3.1418 \(\int \frac{(g \cos (e+f x))^{3/2}}{\sqrt{d \sin (e+f x)} (a+b \sin (e+f x))} \, dx\)

Optimal. Leaf size=577 \[ \frac{2 \sqrt{2} g^2 \sqrt{b^2-a^2} \sqrt{\cos (e+f x)} \Pi \left (-\frac{a}{b-\sqrt{b^2-a^2}};\left .\sin ^{-1}\left (\frac{\sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{\cos (e+f x)+1}}\right )\right |-1\right )}{a b \sqrt{d} f \sqrt{g \cos (e+f x)}}-\frac{2 \sqrt{2} g^2 \sqrt{b^2-a^2} \sqrt{\cos (e+f x)} \Pi \left (-\frac{a}{b+\sqrt{b^2-a^2}};\left .\sin ^{-1}\left (\frac{\sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{\cos (e+f x)+1}}\right )\right |-1\right )}{a b \sqrt{d} f \sqrt{g \cos (e+f x)}}+\frac{g^2 \sqrt{\sin (2 e+2 f x)} F\left (\left .e+f x-\frac{\pi }{4}\right |2\right )}{a f \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}+\frac{g^{3/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{g} \sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{g \cos (e+f x)}}\right )}{\sqrt{2} b \sqrt{d} f}-\frac{g^{3/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{g} \sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{g \cos (e+f x)}}+1\right )}{\sqrt{2} b \sqrt{d} f}-\frac{g^{3/2} \log \left (-\frac{\sqrt{2} \sqrt{g} \sqrt{d \sin (e+f x)}}{\sqrt{g \cos (e+f x)}}+\sqrt{d} \tan (e+f x)+\sqrt{d}\right )}{2 \sqrt{2} b \sqrt{d} f}+\frac{g^{3/2} \log \left (\frac{\sqrt{2} \sqrt{g} \sqrt{d \sin (e+f x)}}{\sqrt{g \cos (e+f x)}}+\sqrt{d} \tan (e+f x)+\sqrt{d}\right )}{2 \sqrt{2} b \sqrt{d} f} \]

[Out]

(g^(3/2)*ArcTan[1 - (Sqrt[2]*Sqrt[g]*Sqrt[d*Sin[e + f*x]])/(Sqrt[d]*Sqrt[g*Cos[e + f*x]])])/(Sqrt[2]*b*Sqrt[d]
*f) - (g^(3/2)*ArcTan[1 + (Sqrt[2]*Sqrt[g]*Sqrt[d*Sin[e + f*x]])/(Sqrt[d]*Sqrt[g*Cos[e + f*x]])])/(Sqrt[2]*b*S
qrt[d]*f) + (2*Sqrt[2]*Sqrt[-a^2 + b^2]*g^2*Sqrt[Cos[e + f*x]]*EllipticPi[-(a/(b - Sqrt[-a^2 + b^2])), ArcSin[
Sqrt[d*Sin[e + f*x]]/(Sqrt[d]*Sqrt[1 + Cos[e + f*x]])], -1])/(a*b*Sqrt[d]*f*Sqrt[g*Cos[e + f*x]]) - (2*Sqrt[2]
*Sqrt[-a^2 + b^2]*g^2*Sqrt[Cos[e + f*x]]*EllipticPi[-(a/(b + Sqrt[-a^2 + b^2])), ArcSin[Sqrt[d*Sin[e + f*x]]/(
Sqrt[d]*Sqrt[1 + Cos[e + f*x]])], -1])/(a*b*Sqrt[d]*f*Sqrt[g*Cos[e + f*x]]) - (g^(3/2)*Log[Sqrt[d] - (Sqrt[2]*
Sqrt[g]*Sqrt[d*Sin[e + f*x]])/Sqrt[g*Cos[e + f*x]] + Sqrt[d]*Tan[e + f*x]])/(2*Sqrt[2]*b*Sqrt[d]*f) + (g^(3/2)
*Log[Sqrt[d] + (Sqrt[2]*Sqrt[g]*Sqrt[d*Sin[e + f*x]])/Sqrt[g*Cos[e + f*x]] + Sqrt[d]*Tan[e + f*x]])/(2*Sqrt[2]
*b*Sqrt[d]*f) + (g^2*EllipticF[e - Pi/4 + f*x, 2]*Sqrt[Sin[2*e + 2*f*x]])/(a*f*Sqrt[g*Cos[e + f*x]]*Sqrt[d*Sin
[e + f*x]])

