\(\int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{13/2}} \, dx\) [115]

Optimal result

Integrand size = 42, antiderivative size = 414 \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{13/2}} \, dx=\frac {4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{21 f g (c-c \sin (e+f x))^{13/2}}-\frac {44 a^2 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{357 c f g (c-c \sin (e+f x))^{11/2}}+\frac {44 a^3 (g \cos (e+f x))^{5/2}}{663 c^2 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}}-\frac {22 a^3 (g \cos (e+f x))^{5/2}}{1989 c^3 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}-\frac {22 a^3 (g \cos (e+f x))^{5/2}}{3315 c^4 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}-\frac {22 a^3 (g \cos (e+f x))^{5/2}}{3315 c^5 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}+\frac {22 a^3 g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{3315 c^6 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \]

[Out]

4/21*a*(g*cos(f*x+e))^(5/2)*(a+a*sin(f*x+e))^(3/2)/f/g/(c-c*sin(f*x+e))^(13/2)+44/663*a^3*(g*cos(f*x+e))^(5/2)
/c^2/f/g/(c-c*sin(f*x+e))^(9/2)/(a+a*sin(f*x+e))^(1/2)-22/1989*a^3*(g*cos(f*x+e))^(5/2)/c^3/f/g/(c-c*sin(f*x+e
))^(7/2)/(a+a*sin(f*x+e))^(1/2)-22/3315*a^3*(g*cos(f*x+e))^(5/2)/c^4/f/g/(c-c*sin(f*x+e))^(5/2)/(a+a*sin(f*x+e
))^(1/2)-22/3315*a^3*(g*cos(f*x+e))^(5/2)/c^5/f/g/(c-c*sin(f*x+e))^(3/2)/(a+a*sin(f*x+e))^(1/2)-44/357*a^2*(g*
cos(f*x+e))^(5/2)*(a+a*sin(f*x+e))^(1/2)/c/f/g/(c-c*sin(f*x+e))^(11/2)+22/3315*a^3*g*(cos(1/2*f*x+1/2*e)^2)^(1
/2)/cos(1/2*f*x+1/2*e)*EllipticE(sin(1/2*f*x+1/2*e),2^(1/2))*cos(f*x+e)^(1/2)*(g*cos(f*x+e))^(1/2)/c^6/f/(a+a*
sin(f*x+e))^(1/2)/(c-c*sin(f*x+e))^(1/2)

Rubi [A] (verified)

Time = 1.47 (sec) , antiderivative size = 414, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.119, Rules used = {2929, 2931, 2921, 2721, 2719} \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{13/2}} \, dx=\frac {22 a^3 g \sqrt {\cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {g \cos (e+f x)}}{3315 c^6 f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {22 a^3 (g \cos (e+f x))^{5/2}}{3315 c^5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac {22 a^3 (g \cos (e+f x))^{5/2}}{3315 c^4 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}-\frac {22 a^3 (g \cos (e+f x))^{5/2}}{1989 c^3 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}+\frac {44 a^3 (g \cos (e+f x))^{5/2}}{663 c^2 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}-\frac {44 a^2 \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{357 c f g (c-c \sin (e+f x))^{11/2}}+\frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{21 f g (c-c \sin (e+f x))^{13/2}} \]

[In]

Int[((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(5/2))/(c - c*Sin[e + f*x])^(13/2),x]

[Out]

(4*a*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(3/2))/(21*f*g*(c - c*Sin[e + f*x])^(13/2)) - (44*a^2*(g*Cos[
e + f*x])^(5/2)*Sqrt[a + a*Sin[e + f*x]])/(357*c*f*g*(c - c*Sin[e + f*x])^(11/2)) + (44*a^3*(g*Cos[e + f*x])^(
5/2))/(663*c^2*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(9/2)) - (22*a^3*(g*Cos[e + f*x])^(5/2))/(198
9*c^3*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(7/2)) - (22*a^3*(g*Cos[e + f*x])^(5/2))/(3315*c^4*f*g
*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2)) - (22*a^3*(g*Cos[e + f*x])^(5/2))/(3315*c^5*f*g*Sqrt[a +
a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) + (22*a^3*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e
+ f*x)/2, 2])/(3315*c^6*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])

