3.246 \(\int \frac{1}{(a+b \sec (c+d x))^2 \sqrt{e \sin (c+d x)}} \, dx\)

Optimal. Leaf size=838 \[ \frac{3 \Pi \left (\frac{2 a}{a-\sqrt{a^2-b^2}};\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{\sin (c+d x)} b^4}{2 a^2 \left (a^2-b^2\right ) \left (a^2-\sqrt{a^2-b^2} a-b^2\right ) d \sqrt{e \sin (c+d x)}}+\frac{3 \Pi \left (\frac{2 a}{a+\sqrt{a^2-b^2}};\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{\sin (c+d x)} b^4}{2 a^2 \left (a^2-b^2\right ) \left (a^2+\sqrt{a^2-b^2} a-b^2\right ) d \sqrt{e \sin (c+d x)}}-\frac{3 \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt{e}}\right ) b^3}{2 a^{3/2} \left (a^2-b^2\right )^{7/4} d \sqrt{e}}-\frac{3 \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt{e}}\right ) b^3}{2 a^{3/2} \left (a^2-b^2\right )^{7/4} d \sqrt{e}}+\frac{\sqrt{e \sin (c+d x)} b^2}{a \left (a^2-b^2\right ) d e (b+a \cos (c+d x))}+\frac{\text{EllipticF}\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right ),2\right ) \sqrt{\sin (c+d x)} b^2}{a^2 \left (a^2-b^2\right ) d \sqrt{e \sin (c+d x)}}+\frac{2 \Pi \left (\frac{2 a}{a-\sqrt{a^2-b^2}};\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{\sin (c+d x)} b^2}{a^2 \left (a^2-\sqrt{a^2-b^2} a-b^2\right ) d \sqrt{e \sin (c+d x)}}+\frac{2 \Pi \left (\frac{2 a}{a+\sqrt{a^2-b^2}};\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{\sin (c+d x)} b^2}{a^2 \left (a^2+\sqrt{a^2-b^2} a-b^2\right ) d \sqrt{e \sin (c+d x)}}-\frac{2 \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt{e}}\right ) b}{a^{3/2} \left (a^2-b^2\right )^{3/4} d \sqrt{e}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt{e}}\right ) b}{a^{3/2} \left (a^2-b^2\right )^{3/4} d \sqrt{e}}+\frac{2 \text{EllipticF}\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right ),2\right ) \sqrt{\sin (c+d x)}}{a^2 d \sqrt{e \sin (c+d x)}} \]

[Out]

(-3*b^3*ArcTan[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(2*a^(3/2)*(a^2 - b^2)^(7/4)*d*Sqr
t[e]) - (2*b*ArcTan[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(a^(3/2)*(a^2 - b^2)^(3/4)*d*
Sqrt[e]) - (3*b^3*ArcTanh[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(2*a^(3/2)*(a^2 - b^2)^
(7/4)*d*Sqrt[e]) - (2*b*ArcTanh[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(a^(3/2)*(a^2 - b
^2)^(3/4)*d*Sqrt[e]) + (2*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(a^2*d*Sqrt[e*Sin[c + d*x]]) +
(b^2*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(a^2*(a^2 - b^2)*d*Sqrt[e*Sin[c + d*x]]) + (2*b^2*El
lipticPi[(2*a)/(a - Sqrt[a^2 - b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(a^2*(a^2 - b^2 - a*Sqrt[a^2
- b^2])*d*Sqrt[e*Sin[c + d*x]]) + (3*b^4*EllipticPi[(2*a)/(a - Sqrt[a^2 - b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[S
in[c + d*x]])/(2*a^2*(a^2 - b^2)*(a^2 - b^2 - a*Sqrt[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]]) + (2*b^2*EllipticPi[(
2*a)/(a + Sqrt[a^2 - b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(a^2*(a^2 - b^2 + a*Sqrt[a^2 - b^2])*d*
Sqrt[e*Sin[c + d*x]]) + (3*b^4*EllipticPi[(2*a)/(a + Sqrt[a^2 - b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x
]])/(2*a^2*(a^2 - b^2)*(a^2 - b^2 + a*Sqrt[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]]) + (b^2*Sqrt[e*Sin[c + d*x]])/(a
*(a^2 - b^2)*d*e*(b + a*Cos[c + d*x]))

