\(\int (d \tan (e+f x))^n \sqrt {a+i a \tan (e+f x)} \, dx\) [321]

Optimal result

Integrand size = 28, antiderivative size = 87 \[ \int (d \tan (e+f x))^n \sqrt {a+i a \tan (e+f x)} \, dx=-\frac {i \operatorname {AppellF1}\left (\frac {1}{2},-n,1,\frac {3}{2},1+i \tan (e+f x),\frac {1}{2} (1+i \tan (e+f x))\right ) (-i \tan (e+f x))^{-n} (d \tan (e+f x))^n \sqrt {a+i a \tan (e+f x)}}{f} \] Output:

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

Mathematica [F]

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

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

Output:

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

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 89, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4047

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 152

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

\(\Big \downarrow \) 150

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

Input:

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

Output:

(a*AppellF1[1 + n, 1/2, 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))/(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]

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

Input:

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

Output:

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

Fricas [F]

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

integrate((d*tan(f*x+e))^n*(a+I*a*tan(f*x+e))^(1/2),x, algorithm="fricas")
 

Output:

integral(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^(I*f*x + I*e), x)
 

Sympy [F]

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

integrate((d*tan(f*x+e))**n*(a+I*a*tan(f*x+e))**(1/2),x)
 

Output:

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

Maxima [F]

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

integrate((d*tan(f*x+e))^n*(a+I*a*tan(f*x+e))^(1/2),x, algorithm="maxima")
 

Output:

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

Giac [F(-2)]

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

integrate((d*tan(f*x+e))^n*(a+I*a*tan(f*x+e))^(1/2),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. \[ \int (d \tan (e+f x))^n \sqrt {a+i a \tan (e+f x)} \, dx=\int {\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^n\,\sqrt {a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int (d \tan (e+f x))^n \sqrt {a+i a \tan (e+f x)} \, dx=\frac {d^{n} \sqrt {a}\, i \left (-2 \tan \left (f x +e \right )^{n} \sqrt {\tan \left (f x +e \right ) i +1}+2 \left (\int \frac {\tan \left (f x +e \right )^{n} \sqrt {\tan \left (f x +e \right ) i +1}}{\tan \left (f x +e \right )}d x \right ) f n +2 \left (\int \tan \left (f x +e \right )^{n} \sqrt {\tan \left (f x +e \right ) i +1}\, \tan \left (f x +e \right )d x \right ) f n +\left (\int \tan \left (f x +e \right )^{n} \sqrt {\tan \left (f x +e \right ) i +1}\, \tan \left (f x +e \right )d x \right ) f \right )}{f} \] Input:

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

Output:

(d**n*sqrt(a)*i*( - 2*tan(e + f*x)**n*sqrt(tan(e + f*x)*i + 1) + 2*int((ta 
n(e + f*x)**n*sqrt(tan(e + f*x)*i + 1))/tan(e + f*x),x)*f*n + 2*int(tan(e 
+ f*x)**n*sqrt(tan(e + f*x)*i + 1)*tan(e + f*x),x)*f*n + int(tan(e + f*x)* 
*n*sqrt(tan(e + f*x)*i + 1)*tan(e + f*x),x)*f))/f