3.1.48 \(\int \sqrt {-1+\cos ^2(x)} \, dx\) [48]

Optimal. Leaf size=14 \[ -\cot (x) \sqrt {-\sin ^2(x)} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3255, 3286, 2718} \begin {gather*} \sqrt {-\sin ^2(x)} (-\cot (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[-1 + Cos[x]^2],x]

[Out]

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

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

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Mathematica [A]
time = 0.01, size = 14, normalized size = 1.00 \begin {gather*} -\cot (x) \sqrt {-\sin ^2(x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[-1 + Cos[x]^2],x]

[Out]

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

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Maple [A]
time = 0.30, size = 14, normalized size = 1.00

method	result	size
default	\(\frac {\sin \left (x \right ) \cos \left (x \right )}{\sqrt {-\left (\sin ^{2}\left (x \right )\right )}}\)	\(14\)
risch	\(-\frac {i \sqrt {\left ({\mathrm e}^{2 i x}-1\right )^{2} {\mathrm e}^{-2 i x}}\, {\mathrm e}^{2 i x}}{2 \left ({\mathrm e}^{2 i x}-1\right )}-\frac {i \sqrt {\left ({\mathrm e}^{2 i x}-1\right )^{2} {\mathrm e}^{-2 i x}}}{2 \left ({\mathrm e}^{2 i x}-1\right )}\)	\(65\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.49, size = 12, normalized size = 0.86 \begin {gather*} -\frac {1}{\sqrt {-\tan \left (x\right )^{2} - 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/sqrt(-tan(x)^2 - 1)

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Fricas [F]
time = 0.38, size = 1, normalized size = 0.07 \begin {gather*} 0 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

0

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(cos(x)**2 - 1), x)

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Giac [C] Result contains complex when optimal does not.
time = 0.40, size = 28, normalized size = 2.00 \begin {gather*} \frac {2 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} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 2.29, size = 39, normalized size = 2.79 \begin {gather*} -\frac {\sqrt {-4\,{\sin \left (x\right )}^2}\,\left (-{\sin \left (x\right )}^2+\frac {\sin \left (2\,x\right )\,1{}\mathrm {i}}{2}+1\right )}{{\sin \left (x\right )}^2\,2{}\mathrm {i}+\sin \left (2\,x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cos(x)^2 - 1)^(1/2),x)

[Out]

-((-4*sin(x)^2)^(1/2)*((sin(2*x)*1i)/2 - sin(x)^2 + 1))/(sin(2*x) + sin(x)^2*2i)

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