\(\int \frac {(e \cot (c+d x))^{3/2}}{a+a \sec (c+d x)} \, dx\) [243]

Optimal result

Integrand size = 25, antiderivative size = 405 \[ \int \frac {(e \cot (c+d x))^{3/2}}{a+a \sec (c+d x)} \, dx=\frac {2 \cot (c+d x) (e \cot (c+d x))^{3/2} (1-\sec (c+d x))}{5 a d}-\frac {2 (e \cot (c+d x))^{3/2} (5-3 \sec (c+d x)) \tan (c+d x)}{5 a d}+\frac {6 (e \cot (c+d x))^{3/2} E\left (\left .c-\frac {\pi }{4}+d x\right |2\right ) \sin (c+d x) \tan (c+d x)}{5 a d \sqrt {\sin (2 c+2 d x)}}+\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right ) (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)}{\sqrt {2} a d}-\frac {\arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right ) (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)}{\sqrt {2} a d}-\frac {(e \cot (c+d x))^{3/2} \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right ) \tan ^{\frac {3}{2}}(c+d x)}{2 \sqrt {2} a d}+\frac {(e \cot (c+d x))^{3/2} \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right ) \tan ^{\frac {3}{2}}(c+d x)}{2 \sqrt {2} a d}-\frac {6 (e \cot (c+d x))^{3/2} \sin (c+d x) \tan ^2(c+d x)}{5 a d} \]

[Out]

2/5*cot(d*x+c)*(e*cot(d*x+c))^(3/2)*(1-sec(d*x+c))/a/d-2/5*(e*cot(d*x+c))^(3/2)*(5-3*sec(d*x+c))*tan(d*x+c)/a/
d-6/5*(e*cot(d*x+c))^(3/2)*(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))*
sin(d*x+c)*tan(d*x+c)/a/d/sin(2*d*x+2*c)^(1/2)-1/2*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*(e*cot(d*x+c))^(3/2)*ta
n(d*x+c)^(3/2)/a/d*2^(1/2)-1/2*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*(e*cot(d*x+c))^(3/2)*tan(d*x+c)^(3/2)/a/d*2^
(1/2)-1/4*(e*cot(d*x+c))^(3/2)*ln(1-2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))*tan(d*x+c)^(3/2)/a/d*2^(1/2)+1/4*(e*c
ot(d*x+c))^(3/2)*ln(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))*tan(d*x+c)^(3/2)/a/d*2^(1/2)-6/5*(e*cot(d*x+c))^(3/
2)*sin(d*x+c)*tan(d*x+c)^2/a/d

Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 405, normalized size of antiderivative = 1.00, number of steps used = 20, 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} \[ \int \frac {(e \cot (c+d x))^{3/2}}{a+a \sec (c+d x)} \, dx=\frac {\tan ^{\frac {3}{2}}(c+d x) \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right ) (e \cot (c+d x))^{3/2}}{\sqrt {2} a d}-\frac {\tan ^{\frac {3}{2}}(c+d x) \arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right ) (e \cot (c+d x))^{3/2}}{\sqrt {2} a d}+\frac {2 \cot (c+d x) (1-\sec (c+d x)) (e \cot (c+d x))^{3/2}}{5 a d}-\frac {\tan ^{\frac {3}{2}}(c+d x) (e \cot (c+d x))^{3/2} \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} a d}+\frac {\tan ^{\frac {3}{2}}(c+d x) (e \cot (c+d x))^{3/2} \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} a d}-\frac {6 \sin (c+d x) \tan ^2(c+d x) (e \cot (c+d x))^{3/2}}{5 a d}-\frac {2 \tan (c+d x) (5-3 \sec (c+d x)) (e \cot (c+d x))^{3/2}}{5 a d}+\frac {6 \sin (c+d x) \tan (c+d x) E\left (\left .c+d x-\frac {\pi }{4}\right |2\right ) (e \cot (c+d x))^{3/2}}{5 a d \sqrt {\sin (2 c+2 d x)}} \]

[In]

Int[(e*Cot[c + d*x])^(3/2)/(a + a*Sec[c + d*x]),x]

[Out]

