Integrand size = 27, antiderivative size = 170 \[ \int \frac {1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{(i a+b) \sqrt {c-i d} f}-\frac {\text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{(i a-b) \sqrt {c+i d} f}-\frac {2 b^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{\left (a^2+b^2\right ) \sqrt {b c-a d} f} \] Output:
arctanh((c+d*tan(f*x+e))^(1/2)/(c-I*d)^(1/2))/(I*a+b)/(c-I*d)^(1/2)/f-arct anh((c+d*tan(f*x+e))^(1/2)/(c+I*d)^(1/2))/(I*a-b)/(c+I*d)^(1/2)/f-2*b^(3/2 )*arctanh(b^(1/2)*(c+d*tan(f*x+e))^(1/2)/(-a*d+b*c)^(1/2))/(a^2+b^2)/(-a*d +b*c)^(1/2)/f
Time = 0.27 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.93 \[ \int \frac {1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}} \, dx=\frac {\frac {(-i a+b) \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{\sqrt {c-i d}}+\frac {(i a+b) \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{\sqrt {c+i d}}-\frac {2 b^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{\sqrt {b c-a d}}}{\left (a^2+b^2\right ) f} \] Input:
Integrate[1/((a + b*Tan[e + f*x])*Sqrt[c + d*Tan[e + f*x]]),x]
Output:
((((-I)*a + b)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/Sqrt[c - I *d] + ((I*a + b)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/Sqrt[c + I*d] - (2*b^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a *d]])/Sqrt[b*c - a*d])/((a^2 + b^2)*f)
Time = 1.10 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.93, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {3042, 4057, 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.
\(\displaystyle \int \frac {1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx\) |
\(\Big \downarrow \) 4057 |
\(\displaystyle \frac {b^2 \int \frac {\tan ^2(e+f x)+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}+\frac {\int \frac {a-b \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {b^2 \int \frac {\tan (e+f x)^2+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}+\frac {\int \frac {a-b \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}\) |
\(\Big \downarrow \) 4022 |
\(\displaystyle \frac {b^2 \int \frac {\tan (e+f x)^2+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}+\frac {\frac {1}{2} (a-i b) \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}}dx+\frac {1}{2} (a+i b) \int \frac {i \tan (e+f x)+1}{\sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {b^2 \int \frac {\tan (e+f x)^2+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}+\frac {\frac {1}{2} (a-i b) \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}}dx+\frac {1}{2} (a+i b) \int \frac {i \tan (e+f x)+1}{\sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}\) |
\(\Big \downarrow \) 4020 |
\(\displaystyle \frac {b^2 \int \frac {\tan (e+f x)^2+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}+\frac {\frac {i (a+i b) \int -\frac {1}{(1-i \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}d(i \tan (e+f x))}{2 f}-\frac {i (a-i b) \int -\frac {1}{(i \tan (e+f x)+1) \sqrt {c+d \tan (e+f x)}}d(-i \tan (e+f x))}{2 f}}{a^2+b^2}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {b^2 \int \frac {\tan (e+f x)^2+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}+\frac {\frac {i (a-i b) \int \frac {1}{(i \tan (e+f x)+1) \sqrt {c+d \tan (e+f x)}}d(-i \tan (e+f x))}{2 f}-\frac {i (a+i b) \int \frac {1}{(1-i \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}d(i \tan (e+f x))}{2 f}}{a^2+b^2}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {b^2 \int \frac {\tan (e+f x)^2+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}+\frac {\frac {(a-i b) \int \frac {1}{-\frac {i \tan ^2(e+f x)}{d}-\frac {i c}{d}+1}d\sqrt {c+d \tan (e+f x)}}{d f}+\frac {(a+i b) \int \frac {1}{\frac {i \tan ^2(e+f x)}{d}+\frac {i c}{d}+1}d\sqrt {c+d \tan (e+f x)}}{d f}}{a^2+b^2}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {b^2 \int \frac {\tan (e+f x)^2+1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}dx}{a^2+b^2}+\frac {\frac {(a+i b) \arctan \left (\frac {\tan (e+f x)}{\sqrt {c-i d}}\right )}{f \sqrt {c-i d}}+\frac {(a-i b) \arctan \left (\frac {\tan (e+f x)}{\sqrt {c+i d}}\right )}{f \sqrt {c+i d}}}{a^2+b^2}\) |
