3.59 \(\int (-1-\cos ^2(x))^{3/2} \, dx\)

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

[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

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Rubi [A]  time = 0.0941136, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {3180, 3172, 3178, 3177, 3183, 3182} \[ -\frac{1}{3} \sin (x) \cos (x) \sqrt{-\cos ^2(x)-1}-\frac{2 \sqrt{\cos ^2(x)+1} F\left (\left .x+\frac{\pi }{2}\right |-1\right )}{3 \sqrt{-\cos ^2(x)-1}}-\frac{2 \sqrt{-\cos ^2(x)-1} E\left (\left .x+\frac{\pi }{2}\right |-1\right )}{\sqrt{\cos ^2(x)+1}} \]

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 3180

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> -Simp[(b*Cos[e + f*x]*Sin[e + f*x]*(a + b*Sin[
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 3172

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 3178

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 3177

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

Rule 3183

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]

Rule 3182

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

Mathematica [A]  time = 0.0680792, size = 66, normalized size = 0.74 \[ \frac{6 \sin (2 x)+\sin (4 x)-8 \sqrt{\cos (2 x)+3} F\left (x\left |\frac{1}{2}\right .\right )+48 \sqrt{\cos (2 x)+3} E\left (x\left |\frac{1}{2}\right .\right )}{12 \sqrt{2} \sqrt{-\cos (2 x)-3}} \]

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 = 1.364, size = 110, normalized size = 1.2 \begin{align*}{\frac{1}{3\,\sin \left ( x \right ) }\sqrt{- \left ( 1+ \left ( \cos \left ( x \right ) \right ) ^{2} \right ) \left ( \sin \left ( x \right ) \right ) ^{2}} \left ( -\cos \left ( x \right ) \left ( \sin \left ( x \right ) \right ) ^{4}+10\,i\sqrt{- \left ( \sin \left ( x \right ) \right ) ^{2}+2}\sqrt{ \left ( \sin \left ( x \right ) \right ) ^{2}}{\it EllipticF} \left ( i\cos \left ( x \right ) ,i \right ) -6\,i\sqrt{- \left ( \sin \left ( x \right ) \right ) ^{2}+2}\sqrt{ \left ( \sin \left ( x \right ) \right ) ^{2}}{\it EllipticE} \left ( i\cos \left ( x \right ) ,i \right ) +2\,\cos \left ( x \right ) \left ( \sin \left ( x \right ) \right ) ^{2} \right ){\frac{1}{\sqrt{ \left ( \cos \left ( x \right ) \right ) ^{4}-1}}}{\frac{1}{\sqrt{-1- \left ( \cos \left ( x \right ) \right ) ^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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*cos(x)*sin(x)^2)/(cos(x)^4-1)^(1/2)/si
n(x)/(-1-cos(x)^2)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (-\cos \left (x\right )^{2} - 1\right )}^{\frac{3}{2}}\,{d x} \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \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{align*}

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(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (-\cos \left (x\right )^{2} - 1\right )}^{\frac{3}{2}}\,{d x} \end{align*}

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)