\(\int (e \cot (c+d x))^{5/2} (a+a \sec (c+d x))^2 \, dx\) [238]

Optimal result

Integrand size = 25, antiderivative size = 357 \[ \int (e \cot (c+d x))^{5/2} (a+a \sec (c+d x))^2 \, dx=-\frac {4 a^2 (e \cot (c+d x))^{5/2} \tan (c+d x)}{3 d}-\frac {4 a^2 (e \cot (c+d x))^{5/2} \sec (c+d x) \tan (c+d x)}{3 d}-\frac {2 a^2 (e \cot (c+d x))^{5/2} \operatorname {EllipticF}\left (c-\frac {\pi }{4}+d x,2\right ) \sec (c+d x) \sqrt {\sin (2 c+2 d x)} \tan ^2(c+d x)}{3 d}+\frac {a^2 \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right ) (e \cot (c+d x))^{5/2} \tan ^{\frac {5}{2}}(c+d x)}{\sqrt {2} d}-\frac {a^2 \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right ) (e \cot (c+d x))^{5/2} \tan ^{\frac {5}{2}}(c+d x)}{\sqrt {2} d}+\frac {a^2 (e \cot (c+d x))^{5/2} \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right ) \tan ^{\frac {5}{2}}(c+d x)}{2 \sqrt {2} d}-\frac {a^2 (e \cot (c+d x))^{5/2} \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right ) \tan ^{\frac {5}{2}}(c+d x)}{2 \sqrt {2} d} \]

[Out]

-4/3*a^2*(e*cot(d*x+c))^(5/2)*tan(d*x+c)/d-4/3*a^2*(e*cot(d*x+c))^(5/2)*sec(d*x+c)*tan(d*x+c)/d+2/3*a^2*(e*cot
(d*x+c))^(5/2)*(sin(c+1/4*Pi+d*x)^2)^(1/2)/sin(c+1/4*Pi+d*x)*EllipticF(cos(c+1/4*Pi+d*x),2^(1/2))*sec(d*x+c)*s
in(2*d*x+2*c)^(1/2)*tan(d*x+c)^2/d-1/2*a^2*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*(e*cot(d*x+c))^(5/2)*tan(d*x+c)
^(5/2)/d*2^(1/2)-1/2*a^2*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))*(e*cot(d*x+c))^(5/2)*tan(d*x+c)^(5/2)/d*2^(1/2)+1/
4*a^2*(e*cot(d*x+c))^(5/2)*ln(1-2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))*tan(d*x+c)^(5/2)/d*2^(1/2)-1/4*a^2*(e*cot
(d*x+c))^(5/2)*ln(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))*tan(d*x+c)^(5/2)/d*2^(1/2)

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 357, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.680, Rules used = {3985, 3971, 3555, 3557, 335, 217, 1179, 642, 1176, 631, 210, 2689, 2694, 2653, 2720, 2687, 30} \[ \int (e \cot (c+d x))^{5/2} (a+a \sec (c+d x))^2 \, dx=\frac {a^2 \tan ^{\frac {5}{2}}(c+d x) \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right ) (e \cot (c+d x))^{5/2}}{\sqrt {2} d}-\frac {a^2 \tan ^{\frac {5}{2}}(c+d x) \arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right ) (e \cot (c+d x))^{5/2}}{\sqrt {2} d}-\frac {4 a^2 \tan (c+d x) (e \cot (c+d x))^{5/2}}{3 d}+\frac {a^2 \tan ^{\frac {5}{2}}(c+d x) (e \cot (c+d x))^{5/2} \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d}-\frac {a^2 \tan ^{\frac {5}{2}}(c+d x) (e \cot (c+d x))^{5/2} \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d}-\frac {4 a^2 \tan (c+d x) \sec (c+d x) (e \cot (c+d x))^{5/2}}{3 d}-\frac {2 a^2 \sqrt {\sin (2 c+2 d x)} \tan ^2(c+d x) \sec (c+d x) \operatorname {EllipticF}\left (c+d x-\frac {\pi }{4},2\right ) (e \cot (c+d x))^{5/2}}{3 d} \]

