\(\int \frac {1}{\sqrt {e \cot (c+d x)} (a+a \sec (c+d x))^2} \, dx\) [250]

Optimal result

Integrand size = 25, antiderivative size = 413 \[ \int \frac {1}{\sqrt {e \cot (c+d x)} (a+a \sec (c+d x))^2} \, dx=\frac {2 \cot (c+d x)}{a^2 d \sqrt {e \cot (c+d x)}}-\frac {12 \cos (c+d x) \cot (c+d x)}{5 a^2 d \sqrt {e \cot (c+d x)}}-\frac {4 \cot ^3(c+d x)}{5 a^2 d \sqrt {e \cot (c+d x)}}+\frac {4 \cot ^2(c+d x) \csc (c+d x)}{5 a^2 d \sqrt {e \cot (c+d x)}}-\frac {12 \cos (c+d x) E\left (\left .c-\frac {\pi }{4}+d x\right |2\right )}{5 a^2 d \sqrt {e \cot (c+d x)} \sqrt {\sin (2 c+2 d x)}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} a^2 d \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}}+\frac {\arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} a^2 d \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}}+\frac {\log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} a^2 d \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}}-\frac {\log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} a^2 d \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}} \]

[Out]

2*cot(d*x+c)/a^2/d/(e*cot(d*x+c))^(1/2)-12/5*cos(d*x+c)*cot(d*x+c)/a^2/d/(e*cot(d*x+c))^(1/2)-4/5*cot(d*x+c)^3
/a^2/d/(e*cot(d*x+c))^(1/2)+4/5*cot(d*x+c)^2*csc(d*x+c)/a^2/d/(e*cot(d*x+c))^(1/2)+12/5*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^2/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^2/d*2^(1/2)/(e*cot(d*x+c))^(1/2)/tan(d*x+c)^(1/2)+1/2*arct
an(1+2^(1/2)*tan(d*x+c)^(1/2))/a^2/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^2/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^2/d*2^(1/2)/(e*cot(d*x+c))^(1/2)/tan(d*x+c)^(1/2)

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 413, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.760, Rules used = {3985, 3973, 3971, 3555, 3557, 335, 303, 1176, 631, 210, 1179, 642, 2689, 2688, 2695, 2652, 2719, 2687, 30} \[ \int \frac {1}{\sqrt {e \cot (c+d x)} (a+a \sec (c+d x))^2} \, dx=-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} a^2 d \sqrt {\tan (c+d x)} \sqrt {e \cot (c+d x)}}+\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} a^2 d \sqrt {\tan (c+d x)} \sqrt {e \cot (c+d x)}}-\frac {4 \cot ^3(c+d x)}{5 a^2 d \sqrt {e \cot (c+d x)}}+\frac {2 \cot (c+d x)}{a^2 d \sqrt {e \cot (c+d x)}}-\frac {12 \cos (c+d x) \cot (c+d x)}{5 a^2 d \sqrt {e \cot (c+d x)}}+\frac {4 \cot ^2(c+d x) \csc (c+d x)}{5 a^2 d \sqrt {e \cot (c+d x)}}+\frac {\log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} a^2 d \sqrt {\tan (c+d x)} \sqrt {e \cot (c+d x)}}-\frac {\log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} a^2 d \sqrt {\tan (c+d x)} \sqrt {e \cot (c+d x)}}-\frac {12 \cos (c+d x) E\left (\left .c+d x-\frac {\pi }{4}\right |2\right )}{5 a^2 d \sqrt {\sin (2 c+2 d x)} \sqrt {e \cot (c+d x)}} \]

[In]

Int[1/(Sqrt[e*Cot[c + d*x]]*(a + a*Sec[c + d*x])^2),x]

[Out]