________________________________________________________________________________________

Rubi [A]  time = 1.92404, antiderivative size = 838, normalized size of antiderivative = 1., number of steps used = 27, number of rules used = 13, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.52, Rules used = {3872, 2912, 2642, 2641, 2694, 2867, 2702, 2807, 2805, 329, 212, 208, 205} \[ \frac{3 \Pi \left (\frac{2 a}{a-\sqrt{a^2-b^2}};\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{\sin (c+d x)} b^4}{2 a^2 \left (a^2-b^2\right ) \left (a^2-\sqrt{a^2-b^2} a-b^2\right ) d \sqrt{e \sin (c+d x)}}+\frac{3 \Pi \left (\frac{2 a}{a+\sqrt{a^2-b^2}};\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{\sin (c+d x)} b^4}{2 a^2 \left (a^2-b^2\right ) \left (a^2+\sqrt{a^2-b^2} a-b^2\right ) d \sqrt{e \sin (c+d x)}}-\frac{3 \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt{e}}\right ) b^3}{2 a^{3/2} \left (a^2-b^2\right )^{7/4} d \sqrt{e}}-\frac{3 \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt{e}}\right ) b^3}{2 a^{3/2} \left (a^2-b^2\right )^{7/4} d \sqrt{e}}+\frac{\sqrt{e \sin (c+d x)} b^2}{a \left (a^2-b^2\right ) d e (b+a \cos (c+d x))}+\frac{F\left (\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{\sin (c+d x)} b^2}{a^2 \left (a^2-b^2\right ) d \sqrt{e \sin (c+d x)}}+\frac{2 \Pi \left (\frac{2 a}{a-\sqrt{a^2-b^2}};\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{\sin (c+d x)} b^2}{a^2 \left (a^2-\sqrt{a^2-b^2} a-b^2\right ) d \sqrt{e \sin (c+d x)}}+\frac{2 \Pi \left (\frac{2 a}{a+\sqrt{a^2-b^2}};\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{\sin (c+d x)} b^2}{a^2 \left (a^2+\sqrt{a^2-b^2} a-b^2\right ) d \sqrt{e \sin (c+d x)}}-\frac{2 \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt{e}}\right ) b}{a^{3/2} \left (a^2-b^2\right )^{3/4} d \sqrt{e}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt{e}}\right ) b}{a^{3/2} \left (a^2-b^2\right )^{3/4} d \sqrt{e}}+\frac{2 F\left (\left .\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{\sin (c+d x)}}{a^2 d \sqrt{e \sin (c+d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*b^3*ArcTan[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(2*a^(3/2)*(a^2 - b^2)^(7/4)*d*Sqr
t[e]) - (2*b*ArcTan[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(a^(3/2)*(a^2 - b^2)^(3/4)*d*
Sqrt[e]) - (3*b^3*ArcTanh[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(2*a^(3/2)*(a^2 - b^2)^
(7/4)*d*Sqrt[e]) - (2*b*ArcTanh[(Sqrt[a]*Sqrt[e*Sin[c + d*x]])/((a^2 - b^2)^(1/4)*Sqrt[e])])/(a^(3/2)*(a^2 - b
^2)^(3/4)*d*Sqrt[e]) + (2*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(a^2*d*Sqrt[e*Sin[c + d*x]]) +
(b^2*EllipticF[(c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(a^2*(a^2 - b^2)*d*Sqrt[e*Sin[c + d*x]]) + (2*b^2*El
lipticPi[(2*a)/(a - Sqrt[a^2 - b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(a^2*(a^2 - b^2 - a*Sqrt[a^2
- b^2])*d*Sqrt[e*Sin[c + d*x]]) + (3*b^4*EllipticPi[(2*a)/(a - Sqrt[a^2 - b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[S
in[c + d*x]])/(2*a^2*(a^2 - b^2)*(a^2 - b^2 - a*Sqrt[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]]) + (2*b^2*EllipticPi[(
2*a)/(a + Sqrt[a^2 - b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x]])/(a^2*(a^2 - b^2 + a*Sqrt[a^2 - b^2])*d*
Sqrt[e*Sin[c + d*x]]) + (3*b^4*EllipticPi[(2*a)/(a + Sqrt[a^2 - b^2]), (c - Pi/2 + d*x)/2, 2]*Sqrt[Sin[c + d*x
]])/(2*a^2*(a^2 - b^2)*(a^2 - b^2 + a*Sqrt[a^2 - b^2])*d*Sqrt[e*Sin[c + d*x]]) + (b^2*Sqrt[e*Sin[c + d*x]])/(a
*(a^2 - b^2)*d*e*(b + a*Cos[c + d*x]))

Rule 3872

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[((g*C
os[e + f*x])^p*(b + a*Sin[e + f*x])^m)/Sin[e + f*x]^m, x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]

Rule 2912

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*
(x_)])^(m_), x_Symbol] :> Int[ExpandTrig[(g*cos[e + f*x])^p, (d*sin[e + f*x])^n*(a + b*sin[e + f*x])^m, x], x]
/; FreeQ[{a, b, d, e, f, g, n, p}, x] && NeQ[a^2 - b^2, 0] && IntegerQ[m] && (GtQ[m, 0] || IntegerQ[n])

