\(\int \frac {\sqrt {e \cot (c+d x)}}{a+a \sec (c+d x)} \, dx\) [244]

Optimal result

Integrand size = 25, antiderivative size = 325 \[ \int \frac {\sqrt {e \cot (c+d x)}}{a+a \sec (c+d x)} \, dx=\frac {2 \cot (c+d x) \sqrt {e \cot (c+d x)} (1-\sec (c+d x))}{3 a d}-\frac {\sqrt {e \cot (c+d x)} \operatorname {EllipticF}\left (c-\frac {\pi }{4}+d x,2\right ) \sec (c+d x) \sqrt {\sin (2 c+2 d x)}}{3 a d}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right ) \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}}{\sqrt {2} a d}+\frac {\arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right ) \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}}{\sqrt {2} a d}-\frac {\sqrt {e \cot (c+d x)} \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right ) \sqrt {\tan (c+d x)}}{2 \sqrt {2} a d}+\frac {\sqrt {e \cot (c+d x)} \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right ) \sqrt {\tan (c+d x)}}{2 \sqrt {2} a d} \]

[Out]

2/3*cot(d*x+c)*(1-sec(d*x+c))*(e*cot(d*x+c))^(1/2)/a/d+1/3*(sin(c+1/4*Pi+d*x)^2)^(1/2)/sin(c+1/4*Pi+d*x)*Ellip
ticF(cos(c+1/4*Pi+d*x),2^(1/2))*sec(d*x+c)*(e*cot(d*x+c))^(1/2)*sin(2*d*x+2*c)^(1/2)/a/d+1/2*arctan(-1+2^(1/2)
*tan(d*x+c)^(1/2))*(e*cot(d*x+c))^(1/2)*tan(d*x+c)^(1/2)/a/d*2^(1/2)+1/2*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*(e
*cot(d*x+c))^(1/2)*tan(d*x+c)^(1/2)/a/d*2^(1/2)-1/4*ln(1-2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))*(e*cot(d*x+c))^(
1/2)*tan(d*x+c)^(1/2)/a/d*2^(1/2)+1/4*ln(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))*(e*cot(d*x+c))^(1/2)*tan(d*x+c
)^(1/2)/a/d*2^(1/2)

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {3985, 3973, 3967, 3969, 3557, 335, 217, 1179, 642, 1176, 631, 210, 2694, 2653, 2720} \[ \int \frac {\sqrt {e \cot (c+d x)}}{a+a \sec (c+d x)} \, dx=-\frac {\sqrt {\tan (c+d x)} \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right ) \sqrt {e \cot (c+d x)}}{\sqrt {2} a d}+\frac {\sqrt {\tan (c+d x)} \arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right ) \sqrt {e \cot (c+d x)}}{\sqrt {2} a d}+\frac {2 \cot (c+d x) (1-\sec (c+d x)) \sqrt {e \cot (c+d x)}}{3 a d}-\frac {\sqrt {\tan (c+d x)} \sqrt {e \cot (c+d x)} \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} a d}+\frac {\sqrt {\tan (c+d x)} \sqrt {e \cot (c+d x)} \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} a d}-\frac {\sqrt {\sin (2 c+2 d x)} \sec (c+d x) \operatorname {EllipticF}\left (c+d x-\frac {\pi }{4},2\right ) \sqrt {e \cot (c+d x)}}{3 a d} \]

[In]

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

[Out]

(2*Cot[c + d*x]*Sqrt[e*Cot[c + d*x]]*(1 - Sec[c + d*x]))/(3*a*d) - (Sqrt[e*Cot[c + d*x]]*EllipticF[c - Pi/4 +
d*x, 2]*Sec[c + d*x]*Sqrt[Sin[2*c + 2*d*x]])/(3*a*d) - (ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]]*Sqrt[e*Cot[c +
d*x]]*Sqrt[Tan[c + d*x]])/(Sqrt[2]*a*d) + (ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]*Sqrt[e*Cot[c + d*x]]*Sqrt[Ta
n[c + d*x]])/(Sqrt[2]*a*d) - (Sqrt[e*Cot[c + d*x]]*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]*Sqrt[Tan
[c + d*x]])/(2*Sqrt[2]*a*d) + (Sqrt[e*Cot[c + d*x]]*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]*Sqrt[Ta
n[c + d*x]])/(2*Sqrt[2]*a*d)