(2*Cot[c + d*x])/(a^2*d*Sqrt[e*Cot[c + d*x]]) - (12*Cos[c + d*x]*Cot[c + d*x])/(5*a^2*d*Sqrt[e*Cot[c + d*x]])
- (4*Cot[c + d*x]^3)/(5*a^2*d*Sqrt[e*Cot[c + d*x]]) + (4*Cot[c + d*x]^2*Csc[c + d*x])/(5*a^2*d*Sqrt[e*Cot[c +
d*x]]) - (12*Cos[c + d*x]*EllipticE[c - Pi/4 + d*x, 2])/(5*a^2*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^2*d*Sqrt[e*Cot[c + d*x]]*Sqrt[Tan[c + d*x]]) + ArcTan[1 +
Sqrt[2]*Sqrt[Tan[c + d*x]]]/(Sqrt[2]*a^2*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^2*d*Sqrt[e*Cot[c + d*x]]*Sqrt[Tan[c + d*x]]) - Log[1 + Sqrt[2]*Sqrt[Ta
n[c + d*x]] + Tan[c + d*x]]/(2*Sqrt[2]*a^2*d*Sqrt[e*Cot[c + d*x]]*Sqrt[Tan[c + d*x]])

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

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 2687

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(b*x)
^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] &&  !(IntegerQ[(n
- 1)/2] && LtQ[0, n, m - 1])

Rule 2688

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*(n + 1))), x] - Dist[a^2*((m - 2)/(b^2*(n + 1))), Int[(a*Sec[e
+ f*x])^(m - 2)*(b*Tan[e + f*x])^(n + 2), x], x] /; FreeQ[{a, b, e, f}, x] && LtQ[n, -1] && (GtQ[m, 1] || (Eq
Q[m, 1] && EqQ[n, -3/2])) && IntegersQ[2*m, 2*n]

Rule 2689

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(a*Sec[e + f
*x])^m*((b*Tan[e + f*x])^(n + 1)/(b*f*(n + 1))), x] - Dist[(m + n + 1)/(b^2*(n + 1)), Int[(a*Sec[e + f*x])^m*(
b*Tan[e + f*x])^(n + 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && LtQ[n, -1] && 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 3555

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*Tan[c + d*x])^(n + 1)/(b*d*(n + 1)), x] - Dist[
1/b^2, Int[(b*Tan[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1]

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 3971

Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Int[ExpandI
ntegrand[(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]

