\(\int \frac {(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{9/2}}{(a+a \sin (e+f x))^{5/2}} \, dx\) [142]

Optimal result

Integrand size = 42, antiderivative size = 357 \[ \int \frac {(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{9/2}}{(a+a \sin (e+f x))^{5/2}} \, dx=\frac {418 c^5 (g \cos (e+f x))^{5/2}}{5 a^2 f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {1254 c^5 g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{5 a^2 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}+\frac {1254 c^4 (g \cos (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}}{35 a^2 f g \sqrt {a+a \sin (e+f x)}}+\frac {114 c^3 (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{7 a^2 f g \sqrt {a+a \sin (e+f x)}}+\frac {76 c^2 (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{5/2}}{5 a f g (a+a \sin (e+f x))^{3/2}}-\frac {4 c (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{7/2}}{5 f g (a+a \sin (e+f x))^{5/2}} \]

[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)

Rubi [A] (verified)

Time = 1.21 (sec) , antiderivative size = 357, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.119, Rules used = {2929, 2930, 2921, 2721, 2719} \[ \int \frac {(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{9/2}}{(a+a \sin (e+f x))^{5/2}} \, dx=\frac {418 c^5 (g \cos (e+f x))^{5/2}}{5 a^2 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {1254 c^5 g \sqrt {\cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {g \cos (e+f x)}}{5 a^2 f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {1254 c^4 \sqrt {c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{35 a^2 f g \sqrt {a \sin (e+f x)+a}}+\frac {114 c^3 (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{7 a^2 f g \sqrt {a \sin (e+f x)+a}}+\frac {76 c^2 (c-c \sin (e+f x))^{5/2} (g \cos (e+f x))^{5/2}}{5 a f g (a \sin (e+f x)+a)^{3/2}}-\frac {4 c (c-c \sin (e+f x))^{7/2} (g \cos (e+f x))^{5/2}}{5 f g (a \sin (e+f x)+a)^{5/2}} \]

[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 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 2930

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] &&
EqQ[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*} \text {integral}& = -\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] (verified)

Time = 15.40 (sec) , antiderivative size = 356, normalized size of antiderivative = 1.00 \[ \int \frac {(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{9/2}}{(a+a \sin (e+f x))^{5/2}} \, dx=\frac {1254 (g \cos (e+f x))^{3/2} E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 (c-c \sin (e+f x))^{9/2}}{5 f \cos ^{\frac {3}{2}}(e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^9 (a (1+\sin (e+f x)))^{5/2}}+\frac {(g \cos (e+f x))^{3/2} \sec (e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 (c-c \sin (e+f x))^{9/2} \left (\frac {736}{5}+\frac {221}{14} \cos (e+f x)-\frac {1}{14} \cos (3 (e+f x))+\frac {128 \sin \left (\frac {1}{2} (e+f x)\right )}{5 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3}-\frac {64}{5 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2}-\frac {1472 \sin \left (\frac {1}{2} (e+f x)\right )}{5 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )}-\frac {7}{5} \sin (2 (e+f x))\right )}{f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^9 (a (1+\sin (e+f x)))^{5/2}} \]

[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)
)

Maple [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 3.82 (sec) , antiderivative size = 2627, normalized size of antiderivative = 7.36

method	result	size
default	\(\text {Expression too large to display}\)	\(2627\)

[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,method=_RETURNVERBOSE)

[Out]

