\(\int \sqrt {e \cot (c+d x)} (a+a \sec (c+d x)) \, dx\) [235]

Optimal result

Integrand size = 23, antiderivative size = 212 \[ \int \sqrt {e \cot (c+d x)} (a+a \sec (c+d x)) \, dx=\frac {a \sqrt {e \cot (c+d x)} \operatorname {EllipticF}\left (c-\frac {\pi }{4}+d x,2\right ) \sec (c+d x) \sqrt {\sin (2 c+2 d x)}}{d}-\frac {a \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right ) \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}}{\sqrt {2} d}+\frac {a \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right ) \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}}{\sqrt {2} d}+\frac {a \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\tan (c+d x)}}{1+\tan (c+d x)}\right ) \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}}{\sqrt {2} d} \] Output:

a*(e*cot(d*x+c))^(1/2)*InverseJacobiAM(c-1/4*Pi+d*x,2^(1/2))*sec(d*x+c)*si 
n(2*d*x+2*c)^(1/2)/d+1/2*a*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*(e*cot(d*x+ 
c))^(1/2)*tan(d*x+c)^(1/2)*2^(1/2)/d+1/2*a*arctan(1+2^(1/2)*tan(d*x+c)^(1/ 
2))*(e*cot(d*x+c))^(1/2)*tan(d*x+c)^(1/2)*2^(1/2)/d+1/2*a*arctanh(2^(1/2)* 
tan(d*x+c)^(1/2)/(1+tan(d*x+c)))*(e*cot(d*x+c))^(1/2)*tan(d*x+c)^(1/2)*2^( 
1/2)/d
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 1.67 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.80 \[ \int \sqrt {e \cot (c+d x)} (a+a \sec (c+d x)) \, dx=\frac {a (1+\cos (c+d x)) \sqrt {e \cot (c+d x)} \sec ^2\left (\frac {1}{2} (c+d x)\right ) \sec (c+d x) \left (4 \sqrt [4]{-1} \sqrt {\cot (c+d x)} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt [4]{-1} \sqrt {\cot (c+d x)}\right ),-1\right )+\sqrt {\csc ^2(c+d x)} \left (-\arcsin (\cos (c+d x)-\sin (c+d x))+\log \left (\cos (c+d x)+\sin (c+d x)+\sqrt {\sin (2 (c+d x))}\right )\right ) \sqrt {\sin (2 (c+d x))}\right )}{4 d \sqrt {\csc ^2(c+d x)}} \] Input:

Integrate[Sqrt[e*Cot[c + d*x]]*(a + a*Sec[c + d*x]),x]
 

Output:

(a*(1 + Cos[c + d*x])*Sqrt[e*Cot[c + d*x]]*Sec[(c + d*x)/2]^2*Sec[c + d*x] 
*(4*(-1)^(1/4)*Sqrt[Cot[c + d*x]]*EllipticF[I*ArcSinh[(-1)^(1/4)*Sqrt[Cot[ 
c + d*x]]], -1] + Sqrt[Csc[c + d*x]^2]*(-ArcSin[Cos[c + d*x] - Sin[c + d*x 
]] + Log[Cos[c + d*x] + Sin[c + d*x] + Sqrt[Sin[2*(c + d*x)]]])*Sqrt[Sin[2 
*(c + d*x)]]))/(4*d*Sqrt[Csc[c + d*x]^2])
 

Rubi [A] (verified)

Time = 0.79 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.98, number of steps used = 21, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.870, Rules used = {3042, 4388, 3042, 4372, 3042, 3094, 3042, 3053, 3042, 3120, 3957, 266, 755, 1476, 1082, 217, 1479, 25, 27, 1103}

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.

\(\displaystyle \int (a \sec (c+d x)+a) \sqrt {e \cot (c+d x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int (a \sec (c+d x)+a) \sqrt {e \cot (c+d x)}dx\)

\(\Big \downarrow \) 4388

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

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {e \cot (c+d x)} \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) a+a}{\sqrt {-\cot \left (c+d x+\frac {\pi }{2}\right )}}dx\)

\(\Big \downarrow \) 4372

\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {e \cot (c+d x)} \left (a \int \frac {1}{\sqrt {\tan (c+d x)}}dx+a \int \frac {\sec (c+d x)}{\sqrt {\tan (c+d x)}}dx\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {e \cot (c+d x)} \left (a \int \frac {1}{\sqrt {\tan (c+d x)}}dx+a \int \frac {\sec (c+d x)}{\sqrt {\tan (c+d x)}}dx\right )\)

