\(\int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{13/2}} \, dx\) [125]

Optimal result

Integrand size = 42, antiderivative size = 414 \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{13/2}} \, dx=\frac {4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{21 f g (c-c \sin (e+f x))^{13/2}}-\frac {20 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{119 c f g (c-c \sin (e+f x))^{11/2}}+\frac {220 a^3 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{1547 c^2 f g (c-c \sin (e+f x))^{9/2}}-\frac {220 a^4 (g \cos (e+f x))^{5/2}}{1989 c^3 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}+\frac {22 a^4 (g \cos (e+f x))^{5/2}}{663 c^4 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}+\frac {22 a^4 (g \cos (e+f x))^{5/2}}{663 c^5 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}-\frac {22 a^4 g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{663 c^6 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \]

[Out]

4/21*a*(g*cos(f*x+e))^(5/2)*(a+a*sin(f*x+e))^(5/2)/f/g/(c-c*sin(f*x+e))^(13/2)-20/119*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))^(11/2)-220/1989*a^4*(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)+22/663*a^4*(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)+22/663*a^4*(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)+220/1547*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))^(9/2)-22/663*a^4*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 = 1.51 (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} \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{13/2}} \, dx=-\frac {22 a^4 g \sqrt {\cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {g \cos (e+f x)}}{663 c^6 f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {22 a^4 (g \cos (e+f x))^{5/2}}{663 c^5 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}+\frac {22 a^4 (g \cos (e+f x))^{5/2}}{663 c^4 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}-\frac {220 a^4 (g \cos (e+f x))^{5/2}}{1989 c^3 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}+\frac {220 a^3 \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{1547 c^2 f g (c-c \sin (e+f x))^{9/2}}-\frac {20 a^2 (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{119 c f g (c-c \sin (e+f x))^{11/2}}+\frac {4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{21 f g (c-c \sin (e+f x))^{13/2}} \]

[In]

Int[((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(7/2))/(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])^(5/2))/(21*f*g*(c - c*Sin[e + f*x])^(13/2)) - (20*a^2*(g*Cos[
e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(3/2))/(119*c*f*g*(c - c*Sin[e + f*x])^(11/2)) + (220*a^3*(g*Cos[e + f*x]
)^(5/2)*Sqrt[a + a*Sin[e + f*x]])/(1547*c^2*f*g*(c - c*Sin[e + f*x])^(9/2)) - (220*a^4*(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)) + (22*a^4*(g*Cos[e + f*x])^(5/2))/(663*c^4
*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2)) + (22*a^4*(g*Cos[e + f*x])^(5/2))/(663*c^5*f*g*Sqrt[
a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) - (22*a^4*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[
(e + f*x)/2, 2])/(663*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))^{5/2}}{21 f g (c-c \sin (e+f x))^{13/2}}-\frac {(5 a) \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{11/2}} \, dx}{7 c} \\ & = \frac {4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{21 f g (c-c \sin (e+f x))^{13/2}}-\frac {20 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{119 c f g (c-c \sin (e+f x))^{11/2}}+\frac {\left (55 a^2\right ) \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{9/2}} \, dx}{119 c^2} \\ & = \frac {4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{21 f g (c-c \sin (e+f x))^{13/2}}-\frac {20 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{119 c f g (c-c \sin (e+f x))^{11/2}}+\frac {220 a^3 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{1547 c^2 f g (c-c \sin (e+f x))^{9/2}}-\frac {\left (55 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))^{5/2}}{21 f g (c-c \sin (e+f x))^{13/2}}-\frac {20 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{119 c f g (c-c \sin (e+f x))^{11/2}}+\frac {220 a^3 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{1547 c^2 f g (c-c \sin (e+f x))^{9/2}}-\frac {220 a^4 (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 (55 a^4\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))^{5/2}}{21 f g (c-c \sin (e+f x))^{13/2}}-\frac {20 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{119 c f g (c-c \sin (e+f x))^{11/2}}+\frac {220 a^3 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{1547 c^2 f g (c-c \sin (e+f x))^{9/2}}-\frac {220 a^4 (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^4 (g \cos (e+f x))^{5/2}}{663 c^4 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}+\frac {\left (11 a^4\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}{663 c^5} \\ & = \frac {4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{21 f g (c-c \sin (e+f x))^{13/2}}-\frac {20 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{119 c f g (c-c \sin (e+f x))^{11/2}}+\frac {220 a^3 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{1547 c^2 f g (c-c \sin (e+f x))^{9/2}}-\frac {220 a^4 (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^4 (g \cos (e+f x))^{5/2}}{663 c^4 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}+\frac {22 a^4 (g \cos (e+f x))^{5/2}}{663 c^5 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}-\frac {\left (11 a^4\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}{663 c^6} \\ & = \frac {4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{21 f g (c-c \sin (e+f x))^{13/2}}-\frac {20 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{119 c f g (c-c \sin (e+f x))^{11/2}}+\frac {220 a^3 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{1547 c^2 f g (c-c \sin (e+f x))^{9/2}}-\frac {220 a^4 (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^4 (g \cos (e+f x))^{5/2}}{663 c^4 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}+\frac {22 a^4 (g \cos (e+f x))^{5/2}}{663 c^5 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}-\frac {\left (11 a^4 g \cos (e+f x)\right ) \int \sqrt {g \cos (e+f x)} \, dx}{663 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))^{5/2}}{21 f g (c-c \sin (e+f x))^{13/2}}-\frac {20 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{119 c f g (c-c \sin (e+f x))^{11/2}}+\frac {220 a^3 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{1547 c^2 f g (c-c \sin (e+f x))^{9/2}}-\frac {220 a^4 (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^4 (g \cos (e+f x))^{5/2}}{663 c^4 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}+\frac {22 a^4 (g \cos (e+f x))^{5/2}}{663 c^5 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}-\frac {\left (11 a^4 g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)}\right ) \int \sqrt {\cos (e+f x)} \, dx}{663 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))^{5/2}}{21 f g (c-c \sin (e+f x))^{13/2}}-\frac {20 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{119 c f g (c-c \sin (e+f x))^{11/2}}+\frac {220 a^3 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{1547 c^2 f g (c-c \sin (e+f x))^{9/2}}-\frac {220 a^4 (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^4 (g \cos (e+f x))^{5/2}}{663 c^4 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}+\frac {22 a^4 (g \cos (e+f x))^{5/2}}{663 c^5 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}-\frac {22 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 )}{663 c^6 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 12.43 (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))^{7/2}}{(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)))^{7/2}}{663 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))^{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}{663}+\frac {32}{21 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^{10}}-\frac {352}{119 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^8}+\frac {464}{221 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^6}-\frac {1216}{1989 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}+\frac {22}{663 \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 )}{21 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^{11}}-\frac {704 \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 {928 \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 {2432 \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 )}{663 \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 )}{663 \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))^{13/2}} \]

