\(\int \frac {1}{\sqrt {e \cot (c+d x)} (a+a \sec (c+d x))} \, dx\) [245]

Optimal result

Integrand size = 25, antiderivative size = 347 \[ \int \frac {1}{\sqrt {e \cot (c+d x)} (a+a \sec (c+d x))} \, dx=\frac {2 \cot (c+d x) (1-\sec (c+d x))}{a d \sqrt {e \cot (c+d x)}}+\frac {2 \sin (c+d x)}{a d \sqrt {e \cot (c+d x)}}-\frac {2 \cos (c+d x) E\left (\left .c-\frac {\pi }{4}+d x\right |2\right )}{a d \sqrt {e \cot (c+d x)} \sqrt {\sin (2 c+2 d x)}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} a d \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}}+\frac {\arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} a d \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}}+\frac {\log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} a d \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}}-\frac {\log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} a d \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 347, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.640, Rules used = {3985, 3973, 3967, 3969, 3557, 335, 303, 1176, 631, 210, 1179, 642, 2693, 2695, 2652, 2719} \[ \int \frac {1}{\sqrt {e \cot (c+d x)} (a+a \sec (c+d x))} \, dx=-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} a d \sqrt {\tan (c+d x)} \sqrt {e \cot (c+d x)}}+\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} a d \sqrt {\tan (c+d x)} \sqrt {e \cot (c+d x)}}+\frac {2 \sin (c+d x)}{a d \sqrt {e \cot (c+d x)}}+\frac {2 \cot (c+d x) (1-\sec (c+d x))}{a d \sqrt {e \cot (c+d x)}}+\frac {\log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} a d \sqrt {\tan (c+d x)} \sqrt {e \cot (c+d x)}}-\frac {\log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} a d \sqrt {\tan (c+d x)} \sqrt {e \cot (c+d x)}}-\frac {2 \cos (c+d x) E\left (\left .c+d x-\frac {\pi }{4}\right |2\right )}{a d \sqrt {\sin (2 c+2 d x)} \sqrt {e \cot (c+d x)}} \]

[In]

Int[1/(Sqrt[e*Cot[c + d*x]]*(a + a*Sec[c + d*x])),x]

[Out]

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

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 303

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]},
Dist[1/(2*s), Int[(r + s*x^2)/(a + b*x^4), x], x] - Dist[1/(2*s), Int[(r - s*x^2)/(a + b*x^4), x], x]] /; Free
Q[{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 2652

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

Rule 2693

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[a^2*(a*Sec[e
+ f*x])^(m - 2)*((b*Tan[e + f*x])^(n + 1)/(b*f*(m + n - 1))), x] + Dist[a^2*((m - 2)/(m + n - 1)), Int[(a*Sec
[e + f*x])^(m - 2)*(b*Tan[e + f*x])^n, x], x] /; FreeQ[{a, b, e, f, n}, x] && (GtQ[m, 1] || (EqQ[m, 1] && EqQ[
n, 1/2])) && NeQ[m + n - 1, 0] && IntegersQ[2*m, 2*n]

Rule 2695

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

Rule 2719

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(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}& = \frac {\int \frac {\sqrt {\tan (c+d x)}}{a+a \sec (c+d x)} \, dx}{\sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}} \\ & = \frac {\int \frac {-a+a \sec (c+d x)}{\tan ^{\frac {3}{2}}(c+d x)} \, dx}{a^2 \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}} \\ & = \frac {2 \cot (c+d x) (1-\sec (c+d x))}{a d \sqrt {e \cot (c+d x)}}+\frac {2 \int \left (\frac {a}{2}+\frac {1}{2} a \sec (c+d x)\right ) \sqrt {\tan (c+d x)} \, dx}{a^2 \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}} \\ & = \frac {2 \cot (c+d x) (1-\sec (c+d x))}{a d \sqrt {e \cot (c+d x)}}+\frac {\int \sqrt {\tan (c+d x)} \, dx}{a \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}}+\frac {\int \sec (c+d x) \sqrt {\tan (c+d x)} \, dx}{a \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}} \\ & = \frac {2 \cot (c+d x) (1-\sec (c+d x))}{a d \sqrt {e \cot (c+d x)}}+\frac {2 \sin (c+d x)}{a d \sqrt {e \cot (c+d x)}}-\frac {2 \int \cos (c+d x) \sqrt {\tan (c+d x)} \, dx}{a \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}}+\frac {\text {Subst}\left (\int \frac {\sqrt {x}}{1+x^2} \, dx,x,\tan (c+d x)\right )}{a d \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}} \\ & = \frac {2 \cot (c+d x) (1-\sec (c+d x))}{a d \sqrt {e \cot (c+d x)}}+\frac {2 \sin (c+d x)}{a d \sqrt {e \cot (c+d x)}}-\frac {\left (2 \sqrt {\cos (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \sqrt {\sin (c+d x)} \, dx}{a \sqrt {e \cot (c+d x)} \sqrt {\sin (c+d x)}}+\frac {2 \text {Subst}\left (\int \frac {x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{a d \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}} \\ & = \frac {2 \cot (c+d x) (1-\sec (c+d x))}{a d \sqrt {e \cot (c+d x)}}+\frac {2 \sin (c+d x)}{a d \sqrt {e \cot (c+d x)}}-\frac {(2 \cos (c+d x)) \int \sqrt {\sin (2 c+2 d x)} \, dx}{a \sqrt {e \cot (c+d x)} \sqrt {\sin (2 c+2 d x)}}-\frac {\text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{a d \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}}+\frac {\text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{a d \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}} \\ & = \frac {2 \cot (c+d x) (1-\sec (c+d x))}{a d \sqrt {e \cot (c+d x)}}+\frac {2 \sin (c+d x)}{a d \sqrt {e \cot (c+d x)}}-\frac {2 \cos (c+d x) E\left (\left .c-\frac {\pi }{4}+d x\right |2\right )}{a d \sqrt {e \cot (c+d x)} \sqrt {\sin (2 c+2 d x)}}+\frac {\text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 a d \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}}+\frac {\text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 a d \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}}+\frac {\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 \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}}+\frac {\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 \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}} \\ & = \frac {2 \cot (c+d x) (1-\sec (c+d x))}{a d \sqrt {e \cot (c+d x)}}+\frac {2 \sin (c+d x)}{a d \sqrt {e \cot (c+d x)}}-\frac {2 \cos (c+d x) E\left (\left .c-\frac {\pi }{4}+d x\right |2\right )}{a d \sqrt {e \cot (c+d x)} \sqrt {\sin (2 c+2 d x)}}+\frac {\log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} a d \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}}-\frac {\log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} a d \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}}+\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} a d \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}}-\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} a d \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}} \\ & = \frac {2 \cot (c+d x) (1-\sec (c+d x))}{a d \sqrt {e \cot (c+d x)}}+\frac {2 \sin (c+d x)}{a d \sqrt {e \cot (c+d x)}}-\frac {2 \cos (c+d x) E\left (\left .c-\frac {\pi }{4}+d x\right |2\right )}{a d \sqrt {e \cot (c+d x)} \sqrt {\sin (2 c+2 d x)}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} a d \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}}+\frac {\arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} a d \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}}+\frac {\log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} a d \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}}-\frac {\log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} a d \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

