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3.142 \(\int \frac{(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{9/2}}{(a+a \sin (e+f x))^{5/2}} \, dx\)

Optimal. Leaf size=357 \[ \frac{418 c^5 (g \cos (e+f x))^{5/2}}{5 a^2 f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{1254 c^4 \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{35 a^2 f g \sqrt{a \sin (e+f x)+a}}+\frac{114 c^3 (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{7 a^2 f g \sqrt{a \sin (e+f x)+a}}+\frac{1254 c^5 g \sqrt{\cos (e+f x)} E\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{g \cos (e+f x)}}{5 a^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{76 c^2 (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{5 a f g (a \sin (e+f x)+a)^{3/2}}-\frac{4 c (c-c \sin (e+f x))^{7/2} (g \cos (e+f x))^{5/2}}{5 f g (a \sin (e+f x)+a)^{5/2}} \]

[Out]

(418*c^5*(g*Cos[e + f*x])^(5/2))/(5*a^2*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (1254*c^5*g*S
qrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(5*a^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c
*Sin[e + f*x]]) + (1254*c^4*(g*Cos[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f*x]])/(35*a^2*f*g*Sqrt[a + a*Sin[e + f*
x]]) + (114*c^3*(g*Cos[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(3/2))/(7*a^2*f*g*Sqrt[a + a*Sin[e + f*x]]) + (76*
c^2*(g*Cos[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(5/2))/(5*a*f*g*(a + a*Sin[e + f*x])^(3/2)) - (4*c*(g*Cos[e +
f*x])^(5/2)*(c - c*Sin[e + f*x])^(7/2))/(5*f*g*(a + a*Sin[e + f*x])^(5/2))

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Rubi [A]  time = 1.72833, antiderivative size = 357, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 42, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.119, Rules used = {2850, 2851, 2842, 2640, 2639} \[ \frac{418 c^5 (g \cos (e+f x))^{5/2}}{5 a^2 f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{1254 c^4 \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{35 a^2 f g \sqrt{a \sin (e+f x)+a}}+\frac{114 c^3 (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{7 a^2 f g \sqrt{a \sin (e+f x)+a}}+\frac{1254 c^5 g \sqrt{\cos (e+f x)} E\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{g \cos (e+f x)}}{5 a^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{76 c^2 (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{5 a f g (a \sin (e+f x)+a)^{3/2}}-\frac{4 c (c-c \sin (e+f x))^{7/2} (g \cos (e+f x))^{5/2}}{5 f g (a \sin (e+f x)+a)^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[((g*Cos[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(9/2))/(a + a*Sin[e + f*x])^(5/2),x]

[Out]

(418*c^5*(g*Cos[e + f*x])^(5/2))/(5*a^2*f*g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) + (1254*c^5*g*S
qrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(5*a^2*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c
*Sin[e + f*x]]) + (1254*c^4*(g*Cos[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f*x]])/(35*a^2*f*g*Sqrt[a + a*Sin[e + f*
x]]) + (114*c^3*(g*Cos[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(3/2))/(7*a^2*f*g*Sqrt[a + a*Sin[e + f*x]]) + (76*
c^2*(g*Cos[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(5/2))/(5*a*f*g*(a + a*Sin[e + f*x])^(3/2)) - (4*c*(g*Cos[e +
f*x])^(5/2)*(c - c*Sin[e + f*x])^(7/2))/(5*f*g*(a + a*Sin[e + f*x])^(5/2))

Rule 2850

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 2851

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] + Dist[(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] && Eq
Q[a^2 - b^2, 0] && GtQ[m, 0] && NeQ[m + n + p, 0] &&  !LtQ[0, n, m] && IntegersQ[2*m, 2*n, 2*p]

Rule 2842

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 2640

Int[Sqrt[(b_)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[b*Sin[c + d*x]]/Sqrt[Sin[c + d*x]], Int[Sqrt[Si
n[c + d*x]], x], x] /; FreeQ[{b, c, d}, x]

