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3.123 \(\int \frac{(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{9/2}} \, dx\)

Optimal. Leaf size=300 \[ -\frac{308 a^4 (g \cos (e+f x))^{5/2}}{39 c^3 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}+\frac{44 a^3 \sqrt{a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{39 c^2 f g (c-c \sin (e+f x))^{5/2}}+\frac{154 a^4 g \sqrt{\cos (e+f x)} E\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{g \cos (e+f x)}}{13 c^4 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{20 a^2 (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{39 c f g (c-c \sin (e+f x))^{7/2}}+\frac{4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{13 f g (c-c \sin (e+f x))^{9/2}} \]

[Out]

(4*a*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(5/2))/(13*f*g*(c - c*Sin[e + f*x])^(9/2)) - (20*a^2*(g*Cos[e
 + f*x])^(5/2)*(a + a*Sin[e + f*x])^(3/2))/(39*c*f*g*(c - c*Sin[e + f*x])^(7/2)) + (44*a^3*(g*Cos[e + f*x])^(5
/2)*Sqrt[a + a*Sin[e + f*x]])/(39*c^2*f*g*(c - c*Sin[e + f*x])^(5/2)) - (308*a^4*(g*Cos[e + f*x])^(5/2))/(39*c
^3*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) + (154*a^4*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x
]]*EllipticE[(e + f*x)/2, 2])/(13*c^4*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])

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Rubi [A]  time = 1.53357, antiderivative size = 300, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 42, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2850, 2842, 2640, 2639} \[ -\frac{308 a^4 (g \cos (e+f x))^{5/2}}{39 c^3 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}+\frac{44 a^3 \sqrt{a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{39 c^2 f g (c-c \sin (e+f x))^{5/2}}+\frac{154 a^4 g \sqrt{\cos (e+f x)} E\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{g \cos (e+f x)}}{13 c^4 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{20 a^2 (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{39 c f g (c-c \sin (e+f x))^{7/2}}+\frac{4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{13 f g (c-c \sin (e+f x))^{9/2}} \]

Antiderivative was successfully verified.

[In]

Int[((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(7/2))/(c - c*Sin[e + f*x])^(9/2),x]

[Out]

(4*a*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(5/2))/(13*f*g*(c - c*Sin[e + f*x])^(9/2)) - (20*a^2*(g*Cos[e
 + f*x])^(5/2)*(a + a*Sin[e + f*x])^(3/2))/(39*c*f*g*(c - c*Sin[e + f*x])^(7/2)) + (44*a^3*(g*Cos[e + f*x])^(5
/2)*Sqrt[a + a*Sin[e + f*x]])/(39*c^2*f*g*(c - c*Sin[e + f*x])^(5/2)) - (308*a^4*(g*Cos[e + f*x])^(5/2))/(39*c
^3*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) + (154*a^4*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x
]]*EllipticE[(e + f*x)/2, 2])/(13*c^4*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])

