\(\int \frac {\sqrt {c+d \tan (e+f x)}}{(a+b \tan (e+f x))^2} \, dx\) [1233]

Optimal result

Integrand size = 27, antiderivative size = 231 \[ \int \frac {\sqrt {c+d \tan (e+f x)}}{(a+b \tan (e+f x))^2} \, dx=-\frac {i \sqrt {c-i d} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{(a-i b)^2 f}+\frac {i \sqrt {c+i d} \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{(a+i b)^2 f}-\frac {\sqrt {b} \left (4 a b c-3 a^2 d+b^2 d\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{\left (a^2+b^2\right )^2 \sqrt {b c-a d} f}-\frac {b \sqrt {c+d \tan (e+f x)}}{\left (a^2+b^2\right ) f (a+b \tan (e+f x))} \] Output:

-I*(c-I*d)^(1/2)*arctanh((c+d*tan(f*x+e))^(1/2)/(c-I*d)^(1/2))/(a-I*b)^2/f 
+I*(c+I*d)^(1/2)*arctanh((c+d*tan(f*x+e))^(1/2)/(c+I*d)^(1/2))/(a+I*b)^2/f 
-b^(1/2)*(-3*a^2*d+4*a*b*c+b^2*d)*arctanh(b^(1/2)*(c+d*tan(f*x+e))^(1/2)/( 
-a*d+b*c)^(1/2))/(a^2+b^2)^2/(-a*d+b*c)^(1/2)/f-b*(c+d*tan(f*x+e))^(1/2)/( 
a^2+b^2)/f/(a+b*tan(f*x+e))
 

Mathematica [A] (verified)

Time = 1.40 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.19 \[ \int \frac {\sqrt {c+d \tan (e+f x)}}{(a+b \tan (e+f x))^2} \, dx=-\frac {\frac {i \left ((a+i b)^2 \sqrt {c-i d} (b c-a d) \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )+(a-i b)^2 \sqrt {c+i d} (-b c+a d) \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )\right )}{a^2+b^2}+\frac {\sqrt {b} \sqrt {b c-a d} \left (4 a b c-3 a^2 d+b^2 d\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{a^2+b^2}-b d \sqrt {c+d \tan (e+f x)}+\frac {b^2 (c+d \tan (e+f x))^{3/2}}{a+b \tan (e+f x)}}{\left (a^2+b^2\right ) (b c-a d) f} \] Input:

Integrate[Sqrt[c + d*Tan[e + f*x]]/(a + b*Tan[e + f*x])^2,x]
 

Output:

-(((I*((a + I*b)^2*Sqrt[c - I*d]*(b*c - a*d)*ArcTanh[Sqrt[c + d*Tan[e + f* 
x]]/Sqrt[c - I*d]] + (a - I*b)^2*Sqrt[c + I*d]*(-(b*c) + a*d)*ArcTanh[Sqrt 
[c + d*Tan[e + f*x]]/Sqrt[c + I*d]]))/(a^2 + b^2) + (Sqrt[b]*Sqrt[b*c - a* 
d]*(4*a*b*c - 3*a^2*d + b^2*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/ 
Sqrt[b*c - a*d]])/(a^2 + b^2) - b*d*Sqrt[c + d*Tan[e + f*x]] + (b^2*(c + d 
*Tan[e + f*x])^(3/2))/(a + b*Tan[e + f*x]))/((a^2 + b^2)*(b*c - a*d)*f))
 

Rubi [A] (warning: unable to verify)

Time = 1.69 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.02, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.593, Rules used = {3042, 4051, 27, 3042, 4136, 27, 3042, 4022, 3042, 4020, 25, 73, 221, 4117, 73, 221}

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.