-2/35/f*(g*cos(f*x+e))^(1/2)*(-c*(sin(f*x+e)-1))^(1/2)*g*c^4/(cos(f*x+e)-sin(f*x+e)+1)/(a*(1+sin(f*x+e)))^(1/2
)/a^2*(-7189-700*ln(2*(2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)*cos(f*x+e)+2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2
)-cos(f*x+e)+1)/(1+cos(f*x+e)))*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(3/2)*cos(f*x+e)*sin(f*x+e)+700*ln((2*(-cos(f*x
+e)/(1+cos(f*x+e))^2)^(1/2)*cos(f*x+e)+2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)-cos(f*x+e)+1)/(1+cos(f*x+e)))*(-
cos(f*x+e)/(1+cos(f*x+e))^2)^(3/2)*cos(f*x+e)*sin(f*x+e)-700*ln(2*(2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)*cos(
f*x+e)+2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)-cos(f*x+e)+1)/(1+cos(f*x+e)))*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(3/
2)*sec(f*x+e)*tan(f*x+e)+700*ln((2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)*cos(f*x+e)+2*(-cos(f*x+e)/(1+cos(f*x+e
))^2)^(1/2)-cos(f*x+e)+1)/(1+cos(f*x+e)))*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(3/2)*sec(f*x+e)*tan(f*x+e)-280*cos(f
*x+e)^2+49*cos(f*x+e)^2*sin(f*x+e)-224*tan(f*x+e)-448*sec(f*x+e)*tan(f*x+e)+224*sec(f*x+e)+1582*cos(f*x+e)-181
3*sin(f*x+e)-44*cos(f*x+e)^3-700*ln(2*(2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)*cos(f*x+e)+2*(-cos(f*x+e)/(1+cos
(f*x+e))^2)^(1/2)-cos(f*x+e)+1)/(1+cos(f*x+e)))*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(3/2)*cos(f*x+e)^2+700*ln((2*(-
cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)*cos(f*x+e)+2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)-cos(f*x+e)+1)/(1+cos(f*x+
e)))*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(3/2)*cos(f*x+e)^2-2800*ln(2*(2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)*cos(f
*x+e)+2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)-cos(f*x+e)+1)/(1+cos(f*x+e)))*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(3/2
)*cos(f*x+e)+2800*ln((2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)*cos(f*x+e)+2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)
-cos(f*x+e)+1)/(1+cos(f*x+e)))*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(3/2)*cos(f*x+e)-2800*ln(2*(2*(-cos(f*x+e)/(1+co
s(f*x+e))^2)^(1/2)*cos(f*x+e)+2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)-cos(f*x+e)+1)/(1+cos(f*x+e)))*(-cos(f*x+e
)/(1+cos(f*x+e))^2)^(3/2)*sec(f*x+e)+2800*ln((2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)*cos(f*x+e)+2*(-cos(f*x+e)
/(1+cos(f*x+e))^2)^(1/2)-cos(f*x+e)+1)/(1+cos(f*x+e)))*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(3/2)*sec(f*x+e)+448*sec
(f*x+e)^2-8778*I*(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)+8778*I*(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)+
5*cos(f*x+e)^3*sin(f*x+e)-231*cos(f*x+e)*sin(f*x+e)+5*cos(f*x+e)^4-4389*I*(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)*cos(f*x+e)+4389*I*(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)*cos(f*x+e)+4389*I*(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)+4389*I*(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)-4389*I*(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)-4389*I*(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)-4389*I*
(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)*sin(f*x+e)+4
389*I*(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)*sin(f*
x+e)-4200*ln(2*(2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)*cos(f*x+e)+2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)-cos(f
*x+e)+1)/(1+cos(f*x+e)))*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(3/2)+4200*ln((2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)*
cos(f*x+e)+2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)-cos(f*x+e)+1)/(1+cos(f*x+e)))*(-cos(f*x+e)/(1+cos(f*x+e))^2)
^(3/2)+700*ln((2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)*cos(f*x+e)+2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)-cos(f*
x+e)+1)/(1+cos(f*x+e)))*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(3/2)*sec(f*x+e)^2+2100*ln((2*(-cos(f*x+e)/(1+cos(f*x+e
))^2)^(1/2)*cos(f*x+e)+2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)-cos(f*x+e)+1)/(1+cos(f*x+e)))*(-cos(f*x+e)/(1+co
s(f*x+e))^2)^(3/2)*tan(f*x+e)-2100*ln(2*(2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)*cos(f*x+e)+2*(-cos(f*x+e)/(1+c
os(f*x+e))^2)^(1/2)-cos(f*x+e)+1)/(1+cos(f*x+e)))*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(3/2)*tan(f*x+e)-700*ln(2*(2*
(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)*cos(f*x+e)+2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)-cos(f*x+e)+1)/(1+cos(f*
x+e)))*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(3/2)*sec(f*x+e)^2-2100*ln(2*(2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)*cos
(f*x+e)+2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)-cos(f*x+e)+1)/(1+cos(f*x+e)))*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(3
/2)*sin(f*x+e)+2100*ln((2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)*cos(f*x+e)+2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/
2)-cos(f*x+e)+1)/(1+cos(f*x+e)))*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(3/2)*sin(f*x+e))

