\(\int \frac {(g \cos (e+f x))^{3/2} \sqrt {c-c \sin (e+f x)}}{(a+a \sin (e+f x))^{3/2}} \, dx\) [137]
Optimal result
Integrand size = 42, antiderivative size = 123 \[
\int \frac {(g \cos (e+f x))^{3/2} \sqrt {c-c \sin (e+f x)}}{(a+a \sin (e+f x))^{3/2}} \, dx=-\frac {4 c (g \cos (e+f x))^{5/2}}{f g (a+a \sin (e+f x))^{3/2} \sqrt {c-c \sin (e+f x)}}-\frac {6 c g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{a f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}
\]
[Out]
-4*c*(g*cos(f*x+e))^(5/2)/f/g/(a+a*sin(f*x+e))^(3/2)/(c-c*sin(f*x+e))^(1/2)-6*c*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/f/(a+a*sin(f
*x+e))^(1/2)/(c-c*sin(f*x+e))^(1/2)
Rubi [A] (verified)
Time = 0.40 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.00, number of
steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2929, 2921, 2721, 2719}
\[
\int \frac {(g \cos (e+f x))^{3/2} \sqrt {c-c \sin (e+f x)}}{(a+a \sin (e+f x))^{3/2}} \, dx=-\frac {4 c (g \cos (e+f x))^{5/2}}{f g (a \sin (e+f x)+a)^{3/2} \sqrt {c-c \sin (e+f x)}}-\frac {6 c g \sqrt {\cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {g \cos (e+f x)}}{a f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}
\]
[In]
Int[((g*Cos[e + f*x])^(3/2)*Sqrt[c - c*Sin[e + f*x]])/(a + a*Sin[e + f*x])^(3/2),x]
[Out]
(-4*c*(g*Cos[e + f*x])^(5/2))/(f*g*(a + a*Sin[e + f*x])^(3/2)*Sqrt[c - c*Sin[e + f*x]]) - (6*c*g*Sqrt[Cos[e +
f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e + f*x)/2, 2])/(a*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]
Rubi steps \begin{align*}
\text {integral}& = -\frac {4 c (g \cos (e+f x))^{5/2}}{f g (a+a \sin (e+f x))^{3/2} \sqrt {c-c \sin (e+f x)}}-\frac {(3 c) \int \frac {(g \cos (e+f x))^{3/2}}{\sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \, dx}{a} \\ & = -\frac {4 c (g \cos (e+f x))^{5/2}}{f g (a+a \sin (e+f x))^{3/2} \sqrt {c-c \sin (e+f x)}}-\frac {(3 c g \cos (e+f x)) \int \sqrt {g \cos (e+f x)} \, dx}{a \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \\ & = -\frac {4 c (g \cos (e+f x))^{5/2}}{f g (a+a \sin (e+f x))^{3/2} \sqrt {c-c \sin (e+f x)}}-\frac {\left (3 c g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)}\right ) \int \sqrt {\cos (e+f x)} \, dx}{a \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \\ & = -\frac {4 c (g \cos (e+f x))^{5/2}}{f g (a+a \sin (e+f x))^{3/2} \sqrt {c-c \sin (e+f x)}}-\frac {6 c g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right )}{a f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \\
\end{align*}
Mathematica [C] (verified)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 3.18 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.73
\[
\int \frac {(g \cos (e+f x))^{3/2} \sqrt {c-c \sin (e+f x)}}{(a+a \sin (e+f x))^{3/2}} \, dx=\frac {2 g \sqrt {e^{-i (e+f x)} \left (1+e^{2 i (e+f x)}\right ) g} \left (-\left (\left (i+5 e^{i (e+f x)}\right ) \sqrt {1+e^{2 i (e+f x)}}\right )+2 e^{2 i (e+f x)} \left (i+e^{i (e+f x)}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (e+f x)}\right )\right ) \sqrt {c-c \sin (e+f x)}}{a \left (-i+e^{i (e+f x)}\right ) \sqrt {-i a e^{-i (e+f x)} \left (i+e^{i (e+f x)}\right )^2} \sqrt {1+e^{2 i (e+f x)}} f}
\]
[In]
Integrate[((g*Cos[e + f*x])^(3/2)*Sqrt[c - c*Sin[e + f*x]])/(a + a*Sin[e + f*x])^(3/2),x]
[Out]
(2*g*Sqrt[((1 + E^((2*I)*(e + f*x)))*g)/E^(I*(e + f*x))]*(-((I + 5*E^(I*(e + f*x)))*Sqrt[1 + E^((2*I)*(e + f*x
))]) + 2*E^((2*I)*(e + f*x))*(I + E^(I*(e + f*x)))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(e + f*x))])*Sqr
t[c - c*Sin[e + f*x]])/(a*(-I + E^(I*(e + f*x)))*Sqrt[((-I)*a*(I + E^(I*(e + f*x)))^2)/E^(I*(e + f*x))]*Sqrt[1
+ E^((2*I)*(e + f*x))]*f)
Maple [C] (verified)
Result contains complex when optimal does not.
