\(\int \frac {1}{(g \cos (e+f x))^{3/2} (d \sin (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx\) [1440]

Optimal result

Integrand size = 37, antiderivative size = 568 \[ \int \frac {1}{(g \cos (e+f x))^{3/2} (d \sin (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx=-\frac {2 a}{\left (a^2-b^2\right ) d f g \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}}+\frac {2 b^2 (g \cos (e+f x))^{3/2}}{a \left (a^2-b^2\right ) d f g^3 \sqrt {d \sin (e+f x)}}-\frac {2 \sqrt {2} b^3 \operatorname {EllipticPi}\left (-\frac {\sqrt {-a+b}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {1+\sin (e+f x)}}\right ),-1\right ) \sqrt {\sin (e+f x)}}{a (-a+b)^{3/2} (a+b)^{3/2} d f g^{3/2} \sqrt {d \sin (e+f x)}}+\frac {2 \sqrt {2} b^3 \operatorname {EllipticPi}\left (\frac {\sqrt {-a+b}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {1+\sin (e+f x)}}\right ),-1\right ) \sqrt {\sin (e+f x)}}{a (-a+b)^{3/2} (a+b)^{3/2} d f g^{3/2} \sqrt {d \sin (e+f x)}}-\frac {2 b \sqrt {d \sin (e+f x)}}{\left (a^2-b^2\right ) d^2 f g \sqrt {g \cos (e+f x)}}+\frac {4 a (d \sin (e+f x))^{3/2}}{\left (a^2-b^2\right ) d^3 f g \sqrt {g \cos (e+f x)}}-\frac {4 a \sqrt {g \cos (e+f x)} E\left (\left .e-\frac {\pi }{4}+f x\right |2\right ) \sqrt {d \sin (e+f x)}}{\left (a^2-b^2\right ) d^2 f g^2 \sqrt {\sin (2 e+2 f x)}}+\frac {2 b^2 \sqrt {g \cos (e+f x)} E\left (\left .e-\frac {\pi }{4}+f x\right |2\right ) \sqrt {d \sin (e+f x)}}{a \left (a^2-b^2\right ) d^2 f g^2 \sqrt {\sin (2 e+2 f x)}} \]

[Out]

4*a*(d*sin(f*x+e))^(3/2)/(a^2-b^2)/d^3/f/g/(g*cos(f*x+e))^(1/2)+2*b^2*(g*cos(f*x+e))^(3/2)/a/(a^2-b^2)/d/f/g^3
/(d*sin(f*x+e))^(1/2)-2*a/(a^2-b^2)/d/f/g/(g*cos(f*x+e))^(1/2)/(d*sin(f*x+e))^(1/2)-2*b^3*EllipticPi((g*cos(f*
x+e))^(1/2)/g^(1/2)/(1+sin(f*x+e))^(1/2),-(-a+b)^(1/2)/(a+b)^(1/2),I)*2^(1/2)*sin(f*x+e)^(1/2)/a/(-a+b)^(3/2)/
(a+b)^(3/2)/d/f/g^(3/2)/(d*sin(f*x+e))^(1/2)+2*b^3*EllipticPi((g*cos(f*x+e))^(1/2)/g^(1/2)/(1+sin(f*x+e))^(1/2
),(-a+b)^(1/2)/(a+b)^(1/2),I)*2^(1/2)*sin(f*x+e)^(1/2)/a/(-a+b)^(3/2)/(a+b)^(3/2)/d/f/g^(3/2)/(d*sin(f*x+e))^(
1/2)-2*b*(d*sin(f*x+e))^(1/2)/(a^2-b^2)/d^2/f/g/(g*cos(f*x+e))^(1/2)+4*a*(sin(e+1/4*Pi+f*x)^2)^(1/2)/sin(e+1/4
*Pi+f*x)*EllipticE(cos(e+1/4*Pi+f*x),2^(1/2))*(g*cos(f*x+e))^(1/2)*(d*sin(f*x+e))^(1/2)/(a^2-b^2)/d^2/f/g^2/si
n(2*f*x+2*e)^(1/2)-2*b^2*(sin(e+1/4*Pi+f*x)^2)^(1/2)/sin(e+1/4*Pi+f*x)*EllipticE(cos(e+1/4*Pi+f*x),2^(1/2))*(g
*cos(f*x+e))^(1/2)*(d*sin(f*x+e))^(1/2)/a/(a^2-b^2)/d^2/f/g^2/sin(2*f*x+2*e)^(1/2)

Rubi [A] (verified)

