\(\int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{11/2}} \, dx\) [124]

Optimal result

Integrand size = 42, antiderivative size = 357 \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{11/2}} \, dx=\frac {4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {60 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{221 c f g (c-c \sin (e+f x))^{9/2}}+\frac {220 a^3 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{663 c^2 f g (c-c \sin (e+f x))^{7/2}}-\frac {308 a^4 (g \cos (e+f x))^{5/2}}{663 c^3 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}+\frac {154 a^4 (g \cos (e+f x))^{5/2}}{221 c^4 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}-\frac {154 a^4 g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{221 c^5 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \]

[Out]

4/17*a*(g*cos(f*x+e))^(5/2)*(a+a*sin(f*x+e))^(5/2)/f/g/(c-c*sin(f*x+e))^(11/2)-60/221*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))^(9/2)-308/663*a^4*(g*cos(f*x+e))^(5/2)/c^3/f/g/(c-c*sin(f*x+e))
^(5/2)/(a+a*sin(f*x+e))^(1/2)+154/221*a^4*(g*cos(f*x+e))^(5/2)/c^4/f/g/(c-c*sin(f*x+e))^(3/2)/(a+a*sin(f*x+e))
^(1/2)+220/663*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))^(7/2)-154/221*a^4*g*(c
os(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^5/f/(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e))^(1/2)

Rubi [A] (verified)

Time = 1.27 (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, 2931, 2921, 2721, 2719} \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{11/2}} \, dx=-\frac {154 a^4 g \sqrt {\cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {g \cos (e+f x)}}{221 c^5 f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {154 a^4 (g \cos (e+f x))^{5/2}}{221 c^4 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac {308 a^4 (g \cos (e+f x))^{5/2}}{663 c^3 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}+\frac {220 a^3 \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{663 c^2 f g (c-c \sin (e+f x))^{7/2}}-\frac {60 a^2 (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{221 c f g (c-c \sin (e+f x))^{9/2}}+\frac {4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}} \]

[In]

Int[((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(7/2))/(c - c*Sin[e + f*x])^(11/2),x]

[Out]

(4*a*(g*Cos[e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(5/2))/(17*f*g*(c - c*Sin[e + f*x])^(11/2)) - (60*a^2*(g*Cos[
e + f*x])^(5/2)*(a + a*Sin[e + f*x])^(3/2))/(221*c*f*g*(c - c*Sin[e + f*x])^(9/2)) + (220*a^3*(g*Cos[e + f*x])
^(5/2)*Sqrt[a + a*Sin[e + f*x]])/(663*c^2*f*g*(c - c*Sin[e + f*x])^(7/2)) - (308*a^4*(g*Cos[e + f*x])^(5/2))/(
663*c^3*f*g*Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2)) + (154*a^4*(g*Cos[e + f*x])^(5/2))/(221*c^4*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]]*E
llipticE[(e + f*x)/2, 2])/(221*c^5*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))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\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))^{9/2}} \, dx}{17 c} \\ & = \frac {4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {60 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{221 c f g (c-c \sin (e+f x))^{9/2}}+\frac {\left (165 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))^{7/2}} \, dx}{221 c^2} \\ & = \frac {4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {60 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{221 c f g (c-c \sin (e+f x))^{9/2}}+\frac {220 a^3 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{663 c^2 f g (c-c \sin (e+f x))^{7/2}}-\frac {\left (385 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))^{5/2}} \, dx}{663 c^3} \\ & = \frac {4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {60 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{221 c f g (c-c \sin (e+f x))^{9/2}}+\frac {220 a^3 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{663 c^2 f g (c-c \sin (e+f x))^{7/2}}-\frac {308 a^4 (g \cos (e+f x))^{5/2}}{663 c^3 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}+\frac {\left (77 a^4\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}{221 c^4} \\ & = \frac {4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {60 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{221 c f g (c-c \sin (e+f x))^{9/2}}+\frac {220 a^3 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{663 c^2 f g (c-c \sin (e+f x))^{7/2}}-\frac {308 a^4 (g \cos (e+f x))^{5/2}}{663 c^3 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}+\frac {154 a^4 (g \cos (e+f x))^{5/2}}{221 c^4 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}{221 c^5} \\ & = \frac {4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {60 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{221 c f g (c-c \sin (e+f x))^{9/2}}+\frac {220 a^3 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{663 c^2 f g (c-c \sin (e+f x))^{7/2}}-\frac {308 a^4 (g \cos (e+f x))^{5/2}}{663 c^3 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}+\frac {154 a^4 (g \cos (e+f x))^{5/2}}{221 c^4 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}{221 c^5 \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}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {60 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{221 c f g (c-c \sin (e+f x))^{9/2}}+\frac {220 a^3 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{663 c^2 f g (c-c \sin (e+f x))^{7/2}}-\frac {308 a^4 (g \cos (e+f x))^{5/2}}{663 c^3 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}+\frac {154 a^4 (g \cos (e+f x))^{5/2}}{221 c^4 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}{221 c^5 \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}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {60 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{221 c f g (c-c \sin (e+f x))^{9/2}}+\frac {220 a^3 (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{663 c^2 f g (c-c \sin (e+f x))^{7/2}}-\frac {308 a^4 (g \cos (e+f x))^{5/2}}{663 c^3 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}+\frac {154 a^4 (g \cos (e+f x))^{5/2}}{221 c^4 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 )}{221 c^5 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 11.46 (sec) , antiderivative size = 532, normalized size of antiderivative = 1.49 \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{11/2}} \, 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 )^{11} (a (1+\sin (e+f x)))^{7/2}}{221 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))^{11/2}}+\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 )^{11} \left (\frac {154}{221}+\frac {32}{17 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^8}-\frac {864}{221 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^6}+\frac {2096}{663 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}-\frac {288}{221 \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 )}{17 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^9}-\frac {1728 \sin \left (\frac {1}{2} (e+f x)\right )}{221 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7}+\frac {4192 \sin \left (\frac {1}{2} (e+f x)\right )}{663 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5}-\frac {576 \sin \left (\frac {1}{2} (e+f x)\right )}{221 \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 )}{221 \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))^{11/2}} \]

