\(\int \frac {\sqrt {d \tan (e+f x)}}{a+a \tan (e+f x)} \, dx\) [360]

Optimal result

Integrand size = 25, antiderivative size = 89 \[ \int \frac {\sqrt {d \tan (e+f x)}}{a+a \tan (e+f x)} \, dx=-\frac {\sqrt {d} \arctan \left (\frac {\sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{a f}-\frac {\sqrt {d} \arctan \left (\frac {\sqrt {d}-\sqrt {d} \tan (e+f x)}{\sqrt {2} \sqrt {d \tan (e+f x)}}\right )}{\sqrt {2} a f} \] Output:

-d^(1/2)*arctan((d*tan(f*x+e))^(1/2)/d^(1/2))/a/f-1/2*d^(1/2)*arctan(1/2*( 
d^(1/2)-d^(1/2)*tan(f*x+e))*2^(1/2)/(d*tan(f*x+e))^(1/2))*2^(1/2)/a/f
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(283\) vs. \(2(89)=178\).

Time = 0.21 (sec) , antiderivative size = 283, normalized size of antiderivative = 3.18 \[ \int \frac {\sqrt {d \tan (e+f x)}}{a+a \tan (e+f x)} \, dx=-\frac {8 d^{3/2} \arctan \left (\frac {\sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )+4 \left (-d^2\right )^{3/4} \arctan \left (\frac {\sqrt {d \tan (e+f x)}}{\sqrt [4]{-d^2}}\right )+2 \sqrt {2} d^{3/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )-2 \sqrt {2} d^{3/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )-4 \left (-d^2\right )^{3/4} \text {arctanh}\left (\frac {\sqrt {d \tan (e+f x)}}{\sqrt [4]{-d^2}}\right )+\sqrt {2} d^{3/2} \log \left (\sqrt {d}+\sqrt {d} \tan (e+f x)-\sqrt {2} \sqrt {d \tan (e+f x)}\right )-\sqrt {2} d^{3/2} \log \left (\sqrt {d}+\sqrt {d} \tan (e+f x)+\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{8 a d f} \] Input:

Integrate[Sqrt[d*Tan[e + f*x]]/(a + a*Tan[e + f*x]),x]
 

Output:

-1/8*(8*d^(3/2)*ArcTan[Sqrt[d*Tan[e + f*x]]/Sqrt[d]] + 4*(-d^2)^(3/4)*ArcT 
an[Sqrt[d*Tan[e + f*x]]/(-d^2)^(1/4)] + 2*Sqrt[2]*d^(3/2)*ArcTan[1 - (Sqrt 
[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]] - 2*Sqrt[2]*d^(3/2)*ArcTan[1 + (Sqrt[2] 
*Sqrt[d*Tan[e + f*x]])/Sqrt[d]] - 4*(-d^2)^(3/4)*ArcTanh[Sqrt[d*Tan[e + f* 
x]]/(-d^2)^(1/4)] + Sqrt[2]*d^(3/2)*Log[Sqrt[d] + Sqrt[d]*Tan[e + f*x] - S 
qrt[2]*Sqrt[d*Tan[e + f*x]]] - Sqrt[2]*d^(3/2)*Log[Sqrt[d] + Sqrt[d]*Tan[e 
 + f*x] + Sqrt[2]*Sqrt[d*Tan[e + f*x]]])/(a*d*f)
 

Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.03, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {3042, 4055, 3042, 4015, 218, 4117, 27, 73, 216}

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.

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

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\sqrt {d \tan (e+f x)}}{a \tan (e+f x)+a}dx\)

\(\Big \downarrow \) 4055

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4015

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

\(\Big \downarrow \) 218

\(\displaystyle -\frac {1}{2} d \int \frac {\tan (e+f x)^2+1}{\sqrt {d \tan (e+f x)} (\tan (e+f x) a+a)}dx-\frac {\sqrt {d} \arctan \left (\frac {a d-a d \tan (e+f x)}{\sqrt {2} a \sqrt {d} \sqrt {d \tan (e+f x)}}\right )}{\sqrt {2} a f}\)

\(\Big \downarrow \) 4117

\(\displaystyle -\frac {d \int \frac {1}{a \sqrt {d \tan (e+f x)} (\tan (e+f x)+1)}d\tan (e+f x)}{2 f}-\frac {\sqrt {d} \arctan \left (\frac {a d-a d \tan (e+f x)}{\sqrt {2} a \sqrt {d} \sqrt {d \tan (e+f x)}}\right )}{\sqrt {2} a f}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {d \int \frac {1}{\sqrt {d \tan (e+f x)} (\tan (e+f x)+1)}d\tan (e+f x)}{2 a f}-\frac {\sqrt {d} \arctan \left (\frac {a d-a d \tan (e+f x)}{\sqrt {2} a \sqrt {d} \sqrt {d \tan (e+f x)}}\right )}{\sqrt {2} a f}\)