\(\displaystyle \int \frac {\sqrt {c+d \tan (e+f x)}}{(a+b \tan (e+f x))^2} \, dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4051

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4136

\(\displaystyle \frac {\frac {b \left (-3 a^2 d+4 a b c+b^2 d\right ) \int \frac {\tan ^2(e+f x)+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}+\frac {\int \frac {2 \left (c a^2+2 b d a-b^2 c-\left (-d a^2+2 b c a+b^2 d\right ) \tan (e+f x)\right )}{\sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}}{2 \left (a^2+b^2\right )}-\frac {b \sqrt {c+d \tan (e+f x)}}{f \left (a^2+b^2\right ) (a+b \tan (e+f x))}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {b \left (-3 a^2 d+4 a b c+b^2 d\right ) \int \frac {\tan ^2(e+f x)+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}+\frac {2 \int \frac {c a^2+2 b d a-b^2 c-\left (-d a^2+2 b c a+b^2 d\right ) \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}}{2 \left (a^2+b^2\right )}-\frac {b \sqrt {c+d \tan (e+f x)}}{f \left (a^2+b^2\right ) (a+b \tan (e+f x))}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {2 \int \frac {c a^2+2 b d a-b^2 c-\left (-d a^2+2 b c a+b^2 d\right ) \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}+\frac {b \left (-3 a^2 d+4 a b c+b^2 d\right ) \int \frac {\tan (e+f x)^2+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}}{2 \left (a^2+b^2\right )}-\frac {b \sqrt {c+d \tan (e+f x)}}{f \left (a^2+b^2\right ) (a+b \tan (e+f x))}\)

\(\Big \downarrow \) 4022

\(\displaystyle -\frac {b \sqrt {c+d \tan (e+f x)}}{f \left (a^2+b^2\right ) (a+b \tan (e+f x))}+\frac {\frac {b \left (-3 a^2 d+4 a b c+b^2 d\right ) \int \frac {\tan (e+f x)^2+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}+\frac {2 \left (\frac {1}{2} (a-i b)^2 (c+i d) \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}}dx+\frac {1}{2} (a+i b)^2 (c-i d) \int \frac {i \tan (e+f x)+1}{\sqrt {c+d \tan (e+f x)}}dx\right )}{a^2+b^2}}{2 \left (a^2+b^2\right )}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {b \sqrt {c+d \tan (e+f x)}}{f \left (a^2+b^2\right ) (a+b \tan (e+f x))}+\frac {\frac {b \left (-3 a^2 d+4 a b c+b^2 d\right ) \int \frac {\tan (e+f x)^2+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}+\frac {2 \left (\frac {1}{2} (a-i b)^2 (c+i d) \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}}dx+\frac {1}{2} (a+i b)^2 (c-i d) \int \frac {i \tan (e+f x)+1}{\sqrt {c+d \tan (e+f x)}}dx\right )}{a^2+b^2}}{2 \left (a^2+b^2\right )}\)

\(\Big \downarrow \) 4020

\(\displaystyle -\frac {b \sqrt {c+d \tan (e+f x)}}{f \left (a^2+b^2\right ) (a+b \tan (e+f x))}+\frac {\frac {b \left (-3 a^2 d+4 a b c+b^2 d\right ) \int \frac {\tan (e+f x)^2+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}+\frac {2 \left (\frac {i (a+i b)^2 (c-i d) \int -\frac {1}{(1-i \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}d(i \tan (e+f x))}{2 f}-\frac {i (a-i b)^2 (c+i d) \int -\frac {1}{(i \tan (e+f x)+1) \sqrt {c+d \tan (e+f x)}}d(-i \tan (e+f x))}{2 f}\right )}{a^2+b^2}}{2 \left (a^2+b^2\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {b \sqrt {c+d \tan (e+f x)}}{f \left (a^2+b^2\right ) (a+b \tan (e+f x))}+\frac {\frac {b \left (-3 a^2 d+4 a b c+b^2 d\right ) \int \frac {\tan (e+f x)^2+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}+\frac {2 \left (\frac {i (a-i b)^2 (c+i d) \int \frac {1}{(i \tan (e+f x)+1) \sqrt {c+d \tan (e+f x)}}d(-i \tan (e+f x))}{2 f}-\frac {i (a+i b)^2 (c-i d) \int \frac {1}{(1-i \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}d(i \tan (e+f x))}{2 f}\right )}{a^2+b^2}}{2 \left (a^2+b^2\right )}\)

