Integrand size = 25, antiderivative size = 365 \[ \int \frac {1}{(a+a \sec (c+d x))^2 \sqrt {e \tan (c+d x)}} \, dx=-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} a^2 d \sqrt {e}}+\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} a^2 d \sqrt {e}}-\frac {\log \left (\sqrt {e}+\sqrt {e} \tan (c+d x)-\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} a^2 d \sqrt {e}}+\frac {\log \left (\sqrt {e}+\sqrt {e} \tan (c+d x)+\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} a^2 d \sqrt {e}}-\frac {4 e^3}{7 a^2 d (e \tan (c+d x))^{7/2}}+\frac {4 e^3 \sec (c+d x)}{7 a^2 d (e \tan (c+d x))^{7/2}}+\frac {2 e}{3 a^2 d (e \tan (c+d x))^{3/2}}-\frac {20 e \sec (c+d x)}{21 a^2 d (e \tan (c+d x))^{3/2}}-\frac {10 \operatorname {EllipticF}\left (c-\frac {\pi }{4}+d x,2\right ) \sec (c+d x) \sqrt {\sin (2 c+2 d x)}}{21 a^2 d \sqrt {e \tan (c+d x)}} \]
-1/2*arctan(1-2^(1/2)*(e*tan(d*x+c))^(1/2)/e^(1/2))/a^2/d*2^(1/2)/e^(1/2)+ 1/2*arctan(1+2^(1/2)*(e*tan(d*x+c))^(1/2)/e^(1/2))/a^2/d*2^(1/2)/e^(1/2)-1 /4*ln(e^(1/2)-2^(1/2)*(e*tan(d*x+c))^(1/2)+e^(1/2)*tan(d*x+c))/a^2/d*2^(1/ 2)/e^(1/2)+1/4*ln(e^(1/2)+2^(1/2)*(e*tan(d*x+c))^(1/2)+e^(1/2)*tan(d*x+c)) /a^2/d*2^(1/2)/e^(1/2)+10/21*(sin(c+1/4*Pi+d*x)^2)^(1/2)/sin(c+1/4*Pi+d*x) *EllipticF(cos(c+1/4*Pi+d*x),2^(1/2))*sec(d*x+c)*sin(2*d*x+2*c)^(1/2)/a^2/ d/(e*tan(d*x+c))^(1/2)-4/7*e^3/a^2/d/(e*tan(d*x+c))^(7/2)+4/7*e^3*sec(d*x+ c)/a^2/d/(e*tan(d*x+c))^(7/2)+2/3*e/a^2/d/(e*tan(d*x+c))^(3/2)-20/21*e*sec (d*x+c)/a^2/d/(e*tan(d*x+c))^(3/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 12.51 (sec) , antiderivative size = 1281, normalized size of antiderivative = 3.51 \[ \int \frac {1}{(a+a \sec (c+d x))^2 \sqrt {e \tan (c+d x)}} \, dx=\frac {40 e^{-i (c+d x)} \sqrt {-\frac {i \left (-1+e^{2 i (c+d x)}\right )}{1+e^{2 i (c+d x)}}} \left (1+e^{2 i (c+d x)}\right ) \cos ^4\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec (2 c) \sec ^2(c+d x) \sqrt {\tan (c+d x)}}{21 d (a+a \sec (c+d x))^2 \sqrt {e \tan (c+d x)}}+\frac {e^{-2 i c} \sqrt {-\frac {i \left (-1+e^{2 i (c+d x)}\right )}{1+e^{2 i (c+d x)}}} \left (e^{4 i c} \sqrt {-1+e^{4 i (c+d x)}} \arctan \left (\sqrt {-1+e^{4 i (c+d x)}}\right )+2 \sqrt {-1+e^{2 i (c+d x)}} \sqrt {1+e^{2 i (c+d x)}} \text {arctanh}\left (\sqrt {\frac {-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right )\right ) \cos ^4\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec (2 c) \sec ^2(c+d x) \sqrt {\tan (c+d x)}}{d \left (-1+e^{2 i (c+d x)}\right ) (a+a \sec (c+d x))^2 \sqrt {e \tan (c+d x)}}+\frac {e^{-2 i c} \sqrt {-\frac {i \left (-1+e^{2 i (c+d x)}\right )}{1+e^{2 i (c+d x)}}} \left (\sqrt {-1+e^{4 i (c+d x)}} \arctan \left (\sqrt {-1+e^{4 i (c+d x)}}\right )+2 e^{4 i c} \sqrt {-1+e^{2 i (c+d x)}} \sqrt {1+e^{2 i (c+d x)}} \text {arctanh}\left (\sqrt {\frac {-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right )\right ) \cos ^4\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec (2 c) \sec ^2(c+d x) \sqrt {\tan (c+d x)}}{d \left (-1+e^{2 i (c+d x)}\right ) (a+a \sec (c+d x))^2 \sqrt {e \tan (c+d x)}}-\frac {2 e^{-i (2 c+d x)} \sqrt {-\frac {i \left (-1+e^{2 i (c+d x)}\right )}{1+e^{2 i (c+d x)}}} \cos ^4\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (3 \left (-1+e^{4 i (c+d x)}\right )+e^{4 i (c+d x)} \left (-1+e^{2 i c}\right ) \sqrt {1-e^{4 i (c+d x)}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},e^{4 i (c+d x)}\right )\right ) \sec (2 c) \sec ^2(c+d x) \sqrt {\tan (c+d x)}}{3 d \left (-1+e^{2 i (c+d x)}\right ) (a+a \sec (c+d x))^2 \sqrt {e \tan (c+d x)}}+\frac {2 e^{-i d x} \sqrt {-\frac {i \left (-1+e^{2 i (c+d x)}\right )}{1+e^{2 i (c+d x)}}} \cos ^4\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (3-3 e^{4 i (c+d x)}+e^{2 i (c+2 d x)} \left (-1+e^{2 i c}\right ) \sqrt {1-e^{4 i (c+d x)}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},e^{4 i (c+d x)}\right )\right ) \sec (2 c) \sec ^2(c+d x) \sqrt {\tan (c+d x)}}{3 d \left (-1+e^{2 i (c+d x)}\right ) (a+a \sec (c+d x))^2 \sqrt {e \tan (c+d x)}}+\frac {\cos ^4\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec ^2(c+d x) \left (-\frac {104}{21 d}+\frac {4 (21-20 \cos (c)+21 \cos (2 c)) \cos (d x) \sec (2 c)}{21 d}+\frac {64 \sec ^2\left (\frac {c}{2}+\frac {d x}{2}\right )}{21 d}-\frac {2 \sec ^4\left (\frac {c}{2}+\frac {d x}{2}\right )}{7 d}-\frac {4 \sec (2 c) (-20 \sin (c)+21 \sin (2 c)) \sin (d x)}{21 d}\right ) \tan (c+d x)}{(a+a \sec (c+d x))^2 \sqrt {e \tan (c+d x)}}+\frac {80 \sqrt [4]{-1} \cos ^4\left (\frac {c}{2}+\frac {d x}{2}\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt [4]{-1} \sqrt {\tan (c+d x)}\right ),-1\right ) \sec ^5(c+d x) \sqrt {\tan (c+d x)}}{21 d (a+a \sec (c+d x))^2 \sqrt {e \tan (c+d x)} \left (1+\tan ^2(c+d x)\right )^{3/2}} \]
(40*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*(1 + E^((2*I)*(c + d*x)))*Cos[c/2 + (d*x)/2]^4*Sec[2*c]*Sec[c + d*x]^2*Sqrt[Ta n[c + d*x]])/(21*d*E^(I*(c + d*x))*(a + a*Sec[c + d*x])^2*Sqrt[e*Tan[c + d *x]]) + (Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))] *(E^((4*I)*c)*Sqrt[-1 + E^((4*I)*(c + d*x))]*ArcTan[Sqrt[-1 + E^((4*I)*(c + d*x))]] + 2*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[1 + E^((2*I)*(c + d*x))] *ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]])*Cos[ c/2 + (d*x)/2]^4*Sec[2*c]*Sec[c + d*x]^2*Sqrt[Tan[c + d*x]])/(d*E^((2*I)*c )*(-1 + E^((2*I)*(c + d*x)))*(a + a*Sec[c + d*x])^2*Sqrt[e*Tan[c + d*x]]) + (Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*(Sqrt [-1 + E^((4*I)*(c + d*x))]*ArcTan[Sqrt[-1 + E^((4*I)*(c + d*x))]] + 2*E^(( 