Rule 2639

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticE[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ[{
c, d}, x]

Rubi steps

\begin{align*} \int \frac{(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{9/2}}{(a+a \sin (e+f x))^{5/2}} \, dx &=-\frac{4 c (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{7/2}}{5 f g (a+a \sin (e+f x))^{5/2}}-\frac{(19 c) \int \frac{(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{7/2}}{(a+a \sin (e+f x))^{3/2}} \, dx}{5 a}\\ &=\frac{76 c^2 (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2}}{5 a f g (a+a \sin (e+f x))^{3/2}}-\frac{4 c (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{7/2}}{5 f g (a+a \sin (e+f x))^{5/2}}+\frac{\left (57 c^2\right ) \int \frac{(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2}}{\sqrt{a+a \sin (e+f x)}} \, dx}{a^2}\\ &=\frac{114 c^3 (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{7 a^2 f g \sqrt{a+a \sin (e+f x)}}+\frac{76 c^2 (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2}}{5 a f g (a+a \sin (e+f x))^{3/2}}-\frac{4 c (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{7/2}}{5 f g (a+a \sin (e+f x))^{5/2}}+\frac{\left (627 c^3\right ) \int \frac{(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2}}{\sqrt{a+a \sin (e+f x)}} \, dx}{7 a^2}\\ &=\frac{1254 c^4 (g \cos (e+f x))^{5/2} \sqrt{c-c \sin (e+f x)}}{35 a^2 f g \sqrt{a+a \sin (e+f x)}}+\frac{114 c^3 (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{7 a^2 f g \sqrt{a+a \sin (e+f x)}}+\frac{76 c^2 (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2}}{5 a f g (a+a \sin (e+f x))^{3/2}}-\frac{4 c (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{7/2}}{5 f g (a+a \sin (e+f x))^{5/2}}+\frac{\left (627 c^4\right ) \int \frac{(g \cos (e+f x))^{3/2} \sqrt{c-c \sin (e+f x)}}{\sqrt{a+a \sin (e+f x)}} \, dx}{5 a^2}\\ &=\frac{418 c^5 (g \cos (e+f x))^{5/2}}{5 a^2 f g \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}+\frac{1254 c^4 (g \cos (e+f x))^{5/2} \sqrt{c-c \sin (e+f x)}}{35 a^2 f g \sqrt{a+a \sin (e+f x)}}+\frac{114 c^3 (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{7 a^2 f g \sqrt{a+a \sin (e+f x)}}+\frac{76 c^2 (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2}}{5 a f g (a+a \sin (e+f x))^{3/2}}-\frac{4 c (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{7/2}}{5 f g (a+a \sin (e+f x))^{5/2}}+\frac{\left (627 c^5\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}{5 a^2}\\ &=\frac{418 c^5 (g \cos (e+f x))^{5/2}}{5 a^2 f g \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}+\frac{1254 c^4 (g \cos (e+f x))^{5/2} \sqrt{c-c \sin (e+f x)}}{35 a^2 f g \sqrt{a+a \sin (e+f x)}}+\frac{114 c^3 (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{7 a^2 f g \sqrt{a+a \sin (e+f x)}}+\frac{76 c^2 (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2}}{5 a f g (a+a \sin (e+f x))^{3/2}}-\frac{4 c (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{7/2}}{5 f g (a+a \sin (e+f x))^{5/2}}+\frac{\left (627 c^5 g \cos (e+f x)\right ) \int \sqrt{g \cos (e+f x)} \, dx}{5 a^2 \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}\\ &=\frac{418 c^5 (g \cos (e+f x))^{5/2}}{5 a^2 f g \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}+\frac{1254 c^4 (g \cos (e+f x))^{5/2} \sqrt{c-c \sin (e+f x)}}{35 a^2 f g \sqrt{a+a \sin (e+f x)}}+\frac{114 c^3 (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{7 a^2 f g \sqrt{a+a \sin (e+f x)}}+\frac{76 c^2 (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2}}{5 a f g (a+a \sin (e+f x))^{3/2}}-\frac{4 c (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{7/2}}{5 f g (a+a \sin (e+f x))^{5/2}}+\frac{\left (627 c^5 g \sqrt{\cos (e+f x)} \sqrt{g \cos (e+f x)}\right ) \int \sqrt{\cos (e+f x)} \, dx}{5 a^2 \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}\\ &=\frac{418 c^5 (g \cos (e+f x))^{5/2}}{5 a^2 f g \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}+\frac{1254 c^5 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 a^2 f \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}+\frac{1254 c^4 (g \cos (e+f x))^{5/2} \sqrt{c-c \sin (e+f x)}}{35 a^2 f g \sqrt{a+a \sin (e+f x)}}+\frac{114 c^3 (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{7 a^2 f g \sqrt{a+a \sin (e+f x)}}+\frac{76 c^2 (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2}}{5 a f g (a+a \sin (e+f x))^{3/2}}-\frac{4 c (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{7/2}}{5 f g (a+a \sin (e+f x))^{5/2}}\\ \end{align*}

