Integrand size = 42, antiderivative size = 300 \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{7/2}} \, dx=\frac {4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{9 f g (c-c \sin (e+f x))^{7/2}}-\frac {4 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{3 c f g (c-c \sin (e+f x))^{5/2}}+\frac {44 a^3 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{3 c^2 f g (c-c \sin (e+f x))^{3/2}}+\frac {154 a^4 (g \cos (e+f x))^{5/2}}{9 c^3 f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}-\frac {154 a^4 g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{3 c^3 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \] Output:
4/9*a*(g*cos(f*x+e))^(5/2)*(a+a*sin(f*x+e))^(5/2)/f/g/(c-c*sin(f*x+e))^(7/ 2)-4/3*a^2*(g*cos(f*x+e))^(5/2)*(a+a*sin(f*x+e))^(3/2)/c/f/g/(c-c*sin(f*x+ e))^(5/2)+44/3*a^3*(g*cos(f*x+e))^(5/2)*(a+a*sin(f*x+e))^(1/2)/c^2/f/g/(c- c*sin(f*x+e))^(3/2)+154/9*a^4*(g*cos(f*x+e))^(5/2)/c^3/f/g/(a+a*sin(f*x+e) )^(1/2)/(c-c*sin(f*x+e))^(1/2)-154/3*a^4*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^3/f/(a+a*sin(f*x+e))^(1/2)/ (c-c*sin(f*x+e))^(1/2)
Time = 15.56 (sec) , antiderivative size = 406, normalized size of antiderivative = 1.35 \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{7/2}} \, dx=-\frac {154 (g \cos (e+f x))^{3/2} E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (a (1+\sin (e+f x)))^{7/2}}{3 f \cos ^{\frac {3}{2}}(e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (c-c \sin (e+f x))^{7/2}}+\frac {(g \cos (e+f x))^{3/2} \sec (e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 \left (\frac {112}{3}+\frac {2}{3} \cos (e+f x)+\frac {32}{9 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}-\frac {32}{3 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^2}+\frac {64 \sin \left (\frac {1}{2} (e+f x)\right )}{9 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5}-\frac {64 \sin \left (\frac {1}{2} (e+f x)\right )}{3 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^3}+\frac {224 \sin \left (\frac {1}{2} (e+f x)\right )}{3 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )}\right ) (a (1+\sin (e+f x)))^{7/2}}{f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (c-c \sin (e+f x))^{7/2}} \] Input:
Integrate[((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(7/2))/(c - c*Sin[e + f*x])^(7/2),x]
Output:
(-154*(g*Cos[e + f*x])^(3/2)*EllipticE[(e + f*x)/2, 2]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(a*(1 + Sin[e + f*x]))^(7/2))/(3*f*Cos[e + f*x]^(3/2) *(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(7/2)) + ((g *Cos[e + f*x])^(3/2)*Sec[e + f*x]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7* (112/3 + (2*Cos[e + f*x])/3 + 32/(9*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^ 4) - 32/(3*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2) + (64*Sin[(e + f*x)/2] )/(9*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5) - (64*Sin[(e + f*x)/2])/(3*( Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3) + (224*Sin[(e + f*x)/2])/(3*(Cos[( e + f*x)/2] - Sin[(e + f*x)/2])))*(a*(1 + Sin[e + f*x]))^(7/2))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(7/2))
Time = 2.44 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3042, 3329, 3042, 3329, 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.