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 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} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{9/2}} \, dx &=\frac{4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{13 f g (c-c \sin (e+f x))^{9/2}}-\frac{(15 a) \int \frac{(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{7/2}} \, dx}{13 c}\\ &=\frac{4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{13 f g (c-c \sin (e+f x))^{9/2}}-\frac{20 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{39 c f g (c-c \sin (e+f x))^{7/2}}+\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))^{5/2}} \, dx}{39 c^2}\\ &=\frac{4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{13 f g (c-c \sin (e+f x))^{9/2}}-\frac{20 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{39 c f g (c-c \sin (e+f x))^{7/2}}+\frac{44 a^3 (g \cos (e+f x))^{5/2} \sqrt{a+a \sin (e+f x)}}{39 c^2 f g (c-c \sin (e+f x))^{5/2}}-\frac{\left (77 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}{39 c^3}\\ &=\frac{4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{13 f g (c-c \sin (e+f x))^{9/2}}-\frac{20 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{39 c f g (c-c \sin (e+f x))^{7/2}}+\frac{44 a^3 (g \cos (e+f x))^{5/2} \sqrt{a+a \sin (e+f x)}}{39 c^2 f g (c-c \sin (e+f x))^{5/2}}-\frac{308 a^4 (g \cos (e+f x))^{5/2}}{39 c^3 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}+\frac{\left (77 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}{13 c^4}\\ &=\frac{4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{13 f g (c-c \sin (e+f x))^{9/2}}-\frac{20 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{39 c f g (c-c \sin (e+f x))^{7/2}}+\frac{44 a^3 (g \cos (e+f x))^{5/2} \sqrt{a+a \sin (e+f x)}}{39 c^2 f g (c-c \sin (e+f x))^{5/2}}-\frac{308 a^4 (g \cos (e+f x))^{5/2}}{39 c^3 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}+\frac{\left (77 a^4 g \cos (e+f x)\right ) \int \sqrt{g \cos (e+f x)} \, dx}{13 c^4 \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}}{13 f g (c-c \sin (e+f x))^{9/2}}-\frac{20 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{39 c f g (c-c \sin (e+f x))^{7/2}}+\frac{44 a^3 (g \cos (e+f x))^{5/2} \sqrt{a+a \sin (e+f x)}}{39 c^2 f g (c-c \sin (e+f x))^{5/2}}-\frac{308 a^4 (g \cos (e+f x))^{5/2}}{39 c^3 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}+\frac{\left (77 a^4 g \sqrt{\cos (e+f x)} \sqrt{g \cos (e+f x)}\right ) \int \sqrt{\cos (e+f x)} \, dx}{13 c^4 \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}}{13 f g (c-c \sin (e+f x))^{9/2}}-\frac{20 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{39 c f g (c-c \sin (e+f x))^{7/2}}+\frac{44 a^3 (g \cos (e+f x))^{5/2} \sqrt{a+a \sin (e+f x)}}{39 c^2 f g (c-c \sin (e+f x))^{5/2}}-\frac{308 a^4 (g \cos (e+f x))^{5/2}}{39 c^3 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}+\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 )}{13 c^4 f \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}\\ \end{align*}

Mathematica [A]  time = 6.69192, size = 464, normalized size = 1.55 \[ \frac{154 E\left (\left .\frac{1}{2} (e+f x)\right |2\right ) (a (\sin (e+f x)+1))^{7/2} (g \cos (e+f x))^{3/2} \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^9}{13 f \cos ^{\frac{3}{2}}(e+f x) (c-c \sin (e+f x))^{9/2} \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^7}+\frac{\sec (e+f x) (a (\sin (e+f x)+1))^{7/2} (g \cos (e+f x))^{3/2} \left (-\frac{256 \sin \left (\frac{1}{2} (e+f x)\right )}{13 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )}+\frac{160 \sin \left (\frac{1}{2} (e+f x)\right )}{13 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^3}-\frac{448 \sin \left (\frac{1}{2} (e+f x)\right )}{39 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^5}+\frac{64 \sin \left (\frac{1}{2} (e+f x)\right )}{13 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^7}+\frac{80}{13 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^2}-\frac{224}{39 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^4}+\frac{32}{13 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^6}-\frac{128}{13}\right ) \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^9}{f (c-c \sin (e+f x))^{9/2} \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^7} \]

Antiderivative was successfully verified.

