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

[Out]

76/5*c^2*(g*cos(f*x+e))^(5/2)*(c-c*sin(f*x+e))^(5/2)/a/f/g/(a+a*sin(f*x+e))^(3/2)-4/5*c*(g*cos(f*x+e))^(5/2)*(
c-c*sin(f*x+e))^(7/2)/f/g/(a+a*sin(f*x+e))^(5/2)+114/7*c^3*(g*cos(f*x+e))^(5/2)*(c-c*sin(f*x+e))^(3/2)/a^2/f/g
/(a+a*sin(f*x+e))^(1/2)+418/5*c^5*(g*cos(f*x+e))^(5/2)/a^2/f/g/(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e))^(1/2)+1
254/5*c^5*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)/a^2/f/(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e))^(1/2)+1254/35*c^4*(g*cos(f*x+e))^(5/2)
*(c-c*sin(f*x+e))^(1/2)/a^2/f/g/(a+a*sin(f*x+e))^(1/2)

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Rubi [A]  time = 1.73, antiderivative size = 357, normalized size of antiderivative = 1.00, 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 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]

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 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 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]

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*}

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Mathematica [A]  time = 6.71, size = 356, normalized size = 1.00 \[ \frac {1254 E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) (c-c \sin (e+f x))^{9/2} (g \cos (e+f x))^{3/2} \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^5}{5 f \cos ^{\frac {3}{2}}(e+f x) (a (\sin (e+f x)+1))^{5/2} \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^9}+\frac {\sec (e+f x) (c-c \sin (e+f x))^{9/2} (g \cos (e+f x))^{3/2} \left (-\frac {7}{5} \sin (2 (e+f x))+\frac {221}{14} \cos (e+f x)-\frac {1}{14} \cos (3 (e+f x))-\frac {1472 \sin \left (\frac {1}{2} (e+f x)\right )}{5 \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )}-\frac {64}{5 \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^2}+\frac {128 \sin \left (\frac {1}{2} (e+f x)\right )}{5 \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^3}+\frac {736}{5}\right ) \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^5}{f (a (\sin (e+f x)+1))^{5/2} \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^9} \]

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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fricas [F]  time = 0.57, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (c^{4} g \cos \left (f x + e\right )^{5} - 8 \, c^{4} g \cos \left (f x + e\right )^{3} + 8 \, c^{4} g \cos \left (f x + e\right ) + 4 \, {\left (c^{4} g \cos \left (f x + e\right )^{3} - 2 \, c^{4} g \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {g \cos \left (f x + e\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{3 \, a^{3} \cos \left (f x + e\right )^{2} - 4 \, a^{3} + {\left (a^{3} \cos \left (f x + e\right )^{2} - 4 \, a^{3}\right )} \sin \left (f x + e\right )}, x\right ) \]

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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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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]

Timed out

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maple [C]  time = 0.65, size = 3654, normalized size = 10.24 \[ \text {output too large to display} \]

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))*(8554*cos(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*(-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)-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)-1400*cos(f*x+e)^4*(-cos(f*x+e)/(c
os(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)+39*sin(f*x+e)*cos(f*x+e)^5-4389*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)-192*cos(
f*x+e)^5-1575*cos(f*x+e)^3-1582*cos(f*x+e)^4+44*cos(f*x+e)^6+5*cos(f*x+e)^7-700*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*(-c
os(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)-1)/sin(f*x+e)^2)+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)+700*(-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)*sin(f*x+e)*cos
(f*x+e)^4-700*(-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)-co
s(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)^4+1400*c
os(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)-co
s(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)*l
n(-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)+9002*cos(f*x+e)^2*sin(f*x+e)+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)*(-cos(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)+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)-1351*sin(f*x+e)*cos(f*x+e)^3+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)/sin(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)+6300*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)-630
0*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)*co
s(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)+231*sin(f*x+e)*cos(f*x+e)^4-5*cos(f*x+e)^6*sin(f*x
+e)-8778*I*sin(f*x+e)*cos(f*x+e)*EllipticE(I*(-1+cos(f*x+e))/sin(f*x+e),I)*(1/(cos(f*x+e)+1))^(1/2)*(cos(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)*EllipticF
(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)/(cos(f*x+
e)+1))^(1/2)*EllipticF(I*(-1+cos(f*x+e))/sin(f*x+e),I)*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)+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
)+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+cos(f*x+e))/s
in(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)+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)-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)-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)+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))*(g*cos(f*x+e))^(3/2)*(-c*(sin(f*x+e)-1))^(9
/2)/(cos(f*x+e)^6-sin(f*x+e)*cos(f*x+e)^5+5*cos(f*x+e)^5+6*sin(f*x+e)*cos(f*x+e)^4-18*cos(f*x+e)^4+12*sin(f*x+
e)*cos(f*x+e)^3-20*cos(f*x+e)^3-32*cos(f*x+e)^2*sin(f*x+e)+48*cos(f*x+e)^2-16*sin(f*x+e)*cos(f*x+e)+16*cos(f*x
+e)+32*sin(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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {9}{2}}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}}}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (g\,\cos \left (e+f\,x\right )\right )}^{3/2}\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{9/2}}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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]

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)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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