[In]

Integrate[((g*Cos[e + f*x])^(3/2)*(a + a*Sin[e + f*x])^(7/2))/(c - c*Sin[e + f*x])^(11/2),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])^11*(a*(1 + Sin[e
+ f*x]))^(7/2))/(221*f*Cos[e + f*x]^(3/2)*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(11/2))
+ ((g*Cos[e + f*x])^(3/2)*Sec[e + f*x]*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^11*(154/221 + 32/(17*(Cos[(e + f
*x)/2] - Sin[(e + f*x)/2])^8) - 864/(221*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^6) + 2096/(663*(Cos[(e + f*x)/2
] - Sin[(e + f*x)/2])^4) - 288/(221*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^2) + (64*Sin[(e + f*x)/2])/(17*(Cos[
(e + f*x)/2] - Sin[(e + f*x)/2])^9) - (1728*Sin[(e + f*x)/2])/(221*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^7) +
(4192*Sin[(e + f*x)/2])/(663*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5) - (576*Sin[(e + f*x)/2])/(221*(Cos[(e +
f*x)/2] - Sin[(e + f*x)/2])^3) + (308*Sin[(e + f*x)/2])/(221*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])))*(a*(1 + S
in[e + f*x]))^(7/2))/(f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^7*(c - c*Sin[e + f*x])^(11/2))

Maple [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 2.90 (sec) , antiderivative size = 3093, normalized size of antiderivative = 8.66

method	result	size
default	\(\text {Expression too large to display}\)	\(3093\)

[In]

int((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(7/2)/(c-c*sin(f*x+e))^(11/2),x,method=_RETURNVERBOSE)

[Out]

