\(\int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{9/2}} \, dx\) [113]
Optimal result
Integrand size = 42, antiderivative size = 300 \[
\int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{9/2}} \, dx=\frac {4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{13 f g (c-c \sin (e+f x))^{9/2}}-\frac {44 a^2 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{117 c f g (c-c \sin (e+f x))^{7/2}}+\frac {308 a^3 (g \cos (e+f x))^{5/2}}{585 c^2 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}-\frac {154 a^3 (g \cos (e+f x))^{5/2}}{195 c^3 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}+\frac {154 a^3 g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{195 c^4 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}
\]
[Out]
4/13*a*(g*cos(f*x+e))^(5/2)*(a+a*sin(f*x+e))^(3/2)/f/g/(c-c*sin(f*x+e))^(9/2)+308/585*a^3*(g*cos(f*x+e))^(5/2)
/c^2/f/g/(c-c*sin(f*x+e))^(5/2)/(a+a*sin(f*x+e))^(1/2)-154/195*a^3*(g*cos(f*x+e))^(5/2)/c^3/f/g/(c-c*sin(f*x+e
))^(3/2)/(a+a*sin(f*x+e))^(1/2)-44/117*a^2*(g*cos(f*x+e))^(5/2)*(a+a*sin(f*x+e))^(1/2)/c/f/g/(c-c*sin(f*x+e))^
(7/2)+154/195*a^3*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)/c^4/f/(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e))^(1/2)
Rubi [A] (verified)
Time = 1.07 (sec) , antiderivative size = 300, normalized size of antiderivative = 1.00, number of
steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.119, Rules used = {2929, 2931, 2921, 2721, 2719}
\[
\int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{9/2}} \, dx=\frac {154 a^3 g \sqrt {\cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {g \cos (e+f x)}}{195 c^4 f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {154 a^3 (g \cos (e+f x))^{5/2}}{195 c^3 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}+\frac {308 a^3 (g \cos (e+f x))^{5/2}}{585 c^2 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}-\frac {44 a^2 \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{117 c f g (c-c \sin (e+f x))^{7/2}}+\frac {4 a (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{13 f g (c-c \sin (e+f x))^{9/2}}
\]
[In]
Int[((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(5/2))/(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])^(3/2))/(13*f*g*(c - c*Sin[e + f*x])^(9/2)) - (44*a^2*(g*Cos[e
+ f*x])^(5/2)*Sqrt[a + a*Sin[e + f*x]])/(117*c*f*g*(c - c*Sin[e + f*x])^(7/2)) + (308*a^3*(g*Cos[e + f*x])^(5
/2))/(585*c^2*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2)) - (154*a^3*(g*Cos[e + f*x])^(5/2))/(195
*c^3*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(3/2)) + (154*a^3*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f
*x]]*EllipticE[(e + f*x)/2, 2])/(195*c^4*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])
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 2931
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*((c + d*Sin[e + f*x])^
n/(a*f*g*(2*m + p + 1))), x] + Dist[(m + n + p + 1)/(a*(2*m + p + 1)), 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] && E
qQ[a^2 - b^2, 0] && LtQ[m, -1] && NeQ[2*m + p + 1, 0] && !LtQ[m, n, -1] && IntegersQ[2*m, 2*n, 2*p]
Rubi steps \begin{align*}
\text {integral}& = \frac {4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{13 f g (c-c \sin (e+f x))^{9/2}}-\frac {(11 a) \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2}}{(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))^{3/2}}{13 f g (c-c \sin (e+f x))^{9/2}}-\frac {44 a^2 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{117 c f g (c-c \sin (e+f x))^{7/2}}+\frac {\left (77 a^2\right ) \int \frac {(g \cos (e+f x))^{3/2} \sqrt {a+a \sin (e+f x)}}{(c-c \sin (e+f x))^{5/2}} \, dx}{117 c^2} \\ & = \frac {4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{13 f g (c-c \sin (e+f x))^{9/2}}-\frac {44 a^2 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{117 c f g (c-c \sin (e+f x))^{7/2}}+\frac {308 a^3 (g \cos (e+f x))^{5/2}}{585 c^2 f g \sqrt {a+a \sin (e+f x)} (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}{195 c^3} \\ & = \frac {4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{13 f g (c-c \sin (e+f x))^{9/2}}-\frac {44 a^2 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{117 c f g (c-c \sin (e+f x))^{7/2}}+\frac {308 a^3 (g \cos (e+f x))^{5/2}}{585 c^2 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}-\frac {154 a^3 (g \cos (e+f x))^{5/2}}{195 c^3 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}+\frac {\left (77 a^3\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}{195 c^4} \\ & = \frac {4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{13 f g (c-c \sin (e+f x))^{9/2}}-\frac {44 a^2 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{117 c f g (c-c \sin (e+f x))^{7/2}}+\frac {308 a^3 (g \cos (e+f x))^{5/2}}{585 c^2 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}-\frac {154 a^3 (g \cos (e+f x))^{5/2}}{195 c^3 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}+\frac {\left (77 a^3 g \cos (e+f x)\right ) \int \sqrt {g \cos (e+f x)} \, dx}{195 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))^{3/2}}{13 f g (c-c \sin (e+f x))^{9/2}}-\frac {44 a^2 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{117 c f g (c-c \sin (e+f x))^{7/2}}+\frac {308 a^3 (g \cos (e+f x))^{5/2}}{585 c^2 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}-\frac {154 a^3 (g \cos (e+f x))^{5/2}}{195 c^3 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}+\frac {\left (77 a^3 g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)}\right ) \int \sqrt {\cos (e+f x)} \, dx}{195 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))^{3/2}}{13 f g (c-c \sin (e+f x))^{9/2}}-\frac {44 a^2 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{117 c f g (c-c \sin (e+f x))^{7/2}}+\frac {308 a^3 (g \cos (e+f x))^{5/2}}{585 c^2 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}-\frac {154 a^3 (g \cos (e+f x))^{5/2}}{195 c^3 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}+\frac {154 a^3 g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{195 c^4 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \\
\end{align*}
Mathematica [A] (verified)
Time = 10.49 (sec) , antiderivative size = 464, normalized size of antiderivative = 1.55
\[
\int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{9/2}} \, dx=\frac {154 (g \cos (e+f x))^{3/2} E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^9 (a (1+\sin (e+f x)))^{5/2}}{195 f \cos ^{\frac {3}{2}}(e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 (c-c \sin (e+f x))^{9/2}}+\frac {(g \cos (e+f x))^{3/2} \sec (e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^9 \left (-\frac {154}{195}+\frac {16}{13 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^6}-\frac {232}{117 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}+\frac {236}{195 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^2}+\frac {32 \sin \left (\frac {1}{2} (e+f x)\right )}{13 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7}-\frac {464 \sin \left (\frac {1}{2} (e+f x)\right )}{117 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5}+\frac {472 \sin \left (\frac {1}{2} (e+f x)\right )}{195 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^3}-\frac {308 \sin \left (\frac {1}{2} (e+f x)\right )}{195 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )}\right ) (a (1+\sin (e+f x)))^{5/2}}{f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 (c-c \sin (e+f x))^{9/2}}
\]
[In]
Integrate[((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(5/2))/(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]))^(5/2))/(195*f*Cos[e + f*x]^(3/2)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5*(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*(-154/195 + 16/(13*(Cos[(e + f*x)
/2] - Sin[(e + f*x)/2])^6) - 232/(117*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^4) + 236/(195*(Cos[(e + f*x)/2] -
Sin[(e + f*x)/2])^2) + (32*Sin[(e + f*x)/2])/(13*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7) - (464*Sin[(e + f*x)
/2])/(117*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5) + (472*Sin[(e + f*x)/2])/(195*(Cos[(e + f*x)/2] - Sin[(e +
f*x)/2])^3) - (308*Sin[(e + f*x)/2])/(195*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])))*(a*(1 + Sin[e + f*x]))^(5/2)
)/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5*(c - c*Sin[e + f*x])^(9/2))
Maple [C] (warning: unable to verify)
Result contains complex when optimal does not.