Time = 0.91 (sec) , antiderivative size = 568, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.324, Rules used = {2983, 2917, 2650, 2651, 2652, 2719, 2643, 2989, 2985, 2984, 504, 1232} \[ \int \frac {1}{(g \cos (e+f x))^{3/2} (d \sin (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx=\frac {4 a (d \sin (e+f x))^{3/2}}{d^3 f g \left (a^2-b^2\right ) \sqrt {g \cos (e+f x)}}+\frac {2 b^2 E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{a d^2 f g^2 \left (a^2-b^2\right ) \sqrt {\sin (2 e+2 f x)}}-\frac {4 a E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}{d^2 f g^2 \left (a^2-b^2\right ) \sqrt {\sin (2 e+2 f x)}}-\frac {2 b \sqrt {d \sin (e+f x)}}{d^2 f g \left (a^2-b^2\right ) \sqrt {g \cos (e+f x)}}+\frac {2 b^2 (g \cos (e+f x))^{3/2}}{a d f g^3 \left (a^2-b^2\right ) \sqrt {d \sin (e+f x)}}-\frac {2 a}{d f g \left (a^2-b^2\right ) \sqrt {d \sin (e+f x)} \sqrt {g \cos (e+f x)}}-\frac {2 \sqrt {2} b^3 \sqrt {\sin (e+f x)} \operatorname {EllipticPi}\left (-\frac {\sqrt {b-a}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {\sin (e+f x)+1}}\right ),-1\right )}{a d f g^{3/2} (b-a)^{3/2} (a+b)^{3/2} \sqrt {d \sin (e+f x)}}+\frac {2 \sqrt {2} b^3 \sqrt {\sin (e+f x)} \operatorname {EllipticPi}\left (\frac {\sqrt {b-a}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {\sin (e+f x)+1}}\right ),-1\right )}{a d f g^{3/2} (b-a)^{3/2} (a+b)^{3/2} \sqrt {d \sin (e+f x)}} \]

[In]

Int[1/((g*Cos[e + f*x])^(3/2)*(d*Sin[e + f*x])^(3/2)*(a + b*Sin[e + f*x])),x]

[Out]

(-2*a)/((a^2 - b^2)*d*f*g*Sqrt[g*Cos[e + f*x]]*Sqrt[d*Sin[e + f*x]]) + (2*b^2*(g*Cos[e + f*x])^(3/2))/(a*(a^2
- b^2)*d*f*g^3*Sqrt[d*Sin[e + f*x]]) - (2*Sqrt[2]*b^3*EllipticPi[-(Sqrt[-a + b]/Sqrt[a + b]), ArcSin[Sqrt[g*Co
s[e + f*x]]/(Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(a*(-a + b)^(3/2)*(a + b)^(3/2)*d*f*g^(
3/2)*Sqrt[d*Sin[e + f*x]]) + (2*Sqrt[2]*b^3*EllipticPi[Sqrt[-a + b]/Sqrt[a + b], ArcSin[Sqrt[g*Cos[e + f*x]]/(
Sqrt[g]*Sqrt[1 + Sin[e + f*x]])], -1]*Sqrt[Sin[e + f*x]])/(a*(-a + b)^(3/2)*(a + b)^(3/2)*d*f*g^(3/2)*Sqrt[d*S
in[e + f*x]]) - (2*b*Sqrt[d*Sin[e + f*x]])/((a^2 - b^2)*d^2*f*g*Sqrt[g*Cos[e + f*x]]) + (4*a*(d*Sin[e + f*x])^
(3/2))/((a^2 - b^2)*d^3*f*g*Sqrt[g*Cos[e + f*x]]) - (4*a*Sqrt[g*Cos[e + f*x]]*EllipticE[e - Pi/4 + f*x, 2]*Sqr
t[d*Sin[e + f*x]])/((a^2 - b^2)*d^2*f*g^2*Sqrt[Sin[2*e + 2*f*x]]) + (2*b^2*Sqrt[g*Cos[e + f*x]]*EllipticE[e -
Pi/4 + f*x, 2]*Sqrt[d*Sin[e + f*x]])/(a*(a^2 - b^2)*d^2*f*g^2*Sqrt[Sin[2*e + 2*f*x]])

Rule 504

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

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 2643

Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Simp[(a*Sin[e +
f*x])^(m + 1)*((b*Cos[e + f*x])^(n + 1)/(a*b*f*(m + 1))), x] /; FreeQ[{a, b, e, f, m, n}, x] && EqQ[m + n + 2,
0] && NeQ[m, -1]

