Integrand size = 42, antiderivative size = 234 \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2}}{\sqrt {c-c \sin (e+f x)}} \, dx=-\frac {22 a^3 (g \cos (e+f x))^{5/2}}{15 f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {22 a^3 g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{5 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}-\frac {22 a^2 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{35 f g \sqrt {c-c \sin (e+f x)}}-\frac {2 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{7 f g \sqrt {c-c \sin (e+f x)}} \]
-2/7*a*(g*cos(f*x+e))^(5/2)*(a+a*sin(f*x+e))^(3/2)/f/g/(c-c*sin(f*x+e))^(1 /2)-22/15*a^3*(g*cos(f*x+e))^(5/2)/f/g/(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x +e))^(1/2)+22/5*a^3*g*(cos(1/2*f*x+1/2*e)^2)^(1/2)/cos(1/2*f*x+1/2*e)*Elli pticE(sin(1/2*f*x+1/2*e),2^(1/2))*cos(f*x+e)^(1/2)*(g*cos(f*x+e))^(1/2)/f/ (a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e))^(1/2)-22/35*a^2*(g*cos(f*x+e))^(5/ 2)*(a+a*sin(f*x+e))^(1/2)/f/g/(c-c*sin(f*x+e))^(1/2)
Time = 9.05 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.68 \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2}}{\sqrt {c-c \sin (e+f x)}} \, dx=-\frac {(g \cos (e+f x))^{3/2} \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) (a (1+\sin (e+f x)))^{5/2} \left (-924 E\left (\left .\frac {1}{2} (e+f x)\right |2\right )+\sqrt {\cos (e+f x)} (515 \cos (e+f x)-15 \cos (3 (e+f x))+126 \sin (2 (e+f x)))\right )}{210 f \cos ^{\frac {3}{2}}(e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 \sqrt {c-c \sin (e+f x)}} \]
-1/210*((g*Cos[e + f*x])^(3/2)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(a*(1 + Sin[e + f*x]))^(5/2)*(-924*EllipticE[(e + f*x)/2, 2] + Sqrt[Cos[e + f*x ]]*(515*Cos[e + f*x] - 15*Cos[3*(e + f*x)] + 126*Sin[2*(e + f*x)])))/(f*Co s[e + f*x]^(3/2)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5*Sqrt[c - c*Sin[e + f*x]])
Time = 1.80 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.02, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3042, 3330, 3042, 3330, 3042, 3330, 3042, 3321, 3042, 3121, 3042, 3119}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {(a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{3/2}}{\sqrt {c-c \sin (e+f x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{3/2}}{\sqrt {c-c \sin (e+f x)}}dx\) |
\(\Big \downarrow \) 3330 |
\(\displaystyle \frac {11}{7} a \int \frac {(g \cos (e+f x))^{3/2} (\sin (e+f x) a+a)^{3/2}}{\sqrt {c-c \sin (e+f x)}}dx-\frac {2 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {c-c \sin (e+f x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {11}{7} a \int \frac {(g \cos (e+f x))^{3/2} (\sin (e+f x) a+a)^{3/2}}{\sqrt {c-c \sin (e+f x)}}dx-\frac {2 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {c-c \sin (e+f x)}}\) |
\(\Big \downarrow \) 3330 |
\(\displaystyle \frac {11}{7} a \left (\frac {7}{5} a \int \frac {(g \cos (e+f x))^{3/2} \sqrt {\sin (e+f x) a+a}}{\sqrt {c-c \sin (e+f x)}}dx-\frac {2 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {c-c \sin (e+f x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {11}{7} a \left (\frac {7}{5} a \int \frac {(g \cos (e+f x))^{3/2} \sqrt {\sin (e+f x) a+a}}{\sqrt {c-c \sin (e+f x)}}dx-\frac {2 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {c-c \sin (e+f x)}}\) |
\(\Big \downarrow \) 3330 |
\(\displaystyle \frac {11}{7} a \left (\frac {7}{5} a \left (a \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} \sqrt {c-c \sin (e+f x)}}dx-\frac {2 a (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {c-c \sin (e+f x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {11}{7} a \left (\frac {7}{5} a \left (a \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} \sqrt {c-c \sin (e+f x)}}dx-\frac {2 a (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {c-c \sin (e+f x)}}\) |
\(\Big \downarrow \) 3321 |
