\(\int \frac {(d \tan (e+f x))^n}{a-i a \tan (e+f x)} \, dx\) [319]

Optimal result

Integrand size = 26, antiderivative size = 158 \[ \int \frac {(d \tan (e+f x))^n}{a-i a \tan (e+f x)} \, dx=\frac {(1-n) \operatorname {Hypergeometric2F1}\left (1,\frac {1+n}{2},\frac {3+n}{2},-\tan ^2(e+f x)\right ) (d \tan (e+f x))^{1+n}}{2 a d f (1+n)}-\frac {i n \operatorname {Hypergeometric2F1}\left (1,\frac {2+n}{2},\frac {4+n}{2},-\tan ^2(e+f x)\right ) (d \tan (e+f x))^{2+n}}{2 a d^2 f (2+n)}+\frac {(d \tan (e+f x))^{1+n}}{2 d f (a-i a \tan (e+f x))} \] Output:

1/2*(1-n)*hypergeom([1, 1/2+1/2*n],[3/2+1/2*n],-tan(f*x+e)^2)*(d*tan(f*x+e 
))^(1+n)/a/d/f/(1+n)-1/2*I*n*hypergeom([1, 1+1/2*n],[2+1/2*n],-tan(f*x+e)^ 
2)*(d*tan(f*x+e))^(2+n)/a/d^2/f/(2+n)+1/2*(d*tan(f*x+e))^(1+n)/d/f/(a-I*a* 
tan(f*x+e))
 

Mathematica [A] (verified)

Time = 0.53 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.78 \[ \int \frac {(d \tan (e+f x))^n}{a-i a \tan (e+f x)} \, dx=\frac {\tan (e+f x) (d \tan (e+f x))^n \left (-\frac {(-1+n) \operatorname {Hypergeometric2F1}\left (1,\frac {1+n}{2},\frac {3+n}{2},-\tan ^2(e+f x)\right )}{a (1+n)}-\frac {i n \operatorname {Hypergeometric2F1}\left (1,\frac {2+n}{2},\frac {4+n}{2},-\tan ^2(e+f x)\right ) \tan (e+f x)}{a (2+n)}+\frac {1}{a-i a \tan (e+f x)}\right )}{2 f} \] Input:

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

Output:

(Tan[e + f*x]*(d*Tan[e + f*x])^n*(-(((-1 + n)*Hypergeometric2F1[1, (1 + n) 
/2, (3 + n)/2, -Tan[e + f*x]^2])/(a*(1 + n))) - (I*n*Hypergeometric2F1[1, 
(2 + n)/2, (4 + n)/2, -Tan[e + f*x]^2]*Tan[e + f*x])/(a*(2 + n)) + (a - I* 
a*Tan[e + f*x])^(-1)))/(2*f)
 

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.99, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {3042, 4035, 25, 3042, 4021, 3042, 3957, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4035

\(\displaystyle \frac {(d \tan (e+f x))^{n+1}}{2 d f (a-i a \tan (e+f x))}-\frac {\int -(d \tan (e+f x))^n (a d (1-n)-i a d n \tan (e+f x))dx}{2 a^2 d}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int (d \tan (e+f x))^n (a d (1-n)-i a d n \tan (e+f x))dx}{2 a^2 d}+\frac {(d \tan (e+f x))^{n+1}}{2 d f (a-i a \tan (e+f x))}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int (d \tan (e+f x))^n (a d (1-n)-i a d n \tan (e+f x))dx}{2 a^2 d}+\frac {(d \tan (e+f x))^{n+1}}{2 d f (a-i a \tan (e+f x))}\)

\(\Big \downarrow \) 4021

\(\displaystyle \frac {a d (1-n) \int (d \tan (e+f x))^ndx-i a n \int (d \tan (e+f x))^{n+1}dx}{2 a^2 d}+\frac {(d \tan (e+f x))^{n+1}}{2 d f (a-i a \tan (e+f x))}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {a d (1-n) \int (d \tan (e+f x))^ndx-i a n \int (d \tan (e+f x))^{n+1}dx}{2 a^2 d}+\frac {(d \tan (e+f x))^{n+1}}{2 d f (a-i a \tan (e+f x))}\)