Rule 2642

Int[1/Sqrt[(b_)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[Sin[c + d*x]]/Sqrt[b*Sin[c + d*x]], Int[1/Sqr
t[Sin[c + d*x]], x], x] /; FreeQ[{b, c, d}, 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 2694

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> -Simp[(b*(g
*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m + 1))/(f*g*(a^2 - b^2)*(m + 1)), x] + Dist[1/((a^2 - b^2)*(m +
1)), Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m + 1)*(a*(m + 1) - b*(m + p + 2)*Sin[e + f*x]), x], x] /; F
reeQ[{a, b, e, f, g, p}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && IntegersQ[2*m, 2*p]

Rule 2867

Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]))/((a_) + (b_.)*sin[(e_.) + (
f_.)*(x_)]), x_Symbol] :> Dist[d/b, Int[(g*Cos[e + f*x])^p, x], x] + Dist[(b*c - a*d)/b, Int[(g*Cos[e + f*x])^
p/(a + b*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[a^2 - b^2, 0]

Rule 2702

Int[1/(Sqrt[cos[(e_.) + (f_.)*(x_)]*(g_.)]*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> With[{q = Rt[
-a^2 + b^2, 2]}, -Dist[a/(2*q), Int[1/(Sqrt[g*Cos[e + f*x]]*(q + b*Cos[e + f*x])), x], x] + (Dist[(b*g)/f, Sub
st[Int[1/(Sqrt[x]*(g^2*(a^2 - b^2) + b^2*x^2)), x], x, g*Cos[e + f*x]], x] - Dist[a/(2*q), Int[1/(Sqrt[g*Cos[e
+ f*x]]*(q - b*Cos[e + f*x])), x], x])] /; FreeQ[{a, b, e, f, g}, x] && NeQ[a^2 - b^2, 0]

Rule 2807

Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Dist
[Sqrt[(c + d*Sin[e + f*x])/(c + d)]/Sqrt[c + d*Sin[e + f*x]], Int[1/((a + b*Sin[e + f*x])*Sqrt[c/(c + d) + (d*
Sin[e + f*x])/(c + d)]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && N
eQ[c^2 - d^2, 0] &&  !GtQ[c + d, 0]

Rule 2805

Int[1/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp
[(2*EllipticPi[(2*b)/(a + b), (1*(e - Pi/2 + f*x))/2, (2*d)/(c + d)])/(f*(a + b)*Sqrt[c + d]), x] /; FreeQ[{a,
b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[c + d, 0]

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 212

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b), 2]
]}, Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&
!GtQ[a/b, 0]