Time = 1.26 (sec) , antiderivative size = 1190, normalized size of antiderivative =
9.67
| | |
| method | result | size |
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| risch |
\(\text {Expression too large to display}\) |
\(1190\) |
| default |
\(\text {Expression too large to display}\) |
\(1453\) |
| | |
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[In]
int((g*cos(f*x+e))^(3/2)*(c-c*sin(f*x+e))^(1/2)/(a+a*sin(f*x+e))^(3/2),x,method=_RETURNVERBOSE)
[Out]
-(4*I*exp(I*(f*x+e))+exp(I*(f*x+e))^2+3)*(exp(I*(f*x+e))-I)*(exp(I*(f*x+e))+I)/f*2^(1/2)*g/a*(g*(exp(I*(f*x+e)
)^2+1)/exp(I*(f*x+e)))^(1/2)/(exp(I*(f*x+e))^2+1)/(-a*(I*exp(I*(f*x+e))^2-I-2*exp(I*(f*x+e)))/exp(I*(f*x+e)))^
(1/2)*(c*(I*exp(I*(f*x+e))^2-I+2*exp(I*(f*x+e)))/exp(I*(f*x+e)))^(1/2)/(2*I*exp(I*(f*x+e))-exp(I*(f*x+e))^2+1)
+1/f*(-exp(I*(f*x+e))+I)*(exp(I*(f*x+e))+I)*(6*(-I*(-exp(I*(f*x+e))+I))^(1/2)*2^(1/2)*(-I*(exp(I*(f*x+e))+I))^
(1/2)*(-I*exp(I*(f*x+e)))^(1/2)*(-c*g*a*(exp(I*(f*x+e))^2+1)*exp(I*(f*x+e)))^(1/2)*(exp(I*(f*x+e))*(exp(I*(f*x
+e))+I)*(-exp(I*(f*x+e))+I)*a*c*g)^(1/2)*EllipticE((-I*(-exp(I*(f*x+e))+I))^(1/2),1/2*2^(1/2))-3*(-I*(-exp(I*(
f*x+e))+I))^(1/2)*2^(1/2)*(-I*(exp(I*(f*x+e))+I))^(1/2)*(-I*exp(I*(f*x+e)))^(1/2)*(-c*g*a*(exp(I*(f*x+e))^2+1)
*exp(I*(f*x+e)))^(1/2)*(exp(I*(f*x+e))*(exp(I*(f*x+e))+I)*(-exp(I*(f*x+e))+I)*a*c*g)^(1/2)*EllipticF((-I*(-exp
(I*(f*x+e))+I))^(1/2),1/2*2^(1/2))-4*(-exp(I*(f*x+e))^3*a*c*g-exp(I*(f*x+e))*a*c*g)^(1/2)*(exp(I*(f*x+e))*(exp
(I*(f*x+e))+I)*(-exp(I*(f*x+e))+I)*a*c*g)^(1/2)*exp(I*(f*x+e))^2-2*exp(I*(f*x+e))^2*(-c*g*a*(exp(I*(f*x+e))^2+
1)*exp(I*(f*x+e)))^(1/2)*(-exp(I*(f*x+e))^3*a*c*g-exp(I*(f*x+e))*a*c*g)^(1/2)-4*(-exp(I*(f*x+e))^3*a*c*g-exp(I
*(f*x+e))*a*c*g)^(1/2)*(exp(I*(f*x+e))*(exp(I*(f*x+e))+I)*(-exp(I*(f*x+e))+I)*a*c*g)^(1/2))/((2*I*exp(I*(f*x+e
))-exp(I*(f*x+e))^2+1)*(exp(I*(f*x+e))^2+1)*(exp(I*(f*x+e))^2-1+2*I*exp(I*(f*x+e)))*c*exp(I*(f*x+e))*g*a)^(1/2
)/(-exp(I*(f*x+e))^3*a*c*g-exp(I*(f*x+e))*a*c*g)^(1/2)/(exp(I*(f*x+e))*(exp(I*(f*x+e))+I)*(-exp(I*(f*x+e))+I)*
a*c*g)^(1/2)*2^(1/2)*g/a*(g*(exp(I*(f*x+e))^2+1)/exp(I*(f*x+e)))^(1/2)/(exp(I*(f*x+e))^2+1)/(-a*(I*exp(I*(f*x+
e))^2-I-2*exp(I*(f*x+e)))/exp(I*(f*x+e)))^(1/2)*(c*(I*exp(I*(f*x+e))^2-I+2*exp(I*(f*x+e)))/exp(I*(f*x+e)))^(1/
2)/(2*I*exp(I*(f*x+e))-exp(I*(f*x+e))^2+1)*(-c*(exp(I*(f*x+e))^2-1-2*I*exp(I*(f*x+e)))*exp(I*(f*x+e))*g*(exp(I
*(f*x+e))^2+1)*a*(exp(I*(f*x+e))^2-1+2*I*exp(I*(f*x+e))))^(1/2)
Fricas [C] (verification not implemented)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.27