Rule 2719

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]

Rule 2721

Int[((b_)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[(b*Sin[c + d*x])^n/Sin[c + d*x]^n, Int[Sin[c + d*x]
^n, x], x] /; FreeQ[{b, c, d}, x] && LtQ[-1, n, 1] && IntegerQ[2*n]

Rule 2921

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)/(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(c_) + (d_.)*sin[(e_
.) + (f_.)*(x_)]]), x_Symbol] :> Dist[g*(Cos[e + f*x]/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])), In
t[(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]

Rule 2929

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] - Dist[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] && IntegersQ[2*m, 2*n, 2*p]

Rule 2931

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*((c + d*Sin[e + f*x])^
n/(a*f*g*(2*m + p + 1))), x] + Dist[(m + n + p + 1)/(a*(2*m + p + 1)), 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] && E
qQ[a^2 - b^2, 0] && LtQ[m, -1] && NeQ[2*m + p + 1, 0] &&  !LtQ[m, n, -1] && IntegersQ[2*m, 2*n, 2*p]

Rubi steps \begin{align*} \text {integral}& = \frac {4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{21 f g (c-c \sin (e+f x))^{13/2}}-\frac {(11 a) \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{11/2}} \, dx}{21 c} \\ & = \frac {4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{21 f g (c-c \sin (e+f x))^{13/2}}-\frac {44 a^2 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{357 c f g (c-c \sin (e+f x))^{11/2}}+\frac {\left (11 a^2\right ) \int \frac {(g \cos (e+f x))^{3/2} \sqrt {a+a \sin (e+f x)}}{(c-c \sin (e+f x))^{9/2}} \, dx}{51 c^2} \\ & = \frac {4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{21 f g (c-c \sin (e+f x))^{13/2}}-\frac {44 a^2 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{357 c f g (c-c \sin (e+f x))^{11/2}}+\frac {44 a^3 (g \cos (e+f x))^{5/2}}{663 c^2 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}}-\frac {\left (11 a^3\right ) \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}} \, dx}{221 c^3} \\ & = \frac {4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{21 f g (c-c \sin (e+f x))^{13/2}}-\frac {44 a^2 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{357 c f g (c-c \sin (e+f x))^{11/2}}+\frac {44 a^3 (g \cos (e+f x))^{5/2}}{663 c^2 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}}-\frac {22 a^3 (g \cos (e+f x))^{5/2}}{1989 c^3 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}-\frac {\left (11 a^3\right ) \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}} \, dx}{663 c^4} \\ & = \frac {4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{21 f g (c-c \sin (e+f x))^{13/2}}-\frac {44 a^2 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{357 c f g (c-c \sin (e+f x))^{11/2}}+\frac {44 a^3 (g \cos (e+f x))^{5/2}}{663 c^2 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}}-\frac {22 a^3 (g \cos (e+f x))^{5/2}}{1989 c^3 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}-\frac {22 a^3 (g \cos (e+f x))^{5/2}}{3315 c^4 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}-\frac {\left (11 a^3\right ) \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}} \, dx}{3315 c^5} \\ & = \frac {4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{21 f g (c-c \sin (e+f x))^{13/2}}-\frac {44 a^2 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{357 c f g (c-c \sin (e+f x))^{11/2}}+\frac {44 a^3 (g \cos (e+f x))^{5/2}}{663 c^2 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}}-\frac {22 a^3 (g \cos (e+f x))^{5/2}}{1989 c^3 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}-\frac {22 a^3 (g \cos (e+f x))^{5/2}}{3315 c^4 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}-\frac {22 a^3 (g \cos (e+f x))^{5/2}}{3315 c^5 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}+\frac {\left (11 a^3\right ) \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \, dx}{3315 c^6} \\ & = \frac {4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{21 