| \(\displaystyle \int \frac {(a \sin (e+f x)+a)^{7/2} (g \cos (e+f x))^{3/2}}{(c-c \sin (e+f x))^{7/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a \sin (e+f x)+a)^{7/2} (g \cos (e+f x))^{3/2}}{(c-c \sin (e+f x))^{7/2}}dx\) |
| \(\Big \downarrow \) 3329 |
| \(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{9 f g (c-c \sin (e+f x))^{7/2}}-\frac {5 a \int \frac {(g \cos (e+f x))^{3/2} (\sin (e+f x) a+a)^{5/2}}{(c-c \sin (e+f x))^{5/2}}dx}{3 c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{9 f g (c-c \sin (e+f x))^{7/2}}-\frac {5 a \int \frac {(g \cos (e+f x))^{3/2} (\sin (e+f x) a+a)^{5/2}}{(c-c \sin (e+f x))^{5/2}}dx}{3 c}\) |
| \(\Big \downarrow \) 3329 |
| \(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{9 f g (c-c \sin (e+f x))^{7/2}}-\frac {5 a \left (\frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{5 f g (c-c \sin (e+f x))^{5/2}}-\frac {11 a \int \frac {(g \cos (e+f x))^{3/2} (\sin (e+f x) a+a)^{3/2}}{(c-c \sin (e+f x))^{3/2}}dx}{5 c}\right )}{3 c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{9 f g (c-c \sin (e+f x))^{7/2}}-\frac {5 a \left (\frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{5 f g (c-c \sin (e+f x))^{5/2}}-\frac {11 a \int \frac {(g \cos (e+f x))^{3/2} (\sin (e+f x) a+a)^{3/2}}{(c-c \sin (e+f x))^{3/2}}dx}{5 c}\right )}{3 c}\) |
| \(\Big \downarrow \) 3329 |
| \(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{9 f g (c-c \sin (e+f x))^{7/2}}-\frac {5 a \left (\frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{5 f g (c-c \sin (e+f x))^{5/2}}-\frac {11 a \left (\frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{f g (c-c \sin (e+f x))^{3/2}}-\frac {7 a \int \frac {(g \cos (e+f x))^{3/2} \sqrt {\sin (e+f x) a+a}}{\sqrt {c-c \sin (e+f x)}}dx}{c}\right )}{5 c}\right )}{3 c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{9 f g (c-c \sin (e+f x))^{7/2}}-\frac {5 a \left (\frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{5 f g (c-c \sin (e+f x))^{5/2}}-\frac {11 a \left (\frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{f g (c-c \sin (e+f x))^{3/2}}-\frac {7 a \int \frac {(g \cos (e+f x))^{3/2} \sqrt {\sin (e+f x) a+a}}{\sqrt {c-c \sin (e+f x)}}dx}{c}\right )}{5 c}\right )}{3 c}\) |
| \(\Big \downarrow \) 3330 |
| \(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{9 f g (c-c \sin (e+f x))^{7/2}}-\frac {5 a \left (\frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{5 f g (c-c \sin (e+f x))^{5/2}}-\frac {11 a \left (\frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{f g (c-c \sin (e+f x))^{3/2}}-\frac {7 a \left (a \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} \sqrt {c-c \sin (e+f x)}}dx-\frac {2 a (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )}{c}\right )}{5 c}\right )}{3 c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{9 f g (c-c \sin (e+f x))^{7/2}}-\frac {5 a \left (\frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{5 f g (c-c \sin (e+f x))^{5/2}}-\frac {11 a \left (\frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{f g (c-c \sin (e+f x))^{3/2}}-\frac {7 a \left (a \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {\sin (e+f x) a+a} \sqrt {c-c \sin (e+f x)}}dx-\frac {2 a (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )}{c}\right )}{5 c}\right )}{3 c}\) |
| \(\Big \downarrow \) 3321 |
| \(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{9 f g (c-c \sin (e+f x))^{7/2}}-\frac {5 a \left (\frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{5 f g (c-c \sin (e+f x))^{5/2}}-\frac {11 a \left (\frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{f g (c-c \sin (e+f x))^{3/2}}-\frac {7 a \left (\frac {a g \cos (e+f x) \int \sqrt {g \cos (e+f x)}dx}{\sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {2 a (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )}{c}\right )}{5 c}\right )}{3 c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{9 f g (c-c \sin (e+f x))^{7/2}}-\frac {5 a \left (\frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{5 f g (c-c \sin (e+f x))^{5/2}}-\frac {11 a \left (\frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{f g (c-c \sin (e+f x))^{3/2}}-\frac {7 a \left (\frac {a g \cos (e+f x) \int \sqrt {g \sin \left (e+f x+\frac {\pi }{2}\right )}dx}{\sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {2 a (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )}{c}\right )}{5 c}\right )}{3 c}\) |
| \(\Big \downarrow \) 3121 |
| \(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{9 f g (c-c \sin (e+f x))^{7/2}}-\frac {5 a \left (\frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{5 f g (c-c \sin (e+f x))^{5/2}}-\frac {11 a \left (\frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{f g (c-c \sin (e+f x))^{3/2}}-\frac {7 a \left (\frac {a g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\cos (e+f x)}dx}{\sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {2 a (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )}{c}\right )}{5 c}\right )}{3 c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{9 f g (c-c \sin (e+f x))^{7/2}}-\frac {5 a \left (\frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{5 f g (c-c \sin (e+f x))^{5/2}}-\frac {11 a \left (\frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{f g (c-c \sin (e+f x))^{3/2}}-\frac {7 a \left (\frac {a g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} \int \sqrt {\sin \left (e+f x+\frac {\pi }{2}\right )}dx}{\sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {2 a (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )}{c}\right )}{5 c}\right )}{3 c}\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \frac {4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{9 f g (c-c \sin (e+f x))^{7/2}}-\frac {5 a \left (\frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{5 f g (c-c \sin (e+f x))^{5/2}}-\frac {11 a \left (\frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{f g (c-c \sin (e+f x))^{3/2}}-\frac {7 a \left (\frac {2 a g \sqrt {\cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {g \cos (e+f x)}}{f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {2 a (g \cos (e+f x))^{5/2}}{3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}\right )}{c}\right )}{5 c}\right )}{3 c}\) |
Input:
Int[((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(7/2))/(c - c*Sin[e + f*x ])^(7/2),x]
Output:
(4*a*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(5/2))/(9*f*g*(c - c*Sin[ e + f*x])^(7/2)) - (5*a*((4*a*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^ (3/2))/(5*f*g*(c - c*Sin[e + f*x])^(5/2)) - (11*a*((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))/ (5*c)))/(3*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. \(2146\) vs. \(2(260)=520\).