________________________________________________________________________________________

Rubi [A]  time = 0.972302, antiderivative size = 577, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 14, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.378, Rules used = {2900, 2838, 2573, 2641, 2574, 297, 1162, 617, 204, 1165, 628, 2908, 2907, 1218} \[ \frac{2 \sqrt{2} g^2 \sqrt{b^2-a^2} \sqrt{\cos (e+f x)} \Pi \left (-\frac{a}{b-\sqrt{b^2-a^2}};\left .\sin ^{-1}\left (\frac{\sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{\cos (e+f x)+1}}\right )\right |-1\right )}{a b \sqrt{d} f \sqrt{g \cos (e+f x)}}-\frac{2 \sqrt{2} g^2 \sqrt{b^2-a^2} \sqrt{\cos (e+f x)} \Pi \left (-\frac{a}{b+\sqrt{b^2-a^2}};\left .\sin ^{-1}\left (\frac{\sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{\cos (e+f x)+1}}\right )\right |-1\right )}{a b \sqrt{d} f \sqrt{g \cos (e+f x)}}+\frac{g^2 \sqrt{\sin (2 e+2 f x)} F\left (\left .e+f x-\frac{\pi }{4}\right |2\right )}{a f \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}+\frac{g^{3/2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{g} \sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{g \cos (e+f x)}}\right )}{\sqrt{2} b \sqrt{d} f}-\frac{g^{3/2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{g} \sqrt{d \sin (e+f x)}}{\sqrt{d} \sqrt{g \cos (e+f x)}}+1\right )}{\sqrt{2} b \sqrt{d} f}-\frac{g^{3/2} \log \left (-\frac{\sqrt{2} \sqrt{g} \sqrt{d \sin (e+f x)}}{\sqrt{g \cos (e+f x)}}+\sqrt{d} \tan (e+f x)+\sqrt{d}\right )}{2 \sqrt{2} b \sqrt{d} f}+\frac{g^{3/2} \log \left (\frac{\sqrt{2} \sqrt{g} \sqrt{d \sin (e+f x)}}{\sqrt{g \cos (e+f x)}}+\sqrt{d} \tan (e+f x)+\sqrt{d}\right )}{2 \sqrt{2} b \sqrt{d} f} \]

Antiderivative was successfully verified.

[In]

Int[(g*Cos[e + f*x])^(3/2)/(Sqrt[d*Sin[e + f*x]]*(a + b*Sin[e + f*x])),x]

[Out]

(g^(3/2)*ArcTan[1 - (Sqrt[2]*Sqrt[g]*Sqrt[d*Sin[e + f*x]])/(Sqrt[d]*Sqrt[g*Cos[e + f*x]])])/(Sqrt[2]*b*Sqrt[d]
*f) - (g^(3/2)*ArcTan[1 + (Sqrt[2]*Sqrt[g]*Sqrt[d*Sin[e + f*x]])/(Sqrt[d]*Sqrt[g*Cos[e + f*x]])])/(Sqrt[2]*b*S
qrt[d]*f) + (2*Sqrt[2]*Sqrt[-a^2 + b^2]*g^2*Sqrt[Cos[e + f*x]]*EllipticPi[-(a/(b - Sqrt[-a^2 + b^2])), ArcSin[
Sqrt[d*Sin[e + f*x]]/(Sqrt[d]*Sqrt[1 + Cos[e + f*x]])], -1])/(a*b*Sqrt[d]*f*Sqrt[g*Cos[e + f*x]]) - (2*Sqrt[2]
*Sqrt[-a^2 + b^2]*g^2*Sqrt[Cos[e + f*x]]*EllipticPi[-(a/(b + Sqrt[-a^2 + b^2])), ArcSin[Sqrt[d*Sin[e + f*x]]/(
Sqrt[d]*Sqrt[1 + Cos[e + f*x]])], -1])/(a*b*Sqrt[d]*f*Sqrt[g*Cos[e + f*x]]) - (g^(3/2)*Log[Sqrt[d] - (Sqrt[2]*
Sqrt[g]*Sqrt[d*Sin[e + f*x]])/Sqrt[g*Cos[e + f*x]] + Sqrt[d]*Tan[e + f*x]])/(2*Sqrt[2]*b*Sqrt[d]*f) + (g^(3/2)
*Log[Sqrt[d] + (Sqrt[2]*Sqrt[g]*Sqrt[d*Sin[e + f*x]])/Sqrt[g*Cos[e + f*x]] + Sqrt[d]*Tan[e + f*x]])/(2*Sqrt[2]
*b*Sqrt[d]*f) + (g^2*EllipticF[e - Pi/4 + f*x, 2]*Sqrt[Sin[2*e + 2*f*x]])/(a*f*Sqrt[g*Cos[e + f*x]]*Sqrt[d*Sin
[e + f*x]])

