\(\int e^{m x} \tan ^2(x) \, dx\) [549]

Optimal result

Integrand size = 10, antiderivative size = 58 \[ \int e^{m x} \tan ^2(x) \, dx=-\frac {e^{m x}}{m}+\frac {4 e^{(2 i+m) x} \operatorname {Hypergeometric2F1}\left (2,1-\frac {i m}{2},2-\frac {i m}{2},-e^{2 i x}\right )}{2 i+m} \] Output:

-exp(m*x)/m+4*exp((2*I+m)*x)*hypergeom([2, 1-1/2*I*m],[2-1/2*I*m],-exp(2*I 
*x))/(2*I+m)
 

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.67 \[ \int e^{m x} \tan ^2(x) \, dx=\frac {e^{m x} \left (-1+\frac {i e^{2 i x} m^2 \operatorname {Hypergeometric2F1}\left (1,1-\frac {i m}{2},2-\frac {i m}{2},-e^{2 i x}\right )}{2 i+m}-i m \operatorname {Hypergeometric2F1}\left (1,-\frac {i m}{2},1-\frac {i m}{2},-e^{2 i x}\right )+m \tan (x)\right )}{m} \] Input:

Integrate[E^(m*x)*Tan[x]^2,x]
 

Output:

(E^(m*x)*(-1 + (I*E^((2*I)*x)*m^2*Hypergeometric2F1[1, 1 - (I/2)*m, 2 - (I 
/2)*m, -E^((2*I)*x)])/(2*I + m) - I*m*Hypergeometric2F1[1, (-1/2*I)*m, 1 - 
 (I/2)*m, -E^((2*I)*x)] + m*Tan[x]))/m
 

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.47, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4942, 2009}

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.

\(\displaystyle \int e^{m x} \tan ^2(x) \, dx\)

\(\Big \downarrow \) 4942

\(\displaystyle -\int \left (e^{m x}-\frac {4 e^{m x}}{1+e^{2 i x}}+\frac {4 e^{m x}}{\left (1+e^{2 i x}\right )^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {4 e^{m x} \operatorname {Hypergeometric2F1}\left (1,-\frac {i m}{2},1-\frac {i m}{2},-e^{2 i x}\right )}{m}-\frac {4 e^{m x} \operatorname {Hypergeometric2F1}\left (2,-\frac {i m}{2},1-\frac {i m}{2},-e^{2 i x}\right )}{m}-\frac {e^{m x}}{m}\)

Input:

Int[E^(m*x)*Tan[x]^2,x]
 

Output:

-(E^(m*x)/m) + (4*E^(m*x)*Hypergeometric2F1[1, (-1/2*I)*m, 1 - (I/2)*m, -E 
^((2*I)*x)])/m - (4*E^(m*x)*Hypergeometric2F1[2, (-1/2*I)*m, 1 - (I/2)*m, 
-E^((2*I)*x)])/m
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 4942

Int[(F_)^((c_.)*((a_.) + (b_.)*(x_)))*Tan[(d_.) + (e_.)*(x_)]^(n_.), x_Symb 
ol] :> Simp[I^n   Int[ExpandIntegrand[F^(c*(a + b*x))*((1 - E^(2*I*(d + e*x 
)))^n/(1 + E^(2*I*(d + e*x)))^n), x], x], x] /; FreeQ[{F, a, b, c, d, e}, x 
] && IntegerQ[n]
 

Maple [F]

\[\int {\mathrm e}^{m x} \tan \left (x \right )^{2}d x\]

Input:

int(exp(m*x)*tan(x)^2,x)
 

Output:

int(exp(m*x)*tan(x)^2,x)
 

Fricas [F]

\[ \int e^{m x} \tan ^2(x) \, dx=\int { e^{\left (m x\right )} \tan \left (x\right )^{2} \,d x } \] Input:

integrate(exp(m*x)*tan(x)^2,x, algorithm="fricas")
 

Output:

integral(e^(m*x)*tan(x)^2, x)
 

Sympy [F]

\[ \int e^{m x} \tan ^2(x) \, dx=\int e^{m x} \tan ^{2}{\left (x \right )}\, dx \] Input:

integrate(exp(m*x)*tan(x)**2,x)
 

Output:

Integral(exp(m*x)*tan(x)**2, x)
 

Maxima [F]

\[ \int e^{m x} \tan ^2(x) \, dx=\int { e^{\left (m x\right )} \tan \left (x\right )^{2} \,d x } \] Input:

integrate(exp(m*x)*tan(x)^2,x, algorithm="maxima")
 