[In]

Integrate[((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(7/2))/(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]))^(7/2))/(663*f*Cos[e + f*x]^(3/2)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(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*(22/663 + 32/(21*(Cos[(e + f*x
)/2] - Sin[(e + f*x)/2])^10) - 352/(119*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^8) + 464/(221*(Cos[(e + f*x)/2]
- Sin[(e + f*x)/2])^6) - 1216/(1989*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^4) + 22/(663*(Cos[(e + f*x)/2] - Sin
[(e + f*x)/2])^2) + (64*Sin[(e + f*x)/2])/(21*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^11) - (704*Sin[(e + f*x)/2
])/(119*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9) + (928*Sin[(e + f*x)/2])/(221*(Cos[(e + f*x)/2] - Sin[(e + f*
x)/2])^7) - (2432*Sin[(e + f*x)/2])/(1989*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5) + (44*Sin[(e + f*x)/2])/(66
3*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3) + (44*Sin[(e + f*x)/2])/(663*(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])^(13/2))

Maple [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 2.93 (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))^(7/2)/(c-c*sin(f*x+e))^(13/2),x,method=_RETURNVERBOSE)

[Out]

-2/13923*I/f*(a*(1+sin(f*x+e)))^(1/2)*(g*cos(f*x+e))^(1/2)*g*a^3/(1+cos(f*x+e))/(cos(f*x+e)^4+4*cos(f*x+e)^2*s
in(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+c
os(f*x+e)))^(1/2)*EllipticF(I*(csc(f*x+e)-cot(f*x+e)),I)+231*I*cos(f*x+e)^4+6912*I*tan(f*x+e)+10608*I*sec(f*x+
e)^2+14304*I*sec(f*x+e)-8876*I*sin(f*x+e)+4718*I*cos(f*x+e)^3+1715*I*cos(f*x+e)^2-18184*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)*Ellip
ticE(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+c
os(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)^3-231
0*(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)-co
t(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*
(csc(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)*Ellip
ticF(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+co
s(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)^2-4256*I*cos(f*x+e)*sin(f
*x+e)+10608*I*tan(f*x+e)*sec(f*x+e)+1155*I*cos(f*x+e)^2*sin(f*x+e)-11716*I-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))

