\(\int \frac {1}{(e \cot (c+d x))^{9/2} (a+a \sec (c+d x))} \, dx\) [249]

Optimal result

Integrand size = 25, antiderivative size = 371 \[ \int \frac {1}{(e \cot (c+d x))^{9/2} (a+a \sec (c+d x))} \, dx=-\frac {6 \cos (c+d x) \cot ^3(c+d x)}{5 a d (e \cot (c+d x))^{9/2}}-\frac {2 \cot ^3(c+d x) (5-3 \sec (c+d x))}{15 a d (e \cot (c+d x))^{9/2}}+\frac {6 \cos (c+d x) \cot ^4(c+d x) E\left (\left .c-\frac {\pi }{4}+d x\right |2\right )}{5 a d (e \cot (c+d x))^{9/2} \sqrt {\sin (2 c+2 d x)}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} a d (e \cot (c+d x))^{9/2} \tan ^{\frac {9}{2}}(c+d x)}+\frac {\arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} a d (e \cot (c+d x))^{9/2} \tan ^{\frac {9}{2}}(c+d x)}+\frac {\log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} a d (e \cot (c+d x))^{9/2} \tan ^{\frac {9}{2}}(c+d x)}-\frac {\log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} a d (e \cot (c+d x))^{9/2} \tan ^{\frac {9}{2}}(c+d x)} \]

[Out]

-6/5*cos(d*x+c)*cot(d*x+c)^3/a/d/(e*cot(d*x+c))^(9/2)-2/15*cot(d*x+c)^3*(5-3*sec(d*x+c))/a/d/(e*cot(d*x+c))^(9
/2)-6/5*cos(d*x+c)*cot(d*x+c)^4*(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))^(9/2)/sin(2*d*x+2*c)^(1/2)+1/2*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))/a/d/(e*cot(d*x+c))^
(9/2)*2^(1/2)/tan(d*x+c)^(9/2)+1/2*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))/a/d/(e*cot(d*x+c))^(9/2)*2^(1/2)/tan(d*x
+c)^(9/2)+1/4*ln(1-2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/a/d/(e*cot(d*x+c))^(9/2)*2^(1/2)/tan(d*x+c)^(9/2)-1/4*
ln(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/a/d/(e*cot(d*x+c))^(9/2)*2^(1/2)/tan(d*x+c)^(9/2)

Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 371, 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, 3966, 3969, 3557, 335, 303, 1176, 631, 210, 1179, 642, 2693, 2695, 2652, 2719} \[ \int \frac {1}{(e \cot (c+d x))^{9/2} (a+a \sec (c+d x))} \, dx=-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} a d \tan ^{\frac {9}{2}}(c+d x) (e \cot (c+d x))^{9/2}}+\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} a d \tan ^{\frac {9}{2}}(c+d x) (e \cot (c+d x))^{9/2}}-\frac {6 \cos (c+d x) \cot ^3(c+d x)}{5 a d (e \cot (c+d x))^{9/2}}-\frac {2 \cot ^3(c+d x) (5-3 \sec (c+d x))}{15 a d (e \cot (c+d x))^{9/2}}+\frac {\log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} a d \tan ^{\frac {9}{2}}(c+d x) (e \cot (c+d x))^{9/2}}-\frac {\log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} a d \tan ^{\frac {9}{2}}(c+d x) (e \cot (c+d x))^{9/2}}+\frac {6 \cos (c+d x) \cot ^4(c+d x) E\left (\left .c+d x-\frac {\pi }{4}\right |2\right )}{5 a d \sqrt {\sin (2 c+2 d x)} (e \cot (c+d x))^{9/2}} \]

[In]

Int[1/((e*Cot[c + d*x])^(9/2)*(a + a*Sec[c + d*x])),x]

[Out]

(-6*Cos[c + d*x]*Cot[c + d*x]^3)/(5*a*d*(e*Cot[c + d*x])^(9/2)) - (2*Cot[c + d*x]^3*(5 - 3*Sec[c + d*x]))/(15*
a*d*(e*Cot[c + d*x])^(9/2)) + (6*Cos[c + d*x]*Cot[c + d*x]^4*EllipticE[c - Pi/4 + d*x, 2])/(5*a*d*(e*Cot[c + d
*x])^(9/2)*Sqrt[Sin[2*c + 2*d*x]]) - ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]]/(Sqrt[2]*a*d*(e*Cot[c + d*x])^(9/2
)*Tan[c + d*x]^(9/2)) + ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]/(Sqrt[2]*a*d*(e*Cot[c + d*x])^(9/2)*Tan[c + d*x
]^(9/2)) + Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]/(2*Sqrt[2]*a*d*(e*Cot[c + d*x])^(9/2)*Tan[c + d*
x]^(9/2)) - Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]/(2*Sqrt[2]*a*d*(e*Cot[c + d*x])^(9/2)*Tan[c + d
*x]^(9/2))

