Integrand size = 42, antiderivative size = 234 \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2}}{\sqrt {c-c \sin (e+f x)}} \, dx=-\frac {22 a^3 (g \cos (e+f x))^{5/2}}{15 f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {22 a^3 g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{5 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}-\frac {22 a^2 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{35 f g \sqrt {c-c \sin (e+f x)}}-\frac {2 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{7 f g \sqrt {c-c \sin (e+f x)}} \] Output:
-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(f*x+e)^(1/2)*(g*cos(f*x+e))^(1/2)*EllipticE(sin(1/2 *f*x+1/2*e),2^(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 )-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)
Time = 9.28 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.68 \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2}}{\sqrt {c-c \sin (e+f x)}} \, dx=-\frac {(g \cos (e+f x))^{3/2} \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) (a (1+\sin (e+f x)))^{5/2} \left (-924 E\left (\left .\frac {1}{2} (e+f x)\right |2\right )+\sqrt {\cos (e+f x)} (515 \cos (e+f x)-15 \cos (3 (e+f x))+126 \sin (2 (e+f x)))\right )}{210 f \cos ^{\frac {3}{2}}(e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 \sqrt {c-c \sin (e+f x)}} \] Input:
Integrate[((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(5/2))/Sqrt[c - c*S in[e + f*x]],x]
Output:
-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 = 1.85 (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 |
\(\displaystyle \frac {11}{7} a \left (\frac {7}{5} a \left (\frac {2 a g \sqrt {\cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {g \cos (e+f x)}}{f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {2 a (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{5 f g \sqrt {c-c \sin (e+f x)}}\right )-\frac {2 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt {c-c \sin (e+f x)}}\) |
Input:
Int[((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(5/2))/Sqrt[c - c*Sin[e + f*x]],x]
Output:
(-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
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]
Leaf count of result is larger than twice the leaf count of optimal. \(1594\) vs. \(2(202)=404\).
Time = 21.44 (sec) , antiderivative size = 1595, normalized size of antiderivative = 6.82
Input:
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)
Output:
-1/105/f/(2*2^(1/2)-3)/(1+2^(1/2))*a^2/g*(5040*(1+(-cos(1/2*f*x+1/2*e)^2-2 *cos(1/2*f*x+1/2*e)-1)*2^(1/2)+cos(1/2*f*x+1/2*e)^2+2*cos(1/2*f*x+1/2*e))* (g*(-1+2*cos(1/2*f*x+1/2*e)^2)/(cos(1/2*f*x+1/2*e)+1)^2)^(1/2)*g^(3/2)*ln( 2/g^(1/2)*(g^(1/2)*(g*(-1+2*cos(1/2*f*x+1/2*e)^2)/(cos(1/2*f*x+1/2*e)+1)^2 )^(1/2)*cos(1/2*f*x+1/2*e)+g^(1/2)*(g*(-1+2*cos(1/2*f*x+1/2*e)^2)/(cos(1/2 *f*x+1/2*e)+1)^2)^(1/2)-2*cos(1/2*f*x+1/2*e)*g-g)/(cos(1/2*f*x+1/2*e)+1))+ 5040*(-1+2^(1/2)*(cos(1/2*f*x+1/2*e)^2+2*cos(1/2*f*x+1/2*e)+1)-cos(1/2*f*x +1/2*e)^2-2*cos(1/2*f*x+1/2*e))*(g*(-1+2*cos(1/2*f*x+1/2*e)^2)/(cos(1/2*f* x+1/2*e)+1)^2)^(1/2)*g^(3/2)*ln(4/g^(1/2)*(g^(1/2)*(g*(-1+2*cos(1/2*f*x+1/ 2*e)^2)/(cos(1/2*f*x+1/2*e)+1)^2)^(1/2)*cos(1/2*f*x+1/2*e)+g^(1/2)*(g*(-1+ 2*cos(1/2*f*x+1/2*e)^2)/(cos(1/2*f*x+1/2*e)+1)^2)^(1/2)-2*cos(1/2*f*x+1/2* e)*g-g)/(cos(1/2*f*x+1/2*e)+1))+4095*(-2+2^(1/2)*(cos(1/2*f*x+1/2*e)^2+2*c os(1/2*f*x+1/2*e)+1)-2*cos(1/2*f*x+1/2*e)^2-4*cos(1/2*f*x+1/2*e))*g^(3/2)* (g*(-1+2*cos(1/2*f*x+1/2*e)^2)/(cos(1/2*f*x+1/2*e)+1)^2)^(1/2)*ln(4*g^(1/2 )*(g*(-1+2*cos(1/2*f*x+1/2*e)^2)/(cos(1/2*f*x+1/2*e)+1)^2)^(1/2)*2^(1/2)*c os(1/2*f*x+1/2*e)+4*2^(1/2)*g^(1/2)*(g*(-1+2*cos(1/2*f*x+1/2*e)^2)/(cos(1/ 2*f*x+1/2*e)+1)^2)^(1/2)+8*cos(1/2*f*x+1/2*e)*g)+4095*(2+(-cos(1/2*f*x+1/2 *e)^2-2*cos(1/2*f*x+1/2*e)-1)*2^(1/2)+2*cos(1/2*f*x+1/2*e)^2+4*cos(1/2*f*x +1/2*e))*g^(3/2)*(g*(-1+2*cos(1/2*f*x+1/2*e)^2)/(cos(1/2*f*x+1/2*e)+1)^2)^ (1/2)*arctanh(2^(1/2)*g^(1/2)*cos(1/2*f*x+1/2*e)/(cos(1/2*f*x+1/2*e)+1)...