Rule 2900

Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_))/((a_) + (b_.)*sin[(e_.) + (f_.
)*(x_)]), x_Symbol] :> Dist[g^2/(a*b), Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^n*(b - a*Sin[e + f*x]), x
], x] + Dist[(g^2*(a^2 - b^2))/(a*b*d), Int[((g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^(n + 1))/(a + b*Sin[e +
f*x]), x], x] /; FreeQ[{a, b, d, e, f, g}, x] && NeQ[a^2 - b^2, 0] && IntegersQ[2*n, 2*p] && GtQ[p, 1] && (Lt
Q[n, -1] || (EqQ[p, 3/2] && EqQ[n, -2^(-1)]))

Rule 2838

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((a_) + (b_.)*sin[(e_.) + (f_.)
*(x_)]), x_Symbol] :> Dist[a, Int[(g*Cos[e + f*x])^p*(d*Sin[e + f*x])^n, x], x] + Dist[b/d, Int[(g*Cos[e + f*x
])^p*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x]

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 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 2574

Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> With[{k = Denomina
tor[m]}, Dist[(k*a*b)/f, Subst[Int[x^(k*(m + 1) - 1)/(a^2 + b^2*x^(2*k)), x], x, (a*Sin[e + f*x])^(1/k)/(b*Cos
[e + f*x])^(1/k)], x]] /; FreeQ[{a, b, e, f}, x] && EqQ[m + n, 0] && GtQ[m, 0] && LtQ[m, 1]

Rule 297

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 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 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 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 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 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 2908

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

Rule 2907

Int[Sqrt[(d_.)*sin[(e_.) + (f_.)*(x_)]]/(Sqrt[cos[(e_.) + (f_.)*(x_)]]*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]))
, x_Symbol] :> With[{q = Rt[-a^2 + b^2, 2]}, Dist[(2*Sqrt[2]*d*(b + q))/(f*q), Subst[Int[1/((d*(b + q) + a*x^2
)*Sqrt[1 - x^4/d^2]), x], x, Sqrt[d*Sin[e + f*x]]/Sqrt[1 + Cos[e + f*x]]], x] - Dist[(2*Sqrt[2]*d*(b - q))/(f*
q), Subst[Int[1/((d*(b - q) + a*x^2)*Sqrt[1 - x^4/d^2]), x], x, Sqrt[d*Sin[e + f*x]]/Sqrt[1 + Cos[e + f*x]]],
x]] /; FreeQ[{a, b, d, e, f}, x] && NeQ[a^2 - b^2, 0]

Rule 1218

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[-(c/a), 4]}, Simp[(1*Ellipt
icPi[-(e/(d*q^2)), ArcSin[q*x], -1])/(d*Sqrt[a]*q), x]] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && GtQ[a, 0]

Rubi steps

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

Mathematica [C]  time = 11.5558, size = 178, normalized size = 0.31 \[ \frac{2 \sqrt{d \sin (e+f x)} (g \cos (e+f x))^{5/2} \left (a+b \sqrt{\sin ^2(e+f x)}\right ) \left (b F_1\left (\frac{5}{4};\frac{1}{4},1;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right )-a F_1\left (\frac{5}{4};\frac{3}{4},1;\frac{9}{4};\cos ^2(e+f x),\frac{b^2 \cos ^2(e+f x)}{b^2-a^2}\right )\right )}{5 d f g \left (a^2-b^2\right ) \sqrt [4]{\sin ^2(e+f x)} (a+b \sin (e+f x))} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(g*Cos[e + f*x])^(3/2)/(Sqrt[d*Sin[e + f*x]]*(a + b*Sin[e + f*x])),x]

[Out]

(2*(b*AppellF1[5/4, 1/4, 1, 9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)] - a*AppellF1[5/4, 3/4, 1,
9/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*(g*Cos[e + f*x])^(5/2)*Sqrt[d*Sin[e + f*x]]*(a + b*Sq
rt[Sin[e + f*x]^2]))/(5*(a^2 - b^2)*d*f*g*(Sin[e + f*x]^2)^(1/4)*(a + b*Sin[e + f*x]))

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Maple [A]  time = 0.273, size = 930, normalized size = 1.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*cos(f*x+e))^(3/2)/(d*sin(f*x+e))^(1/2)/(a+b*sin(f*x+e)),x)

[Out]

