Integrand size = 42, antiderivative size = 182 \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{3/2}} \, dx=\frac {4 a (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{f g (c-c \sin (e+f x))^{3/2}}+\frac {14 a^2 (g \cos (e+f x))^{5/2}}{3 c f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}-\frac {14 a^2 g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{c f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \] Output:
4*a*(g*cos(f*x+e))^(5/2)*(a+a*sin(f*x+e))^(1/2)/f/g/(c-c*sin(f*x+e))^(3/2) +14/3*a^2*(g*cos(f*x+e))^(5/2)/c/f/g/(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e ))^(1/2)-14*a^2*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))/c/f/(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e))^(1/2)
Time = 9.27 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.14 \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{3/2}} \, dx=-\frac {2 (g \cos (e+f x))^{3/2} \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^2 \left (-21 E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )+\sqrt {\cos (e+f x)} \left (\cos \left (\frac {1}{2} (e+f x)\right ) (12+\cos (e+f x))-(-12+\cos (e+f x)) \sin \left (\frac {1}{2} (e+f x)\right )\right )\right ) (a (1+\sin (e+f x)))^{3/2}}{3 c f \cos ^{\frac {3}{2}}(e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3 (-1+\sin (e+f x)) \sqrt {c-c \sin (e+f x)}} \] Input:
Integrate[((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(3/2))/(c - c*Sin[e + f*x])^(3/2),x]
Output:
(-2*(g*Cos[e + f*x])^(3/2)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2*(-21*El lipticE[(e + f*x)/2, 2]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]) + Sqrt[Cos[e + f*x]]*(Cos[(e + f*x)/2]*(12 + Cos[e + f*x]) - (-12 + Cos[e + f*x])*Sin[ (e + f*x)/2]))*(a*(1 + Sin[e + f*x]))^(3/2))/(3*c*f*Cos[e + f*x]^(3/2)*(Co s[(e + f*x)/2] + Sin[(e + f*x)/2])^3*(-1 + Sin[e + f*x])*Sqrt[c - c*Sin[e + f*x]])
Time = 1.42 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.98, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3042, 3329, 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.
\(\displaystyle \int \frac {(a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{3/2}}{(c-c \sin (e+f x))^{3/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{3/2}}{(c-c \sin (e+f x))^{3/2}}dx\) |
\(\Big \downarrow \) 3329 |
\(\displaystyle \frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{f g (c-c \sin (e+f x))^{3/2}}-\frac {7 a \int \frac {(g \cos (e+f x))^{3/2} \sqrt {\sin (e+f x) a+a}}{\sqrt {c-c \sin (e+f x)}}dx}{c}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{f g (c-c \sin (e+f x))^{3/2}}-\frac {7 a \int \frac {(g \cos (e+f x))^{3/2} \sqrt {\sin (e+f x) a+a}}{\sqrt {c-c \sin (e+f x)}}dx}{c}\) |
\(\Big \downarrow \) 3330 |
\(\displaystyle \frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{f g (c-c \sin (e+f x))^{3/2}}-\frac {7 a \left (a \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} \sqrt {c-c \sin (e+f x)}}dx-\frac {2 a (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )}{c}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{f g (c-c \sin (e+f x))^{3/2}}-\frac {7 a \left (a \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} \sqrt {c-c \sin (e+f x)}}dx-\frac {2 a (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )}{c}\) |
\(\Big \downarrow \) 3321 |
\(\displaystyle \frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{f g (c-c \sin (e+f x))^{3/2}}-\frac {7 a \left (\frac {a g \cos (e+f x) \int \sqrt {g \cos (e+f x)}dx}{\sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {2 a (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )}{c}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{f g (c-c \sin (e+f x))^{3/2}}-\frac {7 a \left (\frac {a g \cos (e+f x) \int \sqrt {g \sin \left (e+f x+\frac {\pi }{2}\right )}dx}{\sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {2 a (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )}{c}\) |
\(\Big \downarrow \) 3121 |
\(\displaystyle \frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{f g (c-c \sin (e+f x))^{3/2}}-\frac {7 a \left (\frac {a g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\cos (e+f x)}dx}{\sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {2 a (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )}{c}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{f g (c-c \sin (e+f x))^{3/2}}-\frac {7 a \left (\frac {a g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\sin \left (e+f x+\frac {\pi }{2}\right )}dx}{\sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {2 a (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )}{c}\) |
\(\Big \downarrow \) 3119 |
\(\displaystyle \frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{f g (c-c \sin (e+f x))^{3/2}}-\frac {7 a \left (\frac {2 a g \sqrt {\cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {g \cos (e+f x)}}{f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {2 a (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )}{c}\) |
Input:
Int[((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(3/2))/(c - c*Sin[e + f*x ])^(3/2),x]
Output:
(4*a*(g*Cos[e + f*x])^(5/2)*Sqrt[a + a*Sin[e + f*x]])/(f*g*(c - c*Sin[e + f*x])^(3/2)) - (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]])))/c
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[-2 *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*(2*n + p + 1))), x] - Simp[b*((2*m + p - 1)/(d*(2*n + p + 1))) Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^( n + 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] && GtQ[m, 0] && LtQ[n, -1] && NeQ[2*n + p + 1, 0] && In tegersQ[2*m, 2*n, 2*p]
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. \(1390\) vs. \(2(162)=324\).