Rule 210

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 217

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Di
st[1/(2*r), Int[(r - s*x^2)/(a + b*x^4), x], x] + Dist[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 335

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[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])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[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 1176

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[2*(d/e), 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[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 1179

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

Rule 2653

Int[1/(Sqrt[cos[(e_.) + (f_.)*(x_)]*(b_.)]*Sqrt[(a_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Dist[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 2694

Int[sec[(e_.) + (f_.)*(x_)]/Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[Sqrt[Sin[e + f*x]]/(Sqrt[Co
s[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 2720

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 3557

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[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 3967

Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[(-(e*Cot[c
+ d*x])^(m + 1))*((a + b*Csc[c + d*x])/(d*e*(m + 1))), x] - Dist[1/(e^2*(m + 1)), Int[(e*Cot[c + d*x])^(m + 2)
*(a*(m + 1) + b*(m + 2)*Csc[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e}, x] && LtQ[m, -1]

Rule 3969

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

Rule 3973

Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Dist[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] && E
qQ[a^2 - b^2, 0] && ILtQ[n, 0]

Rule 3985

Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*((a_) + (b_.)*sec[(c_.) + (d_.)*(x_)])^(n_.), x_Symbol] :> Dist[(e*Co
t[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]

Rubi steps \begin{align*} \text {integral}& = \left (\sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {1}{(a+a \sec (c+d x)) \sqrt {\tan (c+d x)}} \, dx \\ & = \frac {\left (\sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {-a+a \sec (c+d x)}{\tan ^{\frac {5}{2}}(c+d x)} \, dx}{a^2} \\ & = \frac {2 \cot (c+d x) \sqrt {e \cot (c+d x)} (1-\sec (c+d x))}{3 a d}+\frac {\left (2 \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {\frac {3 a}{2}-\frac {1}{2} a \sec (c+d x)}{\sqrt {\tan (c+d x)}} \, dx}{3 a^2} \\ & = \frac {2 \cot (c+d x) \sqrt {e \cot (c+d x)} (1-\sec (c+d x))}{3 a d}-\frac {\left (\sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {\sec (c+d x)}{\sqrt {\tan (c+d x)}} \, dx}{3 a}+\frac {\left (\sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \frac {1}{\sqrt {\tan (c+d x)}} \, dx}{a} \\ & = \frac {2 \cot (c+d x) \sqrt {e \cot (c+d x)} (1-\sec (c+d x))}{3 a d}-\frac {\left (\sqrt {e \cot (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {\sin (c+d x)}} \, dx}{3 a \sqrt {\cos (c+d x)}}+\frac {\left (\sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {x} \left (1+x^2\right )} \, dx,x,\tan (c+d x)\right )}{a d} \\ & = \frac {2 \cot (c+d x) \sqrt {e \cot (c+d x)} (1-\sec (c+d x))}{3 a d}-\frac {\left (\sqrt {e \cot (c+d x)} \sec (c+d x) \sqrt {\sin (2 c+2 d x)}\right ) \int \frac {1}{\sqrt {\sin (2 c+2 d x)}} \, dx}{3 a}+\frac {\left (2 \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{a d} \\ & = \frac {2 \cot (c+d x) \sqrt {e \cot (c+d x)} (1-\sec (c+d x))}{3 a d}-\frac {\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)}}{3 a d}+\frac {\left (\sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{a d}+\frac {\left (\sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{a d} \\ & = \frac {2 \cot (c+d x) \sqrt {e \cot (c+d x)} (1-\sec (c+d x))}{3 a d}-\frac {\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)}}{3 a d}+\frac {\left (\sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 a d}+\frac {\left (\sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 a d}-\frac {\left (\sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 \sqrt {2} a d}-\frac {\left (\sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 \sqrt {2} a d} \\ & = \frac {2 \cot (c+d x) \sqrt {e \cot (c+d x)} (1-\sec (c+d x))}{3 a d}-\frac {\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)}}{3 a d}-\frac {\sqrt {e \cot (c+d x)} \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right ) \sqrt {\tan (c+d x)}}{2 \sqrt {2} a d}+\frac {\sqrt {e \cot (c+d x)} \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right ) \sqrt {\tan (c+d x)}}{2 \sqrt {2} a d}+\frac {\left (\sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} a d}-\frac {\left (\sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} a d} \\ & = \frac {2 \cot (c+d x) \sqrt {e \cot (c+d x)} (1-\sec (c+d x))}{3 a d}-\frac {\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)}}{3 a d}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right ) \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}}{\sqrt {2} a d}+\frac {\arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right ) \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}}{\sqrt {2} a d}-\frac {\sqrt {e \cot (c+d x)} \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right ) \sqrt {\tan (c+d x)}}{2 \sqrt {2} a d}+\frac {\sqrt {e \cot (c+d x)} \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right ) \sqrt {\tan (c+d x)}}{2 \sqrt {2} a d} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 12.31 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.42 \[ \int \frac {\sqrt {e \cot (c+d x)}}{a+a \sec (c+d x)} \, dx=-\frac {4 \sqrt {e \cot (c+d x)} \csc (c+d x) \left (\cot ^2(c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},-\frac {1}{2},\frac {1}{4},-\tan ^2(c+d x)\right )+3 \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\tan ^2(c+d x)\right )+\cot ^2(c+d x) \left (-1+\operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {7}{4},-\cot ^2(c+d x)\right )\right )\right ) \left (1+\sqrt {\sec ^2(c+d x)}\right ) \sin ^2\left (\frac {1}{2} (c+d x)\right )}{3 a d} \]