Rule 208

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

Rule 205

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

Rubi steps

\begin{align*} \int \frac{1}{(a+b \sec (c+d x))^2 \sqrt{e \sin (c+d x)}} \, dx &=\int \frac{\cos ^2(c+d x)}{(-b-a \cos (c+d x))^2 \sqrt{e \sin (c+d x)}} \, dx\\ &=\int \left (\frac{1}{a^2 \sqrt{e \sin (c+d x)}}+\frac{b^2}{a^2 (-b-a \cos (c+d x))^2 \sqrt{e \sin (c+d x)}}+\frac{2 b}{a^2 (-b-a \cos (c+d x)) \sqrt{e \sin (c+d x)}}\right ) \, dx\\ &=\frac{\int \frac{1}{\sqrt{e \sin (c+d x)}} \, dx}{a^2}+\frac{(2 b) \int \frac{1}{(-b-a \cos (c+d x)) \sqrt{e \sin (c+d x)}} \, dx}{a^2}+\frac{b^2 \int \frac{1}{(-b-a \cos (c+d x))^2 \sqrt{e \sin (c+d x)}} \, dx}{a^2}\\ &=\frac{b^2 \sqrt{e \sin (c+d x)}}{a \left (a^2-b^2\right ) d e (b+a \cos (c+d x))}+\frac{b^2 \int \frac{b-\frac{1}{2} a \cos (c+d x)}{(-b-a \cos (c+d x)) \sqrt{e \sin (c+d x)}} \, dx}{a^2 \left (a^2-b^2\right )}+\frac{b^2 \int \frac{1}{\sqrt{e \sin (c+d x)} \left (\sqrt{a^2-b^2}-a \sin (c+d x)\right )} \, dx}{a^2 \sqrt{a^2-b^2}}+\frac{b^2 \int \frac{1}{\sqrt{e \sin (c+d x)} \left (\sqrt{a^2-b^2}+a \sin (c+d x)\right )} \, dx}{a^2 \sqrt{a^2-b^2}}+\frac{(2 b e) \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \left (\left (-a^2+b^2\right ) e^2+a^2 x^2\right )} \, dx,x,e \sin (c+d x)\right )}{a d}+\frac{\sqrt{\sin (c+d x)} \int \frac{1}{\sqrt{\sin (c+d x)}} \, dx}{a^2 \sqrt{e \sin (c+d x)}}\\ &=\frac{2 F\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{a^2 d \sqrt{e \sin (c+d x)}}+\frac{b^2 \sqrt{e \sin (c+d x)}}{a \left (a^2-b^2\right ) d e (b+a \cos (c+d x))}+\frac{b^2 \int \frac{1}{\sqrt{e \sin (c+d x)}} \, dx}{2 a^2 \left (a^2-b^2\right )}+\frac{\left (3 b^3\right ) \int \frac{1}{(-b-a \cos (c+d x)) \sqrt{e \sin (c+d x)}} \, dx}{2 a^2 \left (a^2-b^2\right )}+\frac{(4 b e) \operatorname{Subst}\left (\int \frac{1}{\left (-a^2+b^2\right ) e^2+a^2 x^4} \, dx,x,\sqrt{e \sin (c+d x)}\right )}{a d}+\frac{\left (b^2 \sqrt{\sin (c+d x)}\right ) \int \frac{1}{\sqrt{\sin (c+d x)} \left (\sqrt{a^2-b^2}-a \sin (c+d x)\right )} \, dx}{a^2 \sqrt{a^2-b^2} \sqrt{e \sin (c+d x)}}+\frac{\left (b^2 \sqrt{\sin (c+d x)}\right ) \int \frac{1}{\sqrt{\sin (c+d x)} \left (\sqrt{a^2-b^2}+a \sin (c+d x)\right )} \, dx}{a^2 \sqrt{a^2-b^2} \sqrt{e \sin (c+d x)}}\\ &=\frac{2 F\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{a^2 d \sqrt{e \sin (c+d x)}}+\frac{2 b^2 \Pi \left (\frac{2 a}{a-\sqrt{a^2-b^2}};\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{a^2 \left (a^2-b^2-a \sqrt{a^2-b^2}\right ) d \sqrt{e \sin (c+d x)}}+\frac{2 b^2 \Pi \left (\frac{2 a}{a+\sqrt{a^2-b^2}};\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{a^2 \sqrt{a^2-b^2} \left (a+\sqrt{a^2-b^2}\right ) d \sqrt{e \sin (c+d x)}}+\frac{b^2 \sqrt{e \sin (c+d x)}}{a \left (a^2-b^2\right ) d e (b+a \cos (c+d