Output:

-((m^4 + 20*m^2 + 64)*cos(4*x)^2*e^(m*x) - 4*(m^4 + 12*m^2 - 64)*cos(2*x)^ 
2*e^(m*x) + (m^4 + 20*m^2 + 64)*e^(m*x)*sin(4*x)^2 - 4*(m^4 + 12*m^2 - 64) 
*e^(m*x)*sin(2*x)^2 - 16*(11*m^2 - 16)*cos(2*x)*e^(m*x) + 8*(5*m^3 - 16*m) 
*e^(m*x)*sin(2*x) + 2*(8*(m^2 + 16)*cos(2*x)*e^(m*x) + 4*(m^3 + 16*m)*e^(m 
*x)*sin(2*x) + (m^4 - 28*m^2 + 64)*e^(m*x))*cos(4*x) + (m^4 - 76*m^2 + 64) 
*e^(m*x) - 16*(m^6 + 20*m^4 + (m^6 + 20*m^4 + 64*m^2)*cos(4*x)^2 + 4*(m^6 
+ 20*m^4 + 64*m^2)*cos(2*x)^2 + (m^6 + 20*m^4 + 64*m^2)*sin(4*x)^2 + 4*(m^ 
6 + 20*m^4 + 64*m^2)*sin(4*x)*sin(2*x) + 4*(m^6 + 20*m^4 + 64*m^2)*sin(2*x 
)^2 + 64*m^2 + 2*(m^6 + 20*m^4 + 64*m^2 + 2*(m^6 + 20*m^4 + 64*m^2)*cos(2* 
x))*cos(4*x) + 4*(m^6 + 20*m^4 + 64*m^2)*cos(2*x))*integrate(-(6*m*cos(6*x 
)*e^(m*x) + 18*m*cos(4*x)*e^(m*x) + 18*m*cos(2*x)*e^(m*x) - (m^2 - 8)*e^(m 
*x)*sin(6*x) - 3*(m^2 - 8)*e^(m*x)*sin(4*x) - 3*(m^2 - 8)*e^(m*x)*sin(2*x) 
 + 6*m*e^(m*x))/(m^4 + (m^4 + 20*m^2 + 64)*cos(6*x)^2 + 9*(m^4 + 20*m^2 + 
64)*cos(4*x)^2 + 9*(m^4 + 20*m^2 + 64)*cos(2*x)^2 + (m^4 + 20*m^2 + 64)*si 
n(6*x)^2 + 9*(m^4 + 20*m^2 + 64)*sin(4*x)^2 + 18*(m^4 + 20*m^2 + 64)*sin(4 
*x)*sin(2*x) + 9*(m^4 + 20*m^2 + 64)*sin(2*x)^2 + 20*m^2 + 2*(m^4 + 20*m^2 
 + 3*(m^4 + 20*m^2 + 64)*cos(4*x) + 3*(m^4 + 20*m^2 + 64)*cos(2*x) + 64)*c 
os(6*x) + 6*(m^4 + 20*m^2 + 3*(m^4 + 20*m^2 + 64)*cos(2*x) + 64)*cos(4*x) 
+ 6*(m^4 + 20*m^2 + 64)*cos(2*x) + 6*((m^4 + 20*m^2 + 64)*sin(4*x) + (m^4 
+ 20*m^2 + 64)*sin(2*x))*sin(6*x) + 64), x) - 8*((m^3 + 16*m)*cos(2*x)*...
 

Giac [F]

\[ \int e^{m x} \tan ^2(x) \, dx=\int { e^{\left (m x\right )} \tan \left (x\right )^{2} \,d x } \] Input:

integrate(exp(m*x)*tan(x)^2,x, algorithm="giac")
 

Output:

integrate(e^(m*x)*tan(x)^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int e^{m x} \tan ^2(x) \, dx=\int {\mathrm {e}}^{m\,x}\,{\mathrm {tan}\left (x\right )}^2 \,d x \] Input:

int(exp(m*x)*tan(x)^2,x)
 

Output:

int(exp(m*x)*tan(x)^2, x)
 

Reduce [F]

\[ \int e^{m x} \tan ^2(x) \, dx=\int e^{m x} \tan \left (x \right )^{2}d x \] Input:

int(exp(m*x)*tan(x)^2,x)
 

Output:

int(e**(m*x)*tan(x)**2,x)