[In]

Integrate[((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(7/2))/(c - c*Sin[e + f*x])^(9/2),x]

[Out]

(154*(g*Cos[e + f*x])^(3/2)*EllipticE[(e + f*x)/2, 2]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9*(a*(1 + Sin[e +
f*x]))^(7/2))/(13*f*Cos[e + f*x]^(3/2)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(9/2)) + (
(g*Cos[e + f*x])^(3/2)*Sec[e + f*x]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9*(-128/13 + 32/(13*(Cos[(e + f*x)/2
] - Sin[(e + f*x)/2])^6) - 224/(39*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^4) + 80/(13*(Cos[(e + f*x)/2] - Sin[(
e + f*x)/2])^2) + (64*Sin[(e + f*x)/2])/(13*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7) - (448*Sin[(e + f*x)/2])/
(39*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5) + (160*Sin[(e + f*x)/2])/(13*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2]
)^3) - (256*Sin[(e + f*x)/2])/(13*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])))*(a*(1 + Sin[e + f*x]))^(7/2))/(f*(Co
s[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(9/2))

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Maple [C]  time = 0.402, size = 4829, normalized size = 16.1 \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)*(a+a*sin(f*x+e))^(7/2)/(c-c*sin(f*x+e))^(9/2),x)

[Out]

-2/39/f*(-1+cos(f*x+e))*(-2772*I*sin(f*x+e)*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)+1848*I*(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)*cos(f*x+e)*sin(f*x+e)-1848*I*(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)*cos(f*x+e)*sin(f*x+e)-231*I*sin(f*x+e)*cos(f
*x+e)^5*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*EllipticE(I*(-1+cos(f*x+e))/sin(f*x+e),I)*(1/(cos(f*x+e)+1))^(1/2)+1
155*I*sin(f*x+e)*cos(f*x+e)^4*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*EllipticF(I*(-1+cos(f*x+e))/sin(f*x+e),I)*(1/(
cos(f*x+e)+1))^(1/2)-1155*I*sin(f*x+e)*cos(f*x+e)^4*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*(1/(cos(f*x+e)+1))^(1/2)
*EllipticE(I*(-1+cos(f*x+e))/sin(f*x+e),I)-546*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(3/2)*ln(-2*(2*cos(f*x+e)^2*(-co
s(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)*sin(f*x+e)*cos(f*x+e)^5+1092*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*(-co
s(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)-624*(-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)+624*(-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)-1800*cos(f*x+e)^2-858*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)+858*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)+78*sin(f*x+e)*cos(f*x+e)^6*(-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)-78*sin(f*x+e)*cos(f*x+e)^6*(-cos(f*x+e)/(cos(f*x+e)+1)^2)^(3/2)*ln(-2*(2*cos(f*x+e)^2*(-cos(f*x+e)/(co
s(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)-237*cos
(f*x+e)^5+332*cos(f*x+e)^3-284*sin(f*x+e)*cos(f*x+e)^3+39*sin(f*x+e)*cos(f*x+e)^5-660*sin(f*x+e)*cos(f*x+e)^4+
2184*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)-2184*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)+546*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)-546*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)-2184*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)+2184*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)+624*ln(-2*(2*cos(f*x+e)^2*(-cos(f*x+e)/(co
s(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)-624*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)-2184*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)+2184*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)+1896*cos(f*x+e)^2*sin(f*x+e)-39*cos(f*x+e)
^6+1472*cos(f*x+e)^4+546*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)-546*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)+2496*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*(-c
os(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)-2496*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)+2184*sin(f*x+e)*cos(f*x+e)*ln(-2*(2*c
os(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)-2184*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)+924*I*cos(f*x+e)^3*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*EllipticE(I*(-1+
cos(f*x+e))/sin(f*x+e),I)*(1/(cos(f*x+e)+1))^(1/2)+78*cos(f*x+e)^7*(-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)-78*cos(f*x+e)^7*(-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)-109
2*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)-c
os(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*sin(f*x+e)*cos(f*x+e)^5
*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*EllipticF(I*(-1+cos(f*x+e))/sin(f*x+e),I)*(1/(cos(f*x+e)+1))^(1/2)+2772*I*s
in(f*x+e)*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)-1848*I*(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))/si
n(f*x+e),I)*cos(f*x+e)+1848*I*(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)*cos(f*x+e)+462*I*(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)*cos(f*x+e)^5-462*I*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*Ell
ipticF(I*(-1+cos(f*x+e))/sin(f*x+e),I)*cos(f*x+e)^5-2772*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)+2772*I*cos(f*x+e)^2*(1/(cos(f*x+e)+1))^(1/2)*(c
os(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)^6*(cos(f*x+e)/(cos(
f*x+e)+1))^(1/2)*EllipticF(I*(-1+cos(f*x+e))/sin(f*x+e),I)*(1/(cos(f*x+e)+1))^(1/2)-231*I*cos(f*x+e)^6*(cos(f*
x+e)/(cos(f*x+e)+1))^(1/2)*EllipticE(I*(-1+cos(f*x+e))/sin(f*x+e),I)*(1/(cos(f*x+e)+1))^(1/2)-2541*I*cos(f*x+e
)^4*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*EllipticF(I*(-1+cos(f*x+e))/sin(f*x+e),I)*(1/(cos(f*x+e)+1))^(1/2)+2541*
I*cos(f*x+e)^4*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*EllipticE(I*(-1+cos(f*x+e))/sin(f*x+e),I)*(1/(cos(f*x+e)+1))^
(1/2)-924*I*cos(f*x+e)^3*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*EllipticF(I*(-1+cos(f*x+e))/sin(f*x+e),I)*(1/(cos(f
*x+e)+1))^(1/2)+546*(-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)*sin(f*x+e)*cos(f*x+e)^5)
*(g*cos(f*x+e))^(3/2)*(a*(1+sin(f*x+e)))^(7/2)/(sin(f*x+e)*cos(f*x+e)^4-cos(f*x+e)^5+4*sin(f*x+e)*cos(f*x+e)^3
+5*cos(f*x+e)^4-12*cos(f*x+e)^2*sin(f*x+e)+8*cos(f*x+e)^3-8*sin(f*x+e)*cos(f*x+e)-20*cos(f*x+e)^2+16*sin(f*x+e
)-8*cos(f*x+e)+16)/(-c*(-1+sin(f*x+e)))^(9/2)/sin(f*x+e)/cos(f*x+e)