\[
\int \frac {(g \cos (e+f x))^{3/2} \sqrt {c-c \sin (e+f x)}}{(a+a \sin (e+f x))^{3/2}} \, dx=-\frac {4 \, \sqrt {g \cos \left (f x + e\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c} g + 3 \, \sqrt {a c g} {\left (-i \, \sqrt {2} g \sin \left (f x + e\right ) - i \, \sqrt {2} g\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )\right ) + 3 \, \sqrt {a c g} {\left (i \, \sqrt {2} g \sin \left (f x + e\right ) + i \, \sqrt {2} g\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )\right )}{a^{2} f \sin \left (f x + e\right ) + a^{2} f}
\]
[In]
integrate((g*cos(f*x+e))^(3/2)*(c-c*sin(f*x+e))^(1/2)/(a+a*sin(f*x+e))^(3/2),x, algorithm="fricas")
[Out]
-(4*sqrt(g*cos(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)*g + 3*sqrt(a*c*g)*(-I*sqrt(2)*g*si
n(f*x + e) - I*sqrt(2)*g)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(f*x + e) + I*sin(f*x + e))) +
3*sqrt(a*c*g)*(I*sqrt(2)*g*sin(f*x + e) + I*sqrt(2)*g)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(f
*x + e) - I*sin(f*x + e))))/(a^2*f*sin(f*x + e) + a^2*f)
Sympy [F(-1)]
Timed out. \[
\int \frac {(g \cos (e+f x))^{3/2} \sqrt {c-c \sin (e+f x)}}{(a+a \sin (e+f x))^{3/2}} \, dx=\text {Timed out}
\]
[In]
integrate((g*cos(f*x+e))**(3/2)*(c-c*sin(f*x+e))**(1/2)/(a+a*sin(f*x+e))**(3/2),x)
[Out]
Timed out
Maxima [F]
\[
\int \frac {(g \cos (e+f x))^{3/2} \sqrt {c-c \sin (e+f x)}}{(a+a \sin (e+f x))^{3/2}} \, dx=\int { \frac {\left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} \sqrt {-c \sin \left (f x + e\right ) + c}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}}} \,d x }
\]
[In]
integrate((g*cos(f*x+e))^(3/2)*(c-c*sin(f*x+e))^(1/2)/(a+a*sin(f*x+e))^(3/2),x, algorithm="maxima")
[Out]
integrate((g*cos(f*x + e))^(3/2)*sqrt(-c*sin(f*x + e) + c)/(a*sin(f*x + e) + a)^(3/2), x)
Giac [F(-1)]
Timed out. \[
\int \frac {(g \cos (e+f x))^{3/2} \sqrt {c-c \sin (e+f x)}}{(a+a \sin (e+f x))^{3/2}} \, dx=\text {Timed out}
\]
[In]
integrate((g*cos(f*x+e))^(3/2)*(c-c*sin(f*x+e))^(1/2)/(a+a*sin(f*x+e))^(3/2),x, algorithm="giac")
[Out]
Timed out
Mupad [F(-1)]
Timed out. \[
\int \frac {(g \cos (e+f x))^{3/2} \sqrt {c-c \sin (e+f x)}}{(a+a \sin (e+f x))^{3/2}} \, dx=\int \frac {{\left (g\,\cos \left (e+f\,x\right )\right )}^{3/2}\,\sqrt {c-c\,\sin \left (e+f\,x\right )}}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{3/2}} \,d x
\]
[In]
int(((g*cos(e + f*x))^(3/2)*(c - c*sin(e + f*x))^(1/2))/(a + a*sin(e + f*x))^(3/2),x)
[Out]
int(((g*cos(e + f*x))^(3/2)*(c - c*sin(e + f*x))^(1/2))/(a + a*sin(e + f*x))^(3/2), x)