\(\int \frac {1}{(a+a \sec (c+d x))^2 \sqrt {e \tan (c+d x)}} \, dx\) [134]

Optimal result

Integrand size = 25, antiderivative size = 365 \[ \int \frac {1}{(a+a \sec (c+d x))^2 \sqrt {e \tan (c+d x)}} \, dx=-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} a^2 d \sqrt {e}}+\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} a^2 d \sqrt {e}}-\frac {\log \left (\sqrt {e}+\sqrt {e} \tan (c+d x)-\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} a^2 d \sqrt {e}}+\frac {\log \left (\sqrt {e}+\sqrt {e} \tan (c+d x)+\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} a^2 d \sqrt {e}}-\frac {4 e^3}{7 a^2 d (e \tan (c+d x))^{7/2}}+\frac {4 e^3 \sec (c+d x)}{7 a^2 d (e \tan (c+d x))^{7/2}}+\frac {2 e}{3 a^2 d (e \tan (c+d x))^{3/2}}-\frac {20 e \sec (c+d x)}{21 a^2 d (e \tan (c+d x))^{3/2}}-\frac {10 \operatorname {EllipticF}\left (c-\frac {\pi }{4}+d x,2\right ) \sec (c+d x) \sqrt {\sin (2 c+2 d x)}}{21 a^2 d \sqrt {e \tan (c+d x)}} \]

[Out]

-1/2*arctan(1-2^(1/2)*(e*tan(d*x+c))^(1/2)/e^(1/2))/a^2/d*2^(1/2)/e^(1/2)+1/2*arctan(1+2^(1/2)*(e*tan(d*x+c))^
(1/2)/e^(1/2))/a^2/d*2^(1/2)/e^(1/2)-1/4*ln(e^(1/2)-2^(1/2)*(e*tan(d*x+c))^(1/2)+e^(1/2)*tan(d*x+c))/a^2/d*2^(
1/2)/e^(1/2)+1/4*ln(e^(1/2)+2^(1/2)*(e*tan(d*x+c))^(1/2)+e^(1/2)*tan(d*x+c))/a^2/d*2^(1/2)/e^(1/2)+10/21*(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)*sin(2*d*x+2*c)^(1/2)/
a^2/d/(e*tan(d*x+c))^(1/2)-4/7*e^3/a^2/d/(e*tan(d*x+c))^(7/2)+4/7*e^3*sec(d*x+c)/a^2/d/(e*tan(d*x+c))^(7/2)+2/
3*e/a^2/d/(e*tan(d*x+c))^(3/2)-20/21*e*sec(d*x+c)/a^2/d/(e*tan(d*x+c))^(3/2)

Rubi [A] (verified)

Time = 0.67 (sec) , antiderivative size = 365, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.680, Rules used = {3973, 3971, 3555, 3557, 335, 217, 1179, 642, 1176, 631, 210, 2689, 2694, 2653, 2720, 2687, 32} \[ \int \frac {1}{(a+a \sec (c+d x))^2 \sqrt {e \tan (c+d x)}} \, dx=-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} a^2 d \sqrt {e}}+\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} a^2 d \sqrt {e}}-\frac {4 e^3}{7 a^2 d (e \tan (c+d x))^{7/2}}+\frac {4 e^3 \sec (c+d x)}{7 a^2 d (e \tan (c+d x))^{7/2}}+\frac {2 e}{3 a^2 d (e \tan (c+d x))^{3/2}}-\frac {\log \left (\sqrt {e} \tan (c+d x)-\sqrt {2} \sqrt {e \tan (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} a^2 d \sqrt {e}}+\frac {\log \left (\sqrt {e} \tan (c+d x)+\sqrt {2} \sqrt {e \tan (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} a^2 d \sqrt {e}}-\frac {20 e \sec (c+d x)}{21 a^2 d (e \tan (c+d x))^{3/2}}-\frac {10 \sqrt {\sin (2 c+2 d x)} \sec (c+d x) \operatorname {EllipticF}\left (c+d x-\frac {\pi }{4},2\right )}{21 a^2 d \sqrt {e \tan (c+d x)}} \]

[In]

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

[Out]