\(\Big \downarrow \) 3094

\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {e \cot (c+d x)} \left (a \int \frac {1}{\sqrt {\tan (c+d x)}}dx+\frac {a \sqrt {\sin (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {\sin (c+d x)}}dx}{\sqrt {\cos (c+d x)} \sqrt {\tan (c+d x)}}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {e \cot (c+d x)} \left (a \int \frac {1}{\sqrt {\tan (c+d x)}}dx+\frac {a \sqrt {\sin (c+d x)} \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {\sin (c+d x)}}dx}{\sqrt {\cos (c+d x)} \sqrt {\tan (c+d x)}}\right )\)

\(\Big \downarrow \) 3053

\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {e \cot (c+d x)} \left (a \int \frac {1}{\sqrt {\tan (c+d x)}}dx+\frac {a \sqrt {\sin (2 c+2 d x)} \sec (c+d x) \int \frac {1}{\sqrt {\sin (2 c+2 d x)}}dx}{\sqrt {\tan (c+d x)}}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {e \cot (c+d x)} \left (a \int \frac {1}{\sqrt {\tan (c+d x)}}dx+\frac {a \sqrt {\sin (2 c+2 d x)} \sec (c+d x) \int \frac {1}{\sqrt {\sin (2 c+2 d x)}}dx}{\sqrt {\tan (c+d x)}}\right )\)

\(\Big \downarrow \) 3120

\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {e \cot (c+d x)} \left (a \int \frac {1}{\sqrt {\tan (c+d x)}}dx+\frac {a \sqrt {\sin (2 c+2 d x)} \sec (c+d x) \operatorname {EllipticF}\left (c+d x-\frac {\pi }{4},2\right )}{d \sqrt {\tan (c+d x)}}\right )\)

\(\Big \downarrow \) 3957

\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {e \cot (c+d x)} \left (\frac {a \int \frac {1}{\sqrt {\tan (c+d x)} \left (\tan ^2(c+d x)+1\right )}d\tan (c+d x)}{d}+\frac {a \sqrt {\sin (2 c+2 d x)} \sec (c+d x) \operatorname {EllipticF}\left (c+d x-\frac {\pi }{4},2\right )}{d \sqrt {\tan (c+d x)}}\right )\)

\(\Big \downarrow \) 266

\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {e \cot (c+d x)} \left (\frac {2 a \int \frac {1}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}}{d}+\frac {a \sqrt {\sin (2 c+2 d x)} \sec (c+d x) \operatorname {EllipticF}\left (c+d x-\frac {\pi }{4},2\right )}{d \sqrt {\tan (c+d x)}}\right )\)

\(\Big \downarrow \) 755

\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {e \cot (c+d x)} \left (\frac {2 a \left (\frac {1}{2} \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}+\frac {1}{2} \int \frac {\tan (c+d x)+1}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}\right )}{d}+\frac {a \sqrt {\sin (2 c+2 d x)} \sec (c+d x) \operatorname {EllipticF}\left (c+d x-\frac {\pi }{4},2\right )}{d \sqrt {\tan (c+d x)}}\right )\)

\(\Big \downarrow \) 1476

\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {e \cot (c+d x)} \left (\frac {2 a \left (\frac {1}{2} \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}+\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}+\frac {1}{2} \int \frac {1}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}\right )\right )}{d}+\frac {a \sqrt {\sin (2 c+2 d x)} \sec (c+d x) \operatorname {EllipticF}\left (c+d x-\frac {\pi }{4},2\right )}{d \sqrt {\tan (c+d x)}}\right )\)

\(\Big \downarrow \) 1082

\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {e \cot (c+d x)} \left (\frac {2 a \left (\frac {1}{2} \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}+\frac {1}{2} \left (\frac {\int \frac {1}{-\tan (c+d x)-1}d\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\tan (c+d x)-1}d\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}\right )\right )}{d}+\frac {a \sqrt {\sin (2 c+2 d x)} \sec (c+d x) \operatorname {EllipticF}\left (c+d x-\frac {\pi }{4},2\right )}{d \sqrt {\tan (c+d x)}}\right )\)

\(\Big \downarrow \) 217

\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {e \cot (c+d x)} \left (\frac {2 a \left (\frac {1}{2} \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {a \sqrt {\sin (2 c+2 d x)} \sec (c+d x) \operatorname {EllipticF}\left (c+d x-\frac {\pi }{4},2\right )}{d \sqrt {\tan (c+d x)}}\right )\)