1/1326/f*(a*(1+sin(f*x+e)))^(1/2)*(g*cos(f*x+e))^(1/2)*g*a^3/(1+cos(f*x+e))^3/(cos(f*x+e)^2*sin(f*x+e)-3*cos(f
*x+e)^2-4*sin(f*x+e)+4)/(-cos(f*x+e)/(1+cos(f*x+e))^2)^(3/2)/(-c*(sin(f*x+e)-1))^(1/2)/c^5*(2652*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)))-2652*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)))+1044*cos(f*x+e)^3*(-cos(f*x+e)/(1+cos(f*x+
e))^2)^(1/2)-924*cos(f*x+e)^3*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)*sin(f*x+e)-9888*(-cos(f*x+e)/(1+cos(f*x+e))
^2)^(1/2)-5304*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)))+5304*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)))+5304*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)-5304*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)))*sin(f*x+e)+48
96*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)*sin(f*x+e)-2496*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)*tan(f*x+e)-2496*(
-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)*sec(f*x+e)-14784*I*EllipticE(I*(csc(f*x+e)-cot(f*x+e)),I)*(1/(1+cos(f*x+e)
))^(1/2)*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)*sin(f*x+e)+14784*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)-14784*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)+11840*cos(f*x+e)^2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)+663*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)^4-663*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)^4+4432*cos(f*x+e)*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)*si
n(f*x+e)-2960*sin(f*x+e)*cos(f*x+e)^2*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)+752*(-cos(f*x+e)/(1+cos(f*x+e))^2)^
(1/2)*cos(f*x+e)-5304*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)))+5304*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)))+14784*I*EllipticF(I*(csc(f*x+
e)-cot(f*x+e)),I)*(1/(1+cos(f*x+e)))^(1/2)*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1
/2)*sin(f*x+e)+7392*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)-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)*tan(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)*sec(f*x+e)-7392*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)-6468*I*EllipticE(I*(csc(f*x+e)-co
t(f*x+e)),I)*(1/(1+cos(f*x+e)))^(1/2)*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)*c
os(f*x+e)^3+6468*I*EllipticF(I*(csc(f*x+e)-cot(f*x+e)),I)*(1/(1+cos(f*x+e)))^(1/2)*(cos(f*x+e)/(1+cos(f*x+e)))
^(1/2)*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)*cos(f*x+e)^3-14784*I*EllipticE(I*(csc(f*x+e)-cot(f*x+e)),I)*(1/(1+
cos(f*x+e)))^(1/2)*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)*cos(f*x+e)^2+14784*I
*EllipticF(I*(csc(f*x+e)-cot(f*x+e)),I)*(1/(1+cos(f*x+e)))^(1/2)*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*(-cos(f*x+e
)/(1+cos(f*x+e))^2)^(1/2)*cos(f*x+e)^2+924*I*EllipticE(I*(csc(f*x+e)-cot(f*x+e)),I)*(1/(1+cos(f*x+e)))^(1/2)*(
cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)*cos(f*x+e)^5-924*I*EllipticF(I*(csc(f*x+
e)-cot(f*x+e)),I)*(1/(1+cos(f*x+e)))^(1/2)*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1
/2)*cos(f*x+e)^5+1848*I*EllipticE(I*(csc(f*x+e)-cot(f*x+e)),I)*(1/(1+cos(f*x+e)))^(1/2)*(cos(f*x+e)/(1+cos(f*x
+e)))^(1/2)*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)*cos(f*x+e)^4-1848*I*EllipticF(I*(csc(f*x+e)-cot(f*x+e)),I)*(1
/(1+cos(f*x+e)))^(1/2)*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)*cos(f*x+e)^4-369
6*I*EllipticF(I*(csc(f*x+e)-cot(f*x+e)),I)*(1/(1+cos(f*x+e)))^(1/2)*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*(-cos(f*
x+e)/(1+cos(f*x+e))^2)^(1/2)*cos(f*x+e)^3*sin(f*x+e)+7392*I*EllipticE(I*(csc(f*x+e)-cot(f*x+e)),I)*(1/(1+cos(f
*x+e)))^(1/2)*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)*cos(f*x+e)^2*sin(f*x+e)-7
392*I*EllipticF(I*(csc(f*x+e)-cot(f*x+e)),I)*(1/(1+cos(f*x+e)))^(1/2)*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*(-cos(
f*x+e)/(1+cos(f*x+e))^2)^(1/2)*cos(f*x+e)^2*sin(f*x+e)-3696*I*EllipticE(I*(csc(f*x+e)-cot(f*x+e)),I)*(1/(1+cos
(f*x+e)))^(1/2)*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)*cos(f*x+e)*sin(f*x+e)+3
696*I*EllipticF(I*(csc(f*x+e)-cot(f*x+e)),I)*(1/(1+cos(f*x+e)))^(1/2)*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*(-cos(
f*x+e)/(1+cos(f*x+e))^2)^(1/2)*cos(f*x+e)*sin(f*x+e)+3696*I*EllipticE(I*(csc(f*x+e)-cot(f*x+e)),I)*(1/(1+cos(f
*x+e)))^(1/2)*(cos(f*x+e)/(1+cos(f*x+e)))^(1/2)*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2)*cos(f*x+e)^3*sin(f*x+e)-2
652*cos(f*x+e)^4*(-cos(f*x+e)/(1+cos(f*x+e))^2)^(1/2))

Fricas [C] (verification not implemented)

