\(\int \frac {(g \cos (e+f x))^{5/2}}{(d \sin (e+f x))^{11/2} (a+b \sin (e+f x))} \, dx\) [1429]

Optimal result

Integrand size = 37, antiderivative size = 822 \[ \int \frac {(g \cos (e+f x))^{5/2}}{(d \sin (e+f x))^{11/2} (a+b \sin (e+f x))} \, dx=-\frac {2 g (g \cos (e+f x))^{3/2}}{9 a d f (d \sin (e+f x))^{9/2}}+\frac {2 b g (g \cos (e+f x))^{3/2}}{7 a^2 d^2 f (d \sin (e+f x))^{7/2}}-\frac {4 g (g \cos (e+f x))^{3/2}}{15 a d^3 f (d \sin (e+f x))^{5/2}}+\frac {2 \left (a^2-b^2\right ) g (g \cos (e+f x))^{3/2}}{5 a^3 d^3 f (d \sin (e+f x))^{5/2}}+\frac {8 b g (g \cos (e+f x))^{3/2}}{21 a^2 d^4 f (d \sin (e+f x))^{3/2}}-\frac {2 b \left (a^2-b^2\right ) g (g \cos (e+f x))^{3/2}}{3 a^4 d^4 f (d \sin (e+f x))^{3/2}}-\frac {8 g (g \cos (e+f x))^{3/2}}{15 a d^5 f \sqrt {d \sin (e+f x)}}+\frac {4 \left (a^2-b^2\right ) g (g \cos (e+f x))^{3/2}}{5 a^3 d^5 f \sqrt {d \sin (e+f x)}}+\frac {2 b^2 \left (a^2-b^2\right ) g (g \cos (e+f x))^{3/2}}{a^5 d^5 f \sqrt {d \sin (e+f x)}}-\frac {2 \sqrt {2} b^3 \sqrt {-a+b} \sqrt {a+b} g^{5/2} \operatorname {EllipticPi}\left (-\frac {\sqrt {-a+b}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {1+\sin (e+f x)}}\right ),-1\right ) \sqrt {\sin (e+f x)}}{a^5 d^5 f \sqrt {d \sin (e+f x)}}+\frac {2 \sqrt {2} b^3 \sqrt {-a+b} \sqrt {a+b} g^{5/2} \operatorname {EllipticPi}\left (\frac {\sqrt {-a+b}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {1+\sin (e+f x)}}\right ),-1\right ) \sqrt {\sin (e+f x)}}{a^5 d^5 f \sqrt {d \sin (e+f x)}}-\frac {8 g^2 \sqrt {g \cos (e+f x)} E\left (\left .e-\frac {\pi }{4}+f x\right |2\right ) \sqrt {d \sin (e+f x)}}{15 a d^6 f \sqrt {\sin (2 e+2 f x)}}+\frac {4 \left (a^2-b^2\right ) g^2 \sqrt {g \cos (e+f x)} E\left (\left .e-\frac {\pi }{4}+f x\right |2\right ) \sqrt {d \sin (e+f x)}}{5 a^3 d^6 f \sqrt {\sin (2 e+2 f x)}}+\frac {2 b^2 \left (a^2-b^2\right ) g^2 \sqrt {g \cos (e+f x)} E\left (\left .e-\frac {\pi }{4}+f x\right |2\right ) \sqrt {d \sin (e+f x)}}{a^5 d^6 f \sqrt {\sin (2 e+2 f x)}} \]

[Out]

-2/9*g*(g*cos(f*x+e))^(3/2)/a/d/f/(d*sin(f*x+e))^(9/2)+2/7*b*g*(g*cos(f*x+e))^(3/2)/a^2/d^2/f/(d*sin(f*x+e))^(
7/2)-4/15*g*(g*cos(f*x+e))^(3/2)/a/d^3/f/(d*sin(f*x+e))^(5/2)+2/5*(a^2-b^2)*g*(g*cos(f*x+e))^(3/2)/a^3/d^3/f/(
d*sin(f*x+e))^(5/2)+8/21*b*g*(g*cos(f*x+e))^(3/2)/a^2/d^4/f/(d*sin(f*x+e))^(3/2)-2/3*b*(a^2-b^2)*g*(g*cos(f*x+
e))^(3/2)/a^4/d^4/f/(d*sin(f*x+e))^(3/2)-8/15*g*(g*cos(f*x+e))^(3/2)/a/d^5/f/(d*sin(f*x+e))^(1/2)+4/5*(a^2-b^2
)*g*(g*cos(f*x+e))^(3/2)/a^3/d^5/f/(d*sin(f*x+e))^(1/2)+2*b^2*(a^2-b^2)*g*(g*cos(f*x+e))^(3/2)/a^5/d^5/f/(d*si
n(f*x+e))^(1/2)-2*b^3*g^(5/2)*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)*(-a+b)^(1/2)*(a+b)^(1/2)*sin(f*x+e)^(1/2)/a^5/d^5/f/(d*sin(f*x+e))^(1/2)+2*b^3*g^(5/2)*Ellip
ticPi((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)*(-a+b)^(1/2)*(a+b)
^(1/2)*sin(f*x+e)^(1/2)/a^5/d^5/f/(d*sin(f*x+e))^(1/2)+8/15*g^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/d^6/f/sin(2*f*x+2*e)^(1/2)-4/
5*(a^2-b^2)*g^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^3/d^6/f/sin(2*f*x+2*e)^(1/2)-2*b^2*(a^2-b^2)*g^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^5/d^6/f/sin
(2*f*x+2*e)^(1/2)