Rule 2650

Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(b*Cos[e + f*
x])^(n + 1)*((a*Sin[e + f*x])^(m + 1)/(a*b*f*(m + 1))), x] + Dist[(m + n + 2)/(a^2*(m + 1)), Int[(b*Cos[e + f*
x])^n*(a*Sin[e + f*x])^(m + 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && LtQ[m, -1] && IntegersQ[2*m, 2*n]

Rule 2651

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-(b*Sin[e +
f*x])^(n + 1))*((a*Cos[e + f*x])^(m + 1)/(a*b*f*(m + 1))), x] + Dist[(m + n + 2)/(a^2*(m + 1)), Int[(b*Sin[e +
f*x])^n*(a*Cos[e + f*x])^(m + 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && LtQ[m, -1] && IntegersQ[2*m, 2*n]

Rule 2652

Int[Sqrt[cos[(e_.) + (f_.)*(x_)]*(b_.)]*Sqrt[(a_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[Sqrt[a*Sin[e +
f*x]]*(Sqrt[b*Cos[e + f*x]]/Sqrt[Sin[2*e + 2*f*x]]), Int[Sqrt[Sin[2*e + 2*f*x]], x], x] /; FreeQ[{a, b, e, f},
x]

Rule 2719

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(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 2983

Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_))/((a_) + (b_.)*sin[(e_.) + (f_.
)*(x_)]), x_Symbol] :> Dist[1/(a^2 - b^2), Int[(g*Cos[e + f*x])^p*(d*Sin[e + f*x])^n*(a - b*Sin[e + f*x]), x],
x] - Dist[b^2/(g^2*(a^2 - b^2)), Int[(g*Cos[e + f*x])^(p + 2)*((d*Sin[e + f*x])^n/(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] && LtQ[p, -1]

Rule 2984

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

Rule 2985

Int[Sqrt[cos[(e_.) + (f_.)*(x_)]*(g_.)]/(Sqrt[(d_)*sin[(e_.) + (f_.)*(x_)]]*((a_) + (b_.)*sin[(e_.) + (f_.)*(x
_)])), x_Symbol] :> Dist[Sqrt[Sin[e + f*x]]/Sqrt[d*Sin[e + f*x]], Int[Sqrt[g*Cos[e + f*x]]/(Sqrt[Sin[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 2989