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))^2} \, dx}{\sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}} \\ & = \frac {\int \frac {(-a+a \sec (c+d x))^2}{\tan ^{\frac {7}{2}}(c+d x)} \, dx}{a^4 \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}} \\ & = \frac {\int \left (\frac {a^2}{\tan ^{\frac {7}{2}}(c+d x)}-\frac {2 a^2 \sec (c+d x)}{\tan ^{\frac {7}{2}}(c+d x)}+\frac {a^2 \sec ^2(c+d x)}{\tan ^{\frac {7}{2}}(c+d x)}\right ) \, dx}{a^4 \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}} \\ & = \frac {\int \frac {1}{\tan ^{\frac {7}{2}}(c+d x)} \, dx}{a^2 \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}}+\frac {\int \frac {\sec ^2(c+d x)}{\tan ^{\frac {7}{2}}(c+d x)} \, dx}{a^2 \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}}-\frac {2 \int \frac {\sec (c+d x)}{\tan ^{\frac {7}{2}}(c+d x)} \, dx}{a^2 \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}} \\ & = -\frac {2 \cot ^3(c+d x)}{5 a^2 d \sqrt {e \cot (c+d x)}}+\frac {4 \cot ^2(c+d x) \csc (c+d x)}{5 a^2 d \sqrt {e \cot (c+d x)}}-\frac {\int \frac {1}{\tan ^{\frac {3}{2}}(c+d x)} \, dx}{a^2 \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}}+\frac {6 \int \frac {\sec (c+d x)}{\tan ^{\frac {3}{2}}(c+d x)} \, dx}{5 a^2 \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}}+\frac {\text {Subst}\left (\int \frac {1}{x^{7/2}} \, dx,x,\tan (c+d x)\right )}{a^2 d \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}} \\ & = \frac {2 \cot (c+d x)}{a^2 d \sqrt {e \cot (c+d x)}}-\frac {12 \cos (c+d x) \cot (c+d x)}{5 a^2 d \sqrt {e \cot (c+d x)}}-\frac {4 \cot ^3(c+d x)}{5 a^2 d \sqrt {e \cot (c+d x)}}+\frac {4 \cot ^2(c+d x) \csc (c+d x)}{5 a^2 d \sqrt {e \cot (c+d x)}}+\frac {\int \sqrt {\tan (c+d x)} \, dx}{a^2 \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}}-\frac {12 \int \cos (c+d x) \sqrt {\tan (c+d x)} \, dx}{5 a^2 \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}} \\ & = \frac {2 \cot (c+d x)}{a^2 d \sqrt {e \cot (c+d x)}}-\frac {12 \cos (c+d x) \cot (c+d x)}{5 a^2 d \sqrt {e \cot (c+d x)}}-\frac {4 \cot ^3(c+d x)}{5 a^2 d \sqrt {e \cot (c+d x)}}+\frac {4 \cot ^2(c+d x) \csc (c+d x)}{5 a^2 d \sqrt {e \cot (c+d x)}}-\frac {\left (12 \sqrt {\cos (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \sqrt {\sin (c+d x)} \, dx}{5 a^2 \sqrt {e \cot (c+d x)} \sqrt {\sin (c+d x)}}+\frac {\text {Subst}\left (\int \frac {\sqrt {x}}{1+x^2} \, dx,x,\tan (c+d x)\right )}{a^2 d \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}} \\ & = \frac {2 \cot (c+d x)}{a^2 d \sqrt {e \cot (c+d x)}}-\frac {12 \cos (c+d x) \cot (c+d x)}{5 a^2 d \sqrt {e \cot (c+d x)}}-\frac {4 \cot ^3(c+d x)}{5 a^2 d \sqrt {e \cot (c+d x)}}+\frac {4 \cot ^2(c+d x) \csc (c+d x)}{5 a^2 d \sqrt {e \cot (c+d x)}}-\frac {(12 \cos (c+d x)) \int \sqrt {\sin (2 c+2 d x)} \, dx}{5 a^2 \sqrt {e \cot (c+d x)} \sqrt {\sin (2 c+2 d x)}}+\frac {2 \text {Subst}\left (\int \frac {x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{a^2 d \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}} \\ & = \frac {2 \cot (c+d x)}{a^2 d \sqrt {e \cot (c+d x)}}-\frac {12 \cos (c+d x) \cot (c+d x)}{5 a^2 d \sqrt {e \cot (c+d x)}}-\frac {4 \cot ^3(c+d x)}{5 a^2 d \sqrt {e \cot (c+d x)}}+\frac {4 \cot ^2(c+d x) \csc (c+d x)}{5 a^2 d \sqrt {e \cot (c+d x)}}-\frac {12 \cos (c+d x) E\left (\left .c-\frac {\pi }{4}+d x\right |2\right )}{5 a^2 d \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^2 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^2 d \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}} \\ & = \frac {2 \cot (c+d x)}{a^2 d \sqrt {e \cot (c+d x)}}-\frac {12 \cos (c+d x) \cot (c+d x)}{5 a^2 d \sqrt {e \cot (c+d x)}}-\frac {4 \cot ^3(c+d x)}{5 a^2 d \sqrt {e \cot (c+d x)}}+\frac {4 \cot ^2(c+d x) \csc (c+d x)}{5 a^2 d \sqrt {e \cot (c+d x)}}-\frac {12 \cos (c+d x) E\left (\left .c-\frac {\pi }{4}+d x\right |2\right )}{5 a^2 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^2 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^2 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^2 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^2 d \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}} \\ & = \frac {2 \cot (c+d x)}{a^2 d \sqrt {e \cot (c+d x)}}-\frac {12 \cos (c+d x) \cot (c+d x)}{5 a^2 d \sqrt {e \cot (c+d x)}}-\frac {4 \cot ^3(c+d x)}{5 a^2 d \sqrt {e \cot (c+d x)}}+\frac {4 \cot ^2(c+d x) \csc (c+d x)}{5 a^2 d \sqrt {e \cot (c+d x)}}-\frac {12 \cos (c+d x) E\left (\left .c-\frac {\pi }{4}+d x\right |2\right )}{5 a^2 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^2 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^2 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^2 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^2 d \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}} \\ & = \frac {2 \cot (c+d x)}{a^2 d \sqrt {e \cot (c+d x)}}-\frac {12 \cos (c+d x) \cot (c+d x)}{5 a^2 d \sqrt {e \cot (c+d x)}}-\frac {4 \cot ^3(c+d x)}{5 a^2 d \sqrt {e \cot (c+d x)}}+\frac {4 \cot ^2(c+d x) \csc (c+d x)}{5 a^2 d \sqrt {e \cot (c+d x)}}-\frac {12 \cos (c+d x) E\left (\left .c-\frac {\pi }{4}+d x\right |2\right )}{5 a^2 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^2 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^2 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^2 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^2 d \sqrt {e \cot (c+d x)} \sqrt {\tan (c+d x)}} \\ \end{align*}