\(\Big \downarrow \) 73

\(\displaystyle -\frac {b \sqrt {c+d \tan (e+f x)}}{f \left (a^2+b^2\right ) (a+b \tan (e+f x))}+\frac {\frac {b \left (-3 a^2 d+4 a b c+b^2 d\right ) \int \frac {\tan (e+f x)^2+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}+\frac {2 \left (\frac {(a-i b)^2 (c+i d) \int \frac {1}{-\frac {i \tan ^2(e+f x)}{d}-\frac {i c}{d}+1}d\sqrt {c+d \tan (e+f x)}}{d f}+\frac {(a+i b)^2 (c-i d) \int \frac {1}{\frac {i \tan ^2(e+f x)}{d}+\frac {i c}{d}+1}d\sqrt {c+d \tan (e+f x)}}{d f}\right )}{a^2+b^2}}{2 \left (a^2+b^2\right )}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {b \sqrt {c+d \tan (e+f x)}}{f \left (a^2+b^2\right ) (a+b \tan (e+f x))}+\frac {\frac {b \left (-3 a^2 d+4 a b c+b^2 d\right ) \int \frac {\tan (e+f x)^2+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}+\frac {2 \left (\frac {(a-i b)^2 \sqrt {c+i d} \arctan \left (\frac {\tan (e+f x)}{\sqrt {c+i d}}\right )}{f}+\frac {(a+i b)^2 \sqrt {c-i d} \arctan \left (\frac {\tan (e+f x)}{\sqrt {c-i d}}\right )}{f}\right )}{a^2+b^2}}{2 \left (a^2+b^2\right )}\)

\(\Big \downarrow \) 4117

\(\displaystyle -\frac {b \sqrt {c+d \tan (e+f x)}}{f \left (a^2+b^2\right ) (a+b \tan (e+f x))}+\frac {\frac {b \left (-3 a^2 d+4 a b c+b^2 d\right ) \int \frac {1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}d\tan (e+f x)}{f \left (a^2+b^2\right )}+\frac {2 \left (\frac {(a-i b)^2 \sqrt {c+i d} \arctan \left (\frac {\tan (e+f x)}{\sqrt {c+i d}}\right )}{f}+\frac {(a+i b)^2 \sqrt {c-i d} \arctan \left (\frac {\tan (e+f x)}{\sqrt {c-i d}}\right )}{f}\right )}{a^2+b^2}}{2 \left (a^2+b^2\right )}\)

\(\Big \downarrow \) 73

\(\displaystyle -\frac {b \sqrt {c+d \tan (e+f x)}}{f \left (a^2+b^2\right ) (a+b \tan (e+f x))}+\frac {\frac {2 b \left (-3 a^2 d+4 a b c+b^2 d\right ) \int \frac {1}{a+\frac {b (c+d \tan (e+f x))}{d}-\frac {b c}{d}}d\sqrt {c+d \tan (e+f x)}}{d f \left (a^2+b^2\right )}+\frac {2 \left (\frac {(a-i b)^2 \sqrt {c+i d} \arctan \left (\frac {\tan (e+f x)}{\sqrt {c+i d}}\right )}{f}+\frac {(a+i b)^2 \sqrt {c-i d} \arctan \left (\frac {\tan (e+f x)}{\sqrt {c-i d}}\right )}{f}\right )}{a^2+b^2}}{2 \left (a^2+b^2\right )}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {b \sqrt {c+d \tan (e+f x)}}{f \left (a^2+b^2\right ) (a+b \tan (e+f x))}+\frac {-\frac {2 \sqrt {b} \left (-3 a^2 d+4 a b c+b^2 d\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{f \left (a^2+b^2\right ) \sqrt {b c-a d}}+\frac {2 \left (\frac {(a-i b)^2 \sqrt {c+i d} \arctan \left (\frac {\tan (e+f x)}{\sqrt {c+i d}}\right )}{f}+\frac {(a+i b)^2 \sqrt {c-i d} \arctan \left (\frac {\tan (e+f x)}{\sqrt {c-i d}}\right )}{f}\right )}{a^2+b^2}}{2 \left (a^2+b^2\right )}\)