Mathematica [A]  time = 6.76515, size = 356, normalized size = 1. \[ \frac{1254 E\left (\left .\frac{1}{2} (e+f x)\right |2\right ) (c-c \sin (e+f x))^{9/2} (g \cos (e+f x))^{3/2} \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^5}{5 f \cos ^{\frac{3}{2}}(e+f x) (a (\sin (e+f x)+1))^{5/2} \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^9}+\frac{\sec (e+f x) (c-c \sin (e+f x))^{9/2} (g \cos (e+f x))^{3/2} \left (-\frac{7}{5} \sin (2 (e+f x))+\frac{221}{14} \cos (e+f x)-\frac{1}{14} \cos (3 (e+f x))-\frac{1472 \sin \left (\frac{1}{2} (e+f x)\right )}{5 \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )}-\frac{64}{5 \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^2}+\frac{128 \sin \left (\frac{1}{2} (e+f x)\right )}{5 \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^3}+\frac{736}{5}\right ) \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^5}{f (a (\sin (e+f x)+1))^{5/2} \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^9} \]

Antiderivative was successfully verified.

[In]

Integrate[((g*Cos[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(9/2))/(a + a*Sin[e + f*x])^(5/2),x]

[Out]

(1254*(g*Cos[e + f*x])^(3/2)*EllipticE[(e + f*x)/2, 2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5*(c - c*Sin[e +
f*x])^(9/2))/(5*f*Cos[e + f*x]^(3/2)*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9*(a*(1 + Sin[e + f*x]))^(5/2)) + (
(g*Cos[e + f*x])^(3/2)*Sec[e + f*x]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5*(c - c*Sin[e + f*x])^(9/2)*(736/5
+ (221*Cos[e + f*x])/14 - Cos[3*(e + f*x)]/14 + (128*Sin[(e + f*x)/2])/(5*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]
)^3) - 64/(5*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2) - (1472*Sin[(e + f*x)/2])/(5*(Cos[(e + f*x)/2] + Sin[(e
+ f*x)/2])) - (7*Sin[2*(e + f*x)])/5))/(f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9*(a*(1 + Sin[e + f*x]))^(5/2)
)

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Maple [C]  time = 0.401, size = 3655, normalized size = 10.2 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*cos(f*x+e))^(3/2)*(c-c*sin(f*x+e))^(9/2)/(a+a*sin(f*x+e))^(5/2),x)

[Out]

