Integrand size = 17, antiderivative size = 77 \[ \int (e x)^m \tan ^2(a+i \log (x)) \, dx=-\frac {x (e x)^m}{1+m}+\frac {2 x (e x)^m}{1+\frac {e^{2 i a}}{x^2}}-2 x (e x)^m \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (-1-m),\frac {1-m}{2},-\frac {e^{2 i a}}{x^2}\right ) \] Output:
-x*(e*x)^m/(1+m)+2*x*(e*x)^m/(1+exp(2*I*a)/x^2)-2*x*(e*x)^m*hypergeom([1, -1/2-1/2*m],[1/2-1/2*m],-exp(2*I*a)/x^2)
Time = 0.19 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.12 \[ \int (e x)^m \tan ^2(a+i \log (x)) \, dx=\frac {x (e x)^m \left (-1+4 \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{2},\frac {3+m}{2},-x^2 (\cos (2 a)-i \sin (2 a))\right )-4 \operatorname {Hypergeometric2F1}\left (2,\frac {1+m}{2},\frac {3+m}{2},-x^2 (\cos (2 a)-i \sin (2 a))\right )\right )}{1+m} \] Input:
Integrate[(e*x)^m*Tan[a + I*Log[x]]^2,x]
Output:
(x*(e*x)^m*(-1 + 4*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -(x^2*(Cos[2 *a] - I*Sin[2*a]))] - 4*Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2, -(x^2*( Cos[2*a] - I*Sin[2*a]))]))/(1 + m)
Time = 0.33 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.58, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {5006, 999, 25, 366, 27, 363, 278}
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.
\(\displaystyle \int (e x)^m \tan ^2(a+i \log (x)) \, dx\) |
\(\Big \downarrow \) 5006 |
\(\displaystyle \int \frac {\left (i-\frac {i e^{2 i a}}{x^2}\right )^2 (e x)^m}{\left (1+\frac {e^{2 i a}}{x^2}\right )^2}dx\) |
\(\Big \downarrow \) 999 |
\(\displaystyle -\left (\frac {1}{x}\right )^m (e x)^m \int -\frac {\left (1-\frac {e^{2 i a}}{x^2}\right )^2 \left (\frac {1}{x}\right )^{-m-2}}{\left (1+\frac {e^{2 i a}}{x^2}\right )^2}d\frac {1}{x}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \left (\frac {1}{x}\right )^m (e x)^m \int \frac {\left (1-\frac {e^{2 i a}}{x^2}\right )^2 \left (\frac {1}{x}\right )^{-m-2}}{\left (1+\frac {e^{2 i a}}{x^2}\right )^2}d\frac {1}{x}\) |
\(\Big \downarrow \) 366 |
\(\displaystyle -\left (\frac {1}{x}\right )^m (e x)^m \left (\frac {1}{2} e^{-4 i a} \int -\frac {2 \left (e^{4 i a} (2 m+3)+\frac {e^{6 i a}}{x^2}\right ) \left (\frac {1}{x}\right )^{-m-2}}{1+\frac {e^{2 i a}}{x^2}}d\frac {1}{x}-\frac {2 \left (\frac {1}{x}\right )^{-m-1}}{1+\frac {e^{2 i a}}{x^2}}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\left (\frac {1}{x}\right )^m (e x)^m \left (-e^{-4 i a} \int \frac {\left (e^{4 i a} (2 m+3)+\frac {e^{6 i a}}{x^2}\right ) \left (\frac {1}{x}\right )^{-m-2}}{1+\frac {e^{2 i a}}{x^2}}d\frac {1}{x}-\frac {2 \left (\frac {1}{x}\right )^{-m-1}}{1+\frac {e^{2 i a}}{x^2}}\right )\) |
\(\Big \downarrow \) 363 |
\(\displaystyle -\left (\frac {1}{x}\right )^m (e x)^m \left (-e^{-4 i a} \left (2 e^{4 i a} (m+1) \int \frac {\left (\frac {1}{x}\right )^{-m-2}}{1+\frac {e^{2 i a}}{x^2}}d\frac {1}{x}-\frac {e^{4 i a} \left (\frac {1}{x}\right )^{-m-1}}{m+1}\right )-\frac {2 \left (\frac {1}{x}\right )^{-m-1}}{1+\frac {e^{2 i a}}{x^2}}\right )\) |
\(\Big \downarrow \) 278 |
\(\displaystyle -\left (\frac {1}{x}\right )^m (e x)^m \left (-e^{-4 i a} \left (-2 e^{4 i a} \left (\frac {1}{x}\right )^{-m-1} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (-m-1),\frac {1-m}{2},-\frac {e^{2 i a}}{x^2}\right )-\frac {e^{4 i a} \left (\frac {1}{x}\right )^{-m-1}}{m+1}\right )-\frac {2 \left (\frac {1}{x}\right )^{-m-1}}{1+\frac {e^{2 i a}}{x^2}}\right )\) |
Input:
Int[(e*x)^m*Tan[a + I*Log[x]]^2,x]
Output:
