\(\int (e \cot (c+d x))^{3/2} (a+a \sec (c+d x))^2 \, dx\) [239]

Optimal result

Integrand size = 25, antiderivative size = 343 \[ \int (e \cot (c+d x))^{3/2} (a+a \sec (c+d x))^2 \, dx=-\frac {4 a^2 (e \cot (c+d x))^{3/2} \sin (c+d x)}{d}-\frac {4 a^2 (e \cot (c+d x))^{3/2} \tan (c+d x)}{d}-\frac {4 a^2 (e \cot (c+d x))^{3/2} E\left (\left .c-\frac {\pi }{4}+d x\right |2\right ) \sin (c+d x) \tan (c+d x)}{d \sqrt {\sin (2 c+2 d x)}}+\frac {a^2 \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right ) (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)}{\sqrt {2} d}-\frac {a^2 \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right ) (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)}{\sqrt {2} d}-\frac {a^2 (e \cot (c+d x))^{3/2} \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right ) \tan ^{\frac {3}{2}}(c+d x)}{2 \sqrt {2} d}+\frac {a^2 (e \cot (c+d x))^{3/2} \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right ) \tan ^{\frac {3}{2}}(c+d x)}{2 \sqrt {2} d} \]

[Out]

-4*a^2*(e*cot(d*x+c))^(3/2)*sin(d*x+c)/d-4*a^2*(e*cot(d*x+c))^(3/2)*tan(d*x+c)/d+4*a^2*(e*cot(d*x+c))^(3/2)*(s
in(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)/d/sin(2
*d*x+2*c)^(1/2)-1/2*a^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)/d*2^(1/2)-1/
2*a^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)/d*2^(1/2)-1/4*a^2*(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)/d*2^(1/2)+1/4*a^2*(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)/d*2^(1/2)

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 343, 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, 303, 1176, 631, 210, 1179, 642, 2688, 2695, 2652, 2719, 2687, 30} \[ \int (e \cot (c+d x))^{3/2} (a+a \sec (c+d x))^2 \, dx=\frac {a^2 \tan ^{\frac {3}{2}}(c+d x) \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right ) (e \cot (c+d x))^{3/2}}{\sqrt {2} d}-\frac {a^2 \tan ^{\frac {3}{2}}(c+d x) \arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right ) (e \cot (c+d x))^{3/2}}{\sqrt {2} d}-\frac {4 a^2 \sin (c+d x) (e \cot (c+d x))^{3/2}}{d}-\frac {4 a^2 \tan (c+d x) (e \cot (c+d x))^{3/2}}{d}-\frac {a^2 \tan ^{\frac {3}{2}}(c+d x) (e \cot (c+d x))^{3/2} \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d}+\frac {a^2 \tan ^{\frac {3}{2}}(c+d x) (e \cot (c+d x))^{3/2} \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d}-\frac {4 a^2 \sin (c+d x) \tan (c+d x) E\left (\left .c+d x-\frac {\pi }{4}\right |2\right ) (e \cot (c+d x))^{3/2}}{d \sqrt {\sin (2 c+2 d x)}} \]

[In]

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

[Out]

