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

Optimal. Leaf size=237 \[ -\frac {6 g \sqrt {\cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {g \cos (e+f x)}}{5 a^2 c f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {6 (g \cos (e+f x))^{5/2}}{5 a c f g (a \sin (e+f x)+a)^{3/2} \sqrt {c-c \sin (e+f x)}}-\frac {6 (g \cos (e+f x))^{5/2}}{5 c f g (a \sin (e+f x)+a)^{5/2} \sqrt {c-c \sin (e+f x)}}+\frac {2 (g \cos (e+f x))^{5/2}}{f g (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{3/2}} \]

[Out]

2*(g*cos(f*x+e))^(5/2)/f/g/(a+a*sin(f*x+e))^(5/2)/(c-c*sin(f*x+e))^(3/2)-6/5*(g*cos(f*x+e))^(5/2)/c/f/g/(a+a*s
in(f*x+e))^(5/2)/(c-c*sin(f*x+e))^(1/2)-6/5*(g*cos(f*x+e))^(5/2)/a/c/f/g/(a+a*sin(f*x+e))^(3/2)/(c-c*sin(f*x+e
))^(1/2)-6/5*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/c/f/(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e))^(1/2)

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Rubi [A]  time = 1.16, antiderivative size = 237, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2852, 2842, 2640, 2639} \[ -\frac {6 g \sqrt {\cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {g \cos (e+f x)}}{5 a^2 c f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {6 (g \cos (e+f x))^{5/2}}{5 a c f g (a \sin (e+f x)+a)^{3/2} \sqrt {c-c \sin (e+f x)}}-\frac {6 (g \cos (e+f x))^{5/2}}{5 c f g (a \sin (e+f x)+a)^{5/2} \sqrt {c-c \sin (e+f x)}}+\frac {2 (g \cos (e+f x))^{5/2}}{f g (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(g*Cos[e + f*x])^(3/2)/((a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(3/2)),x]

[Out]

(2*(g*Cos[e + f*x])^(5/2))/(f*g*(a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(3/2)) - (6*(g*Cos[e + f*x])^(
5/2))/(5*c*f*g*(a + a*Sin[e + f*x])^(5/2)*Sqrt[c - c*Sin[e + f*x]]) - (6*(g*Cos[e + f*x])^(5/2))/(5*a*c*f*g*(a
+ a*Sin[e + f*x])^(3/2)*Sqrt[c - c*Sin[e + f*x]]) - (6*g*Sqrt[Cos[e + f*x]]*Sqrt[g*Cos[e + f*x]]*EllipticE[(e
+ f*x)/2, 2])/(5*a^2*c*f*Sqrt[a + a*Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]])

Rule 2639

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticE[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ[{
c, d}, x]

Rule 2640

Int[Sqrt[(b_)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[b*Sin[c + d*x]]/Sqrt[Sin[c + d*x]], Int[Sqrt[Si
n[c + d*x]], x], x] /; FreeQ[{b, c, d}, x]

Rule 2842

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 2852

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*} \int \frac {(g \cos (e+f x))^{3/2}}{(a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}} \, dx &=\frac {2 (g \cos (e+f x))^{5/2}}{f g (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}+\frac {3 \int \frac {(g \cos (e+f x))^{3/2}}{(a+a \sin (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}} \, dx}{c}\\ &=\frac {2 (g \cos (e+f x))^{5/2}}{f g (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}-\frac {6 (g \cos (e+f x))^{5/2}}{5 c f g (a+a \sin (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}}+\frac {3 \int \frac {(g \cos (e+f x))^{3/2}}{(a+a \sin (e+f x))^{3/2} \sqrt {c-c \sin (e+f x)}} \, dx}{5 a c}\\ &=\frac {2 (g \cos (e+f x))^{5/2}}{f g (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}-\frac {6 (g \cos (e+f x))^{5/2}}{5 c f g (a+a \sin (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}}-\frac {6 (g \cos (e+f x))^{5/2}}{5 a c f g (a+a \sin (e+f x))^{3/2} \sqrt {c-c \sin (e+f x)}}-\frac {3 \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 c}\\ &=\frac {2 (g \cos (e+f x))^{5/2}}{f g (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}-\frac {6 (g \cos (e+f x))^{5/2}}{5 c f g (a+a \sin (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}}-\frac {6 (g \cos (e+f x))^{5/2}}{5 a c f g (a+a \sin (e+f x))^{3/2} \sqrt {c-c \sin (e+f x)}}-\frac {(3 g \cos (e+f x)) \int \sqrt {g \cos (e+f x)} \, dx}{5 a^2 c \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}\\ &=\frac {2 (g \cos (e+f x))^{5/2}}{f g (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}-\frac {6 (g \cos (e+f x))^{5/2}}{5 c f g (a+a \sin (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}}-\frac {6 (g \cos (e+f x))^{5/2}}{5 a c f g (a+a \sin (e+f x))^{3/2} \sqrt {c-c \sin (e+f x)}}-\frac {\left (3 g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)}\right ) \int \sqrt {\cos (e+f x)} \, dx}{5 a^2 c \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}\\ &=\frac {2 (g \cos (e+f x))^{5/2}}{f g (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}-\frac {6 (g \cos (e+f x))^{5/2}}{5 c f g (a+a \sin (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}}-\frac {6 (g \cos (e+f x))^{5/2}}{5 a c f g (a+a \sin (e+f x))^{3/2} \sqrt {c-c \sin (e+f x)}}-\frac {6 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 c f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}\\ \end {align*}