\(\Big \downarrow \) 73

\(\displaystyle -\frac {\int \frac {1}{\tan (e+f x)+1}d\sqrt {d \tan (e+f x)}}{a f}-\frac {\sqrt {d} \arctan \left (\frac {a d-a d \tan (e+f x)}{\sqrt {2} a \sqrt {d} \sqrt {d \tan (e+f x)}}\right )}{\sqrt {2} a f}\)

\(\Big \downarrow \) 216

\(\displaystyle -\frac {\sqrt {d} \arctan \left (\frac {\sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{a f}-\frac {\sqrt {d} \arctan \left (\frac {a d-a d \tan (e+f x)}{\sqrt {2} a \sqrt {d} \sqrt {d \tan (e+f x)}}\right )}{\sqrt {2} a f}\)

Input:

Int[Sqrt[d*Tan[e + f*x]]/(a + a*Tan[e + f*x]),x]
 

Output:

-((Sqrt[d]*ArcTan[Sqrt[d*Tan[e + f*x]]/Sqrt[d]])/(a*f)) - (Sqrt[d]*ArcTan[ 
(a*d - a*d*Tan[e + f*x])/(Sqrt[2]*a*Sqrt[d]*Sqrt[d*Tan[e + f*x]])])/(Sqrt[ 
2]*a*f)
 

Defintions of rubi rules used

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 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 216

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A 
rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a 
, 0] || GtQ[b, 0])
 

rule 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

rule 3042

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

rule 4015

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

rule 4055

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

rule 4117

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(303\) vs. \(2(72)=144\).

Time = 1.40 (sec) , antiderivative size = 304, normalized size of antiderivative = 3.42

method	result	size
derivativedivides	\(\frac {2 d^{2} \left (-\frac {\arctan \left (\frac {\sqrt {d \tan \left (f x +e \right )}}{\sqrt {d}}\right )}{2 d^{\frac {3}{2}}}+\frac {\frac {\left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 d}+\frac {\sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (d^{2}\right )^{\frac {1}{4}}}}{2 d}\right )}{f a}\)	\(304\)
default	\(\frac {2 d^{2} \left (-\frac {\arctan \left (\frac {\sqrt {d \tan \left (f x +e \right )}}{\sqrt {d}}\right )}{2 d^{\frac {3}{2}}}+\frac {\frac {\left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 d}+\frac {\sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (d^{2}\right )^{\frac {1}{4}}}}{2 d}\right )}{f a}\)	\(304\)

Input:

int((d*tan(f*x+e))^(1/2)/(a+a*tan(f*x+e)),x,method=_RETURNVERBOSE)
 

Output:

2/f/a*d^2*(-1/2/d^(3/2)*arctan((d*tan(f*x+e))^(1/2)/d^(1/2))+1/2/d*(1/8/d* 
(d^2)^(1/4)*2^(1/2)*(ln((d*tan(f*x+e)+(d^2)^(1/4)*(d*tan(f*x+e))^(1/2)*2^( 
1/2)+(d^2)^(1/2))/(d*tan(f*x+e)-(d^2)^(1/4)*(d*tan(f*x+e))^(1/2)*2^(1/2)+( 
d^2)^(1/2)))+2*arctan(2^(1/2)/(d^2)^(1/4)*(d*tan(f*x+e))^(1/2)+1)-2*arctan 
(-2^(1/2)/(d^2)^(1/4)*(d*tan(f*x+e))^(1/2)+1))+1/8/(d^2)^(1/4)*2^(1/2)*(ln 
((d*tan(f*x+e)-(d^2)^(1/4)*(d*tan(f*x+e))^(1/2)*2^(1/2)+(d^2)^(1/2))/(d*ta 
n(f*x+e)+(d^2)^(1/4)*(d*tan(f*x+e))^(1/2)*2^(1/2)+(d^2)^(1/2)))+2*arctan(2 
^(1/2)/(d^2)^(1/4)*(d*tan(f*x+e))^(1/2)+1)-2*arctan(-2^(1/2)/(d^2)^(1/4)*( 
d*tan(f*x+e))^(1/2)+1))))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 205, normalized size of antiderivative = 2.30 \[ \int \frac {\sqrt {d \tan (e+f x)}}{a+a \tan (e+f x)} \, dx=\left [\frac {\sqrt {\frac {1}{2}} \sqrt {-d} \log \left (\frac {d \tan \left (f x + e\right )^{2} + 4 \, \sqrt {\frac {1}{2}} \sqrt {d \tan \left (f x + e\right )} \sqrt {-d} {\left (\tan \left (f x + e\right ) - 1\right )} - 4 \, d \tan \left (f x + e\right ) + d}{\tan \left (f x + e\right )^{2} + 1}\right ) + \sqrt {-d} \log \left (\frac {d \tan \left (f x + e\right ) - 2 \, \sqrt {d \tan \left (f x + e\right )} \sqrt {-d} - d}{\tan \left (f x + e\right ) + 1}\right )}{2 \, a f}, \frac {\sqrt {\frac {1}{2}} \sqrt {d} \arctan \left (\frac {\sqrt {\frac {1}{2}} \sqrt {d \tan \left (f x + e\right )} {\left (\tan \left (f x + e\right ) - 1\right )}}{\sqrt {d} \tan \left (f x + e\right )}\right ) + \sqrt {d} \arctan \left (\frac {\sqrt {d \tan \left (f x + e\right )}}{\sqrt {d} \tan \left (f x + e\right )}\right )}{a f}\right ] \] Input:

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

Output:

[1/2*(sqrt(1/2)*sqrt(-d)*log((d*tan(f*x + e)^2 + 4*sqrt(1/2)*sqrt(d*tan(f* 
x + e))*sqrt(-d)*(tan(f*x + e) - 1) - 4*d*tan(f*x + e) + d)/(tan(f*x + e)^ 
2 + 1)) + sqrt(-d)*log((d*tan(f*x + e) - 2*sqrt(d*tan(f*x + e))*sqrt(-d) - 
 d)/(tan(f*x + e) + 1)))/(a*f), (sqrt(1/2)*sqrt(d)*arctan(sqrt(1/2)*sqrt(d 
*tan(f*x + e))*(tan(f*x + e) - 1)/(sqrt(d)*tan(f*x + e))) + sqrt(d)*arctan 
(sqrt(d*tan(f*x + e))/(sqrt(d)*tan(f*x + e))))/(a*f)]
 

Sympy [F]

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

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

Output:

Integral(sqrt(d*tan(e + f*x))/(tan(e + f*x) + 1), x)/a
 

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.26 \[ \int \frac {\sqrt {d \tan (e+f x)}}{a+a \tan (e+f x)} \, dx=\frac {\frac {d^{2} {\left (\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} + 2 \, \sqrt {d \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}} + \frac {\sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} - 2 \, \sqrt {d \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}}\right )}}{a} - \frac {2 \, d^{\frac {3}{2}} \arctan \left (\frac {\sqrt {d \tan \left (f x + e\right )}}{\sqrt {d}}\right )}{a}}{2 \, d f} \] Input:

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

Output:

1/2*(d^2*(sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*sqrt(d) + 2*sqrt(d*tan(f*x + 
 e)))/sqrt(d))/sqrt(d) + sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2)*sqrt(d) - 2* 
sqrt(d*tan(f*x + e)))/sqrt(d))/sqrt(d))/a - 2*d^(3/2)*arctan(sqrt(d*tan(f* 
x + e))/sqrt(d))/a)/(d*f)
 

Giac [F(-2)]

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

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [B] (verification not implemented)

Time = 1.21 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.16 \[ \int \frac {\sqrt {d \tan (e+f x)}}{a+a \tan (e+f x)} \, dx=\frac {\sqrt {2}\,\sqrt {d}\,\left (2\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}{2\,\sqrt {d}}\right )+2\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}{2\,\sqrt {d}}+\frac {\sqrt {2}\,{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{3/2}}{2\,d^{3/2}}\right )\right )}{4\,a\,f}-\frac {\sqrt {d}\,\mathrm {atan}\left (\frac {\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}{\sqrt {d}}\right )}{a\,f} \] Input:

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

Output:

(2^(1/2)*d^(1/2)*(2*atan((2^(1/2)*(d*tan(e + f*x))^(1/2))/(2*d^(1/2))) + 2 
*atan((2^(1/2)*(d*tan(e + f*x))^(1/2))/(2*d^(1/2)) + (2^(1/2)*(d*tan(e + f 
*x))^(3/2))/(2*d^(3/2)))))/(4*a*f) - (d^(1/2)*atan((d*tan(e + f*x))^(1/2)/ 
d^(1/2)))/(a*f)
 

Reduce [F]

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

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

Output:

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