f g (c-c \sin (e+f x))^{13/2}}-\frac {44 a^2 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{357 c f g (c-c \sin (e+f x))^{11/2}}+\frac {44 a^3 (g \cos (e+f x))^{5/2}}{663 c^2 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}}-\frac {22 a^3 (g \cos (e+f x))^{5/2}}{1989 c^3 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}-\frac {22 a^3 (g \cos (e+f x))^{5/2}}{3315 c^4 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}-\frac {22 a^3 (g \cos (e+f x))^{5/2}}{3315 c^5 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}+\frac {\left (11 a^3 g \cos (e+f x)\right ) \int \sqrt {g \cos (e+f x)} \, dx}{3315 c^6 \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \\ & = \frac {4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{21 f g (c-c \sin (e+f x))^{13/2}}-\frac {44 a^2 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{357 c f g (c-c \sin (e+f x))^{11/2}}+\frac {44 a^3 (g \cos (e+f x))^{5/2}}{663 c^2 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}}-\frac {22 a^3 (g \cos (e+f x))^{5/2}}{1989 c^3 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}-\frac {22 a^3 (g \cos (e+f x))^{5/2}}{3315 c^4 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}-\frac {22 a^3 (g \cos (e+f x))^{5/2}}{3315 c^5 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}+\frac {\left (11 a^3 g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)}\right ) \int \sqrt {\cos (e+f x)} \, dx}{3315 c^6 \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \\ & = \frac {4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{21 f g (c-c \sin (e+f x))^{13/2}}-\frac {44 a^2 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{357 c f g (c-c \sin (e+f x))^{11/2}}+\frac {44 a^3 (g \cos (e+f x))^{5/2}}{663 c^2 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}}-\frac {22 a^3 (g \cos (e+f x))^{5/2}}{1989 c^3 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}-\frac {22 a^3 (g \cos (e+f x))^{5/2}}{3315 c^4 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}-\frac {22 a^3 (g \cos (e+f x))^{5/2}}{3315 c^5 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}+\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 )}{3315 c^6 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 11.78 (sec) , antiderivative size = 600, normalized size of antiderivative = 1.45 \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{13/2}} \, dx=\frac {22 (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 )^{13} (a (1+\sin (e+f x)))^{5/2}}{3315 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 (c-c \sin (e+f x))^{13/2}}+\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 )^{13} \left (-\frac {22}{3315}+\frac {16}{21 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^{10}}-\frac {120}{119 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^8}+\frac {84}{221 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^6}-\frac {22}{1989 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}-\frac {22}{3315 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^2}+\frac {32 \sin \left (\frac {1}{2} (e+f x)\right )}{21 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^{11}}-\frac {240 \sin \left (\frac {1}{2} (e+f x)\right )}{119 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^9}+\frac {168 \sin \left (\frac {1}{2} (e+f x)\right )}{221 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7}-\frac {44 \sin \left (\frac {1}{2} (e+f x)\right )}{1989 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5}-\frac {44 \sin \left (\frac {1}{2} (e+f x)\right )}{3315 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^3}-\frac {44 \sin \left (\frac {1}{2} (e+f x)\right )}{3315 \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)))^{5/2}}{f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 (c-c \sin (e+f x))^{13/2}} \]

[In]

Integrate[((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(5/2))/(c - c*Sin[e + f*x])^(13/2),x]

[Out]