Int[((cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_))/((a_) + (b_.)*sin[(e_.) + (f_.
)*(x_)]), x_Symbol] :> Dist[1/a, Int[(g*Cos[e + f*x])^p*(d*Sin[e + f*x])^n, x], x] - Dist[b/(a*d), Int[(g*Cos[
e + f*x])^p*((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] && LtQ[-1, p, 1] && LtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {a-b \sin (e+f x)}{(g \cos (e+f x))^{3/2} (d \sin (e+f x))^{3/2}} \, dx}{a^2-b^2}-\frac {b^2 \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx}{\left (a^2-b^2\right ) g^2} \\ & = \frac {a \int \frac {1}{(g \cos (e+f x))^{3/2} (d \sin (e+f x))^{3/2}} \, dx}{a^2-b^2}-\frac {b \int \frac {1}{(g \cos (e+f x))^{3/2} \sqrt {d \sin (e+f x)}} \, dx}{\left (a^2-b^2\right ) d}-\frac {b^2 \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{3/2}} \, dx}{a \left (a^2-b^2\right ) g^2}+\frac {b^3 \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))} \, dx}{a \left (a^2-b^2\right ) d g^2} \\ & = -\frac {2 a}{\left (a^2-b^2\right ) d f g \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}}+\frac {2 b^2 (g \cos (e+f x))^{3/2}}{a \left (a^2-b^2\right ) d f g^3 \sqrt {d \sin (e+f x)}}-\frac {2 b \sqrt {d \sin (e+f x)}}{\left (a^2-b^2\right ) d^2 f g \sqrt {g \cos (e+f x)}}+\frac {(2 a) \int \frac {\sqrt {d \sin (e+f x)}}{(g \cos (e+f x))^{3/2}} \, dx}{\left (a^2-b^2\right ) d^2}+\frac {\left (2 b^2\right ) \int \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)} \, dx}{a \left (a^2-b^2\right ) d^2 g^2}+\frac {\left (b^3 \sqrt {\sin (e+f x)}\right ) \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {\sin (e+f x)} (a+b \sin (e+f x))} \, dx}{a \left (a^2-b^2\right ) d g^2 \sqrt {d \sin (e+f x)}} \\ & = -\frac {2 a}{\left (a^2-b^2\right ) d f g \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}}+\frac {2 b^2 (g \cos (e+f x))^{3/2}}{a \left (a^2-b^2\right ) d f g^3 \sqrt {d \sin (e+f x)}}-\frac {2 b \sqrt {d \sin (e+f x)}}{\left (a^2-b^2\right ) d^2 f g \sqrt {g \cos (e+f x)}}+\frac {4 a (d \sin (e+f x))^{3/2}}{\left (a^2-b^2\right ) d^3 f g \sqrt {g \cos (e+f x)}}-\frac {(4 a) \int \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)} \, dx}{\left (a^2-b^2\right ) d^2 g^2}-\frac {\left (4 \sqrt {2} b^3 \sqrt {\sin (e+f x)}\right ) \text {Subst}\left (\int \frac {x^2}{\left ((a+b) g^2+(a-b) x^4\right ) \sqrt {1-\frac {x^4}{g^2}}} \, dx,x,\frac {\sqrt {g \cos (e+f x)}}{\sqrt {1+\sin (e+f x)}}\right )}{a \left (a^2-b^2\right ) d f g \sqrt {d \sin (e+f x)}}+\frac {\left (2 b^2 \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}\right ) \int \sqrt {\sin (2 e+2 f x)} \, dx}{a \left (a^2-b^2\right ) d^2 g^2 \sqrt {\sin (2 e+2 f x)}} \\ & = -\frac {2 a}{\left (a^2-b^2\right ) d f g \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}}+\frac {2 b^2 (g \cos (e+f x))^{3/2}}{a \left (a^2-b^2\right ) d f g^3 \sqrt {d \sin (e+f x)}}-\frac {2 b \sqrt {d \sin (e+f x)}}{\left (a^2-b^2\right ) d^2 f g \sqrt {g \cos (e+f x)}}+\frac {4 a (d \sin (e+f x))^{3/2}}{\left (a^2-b^2\right ) d^3 f g \sqrt {g \cos (e+f x)}}+\frac {2 b^2 \sqrt {g \cos (e+f x)} E\left (\left .e-\frac {\pi }{4}+f x\right |2\right ) \sqrt {d \sin (e+f x)}}{a \left (a^2-b^2\right ) d^2 f g^2 \sqrt {\sin (2 e+2 f x)}}-\frac {\left (2 \sqrt {2} b^3 \sqrt {\sin (e+f x)}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {a+b} g-\sqrt {-a+b} x^2\right ) \sqrt {1-\frac {x^4}{g^2}}} \, dx,x,\frac {\sqrt {g \cos (e+f x)}}{\sqrt {1+\sin (e+f x)}}\right )}{a \sqrt {-a+b} \left (a^2-b^2\right ) d f g \sqrt {d \sin (e+f x)}}+\frac {\left (2 \sqrt {2} b^3 \sqrt {\sin (e+f x)}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {a+b} g+\sqrt {-a+b} x^2\right ) \sqrt {1-\frac {x^4}{g^2}}} \, dx,x,\frac {\sqrt {g \cos (e+f x)}}{\sqrt {1+\sin (e+f x)}}\right )}{a \sqrt {-a+b} \left (a^2-b^2\right ) d f g \sqrt {d \sin (e+f x)}}-\frac {\left (4 a \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}\right ) \int \sqrt {\sin (2 e+2 f x)} \, dx}{\left (a^2-b^2\right ) d^2 g^2 \sqrt {\sin (2 e+2 f x)}} \\ & = -\frac {2 a}{\left (a^2-b^2\right ) d f g \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}}+\frac {2 b^2 (g \cos (e+f x))^{3/2}}{a \left (a^2-b^2\right ) d f g^3 \sqrt {d \sin (e+f x)}}-\frac {2 \sqrt {2} b^3 \operatorname {EllipticPi}\left (-\frac {\sqrt {-a+b}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {1+\sin (e+f x)}}\right ),-1\right ) \sqrt {\sin (e+f x)}}{a (-a+b)^{3/2} (a+b)^{3/2} d f g^{3/2} \sqrt {d \sin (e+f x)}}+\frac {2 \sqrt {2} b^3 \operatorname {EllipticPi}\left (\frac {\sqrt {-a+b}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {1+\sin (e+f x)}}\right ),-1\right ) \sqrt {\sin (e+f x)}}{a (-a+b)^{3/2} (a+b)^{3/2} d f g^{3/2} \sqrt {d \sin (e+f x)}}-\frac {2 b \sqrt {d \sin (e+f x)}}{\left (a^2-b^2\right ) d^2 f g \sqrt {g \cos (e+f x)}}+\frac {4 a (d \sin (e+f x))^{3/2}}{\left (a^2-b^2\right ) d^3 f g \sqrt {g \cos (e+f x)}}-\frac {4 a \sqrt {g \cos (e+f x)} E\left (\left .e-\frac {\pi }{4}+f x\right |2\right ) \sqrt {d \sin (e+f x)}}{\left (a^2-b^2\right ) d^2 f g^2 \sqrt {\sin (2 e+2 f x)}}+\frac {2 b^2 \sqrt {g \cos (e+f x)} E\left (\left .e-\frac {\pi }{4}+f x\right |2\right ) \sqrt {d \sin (e+f x)}}{a \left (a^2-b^2\right ) d^2 f g^2 \sqrt {\sin (2 e+2 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 = 25.28 (sec) , antiderivative size = 1707, normalized size of antiderivative = 3.01 \[ \int \frac {1}{(g \cos (e+f x))^{3/2} (d \sin (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx=\frac {\cos ^2(e+f x) \sin ^2(e+f x) \left (-\frac {2 \cot (e+f x)}{a}+\frac {2 \sec (e+f x) (-b+a \sin (e+f x))}{a^2-b^2}\right )}{f (g \cos (e+f x))^{3/2} (d \sin (e+f x))^{3/2}}-\frac {\cos ^{\frac {3}{2}}(e+f x) \sin ^{\frac {3}{2}}(e+f x) \left (-\frac {2 \left (4 a^3-2 a b^2\right ) \left (-b \operatorname {AppellF1}\left (\frac {3}{4},-\frac {1}{4},1,\frac {7}{4},\cos ^2(e+f x),\frac {b^2 \cos ^2(e+f x)}{-a^2+b^2}\right )+a \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{4},1,\frac {7}{4},\cos ^2(e+f x),\frac {b^2 \cos ^2(e+f x)}{-a^2+b^2}\right )\right ) \cos ^{\frac {3}{2}}(e+f x) \left (a+b \sqrt {1-\cos ^2(e+f x)}\right ) \sin ^{\frac {3}{2}}(e+f x)}{3 \left (a^2-b^2\right ) \left (1-\cos ^2(e+f x)\right )^{3/4} (a+b \sin (e+f x))}+\frac {\left (2 a^2 b-2 b^3\right ) \sqrt {\tan (e+f x)} \left (\frac {3 \sqrt {2} a^{3/2} \left (-2 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a^2-b^2} \sqrt {\tan (e+f x)}}{\sqrt {a}}\right )+2 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{a^2-b^2} \sqrt {\tan (e+f x)}}{\sqrt {a}}\right )-\log \left (-a+\sqrt {2} \sqrt {a} \sqrt [4]{a^2-b^2} \sqrt {\tan (e+f x)}-\sqrt {a^2-b^2} \tan (e+f x)\right )+\log \left (a+\sqrt {2} \sqrt {a} \sqrt [4]{a^2-b^2} \sqrt {\tan (e+f x)}+\sqrt {a^2-b^2} \tan (e+f x)\right )\right )}{\sqrt [4]{a^2-b^2}}-8 b \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},1,\frac {7}{4},-\tan ^2(e+f x),\frac {\left (-a^2+b^2\right ) \tan ^2(e+f x)}{a^2}\right ) \tan ^{\frac {3}{2}}(e+f x)\right ) \left (b \tan (e+f x)+a \sqrt {1+\tan ^2(e+f x)}\right )}{12 a^2 \cos ^{\frac {3}{2}}(e+f x) \sqrt {\sin (e+f x)} (a+b \sin (e+f x)) \left (1+\tan ^2(e+f x)\right )^{3/2}}+\frac {\left (-2 a^2 b+b^3\right ) \cos (2 (e+f x)) \sqrt {\tan (e+f x)} \left (b \tan (e+f x)+a \sqrt {1+\tan ^2(e+f x)}\right ) \left (56 b \left (-3 a^2+b^2\right ) \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},1,\frac {7}{4},-\tan ^2(e+f x),\left (-1+\frac {b^2}{a^2}\right ) \tan ^2(e+f x)\right ) \tan ^{\frac {3}{2}}(e+f x)+24 b \left (-a^2+b^2\right ) \operatorname {AppellF1}\left (\frac {7}{4},\frac {1}{2},1,\frac {11}{4},-\tan ^2(e+f x),\left (-1+\frac {b^2}{a^2}\right ) \tan ^2(e+f x)\right ) \tan ^{\frac {7}{2}}(e+f x)+21 a^{3/2} \left (4 \sqrt {2} a^{3/2} \arctan \left (1-\sqrt {2} \sqrt {\tan (e+f x)}\right )-4 \sqrt {2} a^{3/2} \arctan \left (1+\sqrt {2} \sqrt {\tan (e+f x)}\right )-\frac {4 \sqrt {2} a^2 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a^2-b^2} \sqrt {\tan (e+f x)}}{\sqrt {a}}\right )}{\sqrt [4]{a^2-b^2}}+\frac {2 \sqrt {2} b^2 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a^2-b^2} \sqrt {\tan (e+f x)}}{\sqrt {a}}\right )}{\sqrt [4]{a^2-b^2}}+\frac {4 \sqrt {2} a^2 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{a^2-b^2} \sqrt {\tan (e+f x)}}{\sqrt {a}}\right )}{\sqrt [4]{a^2-b^2}}-\frac {2 \sqrt {2} b^2 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{a^2-b^2} \sqrt {\tan (e+f x)}}{\sqrt {a}}\right )}{\sqrt [4]{a^2-b^2}}+2 \sqrt {2} a^{3/2} \log \left (1-\sqrt {2} \sqrt {\tan (e+f x)}+\tan (e+f x)\right )-2 \sqrt {2} a^{3/2} \log \left (1+\sqrt {2} \sqrt {\tan (e+f x)}+\tan (e+f x)\right )-\frac {2 \sqrt {2} a^2 \log \left (-a+\sqrt {2} \sqrt {a} \sqrt [4]{a^2-b^2} \sqrt {\tan (e+f x)}-\sqrt {a^2-b^2} \tan (e+f x)\right )}{\sqrt [4]{a^2-b^2}}+\frac {\sqrt {2} b^2 \log \left (-a+\sqrt {2} \sqrt {a} \sqrt [4]{a^2-b^2} \sqrt {\tan (e+f x)}-\sqrt {a^2-b^2} \tan (e+f x)\right )}{\sqrt [4]{a^2-b^2}}+\frac {2 \sqrt {2} a^2 \log \left (a+\sqrt {2} \sqrt {a} \sqrt [4]{a^2-b^2} \sqrt {\tan (e+f x)}+\sqrt {a^2-b^2} \tan (e+f x)\right )}{\sqrt [4]{a^2-b^2}}-\frac {\sqrt {2} b^2 \log \left (a+\sqrt {2} \sqrt {a} \sqrt [4]{a^2-b^2} \sqrt {\tan (e+f x)}+\sqrt {a^2-b^2} \tan (e+f x)\right )}{\sqrt [4]{a^2-b^2}}+\frac {8 \sqrt {a} b \tan ^{\frac {3}{2}}(e+f x)}{\sqrt {1+\tan ^2(e+f x)}}\right )\right )}{84 a^2 b^2 \cos ^{\frac {3}{2}}(e+f x) \sqrt {\sin (e+f x)} (a+b \sin (e+f x)) \left (-1+\tan ^2(e+f x)\right ) \sqrt {1+\tan ^2(e+f x)}}\right )}{a (a-b) (a+b) f (g \cos (e+f x))^{3/2} (d \sin (e+f x))^{3/2}} \]

