3.46 \(\int \frac {\cos ^2(e+f x) \sqrt {c-c \sin (e+f x)}}{\sqrt {a+a \sin (e+f x)}} \, dx\)

Optimal. Leaf size=45 \[ -\frac {\cos (e+f x) (c-c \sin (e+f x))^{3/2}}{2 c f \sqrt {a \sin (e+f x)+a}} \]

[Out]

-1/2*cos(f*x+e)*(c-c*sin(f*x+e))^(3/2)/c/f/(a+a*sin(f*x+e))^(1/2)

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Rubi [A]  time = 0.28, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2841, 2738} \[ -\frac {\cos (e+f x) (c-c \sin (e+f x))^{3/2}}{2 c f \sqrt {a \sin (e+f x)+a}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Cos[e + f*x]*(c - c*Sin[e + f*x])^(3/2))/(2*c*f*Sqrt[a + a*Sin[e + f*x]])

Rule 2738

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[
(-2*b*Cos[e + f*x]*(c + d*Sin[e + f*x])^n)/(f*(2*n + 1)*Sqrt[a + b*Sin[e + f*x]]), x] /; FreeQ[{a, b, c, d, e,
f, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[n, -2^(-1)]

Rule 2841

Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*sin[(e_.) + (f_.)*
(x_)])^(n_.), x_Symbol] :> Dist[1/(a^(p/2)*c^(p/2)), Int[(a + b*Sin[e + f*x])^(m + p/2)*(c + d*Sin[e + f*x])^(
n + p/2), x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[p
/2]

Rubi steps

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

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Mathematica [A]  time = 0.29, size = 62, normalized size = 1.38 \[ \frac {\sec (e+f x) \sqrt {a (\sin (e+f x)+1)} \sqrt {c-c \sin (e+f x)} (4 \sin (e+f x)+\cos (2 (e+f x)))}{4 a f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sec[e + f*x]*Sqrt[a*(1 + Sin[e + f*x])]*(Cos[2*(e + f*x)] + 4*Sin[e + f*x])*Sqrt[c - c*Sin[e + f*x]])/(4*a*f)

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fricas [A]  time = 0.44, size = 59, normalized size = 1.31 \[ \frac {{\left (\cos \left (f x + e\right )^{2} + 2 \, \sin \left (f x + e\right ) - 1\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{2 \, a f \cos \left (f x + e\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(f*x+e)^2*(c-c*sin(f*x+e))^(1/2)/(a+a*sin(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

1/2*(cos(f*x + e)^2 + 2*sin(f*x + e) - 1)*sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)/(a*f*cos(f*x + e)
)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(f*x+e)^2*(c-c*sin(f*x+e))^(1/2)/(a+a*sin(f*x+e))^(1/2),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: Unable to check sign: (4*pi/x/2)>(-4*pi/
x/2)Unable to check sign: (8*pi/x/2)>(-8*pi/x/2)-32*sqrt(2*c)*sign(sin(1/2*(f*x+exp(1))-1/4*pi))*tan(1/2*(1/2*
f*x+1/4*(2*exp(1)-pi)))^4/sqrt(2)/sqrt(a)/(tan(1/2*(1/2*f*x+1/4*(2*exp(1)-pi)))^2+1)^4/f/sign(tan(1/2*(1/2*f*x
+1/4*(2*exp(1)-pi)))^4-1)

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maple [B]  time = 0.37, size = 90, normalized size = 2.00 \[ \frac {\sqrt {-c \left (\sin \left (f x +e \right )-1\right )}\, \sin \left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )+\sin \left (f x +e \right ) \cos \left (f x +e \right )-2 \cos \left (f x +e \right )+\sin \left (f x +e \right )+1\right )}{2 f \sqrt {a \left (1+\sin \left (f x +e \right )\right )}\, \left (-1+\cos \left (f x +e \right )+\sin \left (f x +e \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(f*x+e)^2*(c-c*sin(f*x+e))^(1/2)/(a+a*sin(f*x+e))^(1/2),x)

[Out]

