3.143 \(\int \frac{(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{7/2}}{(a+a \sin (e+f x))^{5/2}} \, dx\)
Optimal. Leaf size=298 \[ \frac{154 c^4 (g \cos (e+f x))^{5/2}}{5 a^2 f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{66 c^3 \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{5 a^2 f g \sqrt{a \sin (e+f x)+a}}+\frac{462 c^4 g \sqrt{\cos (e+f x)} E\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{g \cos (e+f x)}}{5 a^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{12 c^2 (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{a f g (a \sin (e+f x)+a)^{3/2}}-\frac{4 c (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{5 f g (a \sin (e+f x)+a)^{5/2}} \]
[Out]
(154*c^4*(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]]) + (462*c^4*g*Sq
rt[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]]) + (66*c^3*(g*Cos[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f*x]])/(5*a^2*f*g*Sqrt[a + a*Sin[e + f*x]])
+ (12*c^2*(g*Cos[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(3/2))/(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])^(5/2))/(5*f*g*(a + a*Sin[e + f*x])^(5/2))
________________________________________________________________________________________
Rubi [A] time = 1.42436, antiderivative size = 298, normalized size of antiderivative = 1.,
number of steps used = 7, 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} \[ \frac{154 c^4 (g \cos (e+f x))^{5/2}}{5 a^2 f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{66 c^3 \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{5 a^2 f g \sqrt{a \sin (e+f x)+a}}+\frac{462 c^4 g \sqrt{\cos (e+f x)} E\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{g \cos (e+f x)}}{5 a^2 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{12 c^2 (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{a f g (a \sin (e+f x)+a)^{3/2}}-\frac{4 c (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{5 f g (a \sin (e+f x)+a)^{5/2}} \]
Antiderivative was successfully verified.
[In]
Int[((g*Cos[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(7/2))/(a + a*Sin[e + f*x])^(5/2),x]
[Out]
(154*c^4*(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]]) + (462*c^4*g*Sq
rt[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]]) + (66*c^3*(g*Cos[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f*x]])/(5*a^2*f*g*Sqrt[a + a*Sin[e + f*x]])
+ (12*c^2*(g*Cos[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(3/2))/(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])^(5/2))/(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))^{7/2}}{(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))^{5/2}}{5 f g (a+a \sin (e+f x))^{5/2}}-\frac{(3 c) \int \frac{(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{5/2}}{(a+a \sin (e+f x))^{3/2}} \, dx}{a}\\ &=\frac{12 c^2 (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{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))^{5/2}}{5 f g (a+a \sin (e+f x))^{5/2}}+\frac{\left (33 c^2\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}{a^2}\\ &=\frac{66 c^3 (g \cos (e+f x))^{5/2} \sqrt{c-c \sin (e+f x)}}{5 a^2 f g \sqrt{a+a \sin (e+f x)}}+\frac{12 c^2 (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{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))^{5/2}}{5 f g (a+a \sin (e+f x))^{5/2}}+\frac{\left (231 c^3\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{154 c^4 (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{66 c^3 (g \cos (e+f x))^{5/2} \sqrt{c-c \sin (e+f x)}}{5 a^2 f g \sqrt{a+a \sin (e+f x)}}+\frac{12 c^2 (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{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))^{5/2}}{5 f g (a+a \sin (e+f x))^{5/2}}+\frac{\left (231 c^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}{5 a^2}\\ &=\frac{154 c^4 (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{66 c^3 (g \cos (e+f x))^{5/2} \sqrt{c-c \sin (e+f x)}}{5 a^2 f g \sqrt{a+a \sin (e+f x)}}+\frac{12 c^2 (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{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))^{5/2}}{5 f g (a+a \sin (e+f x))^{5/2}}+\frac{\left (231 c^4 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{154 c^4 (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{66 c^3 (g \cos (e+f x))^{5/2} \sqrt{c-c \sin (e+f x)}}{5 a^2 f g \sqrt{a+a \sin (e+f x)}}+\frac{12 c^2 (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{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))^{5/2}}{5 f g (a+a \sin (e+f x))^{5/2}}+\frac{\left (231 c^4 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{154 c^4 (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{462 c^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 )}{5 a^2 f \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}+\frac{66 c^3 (g \cos (e+f x))^{5/2} \sqrt{c-c \sin (e+f x)}}{5 a^2 f g \sqrt{a+a \sin (e+f x)}}+\frac{12 c^2 (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{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))^{5/2}}{5 f g (a+a \sin (e+f x))^{5/2}}\\ \end{align*}
Mathematica [A] time = 5.05606, size = 250, normalized size = 0.84 \[ \frac{c^3 \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{3/2} \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^2 \left (\sqrt{\cos (e+f x)} \left (-487 \sin \left (\frac{1}{2} (e+f x)\right )+633 \sin \left (\frac{3}{2} (e+f x)\right )+17 \sin \left (\frac{5}{2} (e+f x)\right )+\sin \left (\frac{7}{2} (e+f x)\right )+487 \cos \left (\frac{1}{2} (e+f x)\right )+633 \cos \left (\frac{3}{2} (e+f x)\right )-17 \cos \left (\frac{5}{2} (e+f x)\right )+\cos \left (\frac{7}{2} (e+f x)\right )\right )+1848 E\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^3\right )}{20 f \cos ^{\frac{3}{2}}(e+f x) (a (\sin (e+f x)+1))^{5/2} \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )} \]
Antiderivative was successfully verified.