-(ArcTan[1 - (Sqrt[2]*Sqrt[e*Tan[c + d*x]])/Sqrt[e]]/(Sqrt[2]*a^2*d*Sqrt[e])) + ArcTan[1 + (Sqrt[2]*Sqrt[e*Tan
[c + d*x]])/Sqrt[e]]/(Sqrt[2]*a^2*d*Sqrt[e]) - Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] - Sqrt[2]*Sqrt[e*Tan[c + d*x
]]]/(2*Sqrt[2]*a^2*d*Sqrt[e]) + Log[Sqrt[e] + Sqrt[e]*Tan[c + d*x] + Sqrt[2]*Sqrt[e*Tan[c + d*x]]]/(2*Sqrt[2]*
a^2*d*Sqrt[e]) - (4*e^3)/(7*a^2*d*(e*Tan[c + d*x])^(7/2)) + (4*e^3*Sec[c + d*x])/(7*a^2*d*(e*Tan[c + d*x])^(7/
2)) + (2*e)/(3*a^2*d*(e*Tan[c + d*x])^(3/2)) - (20*e*Sec[c + d*x])/(21*a^2*d*(e*Tan[c + d*x])^(3/2)) - (10*Ell
ipticF[c - Pi/4 + d*x, 2]*Sec[c + d*x]*Sqrt[Sin[2*c + 2*d*x]])/(21*a^2*d*Sqrt[e*Tan[c + d*x]])

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[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 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]

