3.12 \(\int \cos ^2(e+f x) (a+a \sin (e+f x))^{3/2} \sqrt {c-c \sin (e+f x)} \, dx\)

Optimal. Leaf size=92 \[ \frac {\cos (e+f x) (a \sin (e+f x)+a)^{5/2} \sqrt {c-c \sin (e+f x)}}{4 a f}+\frac {c \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{6 a f \sqrt {c-c \sin (e+f x)}} \]

[Out]

1/6*c*cos(f*x+e)*(a+a*sin(f*x+e))^(5/2)/a/f/(c-c*sin(f*x+e))^(1/2)+1/4*cos(f*x+e)*(a+a*sin(f*x+e))^(5/2)*(c-c*
sin(f*x+e))^(1/2)/a/f

________________________________________________________________________________________

Rubi [A]  time = 0.39, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.079, Rules used = {2841, 2740, 2738} \[ \frac {\cos (e+f x) (a \sin (e+f x)+a)^{5/2} \sqrt {c-c \sin (e+f x)}}{4 a f}+\frac {c \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{6 a f \sqrt {c-c \sin (e+f x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c*Cos[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(6*a*f*Sqrt[c - c*Sin[e + f*x]]) + (Cos[e + f*x]*(a + a*Sin[e + f*
x])^(5/2)*Sqrt[c - c*Sin[e + f*x]])/(4*a*f)

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 2740

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> -Sim
p[(b*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n)/(f*(m + n)), x] + Dist[(a*(2*m - 1))/(m
+ n), Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && E
qQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IGtQ[m - 1/2, 0] &&  !LtQ[n, -1] &&  !(IGtQ[n - 1/2, 0] && LtQ[n, m])
&&  !(ILtQ[m + n, 0] && GtQ[2*m + n + 1, 0])

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 \cos ^2(e+f x) (a+a \sin (e+f x))^{3/2} \sqrt {c-c \sin (e+f x)} \, dx &=\frac {\int (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2} \, dx}{a c}\\ &=\frac {\cos (e+f x) (a+a \sin (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}}{4 a f}+\frac {\int (a+a \sin (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)} \, dx}{2 a}\\ &=\frac {c \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{6 a f \sqrt {c-c \sin (e+f x)}}+\frac {\cos (e+f x) (a+a \sin (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}}{4 a f}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.39, size = 83, normalized size = 0.90 \[ \frac {a \sec (e+f x) \sqrt {a (\sin (e+f x)+1)} \sqrt {c-c \sin (e+f x)} (8 (9 \sin (e+f x)+\sin (3 (e+f x)))-12 \cos (2 (e+f x))-3 \cos (4 (e+f x)))}{96 f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*Sec[e + f*x]*Sqrt[a*(1 + Sin[e + f*x])]*Sqrt[c - c*Sin[e + f*x]]*(-12*Cos[2*(e + f*x)] - 3*Cos[4*(e + f*x)]
+ 8*(9*Sin[e + f*x] + Sin[3*(e + f*x)])))/(96*f)

________________________________________________________________________________________

fricas [A]  time = 0.42, size = 75, normalized size = 0.82 \[ -\frac {{\left (3 \, a \cos \left (f x + e\right )^{4} - 4 \, {\left (a \cos \left (f x + e\right )^{2} + 2 \, a\right )} \sin \left (f x + e\right ) - 3 \, a\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{12 \, f \cos \left (f x + e\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(f*x+e)^2*(a+a*sin(f*x+e))^(3/2)*(c-c*sin(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

-1/12*(3*a*cos(f*x + e)^4 - 4*(a*cos(f*x + e)^2 + 2*a)*sin(f*x + e) - 3*a)*sqrt(a*sin(f*x + e) + a)*sqrt(-c*si
n(f*x + e) + c)/(f*cos(f*x + e))

________________________________________________________________________________________

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*(a+a*sin(f*x+e))^(3/2)*(c-c*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: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check si
gn: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/
x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check si
gn: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)sqrt(2*a)*sqrt(2*c)*(-24*a*f*sign(sin(1/
2*(f*x+exp(1))-1/4*pi))*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*sin(f*x+exp(1))/(8*f)^2-24*a*f*sign(sin(1/2*(f*x+ex
p(1))-1/4*pi))*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*sin(3*f*x+3*exp(1))/(24*f)^2+32*a*f*sign(sin(1/2*(f*x+exp(1)
)-1/4*pi))*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*cos(2*f*x+2*exp(1))/(32*f)^2+64*a*f*sign(sin(1/2*(f*x+exp(1))-1/
4*pi))*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*cos(4*f*x+4*exp(1))/(64*f)^2+32*a*f*sign(sin(1/2*(f*x+exp(1))-1/4*pi
))*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*cos(-2*f*x-2*exp(1))/(-32*f)^2)

________________________________________________________________________________________

maple [A]  time = 0.39, size = 90, normalized size = 0.98 \[ -\frac {\sqrt {-c \left (\sin \left (f x +e \right )-1\right )}\, \sin \left (f x +e \right ) \left (a \left (1+\sin \left (f x +e \right )\right )\right )^{\frac {3}{2}} \left (-3 \left (\cos ^{4}\left (f x +e \right )\right )+\left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )-4 \left (\cos ^{2}\left (f x +e \right )\right )+5 \sin \left (f x +e \right )-5\right )}{12 f \cos \left (f x +e \right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12/f*(-c*(sin(f*x+e)-1))^(1/2)*sin(f*x+e)*(a*(1+sin(f*x+e)))^(3/2)*(-3*cos(f*x+e)^4+cos(f*x+e)^2*sin(f*x+e)
-4*cos(f*x+e)^2+5*sin(f*x+e)-5)/cos(f*x+e)^3

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(f*x+e)^2*(a+a*sin(f*x+e))^(3/2)*(c-c*sin(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 9.81, size = 97, normalized size = 1.05 \[ -\frac {a\,\sqrt {a\,\left (\sin \left (e+f\,x\right )+1\right )}\,\sqrt {-c\,\left (\sin \left (e+f\,x\right )-1\right )}\,\left (12\,\cos \left (e+f\,x\right )+15\,\cos \left (3\,e+3\,f\,x\right )+3\,\cos \left (5\,e+5\,f\,x\right )-80\,\sin \left (2\,e+2\,f\,x\right )-8\,\sin \left (4\,e+4\,f\,x\right )\right )}{96\,f\,\left (\cos \left (2\,e+2\,f\,x\right )+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(a*(a*(sin(e + f*x) + 1))^(1/2)*(-c*(sin(e + f*x) - 1))^(1/2)*(12*cos(e + f*x) + 15*cos(3*e + 3*f*x) + 3*cos(
5*e + 5*f*x) - 80*sin(2*e + 2*f*x) - 8*sin(4*e + 4*f*x)))/(96*f*(cos(2*e + 2*f*x) + 1))

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________