\(\Big \downarrow \) 1479

\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {e \cot (c+d x)} \left (\frac {2 a \left (\frac {1}{2} \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt {\tan (c+d x)}}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {a \sqrt {\sin (2 c+2 d x)} \sec (c+d x) \operatorname {EllipticF}\left (c+d x-\frac {\pi }{4},2\right )}{d \sqrt {\tan (c+d x)}}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {e \cot (c+d x)} \left (\frac {2 a \left (\frac {1}{2} \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\tan (c+d x)}}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {a \sqrt {\sin (2 c+2 d x)} \sec (c+d x) \operatorname {EllipticF}\left (c+d x-\frac {\pi }{4},2\right )}{d \sqrt {\tan (c+d x)}}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {e \cot (c+d x)} \left (\frac {2 a \left (\frac {1}{2} \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\tan (c+d x)}}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\tan (c+d x)}+1}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {a \sqrt {\sin (2 c+2 d x)} \sec (c+d x) \operatorname {EllipticF}\left (c+d x-\frac {\pi }{4},2\right )}{d \sqrt {\tan (c+d x)}}\right )\)

\(\Big \downarrow \) 1103

\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {e \cot (c+d x)} \left (\frac {2 a \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d}+\frac {a \sqrt {\sin (2 c+2 d x)} \sec (c+d x) \operatorname {EllipticF}\left (c+d x-\frac {\pi }{4},2\right )}{d \sqrt {\tan (c+d x)}}\right )\)

Input:

Int[Sqrt[e*Cot[c + d*x]]*(a + a*Sec[c + d*x]),x]
 

Output:

Sqrt[e*Cot[c + d*x]]*((2*a*((-(ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]]/Sqrt 
[2]) + ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]/Sqrt[2])/2 + (-1/2*Log[1 - S 
qrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]/Sqrt[2] + Log[1 + Sqrt[2]*Sqrt[T 
an[c + d*x]] + Tan[c + d*x]]/(2*Sqrt[2]))/2))/d + (a*EllipticF[c - Pi/4 + 
d*x, 2]*Sec[c + d*x]*Sqrt[Sin[2*c + 2*d*x]])/(d*Sqrt[Tan[c + d*x]]))*Sqrt[ 
Tan[c + d*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 217

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

rule 266

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De 
nominator[m]}, Simp[k/c   Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) 
^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I 
ntBinomialQ[a, b, c, 2, m, p, x]
 

rule 755

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] 
], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r)   Int[(r - s*x^2)/(a + b*x^4) 
, x], x] + Simp[1/(2*r)   Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{a, 
 b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] & 
& AtomQ[SplitProduct[SumBaseQ, b]]))
 

rule 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b 
)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /; Fre 
eQ[{a, b, c}, x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1476

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
2*(d/e), 2]}, Simp[e/(2*c)   Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ 
e/(2*c)   Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] 
 && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
 

rule 1479

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
-2*(d/e), 2]}, Simp[e/(2*c*q)   Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], 
 x] + Simp[e/(2*c*q)   Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F 
reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
 

rule 3042

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

rule 3053

Int[1/(Sqrt[cos[(e_.) + (f_.)*(x_)]*(b_.)]*Sqrt[(a_.)*sin[(e_.) + (f_.)*(x_ 
)]]), x_Symbol] :> Simp[Sqrt[Sin[2*e + 2*f*x]]/(Sqrt[a*Sin[e + f*x]]*Sqrt[b 
*Cos[e + f*x]])   Int[1/Sqrt[Sin[2*e + 2*f*x]], x], x] /; FreeQ[{a, b, e, f 
}, x]
 

rule 3094

Int[sec[(e_.) + (f_.)*(x_)]/Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] 
:> Simp[Sqrt[Sin[e + f*x]]/(Sqrt[Cos[e + f*x]]*Sqrt[b*Tan[e + f*x]])   Int[ 
1/(Sqrt[Cos[e + f*x]]*Sqrt[Sin[e + f*x]]), x], x] /; FreeQ[{b, e, f}, x]
 

rule 3120

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2 
)*(c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
 

rule 3957

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b/d   Subst[Int 
[x^n/(b^2 + x^2), x], x, b*Tan[c + d*x]], x] /; FreeQ[{b, c, d, n}, x] && 
!IntegerQ[n]
 

rule 4372

Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + 
(a_)), x_Symbol] :> Simp[a   Int[(e*Cot[c + d*x])^m, x], x] + Simp[b   Int[ 
(e*Cot[c + d*x])^m*Csc[c + d*x], x], x] /; FreeQ[{a, b, c, d, e, m}, x]
 

rule 4388

Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*((a_) + (b_.)*sec[(c_.) + (d_.)*(x 
_)])^(n_.), x_Symbol] :> Simp[(e*Cot[c + d*x])^m*Tan[c + d*x]^m   Int[(a + 
b*Sec[c + d*x])^n/Tan[c + d*x]^m, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] 
 &&  !IntegerQ[m]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 1.94 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.66