[In]

Int[(e*Cot[c + d*x])^(5/2)*(a + a*Sec[c + d*x])^2,x]

[Out]

(-4*a^2*(e*Cot[c + d*x])^(5/2)*Tan[c + d*x])/(3*d) - (4*a^2*(e*Cot[c + d*x])^(5/2)*Sec[c + d*x]*Tan[c + d*x])/
(3*d) - (2*a^2*(e*Cot[c + d*x])^(5/2)*EllipticF[c - Pi/4 + d*x, 2]*Sec[c + d*x]*Sqrt[Sin[2*c + 2*d*x]]*Tan[c +
d*x]^2)/(3*d) + (a^2*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]]*(e*Cot[c + d*x])^(5/2)*Tan[c + d*x]^(5/2))/(Sqrt[
2]*d) - (a^2*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]*(e*Cot[c + d*x])^(5/2)*Tan[c + d*x]^(5/2))/(Sqrt[2]*d) + (
a^2*(e*Cot[c + d*x])^(5/2)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]*Tan[c + d*x]^(5/2))/(2*Sqrt[2]*d
) - (a^2*(e*Cot[c + d*x])^(5/2)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]*Tan[c + d*x]^(5/2))/(2*Sqrt
[2]*d)

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 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 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 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 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 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 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))^{5/2} \tan ^{\frac {5}{2}}(c+d x)\right ) \int \frac {(a+a \sec (c+d x))^2}{\tan ^{\frac {5}{2}}(c+d x)} \, dx \\ & = \left ((e \cot (c+d x))^{5/2} \tan ^{\frac {5}{2}}(c+d x)\right ) \int \left (\frac {a^2}{\tan ^{\frac {5}{2}}(c+d x)}+\frac {2 a^2 \sec (c+d x)}{\tan ^{\frac {5}{2}}(c+d x)}+\frac {a^2 \sec ^2(c+d x)}{\tan ^{\frac {5}{2}}(c+d x)}\right ) \, dx \\ & = \left (a^2 (e \cot (c+d x))^{5/2} \tan ^{\frac {5}{2}}(c+d x)\right ) \int \frac {1}{\tan ^{\frac {5}{2}}(c+d x)} \, dx+\left (a^2 (e \cot (c+d x))^{5/2} \tan ^{\frac {5}{2}}(c+d x)\right ) \int \frac {\sec ^2(c+d x)}{\tan ^{\frac {5}{2}}(c+d x)} \, dx+\left (2 a^2 (e \cot (c+d x))^{5/2} \tan ^{\frac {5}{2}}(c+d x)\right ) \int \frac {\sec (c+d x)}{\tan ^{\frac {5}{2}}(c+d x)} \, dx \\ & = -\frac {2 a^2 (e \cot (c+d x))^{5/2} \tan (c+d x)}{3 d}-\frac {4 a^2 (e \cot (c+d x))^{5/2} \sec (c+d x) \tan (c+d x)}{3 d}-\frac {1}{3} \left (2 a^2 (e \cot (c+d x))^{5/2} \tan ^{\frac {5}{2}}(c+d x)\right ) \int \frac {\sec (c+d x)}{\sqrt {\tan (c+d x)}} \, dx-\left (a^2 (e \cot (c+d x))^{5/2} \tan ^{\frac {5}{2}}(c+d x)\right ) \int \frac {1}{\sqrt {\tan (c+d x)}} \, dx+\frac {\left (a^2 (e \cot (c+d x))^{5/2} \tan ^{\frac {5}{2}}(c+d x)\right ) \text {Subst}\left (\int \frac {1}{x^{5/2}} \, dx,x,\tan (c+d x)\right )}{d} \\ & = -\frac {4 a^2 (e \cot (c+d x))^{5/2} \tan (c+d x)}{3 d}-\frac {4 a^2 (e \cot (c+d x))^{5/2} \sec (c+d x) \tan (c+d x)}{3 d}-\frac {\left (2 a^2 (e \cot (c+d x))^{5/2} \sin ^{\frac {5}{2}}(c+d x)\right ) \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {\sin (c+d x)}} \, dx}{3 \cos ^{\frac {5}{2}}(c+d x)}-\frac {\left (a^2 (e \cot (c+d x))^{5/2} \tan ^{\frac {5}{2}}(c+d x)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {x} \left (1+x^2\right )} \, dx,x,\tan (c+d x)\right )}{d} \\ & = -\frac {4 a^2 (e \cot (c+d x))^{5/2} \tan (c+d x)}{3 d}-\frac {4 a^2 (e \cot (c+d x))^{5/2} \sec (c+d x) \tan (c+d x)}{3 d}-\frac {1}{3} \left (2 a^2 (e \cot (c+d x))^{5/2} \sec (c+d x) \sqrt {\sin (2 c+2 d x)} \tan ^2(c+d x)\right ) \int \frac {1}{\sqrt {\sin (2 c+2 d x)}} \, dx-\frac {\left (2 a^2 (e \cot (c+d x))^{5/2} \tan ^{\frac {5}{2}}(c+d x)\right ) \text {Subst}\left (\int \frac {1}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d} \\ & = -\frac {4 a^2 (e \cot (c+d x))^{5/2} \tan (c+d x)}{3 d}-\frac {4 a^2 (e \cot (c+d x))^{5/2} \sec (c+d x) \tan (c+d x)}{3 d}-\frac {2 a^2 (e \cot (c+d x))^{5/2} \operatorname {EllipticF}\left (c-\frac {\pi }{4}+d x,2\right ) \sec (c+d x) \sqrt {\sin (2 c+2 d x)} \tan ^2(c+d x)}{3 d}-\frac {\left (a^2 (e \cot (c+d x))^{5/2} \tan ^{\frac {5}{2}}(c+d x)\right ) \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d}-\frac {\left (a^2 (e \cot (c+d x))^{5/2} \tan ^{\frac {5}{2}}(c+d x)\right ) \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d} \\ & = -\frac {4 a^2 (e \cot (c+d x))^{5/2} \tan (c+d x)}{3 d}-\frac {4 a^2 (e \cot (c+d x))^{5/2} \sec (c+d x) \tan (c+d x)}{3 d}-\frac {2 a^2 (e \cot (c+d x))^{5/2} \operatorname {EllipticF}\left (c-\frac {\pi }{4}+d x,2\right ) \sec (c+d x) \sqrt {\sin (2 c+2 d x)} \tan ^2(c+d x)}{3 d}-\frac {\left (a^2 (e \cot (c+d x))^{5/2} \tan ^{\frac {5}{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 d}-\frac {\left (a^2 (e \cot (c+d x))^{5/2} \tan ^{\frac {5}{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 d}+\frac {\left (a^2 (e \cot (c+d x))^{5/2} \tan ^{\frac {5}{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} d}+\frac {\left (a^2 (e \cot (c+d x))^{5/2} \tan ^{\frac {5}{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} d} \\ & = -\frac {4 a^2 (e \cot (c+d x))^{5/2} \tan (c+d x)}{3 d}-\frac {4 a^2 (e \cot (c+d x))^{5/2} \sec (c+d x) \tan (c+d x)}{3 d}-\frac {2 a^2 (e \cot (c+d x))^{5/2} \operatorname {EllipticF}\left (c-\frac {\pi }{4}+d x,2\right ) \sec (c+d x) \sqrt {\sin (2 c+2 d x)} \tan ^2(c+d x)}{3 d}+\frac {a^2 (e \cot (c+d x))^{5/2} \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right ) \tan ^{\frac {5}{2}}(c+d x)}{2 \sqrt {2} d}-\frac {a^2 (e \cot (c+d x))^{5/2} \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right ) \tan ^{\frac {5}{2}}(c+d x)}{2 \sqrt {2} d}-\frac {\left (a^2 (e \cot (c+d x))^{5/2} \tan ^{\frac {5}{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} d}+\frac {\left (a^2 (e \cot (c+d x))^{5/2} \tan ^{\frac {5}{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} d} \\ & = -\frac {4 a^2 (e \cot (c+d x))^{5/2} \tan (c+d x)}{3 d}-\frac {4 a^2 (e \cot (c+d x))^{5/2} \sec (c+d x) \tan (c+d x)}{3 d}-\frac {2 a^2 (e \cot (c+d x))^{5/2} \operatorname {EllipticF}\left (c-\frac {\pi }{4}+d x,2\right ) \sec (c+d x) \sqrt {\sin (2 c+2 d x)} \tan ^2(c+d x)}{3 d}+\frac {a^2 \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right ) (e \cot (c+d x))^{5/2} \tan ^{\frac {5}{2}}(c+d x)}{\sqrt {2} d}-\frac {a^2 \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right ) (e \cot (c+d x))^{5/2} \tan ^{\frac {5}{2}}(c+d x)}{\sqrt {2} d}+\frac {a^2 (e \cot (c+d x))^{5/2} \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right ) \tan ^{\frac {5}{2}}(c+d x)}{2 \sqrt {2} d}-\frac {a^2 (e \cot (c+d x))^{5/2} \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right ) \tan ^{\frac {5}{2}}(c+d x)}{2 \sqrt {2} 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 = 12.83 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.26 \[ \int (e \cot (c+d x))^{5/2} (a+a \sec (c+d x))^2 \, dx=-\frac {2 a^2 e \cos ^4\left (\frac {1}{2} (c+d x)\right ) (e \cot (c+d x))^{3/2} \left (2+2 \operatorname {Hypergeometric2F1}\left (-\frac {3}{4},\frac {1}{2},\frac {1}{4},-\tan ^2(c+d x)\right )-\operatorname {Hypergeometric2F1}\left (\frac {3}{4},1,\frac {7}{4},-\cot ^2(c+d x)\right )\right ) \sec ^4\left (\frac {1}{2} \cot ^{-1}(\cot (c+d x))\right )}{3 d} \]