x))}+\frac{\left (3 b^4\right ) \int \frac{1}{\sqrt{e \sin (c+d x)} \left (\sqrt{a^2-b^2}-a \sin (c+d x)\right )} \, dx}{4 a^2 \left (a^2-b^2\right )^{3/2}}+\frac{\left (3 b^4\right ) \int \frac{1}{\sqrt{e \sin (c+d x)} \left (\sqrt{a^2-b^2}+a \sin (c+d x)\right )} \, dx}{4 a^2 \left (a^2-b^2\right )^{3/2}}-\frac{(2 b) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a^2-b^2} e-a x^2} \, dx,x,\sqrt{e \sin (c+d x)}\right )}{a \sqrt{a^2-b^2} d}-\frac{(2 b) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a^2-b^2} e+a x^2} \, dx,x,\sqrt{e \sin (c+d x)}\right )}{a \sqrt{a^2-b^2} d}+\frac{\left (3 b^3 e\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \left (\left (-a^2+b^2\right ) e^2+a^2 x^2\right )} \, dx,x,e \sin (c+d x)\right )}{2 a \left (a^2-b^2\right ) d}+\frac{\left (b^2 \sqrt{\sin (c+d x)}\right ) \int \frac{1}{\sqrt{\sin (c+d x)}} \, dx}{2 a^2 \left (a^2-b^2\right ) \sqrt{e \sin (c+d x)}}\\ &=-\frac{2 b \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt{e}}\right )}{a^{3/2} \left (a^2-b^2\right )^{3/4} d \sqrt{e}}-\frac{2 b \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt{e}}\right )}{a^{3/2} \left (a^2-b^2\right )^{3/4} d \sqrt{e}}+\frac{2 F\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{a^2 d \sqrt{e \sin (c+d x)}}+\frac{b^2 F\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{a^2 \left (a^2-b^2\right ) d \sqrt{e \sin (c+d x)}}+\frac{2 b^2 \Pi \left (\frac{2 a}{a-\sqrt{a^2-b^2}};\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{a^2 \left (a^2-b^2-a \sqrt{a^2-b^2}\right ) d \sqrt{e \sin (c+d x)}}+\frac{2 b^2 \Pi \left (\frac{2 a}{a+\sqrt{a^2-b^2}};\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{a^2 \sqrt{a^2-b^2} \left (a+\sqrt{a^2-b^2}\right ) d \sqrt{e \sin (c+d x)}}+\frac{b^2 \sqrt{e \sin (c+d x)}}{a \left (a^2-b^2\right ) d e (b+a \cos (c+d x))}+\frac{\left (3 b^3 e\right ) \operatorname{Subst}\left (\int \frac{1}{\left (-a^2+b^2\right ) e^2+a^2 x^4} \, dx,x,\sqrt{e \sin (c+d x)}\right )}{a \left (a^2-b^2\right ) d}+\frac{\left (3 b^4 \sqrt{\sin (c+d x)}\right ) \int \frac{1}{\sqrt{\sin (c+d x)} \left (\sqrt{a^2-b^2}-a \sin (c+d x)\right )} \, dx}{4 a^2 \left (a^2-b^2\right )^{3/2} \sqrt{e \sin (c+d x)}}+\frac{\left (3 b^4 \sqrt{\sin (c+d x)}\right ) \int \frac{1}{\sqrt{\sin (c+d x)} \left (\sqrt{a^2-b^2}+a \sin (c+d x)\right )} \, dx}{4 a^2 \left (a^2-b^2\right )^{3/2} \sqrt{e \sin (c+d x)}}\\ &=-\frac{2 b \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt{e}}\right )}{a^{3/2} \left (a^2-b^2\right )^{3/4} d \sqrt{e}}-\frac{2 b \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt{e}}\right )}{a^{3/2} \left (a^2-b^2\right )^{3/4} d \sqrt{e}}+\frac{2 F\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{a^2 d \sqrt{e \sin (c+d x)}}+\frac{b^2 F\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{a^2 \left (a^2-b^2\right ) d \sqrt{e \sin (c+d x)}}-\frac{3 b^4 \Pi \left (\frac{2 a}{a-\sqrt{a^2-b^2}};\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{2 a^2 \left (a^2-b^2\right )^{3/2} \left (a-\sqrt{a^2-b^2}\right ) d \sqrt{e \sin (c+d x)}}+\frac{2 b^2 \Pi \left (\frac{2 a}{a-\sqrt{a^2-b^2}};\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{a^2 \left (a^2-b^2-a \sqrt{a^2-b^2}\right ) d \sqrt{e \sin (c+d x)}}+\frac{3 b^4 \Pi \left (\frac{2 a}{a+\sqrt{a^2-b^2}};\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{2 a^2 \left (a^2-b^2\right )^{3/2} \left (a+\sqrt{a^2-b^2}\right ) d \sqrt{e \sin (c+d x)}}+\frac{2 b^2 \Pi \left (\frac{2 a}{a+\sqrt{a^2-b^2}};\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{a^2 \sqrt{a^2-b^2} \left (a+\sqrt{a^2-b^2}\right ) d \sqrt{e \sin (c+d x)}}+\frac{b^2 \sqrt{e \sin (c+d x)}}{a \left (a^2-b^2\right ) d e (b+a \cos (c+d x))}-\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a^2-b^2} e-a x^2} \, dx,x,\sqrt{e \sin (c+d x)}\right )}{2 a \left (a^2-b^2\right )^{3/2} d}-\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a^2-b^2} e+a x^2} \, dx,x,\sqrt{e \sin (c+d x)}\right )}{2 a \left (a^2-b^2\right )^{3/2} d}\\ &=-\frac{3 b^3 \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt{e}}\right )}{2 a^{3/2} \left (a^2-b^2\right )^{7/4} d \sqrt{e}}-\frac{2 b \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt{e}}\right )}{a^{3/2} \left (a^2-b^2\right )^{3/4} d \sqrt{e}}-\frac{3 b^3 \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt{e}}\right )}{2 a^{3/2} \left (a^2-b^2\right )^{7/4} d \sqrt{e}}-\frac{2 b \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt{e}}\right )}{a^{3/2} \left (a^2-b^2\right )^{3/4} d \sqrt{e}}+\frac{2 F\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{a^2 d \sqrt{e \sin (c+d x)}}+\frac{b^2 F\left (\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{a^2 \left (a^2-b^2\right ) d \sqrt{e \sin (c+d x)}}-\frac{3 b^4 \Pi \left (\frac{2 a}{a-\sqrt{a^2-b^2}};\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{2 a^2 \left (a^2-b^2\right )^{3/2} \left (a-\sqrt{a^2-b^2}\right ) d \sqrt{e \sin (c+d x)}}+\frac{2 b^2 \Pi \left (\frac{2 a}{a-\sqrt{a^2-b^2}};\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{a^2 \left (a^2-b^2-a \sqrt{a^2-b^2}\right ) d \sqrt{e \sin (c+d x)}}+\frac{3 b^4 \Pi \left (\frac{2 a}{a+\sqrt{a^2-b^2}};\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{2 a^2 \left (a^2-b^2\right )^{3/2} \left (a+\sqrt{a^2-b^2}\right ) d \sqrt{e \sin (c+d x)}}+\frac{2 b^2 \Pi \left (\frac{2 a}{a+\sqrt{a^2-b^2}};\left .\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )\right |2\right ) \sqrt{\sin (c+d x)}}{a^2 \sqrt{a^2-b^2} \left (a+\sqrt{a^2-b^2}\right ) d \sqrt{e \sin (c+d x)}}+\frac{b^2 \sqrt{e \sin (c+d x)}}{a \left (a^2-b^2\right ) d e (b+a \cos (c+d x))}\\ \end{align*}