Time = 11.34 (sec) , antiderivative size = 1391, normalized size of antiderivative = 7.64
Input:
int((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(3/2)/(c-c*sin(f*x+e))^(3/2),x,m ethod=_RETURNVERBOSE)
Output:
1/3/f/(1+2^(1/2))/(2*2^(1/2)-3)*a/g/c*(30*(-1+(1-2*cos(1/2*f*x+1/2*e)^2)*2 ^(1/2)+2*cos(1/2*f*x+1/2*e)^2)*g^(5/2)*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))+30*(1+(-1+2*cos(1/2*f*x+1/2*e)^2 )*2^(1/2)-2*cos(1/2*f*x+1/2*e)^2)*g^(5/2)*ln(4/g^(1/2)*(g^(1/2)*(g*(-1+2*c os(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))+24*(2+(-1+2*cos(1/2*f*x+1/2*e )^2)*2^(1/2)-4*cos(1/2*f*x+1/2*e)^2)*g^(5/2)*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)*cos(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)+24*(-2+(1-2*cos(1/2*f*x+1/2*e)^2)*2^(1/2)+4 *cos(1/2*f*x+1/2*e)^2)*g^(5/2)*arctanh(2^(1/2)*g^(1/2)*cos(1/2*f*x+1/2*e)/ (cos(1/2*f*x+1/2*e)+1)/(g*(-1+2*cos(1/2*f*x+1/2*e)^2)/(cos(1/2*f*x+1/2*e)+ 1)^2)^(1/2))+42*(-2+2^(1/2)*(cos(1/2*f*x+1/2*e)^2+2*cos(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))*((2^(1/2)*cos(1/2*f*x+1/2*e)- 2^(1/2)+2*cos(1/2*f*x+1/2*e)-1)/(cos(1/2*f*x+1/2*e)+1))^(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)*EllipticF((1+2^(1/2))*( csc(1/2*f*x+1/2*e)-cot(1/2*f*x+1/2*e)),-2*2^(1/2)+3)*g^2*(-2*(2^(1/2)*c...
Result contains complex when optimal does not.
