Integrand size = 18, antiderivative size = 130 \[ \int e^{c (a+b x)} \tan ^2(d+e x) \, dx=-\frac {e^{c (a+b x)}}{b c}+\frac {4 e^{c (a+b x)} \operatorname {Hypergeometric2F1}\left (1,-\frac {i b c}{2 e},1-\frac {i b c}{2 e},-e^{2 i (d+e x)}\right )}{b c}-\frac {4 e^{c (a+b x)} \operatorname {Hypergeometric2F1}\left (2,-\frac {i b c}{2 e},1-\frac {i b c}{2 e},-e^{2 i (d+e x)}\right )}{b c} \] Output:
-exp(c*(b*x+a))/b/c+4*exp(c*(b*x+a))*hypergeom([1, -1/2*I*b*c/e],[1-1/2*I* b*c/e],-exp(2*I*(e*x+d)))/b/c-4*exp(c*(b*x+a))*hypergeom([2, -1/2*I*b*c/e] ,[1-1/2*I*b*c/e],-exp(2*I*(e*x+d)))/b/c
Time = 1.09 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.35 \[ \int e^{c (a+b x)} \tan ^2(d+e x) \, dx=e^{c (a+b x)} \left (-\frac {1}{b c}+\frac {2 e^{2 i d} \left (i b c e^{2 i e x} \operatorname {Hypergeometric2F1}\left (1,1-\frac {i b c}{2 e},2-\frac {i b c}{2 e},-e^{2 i (d+e x)}\right )+(-i b c+2 e) \operatorname {Hypergeometric2F1}\left (1,-\frac {i b c}{2 e},1-\frac {i b c}{2 e},-e^{2 i (d+e x)}\right )\right )}{(b c+2 i e) e \left (1+e^{2 i d}\right )}+\frac {\sec (d) \sec (d+e x) \sin (e x)}{e}\right ) \] Input:
Integrate[E^(c*(a + b*x))*Tan[d + e*x]^2,x]
Output:
E^(c*(a + b*x))*(-(1/(b*c)) + (2*E^((2*I)*d)*(I*b*c*E^((2*I)*e*x)*Hypergeo metric2F1[1, 1 - ((I/2)*b*c)/e, 2 - ((I/2)*b*c)/e, -E^((2*I)*(d + e*x))] + ((-I)*b*c + 2*e)*Hypergeometric2F1[1, ((-1/2*I)*b*c)/e, 1 - ((I/2)*b*c)/e , -E^((2*I)*(d + e*x))]))/((b*c + (2*I)*e)*e*(1 + E^((2*I)*d))) + (Sec[d]* Sec[d + e*x]*Sin[e*x])/e)
Time = 0.33 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, 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.
\(\displaystyle \int e^{c (a+b x)} \tan ^2(d+e x) \, dx\) |
\(\Big \downarrow \) 4942 |
\(\displaystyle -\int \left (e^{c (a+b x)}-\frac {4 e^{c (a+b x)}}{1+e^{2 i (d+e x)}}+\frac {4 e^{c (a+b x)}}{\left (1+e^{2 i (d+e x)}\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {4 e^{c (a+b x)} \operatorname {Hypergeometric2F1}\left (1,-\frac {i b c}{2 e},1-\frac {i b c}{2 e},-e^{2 i (d+e x)}\right )}{b c}-\frac {4 e^{c (a+b x)} \operatorname {Hypergeometric2F1}\left (2,-\frac {i b c}{2 e},1-\frac {i b c}{2 e},-e^{2 i (d+e x)}\right )}{b c}-\frac {e^{c (a+b x)}}{b c}\) |
Input:
Int[E^(c*(a + b*x))*Tan[d + e*x]^2,x]
Output:
-(E^(c*(a + b*x))/(b*c)) + (4*E^(c*(a + b*x))*Hypergeometric2F1[1, ((-1/2* I)*b*c)/e, 1 - ((I/2)*b*c)/e, -E^((2*I)*(d + e*x))])/(b*c) - (4*E^(c*(a + b*x))*Hypergeometric2F1[2, ((-1/2*I)*b*c)/e, 1 - ((I/2)*b*c)/e, -E^((2*I)* (d + e*x))])/(b*c)
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]
\[\int {\mathrm e}^{c \left (b x +a \right )} \tan \left (e x +d \right )^{2}d x\]
Input:
int(exp(c*(b*x+a))*tan(e*x+d)^2,x)
Output:
int(exp(c*(b*x+a))*tan(e*x+d)^2,x)
\[ \int e^{c (a+b x)} \tan ^2(d+e x) \, dx=\int { e^{\left ({\left (b x + a\right )} c\right )} \tan \left (e x + d\right )^{2} \,d x } \] Input:
integrate(exp(c*(b*x+a))*tan(e*x+d)^2,x, algorithm="fricas")
Output:
integral(e^(b*c*x + a*c)*tan(e*x + d)^2, x)
\[ \int e^{c (a+b x)} \tan ^2(d+e x) \, dx=e^{a c} \int e^{b c x} \tan ^{2}{\left (d + e x \right )}\, dx \] Input:
integrate(exp(c*(b*x+a))*tan(e*x+d)**2,x)
Output:
exp(a*c)*Integral(exp(b*c*x)*tan(d + e*x)**2, x)
\[ \int e^{c (a+b x)} \tan ^2(d+e x) \, dx=\int { e^{\left ({\left (b x + a\right )} c\right )} \tan \left (e x + d\right )^{2} \,d x } \] Input:
integrate(exp(c*(b*x+a))*tan(e*x+d)^2,x, algorithm="maxima")
Output:
-(e*cos(2*e*x + 2*d)^2*e^(b*c*x + a*c) - 2*b*c*e^(b*c*x + a*c)*sin(2*e*x + 2*d) + e*e^(b*c*x + a*c)*sin(2*e*x + 2*d)^2 + 2*e*cos(2*e*x + 2*d)*e^(b*c *x + a*c) + e*e^(b*c*x + a*c) + 2*(b^2*c^2*e^2*cos(2*e*x + 2*d)^2 + b^2*c^ 2*e^2*sin(2*e*x + 2*d)^2 + 2*b^2*c^2*e^2*cos(2*e*x + 2*d) + b^2*c^2*e^2)*i ntegrate(e^(b*c*x + a*c)*sin(2*e*x + 2*d)/(e^2*cos(2*e*x + 2*d)^2 + e^2*si n(2*e*x + 2*d)^2 + 2*e^2*cos(2*e*x + 2*d) + e^2), x))/(b*c*e*cos(2*e*x + 2 *d)^2 + b*c*e*sin(2*e*x + 2*d)^2 + 2*b*c*e*cos(2*e*x + 2*d) + b*c*e)
\[ \int e^{c (a+b x)} \tan ^2(d+e x) \, dx=\int { e^{\left ({\left (b x + a\right )} c\right )} \tan \left (e x + d\right )^{2} \,d x } \] Input:
integrate(exp(c*(b*x+a))*tan(e*x+d)^2,x, algorithm="giac")
Output:
integrate(e^((b*x + a)*c)*tan(e*x + d)^2, x)
Timed out. \[ \int e^{c (a+b x)} \tan ^2(d+e x) \, dx=\int {\mathrm {e}}^{c\,\left (a+b\,x\right )}\,{\mathrm {tan}\left (d+e\,x\right )}^2 \,d x \] Input:
int(exp(c*(a + b*x))*tan(d + e*x)^2,x)
Output:
int(exp(c*(a + b*x))*tan(d + e*x)^2, x)
\[ \int e^{c (a+b x)} \tan ^2(d+e x) \, dx=e^{a c} \left (\int e^{b c x} \tan \left (e x +d \right )^{2}d x \right ) \] Input:
int(exp(c*(b*x+a))*tan(e*x+d)^2,x)
Output:
e**(a*c)*int(e**(b*c*x)*tan(d + e*x)**2,x)