[In]

Integrate[(e*Cot[c + d*x])^(5/2)*(a + a*Sec[c + d*x])^2,x]

[Out]

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

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 17.61 (sec) , antiderivative size = 496, normalized size of antiderivative = 1.39

method	result	size
default	\(-\frac {a^{2} e^{2} \sqrt {2}\, \left (\cos \left (d x +c \right )+1\right ) \left (-3 i \sin \left (d x +c \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 )}\, \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 \sin \left (d x +c \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 )}\, \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 \sin \left (d x +c \right ) \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \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}\, \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 \sin \left (d x +c \right ) \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \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}\, \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 \sin \left (d x +c \right ) \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \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}\, \operatorname {EllipticF}\left (\sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right )+4 \sqrt {2}\, \cos \left (d x +c \right )\right ) \sqrt {e \cot \left (d x +c \right )}\, \sec \left (d x +c \right ) \csc \left (d x +c \right )}{6 d}\)	\(496\)
parts	\(-\frac {2 a^{2} e \left (\frac {\left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{3}-\frac {e^{2} \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 )}{8 \left (e^{2}\right )^{\frac {1}{4}}}\right )}{d}-\frac {2 a^{2} e \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{3 d}-\frac {2 a^{2} \sqrt {2}\, e^{2} \sqrt {e \cot \left (d x +c \right )}\, \left (-\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 )}\, \operatorname {EllipticF}\left (\sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right ) \cot \left (d x +c \right )^{2}-\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 )}\, \operatorname {EllipticF}\left (\sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right ) \cot \left (d x +c \right ) \csc \left (d x +c \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 )}\, \operatorname {EllipticF}\left (\sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right ) \csc \left (d x +c \right )^{2}+\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 )}\, \operatorname {EllipticF}\left (\sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right ) \sec \left (d x +c \right ) \csc \left (d x +c \right )^{2}+\sqrt {2}\, \csc \left (d x +c \right )\right )}{3 d}\)	\(568\)

