\(\int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))} \, dx\) [1418]

Optimal result

Integrand size = 37, antiderivative size = 577 \[ \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))} \, dx=\frac {g^{3/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}\right )}{\sqrt {2} b \sqrt {d} f}-\frac {g^{3/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}\right )}{\sqrt {2} b \sqrt {d} f}+\frac {2 \sqrt {2} \sqrt {-a^2+b^2} g^2 \sqrt {\cos (e+f x)} \operatorname {EllipticPi}\left (-\frac {a}{b-\sqrt {-a^2+b^2}},\arcsin \left (\frac {\sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {1+\cos (e+f x)}}\right ),-1\right )}{a b \sqrt {d} f \sqrt {g \cos (e+f x)}}-\frac {2 \sqrt {2} \sqrt {-a^2+b^2} g^2 \sqrt {\cos (e+f x)} \operatorname {EllipticPi}\left (-\frac {a}{b+\sqrt {-a^2+b^2}},\arcsin \left (\frac {\sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {1+\cos (e+f x)}}\right ),-1\right )}{a b \sqrt {d} f \sqrt {g \cos (e+f x)}}-\frac {g^{3/2} \log \left (\sqrt {d}-\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}+\sqrt {d} \tan (e+f x)\right )}{2 \sqrt {2} b \sqrt {d} f}+\frac {g^{3/2} \log \left (\sqrt {d}+\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}+\sqrt {d} \tan (e+f x)\right )}{2 \sqrt {2} b \sqrt {d} f}+\frac {g^2 \operatorname {EllipticF}\left (e-\frac {\pi }{4}+f x,2\right ) \sqrt {\sin (2 e+2 f x)}}{a f \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}} \]

[Out]

1/2*g^(3/2)*arctan(1-2^(1/2)*g^(1/2)*(d*sin(f*x+e))^(1/2)/d^(1/2)/(g*cos(f*x+e))^(1/2))/b/f*2^(1/2)/d^(1/2)-1/
2*g^(3/2)*arctan(1+2^(1/2)*g^(1/2)*(d*sin(f*x+e))^(1/2)/d^(1/2)/(g*cos(f*x+e))^(1/2))/b/f*2^(1/2)/d^(1/2)-1/4*
g^(3/2)*ln(d^(1/2)-2^(1/2)*g^(1/2)*(d*sin(f*x+e))^(1/2)/(g*cos(f*x+e))^(1/2)+d^(1/2)*tan(f*x+e))/b/f*2^(1/2)/d
^(1/2)+1/4*g^(3/2)*ln(d^(1/2)+2^(1/2)*g^(1/2)*(d*sin(f*x+e))^(1/2)/(g*cos(f*x+e))^(1/2)+d^(1/2)*tan(f*x+e))/b/
f*2^(1/2)/d^(1/2)+2*g^2*EllipticPi((d*sin(f*x+e))^(1/2)/d^(1/2)/(1+cos(f*x+e))^(1/2),-a/(b-(-a^2+b^2)^(1/2)),I
)*2^(1/2)*(-a^2+b^2)^(1/2)*cos(f*x+e)^(1/2)/a/b/f/d^(1/2)/(g*cos(f*x+e))^(1/2)-2*g^2*EllipticPi((d*sin(f*x+e))
^(1/2)/d^(1/2)/(1+cos(f*x+e))^(1/2),-a/(b+(-a^2+b^2)^(1/2)),I)*2^(1/2)*(-a^2+b^2)^(1/2)*cos(f*x+e)^(1/2)/a/b/f
/d^(1/2)/(g*cos(f*x+e))^(1/2)-g^2*(sin(e+1/4*Pi+f*x)^2)^(1/2)/sin(e+1/4*Pi+f*x)*EllipticF(cos(e+1/4*Pi+f*x),2^
(1/2))*sin(2*f*x+2*e)^(1/2)/a/f/(g*cos(f*x+e))^(1/2)/(d*sin(f*x+e))^(1/2)

Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 577, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.378, Rules used = {2979, 2917, 2653, 2720, 2654, 303, 1176, 631, 210, 1179, 642, 2987, 2986, 1232} \[ \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))} \, dx=\frac {2 \sqrt {2} g^2 \sqrt {b^2-a^2} \sqrt {\cos (e+f x)} \operatorname {EllipticPi}\left (-\frac {a}{b-\sqrt {b^2-a^2}},\arcsin \left (\frac {\sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {\cos (e+f x)+1}}\right ),-1\right )}{a b \sqrt {d} f \sqrt {g \cos (e+f x)}}-\frac {2 \sqrt {2} g^2 \sqrt {b^2-a^2} \sqrt {\cos (e+f x)} \operatorname {EllipticPi}\left (-\frac {a}{b+\sqrt {b^2-a^2}},\arcsin \left (\frac {\sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {\cos (e+f x)+1}}\right ),-1\right )}{a b \sqrt {d} f \sqrt {g \cos (e+f x)}}+\frac {g^2 \sqrt {\sin (2 e+2 f x)} \operatorname {EllipticF}\left (e+f x-\frac {\pi }{4},2\right )}{a f \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}+\frac {g^{3/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}\right )}{\sqrt {2} b \sqrt {d} f}-\frac {g^{3/2} \arctan \left (\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {g \cos (e+f x)}}+1\right )}{\sqrt {2} b \sqrt {d} f}-\frac {g^{3/2} \log \left (-\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}+\sqrt {d} \tan (e+f x)+\sqrt {d}\right )}{2 \sqrt {2} b \sqrt {d} f}+\frac {g^{3/2} \log \left (\frac {\sqrt {2} \sqrt {g} \sqrt {d \sin (e+f x)}}{\sqrt {g \cos (e+f x)}}+\sqrt {d} \tan (e+f x)+\sqrt {d}\right )}{2 \sqrt {2} b \sqrt {d} f} \]

[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 210

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

Rule 303

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

Rule 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 1232

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

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 2654

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 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 2917

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 2979

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 2986

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 2987

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]

Rubi steps \begin{align*} \text {integral}& = \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 ) \text {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 ) \text {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 ) \text {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)} \operatorname {EllipticPi}\left (-\frac {a}{b-\sqrt {-a^2+b^2}},\arcsin \left (\frac {\sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {1+\cos (e+f x)}}\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)} \operatorname {EllipticPi}\left (-\frac {a}{b+\sqrt {-a^2+b^2}},\arcsin \left (\frac {\sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {1+\cos (e+f x)}}\right ),-1\right )}{a b \sqrt {d} f \sqrt {g \cos (e+f x)}}+\frac {g^2 \operatorname {EllipticF}\left (e-\frac {\pi }{4}+f x,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 \text {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 \text {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)} \operatorname {EllipticPi}\left (-\frac {a}{b-\sqrt {-a^2+b^2}},\arcsin \left (\frac {\sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {1+\cos (e+f x)}}\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)} \operatorname {EllipticPi}\left (-\frac {a}{b+\sqrt {-a^2+b^2}},\arcsin \left (\frac {\sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {1+\cos (e+f x)}}\right ),-1\right )}{a b \sqrt {d} f \sqrt {g \cos (e+f x)}}+\frac {g^2 \operatorname {EllipticF}\left (e-\frac {\pi }{4}+f x,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 \text {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 \text {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} \text {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} \text {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)} \operatorname {EllipticPi}\left (-\frac {a}{b-\sqrt {-a^2+b^2}},\arcsin \left (\frac {\sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {1+\cos (e+f x)}}\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)} \operatorname {EllipticPi}\left (-\frac {a}{b+\sqrt {-a^2+b^2}},\arcsin \left (\frac {\sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {1+\cos (e+f x)}}\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 \operatorname {EllipticF}\left (e-\frac {\pi }{4}+f x,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} \text {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} \text {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} \arctan \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} \arctan \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)} \operatorname {EllipticPi}\left (-\frac {a}{b-\sqrt {-a^2+b^2}},\arcsin \left (\frac {\sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {1+\cos (e+f x)}}\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)} \operatorname {EllipticPi}\left (-\frac {a}{b+\sqrt {-a^2+b^2}},\arcsin \left (\frac {\sqrt {d \sin (e+f x)}}{\sqrt {d} \sqrt {1+\cos (e+f x)}}\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 \operatorname {EllipticF}\left (e-\frac {\pi }{4}+f x,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] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.

Time = 12.08 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.31 \[ \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))} \, dx=\frac {2 \left (b \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{4},1,\frac {9}{4},\cos ^2(e+f x),\frac {b^2 \cos ^2(e+f x)}{-a^2+b^2}\right )-a \operatorname {AppellF1}\left (\frac {5}{4},\frac {3}{4},1,\frac {9}{4},\cos ^2(e+f x),\frac {b^2 \cos ^2(e+f x)}{-a^2+b^2}\right )\right ) (g \cos (e+f x))^{5/2} \sqrt {d \sin (e+f x)} \left (a+b \sqrt {\sin ^2(e+f x)}\right )}{5 \left (a^2-b^2\right ) d f g \sqrt [4]{\sin ^2(e+f x)} (a+b \sin (e+f x))} \]

[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]))

Maple [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1178 vs. \(2 (490 ) = 980\).