Rubi [A] (verified)

Time = 1.33 (sec) , antiderivative size = 822, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.270, Rules used = {2978, 2650, 2652, 2719, 2643, 2989, 2985, 2984, 504, 1232} \[ \int \frac {(g \cos (e+f x))^{5/2}}{(d \sin (e+f x))^{11/2} (a+b \sin (e+f x))} \, dx=-\frac {2 \sqrt {2} \sqrt {b-a} \sqrt {a+b} g^{5/2} \operatorname {EllipticPi}\left (-\frac {\sqrt {b-a}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {\sin (e+f x)+1}}\right ),-1\right ) \sqrt {\sin (e+f x)} b^3}{a^5 d^5 f \sqrt {d \sin (e+f x)}}+\frac {2 \sqrt {2} \sqrt {b-a} \sqrt {a+b} g^{5/2} \operatorname {EllipticPi}\left (\frac {\sqrt {b-a}}{\sqrt {a+b}},\arcsin \left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt {\sin (e+f x)+1}}\right ),-1\right ) \sqrt {\sin (e+f x)} b^3}{a^5 d^5 f \sqrt {d \sin (e+f x)}}+\frac {2 \left (a^2-b^2\right ) g (g \cos (e+f x))^{3/2} b^2}{a^5 d^5 f \sqrt {d \sin (e+f x)}}+\frac {2 \left (a^2-b^2\right ) g^2 \sqrt {g \cos (e+f x)} E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)} b^2}{a^5 d^6 f \sqrt {\sin (2 e+2 f x)}}-\frac {2 \left (a^2-b^2\right ) g (g \cos (e+f x))^{3/2} b}{3 a^4 d^4 f (d \sin (e+f x))^{3/2}}+\frac {8 g (g \cos (e+f x))^{3/2} b}{21 a^2 d^4 f (d \sin (e+f x))^{3/2}}+\frac {2 g (g \cos (e+f x))^{3/2} b}{7 a^2 d^2 f (d \sin (e+f x))^{7/2}}+\frac {4 \left (a^2-b^2\right ) g (g \cos (e+f x))^{3/2}}{5 a^3 d^5 f \sqrt {d \sin (e+f x)}}-\frac {8 g (g \cos (e+f x))^{3/2}}{15 a d^5 f \sqrt {d \sin (e+f x)}}+\frac {4 \left (a^2-b^2\right ) g^2 \sqrt {g \cos (e+f x)} E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)}}{5 a^3 d^6 f \sqrt {\sin (2 e+2 f x)}}-\frac {8 g^2 \sqrt {g \cos (e+f x)} E\left (\left .e+f x-\frac {\pi }{4}\right |2\right ) \sqrt {d \sin (e+f x)}}{15 a d^6 f \sqrt {\sin (2 e+2 f x)}}+\frac {2 \left (a^2-b^2\right ) g (g \cos (e+f x))^{3/2}}{5 a^3 d^3 f (d \sin (e+f x))^{5/2}}-\frac {4 g (g \cos (e+f x))^{3/2}}{15 a d^3 f (d \sin (e+f x))^{5/2}}-\frac {2 g (g \cos (e+f x))^{3/2}}{9 a d f (d \sin (e+f x))^{9/2}} \]

[In]

Int[(g*Cos[e + f*x])^(5/2)/((d*Sin[e + f*x])^(11/2)*(a + b*Sin[e + f*x])),x]

[Out]

(-2*g*(g*Cos[e + f*x])^(3/2))/(9*a*d*f*(d*Sin[e + f*x])^(9/2)) + (2*b*g*(g*Cos[e + f*x])^(3/2))/(7*a^2*d^2*f*(
d*Sin[e + f*x])^(7/2)) - (4*g*(g*Cos[e + f*x])^(3/2))/(15*a*d^3*f*(d*Sin[e + f*x])^(5/2)) + (2*(a^2 - b^2)*g*(
g*Cos[e + f*x])^(3/2))/(5*a^3*d^3*f*(d*Sin[e + f*x])^(5/2)) + (8*b*g*(g*Cos[e + f*x])^(3/2))/(21*a^2*d^4*f*(d*
Sin[e + f*x])^(3/2)) - (2*b*(a^2 - b^2)*g*(g*Cos[e + f*x])^(3/2))/(3*a^4*d^4*f*(d*Sin[e + f*x])^(3/2)) - (8*g*
(g*Cos[e + f*x])^(3/2))/(15*a*d^5*f*Sqrt[d*Sin[e + f*x]]) + (4*(a^2 - b^2)*g*(g*Cos[e + f*x])^(3/2))/(5*a^3*d^
5*f*Sqrt[d*Sin[e + f*x]]) + (2*b^2*(a^2 - b^2)*g*(g*Cos[e + f*x])^(3/2))/(a^5*d^5*f*Sqrt[d*Sin[e + f*x]]) - (2
*Sqrt[2]*b^3*Sqrt[-a + b]*Sqrt[a + b]*g^(5/2)*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^5*d^5*f*Sqrt[d*Sin[e + f*x]]) + (2*Sqrt[2]*b
^3*Sqrt[-a + b]*Sqrt[a + b]*g^(5/2)*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^5*d^5*f*Sqrt[d*Sin[e + f*x]]) - (8*g^2*Sqrt[g*Cos[e + f*x
]]*EllipticE[e - Pi/4 + f*x, 2]*Sqrt[d*Sin[e + f*x]])/(15*a*d^6*f*Sqrt[Sin[2*e + 2*f*x]]) + (4*(a^2 - b^2)*g^2
*Sqrt[g*Cos[e + f*x]]*EllipticE[e - Pi/4 + f*x, 2]*Sqrt[d*Sin[e + f*x]])/(5*a^3*d^6*f*Sqrt[Sin[2*e + 2*f*x]])
+ (2*b^2*(a^2 - b^2)*g^2*Sqrt[g*Cos[e + f*x]]*EllipticE[e - Pi/4 + f*x, 2]*Sqrt[d*Sin[e + f*x]])/(a^5*d^6*f*Sq
rt[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 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 2978