Time = 27.21 (sec) , antiderivative size = 2147, normalized size of antiderivative = 7.16
Input:
int((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(7/2)/(c-c*sin(f*x+e))^(7/2),x,m ethod=_RETURNVERBOSE)
Output:
1/9/f/(2*2^(1/2)-3)/(1+2^(1/2))*a^3/g/c^3*(90*(-1+2*cos(1/2*f*x+1/2*e)^2)* ((-4*cos(1/2*f*x+1/2*e)^4+4*cos(1/2*f*x+1/2*e)^2-4*cos(1/2*f*x+1/2*e)*sin( 1/2*f*x+1/2*e)+1)*2^(1/2)+4*cos(1/2*f*x+1/2*e)*sin(1/2*f*x+1/2*e)+4*cos(1/ 2*f*x+1/2*e)^4-4*cos(1/2*f*x+1/2*e)^2-1)*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))+90*(-1+2*cos(1/2*f*x+1 /2*e)^2)*((4*cos(1/2*f*x+1/2*e)^4-4*cos(1/2*f*x+1/2*e)^2+4*cos(1/2*f*x+1/2 *e)*sin(1/2*f*x+1/2*e)-1)*2^(1/2)-4*cos(1/2*f*x+1/2*e)*sin(1/2*f*x+1/2*e)- 4*cos(1/2*f*x+1/2*e)^4+4*cos(1/2*f*x+1/2*e)^2+1)*g^(5/2)*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))+72*(-1+2*cos(1 /2*f*x+1/2*e)^2)*((4*cos(1/2*f*x+1/2*e)^4-4*cos(1/2*f*x+1/2*e)^2+4*cos(1/2 *f*x+1/2*e)*sin(1/2*f*x+1/2*e)-1)*2^(1/2)-8*cos(1/2*f*x+1/2*e)*sin(1/2*f*x +1/2*e)-8*cos(1/2*f*x+1/2*e)^4+8*cos(1/2*f*x+1/2*e)^2+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)+72*(-1+2*cos(1/2*f*x+1/ 2*e)^2)*((-4*cos(1/2*f*x+1/2*e)^4+4*cos(1/2*f*x+1/2*e)^2-4*cos(1/2*f*x+...
Result contains complex when optimal does not.