Time = 1.82 (sec) , antiderivative size = 1179, normalized size of antiderivative = 2.04

method	result	size
default	\(\text {Expression too large to display}\)	\(1179\)

[In]

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

[Out]

(-1/2+1/2*I)/f*(a-b)*(-I*EllipticPi((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),-a/(b+(-a^2+b^2)^(1/2)-a),1/2*2^(1/2))*b^
2+2*I*EllipticPi((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),1/2+1/2*I,1/2*2^(1/2))*(-a^2+b^2)^(1/2)*a+I*EllipticPi((-cot
(f*x+e)+csc(f*x+e)+1)^(1/2),-a/(b+(-a^2+b^2)^(1/2)-a),1/2*2^(1/2))*a^2+I*EllipticPi((-cot(f*x+e)+csc(f*x+e)+1)
^(1/2),a/(-b+(-a^2+b^2)^(1/2)+a),1/2*2^(1/2))*b^2-I*EllipticPi((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),-a/(b+(-a^2+b^
2)^(1/2)-a),1/2*2^(1/2))*(-a^2+b^2)^(1/2)*b-I*EllipticPi((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),a/(-b+(-a^2+b^2)^(1/
2)+a),1/2*2^(1/2))*a^2-I*EllipticPi((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),a/(-b+(-a^2+b^2)^(1/2)+a),1/2*2^(1/2))*(-
a^2+b^2)^(1/2)*b+2*I*EllipticF((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),1/2*2^(1/2))*(-a^2+b^2)^(1/2)*b-I*EllipticPi((
-cot(f*x+e)+csc(f*x+e)+1)^(1/2),a/(-b+(-a^2+b^2)^(1/2)+a),1/2*2^(1/2))*(-a^2+b^2)^(1/2)*a-I*EllipticPi((-cot(f
*x+e)+csc(f*x+e)+1)^(1/2),-a/(b+(-a^2+b^2)^(1/2)-a),1/2*2^(1/2))*(-a^2+b^2)^(1/2)*a+2*EllipticPi((-cot(f*x+e)+
csc(f*x+e)+1)^(1/2),1/2-1/2*I,1/2*2^(1/2))*(-a^2+b^2)^(1/2)*a-EllipticPi((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),a/(-
b+(-a^2+b^2)^(1/2)+a),1/2*2^(1/2))*(-a^2+b^2)^(1/2)*a-EllipticPi((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),a/(-b+(-a^2+
b^2)^(1/2)+a),1/2*2^(1/2))*(-a^2+b^2)^(1/2)*b-EllipticPi((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),a/(-b+(-a^2+b^2)^(1/
2)+a),1/2*2^(1/2))*a^2+EllipticPi((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),a/(-b+(-a^2+b^2)^(1/2)+a),1/2*2^(1/2))*b^2-
EllipticPi((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),-a/(b+(-a^2+b^2)^(1/2)-a),1/2*2^(1/2))*(-a^2+b^2)^(1/2)*a-Elliptic
Pi((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),-a/(b+(-a^2+b^2)^(1/2)-a),1/2*2^(1/2))*(-a^2+b^2)^(1/2)*b+EllipticPi((-cot
(f*x+e)+csc(f*x+e)+1)^(1/2),-a/(b+(-a^2+b^2)^(1/2)-a),1/2*2^(1/2))*a^2-EllipticPi((-cot(f*x+e)+csc(f*x+e)+1)^(
1/2),-a/(b+(-a^2+b^2)^(1/2)-a),1/2*2^(1/2))*b^2+2*EllipticF((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),1/2*2^(1/2))*(-a^
2+b^2)^(1/2)*b)*(-csc(f*x+e)+cot(f*x+e))^(1/2)*(-csc(f*x+e)+cot(f*x+e)+1)^(1/2)*(-cot(f*x+e)+csc(f*x+e)+1)^(1/
2)*g*(g*cos(f*x+e))^(1/2)/(d*sin(f*x+e))^(1/2)*(sec(f*x+e)+1)*2^(1/2)/b/(-a^2+b^2)^(1/2)/(-b+(-a^2+b^2)^(1/2)+
a)/(b+(-a^2+b^2)^(1/2)-a)

Fricas [F(-1)]

Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))} \, dx=\text {Timed out} \]

[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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[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)

Giac [F]

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

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))} \, dx=\int \frac {{\left (g\,\cos \left (e+f\,x\right )\right )}^{3/2}}{\sqrt {d\,\sin \left (e+f\,x\right )}\,\left (a+b\,\sin \left (e+f\,x\right )\right )} \,d x \]

[In]

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

[Out]

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