Rubi steps \begin{align*} \text {integral}& = \frac {e^4 \int \frac {(-a+a \sec (c+d x))^2}{(e \tan (c+d x))^{9/2}} \, dx}{a^4} \\ & = \frac {e^4 \int \left (\frac {a^2}{(e \tan (c+d x))^{9/2}}-\frac {2 a^2 \sec (c+d x)}{(e \tan (c+d x))^{9/2}}+\frac {a^2 \sec ^2(c+d x)}{(e \tan (c+d x))^{9/2}}\right ) \, dx}{a^4} \\ & = \frac {e^4 \int \frac {1}{(e \tan (c+d x))^{9/2}} \, dx}{a^2}+\frac {e^4 \int \frac {\sec ^2(c+d x)}{(e \tan (c+d x))^{9/2}} \, dx}{a^2}-\frac {\left (2 e^4\right ) \int \frac {\sec (c+d x)}{(e \tan (c+d x))^{9/2}} \, dx}{a^2} \\ & = -\frac {2 e^3}{7 a^2 d (e \tan (c+d x))^{7/2}}+\frac {4 e^3 \sec (c+d x)}{7 a^2 d (e \tan (c+d x))^{7/2}}-\frac {e^2 \int \frac {1}{(e \tan (c+d x))^{5/2}} \, dx}{a^2}+\frac {\left (10 e^2\right ) \int \frac {\sec (c+d x)}{(e \tan (c+d x))^{5/2}} \, dx}{7 a^2}+\frac {e^4 \text {Subst}\left (\int \frac {1}{(e x)^{9/2}} \, dx,x,\tan (c+d x)\right )}{a^2 d} \\ & = -\frac {4 e^3}{7 a^2 d (e \tan (c+d x))^{7/2}}+\frac {4 e^3 \sec (c+d x)}{7 a^2 d (e \tan (c+d x))^{7/2}}+\frac {2 e}{3 a^2 d (e \tan (c+d x))^{3/2}}-\frac {20 e \sec (c+d x)}{21 a^2 d (e \tan (c+d x))^{3/2}}-\frac {10 \int \frac {\sec (c+d x)}{\sqrt {e \tan (c+d x)}} \, dx}{21 a^2}+\frac {\int \frac {1}{\sqrt {e \tan (c+d x)}} \, dx}{a^2} \\ & = -\frac {4 e^3}{7 a^2 d (e \tan (c+d x))^{7/2}}+\frac {4 e^3 \sec (c+d x)}{7 a^2 d (e \tan (c+d x))^{7/2}}+\frac {2 e}{3 a^2 d (e \tan (c+d x))^{3/2}}-\frac {20 e \sec (c+d x)}{21 a^2 d (e \tan (c+d x))^{3/2}}+\frac {e \text {Subst}\left (\int \frac {1}{\sqrt {x} \left (e^2+x^2\right )} \, dx,x,e \tan (c+d x)\right )}{a^2 d}-\frac {\left (10 \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {\sin (c+d x)}} \, dx}{21 a^2 \sqrt {\cos (c+d x)} \sqrt {e \tan (c+d x)}} \\ & = -\frac {4 e^3}{7 a^2 d (e \tan (c+d x))^{7/2}}+\frac {4 e^3 \sec (c+d x)}{7 a^2 d (e \tan (c+d x))^{7/2}}+\frac {2 e}{3 a^2 d (e \tan (c+d x))^{3/2}}-\frac {20 e \sec (c+d x)}{21 a^2 d (e \tan (c+d x))^{3/2}}+\frac {(2 e) \text {Subst}\left (\int \frac {1}{e^2+x^4} \, dx,x,\sqrt {e \tan (c+d x)}\right )}{a^2 d}-\frac {\left (10 \sec (c+d x) \sqrt {\sin (2 c+2 d x)}\right ) \int \frac {1}{\sqrt {\sin (2 c+2 d x)}} \, dx}{21 a^2 \sqrt {e \tan (c+d x)}} \\ & = -\frac {4 e^3}{7 a^2 d (e \tan (c+d x))^{7/2}}+\frac {4 e^3 \sec (c+d x)}{7 a^2 d (e \tan (c+d x))^{7/2}}+\frac {2 e}{3 a^2 d (e \tan (c+d x))^{3/2}}-\frac {20 e \sec (c+d x)}{21 a^2 d (e \tan (c+d x))^{3/2}}-\frac {10 \operatorname {EllipticF}\left (c-\frac {\pi }{4}+d x,2\right ) \sec (c+d x) \sqrt {\sin (2 c+2 d x)}}{21 a^2 d \sqrt {e \tan (c+d x)}}+\frac {\text {Subst}\left (\int \frac {e-x^2}{e^2+x^4} \, dx,x,\sqrt {e \tan (c+d x)}\right )}{a^2 d}+\frac {\text {Subst}\left (\int \frac {e+x^2}{e^2+x^4} \, dx,x,\sqrt {e \tan (c+d x)}\right )}{a^2 d} \\ & = -\frac {4 e^3}{7 a^2 d (e \tan (c+d x))^{7/2}}+\frac {4 e^3 \sec (c+d x)}{7 a^2 d (e \tan (c+d x))^{7/2}}+\frac {2 e}{3 a^2 d (e \tan (c+d x))^{3/2}}-\frac {20 e \sec (c+d x)}{21 a^2 d (e \tan (c+d x))^{3/2}}-\frac {10 \operatorname {EllipticF}\left (c-\frac {\pi }{4}+d x,2\right ) \sec (c+d x) \sqrt {\sin (2 c+2 d x)}}{21 a^2 d \sqrt {e \tan (c+d x)}}+\frac {\text {Subst}\left (\int \frac {1}{e-\sqrt {2} \sqrt {e} x+x^2} \, dx,x,\sqrt {e \tan (c+d x)}\right )}{2 a^2 d}+\frac {\text {Subst}\left (\int \frac {1}{e+\sqrt {2} \sqrt {e} x+x^2} \, dx,x,\sqrt {e \tan (c+d x)}\right )}{2 a^2 d}-\frac {\text {Subst}\left (\int \frac {\sqrt {2} \sqrt {e}+2 x}{-e-\sqrt {2} \sqrt {e} x-x^2} \, dx,x,\sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} a^2 d \sqrt {e}}-\frac {\text {Subst}\left (\int \frac {\sqrt {2} \sqrt {e}-2 x}{-e+\sqrt {2} \sqrt {e} x-x^2} \, dx,x,\sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} a^2 d \sqrt {e}} \\ & = -\frac {\log \left (\sqrt {e}+\sqrt {e} \tan (c+d x)-\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} a^2 d \sqrt {e}}+\frac {\log \left (\sqrt {e}+\sqrt {e} \tan (c+d x)+\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} a^2 d \sqrt {e}}-\frac {4 e^3}{7 a^2 d (e \tan (c+d x))^{7/2}}+\frac {4 e^3 \sec (c+d x)}{7 a^2 d (e \tan (c+d x))^{7/2}}+\frac {2 e}{3 a^2 d (e \tan (c+d x))^{3/2}}-\frac {20 e \sec (c+d x)}{21 a^2 d (e \tan (c+d x))^{3/2}}-\frac {10 \operatorname {EllipticF}\left (c-\frac {\pi }{4}+d x,2\right ) \sec (c+d x) \sqrt {\sin (2 c+2 d x)}}{21 a^2 d \sqrt {e \tan (c+d x)}}+\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} a^2 d \sqrt {e}}-\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} a^2 d \sqrt {e}} \\ & = -\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} a^2 d \sqrt {e}}+\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} a^2 d \sqrt {e}}-\frac {\log \left (\sqrt {e}+\sqrt {e} \tan (c+d x)-\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} a^2 d \sqrt {e}}+\frac {\log \left (\sqrt {e}+\sqrt {e} \tan (c+d x)+\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} a^2 d \sqrt {e}}-\frac {4 e^3}{7 a^2 d (e \tan (c+d x))^{7/2}}+\frac {4 e^3 \sec (c+d x)}{7 a^2 d (e \tan (c+d x))^{7/2}}+\frac {2 e}{3 a^2 d (e \tan (c+d x))^{3/2}}-\frac {20 e \sec (c+d x)}{21 a^2 d (e \tan (c+d x))^{3/2}}-\frac {10 \operatorname {EllipticF}\left (c-\frac {\pi }{4}+d x,2\right ) \sec (c+d x) \sqrt {\sin (2 c+2 d x)}}{21 a^2 d \sqrt {e \tan (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 = 12.51 (sec) , antiderivative size = 1281, normalized size of antiderivative = 3.51 \[ \int \frac {1}{(a+a \sec (c+d x))^2 \sqrt {e \tan (c+d x)}} \, dx=\frac {40 e^{-i (c+d x)} \sqrt {-\frac {i \left (-1+e^{2 i (c+d x)}\right )}{1+e^{2 i (c+d x)}}} \left (1+e^{2 i (c+d x)}\right ) \cos ^4\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec (2 c) \sec ^2(c+d x) \sqrt {\tan (c+d x)}}{21 d (a+a \sec (c+d x))^2 \sqrt {e \tan (c+d x)}}+\frac {e^{-2 i c} \sqrt {-\frac {i \left (-1+e^{2 i (c+d x)}\right )}{1+e^{2 i (c+d x)}}} \left (e^{4 i c} \sqrt {-1+e^{4 i (c+d x)}} \arctan \left (\sqrt {-1+e^{4 i (c+d x)}}\right )+2 \sqrt {-1+e^{2 i (c+d x)}} \sqrt {1+e^{2 i (c+d x)}} \text {arctanh}\left (\sqrt {\frac {-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right )\right ) \cos ^4\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec (2 c) \sec ^2(c+d x) \sqrt {\tan (c+d x)}}{d \left (-1+e^{2 i (c+d x)}\right ) (a+a \sec (c+d x))^2 \sqrt {e \tan (c+d x)}}+\frac {e^{-2 i c} \sqrt {-\frac {i \left (-1+e^{2 i (c+d x)}\right )}{1+e^{2 i (c+d x)}}} \left (\sqrt {-1+e^{4 i (c+d x)}} \arctan \left (\sqrt {-1+e^{4 i (c+d x)}}\right )+2 e^{4 i c} \sqrt {-1+e^{2 i (c+d x)}} \sqrt {1+e^{2 i (c+d x)}} \text {arctanh}\left (\sqrt {\frac {-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right )\right ) \cos ^4\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec (2 c) \sec ^2(c+d x) \sqrt {\tan (c+d x)}}{d \left (-1+e^{2 i (c+d x)}\right ) (a+a \sec (c+d x))^2 \sqrt {e \tan (c+d x)}}-\frac {2 e^{-i (2 c+d x)} \sqrt {-\frac {i \left (-1+e^{2 i (c+d x)}\right )}{1+e^{2 i (c+d x)}}} \cos ^4\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (3 \left (-1+e^{4 i (c+d x)}\right )+e^{4 i (c+d x)} \left (-1+e^{2 i c}\right ) \sqrt {1-e^{4 i (c+d x)}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},e^{4 i (c+d x)}\right )\right ) \sec (2 c) \sec ^2(c+d x) \sqrt {\tan (c+d x)}}{3 d \left (-1+e^{2 i (c+d x)}\right ) (a+a \sec (c+d x))^2 \sqrt {e \tan (c+d x)}}+\frac {2 e^{-i d x} \sqrt {-\frac {i \left (-1+e^{2 i (c+d x)}\right )}{1+e^{2 i (c+d x)}}} \cos ^4\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (3-3 e^{4 i (c+d x)}+e^{2 i (c+2 d x)} \left (-1+e^{2 i c}\right ) \sqrt {1-e^{4 i (c+d x)}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},e^{4 i (c+d x)}\right )\right ) \sec (2 c) \sec ^2(c+d x) \sqrt {\tan (c+d x)}}{3 d \left (-1+e^{2 i (c+d x)}\right ) (a+a \sec (c+d x))^2 \sqrt {e \tan (c+d x)}}+\frac {\cos ^4\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec ^2(c+d x) \left (-\frac {104}{21 d}+\frac {4 (21-20 \cos (c)+21 \cos (2 c)) \cos (d x) \sec (2 c)}{21 d}+\frac {64 \sec ^2\left (\frac {c}{2}+\frac {d x}{2}\right )}{21 d}-\frac {2 \sec ^4\left (\frac {c}{2}+\frac {d x}{2}\right )}{7 d}-\frac {4 \sec (2 c) (-20 \sin (c)+21 \sin (2 c)) \sin (d x)}{21 d}\right ) \tan (c+d x)}{(a+a \sec (c+d x))^2 \sqrt {e \tan (c+d x)}}+\frac {80 \sqrt [4]{-1} \cos ^4\left (\frac {c}{2}+\frac {d x}{2}\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt [4]{-1} \sqrt {\tan (c+d x)}\right ),-1\right ) \sec ^5(c+d x) \sqrt {\tan (c+d x)}}{21 d (a+a \sec (c+d x))^2 \sqrt {e \tan (c+d x)} \left (1+\tan ^2(c+d x)\right )^{3/2}} \]