1/f*2^(1/2)/b/(-a^2+b^2)^(1/2)/(a-b+(-a^2+b^2)^(1/2))/(b+(-a^2+b^2)^(1/2)-a)*(a-b)*(I*EllipticPi(((1-cos(f*x+e
)+sin(f*x+e))/sin(f*x+e))^(1/2),1/2+1/2*I,1/2*2^(1/2))*(-a^2+b^2)^(1/2)*a-I*EllipticPi(((1-cos(f*x+e)+sin(f*x+
e))/sin(f*x+e))^(1/2),1/2-1/2*I,1/2*2^(1/2))*(-a^2+b^2)^(1/2)*a+EllipticPi(((1-cos(f*x+e)+sin(f*x+e))/sin(f*x+
e))^(1/2),1/2+1/2*I,1/2*2^(1/2))*(-a^2+b^2)^(1/2)*a+EllipticPi(((1-cos(f*x+e)+sin(f*x+e))/sin(f*x+e))^(1/2),1/
2-1/2*I,1/2*2^(1/2))*(-a^2+b^2)^(1/2)*a+2*EllipticF(((1-cos(f*x+e)+sin(f*x+e))/sin(f*x+e))^(1/2),1/2*2^(1/2))*
(-a^2+b^2)^(1/2)*b-EllipticPi(((1-cos(f*x+e)+sin(f*x+e))/sin(f*x+e))^(1/2),a/(a-b+(-a^2+b^2)^(1/2)),1/2*2^(1/2
))*(-a^2+b^2)^(1/2)*a-EllipticPi(((1-cos(f*x+e)+sin(f*x+e))/sin(f*x+e))^(1/2),a/(a-b+(-a^2+b^2)^(1/2)),1/2*2^(
1/2))*(-a^2+b^2)^(1/2)*b-a^2*EllipticPi(((1-cos(f*x+e)+sin(f*x+e))/sin(f*x+e))^(1/2),a/(a-b+(-a^2+b^2)^(1/2)),
1/2*2^(1/2))+EllipticPi(((1-cos(f*x+e)+sin(f*x+e))/sin(f*x+e))^(1/2),a/(a-b+(-a^2+b^2)^(1/2)),1/2*2^(1/2))*b^2
-EllipticPi(((1-cos(f*x+e)+sin(f*x+e))/sin(f*x+e))^(1/2),-a/(b+(-a^2+b^2)^(1/2)-a),1/2*2^(1/2))*(-a^2+b^2)^(1/
2)*a-EllipticPi(((1-cos(f*x+e)+sin(f*x+e))/sin(f*x+e))^(1/2),-a/(b+(-a^2+b^2)^(1/2)-a),1/2*2^(1/2))*(-a^2+b^2)
^(1/2)*b+a^2*EllipticPi(((1-cos(f*x+e)+sin(f*x+e))/sin(f*x+e))^(1/2),-a/(b+(-a^2+b^2)^(1/2)-a),1/2*2^(1/2))-El
lipticPi(((1-cos(f*x+e)+sin(f*x+e))/sin(f*x+e))^(1/2),-a/(b+(-a^2+b^2)^(1/2)-a),1/2*2^(1/2))*b^2)*((1-cos(f*x+
e)+sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e)+sin(f*x+e))/sin(f*x+e))^(1/2)*((-1+cos(f*x+e))/sin(f*x+e))^(1
/2)*(g*cos(f*x+e))^(3/2)*sin(f*x+e)^2/(-1+cos(f*x+e))/(d*sin(f*x+e))^(1/2)/cos(f*x+e)^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (g \cos \left (f x + e\right )\right )^{\frac{3}{2}}}{{\left (b \sin \left (f x + e\right ) + a\right )} \sqrt{d \sin \left (f x + e\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*cos(f*x+e))^(3/2)/(d*sin(f*x+e))^(1/2)/(a+b*sin(f*x+e)),x, algorithm="maxima")

[Out]

integrate((g*cos(f*x + e))^(3/2)/((b*sin(f*x + e) + a)*sqrt(d*sin(f*x + e))), x)

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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((g*cos(f*x+e))^(3/2)/(d*sin(f*x+e))^(1/2)/(a+b*sin(f*x+e)),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((g*cos(f*x+e))**(3/2)/(d*sin(f*x+e))**(1/2)/(a+b*sin(f*x+e)),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (g \cos \left (f x + e\right )\right )^{\frac{3}{2}}}{{\left (b \sin \left (f x + e\right ) + a\right )} \sqrt{d \sin \left (f x + e\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*cos(f*x+e))^(3/2)/(d*sin(f*x+e))^(1/2)/(a+b*sin(f*x+e)),x, algorithm="giac")

[Out]

integrate((g*cos(f*x + e))^(3/2)/((b*sin(f*x + e) + a)*sqrt(d*sin(f*x + e))), x)