Fricas [C] (verification not implemented)

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

Time = 0.15 (sec) , antiderivative size = 279, normalized size of antiderivative = 0.78 \[ \int \frac {(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{9/2}}{(a+a \sin (e+f x))^{5/2}} \, dx=-\frac {2 \, {\left (5 \, c^{4} g \cos \left (f x + e\right )^{4} - 192 \, c^{4} g \cos \left (f x + e\right )^{2} + 2814 \, c^{4} g + {\left (39 \, c^{4} g \cos \left (f x + e\right )^{2} + 3038 \, c^{4} g\right )} \sin \left (f x + e\right )\right )} \sqrt {g \cos \left (f x + e\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c} - 4389 \, {\left (-i \, \sqrt {2} c^{4} g \cos \left (f x + e\right )^{2} + 2 i \, \sqrt {2} c^{4} g \sin \left (f x + e\right ) + 2 i \, \sqrt {2} c^{4} g\right )} \sqrt {a c g} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) - 4389 \, {\left (i \, \sqrt {2} c^{4} g \cos \left (f x + e\right )^{2} - 2 i \, \sqrt {2} c^{4} g \sin \left (f x + e\right ) - 2 i \, \sqrt {2} c^{4} g\right )} \sqrt {a c g} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right )}{35 \, {\left (a^{3} f \cos \left (f x + e\right )^{2} - 2 \, a^{3} f \sin \left (f x + e\right ) - 2 \, a^{3} f\right )}} \]

[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]

-1/35*(2*(5*c^4*g*cos(f*x + e)^4 - 192*c^4*g*cos(f*x + e)^2 + 2814*c^4*g + (39*c^4*g*cos(f*x + e)^2 + 3038*c^4
*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) - 4389*(-I*sqrt(2)*c
^4*g*cos(f*x + e)^2 + 2*I*sqrt(2)*c^4*g*sin(f*x + e) + 2*I*sqrt(2)*c^4*g)*sqrt(a*c*g)*weierstrassZeta(-4, 0, w
eierstrassPInverse(-4, 0, cos(f*x + e) + I*sin(f*x + e))) - 4389*(I*sqrt(2)*c^4*g*cos(f*x + e)^2 - 2*I*sqrt(2)
*c^4*g*sin(f*x + e) - 2*I*sqrt(2)*c^4*g)*sqrt(a*c*g)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(f*x
+ e) - I*sin(f*x + e))))/(a^3*f*cos(f*x + e)^2 - 2*a^3*f*sin(f*x + e) - 2*a^3*f)

Sympy [F(-1)]

Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{9/2}}{(a+a \sin (e+f x))^{5/2}} \, dx=\text {Timed out} \]

[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

Maxima [F]

\[ \int \frac {(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{9/2}}{(a+a \sin (e+f x))^{5/2}} \, dx=\int { \frac {\left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {9}{2}}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]

[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)

Giac [F(-1)]

Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{9/2}}{(a+a \sin (e+f x))^{5/2}} \, dx=\text {Timed out} \]

[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

Mupad [F(-1)]

Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{9/2}}{(a+a \sin (e+f x))^{5/2}} \, dx=\int \frac {{\left (g\,\cos \left (e+f\,x\right )\right )}^{3/2}\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{9/2}}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{5/2}} \,d x \]

[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)