[In]

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

[Out]

(40*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*(1 + E^((2*I)*(c + d*x)))*Cos[c/2 + (d*x
)/2]^4*Sec[2*c]*Sec[c + d*x]^2*Sqrt[Tan[c + d*x]])/(21*d*E^(I*(c + d*x))*(a + a*Sec[c + d*x])^2*Sqrt[e*Tan[c +
d*x]]) + (Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*(E^((4*I)*c)*Sqrt[-1 + E^((4*I)*(
c + d*x))]*ArcTan[Sqrt[-1 + E^((4*I)*(c + d*x))]] + 2*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[1 + E^((2*I)*(c + d*
x))]*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]])*Cos[c/2 + (d*x)/2]^4*Sec[2*c]*Sec[c
+ d*x]^2*Sqrt[Tan[c + d*x]])/(d*E^((2*I)*c)*(-1 + E^((2*I)*(c + d*x)))*(a + a*Sec[c + d*x])^2*Sqrt[e*Tan[c + d
*x]]) + (Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*(Sqrt[-1 + E^((4*I)*(c + d*x))]*Arc
Tan[Sqrt[-1 + E^((4*I)*(c + d*x))]] + 2*E^((4*I)*c)*Sqrt[-1 + E^((2*I)*(c + d*x))]*Sqrt[1 + E^((2*I)*(c + d*x)
)]*ArcTanh[Sqrt[(-1 + E^((2*I)*(c + d*x)))/(1 + E^((2*I)*(c + d*x)))]])*Cos[c/2 + (d*x)/2]^4*Sec[2*c]*Sec[c +
d*x]^2*Sqrt[Tan[c + d*x]])/(d*E^((2*I)*c)*(-1 + E^((2*I)*(c + d*x)))*(a + a*Sec[c + d*x])^2*Sqrt[e*Tan[c + d*x
]]) - (2*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 + E^((2*I)*(c + d*x)))]*Cos[c/2 + (d*x)/2]^4*(3*(-1 + E^((4
*I)*(c + d*x))) + E^((4*I)*(c + d*x))*(-1 + E^((2*I)*c))*Sqrt[1 - E^((4*I)*(c + d*x))]*Hypergeometric2F1[1/2,
3/4, 7/4, E^((4*I)*(c + d*x))])*Sec[2*c]*Sec[c + d*x]^2*Sqrt[Tan[c + d*x]])/(3*d*E^(I*(2*c + d*x))*(-1 + E^((2
*I)*(c + d*x)))*(a + a*Sec[c + d*x])^2*Sqrt[e*Tan[c + d*x]]) + (2*Sqrt[((-I)*(-1 + E^((2*I)*(c + d*x))))/(1 +
E^((2*I)*(c + d*x)))]*Cos[c/2 + (d*x)/2]^4*(3 - 3*E^((4*I)*(c + d*x)) + E^((2*I)*(c + 2*d*x))*(-1 + E^((2*I)*c
))*Sqrt[1 - E^((4*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, E^((4*I)*(c + d*x))])*Sec[2*c]*Sec[c + d*x]^
2*Sqrt[Tan[c + d*x]])/(3*d*E^(I*d*x)*(-1 + E^((2*I)*(c + d*x)))*(a + a*Sec[c + d*x])^2*Sqrt[e*Tan[c + d*x]]) +
(Cos[c/2 + (d*x)/2]^4*Sec[c + d*x]^2*(-104/(21*d) + (4*(21 - 20*Cos[c] + 21*Cos[2*c])*Cos[d*x]*Sec[2*c])/(21*
d) + (64*Sec[c/2 + (d*x)/2]^2)/(21*d) - (2*Sec[c/2 + (d*x)/2]^4)/(7*d) - (4*Sec[2*c]*(-20*Sin[c] + 21*Sin[2*c]
)*Sin[d*x])/(21*d))*Tan[c + d*x])/((a + a*Sec[c + d*x])^2*Sqrt[e*Tan[c + d*x]]) + (80*(-1)^(1/4)*Cos[c/2 + (d*
x)/2]^4*EllipticF[I*ArcSinh[(-1)^(1/4)*Sqrt[Tan[c + d*x]]], -1]*Sec[c + d*x]^5*Sqrt[Tan[c + d*x]])/(21*d*(a +
a*Sec[c + d*x])^2*Sqrt[e*Tan[c + d*x]]*(1 + Tan[c + d*x]^2)^(3/2))