4*I)*c)*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTa nh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]])*Cos[c/2 + (d*x)/2]^4*Sec[2*c]*Sec[c + d*x]^2*Sqrt[Tan[c + d*x]])/(d*E^((2*I)*c)*(-1 + E^((2*I)*(c + d*x)))*(a + a*Sec[c + d*x])^2*Sqrt[e*Tan[c + d*x]]) - (2*S qrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*Cos[c/2 + (d*x)/2]^4*(3*(-1 + E^((4*I)*(c + d*x))) + E^((4*I)*(c + d*x))*(-1 + E^(( 2*I)*c))*Sqrt[1 - E^((4*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, E^ ((4*I)*(c + d*x))])*Sec[2*c]*Sec[c + d*x]^2*Sqrt[Tan[c + d*x]])/(3*d*E^(I* (2*c + d*x))*(-1 + E^((2*I)*(c + d*x)))*(a + a*Sec[c + d*x])^2*Sqrt[e*T...
Time = 0.85 (sec) , antiderivative size = 379, normalized size of antiderivative = 1.04, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3042, 4376, 3042, 4374, 2009}
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.
| \(\displaystyle \int \frac {1}{(a \sec (c+d x)+a)^2 \sqrt {e \tan (c+d x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^2 \sqrt {-e \cot \left (c+d x+\frac {\pi }{2}\right )}}dx\) |
| \(\Big \downarrow \) 4376 |
| \(\displaystyle \frac {e^4 \int \frac {(a-a \sec (c+d x))^2}{(e \tan (c+d x))^{9/2}}dx}{a^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {e^4 \int \frac {\left (a-a \csc \left (c+d x+\frac {\pi }{2}\right )\right )^2}{\left (-e \cot \left (c+d x+\frac {\pi }{2}\right )\right )^{9/2}}dx}{a^4}\) |
| \(\Big \downarrow \) 4374 |
| \(\displaystyle \frac {e^4 \int \left (\frac {\sec ^2(c+d x) a^2}{(e \tan (c+d x))^{9/2}}-\frac {2 \sec (c+d x) a^2}{(e \tan (c+d x))^{9/2}}+\frac {a^2}{(e \tan (c+d x))^{9/2}}\right )dx}{a^4}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {e^4 \left (-\frac {a^2 \arctan \left (1-\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d e^{9/2}}+\frac {a^2 \arctan \left (\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} d e^{9/2}}-\frac {a^2 \log \left (\sqrt {e} \tan (c+d x)-\sqrt {2} \sqrt {e \tan (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} d e^{9/2}}+\frac {a^2 \log \left (\sqrt {e} \tan (c+d x)+\sqrt {2} \sqrt {e \tan (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} d e^{9/2}}-\frac {10 a^2 \sqrt {\sin (2 c+2 d x)} \sec (c+d x) \operatorname {EllipticF}\left (c+d x-\frac {\pi }{4},2\right )}{21 d e^4 \sqrt {e \tan (c+d x)}}+\frac {2 a^2}{3 d e^3 (e \tan (c+d x))^{3/2}}-\frac {20 a^2 \sec (c+d x)}{21 d e^3 (e \tan (c+d x))^{3/2}}-\frac {4 a^2}{7 d e (e \tan (c+d x))^{7/2}}+\frac {4 a^2 \sec (c+d x)}{7 d e (e \tan (c+d x))^{7/2}}\right )}{a^4}\) |