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, Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^n, x], x] + (-Dist[b*(g^2/(a^
2*d)), Int[(g*Cos[e + f*x])^(p - 2)*(d*Sin[e + f*x])^(n + 1), x], x] - Dist[g^2*((a^2 - b^2)/(a^2*d^2)), Int[(
g*Cos[e + f*x])^(p - 2)*((d*Sin[e + f*x])^(n + 2)/(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] && (LeQ[n, -2] || (EqQ[n, -3/2] && EqQ[p, 3/2]))

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 {g^2 \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{11/2}} \, dx}{a}-\frac {\left (\left (a^2-b^2\right ) g^2\right ) \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{7/2} (a+b \sin (e+f x))} \, dx}{a^2 d^2}-\frac {\left (b g^2\right ) \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{9/2}} \, dx}{a^2 d} \\ & = -\frac {2 g (g \cos (e+f x))^{3/2}}{9 a d f (d \sin (e+f x))^{9/2}}+\frac {2 b g (g \cos (e+f x))^{3/2}}{7 a^2 d^2 f (d \sin (e+f x))^{7/2}}-\frac {\left (4 b g^2\right ) \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{5/2}} \, dx}{7 a^2 d^3}+\frac {\left (b \left (a^2-b^2\right ) g^2\right ) \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{5/2} (a+b \sin (e+f x))} \, dx}{a^3 d^3}+\frac {\left (2 g^2\right ) \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{7/2}} \, dx}{3 a d^2}-\frac {\left (\left (a^2-b^2\right ) g^2\right ) \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{7/2}} \, dx}{a^3 d^2} \\ & = -\frac {2 g (g \cos (e+f x))^{3/2}}{9 a d f (d \sin (e+f x))^{9/2}}+\frac {2 b g (g \cos (e+f x))^{3/2}}{7 a^2 d^2 f (d \sin (e+f x))^{7/2}}-\frac {4 g (g \cos (e+f x))^{3/2}}{15 a d^3 f (d \sin (e+f x))^{5/2}}+\frac {2 \left (a^2-b^2\right ) g (g \cos (e+f x))^{3/2}}{5 a^3 d^3 f (d \sin (e+f x))^{5/2}}+\frac {8 b g (g \cos (e+f x))^{3/2}}{21 a^2 d^4 f (d \sin (e+f x))^{3/2}}+\frac {\left (4 g^2\right ) \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{3/2}} \, dx}{15 a d^4}-\frac {\left (2 \left (a^2-b^2\right ) g^2\right ) \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{3/2}} \, dx}{5 a^3 d^4}-\frac {\left (b^2 \left (a^2-b^2\right ) g^2\right ) \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{3/2} (a+b \sin (e+f x))} \, dx}{a^4 d^4}+\frac {\left (b \left (a^2-b^2\right ) g^2\right ) \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{5/2}} \, dx}{a^4 d^3} \\ & = -\frac {2 g (g \cos (e+f x))^{3/2}}{9 a d f (d \sin (e+f x))^{9/2}}+\frac {2 b g (g \cos (e+f x))^{3/2}}{7 a^2 d^2 f (d \sin (e+f x))^{7/2}}-\frac {4 g (g \cos (e+f x))^{3/2}}{15 a d^3 f (d \sin (e+f x))^{5/2}}+\frac {2 \left (a^2-b^2\right ) g (g \cos (e+f x))^{3/2}}{5 a^3 d^3 f (d \sin (e+f x))^{5/2}}+\frac {8 b g (g \cos (e+f x))^{3/2}}{21 a^2 d^4 f (d \sin (e+f x))^{3/2}}-\frac {2 b \left (a^2-b^2\right ) g (g \cos (e+f x))^{3/2}}{3 a^4 d^4 f (d \sin (e+f x))^{3/2}}-\frac {8 g (g \cos (e+f x))^{3/2}}{15 a d^5 f \sqrt {d \sin (e+f x)}}+\frac {4 \left (a^2-b^2\right ) g (g \cos (e+f x))^{3/2}}{5 a^3 d^5 f \sqrt {d \sin (e+f x)}}-\frac {\left (8 g^2\right ) \int \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)} \, dx}{15 a d^6}+\frac {\left (4 \left (a^2-b^2\right ) g^2\right ) \int \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)} \, dx}{5 a^3 d^6}+\frac {\left (b^3 \left (a^2-b^2\right ) g^2\right ) \int \frac {\sqrt {g \cos (e+f x)}}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))} \, dx}{a^5 d^5}-\frac {\left (b^2 \left (a^2-b^2\right ) g^2\right ) \int \frac {\sqrt {g \cos (e+f x)}}{(d \sin (e+f x))^{3/2}} \, dx}{a^5 d^4} \\ & = -\frac {2 g (g \cos (e+f x))^{3/2}}{9 a d f (d \sin (e+f x))^{9/2}}+\frac {2 b g (g \cos (e+f x))^{3/2}}{7 a^2 d^2 f (d \sin (e+f x))^{7/2}}-\frac {4 g (g \cos (e+f x))^{3/2}}{15 a d^3 f (d \sin (e+f x))^{5/2}}+\frac {2 \left (a^2-b^2\right ) g (g \cos (e+f x))^{3/2}}{5 a^3 d^3 f (d \sin (e+f x))^{5/2}}+\frac {8 b g (g \cos (e+f x))^{3/2}}{21 a^2 d^4 f (d \sin (e+f x))^{3/2}}-\frac {2 b \left (a^2-b^2\right ) g (g \cos (e+f x))^{3/2}}{3 a^4 d^4 f (d \sin (e+f x))^{3/2}}-\frac {8 g (g \cos (e+f x))^{3/2}}{15 a d^5 f \sqrt {d \sin (e+f x)}}+\frac {4 \left (a^2-b^2\right ) g (g \cos (e+f x))^{3/2}}{5 a^3 d^5 f \sqrt {d \sin (e+f x)}}+\frac {2 b^2 \left (a^2-b^2\right ) g (g \cos (e+f x))^{3/2}}{a^5 d^5 f \sqrt {d \sin (e+f x)}}+\frac {\left (2 b^2 \left (a^2-b^2\right ) g^2\right ) \int \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)} \, dx}{a^5 d^6}+\frac {\left (b^3 \left (a^2-b^2\right ) g^2 \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^5 d^5 \sqrt {d \sin (e+f x)}}-\frac {\left (8 g^2 \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}\right ) \int \sqrt {\sin (2 e+2 f x)} \, dx}{15 a d^6 \sqrt {\sin (2 e+2 f x)}}+\frac {\left (4 \left (a^2-b^2\right ) g^2 \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}\right ) \int \sqrt {\sin (2 e+2 f x)} \, dx}{5 a^3 d^6 \sqrt {\sin (2 e+2 f x)}} \\ & = -\frac {2 g (g \cos (e+f x))^{3/2}}{9 a d f (d \sin (e+f x))^{9/2}}+\frac {2 b g (g \cos (e+f x))^{3/2}}{7 a^2 d^2 f (d \sin (e+f x))^{7/2}}-\frac {4 