2/35/f*(-1+cos(f*x+e))*(-700*cos(f*x+e)^5*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(3/2)*ln(-2*(2*cos(f*x+e)^2*(-cos(f*x
+e)/(cos(f*x+e)+1)^2)^(1/2)-cos(f*x+e)^2+2*cos(f*x+e)-2*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)-1)/sin(f*x+e)^2)+
1400*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(3/2)*ln(-2*(2*cos(f*x+e)^2*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)-cos(f*x+e
)^2+2*cos(f*x+e)-2*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)-1)/sin(f*x+e)^2)-1400*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(
3/2)*ln(-(2*cos(f*x+e)^2*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)-cos(f*x+e)^2+2*cos(f*x+e)-2*(-cos(f*x+e)/(cos(f*
x+e)+1)^2)^(1/2)-1)/sin(f*x+e)^2)+8554*cos(f*x+e)^2+700*sin(f*x+e)*cos(f*x+e)^4*(-cos(f*x+e)/(cos(f*x+e)+1)^2)
^(3/2)*ln(-2*(2*cos(f*x+e)^2*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)-cos(f*x+e)^2+2*cos(f*x+e)-2*(-cos(f*x+e)/(co
s(f*x+e)+1)^2)^(1/2)-1)/sin(f*x+e)^2)-700*sin(f*x+e)*cos(f*x+e)^4*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(3/2)*ln(-(2*
cos(f*x+e)^2*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)-cos(f*x+e)^2+2*cos(f*x+e)-2*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(
1/2)-1)/sin(f*x+e)^2)-192*cos(f*x+e)^5-1575*cos(f*x+e)^3-1351*sin(f*x+e)*cos(f*x+e)^3+39*sin(f*x+e)*cos(f*x+e)
^5-4389*I*cos(f*x+e)^4*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*EllipticF(I*(-1+cos(f*x+e))/
sin(f*x+e),I)+4389*I*cos(f*x+e)^4*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*EllipticE(I*(-1+c
os(f*x+e))/sin(f*x+e),I)+13167*I*cos(f*x+e)^2*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*Ellip
ticF(I*(-1+cos(f*x+e))/sin(f*x+e),I)-13167*I*cos(f*x+e)^2*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))
^(1/2)*EllipticE(I*(-1+cos(f*x+e))/sin(f*x+e),I)+8778*I*cos(f*x+e)*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f
*x+e)+1))^(1/2)*EllipticF(I*(-1+cos(f*x+e))/sin(f*x+e),I)-8778*I*cos(f*x+e)*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+
e)/(cos(f*x+e)+1))^(1/2)*EllipticE(I*(-1+cos(f*x+e))/sin(f*x+e),I)+231*sin(f*x+e)*cos(f*x+e)^4+5*cos(f*x+e)^7-
1400*cos(f*x+e)^4*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(3/2)*ln(-2*(2*cos(f*x+e)^2*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1
/2)-cos(f*x+e)^2+2*cos(f*x+e)-2*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)-1)/sin(f*x+e)^2)+1400*cos(f*x+e)^4*(-cos(
f*x+e)/(cos(f*x+e)+1)^2)^(3/2)*ln(-(2*cos(f*x+e)^2*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)-cos(f*x+e)^2+2*cos(f*x
+e)-2*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)-1)/sin(f*x+e)^2)+1400*cos(f*x+e)^3*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(
3/2)*ln(-2*(2*cos(f*x+e)^2*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)-cos(f*x+e)^2+2*cos(f*x+e)-2*(-cos(f*x+e)/(cos(
f*x+e)+1)^2)^(1/2)-1)/sin(f*x+e)^2)-1400*cos(f*x+e)^3*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(3/2)*ln(-(2*cos(f*x+e)^2
*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)-cos(f*x+e)^2+2*cos(f*x+e)-2*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)-1)/sin(
f*x+e)^2)+5600*cos(f*x+e)^2*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(3/2)*ln(-2*(2*cos(f*x+e)^2*(-cos(f*x+e)/(cos(f*x+e
)+1)^2)^(1/2)-cos(f*x+e)^2+2*cos(f*x+e)-2*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)-1)/sin(f*x+e)^2)-5600*cos(f*x+e
)^2*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(3/2)*ln(-(2*cos(f*x+e)^2*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)-cos(f*x+e)^2
+2*cos(f*x+e)-2*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)-1)/sin(f*x+e)^2)+1400*ln(-2*(2*cos(f*x+e)^2*(-cos(f*x+e)/
(cos(f*x+e)+1)^2)^(1/2)-cos(f*x+e)^2+2*cos(f*x+e)-2*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)-1)/sin(f*x+e)^2)*(-co
s(f*x+e)/(cos(f*x+e)+1)^2)^(3/2)*sin(f*x+e)-1400*ln(-(2*cos(f*x+e)^2*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)-cos(