Input:

Int[Sqrt[c + d*Tan[e + f*x]]/(a + b*Tan[e + f*x])^2,x]
 

Output:

((2*(((a + I*b)^2*Sqrt[c - I*d]*ArcTan[Tan[e + f*x]/Sqrt[c - I*d]])/f + (( 
a - I*b)^2*Sqrt[c + I*d]*ArcTan[Tan[e + f*x]/Sqrt[c + I*d]])/f))/(a^2 + b^ 
2) - (2*Sqrt[b]*(4*a*b*c - 3*a^2*d + b^2*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*Ta 
n[e + f*x]])/Sqrt[b*c - a*d]])/((a^2 + b^2)*Sqrt[b*c - a*d]*f))/(2*(a^2 + 
b^2)) - (b*Sqrt[c + d*Tan[e + f*x]])/((a^2 + b^2)*f*(a + b*Tan[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 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 221

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

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]
 

rule 4022

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

rule 4051

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

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]
 

rule 4136

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1889\) vs. \(2(199)=398\).

Time = 0.39 (sec) , antiderivative size = 1890, normalized size of antiderivative = 8.18

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1890\)
default	\(\text {Expression too large to display}\)	\(1890\)

Input:

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

Output:

-1/f*d*b/(a^2+b^2)^2*(c+d*tan(f*x+e))^(1/2)/(tan(f*x+e)*b*d+a*d)*a^2-1/f*d 
*b^3/(a^2+b^2)^2*(c+d*tan(f*x+e))^(1/2)/(tan(f*x+e)*b*d+a*d)-3/f*d*b/(a^2+ 
b^2)^2/((a*d-b*c)*b)^(1/2)*arctan(b*(c+d*tan(f*x+e))^(1/2)/((a*d-b*c)*b)^( 
1/2))*a^2+4/f*b^2/(a^2+b^2)^2/((a*d-b*c)*b)^(1/2)*arctan(b*(c+d*tan(f*x+e) 
)^(1/2)/((a*d-b*c)*b)^(1/2))*a*c+1/f*d*b^3/(a^2+b^2)^2/((a*d-b*c)*b)^(1/2) 
*arctan(b*(c+d*tan(f*x+e))^(1/2)/((a*d-b*c)*b)^(1/2))-1/f*d/(a^2+b^2)^2/(2 
*(c^2+d^2)^(1/2)-2*c)^(1/2)*arctan(((2*(c^2+d^2)^(1/2)+2*c)^(1/2)-2*(c+d*t 
an(f*x+e))^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2))*a^2+1/f*d/(a^2+b^2)^2/(2* 
(c^2+d^2)^(1/2)-2*c)^(1/2)*arctan(((2*(c^2+d^2)^(1/2)+2*c)^(1/2)-2*(c+d*ta 
n(f*x+e))^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2))*b^2+1/f*d/(a^2+b^2)^2/(2*( 
c^2+d^2)^(1/2)-2*c)^(1/2)*arctan((2*(c+d*tan(f*x+e))^(1/2)+(2*(c^2+d^2)^(1 
/2)+2*c)^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2))*a^2-1/f*d/(a^2+b^2)^2/(2*(c 
^2+d^2)^(1/2)-2*c)^(1/2)*arctan((2*(c+d*tan(f*x+e))^(1/2)+(2*(c^2+d^2)^(1/ 
2)+2*c)^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2))*b^2+2/f/(a^2+b^2)^2/(2*(c^2+ 
d^2)^(1/2)-2*c)^(1/2)*arctan((2*(c+d*tan(f*x+e))^(1/2)+(2*(c^2+d^2)^(1/2)+ 
2*c)^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2))*(c^2+d^2)^(1/2)*a*b-2/f/(a^2+b^ 
2)^2/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*arctan((2*(c+d*tan(f*x+e))^(1/2)+(2*(c^ 
2+d^2)^(1/2)+2*c)^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2))*a*b*c-2/f/(a^2+b^2 
)^2/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*arctan(((2*(c^2+d^2)^(1/2)+2*c)^(1/2)-2* 
(c+d*tan(f*x+e))^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2))*(c^2+d^2)^(1/2)*...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3808 vs. \(2 (191) = 382\).