g (g \cos (e+f x))^{3/2}}{15 a d^3 f (d \sin (e+f x))^{5/2}}+\frac {2 \left (a^2-b^2\right ) g (g \cos (e+f x))^{3/2}}{5 a^3 d^3 f (d \sin (e+f x))^{5/2}}+\frac {8 b g (g \cos (e+f x))^{3/2}}{21 a^2 d^4 f (d \sin (e+f x))^{3/2}}-\frac {2 b \left (a^2-b^2\right ) g (g \cos (e+f x))^{3/2}}{3 a^4 d^4 f (d \sin (e+f x))^{3/2}}-\frac {8 g (g \cos (e+f x))^{3/2}}{15 a d^5 f \sqrt {d \sin (e+f x)}}+\frac {4 \left (a^2-b^2\right ) g (g \cos (e+f x))^{3/2}}{5 a^3 d^5 f \sqrt {d \sin (e+f x)}}+\frac {2 b^2 \left (a^2-b^2\right ) g (g \cos (e+f x))^{3/2}}{a^5 d^5 f \sqrt {d \sin (e+f x)}}-\frac {8 g^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)}}{15 a d^6 f \sqrt {\sin (2 e+2 f x)}}+\frac {4 \left (a^2-b^2\right ) g^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)}}{5 a^3 d^6 f \sqrt {\sin (2 e+2 f x)}}-\frac {\left (4 \sqrt {2} b^3 \left (a^2-b^2\right ) g^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^5 d^5 f \sqrt {d \sin (e+f x)}}+\frac {\left (2 b^2 \left (a^2-b^2\right ) g^2 \sqrt {g \cos (e+f x)} \sqrt {d \sin (e+f x)}\right ) \int \sqrt {\sin (2 e+2 f x)} \, dx}{a^5 d^6 \sqrt {\sin (2 e+2 f x)}} \\ & = -\frac {2 g (g \cos (e+f x))^{3/2}}{9 a d f (d \sin (e+f x))^{9/2}}+\frac {2 b g (g \cos (e+f x))^{3/2}}{7 a^2 d^2 f (d \sin (e+f x))^{7/2}}-\frac {4 g (g \cos (e+f x))^{3/2}}{15 a d^3 f (d \sin (e+f x))^{5/2}}+\frac {2 \left (a^2-b^2\right ) g (g \cos (e+f x))^{3/2}}{5 a^3 d^3 f (d \sin (e+f x))^{5/2}}+\frac {8 b g (g \cos (e+f x))^{3/2}}{21 a^2 d^4 f (d \sin (e+f x))^{3/2}}-\frac {2 b \left (a^2-b^2\right ) g (g \cos (e+f x))^{3/2}}{3 a^4 d^4 f (d \sin (e+f x))^{3/2}}-\frac {8 g (g \cos (e+f x))^{3/2}}{15 a d^5 f \sqrt {d \sin (e+f x)}}+\frac {4 \left (a^2-b^2\right ) g (g \cos (e+f x))^{3/2}}{5 a^3 d^5 f \sqrt {d \sin (e+f x)}}+\frac {2 b^2 \left (a^2-b^2\right ) g (g \cos (e+f x))^{3/2}}{a^5 d^5 f \sqrt {d \sin (e+f x)}}-\frac {8 g^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)}}{15 a d^6 f \sqrt {\sin (2 e+2 f x)}}+\frac {4 \left (a^2-b^2\right ) g^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)}}{5 a^3 d^6 f \sqrt {\sin (2 e+2 f x)}}+\frac {2 b^2 \left (a^2-b^2\right ) g^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^5 d^6 f \sqrt {\sin (2 e+2 f x)}}-\frac {\left (2 \sqrt {2} b^3 \left (a^2-b^2\right ) g^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^5 \sqrt {-a+b} d^5 f \sqrt {d \sin (e+f x)}}+\frac {\left (2 \sqrt {2} b^3 \left (a^2-b^2\right ) g^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^5 \sqrt {-a+b} d^5 f \sqrt {d \sin (e+f x)}} \\ & = -\frac {2 g (g \cos (e+f x))^{3/2}}{9 a d f (d \sin (e+f x))^{9/2}}+\frac {2 b g (g \cos (e+f x))^{3/2}}{7 a^2 d^2 f (d \sin (e+f x))^{7/2}}-\frac {4 g (g \cos (e+f x))^{3/2}}{15 a d^3 f (d \sin (e+f x))^{5/2}}+\frac {2 \left (a^2-b^2\right ) g (g \cos (e+f x))^{3/2}}{5 a^3 d^3 f (d \sin (e+f x))^{5/2}}+\frac {8 b g (g \cos (e+f x))^{3/2}}{21 a^2 d^4 f (d \sin (e+f x))^{3/2}}-\frac {2 b \left (a^2-b^2\right ) g (g \cos (e+f x))^{3/2}}{3 a^4 d^4 f (d \sin (e+f x))^{3/2}}-\frac {8 g (g \cos (e+f x))^{3/2}}{15 a d^5 f \sqrt {d \sin (e+f x)}}+\frac {4 \left (a^2-b^2\right ) g (g \cos (e+f x))^{3/2}}{5 a^3 d^5 f \sqrt {d \sin (e+f x)}}+\frac {2 b^2 \left (a^2-b^2\right ) g (g \cos (e+f x))^{3/2}}{a^5 d^5 f \sqrt {d \sin (e+f x)}}-\frac {2 \sqrt {2} b^3 \sqrt {-a+b} \sqrt {a+b} g^{5/2} \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^5 d^5 f \sqrt {d \sin (e+f x)}}+\frac {2 \sqrt {2} b^3 \sqrt {-a+b} \sqrt {a+b} g^{5/2} \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^5 d^5 f \sqrt {d \sin (e+f x)}}-\frac {8 g^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)}}{15 a d^6 f \sqrt {\sin (2 e+2 f x)}}+\frac {4 \left (a^2-b^2\right ) g^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)}}{5 a^3 d^6 f \sqrt {\sin (2 e+2 f x)}}+\frac {2 b^2 \left (a^2-b^2\right ) g^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^5 d^6 f \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 = 24.08 (sec) , antiderivative size = 1850, normalized size of antiderivative = 2.25 \[ \int \frac {(g \cos (e+f x))^{5/2}}{(d \sin (e+f x))^{11/2} (a+b \sin (e+f x))} \, dx=\frac {(g \cos (e+f x))^{5/2} \left (\frac {2 \left (2 a^4 \cos (e+f x)+9 a^2 b^2 \cos (e+f x)-15 b^4 \cos (e+f x)\right ) \csc (e+f x)}{15 a^5}-\frac {2 \left (3 a^2 b \cos (e+f x)-7 b^3 \cos (e+f x)\right ) \csc ^2(e+f x)}{21 a^4}+\frac {2 \left (a^2 \cos (e+f x)-3 b^2 \cos (e+f x)\right ) \csc ^3(e+f x)}{15 a^3}+\frac {2 b \cot (e+f x) \csc ^3(e+f x)}{7 a^2}-\frac {2 \cot (e+f x) \csc ^4(e+f x)}{9 a}\right ) \sin ^4(e+f x) \tan ^2(e+f x)}{f (d \sin (e+f x))^{11/2}}+\frac {(g \cos (e+f x))^{5/2} \sin ^{\frac {11}{2}}(e+f x) \left (-\frac {2 \left (4 a^5+18 a^3 b^2-30 a b^4\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^4 b+24 a^2 b^3-30 b^5\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^4 b-9 a^2 b^3+15 b^5\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 )}{15 a^5 f \cos ^{\frac {5}{2}}(e+f x) (d \sin (e+f x))^{11/2}} \]