[In]

int((e*cot(d*x+c))^(5/2)*(a+a*sec(d*x+c))^2,x,method=_RETURNVERBOSE)

[Out]

-1/6*a^2*e^2/d*2^(1/2)*(cos(d*x+c)+1)*(-3*I*sin(d*x+c)*(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)*EllipticPi((csc(d*x+c)-cot(d*x+c)+1)^(1/2),1/2+1/2*I,1/2*2^(1/2))+3*I*s
in(d*x+c)*(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)*Ellipt
icPi((csc(d*x+c)-cot(d*x+c)+1)^(1/2),1/2-1/2*I,1/2*2^(1/2))+3*sin(d*x+c)*(cot(d*x+c)-csc(d*x+c))^(1/2)*(csc(d*
x+c)-cot(d*x+c)+1)^(1/2)*(cot(d*x+c)-csc(d*x+c)+1)^(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*sin(d*x+c)*(cot(d*x+c)-csc(d*x+c))^(1/2)*(csc(d*x+c)-cot(d*x+c)+1)^(1/2)*(cot(d*x+c)-csc(d*x+c)
+1)^(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*sin(d*x+c)*(cot(d*x+c)-csc(d*x+c
))^(1/2)*(csc(d*x+c)-cot(d*x+c)+1)^(1/2)*(cot(d*x+c)-csc(d*x+c)+1)^(1/2)*EllipticF((csc(d*x+c)-cot(d*x+c)+1)^(
1/2),1/2*2^(1/2))+4*2^(1/2)*cos(d*x+c))*(e*cot(d*x+c))^(1/2)*sec(d*x+c)*csc(d*x+c)

Fricas [F(-1)]

Timed out. \[ \int (e \cot (c+d x))^{5/2} (a+a \sec (c+d x))^2 \, dx=\text {Timed out} \]

[In]

integrate((e*cot(d*x+c))^(5/2)*(a+a*sec(d*x+c))^2,x, algorithm="fricas")

[Out]

Timed out

Sympy [F(-1)]

Timed out. \[ \int (e \cot (c+d x))^{5/2} (a+a \sec (c+d x))^2 \, dx=\text {Timed out} \]

[In]

integrate((e*cot(d*x+c))**(5/2)*(a+a*sec(d*x+c))**2,x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int (e \cot (c+d x))^{5/2} (a+a \sec (c+d x))^2 \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((e*cot(d*x+c))^(5/2)*(a+a*sec(d*x+c))^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more
details)Is e

Giac [F]

\[ \int (e \cot (c+d x))^{5/2} (a+a \sec (c+d x))^2 \, dx=\int { \left (e \cot \left (d x + c\right )\right )^{\frac {5}{2}} {\left (a \sec \left (d x + c\right ) + a\right )}^{2} \,d x } \]

[In]

integrate((e*cot(d*x+c))^(5/2)*(a+a*sec(d*x+c))^2,x, algorithm="giac")

[Out]

integrate((e*cot(d*x + c))^(5/2)*(a*sec(d*x + c) + a)^2, x)

Mupad [F(-1)]

Timed out. \[ \int (e \cot (c+d x))^{5/2} (a+a \sec (c+d x))^2 \, dx=\int {\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{5/2}\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^2 \,d x \]

[In]

int((e*cot(c + d*x))^(5/2)*(a + a/cos(c + d*x))^2,x)

[Out]

int((e*cot(c + d*x))^(5/2)*(a + a/cos(c + d*x))^2, x)