(2*Cot[c + d*x]*(e*Cot[c + d*x])^(3/2)*(1 - Sec[c + d*x]))/(5*a*d) - (2*(e*Cot[c + d*x])^(3/2)*(5 - 3*Sec[c +
d*x])*Tan[c + d*x])/(5*a*d) + (6*(e*Cot[c + d*x])^(3/2)*EllipticE[c - Pi/4 + d*x, 2]*Sin[c + d*x]*Tan[c + d*x]
)/(5*a*d*Sqrt[Sin[2*c + 2*d*x]]) + (ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]]*(e*Cot[c + d*x])^(3/2)*Tan[c + d*x]
^(3/2))/(Sqrt[2]*a*d) - (ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]*(e*Cot[c + d*x])^(3/2)*Tan[c + d*x]^(3/2))/(Sq
rt[2]*a*d) - ((e*Cot[c + d*x])^(3/2)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]*Tan[c + d*x]^(3/2))/(2
*Sqrt[2]*a*d) + ((e*Cot[c + d*x])^(3/2)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]*Tan[c + d*x]^(3/2))
/(2*Sqrt[2]*a*d) - (6*(e*Cot[c + d*x])^(3/2)*Sin[c + d*x]*Tan[c + d*x]^2)/(5*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 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}& = \left ((e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)\right ) \int \frac {1}{(a+a \sec (c+d x)) \tan ^{\frac {3}{2}}(c+d x)} \, dx \\ & = \frac {\left ((e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)\right ) \int \frac {-a+a \sec (c+d x)}{\tan ^{\frac {7}{2}}(c+d x)} \, dx}{a^2} \\ & = \frac {2 \cot (c+d x) (e \cot (c+d x))^{3/2} (1-\sec (c+d x))}{5 a d}+\frac {\left (2 (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)\right ) \int \frac {\frac {5 a}{2}-\frac {3}{2} a \sec (c+d x)}{\tan ^{\frac {3}{2}}(c+d x)} \, dx}{5 a^2} \\ & = \frac {2 \cot (c+d x) (e \cot (c+d x))^{3/2} (1-\sec (c+d x))}{5 a d}-\frac {2 (e \cot (c+d x))^{3/2} (5-3 \sec (c+d x)) \tan (c+d x)}{5 a d}+\frac {\left (4 (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)\right ) \int \left (-\frac {5 a}{4}-\frac {3}{4} a \sec (c+d x)\right ) \sqrt {\tan (c+d x)} \, dx}{5 a^2} \\ & = \frac {2 \cot (c+d x) (e \cot (c+d x))^{3/2} (1-\sec (c+d x))}{5 a d}-\frac {2 (e \cot (c+d x))^{3/2} (5-3 \sec (c+d x)) \tan (c+d x)}{5 a d}-\frac {\left (3 (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)\right ) \int \sec (c+d x) \sqrt {\tan (c+d x)} \, dx}{5 a}-\frac {\left ((e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)\right ) \int \sqrt {\tan (c+d x)} \, dx}{a} \\ & = \frac {2 \cot (c+d x) (e \cot (c+d x))^{3/2} (1-\sec (c+d x))}{5 a d}-\frac {2 (e \cot (c+d x))^{3/2} (5-3 \sec (c+d x)) \tan (c+d x)}{5 a d}-\frac {6 (e \cot (c+d x))^{3/2} \sin (c+d x) \tan ^2(c+d x)}{5 a d}+\frac {\left (6 (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)\right ) \int \cos (c+d x) \sqrt {\tan (c+d x)} \, dx}{5 a}-\frac {\left ((e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)\right ) \text {Subst}\left (\int \frac {\sqrt {x}}{1+x^2} \, dx,x,\tan (c+d x)\right )}{a d} \\ & = \frac {2 \cot (c+d x) (e \cot (c+d x))^{3/2} (1-\sec (c+d x))}{5 a d}-\frac {2 (e \cot (c+d x))^{3/2} (5-3 \sec (c+d x)) \tan (c+d x)}{5 a d}-\frac {6 (e \cot (c+d x))^{3/2} \sin (c+d x) \tan ^2(c+d x)}{5 a d}+\frac {\left (6 (e \cot (c+d x))^{3/2} \sin ^{\frac {3}{2}}(c+d x)\right ) \int \sqrt {\cos (c+d x)} \sqrt {\sin (c+d x)} \, dx}{5 a \cos ^{\frac {3}{2}}(c+d x)}-\frac {\left (2 (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)\right ) \text {Subst}\left (\int \frac {x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{a d} \\ & = \frac {2 \cot (c+d x) (e \cot (c+d x))^{3/2} (1-\sec (c+d x))}{5 a d}-\frac {2 (e \cot (c+d x))^{3/2} (5-3 \sec (c+d x)) \tan (c+d x)}{5 a d}-\frac {6 (e \cot (c+d x))^{3/2} \sin (c+d x) \tan ^2(c+d x)}{5 a d}+\frac {\left (6 (e \cot (c+d x))^{3/2} \sin (c+d x) \tan (c+d x)\right ) \int \sqrt {\sin (2 c+2 d x)} \, dx}{5 a \sqrt {\sin (2 c+2 d x)}}+\frac {\left ((e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(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 ((e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(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) (e \cot (c+d x))^{3/2} (1-\sec (c+d x))}{5 a d}-\frac {2 (e \cot (c+d x))^{3/2} (5-3 \sec (c+d x)) \tan (c+d x)}{5 a d}+\frac {6 (e \cot (c+d x))^{3/2} E\left (\left .c-\frac {\pi }{4}+d x\right |2\right ) \sin (c+d x) \tan (c+d x)}{5 a d \sqrt {\sin (2 c+2 d x)}}-\frac {6 (e \cot (c+d x))^{3/2} \sin (c+d x) \tan ^2(c+d x)}{5 a d}-\frac {\left ((e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(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 ((e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(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 ((e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(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 ((e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(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) (e \cot (c+d x))^{3/2} (1-\sec (c+d x))}{5 a d}-\frac {2 (e \cot (c+d x))^{3/2} (5-3 \sec (c+d x)) \tan (c+d x)}{5 a d}+\frac {6 (e \cot (c+d x))^{3/2} E\left (\left .c-\frac {\pi }{4}+d x\right |2\right ) \sin (c+d x) \tan (c+d x)}{5 a d \sqrt {\sin (2 c+2 d x)}}-\frac {(e \cot (c+d x))^{3/2} \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right ) \tan ^{\frac {3}{2}}(c+d x)}{2 \sqrt {2} a d}+\frac {(e \cot (c+d x))^{3/2} \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right ) \tan ^{\frac {3}{2}}(c+d x)}{2 \sqrt {2} a d}-\frac {6 (e \cot (c+d x))^{3/2} \sin (c+d x) \tan ^2(c+d x)}{5 a d}-\frac {\left ((e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(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 ((e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(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) (e \cot (c+d x))^{3/2} (1-\sec (c+d x))}{5 a d}-\frac {2 (e \cot (c+d x))^{3/2} (5-3 \sec (c+d x)) \tan (c+d x)}{5 a d}+\frac {6 (e \cot (c+d x))^{3/2} E\left (\left .c-\frac {\pi }{4}+d x\right |2\right ) \sin (c+d x) \tan (c+d x)}{5 a d \sqrt {\sin (2 c+2 d x)}}+\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right ) (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)}{\sqrt {2} a d}-\frac {\arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right ) (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)}{\sqrt {2} a d}-\frac {(e \cot (c+d x))^{3/2} \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right ) \tan ^{\frac {3}{2}}(c+d x)}{2 \sqrt {2} a d}+\frac {(e \cot (c+d x))^{3/2} \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right ) \tan ^{\frac {3}{2}}(c+d x)}{2 \sqrt {2} a d}-\frac {6 (e \cot (c+d x))^{3/2} \sin (c+d x) \tan ^2(c+d x)}{5 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 = 14.10 (sec) , antiderivative size = 316, normalized size of antiderivative = 0.78 \[ \int \frac {(e \cot (c+d x))^{3/2}}{a+a \sec (c+d x)} \, dx=-\frac {e \sqrt {e \cot (c+d x)} \left (30 \sqrt {2} \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right ) \cot ^{\frac {3}{2}}(c+d x)-30 \sqrt {2} \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right ) \cot ^{\frac {3}{2}}(c+d x)+120 \cot ^2(c+d x)-24 \cot ^4(c+d x)+24 \cot ^4(c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},-\frac {1}{2},-\frac {1}{4},-\tan ^2(c+d x)\right )-120 \cot ^2(c+d x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-\frac {1}{4},\frac {3}{4},-\tan ^2(c+d x)\right )-40 \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-\tan ^2(c+d x)\right )+15 \sqrt {2} \cot ^{\frac {3}{2}}(c+d x) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )-15 \sqrt {2} \cot ^{\frac {3}{2}}(c+d x) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )\right ) \sec (c+d x) \left (1+\sqrt {\sec ^2(c+d x)}\right ) \sin ^2\left (\frac {1}{2} (c+d x)\right )}{30 a d} \]