Time = 4.70 (sec) , antiderivative size = 7655, normalized size of antiderivative = 33.14 \[ \int \frac {\sqrt {c+d \tan (e+f x)}}{(a+b \tan (e+f x))^2} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F]

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

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

Output:

Integral(sqrt(c + d*tan(e + f*x))/(a + b*tan(e + f*x))**2, x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\sqrt {c+d \tan (e+f x)}}{(a+b \tan (e+f x))^2} \, dx=\text {Exception raised: ValueError} \] Input:

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for m 
ore detail
 

Giac [F(-2)]

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

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro 
unding error%%%{%%%{1,[0,14,6]%%%}+%%%{6,[0,12,6]%%%}+%%%{15,[0,10,6]%%%}+ 
%%%{20,[0
                                                                                    
                                                                                    
 

Mupad [B] (verification not implemented)

Time = 9.09 (sec) , antiderivative size = 28314, normalized size of antiderivative = 122.57 \[ \int \frac {\sqrt {c+d \tan (e+f x)}}{(a+b \tan (e+f x))^2} \, dx=\text {Too large to display} \] Input:

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

Output:

(atan((((-b*(a*d - b*c))^(1/2)*((16*(c + d*tan(e + f*x))^(1/2)*(3*b^9*d^12 
 - 3*a^2*b^7*d^12 + 17*a^4*b^5*d^12 - 9*a^6*b^3*d^12 + 3*b^9*c^2*d^10 + 2* 
b^9*c^4*d^8 - 8*a*b^8*c^3*d^9 - 56*a^3*b^6*c*d^11 + 60*a^5*b^4*c*d^11 + 63 
*a^2*b^7*c^2*d^10 - 12*a^2*b^7*c^4*d^8 + 96*a^3*b^6*c^3*d^9 - 123*a^4*b^5* 
c^2*d^10 + 18*a^4*b^5*c^4*d^8 - 24*a^5*b^4*c^3*d^9 + 9*a^6*b^3*c^2*d^10 + 
12*a*b^8*c*d^11))/(a^8*f^4 + b^8*f^4 + 4*a^2*b^6*f^4 + 6*a^4*b^4*f^4 + 4*a 
^6*b^2*f^4) - (((8*(20*b^11*c*d^11*f^2 - 52*a*b^10*d^12*f^2 + 128*a^3*b^8* 
d^12*f^2 + 24*a^5*b^6*d^12*f^2 - 160*a^7*b^4*d^12*f^2 - 4*a^9*b^2*d^12*f^2 
 + 20*b^11*c^3*d^9*f^2 - 256*a^2*b^9*c^3*d^9*f^2 - 128*a^3*b^8*c^4*d^8*f^2 
 + 72*a^4*b^7*c^3*d^9*f^2 - 168*a^5*b^6*c^2*d^10*f^2 - 192*a^5*b^6*c^4*d^8 
*f^2 + 352*a^6*b^5*c^3*d^9*f^2 - 160*a^7*b^4*c^2*d^10*f^2 + 4*a^8*b^3*c^3* 
d^9*f^2 - 4*a^9*b^2*c^2*d^10*f^2 + 12*a*b^10*c^2*d^10*f^2 + 64*a*b^10*c^4* 
d^8*f^2 - 256*a^2*b^9*c*d^11*f^2 + 72*a^4*b^7*c*d^11*f^2 + 352*a^6*b^5*c*d 
^11*f^2 + 4*a^8*b^3*c*d^11*f^2))/(a^8*f^5 + b^8*f^5 + 4*a^2*b^6*f^5 + 6*a^ 
4*b^4*f^5 + 4*a^6*b^2*f^5) + ((-b*(a*d - b*c))^(1/2)*((16*(c + d*tan(e + f 
*x))^(1/2)*(68*a*b^12*d^11*f^2 - 8*b^13*c*d^10*f^2 + 20*a^3*b^10*d^11*f^2 
- 88*a^5*b^8*d^11*f^2 + 40*a^7*b^6*d^11*f^2 + 84*a^9*b^4*d^11*f^2 + 4*a^11 
*b^2*d^11*f^2 - 20*b^13*c^3*d^8*f^2 + 116*a^2*b^11*c^3*d^8*f^2 + 204*a^3*b 
^10*c^2*d^9*f^2 + 216*a^4*b^9*c^3*d^8*f^2 + 168*a^5*b^8*c^2*d^9*f^2 + 8*a^ 
6*b^7*c^3*d^8*f^2 + 184*a^7*b^6*c^2*d^9*f^2 - 68*a^8*b^5*c^3*d^8*f^2 + ...
 