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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 (a \sin \left (f x + e\right ) + a\right )}^{\frac{7}{2}}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac{9}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(7/2)/(c-c*sin(f*x+e))^(9/2),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)^(9/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{{\left (3 \, a^{3} g \cos \left (f x + e\right )^{3} - 4 \, a^{3} g \cos \left (f x + e\right ) +{\left (a^{3} g \cos \left (f x + e\right )^{3} - 4 \, a^{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}}{5 \, c^{5} \cos \left (f x + e\right )^{4} - 20 \, c^{5} \cos \left (f x + e\right )^{2} + 16 \, c^{5} -{\left (c^{5} \cos \left (f x + e\right )^{4} - 12 \, c^{5} \cos \left (f x + e\right )^{2} + 16 \, c^{5}\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)*(a+a*sin(f*x+e))^(7/2)/(c-c*sin(f*x+e))^(9/2),x, algorithm="fricas")

[Out]

integral(-(3*a^3*g*cos(f*x + e)^3 - 4*a^3*g*cos(f*x + e) + (a^3*g*cos(f*x + e)^3 - 4*a^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)/(5*c^5*cos(f*x + e)^4 - 20*c^
5*cos(f*x + e)^2 + 16*c^5 - (c^5*cos(f*x + e)^4 - 12*c^5*cos(f*x + e)^2 + 16*c^5)*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)*(a+a*sin(f*x+e))**(7/2)/(c-c*sin(f*x+e))**(9/2),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 (a \sin \left (f x + e\right ) + a\right )}^{\frac{7}{2}}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac{9}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(7/2)/(c-c*sin(f*x+e))^(9/2),x, algorithm="giac")

[Out]

integrate((g*cos(f*x + e))^(3/2)*(a*sin(f*x + e) + a)^(7/2)/(-c*sin(f*x + e) + c)^(9/2), x)

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