Time = 0.10 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.92 \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{3/2}} \, dx=-\frac {2 \, {\left (21 \, \sqrt {\frac {1}{2}} \sqrt {a c g} {\left (-i \, a g \sin \left (f x + e\right ) + i \, a g\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) + 21 \, \sqrt {\frac {1}{2}} \sqrt {a c g} {\left (i \, a g \sin \left (f x + e\right ) - i \, a g\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right ) - {\left (a g \sin \left (f x + e\right ) - 13 \, a g\right )} \sqrt {g \cos \left (f x + e\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}\right )}}{3 \, {\left (c^{2} f \sin \left (f x + e\right ) - c^{2} f\right )}} \] Input:
integrate((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(3/2)/(c-c*sin(f*x+e))^(3/ 2),x, algorithm="fricas")
Output:
-2/3*(21*sqrt(1/2)*sqrt(a*c*g)*(-I*a*g*sin(f*x + e) + I*a*g)*weierstrassZe ta(-4, 0, weierstrassPInverse(-4, 0, cos(f*x + e) + I*sin(f*x + e))) + 21* sqrt(1/2)*sqrt(a*c*g)*(I*a*g*sin(f*x + e) - I*a*g)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(f*x + e) - I*sin(f*x + e))) - (a*g*sin(f*x + e) - 13*a*g)*sqrt(g*cos(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f *x + e) + c))/(c^2*f*sin(f*x + e) - c^2*f)
Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{3/2}} \, dx=\text {Timed out} \] Input:
integrate((g*cos(f*x+e))**(3/2)*(a+a*sin(f*x+e))**(3/2)/(c-c*sin(f*x+e))** (3/2),x)
Output:
Timed out
\[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{3/2}} \, dx=\int { \frac {\left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(3/2)/(c-c*sin(f*x+e))^(3/ 2),x, algorithm="maxima")
Output:
integrate((g*cos(f*x + e))^(3/2)*(a*sin(f*x + e) + a)^(3/2)/(-c*sin(f*x + e) + c)^(3/2), x)
Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{3/2}} \, dx=\text {Timed out} \] Input:
integrate((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(3/2)/(c-c*sin(f*x+e))^(3/ 2),x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{3/2}} \, dx=\int \frac {{\left (g\,\cos \left (e+f\,x\right )\right )}^{3/2}\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{3/2}}{{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{3/2}} \,d x \] Input:
int(((g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^(3/2))/(c - c*sin(e + f*x ))^(3/2),x)
Output:
int(((g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^(3/2))/(c - c*sin(e + f*x ))^(3/2), x)
\[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{3/2}} \, dx =\text {Too large to display} \] Input:
int((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(3/2)/(c-c*sin(f*x+e))^(3/2),x)
Output:
(sqrt(g)*sqrt(c)*sqrt(a)*a*g*(2*sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x ) + 1)*sqrt(cos(e + f*x))*sin(e + f*x) - 4*sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)*sqrt(cos(e + f*x)) + 6*int((sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)*sqrt(cos(e + f*x))*cos(e + f*x)*sin(e + f*x))/(sin(e + f*x)**3 - sin(e + f*x)**2 - sin(e + f*x) + 1),x)*sin(e + f*x)*f - 6*int( (sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)*sqrt(cos(e + f*x))*cos(e + f*x)*sin(e + f*x))/(sin(e + f*x)**3 - sin(e + f*x)**2 - sin(e + f*x) + 1),x)*f + int((sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)*sqrt(cos(e + f*x))*sin(e + f*x)**4)/(cos(e + f*x)*sin(e + f*x)**3 - cos(e + f*x)*sin (e + f*x)**2 - cos(e + f*x)*sin(e + f*x) + cos(e + f*x)),x)*sin(e + f*x)*f - int((sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)*sqrt(cos(e + f*x) )*sin(e + f*x)**4)/(cos(e + f*x)*sin(e + f*x)**3 - cos(e + f*x)*sin(e + f* x)**2 - cos(e + f*x)*sin(e + f*x) + cos(e + f*x)),x)*f - 2*int((sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)*sqrt(cos(e + f*x))*sin(e + f*x)**3) /(cos(e + f*x)*sin(e + f*x)**3 - cos(e + f*x)*sin(e + f*x)**2 - cos(e + f* x)*sin(e + f*x) + cos(e + f*x)),x)*sin(e + f*x)*f + 2*int((sqrt(sin(e + f* x) + 1)*sqrt( - sin(e + f*x) + 1)*sqrt(cos(e + f*x))*sin(e + f*x)**3)/(cos (e + f*x)*sin(e + f*x)**3 - cos(e + f*x)*sin(e + f*x)**2 - cos(e + f*x)*si n(e + f*x) + cos(e + f*x)),x)*f - int((sqrt(sin(e + f*x) + 1)*sqrt( - sin( e + f*x) + 1)*sqrt(cos(e + f*x))*sin(e + f*x)**2)/(cos(e + f*x)*sin(e +...