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 3966

Int[(cot[(c_.) + (d_.)*(x_)]*(e_.))^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[(-e)*(e*Cot
[c + d*x])^(m - 1)*((a*m + b*(m - 1)*Csc[c + d*x])/(d*m*(m - 1))), x] - Dist[e^2/m, Int[(e*Cot[c + d*x])^(m -
2)*(a*m + b*(m - 1)*Csc[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e}, x] && GtQ[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 {\tan ^{\frac {9}{2}}(c+d x)}{a+a \sec (c+d x)} \, dx}{(e \cot (c+d x))^{9/2} \tan ^{\frac {9}{2}}(c+d x)} \\ & = \frac {\int (-a+a \sec (c+d x)) \tan ^{\frac {5}{2}}(c+d x) \, dx}{a^2 (e \cot (c+d x))^{9/2} \tan ^{\frac {9}{2}}(c+d x)} \\ & = -\frac {2 \cot ^3(c+d x) (5-3 \sec (c+d x))}{15 a d (e \cot (c+d x))^{9/2}}-\frac {2 \int \left (-\frac {5 a}{2}+\frac {3}{2} a \sec (c+d x)\right ) \sqrt {\tan (c+d x)} \, dx}{5 a^2 (e \cot (c+d x))^{9/2} \tan ^{\frac {9}{2}}(c+d x)} \\ & = -\frac {2 \cot ^3(c+d x) (5-3 \sec (c+d x))}{15 a d (e \cot (c+d x))^{9/2}}-\frac {3 \int \sec (c+d x) \sqrt {\tan (c+d x)} \, dx}{5 a (e \cot (c+d x))^{9/2} \tan ^{\frac {9}{2}}(c+d x)}+\frac {\int \sqrt {\tan (c+d x)} \, dx}{a (e \cot (c+d x))^{9/2} \tan ^{\frac {9}{2}}(c+d x)} \\ & = -\frac {6 \cos (c+d x) \cot ^3(c+d x)}{5 a d (e \cot (c+d x))^{9/2}}-\frac {2 \cot ^3(c+d x) (5-3 \sec (c+d x))}{15 a d (e \cot (c+d x))^{9/2}}+\frac {6 \int \cos (c+d x) \sqrt {\tan (c+d x)} \, dx}{5 a (e \cot (c+d x))^{9/2} \tan ^{\frac {9}{2}}(c+d x)}+\frac {\text {Subst}\left (\int \frac {\sqrt {x}}{1+x^2} \, dx,x,\tan (c+d x)\right )}{a d (e \cot (c+d x))^{9/2} \tan ^{\frac {9}{2}}(c+d x)} \\ & = -\frac {6 \cos (c+d x) \cot ^3(c+d x)}{5 a d (e \cot (c+d x))^{9/2}}-\frac {2 \cot ^3(c+d x) (5-3 \sec (c+d x))}{15 a d (e \cot (c+d x))^{9/2}}+\frac {\left (6 \cos ^{\frac {9}{2}}(c+d x)\right ) \int \sqrt {\cos (c+d x)} \sqrt {\sin (c+d x)} \, dx}{5 a (e \cot (c+d x))^{9/2} \sin ^{\frac {9}{2}}(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 (e \cot (c+d x))^{9/2} \tan ^{\frac {9}{2}}(c+d x)} \\ & = -\frac {6 \cos (c+d x) \cot ^3(c+d x)}{5 a d (e \cot (c+d x))^{9/2}}-\frac {2 \cot ^3(c+d x) (5-3 \sec (c+d x))}{15 a d (e \cot (c+d x))^{9/2}}+\frac {\left (6 \cos (c+d x) \cot ^4(c+d x)\right ) \int \sqrt {\sin (2 c+2 d x)} \, dx}{5 a (e \cot (c+d x))^{9/2} \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 (e \cot (c+d x))^{9/2} \tan ^{\frac {9}{2}}(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 (e \cot (c+d x))^{9/2} \tan ^{\frac {9}{2}}(c+d x)} \\ & = -\frac {6 \cos (c+d x) \cot ^3(c+d x)}{5 a d (e \cot (c+d x))^{9/2}}-\frac {2 \cot ^3(c+d x) (5-3 \sec (c+d x))}{15 a d (e \cot (c+d x))^{9/2}}+\frac {6 \cos (c+d x) \cot ^4(c+d x) E\left (\left .c-\frac {\pi }{4}+d x\right |2\right )}{5 a d (e \cot (c+d x))^{9/2} \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 (e \cot (c+d x))^{9/2} \tan ^{\frac {9}{2}}(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 (e \cot (c+d x))^{9/2} \tan ^{\frac {9}{2}}(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 (e \cot (c+d x))^{9/2} \tan ^{\frac {9}{2}}(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 (e \cot (c+d x))^{9/2} \tan ^{\frac {9}{2}}(c+d x)} \\ & = -\frac {6 \cos (c+d x) \cot ^3(c+d x)}{5 a d (e \cot (c+d x))^{9/2}}-\frac {2 \cot ^3(c+d x) (5-3 \sec (c+d x))}{15 a d (e \cot (c+d x))^{9/2}}+\frac {6 \cos (c+d x) \cot ^4(c+d x) E\left (\left .c-\frac {\pi }{4}+d x\right |2\right )}{5 a d (e \cot (c+d x))^{9/2} \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 (e \cot (c+d x))^{9/2} \tan ^{\frac {9}{2}}(c+d x)}-\frac {\log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} a d (e \cot (c+d x))^{9/2} \tan ^{\frac {9}{2}}(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 (e \cot (c+d x))^{9/2} \tan ^{\frac {9}{2}}(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 (e \cot (c+d x))^{9/2} \tan ^{\frac {9}{2}}(c+d x)} \\ & = -\frac {6 \cos (c+d x) \cot ^3(c+d x)}{5 a d (e \cot (c+d x))^{9/2}}-\frac {2 \cot ^3(c+d x) (5-3 \sec (c+d x))}{15 a d (e \cot (c+d x))^{9/2}}+\frac {6 \cos (c+d x) \cot ^4(c+d x) E\left (\left .c-\frac {\pi }{4}+d x\right |2\right )}{5 a d (e \cot (c+d x))^{9/2} \sqrt {\sin (2 c+2 d x)}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} a d (e \cot (c+d x))^{9/2} \tan ^{\frac {9}{2}}(c+d x)}+\frac {\arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} a d (e \cot (c+d x))^{9/2} \tan ^{\frac {9}{2}}(c+d x)}+\frac {\log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} a d (e \cot (c+d x))^{9/2} \tan ^{\frac {9}{2}}(c+d x)}-\frac {\log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} a d (e \cot (c+d x))^{9/2} \tan ^{\frac {9}{2}}(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 = 14.81 (sec) , antiderivative size = 261, normalized size of antiderivative = 0.70 \[ \int \frac {1}{(e \cot (c+d x))^{9/2} (a+a \sec (c+d x))} \, dx=\frac {\sqrt {e \cot (c+d x)} \left (-8+6 \sqrt {2} \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right ) \cot ^{\frac {3}{2}}(c+d x)-6 \sqrt {2} \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right ) \cot ^{\frac {3}{2}}(c+d x)+8 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {3}{4},\frac {7}{4},-\tan ^2(c+d x)\right )-8 \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-\tan ^2(c+d x)\right )+3 \sqrt {2} \cot ^{\frac {3}{2}}(c+d x) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )-3 \sqrt {2} \cot ^{\frac {3}{2}}(c+d x) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )\right ) \sec (c+d x) \left (1+\sqrt {\sec ^2(c+d x)}\right ) \sin ^2\left (\frac {1}{2} (c+d x)\right )}{6 a d e^5} \]

