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

Optimal. Leaf size=375 \[ \frac {2 a^2 \cot (c+d x)}{d (e \cot (c+d x))^{3/2}}+\frac {a^2 \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d \tan ^{\frac {3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}-\frac {a^2 \tan ^{-1}\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d \tan ^{\frac {3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}+\frac {2 a^2 \tan (c+d x)}{5 d (e \cot (c+d x))^{3/2}}+\frac {4 a^2 \csc (c+d x)}{3 d (e \cot (c+d x))^{3/2}}+\frac {a^2 \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \tan ^{\frac {3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}-\frac {a^2 \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \tan ^{\frac {3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}-\frac {2 a^2 \sqrt {\sin (2 c+2 d x)} \cot (c+d x) \csc (c+d x) F\left (\left .c+d x-\frac {\pi }{4}\right |2\right )}{3 d (e \cot (c+d x))^{3/2}} \]

[Out]

2*a^2*cot(d*x+c)/d/(e*cot(d*x+c))^(3/2)+4/3*a^2*csc(d*x+c)/d/(e*cot(d*x+c))^(3/2)+2/3*a^2*cot(d*x+c)*csc(d*x+c
)*(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))*sin(2*d*x+2*c)^(1/2)/d/(e
*cot(d*x+c))^(3/2)-1/2*a^2*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))/d/(e*cot(d*x+c))^(3/2)*2^(1/2)/tan(d*x+c)^(3/2)
-1/2*a^2*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))/d/(e*cot(d*x+c))^(3/2)*2^(1/2)/tan(d*x+c)^(3/2)+1/4*a^2*ln(1-2^(1/
2)*tan(d*x+c)^(1/2)+tan(d*x+c))/d/(e*cot(d*x+c))^(3/2)*2^(1/2)/tan(d*x+c)^(3/2)-1/4*a^2*ln(1+2^(1/2)*tan(d*x+c
)^(1/2)+tan(d*x+c))/d/(e*cot(d*x+c))^(3/2)*2^(1/2)/tan(d*x+c)^(3/2)+2/5*a^2*tan(d*x+c)/d/(e*cot(d*x+c))^(3/2)

________________________________________________________________________________________

Rubi [A]  time = 0.33, antiderivative size = 375, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 17, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.680, Rules used = {3900, 3886, 3473, 3476, 329, 211, 1165, 628, 1162, 617, 204, 2611, 2614, 2573, 2641, 2607, 30} \[ \frac {2 a^2 \cot (c+d x)}{d (e \cot (c+d x))^{3/2}}+\frac {a^2 \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d \tan ^{\frac {3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}-\frac {a^2 \tan ^{-1}\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d \tan ^{\frac {3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}+\frac {2 a^2 \tan (c+d x)}{5 d (e \cot (c+d x))^{3/2}}+\frac {4 a^2 \csc (c+d x)}{3 d (e \cot (c+d x))^{3/2}}+\frac {a^2 \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \tan ^{\frac {3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}-\frac {a^2 \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d \tan ^{\frac {3}{2}}(c+d x) (e \cot (c+d x))^{3/2}}-\frac {2 a^2 \sqrt {\sin (2 c+2 d x)} \cot (c+d x) \csc (c+d x) F\left (\left .c+d x-\frac {\pi }{4}\right |2\right )}{3 d (e \cot (c+d x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a^2*Cot[c + d*x])/(d*(e*Cot[c + d*x])^(3/2)) + (4*a^2*Csc[c + d*x])/(3*d*(e*Cot[c + d*x])^(3/2)) - (2*a^2*C
ot[c + d*x]*Csc[c + d*x]*EllipticF[c - Pi/4 + d*x, 2]*Sqrt[Sin[2*c + 2*d*x]])/(3*d*(e*Cot[c + d*x])^(3/2)) + (
a^2*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d*(e*Cot[c + d*x])^(3/2)*Tan[c + d*x]^(3/2)) - (a^2*ArcTa
n[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(Sqrt[2]*d*(e*Cot[c + d*x])^(3/2)*Tan[c + d*x]^(3/2)) + (a^2*Log[1 - Sqrt[2
]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d*(e*Cot[c + d*x])^(3/2)*Tan[c + d*x]^(3/2)) - (a^2*Log[1 + S
qrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(2*Sqrt[2]*d*(e*Cot[c + d*x])^(3/2)*Tan[c + d*x]^(3/2)) + (2*a^2*Ta
n[c + d*x])/(5*d*(e*Cot[c + d*x])^(3/2))

