3.1.59 \(\int (-1-\cos ^2(x))^{3/2} \, dx\) [59]

Optimal. Leaf size=89 \[ -\frac {2 \sqrt {-1-\cos ^2(x)} E\left (\left .\frac {\pi }{2}+x\right |-1\right )}{\sqrt {1+\cos ^2(x)}}-\frac {2 \sqrt {1+\cos ^2(x)} F\left (\left .\frac {\pi }{2}+x\right |-1\right )}{3 \sqrt {-1-\cos ^2(x)}}-\frac {1}{3} \cos (x) \sqrt {-1-\cos ^2(x)} \sin (x) \]

[Out]

-1/3*cos(x)*sin(x)*(-1-cos(x)^2)^(1/2)+2*(sin(x)^2)^(1/2)/sin(x)*EllipticE(cos(x),I)*(-1-cos(x)^2)^(1/2)/(1+co
s(x)^2)^(1/2)+2/3*(sin(x)^2)^(1/2)/sin(x)*EllipticF(cos(x),I)*(1+cos(x)^2)^(1/2)/(-1-cos(x)^2)^(1/2)

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Rubi [A]
time = 0.06, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3259, 3251, 3257, 3256, 3262, 3261} \begin {gather*} -\frac {1}{3} \sin (x) \cos (x) \sqrt {-\cos ^2(x)-1}-\frac {2 \sqrt {\cos ^2(x)+1} F\left (\left .x+\frac {\pi }{2}\right |-1\right )}{3 \sqrt {-\cos ^2(x)-1}}-\frac {2 \sqrt {-\cos ^2(x)-1} E\left (\left .x+\frac {\pi }{2}\right |-1\right )}{\sqrt {\cos ^2(x)+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 - Cos[x]^2)^(3/2),x]

[Out]

(-2*Sqrt[-1 - Cos[x]^2]*EllipticE[Pi/2 + x, -1])/Sqrt[1 + Cos[x]^2] - (2*Sqrt[1 + Cos[x]^2]*EllipticF[Pi/2 + x
, -1])/(3*Sqrt[-1 - Cos[x]^2]) - (Cos[x]*Sqrt[-1 - Cos[x]^2]*Sin[x])/3

Rule 3251

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)]^2)/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2], x_Symbol] :> Dist[
B/b, Int[Sqrt[a + b*Sin[e + f*x]^2], x], x] + Dist[(A*b - a*B)/b, Int[1/Sqrt[a + b*Sin[e + f*x]^2], x], x] /;
FreeQ[{a, b, e, f, A, B}, x]

Rule 3256

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2], x_Symbol] :> Simp[(Sqrt[a]/f)*EllipticE[e + f*x, -b/a], x] /
; FreeQ[{a, b, e, f}, x] && GtQ[a, 0]

Rule 3257

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2], x_Symbol] :> Dist[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[1 + b*(Sin
[e + f*x]^2/a)], Int[Sqrt[1 + (b*Sin[e + f*x]^2)/a], x], x] /; FreeQ[{a, b, e, f}, x] &&  !GtQ[a, 0]

Rule 3259

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Simp[(-b)*Cos[e + f*x]*Sin[e + f*x]*((a + b*Si
n[e + f*x]^2)^(p - 1)/(2*f*p)), x] + Dist[1/(2*p), Int[(a + b*Sin[e + f*x]^2)^(p - 2)*Simp[a*(b + 2*a*p) + b*(
2*a + b)*(2*p - 1)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a + b, 0] && GtQ[p, 1]

Rule 3261

Int[1/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2], x_Symbol] :> Simp[(1/(Sqrt[a]*f))*EllipticF[e + f*x, -b/a]
, x] /; FreeQ[{a, b, e, f}, x] && GtQ[a, 0]

Rule 3262

Int[1/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2], x_Symbol] :> Dist[Sqrt[1 + b*(Sin[e + f*x]^2/a)]/Sqrt[a +
b*Sin[e + f*x]^2], Int[1/Sqrt[1 + (b*Sin[e + f*x]^2)/a], x], x] /; FreeQ[{a, b, e, f}, x] &&  !GtQ[a, 0]

Rubi steps

\begin {align*} \int \left (-1-\cos ^2(x)\right )^{3/2} \, dx &=-\frac {1}{3} \cos (x) \sqrt {-1-\cos ^2(x)} \sin (x)+\frac {1}{3} \int \frac {4+6 \cos ^2(x)}{\sqrt {-1-\cos ^2(x)}} \, dx\\ &=-\frac {1}{3} \cos (x) \sqrt {-1-\cos ^2(x)} \sin (x)-\frac {2}{3} \int \frac {1}{\sqrt {-1-\cos ^2(x)}} \, dx-2 \int \sqrt {-1-\cos ^2(x)} \, dx\\ &=-\frac {1}{3} \cos (x) \sqrt {-1-\cos ^2(x)} \sin (x)-\frac {\left (2 \sqrt {-1-\cos ^2(x)}\right ) \int \sqrt {1+\cos ^2(x)} \, dx}{\sqrt {1+\cos ^2(x)}}-\frac {\left (2 \sqrt {1+\cos ^2(x)}\right ) \int \frac {1}{\sqrt {1+\cos ^2(x)}} \, dx}{3 \sqrt {-1-\cos ^2(x)}}\\ &=-\frac {2 \sqrt {-1-\cos ^2(x)} E\left (\left .\frac {\pi }{2}+x\right |-1\right )}{\sqrt {1+\cos ^2(x)}}-\frac {2 \sqrt {1+\cos ^2(x)} F\left (\left .\frac {\pi }{2}+x\right |-1\right )}{3 \sqrt {-1-\cos ^2(x)}}-\frac {1}{3} \cos (x) \sqrt {-1-\cos ^2(x)} \sin (x)\\ \end {align*}