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

Time = 0.18 (sec) , antiderivative size = 431, normalized size of antiderivative = 1.21 \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{11/2}} \, dx=\frac {2 \, {\left (231 \, a^{3} g \cos \left (f x + e\right )^{4} - 1600 \, a^{3} g \cos \left (f x + e\right )^{2} + 1544 \, a^{3} g + 4 \, {\left (123 \, a^{3} g \cos \left (f x + e\right )^{2} - 230 \, a^{3} g\right )} \sin \left (f x + e\right )\right )} \sqrt {g \cos \left (f x + e\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c} + 231 \, {\left (5 i \, \sqrt {2} a^{3} g \cos \left (f x + e\right )^{4} - 20 i \, \sqrt {2} a^{3} g \cos \left (f x + e\right )^{2} + 16 i \, \sqrt {2} a^{3} g + {\left (-i \, \sqrt {2} a^{3} g \cos \left (f x + e\right )^{4} + 12 i \, \sqrt {2} a^{3} g \cos \left (f x + e\right )^{2} - 16 i \, \sqrt {2} a^{3} g\right )} \sin \left (f x + e\right )\right )} \sqrt {a c g} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) + 231 \, {\left (-5 i \, \sqrt {2} a^{3} g \cos \left (f x + e\right )^{4} + 20 i \, \sqrt {2} a^{3} g \cos \left (f x + e\right )^{2} - 16 i \, \sqrt {2} a^{3} g + {\left (i \, \sqrt {2} a^{3} g \cos \left (f x + e\right )^{4} - 12 i \, \sqrt {2} a^{3} g \cos \left (f x + e\right )^{2} + 16 i \, \sqrt {2} a^{3} g\right )} \sin \left (f x + e\right )\right )} \sqrt {a c g} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right )}{663 \, {\left (5 \, c^{6} f \cos \left (f x + e\right )^{4} - 20 \, c^{6} f \cos \left (f x + e\right )^{2} + 16 \, c^{6} f - {\left (c^{6} f \cos \left (f x + e\right )^{4} - 12 \, c^{6} f \cos \left (f x + e\right )^{2} + 16 \, c^{6} f\right )} \sin \left (f x + e\right )\right )}} \]

[In]

integrate((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(7/2)/(c-c*sin(f*x+e))^(11/2),x, algorithm="fricas")

[Out]

1/663*(2*(231*a^3*g*cos(f*x + e)^4 - 1600*a^3*g*cos(f*x + e)^2 + 1544*a^3*g + 4*(123*a^3*g*cos(f*x + e)^2 - 23
0*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*(5*I*sqrt
(2)*a^3*g*cos(f*x + e)^4 - 20*I*sqrt(2)*a^3*g*cos(f*x + e)^2 + 16*I*sqrt(2)*a^3*g + (-I*sqrt(2)*a^3*g*cos(f*x
+ e)^4 + 12*I*sqrt(2)*a^3*g*cos(f*x + e)^2 - 16*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*(-5*I*sqrt(2)*a^3*g*cos(f*x + e)^4 + 20*I
*sqrt(2)*a^3*g*cos(f*x + e)^2 - 16*I*sqrt(2)*a^3*g + (I*sqrt(2)*a^3*g*cos(f*x + e)^4 - 12*I*sqrt(2)*a^3*g*cos(
f*x + e)^2 + 16*I*sqrt(2)*a^3*g)*sin(f*x + e))*sqrt(a*c*g)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, c
os(f*x + e) - I*sin(f*x + e))))/(5*c^6*f*cos(f*x + e)^4 - 20*c^6*f*cos(f*x + e)^2 + 16*c^6*f - (c^6*f*cos(f*x
+ e)^4 - 12*c^6*f*cos(f*x + e)^2 + 16*c^6*f)*sin(f*x + e))

Sympy [F(-1)]

Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{11/2}} \, dx=\text {Timed out} \]

[In]

integrate((g*cos(f*x+e))**(3/2)*(a+a*sin(f*x+e))**(7/2)/(c-c*sin(f*x+e))**(11/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{11/2}} \, dx=\int { \frac {\left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {7}{2}}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {11}{2}}} \,d x } \]

[In]

integrate((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(7/2)/(c-c*sin(f*x+e))^(11/2),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)^(11/2), x)

Giac [F(-1)]

Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{11/2}} \, dx=\text {Timed out} \]

[In]

integrate((g*cos(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(7/2)/(c-c*sin(f*x+e))^(11/2),x, algorithm="giac")

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{11/2}} \, dx=\int \frac {{\left (g\,\cos \left (e+f\,x\right )\right )}^{3/2}\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{7/2}}{{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{11/2}} \,d x \]

[In]

int(((g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^(7/2))/(c - c*sin(e + f*x))^(11/2),x)

[Out]

int(((g*cos(e + f*x))^(3/2)*(a + a*sin(e + f*x))^(7/2))/(c - c*sin(e + f*x))^(11/2), x)