Mathematica [C]  time = 12.8795, size = 1246, normalized size = 1.49 \[ \frac{(b+a \cos (c+d x)) \sec (c+d x) \tan (c+d x) b^2}{a \left (a^2-b^2\right ) d (a+b \sec (c+d x))^2 \sqrt{e \sin (c+d x)}}+\frac{(b+a \cos (c+d x))^2 \sec ^2(c+d x) \sqrt{\sin (c+d x)} \left (\frac{2 \left (b^2-2 a^2\right ) \left (\sqrt{1-\sin ^2(c+d x)} a+b\right ) \left (\frac{b \left (-2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{a} \sqrt{\sin (c+d x)}}{\sqrt [4]{b^2-a^2}}\right )+2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{\sin (c+d x)}}{\sqrt [4]{b^2-a^2}}+1\right )-\log \left (a \sin (c+d x)-\sqrt{2} \sqrt{a} \sqrt [4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}\right )+\log \left (a \sin (c+d x)+\sqrt{2} \sqrt{a} \sqrt [4]{b^2-a^2} \sqrt{\sin (c+d x)}+\sqrt{b^2-a^2}\right )\right )}{4 \sqrt{2} \sqrt{a} \left (b^2-a^2\right )^{3/4}}-\frac{5 a \left (a^2-b^2\right ) F_1\left (\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right ) \sqrt{\sin (c+d x)} \sqrt{1-\sin ^2(c+d x)}}{\left (2 \left (2 F_1\left (\frac{5}{4};-\frac{1}{2},2;\frac{9}{4};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right ) a^2+\left (b^2-a^2\right ) F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right )\right ) \sin ^2(c+d x)+5 \left (a^2-b^2\right ) F_1\left (\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right )\right ) \left (\left (\sin ^2(c+d x)-1\right ) a^2+b^2\right )}\right ) \cos ^2(c+d x)}{(b+a \cos (c+d x)) \left (1-\sin ^2(c+d x)\right )}+\frac{4 a b \left (\sqrt{1-\sin ^2(c+d x)} a+b\right ) \left (\frac{5 b \left (a^2-b^2\right ) F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right ) \sqrt{\sin (c+d x)}}{\sqrt{1-\sin ^2(c+d x)} \left (2 \left (2 F_1\left (\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right ) a^2+\left (a^2-b^2\right ) F_1\left (\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right )\right ) \sin ^2(c+d x)+5 \left (a^2-b^2\right ) F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\sin ^2(c+d x),\frac{a^2 \sin ^2(c+d x)}{a^2-b^2}\right )\right ) \left (\left (\sin ^2(c+d x)-1\right ) a^2+b^2\right )}-\frac{\left (\frac{1}{8}-\frac{i}{8}\right ) \sqrt{a} \left (2 \tan ^{-1}\left (1-\frac{(1+i) \sqrt{a} \sqrt{\sin (c+d x)}}{\sqrt [4]{a^2-b^2}}\right )-2 \tan ^{-1}\left (\frac{(1+i) \sqrt{a} \sqrt{\sin (c+d x)}}{\sqrt [4]{a^2-b^2}}+1\right )+\log \left (i a \sin (c+d x)-(1+i) \sqrt{a} \sqrt [4]{a^2-b^2} \sqrt{\sin (c+d x)}+\sqrt{a^2-b^2}\right )-\log \left (i a \sin (c+d x)+(1+i) \sqrt{a} \sqrt [4]{a^2-b^2} \sqrt{\sin (c+d x)}+\sqrt{a^2-b^2}\right )\right )}{\left (a^2-b^2\right )^{3/4}}\right ) \cos (c+d x)}{(b+a \cos (c+d x)) \sqrt{1-\sin ^2(c+d x)}}\right )}{2 a (b-a) (a+b) d (a+b \sec (c+d x))^2 \sqrt{e \sin (c+d x)}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((b + a*Cos[c + d*x])^2*Sec[c + d*x]^2*Sqrt[Sin[c + d*x]]*((2*(-2*a^2 + b^2)*Cos[c + d*x]^2*(b + a*Sqrt[1 - Si
n[c + d*x]^2])*((b*(-2*ArcTan[1 - (Sqrt[2]*Sqrt[a]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] + 2*ArcTan[1 + (Sqr
t[2]*Sqrt[a]*Sqrt[Sin[c + d*x]])/(-a^2 + b^2)^(1/4)] - Log[Sqrt[-a^2 + b^2] - Sqrt[2]*Sqrt[a]*(-a^2 + b^2)^(1/
4)*Sqrt[Sin[c + d*x]] + a*Sin[c + d*x]] + Log[Sqrt[-a^2 + b^2] + Sqrt[2]*Sqrt[a]*(-a^2 + b^2)^(1/4)*Sqrt[Sin[c
+ d*x]] + a*Sin[c + d*x]]))/(4*Sqrt[2]*Sqrt[a]*(-a^2 + b^2)^(3/4)) - (5*a*(a^2 - b^2)*AppellF1[1/4, -1/2, 1,
5/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)]*Sqrt[Sin[c + d*x]]*Sqrt[1 - Sin[c + d*x]^2])/((5*(a^2 -
b^2)*AppellF1[1/4, -1/2, 1, 5/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)] + 2*(2*a^2*AppellF1[5/4, -
1/2, 2, 9/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)] + (-a^2 + b^2)*AppellF1[5/4, 1/2, 1, 9/4, Sin[c
+ d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)])*Sin[c + d*x]^2)*(b^2 + a^2*(-1 + Sin[c + d*x]^2)))))/((b + a*Cos
[c + d*x])*(1 - Sin[c + d*x]^2)) + (4*a*b*Cos[c + d*x]*(b + a*Sqrt[1 - Sin[c + d*x]^2])*(((-1/8 + I/8)*Sqrt[a]
*(2*ArcTan[1 - ((1 + I)*Sqrt[a]*Sqrt[Sin[c + d*x]])/(a^2 - b^2)^(1/4)] - 2*ArcTan[1 + ((1 + I)*Sqrt[a]*Sqrt[Si
n[c + d*x]])/(a^2 - b^2)^(1/4)] + Log[Sqrt[a^2 - b^2] - (1 + I)*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] +
I*a*Sin[c + d*x]] - Log[Sqrt[a^2 - b^2] + (1 + I)*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Sin[c + d*x]] + I*a*Sin[c +
d*x]]))/(a^2 - b^2)^(3/4) + (5*b*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(
a^2 - b^2)]*Sqrt[Sin[c + d*x]])/(Sqrt[1 - Sin[c + d*x]^2]*(5*(a^2 - b^2)*AppellF1[1/4, 1/2, 1, 5/4, Sin[c + d*
x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)] + 2*(2*a^2*AppellF1[5/4, 1/2, 2, 9/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x
]^2)/(a^2 - b^2)] + (a^2 - b^2)*AppellF1[5/4, 3/2, 1, 9/4, Sin[c + d*x]^2, (a^2*Sin[c + d*x]^2)/(a^2 - b^2)])*
Sin[c + d*x]^2)*(b^2 + a^2*(-1 + Sin[c + d*x]^2)))))/((b + a*Cos[c + d*x])*Sqrt[1 - Sin[c + d*x]^2])))/(2*a*(-
a + b)*(a + b)*d*(a + b*Sec[c + d*x])^2*Sqrt[e*Sin[c + d*x]]) + (b^2*(b + a*Cos[c + d*x])*Sec[c + d*x]*Tan[c +
d*x])/(a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2*Sqrt[e*Sin[c + d*x]])