Fricas [C] (verification not implemented)

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

Time = 0.16 (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))^{7/2}}{(c-c \sin (e+f x))^{13/2}} \, dx=-\frac {2 \, {\left (1386 \, a^{3} g \cos \left (f x + e\right )^{4} - 8316 \, a^{3} g \cos \left (f x + e\right )^{2} + 7768 \, a^{3} g - {\left (231 \, a^{3} g \cos \left (f x + e\right )^{4} + 560 \, a^{3} g \cos \left (f x + e\right )^{2} - 2840 \, 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} + 231 \, {\left (-i \, \sqrt {2} a^{3} g \cos \left (f x + e\right )^{6} + 18 i \, \sqrt {2} a^{3} g \cos \left (f x + e\right )^{4} - 48 i \, \sqrt {2} a^{3} g \cos \left (f x + e\right )^{2} + 32 i \, \sqrt {2} a^{3} g + 2 \, {\left (-3 i \, \sqrt {2} a^{3} g \cos \left (f x + e\right )^{4} + 16 i \, \sqrt {2} a^{3} g \cos \left (f x + e\right )^{2} - 16 i \, \sqrt {2} 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 \, {\left (i \, \sqrt {2} a^{3} g \cos \left (f x + e\right )^{6} - 18 i \, \sqrt {2} a^{3} g \cos \left (f x + e\right )^{4} + 48 i \, \sqrt {2} a^{3} g \cos \left (f x + e\right )^{2} - 32 i \, \sqrt {2} a^{3} g + 2 \, {\left (3 i \, \sqrt {2} a^{3} g \cos \left (f x + e\right )^{4} - 16 i \, \sqrt {2} a^{3} g \cos \left (f x + e\right )^{2} + 16 i \, \sqrt {2} 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 )}{13923 \, {\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))^(7/2)/(c-c*sin(f*x+e))^(13/2),x, algorithm="fricas")

[Out]

-1/13923*(2*(1386*a^3*g*cos(f*x + e)^4 - 8316*a^3*g*cos(f*x + e)^2 + 7768*a^3*g - (231*a^3*g*cos(f*x + e)^4 +
560*a^3*g*cos(f*x + e)^2 - 2840*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) + 231*(-I*sqrt(2)*a^3*g*cos(f*x + e)^6 + 18*I*sqrt(2)*a^3*g*cos(f*x + e)^4 - 48*I*sqrt(2)*a^3*g
*cos(f*x + e)^2 + 32*I*sqrt(2)*a^3*g + 2*(-3*I*sqrt(2)*a^3*g*cos(f*x + e)^4 + 16*I*sqrt(2)*a^3*g*cos(f*x + e)^
2 - 16*I*sqrt(2)*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))) + 231*(I*sqrt(2)*a^3*g*cos(f*x + e)^6 - 18*I*sqrt(2)*a^3*g*cos(f*x + e)^4 + 48*I*sqrt(2)
*a^3*g*cos(f*x + e)^2 - 32*I*sqrt(2)*a^3*g + 2*(3*I*sqrt(2)*a^3*g*cos(f*x + e)^4 - 16*I*sqrt(2)*a^3*g*cos(f*x
+ e)^2 + 16*I*sqrt(2)*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))))/(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. \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{13/2}} \, dx=\text {Timed out} \]

[In]

integrate((g*cos(f*x+e))**(3/2)*(a+a*sin(f*x+e))**(7/2)/(c-c*sin(f*x+e))**(13/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{13/2}} \, dx=\int { \frac {\left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {7}{2}}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {13}{2}}} \,d x } \]

[In]

integrate((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(7/2)/(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)^(7/2)/(-c*sin(f*x + e) + c)^(13/2), x)

Giac [F(-1)]

Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{13/2}} \, dx=\text {Timed out} \]

[In]

integrate((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(7/2)/(c-c*sin(f*x+e))^(13/2),x, algorithm="giac")

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{13/2}} \, dx=\int \frac {{\left (g\,\cos \left (e+f\,x\right )\right )}^{3/2}\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{7/2}}{{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{13/2}} \,d x \]

[In]

int(((g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^(7/2))/(c - c*sin(e + f*x))^(13/2),x)

[Out]

int(((g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^(7/2))/(c - c*sin(e + f*x))^(13/2), x)