[In]

Integrate[(g*Cos[e + f*x])^(5/2)/((d*Sin[e + f*x])^(11/2)*(a + b*Sin[e + f*x])),x]

[Out]

((g*Cos[e + f*x])^(5/2)*((2*(2*a^4*Cos[e + f*x] + 9*a^2*b^2*Cos[e + f*x] - 15*b^4*Cos[e + f*x])*Csc[e + f*x])/
(15*a^5) - (2*(3*a^2*b*Cos[e + f*x] - 7*b^3*Cos[e + f*x])*Csc[e + f*x]^2)/(21*a^4) + (2*(a^2*Cos[e + f*x] - 3*
b^2*Cos[e + f*x])*Csc[e + f*x]^3)/(15*a^3) + (2*b*Cot[e + f*x]*Csc[e + f*x]^3)/(7*a^2) - (2*Cot[e + f*x]*Csc[e
+ f*x]^4)/(9*a))*Sin[e + f*x]^4*Tan[e + f*x]^2)/(f*(d*Sin[e + f*x])^(11/2)) + ((g*Cos[e + f*x])^(5/2)*Sin[e +
f*x]^(11/2)*((-2*(4*a^5 + 18*a^3*b^2 - 30*a*b^4)*(-(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)])*Co
s[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^4*b + 24*a^2*b^3 - 30*b^5)*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*Cos[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^4*b - 9*a^2*b^3 + 15*b^5)*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)*S
qrt[Tan[e + f*x]])/Sqrt[a]])/(a^2 - b^2)^(1/4) + 2*Sqrt[2]*a^(3/2)*Log[1 - Sqrt[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])))/(15*a^5*f*Cos[e + f*x]^(5/2)*(d*Sin[e + f*x])^(11/2))

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(4944\) vs. \(2(779)=1558\).

Time = 2.53 (sec) , antiderivative size = 4945, normalized size of antiderivative = 6.02

method	result	size
default	\(\text {Expression too large to display}\)	\(4945\)

[In]

int((g*cos(f*x+e))^(5/2)/(d*sin(f*x+e))^(11/2)/(a+b*sin(f*x+e)),x,method=_RETURNVERBOSE)

[Out]