Time = 2.42 (sec) , antiderivative size = 2675, normalized size of antiderivative =
8.92
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\(\text {Expression too large to display}\) |
\(2675\) |
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[In]
int((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(5/2)/(c-c*sin(f*x+e))^(9/2),x,method=_RETURNVERBOSE)
[Out]
1/1170/f*(a*(1+sin(f*x+e)))^(1/2)*(g*cos(f*x+e))^(1/2)*g*a^2/(1+cos(f*x+e))^3/(cos(f*x+e)^2+2*sin(f*x+e)-2)/(-
cos(f*x+e)/(1+cos(f*x+e))^2)^(3/2)/(-c*(sin(f*x+e)-1))^(1/2)/c^4*(585*cos(f*x+e)^2*sin(f*x+e)*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)))-585
*cos(f*x+e)^2*sin(f*x+e)*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)))+924*cos(f*x+e)^3*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)-2256*(-cos(f*x+e
)/(1+cos(f*x+e))^2)^(1/2)-2340*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)))+2340*ln((2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)*cos(f*x+e)+2*(-c
os(f*x+e)/(1+cos(f*x+e))^2)^(1/2)-cos(f*x+e)+1)/(1+cos(f*x+e)))+2340*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)))*sin(f*x+e)-2340*ln((2*(-co
s(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)
))*sin(f*x+e)+5136*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)*sin(f*x+e)+1440*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)*t
an(f*x+e)+1440*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)*sec(f*x+e)+7392*I*sin(f*x+e)*(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)/(1+cos(f*x+e))^2)^(1/2)+1848
*I*cos(f*x+e)^2*sin(f*x+e)*(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)/(1+cos(f*x+e))^2)^(1/2)-1848*I*cos(f*x+e)^2*sin(f*x+e)*(1/(1+cos(f*x+e)))^(1/2)*(c
os(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)/(1+cos(f*x+e))^2)^(1/2)-27
72*I*cos(f*x+e)*sin(f*x+e)*(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)/(1+cos(f*x+e))^2)^(1/2)+924*I*cos(f*x+e)^3*sin(f*x+e)*(1/(1+cos(f*x+e)))^(1/2)*(co
s(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)/(1+cos(f*x+e))^2)^(1/2)+277
2*I*cos(f*x+e)*sin(f*x+e)*(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)-c
ot(f*x+e)),I)*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)+1868*cos(f*x+e)^2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)+432*
cos(f*x+e)*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)*sin(f*x+e)-2340*sin(f*x+e)*cos(f*x+e)^2*(-cos(f*x+e)/(1+cos(f*
x+e))^2)^(1/2)-2752*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)*cos(f*x+e)-924*I*cos(f*x+e)^3*sin(f*x+e)*(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)/(1+cos(f*x+
e))^2)^(1/2)-1755*cos(f*x+e)^2*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)))+1755*cos(f*x+e)^2*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)))+7392*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)/(1+cos(f*x+e))^2)^(1/2)-7
392*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)/(1+cos(f*x+e))^2)^(1/2)-3696*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)/(1+cos(f*x+e))^2)^(1/2)*sec(f*x+e)-2772*I*cos(f*x+e)^3*(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)/(1+cos(f*x+e))^
2)^(1/2)+2772*I*cos(f*x+e)^3*(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)/(1+cos(f*x+e))^2)^(1/2)-5544*I*cos(f*x+e)^2*(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)/(1+cos(f*x+e))^2)^(1/2)+5544*I*cos(
f*x+e)^2*(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)*(-c
os(f*x+e)/(1+cos(f*x+e))^2)^(1/2)-7392*I*sin(f*x+e)*(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)/(1+cos(f*x+e))^2)^(1/2)+924*I*cos(f*x+e)*(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)/(1+cos(f*x+e))^
2)^(1/2)-924*I*cos(f*x+e)*(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)-c
ot(f*x+e)),I)*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)-3696*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)/(1+cos(f*x+e))^2)^(1/2)*tan(f*x+e)+3696*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)/(1+cos(f