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Mathematica [A]
time = 0.08, size = 66, normalized size = 0.74 \begin {gather*} \frac {48 \sqrt {3+\cos (2 x)} E\left (x\left |\frac {1}{2}\right .\right )-8 \sqrt {3+\cos (2 x)} F\left (x\left |\frac {1}{2}\right .\right )+6 \sin (2 x)+\sin (4 x)}{12 \sqrt {2} \sqrt {-3-\cos (2 x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 - Cos[x]^2)^(3/2),x]

[Out]

(48*Sqrt[3 + Cos[2*x]]*EllipticE[x, 1/2] - 8*Sqrt[3 + Cos[2*x]]*EllipticF[x, 1/2] + 6*Sin[2*x] + Sin[4*x])/(12
*Sqrt[2]*Sqrt[-3 - Cos[2*x]])

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Maple [A]
time = 0.47, size = 110, normalized size = 1.24

method	result	size
default	\(\frac {\sqrt {-\left (1+\cos ^{2}\left (x \right )\right ) \left (\sin ^{2}\left (x \right )\right )}\, \left (-\cos \left (x \right ) \left (\sin ^{4}\left (x \right )\right )+10 i \sqrt {-\left (\sin ^{2}\left (x \right )\right )+2}\, \sqrt {\frac {1}{2}-\frac {\cos \left (2 x \right )}{2}}\, \EllipticF \left (i \cos \left (x \right ), i\right )-6 i \sqrt {-\left (\sin ^{2}\left (x \right )\right )+2}\, \sqrt {\frac {1}{2}-\frac {\cos \left (2 x \right )}{2}}\, \EllipticE \left (i \cos \left (x \right ), i\right )+2 \left (\sin ^{2}\left (x \right )\right ) \cos \left (x \right )\right )}{3 \sqrt {\cos ^{4}\left (x \right )-1}\, \sin \left (x \right ) \sqrt {-1-\left (\cos ^{2}\left (x \right )\right )}}\)	\(110\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-1-cos(x)^2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/3*(-(1+cos(x)^2)*sin(x)^2)^(1/2)*(-cos(x)*sin(x)^4+10*I*(-sin(x)^2+2)^(1/2)*(sin(x)^2)^(1/2)*EllipticF(I*cos
(x),I)-6*I*(-sin(x)^2+2)^(1/2)*(sin(x)^2)^(1/2)*EllipticE(I*cos(x),I)+2*sin(x)^2*cos(x))/(cos(x)^4-1)^(1/2)/si
n(x)/(-1-cos(x)^2)^(1/2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1-cos(x)^2)^(3/2),x, algorithm="maxima")

[Out]

integrate((-cos(x)^2 - 1)^(3/2), x)

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Fricas [F]
time = 0.09, size = 145, normalized size = 1.63 \begin {gather*} \frac {24 \, {\left (e^{\left (4 i \, x\right )} - e^{\left (3 i \, x\right )}\right )} {\rm integral}\left (-\frac {4 \, \sqrt {e^{\left (4 i \, x\right )} + 6 \, e^{\left (2 i \, x\right )} + 1} {\left (5 \, e^{\left (2 i \, x\right )} + 2 \, e^{\left (i \, x\right )} + 5\right )}}{3 \, {\left (e^{\left (6 i \, x\right )} - 2 \, e^{\left (5 i \, x\right )} + 7 \, e^{\left (4 i \, x\right )} - 12 \, e^{\left (3 i \, x\right )} + 7 \, e^{\left (2 i \, x\right )} - 2 \, e^{\left (i \, x\right )} + 1\right )}}, x\right ) - {\left (e^{\left (5 i \, x\right )} - e^{\left (4 i \, x\right )} + 24 \, e^{\left (3 i \, x\right )} + 24 \, e^{\left (2 i \, x\right )} - e^{\left (i \, x\right )} + 1\right )} \sqrt {e^{\left (4 i \, x\right )} + 6 \, e^{\left (2 i \, x\right )} + 1}}{24 \, {\left (e^{\left (4 i \, x\right )} - e^{\left (3 i \, x\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1-cos(x)^2)^(3/2),x, algorithm="fricas")

[Out]

1/24*(24*(e^(4*I*x) - e^(3*I*x))*integral(-4/3*sqrt(e^(4*I*x) + 6*e^(2*I*x) + 1)*(5*e^(2*I*x) + 2*e^(I*x) + 5)
/(e^(6*I*x) - 2*e^(5*I*x) + 7*e^(4*I*x) - 12*e^(3*I*x) + 7*e^(2*I*x) - 2*e^(I*x) + 1), x) - (e^(5*I*x) - e^(4*
I*x) + 24*e^(3*I*x) + 24*e^(2*I*x) - e^(I*x) + 1)*sqrt(e^(4*I*x) + 6*e^(2*I*x) + 1))/(e^(4*I*x) - e^(3*I*x))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (- \cos ^{2}{\left (x \right )} - 1\right )^{\frac {3}{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1-cos(x)**2)**(3/2),x)

[Out]

Integral((-cos(x)**2 - 1)**(3/2), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1-cos(x)^2)^(3/2),x, algorithm="giac")

[Out]

integrate((-cos(x)^2 - 1)^(3/2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (-{\cos \left (x\right )}^2-1\right )}^{3/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((- cos(x)^2 - 1)^(3/2),x)

[Out]

int((- cos(x)^2 - 1)^(3/2), x)

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