f*x+e)^2+2*cos(f*x+e)-2*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)-1)/sin(f*x+e)^2)*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(
3/2)*sin(f*x+e)+4900*cos(f*x+e)*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(3/2)*ln(-2*(2*cos(f*x+e)^2*(-cos(f*x+e)/(cos(f
*x+e)+1)^2)^(1/2)-cos(f*x+e)^2+2*cos(f*x+e)-2*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)-1)/sin(f*x+e)^2)-4900*cos(f
*x+e)*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(3/2)*ln(-(2*cos(f*x+e)^2*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)-cos(f*x+e)
^2+2*cos(f*x+e)-2*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)-1)/sin(f*x+e)^2)-5*cos(f*x+e)^6*sin(f*x+e)-13167*I*cos(
f*x+e)^2*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*EllipticE(I*(-1+cos(f*x+e))/sin(f*x+e),I)*
sin(f*x+e)+8778*I*sin(f*x+e)*cos(f*x+e)*EllipticF(I*(-1+cos(f*x+e))/sin(f*x+e),I)*(1/(cos(f*x+e)+1))^(1/2)*(co
s(f*x+e)/(cos(f*x+e)+1))^(1/2)-8778*I*cos(f*x+e)*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*El
lipticE(I*(-1+cos(f*x+e))/sin(f*x+e),I)*sin(f*x+e)+4389*I*cos(f*x+e)^3*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(c
os(f*x+e)+1))^(1/2)*EllipticF(I*(-1+cos(f*x+e))/sin(f*x+e),I)*sin(f*x+e)+9002*cos(f*x+e)^2*sin(f*x+e)+44*cos(f
*x+e)^6-1582*cos(f*x+e)^4+3500*sin(f*x+e)*cos(f*x+e)^3*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(3/2)*ln(-2*(2*cos(f*x+e
)^2*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)-cos(f*x+e)^2+2*cos(f*x+e)-2*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)-1)/s
in(f*x+e)^2)-3500*sin(f*x+e)*cos(f*x+e)^3*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(3/2)*ln(-(2*cos(f*x+e)^2*(-cos(f*x+e
)/(cos(f*x+e)+1)^2)^(1/2)-cos(f*x+e)^2+2*cos(f*x+e)-2*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)-1)/sin(f*x+e)^2)+63
00*sin(f*x+e)*cos(f*x+e)^2*ln(-2*(2*cos(f*x+e)^2*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)-cos(f*x+e)^2+2*cos(f*x+e
)-2*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)-1)/sin(f*x+e)^2)*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(3/2)-6300*sin(f*x+e)
*cos(f*x+e)^2*ln(-(2*cos(f*x+e)^2*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)-cos(f*x+e)^2+2*cos(f*x+e)-2*(-cos(f*x+e
)/(cos(f*x+e)+1)^2)^(1/2)-1)/sin(f*x+e)^2)*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(3/2)+4900*sin(f*x+e)*cos(f*x+e)*ln(
-2*(2*cos(f*x+e)^2*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)-cos(f*x+e)^2+2*cos(f*x+e)-2*(-cos(f*x+e)/(cos(f*x+e)+1
)^2)^(1/2)-1)/sin(f*x+e)^2)*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(3/2)-4900*sin(f*x+e)*cos(f*x+e)*ln(-(2*cos(f*x+e)^
2*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)-cos(f*x+e)^2+2*cos(f*x+e)-2*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)-1)/sin
(f*x+e)^2)*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(3/2)-4389*I*cos(f*x+e)^3*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(
f*x+e)+1))^(1/2)*EllipticE(I*(-1+cos(f*x+e))/sin(f*x+e),I)*sin(f*x+e)+700*cos(f*x+e)^5*(-cos(f*x+e)/(cos(f*x+e
)+1)^2)^(3/2)*ln(-(2*cos(f*x+e)^2*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)-cos(f*x+e)^2+2*cos(f*x+e)-2*(-cos(f*x+e
)/(cos(f*x+e)+1)^2)^(1/2)-1)/sin(f*x+e)^2)+13167*I*cos(f*x+e)^2*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+
e)+1))^(1/2)*EllipticF(I*(-1+cos(f*x+e))/sin(f*x+e),I)*sin(f*x+e))*(g*cos(f*x+e))^(3/2)*(-c*(-1+sin(f*x+e)))^(
9/2)/(sin(f*x+e)*cos(f*x+e)^5-cos(f*x+e)^6-6*sin(f*x+e)*cos(f*x+e)^4-5*cos(f*x+e)^5-12*sin(f*x+e)*cos(f*x+e)^3
+18*cos(f*x+e)^4+32*cos(f*x+e)^2*sin(f*x+e)+20*cos(f*x+e)^3+16*sin(f*x+e)*cos(f*x+e)-48*cos(f*x+e)^2-32*sin(f*
x+e)-16*cos(f*x+e)+32)/sin(f*x+e)/cos(f*x+e)/(a*(1+sin(f*x+e)))^(5/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (g \cos \left (f x + e\right )\right )^{\frac{3}{2}}{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac{9}{2}}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*cos(f*x+e))^(3/2)*(c-c*sin(f*x+e))^(9/2)/(a+a*sin(f*x+e))^(5/2),x, algorithm="maxima")