Rule 30

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

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 211

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 329

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 617

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 628

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 1162

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 1165

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 2573

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 2607

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 2611

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(a*Sec[e
+ f*x])^m*(b*Tan[e + f*x])^(n - 1))/(f*(m + n - 1)), x] - Dist[(b^2*(n - 1))/(m + 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] && GtQ[n, 1] && NeQ[m + n - 1, 0] && Integers
Q[2*m, 2*n]

Rule 2614

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 2641

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ
[{c, d}, x]

Rule 3473

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

Rule 3476

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 3886

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 3900

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

________________________________________________________________________________________

Mathematica [C]  time = 6.27, size = 127, normalized size = 0.34 \[ \frac {a^2 \sin ^2(c+d x) (\sec (c+d x)+1)^2 \sec ^4\left (\frac {1}{2} \cot ^{-1}(\cot (c+d x))\right ) \left (\, _2F_1\left (-\frac {5}{4},1;-\frac {1}{4};-\cot ^2(c+d x)\right )+2 \left (\, _2F_1\left (\frac {1}{2},\frac {5}{4};\frac {9}{4};-\tan ^2(c+d x)\right )+5 \cot ^2(c+d x) \, _2F_1\left (-\frac {1}{4},1;\frac {3}{4};-\cot ^2(c+d x)\right )\right )\right )}{10 d e \sqrt {e \cot (c+d x)}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(a^2*(Hypergeometric2F1[-5/4, 1, -1/4, -Cot[c + d*x]^2] + 2*(5*Cot[c + d*x]^2*Hypergeometric2F1[-1/4, 1, 3/4,
-Cot[c + d*x]^2] + Hypergeometric2F1[1/2, 5/4, 9/4, -Tan[c + d*x]^2]))*(1 + Sec[c + d*x])^2*Sec[ArcCot[Cot[c +
d*x]]/2]^4*Sin[c + d*x]^2)/(10*d*e*Sqrt[e*Cot[c + d*x]])

________________________________________________________________________________________

fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a \sec \left (d x + c\right ) + a\right )}^{2}}{\left (e \cot \left (d x + c\right )\right )^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [C]  time = 2.28, size = 721, normalized size = 1.92 \[ -\frac {a^{2} \left (-1+\cos \left (d x +c \right )\right ) \left (15 i \sin \left (d x +c \right ) \EllipticPi \left (\sqrt {\frac {1-\cos \left (d x +c \right )+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right ) \sqrt {\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \sqrt {\frac {-1+\cos \left (d x +c \right )+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \sqrt {\frac {1-\cos \left (d x +c \right )+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \left (\cos ^{2}\left (d x +c \right )\right )-15 i \sin \left (d x +c \right ) \EllipticPi \left (\sqrt {\frac {1-\cos \left (d x +c \right )+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right ) \sqrt {\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \sqrt {\frac {-1+\cos \left (d x +c \right )+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \sqrt {\frac {1-\cos \left (d x +c \right )+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \left (\cos ^{2}\left (d x +c \right )\right )-15 \sin \left (d x +c \right ) \EllipticPi \left (\sqrt {\frac {1-\cos \left (d x +c \right )+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right ) \sqrt {\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \sqrt {\frac {-1+\cos \left (d x +c \right )+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \sqrt {\frac {1-\cos \left (d x +c \right )+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \left (\cos ^{2}\left (d x +c \right )\right )+10 \sin \left (d x +c \right ) \EllipticF \left (\sqrt {\frac {1-\cos \left (d x +c \right )+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}, \frac {\sqrt {2}}{2}\right ) \sqrt {\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \sqrt {\frac {-1+\cos \left (d x +c \right )+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \sqrt {\frac {1-\cos \left (d x +c \right )+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \left (\cos ^{2}\left (d x +c \right )\right )-15 \sin \left (d x +c \right ) \EllipticPi \left (\sqrt {\frac {1-\cos \left (d x +c \right )+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}, \frac {1}{2}-\frac {i}{2}, \frac {\sqrt {2}}{2}\right ) \sqrt {\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \sqrt {\frac {-1+\cos \left (d x +c \right )+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \sqrt {\frac {1-\cos \left (d x +c \right )+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \left (\cos ^{2}\left (d x +c \right )\right )-24 \sqrt {2}\, \left (\cos ^{3}\left (d x +c \right )\right )+4 \left (\cos ^{2}\left (d x +c \right )\right ) \sqrt {2}+14 \cos \left (d x +c \right ) \sqrt {2}+6 \sqrt {2}\right ) \left (1+\cos \left (d x +c \right )\right )^{2} \sqrt {2}}{30 d \left (\frac {e \cos \left (d x +c \right )}{\sin \left (d x +c \right )}\right )^{\frac {3}{2}} \sin \left (d x +c \right )^{5} \cos \left (d x +c \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/30*a^2/d*(-1+cos(d*x+c))*(15*I*sin(d*x+c)*EllipticPi(((1-cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2),1/2+1/2*I
,1/2*2^(1/2))*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*((-1+cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2)*((1-cos(d*x+c)+
sin(d*x+c))/sin(d*x+c))^(1/2)*cos(d*x+c)^2-15*I*sin(d*x+c)*EllipticPi(((1-cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(
1/2),1/2-1/2*I,1/2*2^(1/2))*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*((-1+cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2)*(
(1-cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2)*cos(d*x+c)^2-15*sin(d*x+c)*EllipticPi(((1-cos(d*x+c)+sin(d*x+c))/s
in(d*x+c))^(1/2),1/2+1/2*I,1/2*2^(1/2))*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*((-1+cos(d*x+c)+sin(d*x+c))/sin(d*x
+c))^(1/2)*((1-cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2)*cos(d*x+c)^2+10*sin(d*x+c)*EllipticF(((1-cos(d*x+c)+si
n(d*x+c))/sin(d*x+c))^(1/2),1/2*2^(1/2))*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*((-1+cos(d*x+c)+sin(d*x+c))/sin(d*
x+c))^(1/2)*((1-cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2)*cos(d*x+c)^2-15*sin(d*x+c)*EllipticPi(((1-cos(d*x+c)+
sin(d*x+c))/sin(d*x+c))^(1/2),1/2-1/2*I,1/2*2^(1/2))*((-1+cos(d*x+c))/sin(d*x+c))^(1/2)*((-1+cos(d*x+c)+sin(d*
x+c))/sin(d*x+c))^(1/2)*((1-cos(d*x+c)+sin(d*x+c))/sin(d*x+c))^(1/2)*cos(d*x+c)^2-24*2^(1/2)*cos(d*x+c)^3+4*co
s(d*x+c)^2*2^(1/2)+14*cos(d*x+c)*2^(1/2)+6*2^(1/2))*(1+cos(d*x+c))^2/(e*cos(d*x+c)/sin(d*x+c))^(3/2)/sin(d*x+c
)^5/cos(d*x+c)*2^(1/2)