(-4*a^2*(e*Cot[c + d*x])^(3/2)*Sin[c + d*x])/d - (4*a^2*(e*Cot[c + d*x])^(3/2)*Tan[c + d*x])/d - (4*a^2*(e*Cot
[c + d*x])^(3/2)*EllipticE[c - Pi/4 + d*x, 2]*Sin[c + d*x]*Tan[c + d*x])/(d*Sqrt[Sin[2*c + 2*d*x]]) + (a^2*Arc
Tan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]]*(e*Cot[c + d*x])^(3/2)*Tan[c + d*x]^(3/2))/(Sqrt[2]*d) - (a^2*ArcTan[1 + S
qrt[2]*Sqrt[Tan[c + d*x]]]*(e*Cot[c + d*x])^(3/2)*Tan[c + d*x]^(3/2))/(Sqrt[2]*d) - (a^2*(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]*d) + (a^2*(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]*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 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 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 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 {(a+a \sec (c+d x))^2}{\tan ^{\frac {3}{2}}(c+d x)} \, dx \\ & = \left ((e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)\right ) \int \left (\frac {a^2}{\tan ^{\frac {3}{2}}(c+d x)}+\frac {2 a^2 \sec (c+d x)}{\tan ^{\frac {3}{2}}(c+d x)}+\frac {a^2 \sec ^2(c+d x)}{\tan ^{\frac {3}{2}}(c+d x)}\right ) \, dx \\ & = \left (a^2 (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)\right ) \int \frac {1}{\tan ^{\frac {3}{2}}(c+d x)} \, dx+\left (a^2 (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)\right ) \int \frac {\sec ^2(c+d x)}{\tan ^{\frac {3}{2}}(c+d x)} \, dx+\left (2 a^2 (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)\right ) \int \frac {\sec (c+d x)}{\tan ^{\frac {3}{2}}(c+d x)} \, dx \\ & = -\frac {4 a^2 (e \cot (c+d x))^{3/2} \sin (c+d x)}{d}-\frac {2 a^2 (e \cot (c+d x))^{3/2} \tan (c+d x)}{d}-\left (a^2 (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)\right ) \int \sqrt {\tan (c+d x)} \, dx-\left (4 a^2 (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+\frac {\left (a^2 (e \cot (c+d x))^{3/2} \tan ^{\frac {3}{2}}(c+d x)\right ) \text {Subst}\left (\int \frac {1}{x^{3/2}} \, dx,x,\tan (c+d x)\right )}{d} \\ & = -\frac {4 a^2 (e \cot (c+d x))^{3/2} \sin (c+d x)}{d}-\frac {4 a^2 (e \cot (c+d x))^{3/2} \tan (c+d x)}{d}-\frac {\left (4 a^2 (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}{\cos ^{\frac {3}{2}}(c+d x)}-\frac {\left (a^2 (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 )}{d} \\ & = -\frac {4 a^2 (e \cot (c+d x))^{3/2} \sin (c+d x)}{d}-\frac {4 a^2 (e \cot (c+d x))^{3/2} \tan (c+d x)}{d}-\frac {\left (4 a^2 (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}{\sqrt {\sin (2 c+2 d x)}}-\frac {\left (2 a^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 )}{d} \\ & = -\frac {4 a^2 (e \cot (c+d x))^{3/2} \sin (c+d x)}{d}-\frac {4 a^2 (e \cot (c+d x))^{3/2} \tan (c+d x)}{d}-\frac {4 a^2 (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)}{d \sqrt {\sin (2 c+2 d x)}}+\frac {\left (a^2 (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 )}{d}-\frac {\left (a^2 (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 )}{d} \\ & = -\frac {4 a^2 (e \cot (c+d x))^{3/2} \sin (c+d x)}{d}-\frac {4 a^2 (e \cot (c+d x))^{3/2} \tan (c+d x)}{d}-\frac {4 a^2 (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)}{d \sqrt {\sin (2 c+2 d x)}}-\frac {\left (a^2 (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 d}-\frac {\left (a^2 (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 d}-\frac {\left (a^2 (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} d}-\frac {\left (a^2 (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} d} \\ & = -\frac {4 a^2 (e \cot (c+d x))^{3/2} \sin (c+d x)}{d}-\frac {4 a^2 (e \cot (c+d x))^{3/2} \tan (c+d x)}{d}-\frac {4 a^2 (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)}{d \sqrt {\sin (2 c+2 d x)}}-\frac {a^2 (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} d}+\frac {a^2 (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} d}-\frac {\left (a^2 (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} d}+\frac {\left (a^2 (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} d} \\ & = -\frac {4 a^2 (e \cot (c+d x))^{3/2} \sin (c+d x)}{d}-\frac {4 a^2 (e \cot (c+d x))^{3/2} \tan (c+d x)}{d}-\frac {4 a^2 (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)}{d \sqrt {\sin (2 c+2 d x)}}+\frac {a^2 \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} d}-\frac {a^2 \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} d}-\frac {a^2 (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} d}+\frac {a^2 (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} 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 = 14.44 (sec) , antiderivative size = 220, normalized size of antiderivative = 0.64 \[ \int (e \cot (c+d x))^{3/2} (a+a \sec (c+d x))^2 \, dx=-\frac {a^2 \cos ^4\left (\frac {1}{2} (c+d x)\right ) (e \cot (c+d x))^{3/2} \left (2 \sqrt {2} \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )-2 \sqrt {2} \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )+16 \sqrt {\cot (c+d x)}+16 \sqrt {\cot (c+d x)} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{2},\frac {3}{4},-\tan ^2(c+d x)\right )+\sqrt {2} \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )-\sqrt {2} \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )\right ) \sec ^4\left (\frac {1}{2} \cot ^{-1}(\cot (c+d x))\right )}{4 d \cot ^{\frac {3}{2}}(c+d x)} \]

[In]

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

[Out]

-1/4*(a^2*Cos[(c + d*x)/2]^4*(e*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]]] + 16*Sqrt[Cot[c + d*x]] + 16*Sqrt[Cot[c + d*x]]*Hypergeometric2F1[-
1/4, 1/2, 3/4, -Tan[c + d*x]^2] + 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]])*Sec[ArcCot[Cot[c + d*x]]/2]^4)/(d*Cot[c + d*x]^(3/2))

Maple [A] (verified)

Time = 10.33 (sec) , antiderivative size = 524, normalized size of antiderivative = 1.53

method	result	size
parts	\(-\frac {2 a^{2} e \left (\sqrt {e \cot \left (d x +c \right )}-\frac {\left (e^{2}\right )^{\frac {1}{4}} \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}\right )}{d}-\frac {2 a^{2} e \sqrt {e \cot \left (d x +c \right )}}{d}-\frac {2 a^{2} \sqrt {2}\, e \sqrt {e \cot \left (d x +c \right )}\, \left (-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 {EllipticE}\left (\sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\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 )-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 {EllipticE}\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 )+\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 )+\sqrt {2}\right )}{d}\)	\(524\)
default	\(\text {Expression too large to display}\)	\(1037\)

[In]

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

[Out]

-2*a^2/d*e*((e*cot(d*x+c))^(1/2)-1/8*(e^2)^(1/4)*2^(1/2)*(ln((e*cot(d*x+c)+(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)*2^
(1/2)+(e^2)^(1/2))/(e*cot(d*x+c)-(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)*2^(1/2)+(e^2)^(1/2)))+2*arctan(2^(1/2)/(e^2)
^(1/4)*(e*cot(d*x+c))^(1/2)+1)-2*arctan(-2^(1/2)/(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)+1)))-2*a^2/d*e*(e*cot(d*x+c)
)^(1/2)-2*a^2/d*2^(1/2)*e*(e*cot(d*x+c))^(1/2)*(-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)*EllipticE((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)*EllipticF((csc(d*x+c)-cot(d*x+c)+1)^
(1/2),1/2*2^(1/2))-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)*EllipticE((csc(d*x+c)-cot(d*x+c)+1)^(1/2),1/2*2^(1/2))*sec(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)*EllipticF((csc(d*x+c)-cot(d*x+c)+1)^(1/2),1/2*2^(1/2))*
sec(d*x+c)+2^(1/2))

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int (e \cot (c+d x))^{3/2} (a+a \sec (c+d x))^2 \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((e*cot(d*x+c))^(3/2)*(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]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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