\(\int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{7/2}} \, dx\) [122]

Optimal result

Integrand size = 42, antiderivative size = 300 \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{7/2}} \, dx=\frac {4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{9 f g (c-c \sin (e+f x))^{7/2}}-\frac {4 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{3 c f g (c-c \sin (e+f x))^{5/2}}+\frac {44 a^3 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{3 c^2 f g (c-c \sin (e+f x))^{3/2}}+\frac {154 a^4 (g \cos (e+f x))^{5/2}}{9 c^3 f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}-\frac {154 a^4 g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{3 c^3 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \]

[Out]

4/9*a*(g*cos(f*x+e))^(5/2)*(a+a*sin(f*x+e))^(5/2)/f/g/(c-c*sin(f*x+e))^(7/2)-4/3*a^2*(g*cos(f*x+e))^(5/2)*(a+a
*sin(f*x+e))^(3/2)/c/f/g/(c-c*sin(f*x+e))^(5/2)+44/3*a^3*(g*cos(f*x+e))^(5/2)*(a+a*sin(f*x+e))^(1/2)/c^2/f/g/(
c-c*sin(f*x+e))^(3/2)+154/9*a^4*(g*cos(f*x+e))^(5/2)/c^3/f/g/(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e))^(1/2)-154
/3*a^4*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^3/f/(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e))^(1/2)

Rubi [A] (verified)

Time = 1.03 (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, 2930, 2921, 2721, 2719} \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{7/2}} \, dx=\frac {154 a^4 (g \cos (e+f x))^{5/2}}{9 c^3 f g \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {154 a^4 g \sqrt {\cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {g \cos (e+f x)}}{3 c^3 f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {44 a^3 \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{3 c^2 f g (c-c \sin (e+f x))^{3/2}}-\frac {4 a^2 (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{3 c f g (c-c \sin (e+f x))^{5/2}}+\frac {4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{9 f g (c-c \sin (e+f x))^{7/2}} \]

[In]

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

[Out]

(4*a*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(5/2))/(9*f*g*(c - c*Sin[e + f*x])^(7/2)) - (4*a^2*(g*Cos[e +
f*x])^(5/2)*(a + a*Sin[e + f*x])^(3/2))/(3*c*f*g*(c - c*Sin[e + f*x])^(5/2)) + (44*a^3*(g*Cos[e + f*x])^(5/2)
*Sqrt[a + a*Sin[e + f*x]])/(3*c^2*f*g*(c - c*Sin[e + f*x])^(3/2)) + (154*a^4*(g*Cos[e + f*x])^(5/2))/(9*c^3*f*
g*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]) - (154*a^4*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*Elli
pticE[(e + f*x)/2, 2])/(3*c^3*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 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 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{9 f g (c-c \sin (e+f x))^{7/2}}-\frac {(5 a) \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{5/2}} \, dx}{3 c} \\ & = \frac {4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{9 f g (c-c \sin (e+f x))^{7/2}}-\frac {4 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{3 c f g (c-c \sin (e+f x))^{5/2}}+\frac {\left (11 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))^{3/2}} \, dx}{3 c^2} \\ & = \frac {4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{9 f g (c-c \sin (e+f x))^{7/2}}-\frac {4 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{3 c f g (c-c \sin (e+f x))^{5/2}}+\frac {44 a^3 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{3 c^2 f g (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}{3 c^3} \\ & = \frac {4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{9 f g (c-c \sin (e+f x))^{7/2}}-\frac {4 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{3 c f g (c-c \sin (e+f x))^{5/2}}+\frac {44 a^3 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{3 c^2 f g (c-c \sin (e+f x))^{3/2}}+\frac {154 a^4 (g \cos (e+f x))^{5/2}}{9 c^3 f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}-\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}{3 c^3} \\ & = \frac {4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{9 f g (c-c \sin (e+f x))^{7/2}}-\frac {4 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{3 c f g (c-c \sin (e+f x))^{5/2}}+\frac {44 a^3 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{3 c^2 f g (c-c \sin (e+f x))^{3/2}}+\frac {154 a^4 (g \cos (e+f x))^{5/2}}{9 c^3 f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}-\frac {\left (77 a^4 g \cos (e+f x)\right ) \int \sqrt {g \cos (e+f x)} \, dx}{3 c^3 \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}}{9 f g (c-c \sin (e+f x))^{7/2}}-\frac {4 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{3 c f g (c-c \sin (e+f x))^{5/2}}+\frac {44 a^3 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{3 c^2 f g (c-c \sin (e+f x))^{3/2}}+\frac {154 a^4 (g \cos (e+f x))^{5/2}}{9 c^3 f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}-\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}{3 c^3 \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}}{9 f g (c-c \sin (e+f x))^{7/2}}-\frac {4 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{3 c f g (c-c \sin (e+f x))^{5/2}}+\frac {44 a^3 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{3 c^2 f g (c-c \sin (e+f x))^{3/2}}+\frac {154 a^4 (g \cos (e+f x))^{5/2}}{9 c^3 f g \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}-\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 )}{3 c^3 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 14.82 (sec) , antiderivative size = 406, normalized size of antiderivative = 1.35 \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{7/2}} \, dx=-\frac {154 (g \cos (e+f x))^{3/2} E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (a (1+\sin (e+f x)))^{7/2}}{3 f \cos ^{\frac {3}{2}}(e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (c-c \sin (e+f x))^{7/2}}+\frac {(g \cos (e+f x))^{3/2} \sec (e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 \left (\frac {112}{3}+\frac {2}{3} \cos (e+f x)+\frac {32}{9 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}-\frac {32}{3 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^2}+\frac {64 \sin \left (\frac {1}{2} (e+f x)\right )}{9 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5}-\frac {64 \sin \left (\frac {1}{2} (e+f x)\right )}{3 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^3}+\frac {224 \sin \left (\frac {1}{2} (e+f x)\right )}{3 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )}\right ) (a (1+\sin (e+f x)))^{7/2}}{f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^7 (c-c \sin (e+f x))^{7/2}} \]

