\(\int \frac {(d \tan (e+f x))^n}{(a+i a \tan (e+f x))^{3/2}} \, dx\) [323]

Optimal result

Integrand size = 28, antiderivative size = 89 \[ \int \frac {(d \tan (e+f x))^n}{(a+i a \tan (e+f x))^{3/2}} \, dx=\frac {i \operatorname {AppellF1}\left (-\frac {3}{2},-n,1,-\frac {1}{2},1+i \tan (e+f x),\frac {1}{2} (1+i \tan (e+f x))\right ) (-i \tan (e+f x))^{-n} (d \tan (e+f x))^n}{3 f (a+i a \tan (e+f x))^{3/2}} \] Output:

1/3*I*AppellF1(-3/2,-n,1,-1/2,1+I*tan(f*x+e),1/2+1/2*I*tan(f*x+e))*(d*tan( 
f*x+e))^n/f/((-I*tan(f*x+e))^n)/(a+I*a*tan(f*x+e))^(3/2)
 

Mathematica [F]

\[ \int \frac {(d \tan (e+f x))^n}{(a+i a \tan (e+f x))^{3/2}} \, dx=\int \frac {(d \tan (e+f x))^n}{(a+i a \tan (e+f x))^{3/2}} \, dx \] Input:

Integrate[(d*Tan[e + f*x])^n/(a + I*a*Tan[e + f*x])^(3/2),x]
 

Output:

Integrate[(d*Tan[e + f*x])^n/(a + I*a*Tan[e + f*x])^(3/2), x]
 

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.02, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3042, 4047, 25, 27, 152, 150}

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.

\(\displaystyle \int \frac {(d \tan (e+f x))^n}{(a+i a \tan (e+f x))^{3/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {(d \tan (e+f x))^n}{(a+i a \tan (e+f x))^{3/2}}dx\)

\(\Big \downarrow \) 4047

\(\displaystyle \frac {i a^2 \int -\frac {(d \tan (e+f x))^n}{a (a-i a \tan (e+f x)) (i \tan (e+f x) a+a)^{5/2}}d(i a \tan (e+f x))}{f}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {i a^2 \int \frac {(d \tan (e+f x))^n}{a (a-i a \tan (e+f x)) (i \tan (e+f x) a+a)^{5/2}}d(i a \tan (e+f x))}{f}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {i a \int \frac {(d \tan (e+f x))^n}{(a-i a \tan (e+f x)) (i \tan (e+f x) a+a)^{5/2}}d(i a \tan (e+f x))}{f}\)

\(\Big \downarrow \) 152

\(\displaystyle -\frac {i \sqrt {1+i \tan (e+f x)} \int \frac {(d \tan (e+f x))^n}{(i \tan (e+f x)+1)^{5/2} (a-i a \tan (e+f x))}d(i a \tan (e+f x))}{a f \sqrt {a+i a \tan (e+f x)}}\)

\(\Big \downarrow \) 150

\(\displaystyle \frac {\sqrt {1+i \tan (e+f x)} \operatorname {AppellF1}\left (n+1,\frac {5}{2},1,n+2,-i \tan (e+f x),i \tan (e+f x)\right ) (d \tan (e+f x))^{n+1}}{a d f (n+1) \sqrt {a+i a \tan (e+f x)}}\)

Input:

Int[(d*Tan[e + f*x])^n/(a + I*a*Tan[e + f*x])^(3/2),x]
 

Output:

(AppellF1[1 + n, 5/2, 1, 2 + n, (-I)*Tan[e + f*x], I*Tan[e + f*x]]*Sqrt[1 
+ I*Tan[e + f*x]]*(d*Tan[e + f*x])^(1 + n))/(a*d*f*(1 + n)*Sqrt[a + I*a*Ta 
n[e + f*x]])
 