Time = 0.12 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.00 \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{7/2}} \, dx=\frac {2 \, {\left (231 \, \sqrt {\frac {1}{2}} {\left (3 i \, a^{3} g \cos \left (f x + e\right )^{2} - 4 i \, a^{3} g + {\left (-i \, a^{3} g \cos \left (f x + e\right )^{2} + 4 i \, a^{3} g\right )} \sin \left (f x + e\right )\right )} \sqrt {a c g} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) + 231 \, \sqrt {\frac {1}{2}} {\left (-3 i \, a^{3} g \cos \left (f x + e\right )^{2} + 4 i \, a^{3} g + {\left (i \, a^{3} g \cos \left (f x + e\right )^{2} - 4 i \, a^{3} g\right )} \sin \left (f x + e\right )\right )} \sqrt {a c g} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right ) + {\left (177 \, a^{3} g \cos \left (f x + e\right )^{2} - 316 \, a^{3} g - 3 \, {\left (a^{3} g \cos \left (f x + e\right )^{2} - 100 \, a^{3} g\right )} \sin \left (f x + e\right )\right )} \sqrt {g \cos \left (f x + e\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}\right )}}{9 \, {\left (3 \, c^{4} f \cos \left (f x + e\right )^{2} - 4 \, c^{4} f - {\left (c^{4} f \cos \left (f x + e\right )^{2} - 4 \, c^{4} f\right )} \sin \left (f x + e\right )\right )}} \] Input:
integrate((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(7/2)/(c-c*sin(f*x+e))^(7/ 2),x, algorithm="fricas")
Output:
2/9*(231*sqrt(1/2)*(3*I*a^3*g*cos(f*x + e)^2 - 4*I*a^3*g + (-I*a^3*g*cos(f *x + e)^2 + 4*I*a^3*g)*sin(f*x + e))*sqrt(a*c*g)*weierstrassZeta(-4, 0, we ierstrassPInverse(-4, 0, cos(f*x + e) + I*sin(f*x + e))) + 231*sqrt(1/2)*( -3*I*a^3*g*cos(f*x + e)^2 + 4*I*a^3*g + (I*a^3*g*cos(f*x + e)^2 - 4*I*a^3* g)*sin(f*x + e))*sqrt(a*c*g)*weierstrassZeta(-4, 0, weierstrassPInverse(-4 , 0, cos(f*x + e) - I*sin(f*x + e))) + (177*a^3*g*cos(f*x + e)^2 - 316*a^3 *g - 3*(a^3*g*cos(f*x + e)^2 - 100*a^3*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))/(3*c^4*f*cos(f*x + e)^2 - 4*c^4*f - (c^4*f*cos(f*x + e)^2 - 4*c^4*f)*sin(f*x + e))
Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{7/2}} \, dx=\text {Timed out} \] Input:
integrate((g*cos(f*x+e))**(3/2)*(a+a*sin(f*x+e))**(7/2)/(c-c*sin(f*x+e))** (7/2),x)
Output:
Timed out
\[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{7/2}} \, dx=\int { \frac {\left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {7}{2}}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {7}{2}}} \,d x } \] Input:
integrate((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(7/2)/(c-c*sin(f*x+e))^(7/ 2),x, algorithm="maxima")
Output:
integrate((g*cos(f*x + e))^(3/2)*(a*sin(f*x + e) + a)^(7/2)/(-c*sin(f*x + e) + c)^(7/2), x)
Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{7/2}} \, dx=\text {Timed out} \] Input:
integrate((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(7/2)/(c-c*sin(f*x+e))^(7/ 2),x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{7/2}} \, dx=\int \frac {{\left (g\,\cos \left (e+f\,x\right )\right )}^{3/2}\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{7/2}}{{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{7/2}} \,d x \] Input:
int(((g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^(7/2))/(c - c*sin(e + f*x ))^(7/2),x)
Output:
int(((g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^(7/2))/(c - c*sin(e + f*x ))^(7/2), x)
\[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{7/2}} \, dx=\text {too large to display} \] Input:
int((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(7/2)/(c-c*sin(f*x+e))^(7/2),x)
Output:
(sqrt(g)*sqrt(c)*sqrt(a)*a**3*g*( - 60*sqrt(sin(e + f*x) + 1)*sqrt( - sin( e + f*x) + 1)*sqrt(cos(e + f*x))*sin(e + f*x)**3 + 20*sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)*sqrt(cos(e + f*x))*sin(e + f*x)**2 - 240*sqrt (sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)*sqrt(cos(e + f*x))*sin(e + f* x) + 154*sqrt(sin(e + f*x) + 1)*sqrt( - sin(e + f*x) + 1)*sqrt(cos(e + f*x )) + 60*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)**4)/(sin(e + f*x)**5 - 3*sin(e + f*x)**4 + 2*sin(e + f*x)**3 + 2*sin(e + f*x)**2 - 3*sin(e + f*x) + 1),x)*sin(e + f *x)**3*f - 180*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)**4)/(sin(e + f*x)**5 - 3*sin(e + f *x)**4 + 2*sin(e + f*x)**3 + 2*sin(e + f*x)**2 - 3*sin(e + f*x) + 1),x)*si n(e + f*x)**2*f + 180*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)**4)/(sin(e + f*x)**5 - 3*si n(e + f*x)**4 + 2*sin(e + f*x)**3 + 2*sin(e + f*x)**2 - 3*sin(e + f*x) + 1 ),x)*sin(e + f*x)*f - 60*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)**4)/(sin(e + f*x)**5 - 3 *sin(e + f*x)**4 + 2*sin(e + f*x)**3 + 2*sin(e + f*x)**2 - 3*sin(e + f*x) + 1),x)*f + 60*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)**3)/(sin(e + f*x)**4 - 4*sin(e + f *x)**3 + 6*sin(e + f*x)**2 - 4*sin(e + f*x) + 1),x)*sin(e + f*x)**3*f -...