[In]

Integrate[((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(7/2))/(c - c*Sin[e + f*x])^(7/2),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])^7*(a*(1 + Sin[e +
f*x]))^(7/2))/(3*f*Cos[e + f*x]^(3/2)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(7/2)) + (
(g*Cos[e + f*x])^(3/2)*Sec[e + f*x]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7*(112/3 + (2*Cos[e + f*x])/3 + 32/(
9*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^4) - 32/(3*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2) + (64*Sin[(e + f*x
)/2])/(9*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5) - (64*Sin[(e + f*x)/2])/(3*(Cos[(e + f*x)/2] - Sin[(e + f*x)
/2])^3) + (224*Sin[(e + f*x)/2])/(3*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])))*(a*(1 + Sin[e + f*x]))^(7/2))/(f*(
Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(7/2))

Maple [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 2.86 (sec) , antiderivative size = 3173, normalized size of antiderivative = 10.58

method	result	size
default	\(\text {Expression too large to display}\)	\(3173\)

[In]

int((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(7/2)/(c-c*sin(f*x+e))^(7/2),x,method=_RETURNVERBOSE)

[Out]

-2/9/f*(g*cos(f*x+e))^(1/2)*(a*(1+sin(f*x+e)))^(1/2)*g*a^3/(cos(f*x+e)-sin(f*x+e)+1)/(-c*(sin(f*x+e)-1))^(1/2)
/c^3*(105+54*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)-co
s(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)-54*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)+378*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)*se
c(f*x+e)*tan(f*x+e)-378*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)+3*cos(f*x+e)^2
+510*tan(f*x+e)-16*sec(f*x+e)*tan(f*x+e)-398*sec(f*x+e)+60*cos(f*x+e)-57*sin(f*x+e)-108*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*tan(f*x+e)+108*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)*se
c(f*x+e)^2*tan(f*x+e)+54*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)^2-54*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)))*(-co
s(f*x+e)/(1+cos(f*x+e))^2)^(3/2)*cos(f*x+e)^2+108*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)-108*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)-432*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)+432*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)-16*sec(f*x+e)^2-32*s
ec(f*x+e)^2*tan(f*x+e)+3*cos(f*x+e)*sin(f*x+e)-32*sec(f*x+e)^3-693*I*(1/(1+cos(f*x+e)))^(1/2)*(cos(f*x+e)/(1+c
os(f*x+e)))^(1/2)*EllipticF(I*(csc(f*x+e)-cot(f*x+e)),I)*sec(f*x+e)+693*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)-462*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)^2+462*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)^2+693*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)-693*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)-108*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)+108*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)+231*I*(1/(1+c
os(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)+378*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)^2-486*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)*tan(f*x+e)+486*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)*tan(f*x+e)-378*ln(2*(2*(-cos(f*x+e)/(1+c
