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

Optimal. Leaf size=45 \[ -\frac {\cos (e+f x) (c-c \sin (e+f x))^{7/2}}{4 c f \sqrt {a \sin (e+f x)+a}} \]

[Out]

-1/4*cos(f*x+e)*(c-c*sin(f*x+e))^(7/2)/c/f/(a+a*sin(f*x+e))^(1/2)

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Rubi [A]  time = 0.31, 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} \[ -\frac {\cos (e+f x) (c-c \sin (e+f x))^{7/2}}{4 c f \sqrt {a \sin (e+f x)+a}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Cos[e + f*x]*(c - c*Sin[e + f*x])^(7/2))/(4*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) (c-c \sin (e+f x))^{5/2}}{\sqrt {a+a \sin (e+f x)}} \, dx &=\frac {\int \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2} \, dx}{a c}\\ &=-\frac {\cos (e+f x) (c-c \sin (e+f x))^{7/2}}{4 c f \sqrt {a+a \sin (e+f x)}}\\ \end {align*}

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Mathematica [B]  time = 0.90, size = 134, normalized size = 2.98 \[ \frac {c^2 (\sin (e+f x)-1)^2 \sqrt {c-c \sin (e+f x)} \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right ) (56 \sin (e+f x)-8 \sin (3 (e+f x))+28 \cos (2 (e+f x))-\cos (4 (e+f x)))}{32 f \sqrt {a (\sin (e+f x)+1)} \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c^2*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*(-1 + Sin[e + f*x])^2*Sqrt[c - c*Sin[e + f*x]]*(28*Cos[2*(e + f*x)]
- Cos[4*(e + f*x)] + 56*Sin[e + f*x] - 8*Sin[3*(e + f*x)]))/(32*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^5*Sqr
t[a*(1 + Sin[e + f*x])])

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fricas [B]  time = 0.45, size = 98, normalized size = 2.18 \[ -\frac {{\left (c^{2} \cos \left (f x + e\right )^{4} - 8 \, c^{2} \cos \left (f x + e\right )^{2} + 7 \, c^{2} + 4 \, {\left (c^{2} \cos \left (f x + e\right )^{2} - 2 \, c^{2}\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{4 \, a f \cos \left (f x + e\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(f*x+e)^2*(c-c*sin(f*x+e))^(5/2)/(a+a*sin(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

-1/4*(c^2*cos(f*x + e)^4 - 8*c^2*cos(f*x + e)^2 + 7*c^2 + 4*(c^2*cos(f*x + e)^2 - 2*c^2)*sin(f*x + e))*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(-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*(c-c*sin(f*x+e))^(5/2)/(a+a*sin(f*x+e))^(1/2),x, algorithm="giac")

[Out]

Timed out

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maple [B]  time = 0.38, size = 195, normalized size = 4.33 \[ \frac {\sin \left (f x +e \right ) \left (-c \left (\sin \left (f x +e \right )-1\right )\right )^{\frac {5}{2}} \left (\sin \left (f x +e \right ) \left (\cos ^{3}\left (f x +e \right )\right )+\cos ^{4}\left (f x +e \right )+3 \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )-4 \left (\cos ^{3}\left (f x +e \right )\right )-7 \sin \left (f x +e \right ) \cos \left (f x +e \right )-4 \left (\cos ^{2}\left (f x +e \right )\right )-\sin \left (f x +e \right )+8 \cos \left (f x +e \right )-1\right )}{4 f \sqrt {a \left (1+\sin \left (f x +e \right )\right )}\, \left (\left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )+\cos ^{3}\left (f x +e \right )+2 \sin \left (f x +e \right ) \cos \left (f x +e \right )-3 \left (\cos ^{2}\left (f x +e \right )\right )-4 \sin \left (f x +e \right )-2 \cos \left (f x +e \right )+4\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4/f*sin(f*x+e)*(-c*(sin(f*x+e)-1))^(5/2)*(sin(f*x+e)*cos(f*x+e)^3+cos(f*x+e)^4+3*cos(f*x+e)^2*sin(f*x+e)-4*c
os(f*x+e)^3-7*sin(f*x+e)*cos(f*x+e)-4*cos(f*x+e)^2-sin(f*x+e)+8*cos(f*x+e)-1)/(a*(1+sin(f*x+e)))^(1/2)/(cos(f*
x+e)^2*sin(f*x+e)+cos(f*x+e)^3+2*sin(f*x+e)*cos(f*x+e)-3*cos(f*x+e)^2-4*sin(f*x+e)-2*cos(f*x+e)+4)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(f*x+e)^2*(c-c*sin(f*x+e))^(5/2)/(a+a*sin(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 2.00, size = 96, normalized size = 2.13 \[ -\frac {c^2\,\sqrt {-c\,\left (\sin \left (e+f\,x\right )-1\right )}\,\left (28\,\cos \left (e+f\,x\right )+27\,\cos \left (3\,e+3\,f\,x\right )-\cos \left (5\,e+5\,f\,x\right )+48\,\sin \left (2\,e+2\,f\,x\right )-8\,\sin \left (4\,e+4\,f\,x\right )\right )}{64\,f\,\sqrt {a\,\left (\sin \left (e+f\,x\right )+1\right )}\,\left (\sin \left (e+f\,x\right )-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(c^2*(-c*(sin(e + f*x) - 1))^(1/2)*(28*cos(e + f*x) + 27*cos(3*e + 3*f*x) - cos(5*e + 5*f*x) + 48*sin(2*e + 2
*f*x) - 8*sin(4*e + 4*f*x)))/(64*f*(a*(sin(e + f*x) + 1))^(1/2)*(sin(e + f*x) - 1))

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

[Out]

Timed out

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