(e^4*(-((a^2*ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sqrt[2]* d*e^(9/2))) + (a^2*ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]])/(Sq rt[2]*d*e^(9/2)) - (a^2*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[ e*Tan[c + d*x]]])/(2*Sqrt[2]*d*e^(9/2)) + (a^2*Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]])/(2*Sqrt[2]*d*e^(9/2)) - (4*a^2)/( 7*d*e*(e*Tan[c + d*x])^(7/2)) + (4*a^2*Sec[c + d*x])/(7*d*e*(e*Tan[c + d*x ])^(7/2)) + (2*a^2)/(3*d*e^3*(e*Tan[c + d*x])^(3/2)) - (20*a^2*Sec[c + d*x ])/(21*d*e^3*(e*Tan[c + d*x])^(3/2)) - (10*a^2*EllipticF[c - Pi/4 + d*x, 2 ]*Sec[c + d*x]*Sqrt[Sin[2*c + 2*d*x]])/(21*d*e^4*Sqrt[e*Tan[c + d*x]])))/a ^4
3.2.34.3.1 Defintions of rubi rules used
Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + ( a_))^(n_), x_Symbol] :> Int[ExpandIntegrand[(e*Cot[c + d*x])^m, (a + b*Csc[ c + d*x])^n, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n, 0]
Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + ( a_))^(n_), x_Symbol] :> Simp[a^(2*n)/e^(2*n) Int[(e*Cot[c + d*x])^(m + 2* n)/(-a + b*Csc[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && EqQ[a ^2 - b^2, 0] && ILtQ[n, 0]
Result contains complex when optimal does not.
Time = 5.60 (sec) , antiderivative size = 633, normalized size of antiderivative = 1.73
| method | result | size |
| default | \(-\frac {\sqrt {2}\, \left (21 i \sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}\, \sqrt {2-2 \csc \left (d x +c \right )+2 \cot \left (d x +c \right )}\, \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \operatorname {EllipticPi}\left (\sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right )-21 i \sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}\, \sqrt {2-2 \csc \left (d x +c \right )+2 \cot \left (d x +c \right )}\, \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \operatorname {EllipticPi}\left (\sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right )-3 \left (1-\cos \left (d x +c \right )\right )^{5} \csc \left (d x +c \right )^{5}-62 \sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}\, \sqrt {2-2 \csc \left (d x +c \right )+2 \cot \left (d x +c \right )}\, \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \operatorname {EllipticF}\left (\sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right )+21 \sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}\, \sqrt {2-2 \csc \left (d x +c \right )+2 \cot \left (d x +c \right )}\, \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \operatorname {EllipticPi}\left (\sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right )+21 \sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}\, \sqrt {2-2 \csc \left (d x +c \right )+2 \cot \left (d x +c \right )}\, \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \operatorname {EllipticPi}\left (\sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right )+26 \left (1-\cos \left (d x +c \right )\right )^{3} \csc \left (d x +c \right )^{3}-23 \csc \left (d x +c \right )+23 \cot \left (d x +c \right )\right ) \left (1-\cos \left (d