-1/2520/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)))^(11/2)*(1-cos(f*x+e))/((1-c
os(f*x+e))^2*csc(f*x+e)^2+1)^3*(-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))^(5/2)*
(5040*csc(f*x+e)^4*a^2*b^4*(-a^2+b^2)^(1/2)*(1-cos(f*x+e))^4-5040*csc(f*x+e)^4*a*b^5*(-a^2+b^2)^(1/2)*(1-cos(f
*x+e))^4-35*csc(f*x+e)^10*a^5*b*(-a^2+b^2)^(1/2)*(1-cos(f*x+e))^10-90*csc(f*x+e)^9*a^5*b*(-a^2+b^2)^(1/2)*(1-c
os(f*x+e))^9+90*csc(f*x+e)^9*a^4*b^2*(-a^2+b^2)^(1/2)*(1-cos(f*x+e))^9+49*csc(f*x+e)^8*a^5*b*(-a^2+b^2)^(1/2)*
(1-cos(f*x+e))^8-35*(-a^2+b^2)^(1/2)*a^5*b+252*csc(f*x+e)^8*a^4*b^2*(-a^2+b^2)^(1/2)*(1-cos(f*x+e))^8-252*csc(
f*x+e)^8*a^3*b^3*(-a^2+b^2)^(1/2)*(1-cos(f*x+e))^8+360*csc(f*x+e)^7*a^5*b*(-a^2+b^2)^(1/2)*(1-cos(f*x+e))^7-36
0*csc(f*x+e)^7*a^4*b^2*(-a^2+b^2)^(1/2)*(1-cos(f*x+e))^7-840*csc(f*x+e)^7*a^3*b^3*(-a^2+b^2)^(1/2)*(1-cos(f*x+
e))^7+840*csc(f*x+e)^7*a^2*b^4*(-a^2+b^2)^(1/2)*(1-cos(f*x+e))^7-686*csc(f*x+e)^6*a^5*b*(-a^2+b^2)^(1/2)*(1-co
s(f*x+e))^6+360*csc(f*x+e)^3*a^5*b*(-a^2+b^2)^(1/2)*(1-cos(f*x+e))^3-360*csc(f*x+e)^3*a^4*b^2*(-a^2+b^2)^(1/2)
*(1-cos(f*x+e))^3-840*csc(f*x+e)^3*a^3*b^3*(-a^2+b^2)^(1/2)*(1-cos(f*x+e))^3+840*csc(f*x+e)^3*a^2*b^4*(-a^2+b^
2)^(1/2)*(1-cos(f*x+e))^3-90*a^5*b*(-a^2+b^2)^(1/2)*(csc(f*x+e)-cot(f*x+e))+90*a^4*b^2*(-a^2+b^2)^(1/2)*(csc(f
*x+e)-cot(f*x+e))+35*csc(f*x+e)^10*a^6*(-a^2+b^2)^(1/2)*(1-cos(f*x+e))^10-49*csc(f*x+e)^8*a^6*(-a^2+b^2)^(1/2)
*(1-cos(f*x+e))^8+686*csc(f*x+e)^6*a^6*(-a^2+b^2)^(1/2)*(1-cos(f*x+e))^6-658*csc(f*x+e)^4*a^6*(-a^2+b^2)^(1/2)
*(1-cos(f*x+e))^4-49*csc(f*x+e)^2*a^6*(-a^2+b^2)^(1/2)*(1-cos(f*x+e))^2+49*csc(f*x+e)^2*a^5*b*(-a^2+b^2)^(1/2)
*(1-cos(f*x+e))^2+252*csc(f*x+e)^2*a^4*b^2*(-a^2+b^2)^(1/2)*(1-cos(f*x+e))^2-252*csc(f*x+e)^2*a^3*b^3*(-a^2+b^
2)^(1/2)*(1-cos(f*x+e))^2+2772*csc(f*x+e)^6*a^4*b^2*(-a^2+b^2)^(1/2)*(1-cos(f*x+e))^6-2772*csc(f*x+e)^6*a^3*b^
3*(-a^2+b^2)^(1/2)*(1-cos(f*x+e))^6-5040*csc(f*x+e)^6*a^2*b^4*(-a^2+b^2)^(1/2)*(1-cos(f*x+e))^6+5040*csc(f*x+e
)^6*a*b^5*(-a^2+b^2)^(1/2)*(1-cos(f*x+e))^6-540*csc(f*x+e)^5*a^5*b*(-a^2+b^2)^(1/2)*(1-cos(f*x+e))^5+540*csc(f
*x+e)^5*a^4*b^2*(-a^2+b^2)^(1/2)*(1-cos(f*x+e))^5+1680*csc(f*x+e)^5*a^3*b^3*(-a^2+b^2)^(1/2)*(1-cos(f*x+e))^5-
1680*csc(f*x+e)^5*a^2*b^4*(-a^2+b^2)^(1/2)*(1-cos(f*x+e))^5+658*csc(f*x+e)^4*a^5*b*(-a^2+b^2)^(1/2)*(1-cos(f*x
+e))^4-3276*csc(f*x+e)^4*a^4*b^2*(-a^2+b^2)^(1/2)*(1-cos(f*x+e))^4+3276*csc(f*x+e)^4*a^3*b^3*(-a^2+b^2)^(1/2)*
(1-cos(f*x+e))^4-2520*csc(f*x+e)^4*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^4*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)*(1-cos(f*x+e))^4+5040*csc(f*x+e)^4*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^5*(-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)*(1-cos(f*x+e))^4-2520*csc(f*x+e)^4*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^6*(-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)*(-a^2+b^2)^(1/2)*(1-cos(f*x+e))^4+35*(-a^2+b^2)^(1/2)*a^6-1344*csc(f*x+e)^4*EllipticE((-cot(f*x+
e)+csc(f*x+e)+1)^(1/2),1/2*2^(1/2))*a^5*b*(-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)*(-a^2+b^2)^(1/2)*(1-cos(f*x+e))^4+2520*csc(f*x+e)^4*EllipticPi((-cot(f*x+e)+cs
c(f*x+e)+1)^(1/2),-a/(b+(-a^2+b^2)^(1/2)-a),1/2*2^(1/2))*a^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)*(-a^2+b^2)^(1/2)*(1-cos(f*x+e))^4+2520*csc(f*x+e)^4*Ell
ipticPi((-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^5*(-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)*(-a^2+b^2)^(1/2)*(1-cos(f*x+e))^4-
2520*csc(f*x+e)^4*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^3*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)*(-a^2+b^2)^