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Maple [A]  time = 7.002, size = 1475, normalized size = 1.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*sec(d*x+c))^2/(e*sin(d*x+c))^(1/2),x)

[Out]

1/d/a*e*b^3/(a^2-b^2)*(e*sin(d*x+c))^(1/2)/(-a^2*cos(d*x+c)^2*e^2+b^2*e^2)+2/d*a*e*b/(a^2-b^2)*(e^2*(a^2-b^2)/
a^2)^(1/4)/(-a^2*e^2+b^2*e^2)*arctan((e*sin(d*x+c))^(1/2)/(e^2*(a^2-b^2)/a^2)^(1/4))-1/2/d/a*e*b^3/(a^2-b^2)*(
e^2*(a^2-b^2)/a^2)^(1/4)/(-a^2*e^2+b^2*e^2)*arctan((e*sin(d*x+c))^(1/2)/(e^2*(a^2-b^2)/a^2)^(1/4))+1/d*a*e*b/(
a^2-b^2)*(e^2*(a^2-b^2)/a^2)^(1/4)/(-a^2*e^2+b^2*e^2)*ln(((e*sin(d*x+c))^(1/2)+(e^2*(a^2-b^2)/a^2)^(1/4))/((e*
sin(d*x+c))^(1/2)-(e^2*(a^2-b^2)/a^2)^(1/4)))-1/4/d/a*e*b^3/(a^2-b^2)*(e^2*(a^2-b^2)/a^2)^(1/4)/(-a^2*e^2+b^2*
e^2)*ln(((e*sin(d*x+c))^(1/2)+(e^2*(a^2-b^2)/a^2)^(1/4))/((e*sin(d*x+c))^(1/2)-(e^2*(a^2-b^2)/a^2)^(1/4)))-1/d
/cos(d*x+c)/(e*sin(d*x+c))^(1/2)/a^2*(-sin(d*x+c)+1)^(1/2)*(2+2*sin(d*x+c))^(1/2)*sin(d*x+c)^(1/2)*EllipticF((
-sin(d*x+c)+1)^(1/2),1/2*2^(1/2))+3/2/d/cos(d*x+c)/(e*sin(d*x+c))^(1/2)/a^3*b^2/(a^2-b^2)^(1/2)*(-sin(d*x+c)+1
)^(1/2)*(2+2*sin(d*x+c))^(1/2)*sin(d*x+c)^(1/2)/(1-(a^2-b^2)^(1/2)/a)*EllipticPi((-sin(d*x+c)+1)^(1/2),1/(1-(a
^2-b^2)^(1/2)/a),1/2*2^(1/2))-3/2/d/cos(d*x+c)/(e*sin(d*x+c))^(1/2)/a^3*b^2/(a^2-b^2)^(1/2)*(-sin(d*x+c)+1)^(1
/2)*(2+2*sin(d*x+c))^(1/2)*sin(d*x+c)^(1/2)/(1+(a^2-b^2)^(1/2)/a)*EllipticPi((-sin(d*x+c)+1)^(1/2),1/(1+(a^2-b
^2)^(1/2)/a),1/2*2^(1/2))-1/d*sin(d*x+c)*cos(d*x+c)/(e*sin(d*x+c))^(1/2)*b^2/(a^2-b^2)/(-cos(d*x+c)^2*a^2+b^2)
-1/2/d/cos(d*x+c)/(e*sin(d*x+c))^(1/2)/a^2*b^2/(a^2-b^2)*(-sin(d*x+c)+1)^(1/2)*(2+2*sin(d*x+c))^(1/2)*sin(d*x+
c)^(1/2)*EllipticF((-sin(d*x+c)+1)^(1/2),1/2*2^(1/2))-1/2/d/cos(d*x+c)/(e*sin(d*x+c))^(1/2)/a*b^2/(a^2-b^2)^(3
/2)*(-sin(d*x+c)+1)^(1/2)*(2+2*sin(d*x+c))^(1/2)*sin(d*x+c)^(1/2)/(1-(a^2-b^2)^(1/2)/a)*EllipticPi((-sin(d*x+c
)+1)^(1/2),1/(1-(a^2-b^2)^(1/2)/a),1/2*2^(1/2))+5/4/d/cos(d*x+c)/(e*sin(d*x+c))^(1/2)/a^3*b^4/(a^2-b^2)^(3/2)*
(-sin(d*x+c)+1)^(1/2)*(2+2*sin(d*x+c))^(1/2)*sin(d*x+c)^(1/2)/(1-(a^2-b^2)^(1/2)/a)*EllipticPi((-sin(d*x+c)+1)
^(1/2),1/(1-(a^2-b^2)^(1/2)/a),1/2*2^(1/2))+1/2/d/cos(d*x+c)/(e*sin(d*x+c))^(1/2)/a*b^2/(a^2-b^2)^(3/2)*(-sin(
d*x+c)+1)^(1/2)*(2+2*sin(d*x+c))^(1/2)*sin(d*x+c)^(1/2)/(1+(a^2-b^2)^(1/2)/a)*EllipticPi((-sin(d*x+c)+1)^(1/2)
,1/(1+(a^2-b^2)^(1/2)/a),1/2*2^(1/2))-5/4/d/cos(d*x+c)/(e*sin(d*x+c))^(1/2)/a^3*b^4/(a^2-b^2)^(3/2)*(-sin(d*x+
c)+1)^(1/2)*(2+2*sin(d*x+c))^(1/2)*sin(d*x+c)^(1/2)/(1+(a^2-b^2)^(1/2)/a)*EllipticPi((-sin(d*x+c)+1)^(1/2),1/(
1+(a^2-b^2)^(1/2)/a),1/2*2^(1/2))

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Maxima [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b \sec \left (d x + c\right ) + a\right )}^{2} \sqrt{e \sin \left (d x + c\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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