(22*(g*Cos[e + f*x])^(3/2)*EllipticE[(e + f*x)/2, 2]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^13*(a*(1 + Sin[e +
f*x]))^(5/2))/(3315*f*Cos[e + f*x]^(3/2)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5*(c - c*Sin[e + f*x])^(13/2))
+ ((g*Cos[e + f*x])^(3/2)*Sec[e + f*x]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^13*(-22/3315 + 16/(21*(Cos[(e + f
*x)/2] - Sin[(e + f*x)/2])^10) - 120/(119*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^8) + 84/(221*(Cos[(e + f*x)/2]
- Sin[(e + f*x)/2])^6) - 22/(1989*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^4) - 22/(3315*(Cos[(e + f*x)/2] - Sin
[(e + f*x)/2])^2) + (32*Sin[(e + f*x)/2])/(21*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^11) - (240*Sin[(e + f*x)/2
])/(119*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9) + (168*Sin[(e + f*x)/2])/(221*(Cos[(e + f*x)/2] - Sin[(e + f*
x)/2])^7) - (44*Sin[(e + f*x)/2])/(1989*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5) - (44*Sin[(e + f*x)/2])/(3315
*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3) - (44*Sin[(e + f*x)/2])/(3315*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]))
)*(a*(1 + Sin[e + f*x]))^(5/2))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5*(c - c*Sin[e + f*x])^(13/2))

Maple [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 5.08 (sec) , antiderivative size = 1991, normalized size of antiderivative = 4.81

method	result	size
default	\(\text {Expression too large to display}\)	\(1991\)

[In]

int((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(5/2)/(c-c*sin(f*x+e))^(13/2),x,method=_RETURNVERBOSE)

[Out]

2/69615*I/f*(a*(1+sin(f*x+e)))^(1/2)*(g*cos(f*x+e))^(1/2)*g*a^2/(1+cos(f*x+e))/(cos(f*x+e)^4+4*cos(f*x+e)^2*si
n(f*x+e)-8*cos(f*x+e)^2-8*sin(f*x+e)+8)/(-c*(sin(f*x+e)-1))^(1/2)/c^6*(-924*(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)+924*(1/(1+cos(f*x+e)))^(1/2)*(cos(f*x+e)/(1+co
s(f*x+e)))^(1/2)*EllipticF(I*(csc(f*x+e)-cot(f*x+e)),I)-26520*I*sec(f*x+e)^2+9688*I*sin(f*x+e)-30216*I*tan(f*x
+e)-22824*I*sec(f*x+e)+231*I*cos(f*x+e)^4+77*I*cos(f*x+e)^3-2926*I*cos(f*x+e)^2+18944*I*cos(f*x+e)-231*sin(f*x
+e)*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*EllipticE(I*(csc(f*x+e)-cot(f*x+e)),I)*(1/(1+cos(f*x+e)))^(1/2)*cos(f*x+
e)^4+231*sin(f*x+e)*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*EllipticF(I*(csc(f*x+e)-cot(f*x+e)),I)*(1/(1+cos(f*x+e))
)^(1/2)*cos(f*x+e)^4-462*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*EllipticE(I*(csc(f*x+e)-cot(f*x+e)),I)*(1/(1+cos(f*
x+e)))^(1/2)*cos(f*x+e)^3*sin(f*x+e)+462*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*EllipticF(I*(csc(f*x+e)-cot(f*x+e))
,I)*(1/(1+cos(f*x+e)))^(1/2)*cos(f*x+e)^3*sin(f*x+e)+2541*sin(f*x+e)*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*Ellipti
cE(I*(csc(f*x+e)-cot(f*x+e)),I)*(1/(1+cos(f*x+e)))^(1/2)*cos(f*x+e)^2-2541*sin(f*x+e)*(cos(f*x+e)/(1+cos(f*x+e
)))^(1/2)*EllipticF(I*(csc(f*x+e)-cot(f*x+e)),I)*(1/(1+cos(f*x+e)))^(1/2)*cos(f*x+e)^2+1155*(cos(f*x+e)/(1+cos
(f*x+e)))^(1/2)*EllipticE(I*(csc(f*x+e)-cot(f*x+e)),I)*(1/(1+cos(f*x+e)))^(1/2)*cos(f*x+e)^4-1155*(cos(f*x+e)/
(1+cos(f*x+e)))^(1/2)*EllipticF(I*(csc(f*x+e)-cot(f*x+e)),I)*(1/(1+cos(f*x+e)))^(1/2)*cos(f*x+e)^4+2310*(cos(f
*x+e)/(1+cos(f*x+e)))^(1/2)*EllipticE(I*(csc(f*x+e)-cot(f*x+e)),I)*(1/(1+cos(f*x+e)))^(1/2)*cos(f*x+e)^3-2310*
(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*EllipticF(I*(csc(f*x+e)-cot(f*x+e)),I)*(1/(1+cos(f*x+e)))^(1/2)*cos(f*x+e)^3
+5544*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*EllipticE(I*(csc(f*x+e)-cot(f*x+e)),I)*(1/(1+cos(f*x+e)))^(1/2)*cos(f*
x+e)*sin(f*x+e)-5544*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*EllipticF(I*(csc(f*x+e)-cot(f*x+e)),I)*(1/(1+cos(f*x+e)
))^(1/2)*cos(f*x+e)*sin(f*x+e)-3465*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*EllipticE(I*(csc(f*x+e)-cot(f*x+e)),I)*(
1/(1+cos(f*x+e)))^(1/2)*cos(f*x+e)^2+3465*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*EllipticF(I*(csc(f*x+e)-cot(f*x+e)
),I)*(1/(1+cos(f*x+e)))^(1/2)*cos(f*x+e)^2-9240*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*EllipticE(I*(csc(f*x+e)-cot(
f*x+e)),I)*(1/(1+cos(f*x+e)))^(1/2)*cos(f*x+e)-924*sin(f*x+e)*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*EllipticE(I*(c
sc(f*x+e)-cot(f*x+e)),I)*(1/(1+cos(f*x+e)))^(1/2)+9240*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*EllipticF(I*(csc(f*x+
e)-cot(f*x+e)),I)*(1/(1+cos(f*x+e)))^(1/2)*cos(f*x+e)+924*sin(f*x+e)*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*Ellipti
cF(I*(csc(f*x+e)-cot(f*x+e)),I)*(1/(1+cos(f*x+e)))^(1/2)-7392*(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)*tan(f*x+e)+7392*(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)*tan(f*x+e)+7392*(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)-7392*(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)+3696*(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)^2-3696*(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+1155*I*cos(f*x+e)^2*sin(f
*x+e)+14308*I*cos(f*x+e)*sin(f*x+e)-26520*I*tan(f*x+e)*sec(f*x+e)-3696*(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)*tan(f*x+e)*sec(f*x+e)+3696*(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)*tan(f*x+e)*sec(f*x+e)+25412*I)