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 150

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_ 
] :> Simp[c^n*e^p*((b*x)^(m + 1)/(b*(m + 1)))*AppellF1[m + 1, -n, -p, m + 2 
, (-d)*(x/c), (-f)*(x/e)], x] /; FreeQ[{b, c, d, e, f, m, n, p}, x] &&  !In 
tegerQ[m] &&  !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])
 

rule 152

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_ 
] :> Simp[c^IntPart[n]*((c + d*x)^FracPart[n]/(1 + d*(x/c))^FracPart[n]) 
Int[(b*x)^m*(1 + d*(x/c))^n*(e + f*x)^p, x], x] /; FreeQ[{b, c, d, e, f, m, 
 n, p}, x] &&  !IntegerQ[m] &&  !IntegerQ[n] &&  !GtQ[c, 0]
 

rule 3042

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

rule 4047

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

Maple [F]

\[\int \frac {\left (d \tan \left (f x +e \right )\right )^{n}}{\left (a +i a \tan \left (f x +e \right )\right )^{\frac {3}{2}}}d x\]

Input:

int((d*tan(f*x+e))^n/(a+I*a*tan(f*x+e))^(3/2),x)
 

Output:

int((d*tan(f*x+e))^n/(a+I*a*tan(f*x+e))^(3/2),x)
 

Fricas [F]

\[ \int \frac {(d \tan (e+f x))^n}{(a+i a \tan (e+f x))^{3/2}} \, dx=\int { \frac {\left (d \tan \left (f x + e\right )\right )^{n}}{{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate((d*tan(f*x+e))^n/(a+I*a*tan(f*x+e))^(3/2),x, algorithm="fricas")
 

Output:

integral(1/4*sqrt(2)*((-I*d*e^(2*I*f*x + 2*I*e) + I*d)/(e^(2*I*f*x + 2*I*e 
) + 1))^n*sqrt(a/(e^(2*I*f*x + 2*I*e) + 1))*(e^(4*I*f*x + 4*I*e) + 2*e^(2* 
I*f*x + 2*I*e) + 1)*e^(-3*I*f*x - 3*I*e)/a^2, x)
 

Sympy [F]

\[ \int \frac {(d \tan (e+f x))^n}{(a+i a \tan (e+f x))^{3/2}} \, dx=\int \frac {\left (d \tan {\left (e + f x \right )}\right )^{n}}{\left (i a \left (\tan {\left (e + f x \right )} - i\right )\right )^{\frac {3}{2}}}\, dx \] Input:

integrate((d*tan(f*x+e))**n/(a+I*a*tan(f*x+e))**(3/2),x)
 

Output:

Integral((d*tan(e + f*x))**n/(I*a*(tan(e + f*x) - I))**(3/2), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(d \tan (e+f x))^n}{(a+i a \tan (e+f x))^{3/2}} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate((d*tan(f*x+e))^n/(a+I*a*tan(f*x+e))^(3/2),x, algorithm="maxima")
 

Output:

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

Giac [F(-2)]

Exception generated. \[ \int \frac {(d \tan (e+f x))^n}{(a+i a \tan (e+f x))^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((d*tan(f*x+e))^n/(a+I*a*tan(f*x+e))^(3/2),x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument TypeError: Bad 
Argument TypeDone
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(d \tan (e+f x))^n}{(a+i a \tan (e+f x))^{3/2}} \, dx=\int \frac {{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^n}{{\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{3/2}} \,d x \] Input:

int((d*tan(e + f*x))^n/(a + a*tan(e + f*x)*1i)^(3/2),x)
 

Output:

int((d*tan(e + f*x))^n/(a + a*tan(e + f*x)*1i)^(3/2), x)
 

Reduce [F]

\[ \int \frac {(d \tan (e+f x))^n}{(a+i a \tan (e+f x))^{3/2}} \, dx=\int \frac {\left (d \tan \left (f x +e \right )\right )^{n}}{\left (a +i a \tan \left (f x +e \right )\right )^{\frac {3}{2}}}d x \] Input:

int((d*tan(f*x+e))^n/(a+I*a*tan(f*x+e))^(3/2),x)
                                                                                    
                                                                                    
 

Output:

int((d*tan(f*x+e))^n/(a+I*a*tan(f*x+e))^(3/2),x)