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Mathematica [A]  time = 1.06, size = 133, normalized size = 0.56 \[ \frac {\sqrt {\cos (e+f x)} (g \cos (e+f x))^{3/2} \left (\sqrt {\cos (e+f x)} (-6 \sin (e+f x)+3 \cos (2 (e+f x))-1)+3 E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) (\sin (2 (e+f x))+2 \cos (e+f x))\right )}{5 c f (\sin (e+f x)-1) (a (\sin (e+f x)+1))^{5/2} \sqrt {c-c \sin (e+f x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[(g*Cos[e + f*x])^(3/2)/((a + a*Sin[e + f*x])^(5/2)*(c - c*Sin[e + f*x])^(3/2)),x]

[Out]

(Sqrt[Cos[e + f*x]]*(g*Cos[e + f*x])^(3/2)*(Sqrt[Cos[e + f*x]]*(-1 + 3*Cos[2*(e + f*x)] - 6*Sin[e + f*x]) + 3*
EllipticE[(e + f*x)/2, 2]*(2*Cos[e + f*x] + Sin[2*(e + f*x)])))/(5*c*f*(-1 + Sin[e + f*x])*(a*(1 + Sin[e + f*x
]))^(5/2)*Sqrt[c - c*Sin[e + f*x]])

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fricas [F]  time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {g \cos \left (f x + e\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c} g}{a^{3} c^{2} \cos \left (f x + e\right )^{3} \sin \left (f x + e\right ) + a^{3} c^{2} \cos \left (f x + e\right )^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*cos(f*x+e))^(3/2)/(a+a*sin(f*x+e))^(5/2)/(c-c*sin(f*x+e))^(3/2),x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*cos(f*x+e))^(3/2)/(a+a*sin(f*x+e))^(5/2)/(c-c*sin(f*x+e))^(3/2),x, algorithm="giac")

[Out]

integrate((g*cos(f*x + e))^(3/2)/((a*sin(f*x + e) + a)^(5/2)*(-c*sin(f*x + e) + c)^(3/2)), x)

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maple [C]  time = 0.56, size = 877, normalized size = 3.70 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*cos(f*x+e))^(3/2)/(a+a*sin(f*x+e))^(5/2)/(c-c*sin(f*x+e))^(3/2),x)

[Out]

-2/5/f*(cos(f*x+e)+1)^2*(g*cos(f*x+e))^(3/2)*(-1+cos(f*x+e))^2*(1+sin(f*x+e))*(sin(f*x+e)-1)*(3*I*EllipticE(I*
(-1+cos(f*x+e))/sin(f*x+e),I)*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*sin(f*x+e)+3*I*(1/(co
s(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*cos(f*x+e)^3*EllipticF(I*(-1+cos(f*x+e))/sin(f*x+e),I)-3*
I*EllipticF(I*(-1+cos(f*x+e))/sin(f*x+e),I)*sin(f*x+e)*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1
/2)-3*I*EllipticF(I*(-1+cos(f*x+e))/sin(f*x+e),I)*(1/(cos(f*x+e)+1))^(1/2)*sin(f*x+e)*cos(f*x+e)*(cos(f*x+e)/(
cos(f*x+e)+1))^(1/2)-3*I*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*EllipticF(I*(-1+cos(f*x+e)
)/sin(f*x+e),I)*cos(f*x+e)-3*I*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*cos(f*x+e)^3*Ellipti
cE(I*(-1+cos(f*x+e))/sin(f*x+e),I)+3*I*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*cos(f*x+e)^2
*EllipticF(I*(-1+cos(f*x+e))/sin(f*x+e),I)+3*I*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*Elli
pticE(I*(-1+cos(f*x+e))/sin(f*x+e),I)+3*I*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*EllipticE
(I*(-1+cos(f*x+e))/sin(f*x+e),I)*cos(f*x+e)+3*I*sin(f*x+e)*cos(f*x+e)*EllipticE(I*(-1+cos(f*x+e))/sin(f*x+e),I
)*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)-3*I*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x
+e)+1))^(1/2)*cos(f*x+e)^2*EllipticE(I*(-1+cos(f*x+e))/sin(f*x+e),I)-3*I*EllipticF(I*(-1+cos(f*x+e))/sin(f*x+e
),I)*(1/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)-3*sin(f*x+e)*cos(f*x+e)+2*sin(f*x+e)-3*cos(f*x
+e)+3)/(a*(1+sin(f*x+e)))^(5/2)/(-c*(sin(f*x+e)-1))^(3/2)/cos(f*x+e)/sin(f*x+e)^5

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*cos(f*x+e))^(3/2)/(a+a*sin(f*x+e))^(5/2)/(c-c*sin(f*x+e))^(3/2),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)^(3/2)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (g\,\cos \left (e+f\,x\right )\right )}^{3/2}}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{5/2}\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*cos(e + f*x))^(3/2)/((a + a*sin(e + f*x))^(5/2)*(c - c*sin(e + f*x))^(3/2)),x)

[Out]

int((g*cos(e + f*x))^(3/2)/((a + a*sin(e + f*x))^(5/2)*(c - c*sin(e + f*x))^(3/2)), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*cos(f*x+e))**(3/2)/(a+a*sin(f*x+e))**(5/2)/(c-c*sin(f*x+e))**(3/2),x)

[Out]

Timed out

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