[In]

Integrate[1/((g*Cos[e + f*x])^(3/2)*(d*Sin[e + f*x])^(3/2)*(a + b*Sin[e + f*x])),x]

[Out]

(Cos[e + f*x]^2*Sin[e + f*x]^2*((-2*Cot[e + f*x])/a + (2*Sec[e + f*x]*(-b + a*Sin[e + f*x]))/(a^2 - b^2)))/(f*
(g*Cos[e + f*x])^(3/2)*(d*Sin[e + f*x])^(3/2)) - (Cos[e + f*x]^(3/2)*Sin[e + f*x]^(3/2)*((-2*(4*a^3 - 2*a*b^2)
*(-(b*AppellF1[3/4, -1/4, 1, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)]) + a*AppellF1[3/4, 1/4, 1
, 7/4, Cos[e + f*x]^2, (b^2*Cos[e + f*x]^2)/(-a^2 + b^2)])*Cos[e + f*x]^(3/2)*(a + b*Sqrt[1 - Cos[e + f*x]^2])
*Sin[e + f*x]^(3/2))/(3*(a^2 - b^2)*(1 - Cos[e + f*x]^2)^(3/4)*(a + b*Sin[e + f*x])) + ((2*a^2*b - 2*b^3)*Sqrt
[Tan[e + f*x]]*((3*Sqrt[2]*a^(3/2)*(-2*ArcTan[1 - (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]] + 2*
ArcTan[1 + (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]] - Log[-a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4
)*Sqrt[Tan[e + f*x]] - Sqrt[a^2 - b^2]*Tan[e + f*x]] + Log[a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e +
f*x]] + Sqrt[a^2 - b^2]*Tan[e + f*x]]))/(a^2 - b^2)^(1/4) - 8*b*AppellF1[3/4, 1/2, 1, 7/4, -Tan[e + f*x]^2, ((
-a^2 + b^2)*Tan[e + f*x]^2)/a^2]*Tan[e + f*x]^(3/2))*(b*Tan[e + f*x] + a*Sqrt[1 + Tan[e + f*x]^2]))/(12*a^2*Co
s[e + f*x]^(3/2)*Sqrt[Sin[e + f*x]]*(a + b*Sin[e + f*x])*(1 + Tan[e + f*x]^2)^(3/2)) + ((-2*a^2*b + b^3)*Cos[2
*(e + f*x)]*Sqrt[Tan[e + f*x]]*(b*Tan[e + f*x] + a*Sqrt[1 + Tan[e + f*x]^2])*(56*b*(-3*a^2 + b^2)*AppellF1[3/4
, 1/2, 1, 7/4, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Tan[e + f*x]^(3/2) + 24*b*(-a^2 + b^2)*AppellF1
[7/4, 1/2, 1, 11/4, -Tan[e + f*x]^2, (-1 + b^2/a^2)*Tan[e + f*x]^2]*Tan[e + f*x]^(7/2) + 21*a^(3/2)*(4*Sqrt[2]
*a^(3/2)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[e + f*x]]] - 4*Sqrt[2]*a^(3/2)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[e + f*x]]] - (
4*Sqrt[2]*a^2*ArcTan[1 - (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) + (2*Sqrt[
2]*b^2*ArcTan[1 - (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) + (4*Sqrt[2]*a^2*
ArcTan[1 + (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) - (2*Sqrt[2]*b^2*ArcTan[
1 + (Sqrt[2]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) + 2*Sqrt[2]*a^(3/2)*Log[1 - Sqr
t[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]] - 2*Sqrt[2]*a^(3/2)*Log[1 + Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]]
- (2*Sqrt[2]*a^2*Log[-a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] - Sqrt[a^2 - b^2]*Tan[e + f*x]
])/(a^2 - b^2)^(1/4) + (Sqrt[2]*b^2*Log[-a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e + f*x]] - Sqrt[a^2 -
b^2]*Tan[e + f*x]])/(a^2 - b^2)^(1/4) + (2*Sqrt[2]*a^2*Log[a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(1/4)*Sqrt[Tan[e +
f*x]] + Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(1/4) - (Sqrt[2]*b^2*Log[a + Sqrt[2]*Sqrt[a]*(a^2 - b^2)^(
1/4)*Sqrt[Tan[e + f*x]] + Sqrt[a^2 - b^2]*Tan[e + f*x]])/(a^2 - b^2)^(1/4) + (8*Sqrt[a]*b*Tan[e + f*x]^(3/2))/
Sqrt[1 + Tan[e + f*x]^2])))/(84*a^2*b^2*Cos[e + f*x]^(3/2)*Sqrt[Sin[e + f*x]]*(a + b*Sin[e + f*x])*(-1 + Tan[e
+ f*x]^2)*Sqrt[1 + Tan[e + f*x]^2])))/(a*(a - b)*(a + b)*f*(g*Cos[e + f*x])^(3/2)*(d*Sin[e + f*x])^(3/2))

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(1486\) vs. \(2(546)=1092\).

Time = 2.46 (sec) , antiderivative size = 1487, normalized size of antiderivative = 2.62

method	result	size
default	\(\text {Expression too large to display}\)	\(1487\)

[In]

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

[Out]