*x+e))^2)^(1/2)*tan(f*x+e)+3696*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)/(1+cos(f*x+e))^2)^(1/2)*sec(f*x+e))
Fricas [C] (verification not implemented)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.14 (sec) , antiderivative size = 370, normalized size of antiderivative = 1.23
\[
\int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{9/2}} \, dx=\frac {2 \, {\left (339 \, a^{2} g \cos \left (f x + e\right )^{2} - 436 \, a^{2} g - {\left (231 \, a^{2} g \cos \left (f x + e\right )^{2} - 796 \, a^{2} g\right )} \sin \left (f x + e\right )\right )} \sqrt {g \cos \left (f x + e\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c} - 231 \, {\left (i \, \sqrt {2} a^{2} g \cos \left (f x + e\right )^{4} - 8 i \, \sqrt {2} a^{2} g \cos \left (f x + e\right )^{2} + 8 i \, \sqrt {2} a^{2} g + 4 \, {\left (i \, \sqrt {2} a^{2} g \cos \left (f x + e\right )^{2} - 2 i \, \sqrt {2} a^{2} g\right )} \sin \left (f x + e\right )\right )} \sqrt {a c g} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) - 231 \, {\left (-i \, \sqrt {2} a^{2} g \cos \left (f x + e\right )^{4} + 8 i \, \sqrt {2} a^{2} g \cos \left (f x + e\right )^{2} - 8 i \, \sqrt {2} a^{2} g + 4 \, {\left (-i \, \sqrt {2} a^{2} g \cos \left (f x + e\right )^{2} + 2 i \, \sqrt {2} a^{2} g\right )} \sin \left (f x + e\right )\right )} \sqrt {a c g} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right )}{585 \, {\left (c^{5} f \cos \left (f x + e\right )^{4} - 8 \, c^{5} f \cos \left (f x + e\right )^{2} + 8 \, c^{5} f + 4 \, {\left (c^{5} f \cos \left (f x + e\right )^{2} - 2 \, c^{5} f\right )} \sin \left (f x + e\right )\right )}}
\]
[In]
integrate((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(5/2)/(c-c*sin(f*x+e))^(9/2),x, algorithm="fricas")
[Out]
1/585*(2*(339*a^2*g*cos(f*x + e)^2 - 436*a^2*g - (231*a^2*g*cos(f*x + e)^2 - 796*a^2*g)*sin(f*x + e))*sqrt(g*c
os(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c) - 231*(I*sqrt(2)*a^2*g*cos(f*x + e)^4 - 8*I*sq
rt(2)*a^2*g*cos(f*x + e)^2 + 8*I*sqrt(2)*a^2*g + 4*(I*sqrt(2)*a^2*g*cos(f*x + e)^2 - 2*I*sqrt(2)*a^2*g)*sin(f*
x + e))*sqrt(a*c*g)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(f*x + e) + I*sin(f*x + e))) - 231*(-
I*sqrt(2)*a^2*g*cos(f*x + e)^4 + 8*I*sqrt(2)*a^2*g*cos(f*x + e)^2 - 8*I*sqrt(2)*a^2*g + 4*(-I*sqrt(2)*a^2*g*co
s(f*x + e)^2 + 2*I*sqrt(2)*a^2*g)*sin(f*x + e))*sqrt(a*c*g)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0,
cos(f*x + e) - I*sin(f*x + e))))/(c^5*f*cos(f*x + e)^4 - 8*c^5*f*cos(f*x + e)^2 + 8*c^5*f + 4*(c^5*f*cos(f*x +
e)^2 - 2*c^5*f)*sin(f*x + e))
Sympy [F(-1)]
Timed out. \[
\int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{9/2}} \, dx=\text {Timed out}
\]
[In]
integrate((g*cos(f*x+e))**(3/2)*(a+a*sin(f*x+e))**(5/2)/(c-c*sin(f*x+e))**(9/2),x)
[Out]
Timed out
Maxima [F]
\[
\int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{9/2}} \, dx=\int { \frac {\left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {9}{2}}} \,d x }
\]
[In]
integrate((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(5/2)/(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)^(5/2)/(-c*sin(f*x + e) + c)^(9/2), x)
Giac [F(-1)]
Timed out. \[
\int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{9/2}} \, dx=\text {Timed out}
\]
[In]
integrate((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(5/2)/(c-c*sin(f*x+e))^(9/2),x, algorithm="giac")
[Out]
Timed out
Mupad [F(-1)]
Timed out. \[
\int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{9/2}} \, dx=\int \frac {{\left (g\,\cos \left (e+f\,x\right )\right )}^{3/2}\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{5/2}}{{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{9/2}} \,d x
\]
[In]
int(((g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^(5/2))/(c - c*sin(e + f*x))^(9/2),x)
[Out]
int(((g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^(5/2))/(c - c*sin(e + f*x))^(9/2), x)