Integrand size = 42, antiderivative size = 235 \[ \int (g \cos (e+f x))^{3/2} \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2} \, dx=\frac {2 a c^2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {6 a c^2 g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{5 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {6 a c (g \cos (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}}{35 f g \sqrt {a+a \sin (e+f x)}}-\frac {2 a (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{7 f g \sqrt {a+a \sin (e+f x)}} \]
-2/7*a*(g*cos(f*x+e))^(5/2)*(c-c*sin(f*x+e))^(3/2)/f/g/(a+a*sin(f*x+e))^(1 /2)+2/5*a*c^2*(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)+6/5*a*c^2*g*(cos(1/2*f*x+1/2*e)^2)^(1/2)/cos(1/2*f*x+1/2*e)*Ell ipticE(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)+6/35*a*c*(g*cos(f*x+e))^(5/ 2)*(c-c*sin(f*x+e))^(1/2)/f/g/(a+a*sin(f*x+e))^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 3.02 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.09 \[ \int (g \cos (e+f x))^{3/2} \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2} \, dx=\frac {c^2 e^{-3 i (e+f x)} \left (-i+e^{i (e+f x)}\right ) g \sqrt {g \cos (e+f x)} \left (\sqrt {1+e^{2 i (e+f x)}} \left (5 i-14 e^{i (e+f x)}+15 i e^{2 i (e+f x)}-168 e^{3 i (e+f x)}+15 i e^{4 i (e+f x)}+14 e^{5 i (e+f x)}+5 i e^{6 i (e+f x)}\right )+112 e^{5 i (e+f x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (e+f x)}\right )\right ) \sqrt {a (1+\sin (e+f x))}}{140 \left (i+e^{i (e+f x)}\right ) \sqrt {1+e^{2 i (e+f x)}} f \sqrt {c-c \sin (e+f x)}} \]
(c^2*(-I + E^(I*(e + f*x)))*g*Sqrt[g*Cos[e + f*x]]*(Sqrt[1 + E^((2*I)*(e + f*x))]*(5*I - 14*E^(I*(e + f*x)) + (15*I)*E^((2*I)*(e + f*x)) - 168*E^((3 *I)*(e + f*x)) + (15*I)*E^((4*I)*(e + f*x)) + 14*E^((5*I)*(e + f*x)) + (5* I)*E^((6*I)*(e + f*x))) + 112*E^((5*I)*(e + f*x))*Hypergeometric2F1[1/2, 3 /4, 7/4, -E^((2*I)*(e + f*x))])*Sqrt[a*(1 + Sin[e + f*x])])/(140*E^((3*I)* (e + f*x))*(I + E^(I*(e + f*x)))*Sqrt[1 + E^((2*I)*(e + f*x))]*f*Sqrt[c - c*Sin[e + f*x]])
Time = 1.73 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.01, 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.
| \(\displaystyle \int \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{3/2}dx\) |
| \(\Big \downarrow \) 3330 |
| \(\displaystyle \frac {3}{7} a \int \frac {(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a}}dx-\frac {2 a (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {a \sin (e+f x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3}{7} a \int \frac {(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a}}dx-\frac {2 a (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {a \sin (e+f x)+a}}\) |
| \(\Big \downarrow \) 3330 |
| \(\displaystyle \frac {3}{7} a \left (\frac {7}{5} c \int \frac {(g \cos (e+f x))^{3/2} \sqrt {c-c \sin (e+f x)}}{\sqrt {\sin (e+f x) a+a}}dx+\frac {2 c \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a}}\right )-\frac {2 a (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {a \sin (e+f x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3}{7} a \left (\frac {7}{5} c \int \frac {(g \cos (e+f x))^{3/2} \sqrt {c-c \sin (e+f x)}}{\sqrt {\sin (e+f x) a+a}}dx+\frac {2 c \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a}}\right )-\frac {2 a (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {a \sin (e+f x)+a}}\) |
| \(\Big \downarrow \) 3330 |
| \(\displaystyle \frac {3}{7} a \left (\frac {7}{5} c \left (c \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} \sqrt {c-c \sin (e+f x)}}dx+\frac {2 c (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a}}\right )-\frac {2 a (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {a \sin (e+f x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3}{7} a \left (\frac {7}{5} c \left (c \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} \sqrt {c-c \sin (e+f x)}}dx+\frac {2 c (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a}}\right )-\frac {2 a (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {a \sin (e+f x)+a}}\) |
| \(\Big \downarrow \) 3321 |
| \(\displaystyle \frac {3}{7} a \left (\frac {7}{5} c \left (\frac {c g \cos (e+f x) \int \sqrt {g \cos (e+f x)}dx}{\sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {2 c (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a}}\right )-\frac {2 a (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {a \sin (e+f x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3}{7} a \left (\frac {7}{5} c \left (\frac {c g \cos (e+f x) \int \sqrt {g \sin \left (e+f x+\frac {\pi }{2}\right )}dx}{\sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {2 c (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a}}\right )-\frac {2 a (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {a \sin (e+f x)+a}}\) |
| \(\Big \downarrow \) 3121 |
| \(\displaystyle \frac {3}{7} a \left (\frac {7}{5} c \left (\frac {c g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\cos (e+f x)}dx}{\sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {2 c (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a}}\right )-\frac {2 a (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {a \sin (e+f x)+a}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {3}{7} a \left (\frac {7}{5} c \left (\frac {c g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\sin \left (e+f x+\frac {\pi }{2}\right )}dx}{\sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {2 c (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )+\frac {2 c \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a}}\right )-\frac {2 a (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {a \sin (e+f x)+a}}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {3}{7} a \left (\frac {2 c \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {a \sin (e+f x)+a}}+\frac {7}{5} c \left (\frac {2 c (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {2 c g \sqrt {\cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {g \cos (e+f x)}}{f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )\right )-\frac {2 a (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {a \sin (e+f x)+a}}\) |