1/f*csc(f*x+e)/(d/((1-cos(f*x+e))^2*csc(f*x+e)^2+1)*(csc(f*x+e)-cot(f*x+e)))^(3/2)*(1-cos(f*x+e))/((1-cos(f*x+
e))^2*csc(f*x+e)^2+1)^3*(8*(-a^2+b^2)^(1/2)*(-cot(f*x+e)+csc(f*x+e)+1)^(1/2)*(2+2*cot(f*x+e)-2*csc(f*x+e))^(1/
2)*(-csc(f*x+e)+cot(f*x+e))^(1/2)*EllipticE((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),1/2*2^(1/2))*a^3-4*EllipticE((-co
t(f*x+e)+csc(f*x+e)+1)^(1/2),1/2*2^(1/2))*a*b^2*(-a^2+b^2)^(1/2)*(-cot(f*x+e)+csc(f*x+e)+1)^(1/2)*(2+2*cot(f*x
+e)-2*csc(f*x+e))^(1/2)*(-csc(f*x+e)+cot(f*x+e))^(1/2)-4*(-a^2+b^2)^(1/2)*(-cot(f*x+e)+csc(f*x+e)+1)^(1/2)*(2+
2*cot(f*x+e)-2*csc(f*x+e))^(1/2)*(-csc(f*x+e)+cot(f*x+e))^(1/2)*EllipticF((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),1/2
*2^(1/2))*a^3+2*EllipticF((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),1/2*2^(1/2))*a*b^2*(-a^2+b^2)^(1/2)*(-cot(f*x+e)+cs
c(f*x+e)+1)^(1/2)*(2+2*cot(f*x+e)-2*csc(f*x+e))^(1/2)*(-csc(f*x+e)+cot(f*x+e))^(1/2)-2*EllipticF((-cot(f*x+e)+
csc(f*x+e)+1)^(1/2),1/2*2^(1/2))*b^3*(-a^2+b^2)^(1/2)*(-cot(f*x+e)+csc(f*x+e)+1)^(1/2)*(2+2*cot(f*x+e)-2*csc(f
*x+e))^(1/2)*(-csc(f*x+e)+cot(f*x+e))^(1/2)-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*b^3*(-cot(f*x+e)+csc(f*x+e)+1)^(1/2)*(2+2*cot(f*x+e)-2*csc(f*x+e))^(1/2)*(-csc(f*x+e)+cot(f
*x+e))^(1/2)-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^4*(-cot(f*x+
e)+csc(f*x+e)+1)^(1/2)*(2+2*cot(f*x+e)-2*csc(f*x+e))^(1/2)*(-csc(f*x+e)+cot(f*x+e))^(1/2)+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^3*(-a^2+b^2)^(1/2)*(-cot(f*x+e)+csc(f*x+e)+1)
^(1/2)*(2+2*cot(f*x+e)-2*csc(f*x+e))^(1/2)*(-csc(f*x+e)+cot(f*x+e))^(1/2)+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*b^3*(-cot(f*x+e)+csc(f*x+e)+1)^(1/2)*(2+2*cot(f*x+e)-2*csc(f*
x+e))^(1/2)*(-csc(f*x+e)+cot(f*x+e))^(1/2)+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^4*(-cot(f*x+e)+csc(f*x+e)+1)^(1/2)*(2+2*cot(f*x+e)-2*csc(f*x+e))^(1/2)*(-csc(f*x+e)+cot(f*x+
e))^(1/2)+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^3*(-a^2+b^2)^(1
/2)*(-cot(f*x+e)+csc(f*x+e)+1)^(1/2)*(2+2*cot(f*x+e)-2*csc(f*x+e))^(1/2)*(-csc(f*x+e)+cot(f*x+e))^(1/2)+6*csc(
f*x+e)^2*a^3*(-a^2+b^2)^(1/2)*(1-cos(f*x+e))^2-2*csc(f*x+e)^2*a*b^2*(-a^2+b^2)^(1/2)*(1-cos(f*x+e))^2-4*a^2*b*
(-a^2+b^2)^(1/2)*(csc(f*x+e)-cot(f*x+e))-2*(-a^2+b^2)^(1/2)*a^3+2*(-a^2+b^2)^(1/2)*a*b^2)*((1-cos(f*x+e))^2*cs
c(f*x+e)^2-1)/(-g*((1-cos(f*x+e))^2*csc(f*x+e)^2-1)/((1-cos(f*x+e))^2*csc(f*x+e)^2+1))^(3/2)*2^(1/2)/(a+b)/(-a
^2+b^2)^(1/2)/(-b+(-a^2+b^2)^(1/2)+a)/(b+(-a^2+b^2)^(1/2)-a)/a

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{(g \cos (e+f x))^{3/2} (d \sin (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(g \cos (e+f x))^{3/2} (d \sin (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \frac {1}{(g \cos (e+f x))^{3/2} (d \sin (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx=\int { \frac {1}{\left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} {\left (b \sin \left (f x + e\right ) + a\right )} \left (d \sin \left (f x + e\right )\right )^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {1}{(g \cos (e+f x))^{3/2} (d \sin (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx=\int { \frac {1}{\left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} {\left (b \sin \left (f x + e\right ) + a\right )} \left (d \sin \left (f x + e\right )\right )^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(g \cos (e+f x))^{3/2} (d \sin (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx=\int \frac {1}{{\left (g\,\cos \left (e+f\,x\right )\right )}^{3/2}\,{\left (d\,\sin \left (e+f\,x\right )\right )}^{3/2}\,\left (a+b\,\sin \left (e+f\,x\right )\right )} \,d x \]

[In]

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

[Out]

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