(1/2)*(1-cos(f*x+e))^4+2520*csc(f*x+e)^4*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^5*(-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)*(-a^2+b^2)^(1/2)*(1-cos(f*x+e))^4+672*csc(f*x+e)^4*EllipticF((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),1/2*2^
(1/2))*a^5*b*(-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)*(-a^2+b^2)^(1/2)*(1-cos(f*x+e))^4-3024*csc(f*x+e)^4*EllipticF((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),1/2*2^(1/2))
*a^4*b^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)*(
-a^2+b^2)^(1/2)*(1-cos(f*x+e))^4+8064*csc(f*x+e)^4*EllipticF((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),1/2*2^(1/2))*a^3
*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)*(-a^2
+b^2)^(1/2)*(1-cos(f*x+e))^4-10080*csc(f*x+e)^4*EllipticF((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),1/2*2^(1/2))*a*b^5*
(-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)*(-a^2+b^2)
^(1/2)*(1-cos(f*x+e))^4+6048*csc(f*x+e)^4*EllipticE((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),1/2*2^(1/2))*a^4*b^2*(-co
t(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)*(-a^2+b^2)^(1/
2)*(1-cos(f*x+e))^4-6048*csc(f*x+e)^4*EllipticE((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),1/2*2^(1/2))*a^3*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)*(-a^2+b^2)^(1/2)*(
1-cos(f*x+e))^4-10080*csc(f*x+e)^4*EllipticE((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),1/2*2^(1/2))*a^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)*(-a^2+b^2)^(1/2)*(1-c
os(f*x+e))^4+10080*csc(f*x+e)^4*EllipticE((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),1/2*2^(1/2))*a*b^5*(-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)*(-a^2+b^2)^(1/2)*(1-cos(f*
x+e))^4-2520*csc(f*x+e)^4*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
^3*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)*(-a
^2+b^2)^(1/2)*(1-cos(f*x+e))^4+2520*csc(f*x+e)^4*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^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)*(-a^2+b^2)^(1/2)*(1-cos(f*x+e))^4-672*csc(f*x+e)^4*EllipticF((-cot(f*x+e)+csc(f*x+e)+1)^(1
/2),1/2*2^(1/2))*a^6*(-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)*(-a^2+b^2)^(1/2)*(1-cos(f*x+e))^4+5040*csc(f*x+e)^4*EllipticF((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),1/2*
2^(1/2))*b^6*(-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)*(-a^2+b^2)^(1/2)*(1-cos(f*x+e))^4+1344*csc(f*x+e)^4*EllipticE((-cot(f*x+e)+csc(f*x+e)+1)^(1/2),1/2*2^(1/2))
*a^6*(-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)*(-a^2
+b^2)^(1/2)*(1-cos(f*x+e))^4+2520*csc(f*x+e)^4*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^4*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)*(1-cos(f*x+e))^4-5040*csc(f*x+e)^4*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^5*(-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)*(1-cos(f*x+e))^4-2520*csc(f*x+e)^4*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^6*(-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)*(-a^2+b^2)^(1/2)*(1-cos(f*x+e))^4+2520*csc(f*x+e)^4*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^7*(-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)*(1-cos(f*x+e))^4-2520*csc(f*x+e)^4*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^7*(-cot(f*x+e)+csc(f*x+e)+1)^(1/2)*(2+2*cot(f*x+e)-2*cs
c(f*x+e))^(1/2)*(-csc(f*x+e)+cot(f*x+e))^(1/2)*(1-cos(f*x+e))^4)/((1-cos(f*x+e))^2*csc(f*x+e)^2-1)^3*2^(1/2)/a
^5/(-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. \[ \int \frac {(g \cos (e+f x))^{5/2}}{(d \sin (e+f x))^{11/2} (a+b \sin (e+f x))} \, dx=\text {Timed out} \]