\(\Big \downarrow \) 3957

\(\displaystyle \frac {\frac {a d^2 (1-n) \int \frac {(d \tan (e+f x))^n}{\tan ^2(e+f x) d^2+d^2}d(d \tan (e+f x))}{f}-\frac {i a d n \int \frac {(d \tan (e+f x))^{n+1}}{\tan ^2(e+f x) d^2+d^2}d(d \tan (e+f x))}{f}}{2 a^2 d}+\frac {(d \tan (e+f x))^{n+1}}{2 d f (a-i a \tan (e+f x))}\)

\(\Big \downarrow \) 278

\(\displaystyle \frac {\frac {a (1-n) (d \tan (e+f x))^{n+1} \operatorname {Hypergeometric2F1}\left (1,\frac {n+1}{2},\frac {n+3}{2},-\tan ^2(e+f x)\right )}{f (n+1)}-\frac {i a n (d \tan (e+f x))^{n+2} \operatorname {Hypergeometric2F1}\left (1,\frac {n+2}{2},\frac {n+4}{2},-\tan ^2(e+f x)\right )}{d f (n+2)}}{2 a^2 d}+\frac {(d \tan (e+f x))^{n+1}}{2 d f (a-i a \tan (e+f x))}\)

Input:

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

Output:

(d*Tan[e + f*x])^(1 + n)/(2*d*f*(a - I*a*Tan[e + f*x])) + ((a*(1 - n)*Hype 
rgeometric2F1[1, (1 + n)/2, (3 + n)/2, -Tan[e + f*x]^2]*(d*Tan[e + f*x])^( 
1 + n))/(f*(1 + n)) - (I*a*n*Hypergeometric2F1[1, (2 + n)/2, (4 + n)/2, -T 
an[e + f*x]^2]*(d*Tan[e + f*x])^(2 + n))/(d*f*(2 + n)))/(2*a^2*d)
 

Defintions of rubi rules used

rule 25

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

rule 278

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])
 

rule 3042

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

rule 3957

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b/d   Subst[Int 
[x^n/(b^2 + x^2), x], x, b*Tan[c + d*x]], x] /; FreeQ[{b, c, d, n}, x] && 
!IntegerQ[n]
 

rule 4021

Int[((b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x 
_)]), x_Symbol] :> Simp[c   Int[(b*Tan[e + f*x])^m, x], x] + Simp[d/b   Int 
[(b*Tan[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x] && NeQ[c^ 
2 + d^2, 0] &&  !IntegerQ[2*m]
 

rule 4035

Int[((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)/((a_) + (b_.)*tan[(e_.) + 
(f_.)*(x_)]), x_Symbol] :> Simp[(-a)*((c + d*Tan[e + f*x])^(n + 1)/(2*f*(b* 
c - a*d)*(a + b*Tan[e + f*x]))), x] + Simp[1/(2*a*(b*c - a*d))   Int[(c + d 
*Tan[e + f*x])^n*Simp[b*c + a*d*(n - 1) - b*d*n*Tan[e + f*x], x], x], x] /; 
 FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] 
&& NeQ[c^2 + d^2, 0] &&  !GtQ[n, 0]
 

Maple [F]

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

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]

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

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

Output:

integral(1/2*((-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) + 1)/a, x)
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {(d \tan (e+f x))^n}{a-i a \tan (e+f x)} \, dx=\text {Exception raised: RuntimeError} \] Input:

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

Output:

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is un 
defined.
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

\[ \int \frac {(d \tan (e+f x))^n}{a-i a \tan (e+f x)} \, dx=\frac {d^{n} \left (\tan \left (f x +e \right )^{n} i +\left (\int \frac {\tan \left (f x +e \right )^{n}}{\tan \left (f x +e \right )^{2}+\tan \left (f x +e \right ) i}d x \right ) f n -\left (\int \frac {\tan \left (f x +e \right )^{n} \tan \left (f x +e \right )^{2}}{\tan \left (f x +e \right )+i}d x \right ) f i n +\left (\int \frac {\tan \left (f x +e \right )^{n} \tan \left (f x +e \right )}{\tan \left (f x +e \right )+i}d x \right ) f n \right )}{a f n} \] Input:

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

Output:

(d**n*(tan(e + f*x)**n*i + int(tan(e + f*x)**n/(tan(e + f*x)**2 + tan(e + 
f*x)*i),x)*f*n - int((tan(e + f*x)**n*tan(e + f*x)**2)/(tan(e + f*x) + i), 
x)*f*i*n + int((tan(e + f*x)**n*tan(e + f*x))/(tan(e + f*x) + i),x)*f*n))/ 
(a*f*n)