1/2/f*(-c*(sin(f*x+e)-1))^(1/2)*sin(f*x+e)*(cos(f*x+e)^2+sin(f*x+e)*cos(f*x+e)-2*cos(f*x+e)+sin(f*x+e)+1)/(a*(
1+sin(f*x+e)))^(1/2)/(-1+cos(f*x+e)+sin(f*x+e))

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maxima [B]  time = 1.63, size = 388, normalized size = 8.62 \[ -\frac {\frac {2 \, \sqrt {a} \sqrt {c} + \frac {\sqrt {a} \sqrt {c} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {\sqrt {a} \sqrt {c} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {\sqrt {a} \sqrt {c} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}}{a + \frac {2 \, a \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {a \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}}} - \frac {2 \, \sqrt {a} \sqrt {c} - \frac {\sqrt {a} \sqrt {c} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {3 \, \sqrt {a} \sqrt {c} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {\sqrt {a} \sqrt {c} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}}{a + \frac {2 \, a \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {a \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}}} + \frac {2 \, {\left (\frac {\sqrt {a} \sqrt {c} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {\sqrt {a} \sqrt {c} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {\sqrt {a} \sqrt {c} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}\right )}}{a + \frac {2 \, a \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {a \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}}}}{2 \, f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(f*x+e)^2*(c-c*sin(f*x+e))^(1/2)/(a+a*sin(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

-1/2*((2*sqrt(a)*sqrt(c) + sqrt(a)*sqrt(c)*sin(f*x + e)/(cos(f*x + e) + 1) + sqrt(a)*sqrt(c)*sin(f*x + e)^2/(c
os(f*x + e) + 1)^2 + sqrt(a)*sqrt(c)*sin(f*x + e)^3/(cos(f*x + e) + 1)^3)/(a + 2*a*sin(f*x + e)^2/(cos(f*x + e
) + 1)^2 + a*sin(f*x + e)^4/(cos(f*x + e) + 1)^4) - (2*sqrt(a)*sqrt(c) - sqrt(a)*sqrt(c)*sin(f*x + e)/(cos(f*x
+ e) + 1) + 3*sqrt(a)*sqrt(c)*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - sqrt(a)*sqrt(c)*sin(f*x + e)^3/(cos(f*x +
e) + 1)^3)/(a + 2*a*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + a*sin(f*x + e)^4/(cos(f*x + e) + 1)^4) + 2*(sqrt(a)
*sqrt(c)*sin(f*x + e)/(cos(f*x + e) + 1) - sqrt(a)*sqrt(c)*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + sqrt(a)*sqrt(
c)*sin(f*x + e)^3/(cos(f*x + e) + 1)^3)/(a + 2*a*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + a*sin(f*x + e)^4/(cos(f
*x + e) + 1)^4))/f

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mupad [B]  time = 0.93, size = 67, normalized size = 1.49 \[ -\frac {\sqrt {-c\,\left (\sin \left (e+f\,x\right )-1\right )}\,\left (\cos \left (e+f\,x\right )+\cos \left (3\,e+3\,f\,x\right )+4\,\sin \left (2\,e+2\,f\,x\right )\right )}{8\,f\,\sqrt {a\,\left (\sin \left (e+f\,x\right )+1\right )}\,\left (\sin \left (e+f\,x\right )-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cos(e + f*x)^2*(c - c*sin(e + f*x))^(1/2))/(a + a*sin(e + f*x))^(1/2),x)

[Out]

-((-c*(sin(e + f*x) - 1))^(1/2)*(cos(e + f*x) + cos(3*e + 3*f*x) + 4*sin(2*e + 2*f*x)))/(8*f*(a*(sin(e + f*x)
+ 1))^(1/2)*(sin(e + f*x) - 1))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {- c \left (\sin {\left (e + f x \right )} - 1\right )} \cos ^{2}{\left (e + f x \right )}}{\sqrt {a \left (\sin {\left (e + f x \right )} + 1\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(f*x+e)**2*(c-c*sin(f*x+e))**(1/2)/(a+a*sin(f*x+e))**(1/2),x)

[Out]

Integral(sqrt(-c*(sin(e + f*x) - 1))*cos(e + f*x)**2/sqrt(a*(sin(e + f*x) + 1)), x)

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