Integrand size = 15, antiderivative size = 162 \[ \int (e x)^m \tan ^p(a+b \log (x)) \, dx=\frac {(e x)^{1+m} \left (1-e^{2 i a} x^{2 i b}\right )^{-p} \left (\frac {i \left (1-e^{2 i a} x^{2 i b}\right )}{1+e^{2 i a} x^{2 i b}}\right )^p \left (1+e^{2 i a} x^{2 i b}\right )^p \operatorname {AppellF1}\left (-\frac {i (1+m)}{2 b},-p,p,1-\frac {i (1+m)}{2 b},e^{2 i a} x^{2 i b},-e^{2 i a} x^{2 i b}\right )}{e (1+m)} \]
(e*x)^(1+m)*(I*(1-exp(2*I*a)*x^(2*I*b))/(1+exp(2*I*a)*x^(2*I*b)))^p*(1+exp (2*I*a)*x^(2*I*b))^p*AppellF1(-1/2*I*(1+m)/b,-p,p,1-1/2*I*(1+m)/b,exp(2*I* a)*x^(2*I*b),-exp(2*I*a)*x^(2*I*b))/e/(1+m)/((1-exp(2*I*a)*x^(2*I*b))^p)
Time = 0.70 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.97 \[ \int (e x)^m \tan ^p(a+b \log (x)) \, dx=\frac {x (e x)^m \left (1-e^{2 i a} x^{2 i b}\right )^{-p} \left (-\frac {i \left (-1+e^{2 i a} x^{2 i b}\right )}{1+e^{2 i a} x^{2 i b}}\right )^p \left (1+e^{2 i a} x^{2 i b}\right )^p \operatorname {AppellF1}\left (-\frac {i (1+m)}{2 b},-p,p,1-\frac {i (1+m)}{2 b},e^{2 i a} x^{2 i b},-e^{2 i a} x^{2 i b}\right )}{1+m} \]
(x*(e*x)^m*(((-I)*(-1 + E^((2*I)*a)*x^((2*I)*b)))/(1 + E^((2*I)*a)*x^((2*I )*b)))^p*(1 + E^((2*I)*a)*x^((2*I)*b))^p*AppellF1[((-1/2*I)*(1 + m))/b, -p , p, 1 - ((I/2)*(1 + m))/b, E^((2*I)*a)*x^((2*I)*b), -(E^((2*I)*a)*x^((2*I )*b))])/((1 + m)*(1 - E^((2*I)*a)*x^((2*I)*b))^p)
Time = 0.34 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {5006, 2058, 1013, 1012}
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.
\(\displaystyle \int (e x)^m \tan ^p(a+b \log (x)) \, dx\) |
\(\Big \downarrow \) 5006 |
\(\displaystyle \int (e x)^m \left (\frac {i-i e^{2 i a} x^{2 i b}}{1+e^{2 i a} x^{2 i b}}\right )^pdx\) |
\(\Big \downarrow \) 2058 |
\(\displaystyle \left (i-i e^{2 i a} x^{2 i b}\right )^{-p} \left (\frac {i \left (1-e^{2 i a} x^{2 i b}\right )}{1+e^{2 i a} x^{2 i b}}\right )^p \left (1+e^{2 i a} x^{2 i b}\right )^p \int (e x)^m \left (i-i e^{2 i a} x^{2 i b}\right )^p \left (e^{2 i a} x^{2 i b}+1\right )^{-p}dx\) |
\(\Big \downarrow \) 1013 |
\(\displaystyle \left (1-e^{2 i a} x^{2 i b}\right )^{-p} \left (\frac {i \left (1-e^{2 i a} x^{2 i b}\right )}{1+e^{2 i a} x^{2 i b}}\right )^p \left (1+e^{2 i a} x^{2 i b}\right )^p \int (e x)^m \left (1-e^{2 i a} x^{2 i b}\right )^p \left (e^{2 i a} x^{2 i b}+1\right )^{-p}dx\) |
\(\Big \downarrow \) 1012 |
\(\displaystyle \frac {(e x)^{m+1} \left (1-e^{2 i a} x^{2 i b}\right )^{-p} \left (\frac {i \left (1-e^{2 i a} x^{2 i b}\right )}{1+e^{2 i a} x^{2 i b}}\right )^p \left (1+e^{2 i a} x^{2 i b}\right )^p \operatorname {AppellF1}\left (-\frac {i (m+1)}{2 b},-p,p,1-\frac {i (m+1)}{2 b},e^{2 i a} x^{2 i b},-e^{2 i a} x^{2 i b}\right )}{e (m+1)}\) |
((e*x)^(1 + m)*((I*(1 - E^((2*I)*a)*x^((2*I)*b)))/(1 + E^((2*I)*a)*x^((2*I )*b)))^p*(1 + E^((2*I)*a)*x^((2*I)*b))^p*AppellF1[((-1/2*I)*(1 + m))/b, -p , p, 1 - ((I/2)*(1 + m))/b, E^((2*I)*a)*x^((2*I)*b), -(E^((2*I)*a)*x^((2*I )*b))])/(e*(1 + m)*(1 - E^((2*I)*a)*x^((2*I)*b))^p)
3.2.54.3.1 Defintions of rubi rules used
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ ))^(q_), x_Symbol] :> Simp[a^p*c^q*((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, (-b)*(x^n/a), (-d)*(x^n/c)], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
Int[((e_.)*(x_))^(m_.)*((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[(e*x)^m*(1 + b*(x^n/a))^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] & & NeQ[m, 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[((e_.)*(x_))^(m_.)*Tan[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_.), x_Symbol] :> Int[(e*x)^m*((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, e, m, p}, x]
\[\int \left (e x \right )^{m} \tan \left (a +b \ln \left (x \right )\right )^{p}d x\]
\[ \int (e x)^m \tan ^p(a+b \log (x)) \, dx=\int { \left (e x\right )^{m} \tan \left (b \log \left (x\right ) + a\right )^{p} \,d x } \]
\[ \int (e x)^m \tan ^p(a+b \log (x)) \, dx=\int \left (e x\right )^{m} \tan ^{p}{\left (a + b \log {\left (x \right )} \right )}\, dx \]
\[ \int (e x)^m \tan ^p(a+b \log (x)) \, dx=\int { \left (e x\right )^{m} \tan \left (b \log \left (x\right ) + a\right )^{p} \,d x } \]
\[ \int (e x)^m \tan ^p(a+b \log (x)) \, dx=\int { \left (e x\right )^{m} \tan \left (b \log \left (x\right ) + a\right )^{p} \,d x } \]
Timed out. \[ \int (e x)^m \tan ^p(a+b \log (x)) \, dx=\int {\mathrm {tan}\left (a+b\,\ln \left (x\right )\right )}^p\,{\left (e\,x\right )}^m \,d x \]