method	result	size
default	\(\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) a \sqrt {e \cot \left (d x +c \right )}\, \sqrt {-\cot \left (d x +c \right )+\csc \left (d x +c \right )+1}\, \sqrt {2 \cot \left (d x +c \right )-2 \csc \left (d x +c \right )+2}\, \sqrt {-\csc \left (d x +c \right )+\cot \left (d x +c \right )}\, \left (\operatorname {EllipticPi}\left (\sqrt {-\cot \left (d x +c \right )+\csc \left (d x +c \right )+1}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right )+i \operatorname {EllipticPi}\left (\sqrt {-\cot \left (d x +c \right )+\csc \left (d x +c \right )+1}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right )\right ) \left (1+\sec \left (d x +c \right )\right )}{d}\)	\(140\)
parts	\(-\frac {a e \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 d \left (e^{2}\right )^{\frac {1}{4}}}+\frac {a \sqrt {e \cot \left (d x +c \right )}\, \sqrt {-\cot \left (d x +c \right )+\csc \left (d x +c \right )+1}\, \sqrt {2 \cot \left (d x +c \right )-2 \csc \left (d x +c \right )+2}\, \sqrt {-\csc \left (d x +c \right )+\cot \left (d x +c \right )}\, \operatorname {EllipticF}\left (\sqrt {-\cot \left (d x +c \right )+\csc \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right ) \left (1+\sec \left (d x +c \right )\right )}{d}\)	\(240\)

Input:

int((e*cot(d*x+c))^(1/2)*(a+a*sec(d*x+c)),x,method=_RETURNVERBOSE)
 

Output:

(1/2-1/2*I)*a/d*(e*cot(d*x+c))^(1/2)*(-cot(d*x+c)+csc(d*x+c)+1)^(1/2)*(2*c 
ot(d*x+c)-2*csc(d*x+c)+2)^(1/2)*(-csc(d*x+c)+cot(d*x+c))^(1/2)*(EllipticPi 
((-cot(d*x+c)+csc(d*x+c)+1)^(1/2),1/2+1/2*I,1/2*2^(1/2))+I*EllipticPi((-co 
t(d*x+c)+csc(d*x+c)+1)^(1/2),1/2-1/2*I,1/2*2^(1/2)))*(1+sec(d*x+c))
 

Fricas [F(-1)]

Timed out. \[ \int \sqrt {e \cot (c+d x)} (a+a \sec (c+d x)) \, dx=\text {Timed out} \] Input:

integrate((e*cot(d*x+c))^(1/2)*(a+a*sec(d*x+c)),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \sqrt {e \cot (c+d x)} (a+a \sec (c+d x)) \, dx=a \left (\int \sqrt {e \cot {\left (c + d x \right )}}\, dx + \int \sqrt {e \cot {\left (c + d x \right )}} \sec {\left (c + d x \right )}\, dx\right ) \] Input:

integrate((e*cot(d*x+c))**(1/2)*(a+a*sec(d*x+c)),x)
 

Output:

a*(Integral(sqrt(e*cot(c + d*x)), x) + Integral(sqrt(e*cot(c + d*x))*sec(c 
 + d*x), x))
 

Maxima [F(-2)]

Exception generated. \[ \int \sqrt {e \cot (c+d x)} (a+a \sec (c+d x)) \, dx=\text {Exception raised: ValueError} \] Input:

integrate((e*cot(d*x+c))^(1/2)*(a+a*sec(d*x+c)),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(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [F]

\[ \int \sqrt {e \cot (c+d x)} (a+a \sec (c+d x)) \, dx=\int { \sqrt {e \cot \left (d x + c\right )} {\left (a \sec \left (d x + c\right ) + a\right )} \,d x } \] Input:

integrate((e*cot(d*x+c))^(1/2)*(a+a*sec(d*x+c)),x, algorithm="giac")
 

Output:

integrate(sqrt(e*cot(d*x + c))*(a*sec(d*x + c) + a), x)
 

Mupad [F(-1)]

Timed out. \[ \int \sqrt {e \cot (c+d x)} (a+a \sec (c+d x)) \, dx=\int \sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right ) \,d x \] Input:

int((e*cot(c + d*x))^(1/2)*(a + a/cos(c + d*x)),x)
 

Output:

int((e*cot(c + d*x))^(1/2)*(a + a/cos(c + d*x)), x)
 

Reduce [F]

\[ \int \sqrt {e \cot (c+d x)} (a+a \sec (c+d x)) \, dx=\sqrt {e}\, a \left (\int \sqrt {\cot \left (d x +c \right )}d x +\int \sqrt {\cot \left (d x +c \right )}\, \sec \left (d x +c \right )d x \right ) \] Input:

int((e*cot(d*x+c))^(1/2)*(a+a*sec(d*x+c)),x)
 

Output:

sqrt(e)*a*(int(sqrt(cot(c + d*x)),x) + int(sqrt(cot(c + d*x))*sec(c + d*x) 
,x))