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.19 (sec) , antiderivative size = 495, normalized size of antiderivative = 1.20 \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{13/2}} \, dx=\frac {2 \, {\left (1386 \, a^{2} g \cos \left (f x + e\right )^{4} + 5607 \, a^{2} g \cos \left (f x + e\right )^{2} - 10796 \, a^{2} g - {\left (231 \, a^{2} g \cos \left (f x + e\right )^{4} - 4081 \, a^{2} g \cos \left (f x + e\right )^{2} + 15724 \, a^{2} 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} - 231 \, {\left (i \, \sqrt {2} a^{2} g \cos \left (f x + e\right )^{6} - 18 i \, \sqrt {2} a^{2} g \cos \left (f x + e\right )^{4} + 48 i \, \sqrt {2} a^{2} g \cos \left (f x + e\right )^{2} - 32 i \, \sqrt {2} a^{2} g + 2 \, {\left (3 i \, \sqrt {2} a^{2} g \cos \left (f x + e\right )^{4} - 16 i \, \sqrt {2} a^{2} g \cos \left (f x + e\right )^{2} + 16 i \, \sqrt {2} a^{2} 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 \, {\left (-i \, \sqrt {2} a^{2} g \cos \left (f x + e\right )^{6} + 18 i \, \sqrt {2} a^{2} g \cos \left (f x + e\right )^{4} - 48 i \, \sqrt {2} a^{2} g \cos \left (f x + e\right )^{2} + 32 i \, \sqrt {2} a^{2} g + 2 \, {\left (-3 i \, \sqrt {2} a^{2} g \cos \left (f x + e\right )^{4} + 16 i \, \sqrt {2} a^{2} g \cos \left (f x + e\right )^{2} - 16 i \, \sqrt {2} a^{2} 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 )}{69615 \, {\left (c^{7} f \cos \left (f x + e\right )^{6} - 18 \, c^{7} f \cos \left (f x + e\right )^{4} + 48 \, c^{7} f \cos \left (f x + e\right )^{2} - 32 \, c^{7} f + 2 \, {\left (3 \, c^{7} f \cos \left (f x + e\right )^{4} - 16 \, c^{7} f \cos \left (f x + e\right )^{2} + 16 \, c^{7} f\right )} \sin \left (f x + e\right )\right )}} \]

