3.1.49 \(\int (1-\cos ^2(x))^{3/2} \, dx\) [49]

Optimal. Leaf size=29 \[ -\frac {2}{3} \cot (x) \sqrt {\sin ^2(x)}-\frac {1}{3} \cot (x) \sin ^2(x)^{3/2} \]

[Out]

-1/3*cot(x)*(sin(x)^2)^(3/2)-2/3*cot(x)*(sin(x)^2)^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3255, 3282, 3286, 2718} \begin {gather*} -\frac {1}{3} \sin ^2(x)^{3/2} \cot (x)-\frac {2}{3} \sqrt {\sin ^2(x)} \cot (x) \end {gather*}

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 2718

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

Rule 3255

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 3282

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 3286

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
]])

Rubi steps

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

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Mathematica [A]
time = 0.03, size = 23, normalized size = 0.79 \begin {gather*} \frac {1}{12} (-9 \cos (x)+\cos (3 x)) \csc (x) \sqrt {\sin ^2(x)} \end {gather*}

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 = 0.24, size = 19, normalized size = 0.66

method	result	size
default	\(\frac {2 \sin \left (x \right ) \cos \left (x \right ) \left (\cos ^{2}\left (x \right )-3\right )}{3 \sqrt {2-2 \cos \left (2 x \right )}}\)	\(19\)
risch	\(\frac {i {\mathrm e}^{4 i x} \sqrt {-\left ({\mathrm e}^{2 i x}-1\right )^{2} {\mathrm e}^{-2 i x}}}{24 \,{\mathrm e}^{2 i x}-24}-\frac {3 i \sqrt {-\left ({\mathrm e}^{2 i x}-1\right )^{2} {\mathrm e}^{-2 i x}}\, {\mathrm e}^{2 i x}}{8 \left ({\mathrm e}^{2 i x}-1\right )}-\frac {3 i \sqrt {-\left ({\mathrm e}^{2 i x}-1\right )^{2} {\mathrm e}^{-2 i x}}}{8 \left ({\mathrm e}^{2 i x}-1\right )}+\frac {i {\mathrm e}^{-2 i x} \sqrt {-\left ({\mathrm e}^{2 i x}-1\right )^{2} {\mathrm e}^{-2 i x}}}{24 \,{\mathrm e}^{2 i x}-24}\)	\(137\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*sin(x)*cos(x)*(cos(x)^2-3)/(sin(x)^2)^(1/2)

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Maxima [A]
time = 0.52, size = 11, normalized size = 0.38 \begin {gather*} -\frac {1}{12} \, \cos \left (3 \, x\right ) + \frac {3}{4} \, \cos \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*cos(3*x) + 3/4*cos(x)

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Fricas [A]
time = 0.39, size = 11, normalized size = 0.38 \begin {gather*} \frac {1}{3} \, \cos \left (x\right )^{3} - \cos \left (x\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]

1/3*cos(x)^3 - cos(x)

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 45 vs. \(2 (21) = 42\).
time = 0.41, size = 45, normalized size = 1.55 \begin {gather*} -\frac {4 \, {\left (3 \, \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} + \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right )^{3} + \tan \left (\frac {1}{2} \, x\right )\right )\right )}}{3 \, {\left (\tan \left (\frac {1}{2} \, x\right )^{2} + 1\right )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int {\left (1-{\cos \left (x\right )}^2\right )}^{3/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - cos(x)^2)^(3/2),x)

[Out]

int((1 - cos(x)^2)^(3/2), x)

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