[In]

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

[Out]

(-4*Sqrt[e*Cot[c + d*x]]*Csc[c + d*x]*(Cot[c + d*x]^2*Hypergeometric2F1[-3/4, -1/2, 1/4, -Tan[c + d*x]^2] + 3*
Hypergeometric2F1[1/4, 1/2, 5/4, -Tan[c + d*x]^2] + Cot[c + d*x]^2*(-1 + Hypergeometric2F1[3/4, 1, 7/4, -Cot[c
+ d*x]^2]))*(1 + Sqrt[Sec[c + d*x]^2])*Sin[(c + d*x)/2]^2)/(3*a*d)

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 8.16 (sec) , antiderivative size = 611, normalized size of antiderivative = 1.88

method	result	size
default	\(-\frac {\sqrt {2}\, \sqrt {-\frac {e \left (\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )-\sin \left (d x +c \right )\right )}{1-\cos \left (d x +c \right )}}\, \left (1-\cos \left (d x +c \right )\right ) \left (3 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 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 )-8 \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 )+3 \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 \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 )+2 \left (1-\cos \left (d x +c \right )\right )^{3} \csc \left (d x +c \right )^{3}-2 \csc \left (d x +c \right )+2 \cot \left (d x +c \right )\right ) \csc \left (d x +c \right )}{6 a d \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 {\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 )}}\)	\(611\)

[In]

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

[Out]

-1/6/a/d*2^(1/2)*(-e/(1-cos(d*x+c))*((1-cos(d*x+c))^2*csc(d*x+c)-sin(d*x+c)))^(1/2)*(1-cos(d*x+c))*(3*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)-cot(d*x+c)+1)^(1/2),1/2-1/2*I,1/2*2^(1/2))-3*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)-cot(d*x+c)+1)^(1/2),1/2+1/2*I,1/2*2^(1/2))-8*(c
sc(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))+3*(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))+3*(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))+2*(1-cos(d*x+c))^3*csc(d*x+c)^3-2*csc(d*x+c)+2*cot(d*x+c))/((1-cos(
d*x+c))*((1-cos(d*x+c))^2*csc(d*x+c)^2-1)*csc(d*x+c))^(1/2)/((1-cos(d*x+c))^3*csc(d*x+c)^3+cot(d*x+c)-csc(d*x+
c))^(1/2)*csc(d*x+c)

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F]

\[ \int \frac {\sqrt {e \cot (c+d x)}}{a+a \sec (c+d x)} \, dx=\frac {\int \frac {\sqrt {e \cot {\left (c + d x \right )}}}{\sec {\left (c + d x \right )} + 1}\, dx}{a} \]

[In]

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

[Out]

Integral(sqrt(e*cot(c + d*x))/(sec(c + d*x) + 1), x)/a

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {e \cot (c+d x)}}{a+a \sec (c+d x)} \, dx=\int \frac {\cos \left (c+d\,x\right )\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{a\,\left (\cos \left (c+d\,x\right )+1\right )} \,d x \]

[In]

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

[Out]

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