-((x^(-1))^m*(e*x)^m*((-2*(x^(-1))^(-1 - m))/(1 + E^((2*I)*a)/x^2) - (-((E ^((4*I)*a)*(x^(-1))^(-1 - m))/(1 + m)) - 2*E^((4*I)*a)*(x^(-1))^(-1 - m)*H ypergeometric2F1[1, (-1 - m)/2, (1 - m)/2, -(E^((2*I)*a)/x^2)])/E^((4*I)*a )))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*(( c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/2, (m + 1)/2 + 1, ( -b)*(x^2/a)], x] /; FreeQ[{a, b, c, m, p}, x] && !IGtQ[p, 0] && (ILtQ[p, 0 ] || GtQ[a, 0])
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x _Symbol] :> Simp[d*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(b*e*(m + 2*p + 3))), x] - Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(b*(m + 2*p + 3)) Int[(e*x)^ m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b*c - a*d , 0] && NeQ[m + 2*p + 3, 0]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^2, x_Symbol] :> Simp[(-(b*c - a*d)^2)*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(2*a* b^2*e*(p + 1))), x] + Simp[1/(2*a*b^2*(p + 1)) Int[(e*x)^m*(a + b*x^2)^(p + 1)*Simp[(b*c - a*d)^2*(m + 1) + 2*b^2*c^2*(p + 1) + 2*a*b*d^2*(p + 1)*x^ 2, x], x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && LtQ[p , -1]
Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_) )^(q_), x_Symbol] :> Simp[(-(e*x)^m)*(x^(-1))^m Subst[Int[(a + b/x^n)^p*( (c + d/x^n)^q/x^(m + 2)), x], x, 1/x], x] /; FreeQ[{a, b, c, d, e, m, p, q} , x] && NeQ[b*c - a*d, 0] && ILtQ[n, 0] && !RationalQ[m]
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]
\[\int \left (e x \right )^{m} \tan \left (a +i \ln \left (x \right )\right )^{2}d x\]
Input:
int((e*x)^m*tan(a+I*ln(x))^2,x)
Output:
int((e*x)^m*tan(a+I*ln(x))^2,x)
\[ \int (e x)^m \tan ^2(a+i \log (x)) \, dx=\int { \left (e x\right )^{m} \tan \left (a + i \, \log \left (x\right )\right )^{2} \,d x } \] Input:
integrate((e*x)^m*tan(a+I*log(x))^2,x, algorithm="fricas")
Output:
integral(-(x^4 - 2*x^2*e^(2*I*a) + e^(4*I*a))*e^(m*log(e) + m*log(x))/(x^4 + 2*x^2*e^(2*I*a) + e^(4*I*a)), x)
\[ \int (e x)^m \tan ^2(a+i \log (x)) \, dx=\int \left (e x\right )^{m} \tan ^{2}{\left (a + i \log {\left (x \right )} \right )}\, dx \] Input:
integrate((e*x)**m*tan(a+I*ln(x))**2,x)
Output:
Integral((e*x)**m*tan(a + I*log(x))**2, x)
\[ \int (e x)^m \tan ^2(a+i \log (x)) \, dx=\int { \left (e x\right )^{m} \tan \left (a + i \, \log \left (x\right )\right )^{2} \,d x } \] Input:
integrate((e*x)^m*tan(a+I*log(x))^2,x, algorithm="maxima")
Output:
integrate((e*x)^m*tan(a + I*log(x))^2, x)
\[ \int (e x)^m \tan ^2(a+i \log (x)) \, dx=\int { \left (e x\right )^{m} \tan \left (a + i \, \log \left (x\right )\right )^{2} \,d x } \] Input:
integrate((e*x)^m*tan(a+I*log(x))^2,x, algorithm="giac")
Output:
integrate((e*x)^m*tan(a + I*log(x))^2, x)
Timed out. \[ \int (e x)^m \tan ^2(a+i \log (x)) \, dx=\int {\mathrm {tan}\left (a+\ln \left (x\right )\,1{}\mathrm {i}\right )}^2\,{\left (e\,x\right )}^m \,d x \] Input:
int(tan(a + log(x)*1i)^2*(e*x)^m,x)
Output:
int(tan(a + log(x)*1i)^2*(e*x)^m, x)
\[ \int (e x)^m \tan ^2(a+i \log (x)) \, dx=\frac {e^{m} \left (-x^{m} \tan \left (\mathrm {log}\left (x \right ) i +a \right ) i m x -x^{m} \tan \left (\mathrm {log}\left (x \right ) i +a \right ) i x -x^{m} x +\left (\int x^{m} \tan \left (\mathrm {log}\left (x \right ) i +a \right )d x \right ) i \,m^{2}+2 \left (\int x^{m} \tan \left (\mathrm {log}\left (x \right ) i +a \right )d x \right ) i m +\left (\int x^{m} \tan \left (\mathrm {log}\left (x \right ) i +a \right )d x \right ) i \right )}{m +1} \] Input:
int((e*x)^m*tan(a+I*log(x))^2,x)
Output:
(e**m*( - x**m*tan(log(x)*i + a)*i*m*x - x**m*tan(log(x)*i + a)*i*x - x**m *x + int(x**m*tan(log(x)*i + a),x)*i*m**2 + 2*int(x**m*tan(log(x)*i + a),x )*i*m + int(x**m*tan(log(x)*i + a),x)*i))/(m + 1)