Mathematica [F]

\[ \int \frac {1}{\sqrt {e \cot (c+d x)} (a+a \sec (c+d x))^2} \, dx=\int \frac {1}{\sqrt {e \cot (c+d x)} (a+a \sec (c+d x))^2} \, dx \]

[In]

Integrate[1/(Sqrt[e*Cot[c + d*x]]*(a + a*Sec[c + d*x])^2),x]

[Out]

Integrate[1/(Sqrt[e*Cot[c + d*x]]*(a + a*Sec[c + d*x])^2), x]

Maple [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 8.55 (sec) , antiderivative size = 705, normalized size of antiderivative = 1.71

method	result	size
default	\(\frac {\sqrt {2}\, \left (\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1\right ) \left (5 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 )-5 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 )+24 \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 {EllipticE}\left (\sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right )-12 \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 )-5 \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 )-5 \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 )^{4} \csc \left (d x +c \right )^{4}-2 \left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}\right )}{10 a^{2} d \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 )}}\, \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 )}}\)	\(705\)

[In]

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

[Out]

1/10/a^2/d*2^(1/2)*((1-cos(d*x+c))^2*csc(d*x+c)^2-1)*(5*I*(csc(d*x+c)-cot(d*x+c)+1)^(1/2)*(2-2*csc(d*x+c)+2*co
t(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)
)-5*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)*Ellipt
icPi((csc(d*x+c)-cot(d*x+c)+1)^(1/2),1/2+1/2*I,1/2*2^(1/2))+24*(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)*EllipticE((csc(d*x+c)-cot(d*x+c)+1)^(1/2),1/2*2^(1/2))-12*(
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((cs
c(d*x+c)-cot(d*x+c)+1)^(1/2),1/2*2^(1/2))-5*(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))-5*(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))^4*csc(d*x+c)^4-2*(1-cos(d*x+c))^2*csc(d*x+c)^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))*((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)

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {e \cot (c+d x)} (a+a \sec (c+d x))^2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Sympy [F]

\[ \int \frac {1}{\sqrt {e \cot (c+d x)} (a+a \sec (c+d x))^2} \, dx=\frac {\int \frac {1}{\sqrt {e \cot {\left (c + d x \right )}} \sec ^{2}{\left (c + d x \right )} + 2 \sqrt {e \cot {\left (c + d x \right )}} \sec {\left (c + d x \right )} + \sqrt {e \cot {\left (c + d x \right )}}}\, dx}{a^{2}} \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {1}{\sqrt {e \cot (c+d x)} (a+a \sec (c+d x))^2} \, dx=\int { \frac {1}{\sqrt {e \cot \left (d x + c\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{2}} \,d x } \]

[In]

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

Giac [F]

\[ \int \frac {1}{\sqrt {e \cot (c+d x)} (a+a \sec (c+d x))^2} \, dx=\int { \frac {1}{\sqrt {e \cot \left (d x + c\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{2}} \,d x } \]

[In]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {e \cot (c+d x)} (a+a \sec (c+d x))^2} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^2}{a^2\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,{\left (\cos \left (c+d\,x\right )+1\right )}^2} \,d x \]

[In]

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

[Out]

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