\(\displaystyle \frac {11}{7} a \left (\frac {7}{5} a \left (\frac {a g \cos (e+f x) \int \sqrt {g \cos (e+f x)}dx}{\sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {2 a (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {c-c \sin (e+f x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {11}{7} a \left (\frac {7}{5} a \left (\frac {a g \cos (e+f x) \int \sqrt {g \sin \left (e+f x+\frac {\pi }{2}\right )}dx}{\sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {2 a (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {c-c \sin (e+f x)}}\) |
\(\Big \downarrow \) 3121 |
\(\displaystyle \frac {11}{7} a \left (\frac {7}{5} a \left (\frac {a g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\cos (e+f x)}dx}{\sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {2 a (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {c-c \sin (e+f x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {11}{7} a \left (\frac {7}{5} a \left (\frac {a g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\sin \left (e+f x+\frac {\pi }{2}\right )}dx}{\sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {2 a (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {c-c \sin (e+f x)}}\) |
\(\Big \downarrow \) 3119 |
\(\displaystyle \frac {11}{7} a \left (\frac {7}{5} a \left (\frac {2 a g \sqrt {\cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {g \cos (e+f x)}}{f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {2 a (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {c-c \sin (e+f x)}}\) |
(-2*a*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(3/2))/(7*f*g*Sqrt[c - c *Sin[e + f*x]]) + (11*a*((-2*a*(g*Cos[e + f*x])^(5/2)*Sqrt[a + a*Sin[e + f *x]])/(5*f*g*Sqrt[c - c*Sin[e + f*x]]) + (7*a*((-2*a*(g*Cos[e + f*x])^(5/2 ))/(3*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (2*a*g*Sqrt [Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])))/5))/7
3.2.9.3.1 Defintions of rubi rules used
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]
Int[((b_)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*Sin[c + d*x]) ^n/Sin[c + d*x]^n Int[Sin[c + d*x]^n, x], x] /; FreeQ[{b, c, d}, x] && Lt Q[-1, n, 1] && IntegerQ[2*n]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)/(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_ .)*(x_)]]*Sqrt[(c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Simp[g* (Cos[e + f*x]/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])) Int[(g *Cos[e + f*x])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && EqQ [b*c + a*d, 0] && EqQ[a^2 - b^2, 0]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(- b)*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m - 1)*((c + d*Sin[e + f* x])^n/(f*g*(m + n + p))), x] + Simp[a*((2*m + p - 1)/(m + n + p)) Int[(g* Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && GtQ[m, 0] && NeQ[m + n + p, 0] && !LtQ[0, n, m] && IntegersQ[2 *m, 2*n, 2*p]
Result contains complex when optimal does not.
Time = 2.47 (sec) , antiderivative size = 510, normalized size of antiderivative = 2.18
method | result | size |
default | \(\frac {2 \left (-231 i \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, F\left (i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right ), i\right ) \left (\cos ^{2}\left (f x +e \right )\right )+231 i \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, E\left (i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right ), i\right ) \left (\cos ^{2}\left (f x +e \right )\right )+15 \left (\cos ^{5}\left (f x +e \right )\right )-462 i \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, F\left (i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right ), i\right ) \cos \left (f x +e \right )+462 i \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, E\left (i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right ), i\right ) \cos \left (f x +e \right )+15 \left (\cos ^{4}\left (f x +e \right )\right )-63 \left (\cos ^{3}\left (f x +e \right )\right ) \sin \left (f x +e \right )-231 i \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, F\left (i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right ), i\right )+231 i \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, E\left (i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right ), i\right )-140 \left (\cos ^{3}\left (f x +e \right )\right )-63 \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )-140 \left (\cos ^{2}\left (f x +e \right )\right )+231 \cos \left (f x +e \right ) \sin \left (f x +e \right )\right ) \sqrt {a \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {g \cos \left (f x +e \right )}\, g \,a^{2}}{105 f \left (1+\sin \left (f x +e \right )\right ) \left (1+\cos \left (f x +e \right )\right ) \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}}\) | \(510\) |
int((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(5/2)/(c-c*sin(f*x+e))^(1/2),x,m ethod=_RETURNVERBOSE)
2/105/f*(-231*I*(1/(1+cos(f*x+e)))^(1/2)*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2) *EllipticF(I*(csc(f*x+e)-cot(f*x+e)),I)*cos(f*x+e)^2+231*I*(1/(1+cos(f*x+e )))^(1/2)*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*EllipticE(I*(csc(f*x+e)-cot(f* x+e)),I)*cos(f*x+e)^2+15*cos(f*x+e)^5-462*I*(1/(1+cos(f*x+e)))^(1/2)*(cos( f*x+e)/(1+cos(f*x+e)))^(1/2)*EllipticF(I*(csc(f*x+e)-cot(f*x+e)),I)*cos(f* x+e)+462*I*(1/(1+cos(f*x+e)))^(1/2)*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*Elli pticE(I*(csc(f*x+e)-cot(f*x+e)),I)*cos(f*x+e)+15*cos(f*x+e)^4-63*cos(f*x+e )^3*sin(f*x+e)-231*I*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*(1/(1+cos(f*x+e)))^ (1/2)*EllipticF(I*(csc(f*x+e)-cot(f*x+e)),I)+231*I*(cos(f*x+e)/(1+cos(f*x+ e)))^(1/2)*(1/(1+cos(f*x+e)))^(1/2)*EllipticE(I*(csc(f*x+e)-cot(f*x+e)),I) -140*cos(f*x+e)^3-63*cos(f*x+e)^2*sin(f*x+e)-140*cos(f*x+e)^2+231*cos(f*x+ e)*sin(f*x+e))*(a*(1+sin(f*x+e)))^(1/2)*(g*cos(f*x+e))^(1/2)*g*a^2/(1+sin( f*x+e))/(1+cos(f*x+e))/(-c*(sin(f*x+e)-1))^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.13 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.65 \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2}}{\sqrt {c-c \sin (e+f x)}} \, dx=\frac {-231 i \, \sqrt {2} \sqrt {a c g} a^{2} g {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) + 231 i \, \sqrt {2} \sqrt {a c g} a^{2} g {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right ) + 2 \, {\left (15 \, a^{2} g \cos \left (f x + e\right )^{2} - 63 \, a^{2} g \sin \left (f x + e\right ) - 140 \, a^{2} g\right )} \sqrt {g \cos \left (f x + e\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{105 \, c f} \]
integrate((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(5/2)/(c-c*sin(f*x+e))^(1/ 2),x, algorithm="fricas")
1/105*(-231*I*sqrt(2)*sqrt(a*c*g)*a^2*g*weierstrassZeta(-4, 0, weierstrass PInverse(-4, 0, cos(f*x + e) + I*sin(f*x + e))) + 231*I*sqrt(2)*sqrt(a*c*g )*a^2*g*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(f*x + e) - I *sin(f*x + e))) + 2*(15*a^2*g*cos(f*x + e)^2 - 63*a^2*g*sin(f*x + e) - 140 *a^2*g)*sqrt(g*cos(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c))/(c*f)
Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2}}{\sqrt {c-c \sin (e+f x)}} \, dx=\text {Timed out} \]
\[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2}}{\sqrt {c-c \sin (e+f x)}} \, dx=\int { \frac {\left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}}}{\sqrt {-c \sin \left (f x + e\right ) + c}} \,d x } \]
integrate((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(5/2)/(c-c*sin(f*x+e))^(1/ 2),x, algorithm="maxima")
Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2}}{\sqrt {c-c \sin (e+f x)}} \, dx=\text {Timed out} \]
integrate((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(5/2)/(c-c*sin(f*x+e))^(1/ 2),x, algorithm="giac")
Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2}}{\sqrt {c-c \sin (e+f x)}} \, dx=\int \frac {{\left (g\,\cos \left (e+f\,x\right )\right )}^{3/2}\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{5/2}}{\sqrt {c-c\,\sin \left (e+f\,x\right )}} \,d x \]