[Out]

integrate((g*cos(f*x + e))^(3/2)*(-c*sin(f*x + e) + c)^(9/2)/(a*sin(f*x + e) + a)^(5/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{{\left (c^{4} g \cos \left (f x + e\right )^{5} - 8 \, c^{4} g \cos \left (f x + e\right )^{3} + 8 \, c^{4} g \cos \left (f x + e\right ) + 4 \,{\left (c^{4} g \cos \left (f x + e\right )^{3} - 2 \, c^{4} g \cos \left (f x + e\right )\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}}{3 \, a^{3} \cos \left (f x + e\right )^{2} - 4 \, a^{3} +{\left (a^{3} \cos \left (f x + e\right )^{2} - 4 \, a^{3}\right )} \sin \left (f x + e\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*cos(f*x+e))^(3/2)*(c-c*sin(f*x+e))^(9/2)/(a+a*sin(f*x+e))^(5/2),x, algorithm="fricas")

[Out]

integral(-(c^4*g*cos(f*x + e)^5 - 8*c^4*g*cos(f*x + e)^3 + 8*c^4*g*cos(f*x + e) + 4*(c^4*g*cos(f*x + e)^3 - 2*
c^4*g*cos(f*x + e))*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*a
^3*cos(f*x + e)^2 - 4*a^3 + (a^3*cos(f*x + e)^2 - 4*a^3)*sin(f*x + e)), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*cos(f*x+e))**(3/2)*(c-c*sin(f*x+e))**(9/2)/(a+a*sin(f*x+e))**(5/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (g \cos \left (f x + e\right )\right )^{\frac{3}{2}}{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac{9}{2}}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*cos(f*x+e))^(3/2)*(c-c*sin(f*x+e))^(9/2)/(a+a*sin(f*x+e))^(5/2),x, algorithm="giac")

[Out]

integrate((g*cos(f*x + e))^(3/2)*(-c*sin(f*x + e) + c)^(9/2)/(a*sin(f*x + e) + a)^(5/2), x)

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