[In]

integrate((g*cos(f*x+e))^(5/2)/(d*sin(f*x+e))^(11/2)/(a+b*sin(f*x+e)),x, algorithm="fricas")

[Out]

Timed out

Sympy [F(-1)]

Timed out. \[ \int \frac {(g \cos (e+f x))^{5/2}}{(d \sin (e+f x))^{11/2} (a+b \sin (e+f x))} \, dx=\text {Timed out} \]

[In]

integrate((g*cos(f*x+e))**(5/2)/(d*sin(f*x+e))**(11/2)/(a+b*sin(f*x+e)),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {(g \cos (e+f x))^{5/2}}{(d \sin (e+f x))^{11/2} (a+b \sin (e+f x))} \, dx=\int { \frac {\left (g \cos \left (f x + e\right )\right )^{\frac {5}{2}}}{{\left (b \sin \left (f x + e\right ) + a\right )} \left (d \sin \left (f x + e\right )\right )^{\frac {11}{2}}} \,d x } \]

[In]

integrate((g*cos(f*x+e))^(5/2)/(d*sin(f*x+e))^(11/2)/(a+b*sin(f*x+e)),x, algorithm="maxima")

[Out]

integrate((g*cos(f*x + e))^(5/2)/((b*sin(f*x + e) + a)*(d*sin(f*x + e))^(11/2)), x)

Giac [F]

\[ \int \frac {(g \cos (e+f x))^{5/2}}{(d \sin (e+f x))^{11/2} (a+b \sin (e+f x))} \, dx=\int { \frac {\left (g \cos \left (f x + e\right )\right )^{\frac {5}{2}}}{{\left (b \sin \left (f x + e\right ) + a\right )} \left (d \sin \left (f x + e\right )\right )^{\frac {11}{2}}} \,d x } \]

[In]

integrate((g*cos(f*x+e))^(5/2)/(d*sin(f*x+e))^(11/2)/(a+b*sin(f*x+e)),x, algorithm="giac")

[Out]

integrate((g*cos(f*x + e))^(5/2)/((b*sin(f*x + e) + a)*(d*sin(f*x + e))^(11/2)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(g \cos (e+f x))^{5/2}}{(d \sin (e+f x))^{11/2} (a+b \sin (e+f x))} \, dx=\int \frac {{\left (g\,\cos \left (e+f\,x\right )\right )}^{5/2}}{{\left (d\,\sin \left (e+f\,x\right )\right )}^{11/2}\,\left (a+b\,\sin \left (e+f\,x\right )\right )} \,d x \]

[In]

int((g*cos(e + f*x))^(5/2)/((d*sin(e + f*x))^(11/2)*(a + b*sin(e + f*x))),x)

[Out]

int((g*cos(e + f*x))^(5/2)/((d*sin(e + f*x))^(11/2)*(a + b*sin(e + f*x))), x)