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

Time = 13.94 (sec) , antiderivative size = 249, normalized size of antiderivative = 0.72 \[ \int \frac {1}{\sqrt {e \cot (c+d x)} (a+a \sec (c+d x))} \, dx=-\frac {\csc (c+d x) \left (24 \cot ^2(c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-\frac {1}{4},\frac {3}{4},-\tan ^2(c+d x)\right )+8 \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-\tan ^2(c+d x)\right )-3 \cot ^{\frac {3}{2}}(c+d x) \left (2 \sqrt {2} \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )-2 \sqrt {2} \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )+8 \sqrt {\cot (c+d x)}+\sqrt {2} \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )-\sqrt {2} \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )\right )\right ) \left (1+\sqrt {\sec ^2(c+d x)}\right ) \sin ^2\left (\frac {1}{2} (c+d x)\right )}{6 a d \sqrt {e \cot (c+d x)}} \]

[In]

Integrate[1/(Sqrt[e*Cot[c + d*x]]*(a + a*Sec[c + d*x])),x]

[Out]

-1/6*(Csc[c + d*x]*(24*Cot[c + d*x]^2*Hypergeometric2F1[-1/2, -1/4, 3/4, -Tan[c + d*x]^2] + 8*Hypergeometric2F
1[1/2, 3/4, 7/4, -Tan[c + d*x]^2] - 3*Cot[c + d*x]^(3/2)*(2*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]] - 2
*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]] + 8*Sqrt[Cot[c + d*x]] + Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Cot[c +
d*x]] + Cot[c + d*x]] - Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]))*(1 + Sqrt[Sec[c + d*x]^2]
)*Sin[(c + d*x)/2]^2)/(a*d*Sqrt[e*Cot[c + d*x]])

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 8.32 (sec) , antiderivative size = 252, normalized size of antiderivative = 0.73

method	result	size
default	\(\frac {\left (-\frac {1}{2}-\frac {i}{2}\right ) \sqrt {2}\, \left (\operatorname {EllipticPi}\left (\sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right )-i \operatorname {EllipticPi}\left (\sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right )+2 i \operatorname {EllipticE}\left (\sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right )-i \operatorname {EllipticF}\left (\sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right )-2 \operatorname {EllipticE}\left (\sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right )+\operatorname {EllipticF}\left (\sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right )\right ) \sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}\, \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )+1}\, \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \left (\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{a d \sqrt {e \cot \left (d x +c \right )}}\)	\(252\)

[In]

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

[Out]

(-1/2-1/2*I)/a/d*2^(1/2)*(EllipticPi((csc(d*x+c)-cot(d*x+c)+1)^(1/2),1/2+1/2*I,1/2*2^(1/2))-I*EllipticPi((csc(
d*x+c)-cot(d*x+c)+1)^(1/2),1/2-1/2*I,1/2*2^(1/2))+2*I*EllipticE((csc(d*x+c)-cot(d*x+c)+1)^(1/2),1/2*2^(1/2))-I
*EllipticF((csc(d*x+c)-cot(d*x+c)+1)^(1/2),1/2*2^(1/2))-2*EllipticE((csc(d*x+c)-cot(d*x+c)+1)^(1/2),1/2*2^(1/2
))+EllipticF((csc(d*x+c)-cot(d*x+c)+1)^(1/2),1/2*2^(1/2)))*(csc(d*x+c)-cot(d*x+c)+1)^(1/2)*(cot(d*x+c)-csc(d*x
+c)+1)^(1/2)*(cot(d*x+c)-csc(d*x+c))^(1/2)/(e*cot(d*x+c))^(1/2)*(cot(d*x+c)+csc(d*x+c))

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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