x +c \right )\right ) \csc \left (d x +c \right )}{42 a^{2} d \sqrt {\left (1-\cos \left (d x +c \right )\right )^{3} \csc \left (d x +c \right )^{3}+\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \sqrt {\left (1-\cos \left (d x +c \right )\right ) \left (\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1\right ) \csc \left (d x +c \right )}\, \sqrt {-\frac {e \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1}}}\) | \(633\) |
-1/42/a^2/d*2^(1/2)*(21*I*(csc(d*x+c)-cot(d*x+c)+1)^(1/2)*(2-2*csc(d*x+c)+ 2*cot(d*x+c))^(1/2)*(cot(d*x+c)-csc(d*x+c))^(1/2)*EllipticPi((csc(d*x+c)-c ot(d*x+c)+1)^(1/2),1/2-1/2*I,1/2*2^(1/2))-21*I*(csc(d*x+c)-cot(d*x+c)+1)^( 1/2)*(2-2*csc(d*x+c)+2*cot(d*x+c))^(1/2)*(cot(d*x+c)-csc(d*x+c))^(1/2)*Ell ipticPi((csc(d*x+c)-cot(d*x+c)+1)^(1/2),1/2+1/2*I,1/2*2^(1/2))-3*(1-cos(d* x+c))^5*csc(d*x+c)^5-62*(csc(d*x+c)-cot(d*x+c)+1)^(1/2)*(2-2*csc(d*x+c)+2* cot(d*x+c))^(1/2)*(cot(d*x+c)-csc(d*x+c))^(1/2)*EllipticF((csc(d*x+c)-cot( d*x+c)+1)^(1/2),1/2*2^(1/2))+21*(csc(d*x+c)-cot(d*x+c)+1)^(1/2)*(2-2*csc(d *x+c)+2*cot(d*x+c))^(1/2)*(cot(d*x+c)-csc(d*x+c))^(1/2)*EllipticPi((csc(d* x+c)-cot(d*x+c)+1)^(1/2),1/2-1/2*I,1/2*2^(1/2))+21*(csc(d*x+c)-cot(d*x+c)+ 1)^(1/2)*(2-2*csc(d*x+c)+2*cot(d*x+c))^(1/2)*(cot(d*x+c)-csc(d*x+c))^(1/2) *EllipticPi((csc(d*x+c)-cot(d*x+c)+1)^(1/2),1/2+1/2*I,1/2*2^(1/2))+26*(1-c os(d*x+c))^3*csc(d*x+c)^3-23*csc(d*x+c)+23*cot(d*x+c))*(1-cos(d*x+c))/((1- cos(d*x+c))^3*csc(d*x+c)^3+cot(d*x+c)-csc(d*x+c))^(1/2)/((1-cos(d*x+c))*(( 1-cos(d*x+c))^2*csc(d*x+c)^2-1)*csc(d*x+c))^(1/2)/(-e/((1-cos(d*x+c))^2*cs c(d*x+c)^2-1)*(-cot(d*x+c)+csc(d*x+c)))^(1/2)*csc(d*x+c)
Timed out. \[ \int \frac {1}{(a+a \sec (c+d x))^2 \sqrt {e \tan (c+d x)}} \, dx=\text {Timed out} \]
\[ \int \frac {1}{(a+a \sec (c+d x))^2 \sqrt {e \tan (c+d x)}} \, dx=\frac {\int \frac {1}{\sqrt {e \tan {\left (c + d x \right )}} \sec ^{2}{\left (c + d x \right )} + 2 \sqrt {e \tan {\left (c + d x \right )}} \sec {\left (c + d x \right )} + \sqrt {e \tan {\left (c + d x \right )}}}\, dx}{a^{2}} \]
Integral(1/(sqrt(e*tan(c + d*x))*sec(c + d*x)**2 + 2*sqrt(e*tan(c + d*x))* sec(c + d*x) + sqrt(e*tan(c + d*x))), x)/a**2
\[ \int \frac {1}{(a+a \sec (c+d x))^2 \sqrt {e \tan (c+d x)}} \, dx=\int { \frac {1}{{\left (a \sec \left (d x + c\right ) + a\right )}^{2} \sqrt {e \tan \left (d x + c\right )}} \,d x } \]
\[ \int \frac {1}{(a+a \sec (c+d x))^2 \sqrt {e \tan (c+d x)}} \, dx=\int { \frac {1}{{\left (a \sec \left (d x + c\right ) + a\right )}^{2} \sqrt {e \tan \left (d x + c\right )}} \,d x } \]
Timed out. \[ \int \frac {1}{(a+a \sec (c+d x))^2 \sqrt {e \tan (c+d x)}} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^2}{a^2\,\sqrt {e\,\mathrm {tan}\left (c+d\,x\right )}\,{\left (\cos \left (c+d\,x\right )+1\right )}^2} \,d x \]