[In]

integrate((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(5/2)/(c-c*sin(f*x+e))^(13/2),x, algorithm="fricas")

[Out]

1/69615*(2*(1386*a^2*g*cos(f*x + e)^4 + 5607*a^2*g*cos(f*x + e)^2 - 10796*a^2*g - (231*a^2*g*cos(f*x + e)^4 -
4081*a^2*g*cos(f*x + e)^2 + 15724*a^2*g)*sin(f*x + e))*sqrt(g*cos(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(-c*s
in(f*x + e) + c) - 231*(I*sqrt(2)*a^2*g*cos(f*x + e)^6 - 18*I*sqrt(2)*a^2*g*cos(f*x + e)^4 + 48*I*sqrt(2)*a^2*
g*cos(f*x + e)^2 - 32*I*sqrt(2)*a^2*g + 2*(3*I*sqrt(2)*a^2*g*cos(f*x + e)^4 - 16*I*sqrt(2)*a^2*g*cos(f*x + e)^
2 + 16*I*sqrt(2)*a^2*g)*sin(f*x + e))*sqrt(a*c*g)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(f*x +
e) + I*sin(f*x + e))) - 231*(-I*sqrt(2)*a^2*g*cos(f*x + e)^6 + 18*I*sqrt(2)*a^2*g*cos(f*x + e)^4 - 48*I*sqrt(2
)*a^2*g*cos(f*x + e)^2 + 32*I*sqrt(2)*a^2*g + 2*(-3*I*sqrt(2)*a^2*g*cos(f*x + e)^4 + 16*I*sqrt(2)*a^2*g*cos(f*
x + e)^2 - 16*I*sqrt(2)*a^2*g)*sin(f*x + e))*sqrt(a*c*g)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos
(f*x + e) - I*sin(f*x + e))))/(c^7*f*cos(f*x + e)^6 - 18*c^7*f*cos(f*x + e)^4 + 48*c^7*f*cos(f*x + e)^2 - 32*c
^7*f + 2*(3*c^7*f*cos(f*x + e)^4 - 16*c^7*f*cos(f*x + e)^2 + 16*c^7*f)*sin(f*x + e))

Sympy [F(-1)]

Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{13/2}} \, dx=\text {Timed out} \]

[In]

integrate((g*cos(f*x+e))**(3/2)*(a+a*sin(f*x+e))**(5/2)/(c-c*sin(f*x+e))**(13/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{13/2}} \, dx=\int { \frac {\left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {13}{2}}} \,d x } \]

[In]

integrate((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(5/2)/(c-c*sin(f*x+e))^(13/2),x, algorithm="maxima")

[Out]

integrate((g*cos(f*x + e))^(3/2)*(a*sin(f*x + e) + a)^(5/2)/(-c*sin(f*x + e) + c)^(13/2), x)

Giac [F(-1)]

Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{13/2}} \, dx=\text {Timed out} \]

[In]

integrate((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(5/2)/(c-c*sin(f*x+e))^(13/2),x, algorithm="giac")

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{13/2}} \, dx=\int \frac {{\left (g\,\cos \left (e+f\,x\right )\right )}^{3/2}\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{5/2}}{{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{13/2}} \,d x \]

[In]

int(((g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^(5/2))/(c - c*sin(e + f*x))^(13/2),x)

[Out]

int(((g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^(5/2))/(c - c*sin(e + f*x))^(13/2), x)