[In]

Integrate[1/((e*Cot[c + d*x])^(9/2)*(a + a*Sec[c + d*x])),x]

[Out]

(Sqrt[e*Cot[c + d*x]]*(-8 + 6*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]]*Cot[c + d*x]^(3/2) - 6*Sqrt[2]*Ar
cTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]]*Cot[c + d*x]^(3/2) + 8*Hypergeometric2F1[-1/2, 3/4, 7/4, -Tan[c + d*x]^2]
- 8*Hypergeometric2F1[1/2, 3/4, 7/4, -Tan[c + d*x]^2] + 3*Sqrt[2]*Cot[c + d*x]^(3/2)*Log[1 - Sqrt[2]*Sqrt[Cot
[c + d*x]] + Cot[c + d*x]] - 3*Sqrt[2]*Cot[c + d*x]^(3/2)*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]])*
Sec[c + d*x]*(1 + Sqrt[Sec[c + d*x]^2])*Sin[(c + d*x)/2]^2)/(6*a*d*e^5)

Maple [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 9.23 (sec) , antiderivative size = 1166, normalized size of antiderivative = 3.14

method	result	size
default	\(\text {Expression too large to display}\)	\(1166\)

[In]

int(1/(e*cot(d*x+c))^(9/2)/(a+a*sec(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