Result contains complex when optimal does not.
Time = 0.10 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.65 \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2}}{\sqrt {c-c \sin (e+f x)}} \, dx=-\frac {2 \, {\left (231 i \, \sqrt {\frac {1}{2}} \sqrt {a c g} a^{2} g {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) - 231 i \, \sqrt {\frac {1}{2}} \sqrt {a c g} a^{2} g {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right ) - {\left (15 \, a^{2} g \cos \left (f x + e\right )^{2} - 63 \, a^{2} g \sin \left (f x + e\right ) - 140 \, a^{2} g\right )} \sqrt {g \cos \left (f x + e\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}\right )}}{105 \, c f} \] Input:
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")
Output:
-2/105*(231*I*sqrt(1/2)*sqrt(a*c*g)*a^2*g*weierstrassZeta(-4, 0, weierstra ssPInverse(-4, 0, cos(f*x + e) + I*sin(f*x + e))) - 231*I*sqrt(1/2)*sqrt(a *c*g)*a^2*g*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(f*x + e) - I*sin(f*x + e))) - (15*a^2*g*cos(f*x + e)^2 - 63*a^2*g*sin(f*x + e) - 1 40*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. \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2}}{\sqrt {c-c \sin (e+f x)}} \, dx=\text {Timed out} \] Input:
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)
Output:
Timed out
\[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2}}{\sqrt {c-c \sin (e+f x)}} \, dx=\int { \frac {\left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}}}{\sqrt {-c \sin \left (f x + e\right ) + c}} \,d x } \] Input:
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")
Output:
integrate((g*cos(f*x + e))^(3/2)*(a*sin(f*x + e) + a)^(5/2)/sqrt(-c*sin(f* x + e) + c), x)
Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2}}{\sqrt {c-c \sin (e+f x)}} \, dx=\text {Timed out} \] Input:
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")
Output:
Timed out
Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2}}{\sqrt {c-c \sin (e+f x)}} \, dx=\int \frac {{\left (g\,\cos \left (e+f\,x\right )\right )}^{3/2}\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{5/2}}{\sqrt {c-c\,\sin \left (e+f\,x\right )}} \,d x \] Input:
int(((g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^(5/2))/(c - c*sin(e + f*x ))^(1/2),x)
Output:
int(((g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^(5/2))/(c - c*sin(e + f*x ))^(1/2), x)
\[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2}}{\sqrt {c-c \sin (e+f x)}} \, dx=\frac {\sqrt {g}\, \sqrt {c}\, \sqrt {a}\, a^{2} g \left (-\left (\int \frac {\sqrt {\sin \left (f x +e \right )+1}\, \sqrt {-\sin \left (f x +e \right )+1}\, \sqrt {\cos \left (f x +e \right )}\, \cos \left (f x +e \right ) \sin \left (f x +e \right )^{2}}{\sin \left (f x +e \right )-1}d x \right )-2 \left (\int \frac {\sqrt {\sin \left (f x +e \right )+1}\, \sqrt {-\sin \left (f x +e \right )+1}\, \sqrt {\cos \left (f x +e \right )}\, \cos \left (f x +e \right ) \sin \left (f x +e \right )}{\sin \left (f x +e \right )-1}d x \right )-\left (\int \frac {\sqrt {\sin \left (f x +e \right )+1}\, \sqrt {-\sin \left (f x +e \right )+1}\, \sqrt {\cos \left (f x +e \right )}\, \cos \left (f x +e \right )}{\sin \left (f x +e \right )-1}d x \right )\right )}{c} \] Input:
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)
Output:
(sqrt(g)*sqrt(c)*sqrt(a)*a**2*g*( - int((sqrt(sin(e + f*x) + 1)*sqrt( - si n(e + f*x) + 1)*sqrt(cos(e + f*x))*cos(e + f*x)*sin(e + f*x)**2)/(sin(e + f*x) - 1),x) - 2*int((sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)*sqr t(cos(e + f*x))*cos(e + f*x)*sin(e + f*x))/(sin(e + f*x) - 1),x) - int((sq rt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)*sqrt(cos(e + f*x))*cos(e + f*x))/(sin(e + f*x) - 1),x)))/c