________________________________________________________________________________________

maxima [A]  time = 0.49, size = 216, normalized size = 0.58 \[ \frac {5 \, a^{2} e {\left (\frac {\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {e} + 2 \, \sqrt {\frac {e}{\tan \left (d x + c\right )}}\right )}}{2 \, \sqrt {e}}\right )}{\sqrt {e}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {e} - 2 \, \sqrt {\frac {e}{\tan \left (d x + c\right )}}\right )}}{2 \, \sqrt {e}}\right )}{\sqrt {e}} - \frac {\sqrt {2} \log \left (\sqrt {2} \sqrt {e} \sqrt {\frac {e}{\tan \left (d x + c\right )}} + e + \frac {e}{\tan \left (d x + c\right )}\right )}{\sqrt {e}} + \frac {\sqrt {2} \log \left (-\sqrt {2} \sqrt {e} \sqrt {\frac {e}{\tan \left (d x + c\right )}} + e + \frac {e}{\tan \left (d x + c\right )}\right )}{\sqrt {e}}}{e^{2}} + \frac {8}{e^{2} \sqrt {\frac {e}{\tan \left (d x + c\right )}}}\right )} + \frac {8 \, a^{2} \tan \left (d x + c\right )}{\left (\frac {e}{\tan \left (d x + c\right )}\right )^{\frac {3}{2}}}}{20 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/20*(5*a^2*e*((2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*sqrt(e) + 2*sqrt(e/tan(d*x + c)))/sqrt(e))/sqrt(e) + 2*s
qrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2)*sqrt(e) - 2*sqrt(e/tan(d*x + c)))/sqrt(e))/sqrt(e) - sqrt(2)*log(sqrt(2)*s
qrt(e)*sqrt(e/tan(d*x + c)) + e + e/tan(d*x + c))/sqrt(e) + sqrt(2)*log(-sqrt(2)*sqrt(e)*sqrt(e/tan(d*x + c))
+ e + e/tan(d*x + c))/sqrt(e))/e^2 + 8/(e^2*sqrt(e/tan(d*x + c)))) + 8*a^2*tan(d*x + c)/(e/tan(d*x + c))^(3/2)
)/d

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^2}{{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ a^{2} \left (\int \frac {1}{\left (e \cot {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx + \int \frac {2 \sec {\left (c + d x \right )}}{\left (e \cot {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx + \int \frac {\sec ^{2}{\left (c + d x \right )}}{\left (e \cot {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________