os(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+231*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)-231*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)-231*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)+270*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)-270*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)-c
os(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)+462*I*(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)*sec(f*x+e)*tan(f*x+e)-462*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)*tan(f*x+e)+1
08*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)^3-108*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)^3)

Fricas [C] (verification not implemented)

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

Time = 0.14 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.07 \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{7/2}} \, dx=\frac {2 \, {\left (177 \, a^{3} g \cos \left (f x + e\right )^{2} - 316 \, a^{3} g - 3 \, {\left (a^{3} g \cos \left (f x + e\right )^{2} - 100 \, a^{3} g\right )} \sin \left (f x + e\right )\right )} \sqrt {g \cos \left (f x + e\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c} + 231 \, {\left (3 i \, \sqrt {2} a^{3} g \cos \left (f x + e\right )^{2} - 4 i \, \sqrt {2} a^{3} g + {\left (-i \, \sqrt {2} a^{3} g \cos \left (f x + e\right )^{2} + 4 i \, \sqrt {2} a^{3} g\right )} \sin \left (f x + e\right )\right )} \sqrt {a c g} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) + 231 \, {\left (-3 i \, \sqrt {2} a^{3} g \cos \left (f x + e\right )^{2} + 4 i \, \sqrt {2} a^{3} g + {\left (i \, \sqrt {2} a^{3} g \cos \left (f x + e\right )^{2} - 4 i \, \sqrt {2} a^{3} g\right )} \sin \left (f x + e\right )\right )} \sqrt {a c g} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right )}{9 \, {\left (3 \, c^{4} f \cos \left (f x + e\right )^{2} - 4 \, c^{4} f - {\left (c^{4} f \cos \left (f x + e\right )^{2} - 4 \, c^{4} f\right )} \sin \left (f x + e\right )\right )}} \]

[In]

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

[Out]

1/9*(2*(177*a^3*g*cos(f*x + e)^2 - 316*a^3*g - 3*(a^3*g*cos(f*x + e)^2 - 100*a^3*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) + 231*(3*I*sqrt(2)*a^3*g*cos(f*x + e)^2 - 4*I*sqrt
(2)*a^3*g + (-I*sqrt(2)*a^3*g*cos(f*x + e)^2 + 4*I*sqrt(2)*a^3*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*(-3*I*sqrt(2)*a^3*g*cos(f*x + e)^2 + 4*I
*sqrt(2)*a^3*g + (I*sqrt(2)*a^3*g*cos(f*x + e)^2 - 4*I*sqrt(2)*a^3*g)*sin(f*x + e))*sqrt(a*c*g)*weierstrassZet
a(-4, 0, weierstrassPInverse(-4, 0, cos(f*x + e) - I*sin(f*x + e))))/(3*c^4*f*cos(f*x + e)^2 - 4*c^4*f - (c^4*
f*cos(f*x + e)^2 - 4*c^4*f)*sin(f*x + e))

Sympy [F(-1)]

Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{7/2}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{7/2}} \, dx=\int { \frac {\left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {7}{2}}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {7}{2}}} \,d x } \]

[In]

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

Giac [F(-1)]

Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{7/2}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{7/2}} \, dx=\int \frac {{\left (g\,\cos \left (e+f\,x\right )\right )}^{3/2}\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{7/2}}{{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{7/2}} \,d x \]

[In]

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

[Out]

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