\(\Big \downarrow \) 4117 |
\(\displaystyle \frac {b^2 \int \frac {1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}}d\tan (e+f x)}{f \left (a^2+b^2\right )}+\frac {\frac {(a+i b) \arctan \left (\frac {\tan (e+f x)}{\sqrt {c-i d}}\right )}{f \sqrt {c-i d}}+\frac {(a-i b) \arctan \left (\frac {\tan (e+f x)}{\sqrt {c+i d}}\right )}{f \sqrt {c+i d}}}{a^2+b^2}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {2 b^2 \int \frac {1}{a+\frac {b (c+d \tan (e+f x))}{d}-\frac {b c}{d}}d\sqrt {c+d \tan (e+f x)}}{d f \left (a^2+b^2\right )}+\frac {\frac {(a+i b) \arctan \left (\frac {\tan (e+f x)}{\sqrt {c-i d}}\right )}{f \sqrt {c-i d}}+\frac {(a-i b) \arctan \left (\frac {\tan (e+f x)}{\sqrt {c+i d}}\right )}{f \sqrt {c+i d}}}{a^2+b^2}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle -\frac {2 b^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{f \left (a^2+b^2\right ) \sqrt {b c-a d}}+\frac {\frac {(a+i b) \arctan \left (\frac {\tan (e+f x)}{\sqrt {c-i d}}\right )}{f \sqrt {c-i d}}+\frac {(a-i b) \arctan \left (\frac {\tan (e+f x)}{\sqrt {c+i d}}\right )}{f \sqrt {c+i d}}}{a^2+b^2}\) |
Input:
Int[1/((a + b*Tan[e + f*x])*Sqrt[c + d*Tan[e + f*x]]),x]
Output:
(((a + I*b)*ArcTan[Tan[e + f*x]/Sqrt[c - I*d]])/(Sqrt[c - I*d]*f) + ((a - I*b)*ArcTan[Tan[e + f*x]/Sqrt[c + I*d]])/(Sqrt[c + I*d]*f))/(a^2 + b^2) - (2*b^(3/2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]])/(( a^2 + b^2)*Sqrt[b*c - a*d]*f)
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]
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]
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]
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]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)/((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[1/(c^2 + d^2) Int[(a + b*Tan[e + f*x])^m *(c - d*Tan[e + f*x]), x], x] + Simp[d^2/(c^2 + d^2) Int[(a + b*Tan[e + f *x])^m*((1 + Tan[e + f*x]^2)/(c + d*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]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(4445\) vs. \(2(142)=284\).
Time = 0.38 (sec) , antiderivative size = 4446, normalized size of antiderivative = 26.15
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(4446\) |
default | \(\text {Expression too large to display}\) | \(4446\) |
Input:
int(1/(a+b*tan(f*x+e))/(c+d*tan(f*x+e))^(1/2),x,method=_RETURNVERBOSE)
Output:
2/f*b^2/(a^2+b^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)/(c^2+d^2)^(1/2)/(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))*a-1/f/d^2/(a^2+b^2)/(c^2+d^2)^(1/2)/(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*c^3+1/f/d^2/(a^2+b^2)/(c^2+d^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))*b*c^4-1/4/f/d/(a^2+b^2)/ (c^2+d^2)*ln((c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)-d*tan(f* x+e)-c-(c^2+d^2)^(1/2))*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^2-1/4/f/d^2/(a^2 +b^2)/(c^2+d^2)*ln((c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)-d* tan(f*x+e)-c-(c^2+d^2)^(1/2))*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c^3-1/f/d^2/ (a^2+b^2)/(c^2+d^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*c ^4+1/4/f/d^2/(a^2+b^2)/(c^2+d^2)*ln(d*tan(f*x+e)+c+(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))*(2*(c^2+d^2)^(1/2)+2*c)^(1/ 2)*b*c^3+1/4/f/d/(a^2+b^2)/(c^2+d^2)*ln(d*tan(f*x+e)+c+(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))*(2*(c^2+d^2)^(1/2)+2*c) ^(1/2)*a*c^2+1/f/d/(a^2+b^2)/(c^2+d^2)^(3/2)/(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^...
Leaf count of result is larger than twice the leaf count of optimal. 3636 vs. \(2 (136) = 272\).
Time = 1.19 (sec) , antiderivative size = 7311, normalized size of antiderivative = 43.01 \[ \int \frac {1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}} \, dx=\text {Too large to display} \] Input:
integrate(1/(a+b*tan(f*x+e))/(c+d*tan(f*x+e))^(1/2),x, algorithm="fricas")
Output:
Too large to include
\[ \int \frac {1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}} \, dx=\int \frac {1}{\left (a + b \tan {\left (e + f x \right )}\right ) \sqrt {c + d \tan {\left (e + f x \right )}}}\, dx \] Input:
integrate(1/(a+b*tan(f*x+e))/(c+d*tan(f*x+e))**(1/2),x)
Output:
Integral(1/((a + b*tan(e + f*x))*sqrt(c + d*tan(e + f*x))), x)
Exception generated. \[ \int \frac {1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(1/(a+b*tan(f*x+e))/(c+d*tan(f*x+e))^(1/2),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
Exception generated. \[ \int \frac {1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(1/(a+b*tan(f*x+e))/(c+d*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:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Time = 7.33 (sec) , antiderivative size = 9908, normalized size of antiderivative = 58.28 \[ \int \frac {1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}} \, dx=\text {Too large to display} \] Input:
int(1/((a + b*tan(e + f*x))*(c + d*tan(e + f*x))^(1/2)),x)
Output:
- atan(((((((32*(16*b^8*d^10*f^2 + 28*a^2*b^6*d^10*f^2 + 8*a^4*b^4*d^10*f^ 2 - 4*a^6*b^2*d^10*f^2 + 12*b^8*c^2*d^8*f^2 + 24*a^2*b^6*c^2*d^8*f^2 + 12* a^4*b^4*c^2*d^8*f^2 - 8*a*b^7*c*d^9*f^2 - 16*a^3*b^5*c*d^9*f^2 - 8*a^5*b^3 *c*d^9*f^2))/f^3 - (32*(1i/(4*(a^2*d*f^2 - a^2*c*f^2*1i + b^2*c*f^2*1i - b ^2*d*f^2 + 2*a*b*c*f^2 + a*b*d*f^2*2i)))^(1/2)*(c + d*tan(e + f*x))^(1/2)* (16*b^9*d^10*f^4 + 16*a^2*b^7*d^10*f^4 - 16*a^4*b^5*d^10*f^4 - 16*a^6*b^3* d^10*f^4 + 24*b^9*c^2*d^8*f^4 + 40*a^2*b^7*c^2*d^8*f^4 + 8*a^4*b^5*c^2*d^8 *f^4 - 8*a^6*b^3*c^2*d^8*f^4 + 8*a*b^8*c*d^9*f^4 + 24*a^3*b^6*c*d^9*f^4 + 24*a^5*b^4*c*d^9*f^4 + 8*a^7*b^2*c*d^9*f^4))/f^4)*(1i/(4*(a^2*d*f^2 - a^2* c*f^2*1i + b^2*c*f^2*1i - b^2*d*f^2 + 2*a*b*c*f^2 + a*b*d*f^2*2i)))^(1/2) + (32*(c + d*tan(e + f*x))^(1/2)*(30*a*b^6*d^9*f^2 + 18*b^7*c*d^8*f^2 - 4* a^3*b^4*d^9*f^2 - 2*a^5*b^2*d^9*f^2 - 12*a^2*b^5*c*d^8*f^2 + 2*a^4*b^3*c*d ^8*f^2))/f^4)*(1i/(4*(a^2*d*f^2 - a^2*c*f^2*1i + b^2*c*f^2*1i - b^2*d*f^2 + 2*a*b*c*f^2 + a*b*d*f^2*2i)))^(1/2) - (32*(5*a*b^5*d^9 + 3*b^6*c*d^8 + a ^3*b^3*d^9 - a^2*b^4*c*d^8))/f^3)*(1i/(4*(a^2*d*f^2 - a^2*c*f^2*1i + b^2*c *f^2*1i - b^2*d*f^2 + 2*a*b*c*f^2 + a*b*d*f^2*2i)))^(1/2) - (96*b^5*d^8*(c + d*tan(e + f*x))^(1/2))/f^4)*(1i/(4*(a^2*d*f^2 - a^2*c*f^2*1i + b^2*c*f^ 2*1i - b^2*d*f^2 + 2*a*b*c*f^2 + a*b*d*f^2*2i)))^(1/2)*1i - (((((32*(16*b^ 8*d^10*f^2 + 28*a^2*b^6*d^10*f^2 + 8*a^4*b^4*d^10*f^2 - 4*a^6*b^2*d^10*f^2 + 12*b^8*c^2*d^8*f^2 + 24*a^2*b^6*c^2*d^8*f^2 + 12*a^4*b^4*c^2*d^8*f^2...
\[ \int \frac {1}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}} \, dx=\frac {2 \sqrt {d \tan \left (f x +e \right )+c}-\left (\int \frac {\sqrt {d \tan \left (f x +e \right )+c}\, \tan \left (f x +e \right )^{3}}{\tan \left (f x +e \right )^{2} b d +\tan \left (f x +e \right ) a d +\tan \left (f x +e \right ) b c +a c}d x \right ) b d f -\left (\int \frac {\sqrt {d \tan \left (f x +e \right )+c}\, \tan \left (f x +e \right )^{2}}{\tan \left (f x +e \right )^{2} b d +\tan \left (f x +e \right ) a d +\tan \left (f x +e \right ) b c +a c}d x \right ) a d f -\left (\int \frac {\sqrt {d \tan \left (f x +e \right )+c}\, \tan \left (f x +e \right )}{\tan \left (f x +e \right )^{2} b d +\tan \left (f x +e \right ) a d +\tan \left (f x +e \right ) b c +a c}d x \right ) b d f}{a d f} \] Input:
int(1/(a+b*tan(f*x+e))/(c+d*tan(f*x+e))^(1/2),x)
Output:
(2*sqrt(tan(e + f*x)*d + c) - int((sqrt(tan(e + f*x)*d + c)*tan(e + f*x)** 3)/(tan(e + f*x)**2*b*d + tan(e + f*x)*a*d + tan(e + f*x)*b*c + a*c),x)*b* d*f - int((sqrt(tan(e + f*x)*d + c)*tan(e + f*x)**2)/(tan(e + f*x)**2*b*d + tan(e + f*x)*a*d + tan(e + f*x)*b*c + a*c),x)*a*d*f - int((sqrt(tan(e + f*x)*d + c)*tan(e + f*x))/(tan(e + f*x)**2*b*d + tan(e + f*x)*a*d + tan(e + f*x)*b*c + a*c),x)*b*d*f)/(a*d*f)