{2 i a} x^{4 i}\right )-4 i e^{2 i a} p x^{4 i} \left (\operatorname {AppellF1}\left (1-\frac {i}{4},1-p,p,2-\frac {i}{4},e^{2 i a} x^{4 i},-e^{2 i a} x^{4 i}\right )+\operatorname {AppellF1}\left (1-\frac {i}{4},-p,1+p,2-\frac {i}{4},e^{2 i a} x^{4 i},-e^{2 i a} x^{4 i}\right )\right )} \]
((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))]))
Time = 0.27 (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.
\(\displaystyle \int \tan ^p(a+2 \log (x)) \, dx\) |
\(\Big \downarrow \) 5002 |
\(\displaystyle \int \left (\frac {i-i e^{2 i a} x^{4 i}}{1+e^{2 i a} x^{4 i}}\right )^pdx\) |
\(\Big \downarrow \) 2058 |
\(\displaystyle \left (i-i e^{2 i a} x^{4 i}\right )^{-p} \left (\frac {i \left (1-e^{2 i a} x^{4 i}\right )}{1+e^{2 i a} x^{4 i}}\right )^p \left (1+e^{2 i a} x^{4 i}\right )^p \int \left (i-i e^{2 i a} x^{4 i}\right )^p \left (e^{2 i a} x^{4 i}+1\right )^{-p}dx\) |
\(\Big \downarrow \) 937 |
\(\displaystyle \left (1-e^{2 i a} x^{4 i}\right )^{-p} \left (\frac {i \left (1-e^{2 i a} x^{4 i}\right )}{1+e^{2 i a} x^{4 i}}\right )^p \left (1+e^{2 i a} x^{4 i}\right )^p \int \left (1-e^{2 i a} x^{4 i}\right )^p \left (e^{2 i a} x^{4 i}+1\right )^{-p}dx\) |
\(\Big \downarrow \) 936 |
\(\displaystyle x \left (1-e^{2 i a} x^{4 i}\right )^{-p} \left (\frac {i \left (1-e^{2 i a} x^{4 i}\right )}{1+e^{2 i a} x^{4 i}}\right )^p \left (1+e^{2 i a} x^{4 i}\right )^p \operatorname {AppellF1}\left (-\frac {i}{4},-p,p,1-\frac {i}{4},e^{2 i a} x^{4 i},-e^{2 i a} x^{4 i}\right )\) |
(((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[-1/4*I, -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
3.2.56.3.1 Defintions of rubi rules used
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])
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])
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]
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]
\[\int \tan \left (a +2 \ln \left (x \right )\right )^{p}d x\]
\[ \int \tan ^p(a+2 \log (x)) \, dx=\int { \tan \left (a + 2 \, \log \left (x\right )\right )^{p} \,d x } \]
\[ \int \tan ^p(a+2 \log (x)) \, dx=\int \tan ^{p}{\left (a + 2 \log {\left (x \right )} \right )}\, dx \]
\[ \int \tan ^p(a+2 \log (x)) \, dx=\int { \tan \left (a + 2 \, \log \left (x\right )\right )^{p} \,d x } \]
Timed out. \[ \int \tan ^p(a+2 \log (x)) \, dx=\text {Timed out} \]
Timed out. \[ \int \tan ^p(a+2 \log (x)) \, dx=\int {\mathrm {tan}\left (a+2\,\ln \left (x\right )\right )}^p \,d x \]