Maple [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 5.60 (sec) , antiderivative size = 633, normalized size of antiderivative = 1.73

method	result	size
default	\(-\frac {\sqrt {2}\, \left (21 i \sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}\, \sqrt {2-2 \csc \left (d x +c \right )+2 \cot \left (d x +c \right )}\, \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 )-21 i \sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}\, \sqrt {2-2 \csc \left (d x +c \right )+2 \cot \left (d x +c \right )}\, \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 \left (1-\cos \left (d x +c \right )\right )^{5} \csc \left (d x +c \right )^{5}-62 \sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}\, \sqrt {2-2 \csc \left (d x +c \right )+2 \cot \left (d x +c \right )}\, \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 )+21 \sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}\, \sqrt {2-2 \csc \left (d x +c \right )+2 \cot \left (d x +c \right )}\, \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 )+21 \sqrt {\csc \left (d x +c \right )-\cot \left (d x +c \right )+1}\, \sqrt {2-2 \csc \left (d x +c \right )+2 \cot \left (d x +c \right )}\, \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 )+26 \left (1-\cos \left (d x +c \right )\right )^{3} \csc \left (d x +c \right )^{3}-23 \csc \left (d x +c \right )+23 \cot \left (d x +c \right )\right ) \left (1-\cos \left (d x +c \right )\right ) \csc \left (d x +c \right )}{42 a^{2} d \sqrt {\left (1-\cos \left (d x +c \right )\right )^{3} \csc \left (d x +c \right )^{3}+\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \sqrt {\left (1-\cos \left (d x +c \right )\right ) \left (\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1\right ) \csc \left (d x +c \right )}\, \sqrt {-\frac {e \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}{\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1}}}\)	\(633\)

[In]

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

[Out]

-1/42/a^2/d*2^(1/2)*(21*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))-21*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))-3*(1-cos(d*x+c))^5*csc(d*x+c)^5-62*(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)*EllipticF((csc(d*x+c)-cot(d*x+c)+1)^(1/2),1/2*2^(1/2))+21*
(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))+21*(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))
+26*(1-cos(d*x+c))^3*csc(d*x+c)^3-23*csc(d*x+c)+23*cot(d*x+c))*(1-cos(d*x+c))/((1-cos(d*x+c))^3*csc(d*x+c)^3+c
ot(d*x+c)-csc(d*x+c))^(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)/(-e/((1-cos(d*
x+c))^2*csc(d*x+c)^2-1)*(-cot(d*x+c)+csc(d*x+c)))^(1/2)*csc(d*x+c)

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F]

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

[In]

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

[Out]

Integral(1/(sqrt(e*tan(c + d*x))*sec(c + d*x)**2 + 2*sqrt(e*tan(c + d*x))*sec(c + d*x) + sqrt(e*tan(c + d*x)))
, x)/a**2

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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