\(\int (d \tan (e+f x))^n (a+i a \tan (e+f x)) \, dx\) [313]

Optimal result

Integrand size = 24, antiderivative size = 43 \[ \int (d \tan (e+f x))^n (a+i a \tan (e+f x)) \, dx=\frac {a \operatorname {Hypergeometric2F1}(1,1+n,2+n,i \tan (e+f x)) (d \tan (e+f x))^{1+n}}{d f (1+n)} \] Output:

a*hypergeom([1, 1+n],[2+n],I*tan(f*x+e))*(d*tan(f*x+e))^(1+n)/d/f/(1+n)
 

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00 \[ \int (d \tan (e+f x))^n (a+i a \tan (e+f x)) \, dx=\frac {a \operatorname {Hypergeometric2F1}(1,1+n,2+n,i \tan (e+f x)) (d \tan (e+f x))^{1+n}}{d f (1+n)} \] Input:

Integrate[(d*Tan[e + f*x])^n*(a + I*a*Tan[e + f*x]),x]
 

Output:

(a*Hypergeometric2F1[1, 1 + n, 2 + n, I*Tan[e + f*x]]*(d*Tan[e + f*x])^(1 
+ n))/(d*f*(1 + n))
 

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {3042, 4020, 25, 27, 74}

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.

\(\displaystyle \int (a+i a \tan (e+f x)) (d \tan (e+f x))^n \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int (a+i a \tan (e+f x)) (d \tan (e+f x))^ndx\)

\(\Big \downarrow \) 4020

\(\displaystyle \frac {i a^2 \int -\frac {(d \tan (e+f x))^n}{a (a-i a \tan (e+f x))}d(i a \tan (e+f x))}{f}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {i a^2 \int \frac {(d \tan (e+f x))^n}{a (a-i a \tan (e+f x))}d(i a \tan (e+f x))}{f}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {i a \int \frac {(d \tan (e+f x))^n}{a-i a \tan (e+f x)}d(i a \tan (e+f x))}{f}\)

\(\Big \downarrow \) 74

\(\displaystyle \frac {a (d \tan (e+f x))^{n+1} \operatorname {Hypergeometric2F1}(1,n+1,n+2,i \tan (e+f x))}{d f (n+1)}\)

Input:

Int[(d*Tan[e + f*x])^n*(a + I*a*Tan[e + f*x]),x]
 

Output:

(a*Hypergeometric2F1[1, 1 + n, 2 + n, I*Tan[e + f*x]]*(d*Tan[e + f*x])^(1 
+ n))/(d*f*(1 + n))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 74

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[c^n*((b*x 
)^(m + 1)/(b*(m + 1)))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*(x/c)], x] 
/; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m] && (IntegerQ[n] || (GtQ[c, 0] 
 &&  !(EqQ[n, -2^(-1)] && EqQ[c^2 - d^2, 0] && GtQ[-d/(b*c), 0])))
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4020

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + 
(f_.)*(x_)]), x_Symbol] :> Simp[c*(d/f)   Subst[Int[(a + (b/d)*x)^m/(d^2 + 
c*x), x], x, d*Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[ 
b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[c^2 + d^2, 0]
 

Maple [F]

\[\int \left (d \tan \left (f x +e \right )\right )^{n} \left (a +i a \tan \left (f x +e \right )\right )d x\]

Input:

int((d*tan(f*x+e))^n*(a+I*a*tan(f*x+e)),x)
 

Output:

int((d*tan(f*x+e))^n*(a+I*a*tan(f*x+e)),x)
 

Fricas [F]

\[ \int (d \tan (e+f x))^n (a+i a \tan (e+f x)) \, dx=\int { {\left (i \, a \tan \left (f x + e\right ) + a\right )} \left (d \tan \left (f x + e\right )\right )^{n} \,d x } \] Input:

integrate((d*tan(f*x+e))^n*(a+I*a*tan(f*x+e)),x, algorithm="fricas")
 

Output:

integral(2*a*((-I*d*e^(2*I*f*x + 2*I*e) + I*d)/(e^(2*I*f*x + 2*I*e) + 1))^ 
n*e^(2*I*f*x + 2*I*e)/(e^(2*I*f*x + 2*I*e) + 1), x)
 

Sympy [F]

\[ \int (d \tan (e+f x))^n (a+i a \tan (e+f x)) \, dx=i a \left (\int \left (- i \left (d \tan {\left (e + f x \right )}\right )^{n}\right )\, dx + \int \left (d \tan {\left (e + f x \right )}\right )^{n} \tan {\left (e + f x \right )}\, dx\right ) \] Input:

integrate((d*tan(f*x+e))**n*(a+I*a*tan(f*x+e)),x)
 

Output:

I*a*(Integral(-I*(d*tan(e + f*x))**n, x) + Integral((d*tan(e + f*x))**n*ta 
n(e + f*x), x))
 

Maxima [F]

\[ \int (d \tan (e+f x))^n (a+i a \tan (e+f x)) \, dx=\int { {\left (i \, a \tan \left (f x + e\right ) + a\right )} \left (d \tan \left (f x + e\right )\right )^{n} \,d x } \] Input:

integrate((d*tan(f*x+e))^n*(a+I*a*tan(f*x+e)),x, algorithm="maxima")
 

Output:

integrate((I*a*tan(f*x + e) + a)*(d*tan(f*x + e))^n, x)
 

Giac [F]

\[ \int (d \tan (e+f x))^n (a+i a \tan (e+f x)) \, dx=\int { {\left (i \, a \tan \left (f x + e\right ) + a\right )} \left (d \tan \left (f x + e\right )\right )^{n} \,d x } \] Input:

integrate((d*tan(f*x+e))^n*(a+I*a*tan(f*x+e)),x, algorithm="giac")
 

Output:

integrate((I*a*tan(f*x + e) + a)*(d*tan(f*x + e))^n, x)
 

Mupad [F(-1)]

Timed out. \[ \int (d \tan (e+f x))^n (a+i a \tan (e+f x)) \, dx=\int {\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^n\,\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right ) \,d x \] Input:

int((d*tan(e + f*x))^n*(a + a*tan(e + f*x)*1i),x)
 

Output:

int((d*tan(e + f*x))^n*(a + a*tan(e + f*x)*1i), x)
 

Reduce [F]

\[ \int (d \tan (e+f x))^n (a+i a \tan (e+f x)) \, dx=\frac {d^{n} a \left (\tan \left (f x +e \right )^{n} i +\left (\int \tan \left (f x +e \right )^{n}d x \right ) f n -\left (\int \frac {\tan \left (f x +e \right )^{n}}{\tan \left (f x +e \right )}d x \right ) f i n \right )}{f n} \] Input:

int((d*tan(f*x+e))^n*(a+I*a*tan(f*x+e)),x)
 

Output:

(d**n*a*(tan(e + f*x)**n*i + int(tan(e + f*x)**n,x)*f*n - int(tan(e + f*x) 
**n/tan(e + f*x),x)*f*i*n))/(f*n)