1/30/a/d*2^(1/2)*(-15*I*(cot(d*x+c)-csc(d*x+c)+1)^(1/2)*(cot(d*x+c)-csc(d*x+c))^(1/2)*(csc(d*x+c)-cot(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))*cos(d*x+c)^2-15*I*(cot(d*x+c)-csc(d*
x+c)+1)^(1/2)*(cot(d*x+c)-csc(d*x+c))^(1/2)*(csc(d*x+c)-cot(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))*cos(d*x+c)^3+15*I*(cot(d*x+c)-csc(d*x+c)+1)^(1/2)*(cot(d*x+c)-csc(d*x+c))^(1/2
)*(csc(d*x+c)-cot(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))*cos(d*x+c)
^3+15*I*(cot(d*x+c)-csc(d*x+c)+1)^(1/2)*(cot(d*x+c)-csc(d*x+c))^(1/2)*(csc(d*x+c)-cot(d*x+c)+1)^(1/2)*Elliptic
Pi((csc(d*x+c)-cot(d*x+c)+1)^(1/2),1/2-1/2*I,1/2*2^(1/2))*cos(d*x+c)^2+18*(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)*EllipticF((csc(d*x+c)-cot(d*x+c)+1)^(1/2),1/2*2^(1/2
))*cos(d*x+c)^3-15*(cot(d*x+c)-csc(d*x+c)+1)^(1/2)*(cot(d*x+c)-csc(d*x+c))^(1/2)*(csc(d*x+c)-cot(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))*cos(d*x+c)^3-15*(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))*(csc(d*
x+c)-cot(d*x+c)+1)^(1/2)*cos(d*x+c)^3-36*(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)*EllipticE((csc(d*x+c)-cot(d*x+c)+1)^(1/2),1/2*2^(1/2))*cos(d*x+c)^3+18*(csc(d*x+c)-co
t(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)*EllipticF((csc(d*x+c)-cot(d*x+
c)+1)^(1/2),1/2*2^(1/2))*cos(d*x+c)^2-15*(cot(d*x+c)-csc(d*x+c)+1)^(1/2)*(cot(d*x+c)-csc(d*x+c))^(1/2)*(csc(d*
x+c)-cot(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))*cos(d*x+c)^2-15*(co
t(d*x+c)-csc(d*x+c)+1)^(1/2)*(cot(d*x+c)-csc(d*x+c))^(1/2)*(csc(d*x+c)-cot(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))*cos(d*x+c)^2-36*(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)*EllipticE((csc(d*x+c)-cot(d*x+c)+1)^(1/2),1/2*2^(1/2))*cos(d*x+
c)^2+28*cos(d*x+c)^3*2^(1/2)-24*2^(1/2)*cos(d*x+c)^2-10*2^(1/2)*cos(d*x+c)+6*2^(1/2))/e^4/(e*cot(d*x+c))^(1/2)
/(cos(d*x+c)^2-1)^2*sin(d*x+c)*tan(d*x+c)^2

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{(e \cot (c+d x))^{9/2} (a+a \sec (c+d x))} \, dx=\text {Timed out} \]

[In]

integrate(1/(e*cot(d*x+c))^(9/2)/(a+a*sec(d*x+c)),x, algorithm="fricas")

[Out]

Timed out

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(e \cot (c+d x))^{9/2} (a+a \sec (c+d x))} \, dx=\text {Timed out} \]

[In]

integrate(1/(e*cot(d*x+c))**(9/2)/(a+a*sec(d*x+c)),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {1}{(e \cot (c+d x))^{9/2} (a+a \sec (c+d x))} \, dx=\int { \frac {1}{\left (e \cot \left (d x + c\right )\right )^{\frac {9}{2}} {\left (a \sec \left (d x + c\right ) + a\right )}} \,d x } \]

[In]

integrate(1/(e*cot(d*x+c))^(9/2)/(a+a*sec(d*x+c)),x, algorithm="maxima")

[Out]

integrate(1/((e*cot(d*x + c))^(9/2)*(a*sec(d*x + c) + a)), x)

Giac [F]

\[ \int \frac {1}{(e \cot (c+d x))^{9/2} (a+a \sec (c+d x))} \, dx=\int { \frac {1}{\left (e \cot \left (d x + c\right )\right )^{\frac {9}{2}} {\left (a \sec \left (d x + c\right ) + a\right )}} \,d x } \]

[In]

integrate(1/(e*cot(d*x+c))^(9/2)/(a+a*sec(d*x+c)),x, algorithm="giac")

[Out]

integrate(1/((e*cot(d*x + c))^(9/2)*(a*sec(d*x + c) + a)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(e \cot (c+d x))^{9/2} (a+a \sec (c+d x))} \, dx=\int \frac {\cos \left (c+d\,x\right )}{a\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{9/2}\,\left (\cos \left (c+d\,x\right )+1\right )} \,d x \]

[In]

int(1/((e*cot(c + d*x))^(9/2)*(a + a/cos(c + d*x))),x)

[Out]

int(cos(c + d*x)/(a*(e*cot(c + d*x))^(9/2)*(cos(c + d*x) + 1)), x)