[In]

Integrate[(e*Cot[c + d*x])^(3/2)/(a + a*Sec[c + d*x]),x]

[Out]

-1/30*(e*Sqrt[e*Cot[c + d*x]]*(30*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]]*Cot[c + d*x]^(3/2) - 30*Sqrt[
2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]]*Cot[c + d*x]^(3/2) + 120*Cot[c + d*x]^2 - 24*Cot[c + d*x]^4 + 24*Cot
[c + d*x]^4*Hypergeometric2F1[-5/4, -1/2, -1/4, -Tan[c + d*x]^2] - 120*Cot[c + d*x]^2*Hypergeometric2F1[-1/2,
-1/4, 3/4, -Tan[c + d*x]^2] - 40*Hypergeometric2F1[1/2, 3/4, 7/4, -Tan[c + d*x]^2] + 15*Sqrt[2]*Cot[c + d*x]^(
3/2)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] - 15*Sqrt[2]*Cot[c + d*x]^(3/2)*Log[1 + Sqrt[2]*Sqrt[C
ot[c + d*x]] + Cot[c + d*x]])*Sec[c + d*x]*(1 + Sqrt[Sec[c + d*x]^2])*Sin[(c + d*x)/2]^2)/(a*d)

Maple [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 8.23 (sec) , antiderivative size = 1116, normalized size of antiderivative = 2.76

method	result	size
default	\(\text {Expression too large to display}\)	\(1116\)

[In]

int((e*cot(d*x+c))^(3/2)/(a+a*sec(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

-1/10/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)))^(3/2)*(1-cos(d*x+c))*(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)*EllipticPi((csc(d
*x+c)-cot(d*x+c)+1)^(1/2),1/2-1/2*I,1/2*2^(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)-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)*E
llipticPi((csc(d*x+c)-cot(d*x+c)+1)^(1/2),1/2+1/2*I,1/2*2^(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)+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)*EllipticE((csc(d*x+c)-cot(d*x+c)+1)^(1/2),1/2*2^(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)-6*(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)-c
sc(d*x+c))^(1/2)*EllipticF((csc(d*x+c)-cot(d*x+c)+1)^(1/2),1/2*2^(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)-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))*((1-cos(d*x+c))*((1-cos(
d*x+c))^2*csc(d*x+c)^2-1)*csc(d*x+c))^(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))*((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))*((1-cos(d*x+c))^2*csc(d*x+c)^2-1)*cs
c(d*x+c))^(1/2)*(1-cos(d*x+c))^4*csc(d*x+c)^4-((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))^2*csc(d*x+c)^2-5*(1-cos(d*x+c))^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)^2+5*((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))^2*csc(d*x+c)^2-
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. \[ \int \frac {(e \cot (c+d x))^{3/2}}{a+a \sec (c+d x)} \, dx=\text {Timed out} \]

[In]

integrate((e*cot(d*x+c))^(3/2)/(a+a*sec(d*x+c)),x, algorithm="fricas")

[Out]

Timed out

Sympy [F]

\[ \int \frac {(e \cot (c+d x))^{3/2}}{a+a \sec (c+d x)} \, dx=\frac {\int \frac {\left (e \cot {\left (c + d x \right )}\right )^{\frac {3}{2}}}{\sec {\left (c + d x \right )} + 1}\, dx}{a} \]

[In]

integrate((e*cot(d*x+c))**(3/2)/(a+a*sec(d*x+c)),x)

[Out]

Integral((e*cot(c + d*x))**(3/2)/(sec(c + d*x) + 1), x)/a

Maxima [F(-1)]

Timed out. \[ \int \frac {(e \cot (c+d x))^{3/2}}{a+a \sec (c+d x)} \, dx=\text {Timed out} \]

[In]

integrate((e*cot(d*x+c))^(3/2)/(a+a*sec(d*x+c)),x, algorithm="maxima")

[Out]

Timed out

Giac [F]

\[ \int \frac {(e \cot (c+d x))^{3/2}}{a+a \sec (c+d x)} \, dx=\int { \frac {\left (e \cot \left (d x + c\right )\right )^{\frac {3}{2}}}{a \sec \left (d x + c\right ) + a} \,d x } \]

[In]

integrate((e*cot(d*x+c))^(3/2)/(a+a*sec(d*x+c)),x, algorithm="giac")

[Out]

integrate((e*cot(d*x + c))^(3/2)/(a*sec(d*x + c) + a), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(e \cot (c+d x))^{3/2}}{a+a \sec (c+d x)} \, dx=\int \frac {\cos \left (c+d\,x\right )\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{3/2}}{a\,\left (\cos \left (c+d\,x\right )+1\right )} \,d x \]

[In]

int((e*cot(c + d*x))^(3/2)/(a + a/cos(c + d*x)),x)

[Out]

int((cos(c + d*x)*(e*cot(c + d*x))^(3/2))/(a*(cos(c + d*x) + 1)), x)