(-2*a*(g*Cos[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(3/2))/(7*f*g*Sqrt[a + a *Sin[e + f*x]]) + (3*a*((2*c*(g*Cos[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f*x ]])/(5*f*g*Sqrt[a + a*Sin[e + f*x]]) + (7*c*((2*c*(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*c*g*Sqrt[Co s[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
3.1.90.3.1 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 = 3.04 (sec) , antiderivative size = 462, normalized size of antiderivative = 1.97
| method | result | size |
| default | \(-\frac {2 \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}\, \sqrt {a \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {g \cos \left (f x +e \right )}\, g c \left (21 i \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, F\left (i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right ), i\right )-21 i \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, E\left (i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right ), i\right )+42 i \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, F\left (i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right ), i\right ) \sec \left (f x +e \right )-42 i \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, E\left (i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right ), i\right ) \sec \left (f x +e \right )-5 \left (\cos ^{3}\left (f x +e \right )\right )+21 i \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, F\left (i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right ), i\right ) \left (\sec ^{2}\left (f x +e \right )\right )-21 i \sqrt {\frac {\cos \left (f x +e \right )}{1+\cos \left (f x +e \right )}}\, \sqrt {\frac {1}{1+\cos \left (f x +e \right )}}\, E\left (i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right ), i\right ) \left (\sec ^{2}\left (f x +e \right )\right )-5 \left (\cos ^{2}\left (f x +e \right )\right )-7 \cos \left (f x +e \right ) \sin \left (f x +e \right )-7 \sin \left (f x +e \right )-21 \tan \left (f x +e \right )\right )}{35 f \left (1+\cos \left (f x +e \right )\right )}\) | \(462\) |
int((g*cos(f*x+e))^(3/2)*(c-c*sin(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(1/2),x,m ethod=_RETURNVERBOSE)
-2/35/f*(-c*(sin(f*x+e)-1))^(1/2)*(a*(1+sin(f*x+e)))^(1/2)*(g*cos(f*x+e))^ (1/2)*g*c/(1+cos(f*x+e))*(21*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)-21*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)+42*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)*sec(f*x+e)-42*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)*sec(f*x+e)-5*cos(f*x+e)^3+21*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)*sec(f*x+e)^2-2 1*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)*sec(f*x+e)^2-5*cos(f*x+e)^2-7*cos(f*x+e)*sin(f *x+e)-7*sin(f*x+e)-21*tan(f*x+e))
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.57 \[ \int (g \cos (e+f x))^{3/2} \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2} \, dx=\frac {-21 i \, \sqrt {2} \sqrt {a c g} c g {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) + 21 i \, \sqrt {2} \sqrt {a c g} c g {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right ) + 2 \, {\left (5 \, c g \cos \left (f x + e\right )^{2} + 7 \, c g \sin \left (f x + e\right )\right )} \sqrt {g \cos \left (f x + e\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{35 \, f} \]
integrate((g*cos(f*x+e))^(3/2)*(c-c*sin(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(1/ 2),x, algorithm="fricas")
1/35*(-21*I*sqrt(2)*sqrt(a*c*g)*c*g*weierstrassZeta(-4, 0, weierstrassPInv erse(-4, 0, cos(f*x + e) + I*sin(f*x + e))) + 21*I*sqrt(2)*sqrt(a*c*g)*c*g *weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(f*x + e) - I*sin(f* x + e))) + 2*(5*c*g*cos(f*x + e)^2 + 7*c*g*sin(f*x + e))*sqrt(g*cos(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c))/f
Timed out. \[ \int (g \cos (e+f x))^{3/2} \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2} \, dx=\text {Timed out} \]
\[ \int (g \cos (e+f x))^{3/2} \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2} \, dx=\int { \left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} \sqrt {a \sin \left (f x + e\right ) + a} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}} \,d x } \]
integrate((g*cos(f*x+e))^(3/2)*(c-c*sin(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(1/ 2),x, algorithm="maxima")
\[ \int (g \cos (e+f x))^{3/2} \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2} \, dx=\int { \left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} \sqrt {a \sin \left (f x + e\right ) + a} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}} \,d x } \]
integrate((g*cos(f*x+e))^(3/2)*(c-c*sin(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(1/ 2),x, algorithm="giac")
Timed out. \[ \int (g \cos (e+f x))^{3/2} \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2} \, dx=\int {\left (g\,\cos \left (e+f\,x\right )\right )}^{3/2}\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{3/2} \,d x \]