[In]
Integrate[((g*Cos[e + f*x])^(3/2)*(c - c*Sin[e + f*x])^(7/2))/(a + a*Sin[e + f*x])^(5/2),x]
[Out]
(c^3*(g*Cos[e + f*x])^(3/2)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^2*Sqrt[c - c*Sin[e + f*x]]*(1848*EllipticE[(
e + f*x)/2, 2]*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 + Sqrt[Cos[e + f*x]]*(487*Cos[(e + f*x)/2] + 633*Cos[(3
*(e + f*x))/2] - 17*Cos[(5*(e + f*x))/2] + Cos[(7*(e + f*x))/2] - 487*Sin[(e + f*x)/2] + 633*Sin[(3*(e + f*x))
/2] + 17*Sin[(5*(e + f*x))/2] + Sin[(7*(e + f*x))/2])))/(20*f*Cos[e + f*x]^(3/2)*(Cos[(e + f*x)/2] - Sin[(e +
f*x)/2])*(a*(1 + Sin[e + f*x]))^(5/2))
________________________________________________________________________________________
Maple [C] time = 0.401, size = 3598, normalized size = 12.1 \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))^(7/2)/(a+a*sin(f*x+e))^(5/2),x)
[Out]
-2/5/f*(-1+cos(f*x+e))*(-40*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)+8
0*(-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)-80*(-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)+446*cos(f*x+e)^2+40*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)/(cos(f*x+e
)+1)^2)^(1/2)-1)/sin(f*x+e)^2)-40*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)-8*cos(f*x+e)^5-85*cos(f*x+e)^3-69*sin(f*x+e)*cos(f*x+e)^3+sin(f*x+e)*cos(f*x+e)^5+231*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)*EllipticF(I*(-1+cos(f*x+e))/sin(f*x+e),I)*sin(
f*x+e)+9*sin(f*x+e)*cos(f*x+e)^4-80*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)+80*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)+80*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)-80*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)+320*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)-320*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)+80*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)*sin(f*x+e)-80*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)+280*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)-280*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)+478*cos(f*x+e)^2*sin(
f*x+e)+cos(f*x+e)^6-78*cos(f*x+e)^4+200*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)/sin(f*x+e)^2)-200*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*(-c
os(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)+360*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*co
s(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)-360*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)+280*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)-280*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)+40*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)-231*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)*Elliptic
E(I*(-1+cos(f*x+e))/sin(f*x+e),I)*sin(f*x+e)-231*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)+231*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+cos(f*x+e))/sin(f*x+e),I)+693*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)-693*I*(1/(cos(f*x+e)+1))^(1/2)*(cos
(f*x+e)/(cos(f*x+e)+1))^(1/2)*cos(f*x+e)^2*EllipticE(I*(-1+cos(f*x+e))/sin(f*x+e),I)+462*I*(1/(cos(f*x+e)+1))^
(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*cos(f*x+e)*EllipticF(I*(-1+cos(f*x+e))/sin(f*x+e),I)-462*I*(1/(cos(f*x
+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*cos(f*x+e)*EllipticE(I*(-1+cos(f*x+e))/sin(f*x+e),I)+693*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)-693*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)+462*I*(1/(cos(f*x+e)+1))^(1/2)*cos(f*x+e)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*E
llipticF(I*(-1+cos(f*x+e))/sin(f*x+e),I)*sin(f*x+e)-462*I*(1/(cos(f*x+e)+1))^(1/2)*cos(f*x+e)*(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))*(g*cos(f*x+e))^(3/2)*(-c*(-1+sin(f*x+e
)))^(7/2)/(cos(f*x+e)^5+sin(f*x+e)*cos(f*x+e)^4-5*cos(f*x+e)^4+4*sin(f*x+e)*cos(f*x+e)^3-8*cos(f*x+e)^3-12*cos
(f*x+e)^2*sin(f*x+e)+20*cos(f*x+e)^2-8*sin(f*x+e)*cos(f*x+e)+8*cos(f*x+e)+16*sin(f*x+e)-16)/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{7}{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))^(7/2)/(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)^(7/2)/(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 (3 \, c^{3} g \cos \left (f x + e\right )^{3} - 4 \, c^{3} g \cos \left (f x + e\right ) -{\left (c^{3} g \cos \left (f x + e\right )^{3} - 4 \, c^{3} 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))^(7/2)/(a+a*sin(f*x+e))^(5/2),x, algorithm="fricas")
[Out]
integral((3*c^3*g*cos(f*x + e)^3 - 4*c^3*g*cos(f*x + e) - (c^3*g*cos(f*x + e)^3 - 4*c^3*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))**(7/2)/(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{7}{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))^(7/2)/(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)^(7/2)/(a*sin(f*x + e) + a)^(5/2), x)