Reduce [F]

\[ \int \frac {\sqrt {c+d \tan (e+f x)}}{(a+b \tan (e+f x))^2} \, dx=\text {too large to display} \] Input:

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

Output:

(2*sqrt(tan(e + f*x)*d + c)*a*d - 2*sqrt(tan(e + f*x)*d + c)*b*c - int(sqr 
t(tan(e + f*x)*d + c)/(tan(e + f*x)**3*a*b**2*d**2 - 2*tan(e + f*x)**3*b** 
3*c*d + 2*tan(e + f*x)**2*a**2*b*d**2 - 3*tan(e + f*x)**2*a*b**2*c*d - 2*t 
an(e + f*x)**2*b**3*c**2 + tan(e + f*x)*a**3*d**2 - 4*tan(e + f*x)*a*b**2* 
c**2 + a**3*c*d - 2*a**2*b*c**2),x)*tan(e + f*x)*a**3*b*d**3*f + 6*int(sqr 
t(tan(e + f*x)*d + c)/(tan(e + f*x)**3*a*b**2*d**2 - 2*tan(e + f*x)**3*b** 
3*c*d + 2*tan(e + f*x)**2*a**2*b*d**2 - 3*tan(e + f*x)**2*a*b**2*c*d - 2*t 
an(e + f*x)**2*b**3*c**2 + tan(e + f*x)*a**3*d**2 - 4*tan(e + f*x)*a*b**2* 
c**2 + a**3*c*d - 2*a**2*b*c**2),x)*tan(e + f*x)*a**2*b**2*c*d**2*f - 12*i 
nt(sqrt(tan(e + f*x)*d + c)/(tan(e + f*x)**3*a*b**2*d**2 - 2*tan(e + f*x)* 
*3*b**3*c*d + 2*tan(e + f*x)**2*a**2*b*d**2 - 3*tan(e + f*x)**2*a*b**2*c*d 
 - 2*tan(e + f*x)**2*b**3*c**2 + tan(e + f*x)*a**3*d**2 - 4*tan(e + f*x)*a 
*b**2*c**2 + a**3*c*d - 2*a**2*b*c**2),x)*tan(e + f*x)*a*b**3*c**2*d*f + 8 
*int(sqrt(tan(e + f*x)*d + c)/(tan(e + f*x)**3*a*b**2*d**2 - 2*tan(e + f*x 
)**3*b**3*c*d + 2*tan(e + f*x)**2*a**2*b*d**2 - 3*tan(e + f*x)**2*a*b**2*c 
*d - 2*tan(e + f*x)**2*b**3*c**2 + tan(e + f*x)*a**3*d**2 - 4*tan(e + f*x) 
*a*b**2*c**2 + a**3*c*d - 2*a**2*b*c**2),x)*tan(e + f*x)*b**4*c**3*f - int 
(sqrt(tan(e + f*x)*d + c)/(tan(e + f*x)**3*a*b**2*d**2 - 2*tan(e + f*x)**3 
*b**3*c*d + 2*tan(e + f*x)**2*a**2*b*d**2 - 3*tan(e + f*x)**2*a*b**2*c*d - 
 2*tan(e + f*x)**2*b**3*c**2 + tan(e + f*x)*a**3*d**2 - 4*tan(e + f*x)*...