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

Optimal. Leaf size=33 \[ \frac{2}{3} \sqrt{-\sin ^2(x)} \cot (x)-\frac{1}{3} \left (-\sin ^2(x)\right )^{3/2} \cot (x) \]

[Out]

(2*Cot[x]*Sqrt[-Sin[x]^2])/3 - (Cot[x]*(-Sin[x]^2)^(3/2))/3

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Rubi [A]  time = 0.0246809, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {3176, 3203, 3207, 2638} \[ \frac{2}{3} \sqrt{-\sin ^2(x)} \cot (x)-\frac{1}{3} \left (-\sin ^2(x)\right )^{3/2} \cot (x) \]

Antiderivative was successfully verified.

[In]

Int[(-1 + Cos[x]^2)^(3/2),x]

[Out]

(2*Cot[x]*Sqrt[-Sin[x]^2])/3 - (Cot[x]*(-Sin[x]^2)^(3/2))/3

Rule 3176

Int[(u_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Int[ActivateTrig[u*(a*cos[e + f*x]^2)^p]
, x] /; FreeQ[{a, b, e, f, p}, x] && EqQ[a + b, 0]

Rule 3203

Int[((b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> -Simp[(Cot[e + f*x]*(b*Sin[e + f*x]^2)^p)/(2*f*p), x]
+ Dist[(b*(2*p - 1))/(2*p), Int[(b*Sin[e + f*x]^2)^(p - 1), x], x] /; FreeQ[{b, e, f}, x] &&  !IntegerQ[p] &&
GtQ[p, 1]

Rule 3207

Int[(u_.)*((b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Di
st[((b*ff^n)^IntPart[p]*(b*Sin[e + f*x]^n)^FracPart[p])/(Sin[e + f*x]/ff)^(n*FracPart[p]), Int[ActivateTrig[u]
*(Sin[e + f*x]/ff)^(n*p), x], x]] /; FreeQ[{b, e, f, n, p}, x] &&  !IntegerQ[p] && IntegerQ[n] && (EqQ[u, 1] |
| MatchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) /; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig
]])

Rule 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int \left (-1+\cos ^2(x)\right )^{3/2} \, dx &=\int \left (-\sin ^2(x)\right )^{3/2} \, dx\\ &=-\frac{1}{3} \cot (x) \left (-\sin ^2(x)\right )^{3/2}-\frac{2}{3} \int \sqrt{-\sin ^2(x)} \, dx\\ &=-\frac{1}{3} \cot (x) \left (-\sin ^2(x)\right )^{3/2}-\frac{1}{3} \left (2 \csc (x) \sqrt{-\sin ^2(x)}\right ) \int \sin (x) \, dx\\ &=\frac{2}{3} \cot (x) \sqrt{-\sin ^2(x)}-\frac{1}{3} \cot (x) \left (-\sin ^2(x)\right )^{3/2}\\ \end{align*}

Mathematica [A]  time = 0.0255911, size = 25, normalized size = 0.76 \[ -\frac{1}{12} \sqrt{-\sin ^2(x)} (\cos (3 x)-9 \cos (x)) \csc (x) \]

Antiderivative was successfully verified.

[In]

Integrate[(-1 + Cos[x]^2)^(3/2),x]

[Out]

-((-9*Cos[x] + Cos[3*x])*Csc[x]*Sqrt[-Sin[x]^2])/12

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Maple [A]  time = 0.464, size = 21, normalized size = 0.6 \begin{align*} -{\frac{\cos \left ( x \right ) \sin \left ( x \right ) \left ( \left ( \sin \left ( x \right ) \right ) ^{2}+2 \right ) }{3}{\frac{1}{\sqrt{- \left ( \sin \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*sin(x)*cos(x)*(sin(x)^2+2)/(-sin(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 [A]  time = 1.54771, size = 4, normalized size = 0.12 \begin{align*} 0 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

0

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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 [C]  time = 1.24418, size = 74, normalized size = 2.24 \begin{align*} -\frac{12 i \, \mathrm{sgn}\left (-\tan \left (\frac{1}{2} \, x\right )^{3} - \tan \left (\frac{1}{2} \, x\right )\right ) \tan \left (\frac{1}{2} \, x\right )^{2} + 4 i \, \mathrm{sgn}\left (-\tan \left (\frac{1}{2} \, x\right )^{3} - \tan \left (\frac{1}{2} \, x\right )\right )}{3 \,{\left (\tan \left (\frac{1}{2} \, x\right )^{2} + 1\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(12*I*sgn(-tan(1/2*x)^3 - tan(1/2*x))*tan(1/2*x)^2 + 4*I*sgn(-tan(1/2*x)^3 - tan(1/2*x)))/(tan(1/2*x)^2 +
1)^3