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

Optimal. Leaf size=36 \[ \frac{\cot (x)}{2 \sqrt{-\sin ^2(x)}}+\frac{\sin (x) \tanh ^{-1}(\cos (x))}{2 \sqrt{-\sin ^2(x)}} \]

[Out]

Cot[x]/(2*Sqrt[-Sin[x]^2]) + (ArcTanh[Cos[x]]*Sin[x])/(2*Sqrt[-Sin[x]^2])

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Rubi [A]  time = 0.025746, antiderivative size = 36, 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, 3204, 3207, 3770} \[ \frac{\cot (x)}{2 \sqrt{-\sin ^2(x)}}+\frac{\sin (x) \tanh ^{-1}(\cos (x))}{2 \sqrt{-\sin ^2(x)}} \]

Antiderivative was successfully verified.

[In]

Int[(-1 + Cos[x]^2)^(-3/2),x]

[Out]

Cot[x]/(2*Sqrt[-Sin[x]^2]) + (ArcTanh[Cos[x]]*Sin[x])/(2*Sqrt[-Sin[x]^2])

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 3204

Int[((b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Simp[(Cot[e + f*x]*(b*Sin[e + f*x]^2)^(p + 1))/(b*f*(
2*p + 1)), x] + Dist[(2*(p + 1))/(b*(2*p + 1)), Int[(b*Sin[e + f*x]^2)^(p + 1), x], x] /; FreeQ[{b, e, f}, x]
&&  !IntegerQ[p] && LtQ[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 3770

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

Rubi steps

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

Mathematica [A]  time = 0.0300801, size = 53, normalized size = 1.47 \[ \frac{\sin (x) \left (\csc ^2\left (\frac{x}{2}\right )-\sec ^2\left (\frac{x}{2}\right )-4 \log \left (\sin \left (\frac{x}{2}\right )\right )+4 \log \left (\cos \left (\frac{x}{2}\right )\right )\right )}{8 \sqrt{-\sin ^2(x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[(-1 + Cos[x]^2)^(-3/2),x]

[Out]

((Csc[x/2]^2 + 4*Log[Cos[x/2]] - 4*Log[Sin[x/2]] - Sec[x/2]^2)*Sin[x])/(8*Sqrt[-Sin[x]^2])

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Maple [A]  time = 0.499, size = 51, normalized size = 1.4 \begin{align*} -{\frac{1}{2\,\cos \left ( x \right ) \sin \left ( x \right ) }\sqrt{- \left ( \cos \left ( x \right ) \right ) ^{2}} \left ( -\arctan \left ({\frac{1}{\sqrt{- \left ( \cos \left ( x \right ) \right ) ^{2}}}} \right ) \left ( \sin \left ( x \right ) \right ) ^{2}+\sqrt{- \left ( \cos \left ( x \right ) \right ) ^{2}} \right ){\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/(-1+cos(x)^2)^(3/2),x)

[Out]

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

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Maxima [B]  time = 1.67064, size = 383, normalized size = 10.64 \begin{align*} \frac{{\left (2 \,{\left (2 \, \cos \left (2 \, x\right ) - 1\right )} \cos \left (4 \, x\right ) - \cos \left (4 \, x\right )^{2} - 4 \, \cos \left (2 \, x\right )^{2} - \sin \left (4 \, x\right )^{2} + 4 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) - 4 \, \sin \left (2 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right ) - 1\right )} \arctan \left (\sin \left (x\right ), \cos \left (x\right ) + 1\right ) -{\left (2 \,{\left (2 \, \cos \left (2 \, x\right ) - 1\right )} \cos \left (4 \, x\right ) - \cos \left (4 \, x\right )^{2} - 4 \, \cos \left (2 \, x\right )^{2} - \sin \left (4 \, x\right )^{2} + 4 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) - 4 \, \sin \left (2 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right ) - 1\right )} \arctan \left (\sin \left (x\right ), \cos \left (x\right ) - 1\right ) + 2 \,{\left (\sin \left (3 \, x\right ) + \sin \left (x\right )\right )} \cos \left (4 \, x\right ) - 2 \,{\left (\cos \left (3 \, x\right ) + \cos \left (x\right )\right )} \sin \left (4 \, x\right ) - 2 \,{\left (2 \, \cos \left (2 \, x\right ) - 1\right )} \sin \left (3 \, x\right ) + 4 \, \cos \left (3 \, x\right ) \sin \left (2 \, x\right ) + 4 \, \cos \left (x\right ) \sin \left (2 \, x\right ) - 4 \, \cos \left (2 \, x\right ) \sin \left (x\right ) + 2 \, \sin \left (x\right )}{2 \,{\left (2 \,{\left (2 \, \cos \left (2 \, x\right ) - 1\right )} \cos \left (4 \, x\right ) - \cos \left (4 \, x\right )^{2} - 4 \, \cos \left (2 \, x\right )^{2} - \sin \left (4 \, x\right )^{2} + 4 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) - 4 \, \sin \left (2 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right ) - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*((2*(2*cos(2*x) - 1)*cos(4*x) - cos(4*x)^2 - 4*cos(2*x)^2 - sin(4*x)^2 + 4*sin(4*x)*sin(2*x) - 4*sin(2*x)^
2 + 4*cos(2*x) - 1)*arctan2(sin(x), cos(x) + 1) - (2*(2*cos(2*x) - 1)*cos(4*x) - cos(4*x)^2 - 4*cos(2*x)^2 - s
in(4*x)^2 + 4*sin(4*x)*sin(2*x) - 4*sin(2*x)^2 + 4*cos(2*x) - 1)*arctan2(sin(x), cos(x) - 1) + 2*(sin(3*x) + s
in(x))*cos(4*x) - 2*(cos(3*x) + cos(x))*sin(4*x) - 2*(2*cos(2*x) - 1)*sin(3*x) + 4*cos(3*x)*sin(2*x) + 4*cos(x
)*sin(2*x) - 4*cos(2*x)*sin(x) + 2*sin(x))/(2*(2*cos(2*x) - 1)*cos(4*x) - cos(4*x)^2 - 4*cos(2*x)^2 - sin(4*x)
^2 + 4*sin(4*x)*sin(2*x) - 4*sin(2*x)^2 + 4*cos(2*x) - 1)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

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

[Out]

Timed out

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Giac [C]  time = 1.29558, size = 122, normalized size = 3.39 \begin{align*} -\frac{i \, \tan \left (\frac{1}{2} \, x\right )^{2}}{8 \, \mathrm{sgn}\left (-\tan \left (\frac{1}{2} \, x\right )^{3} - \tan \left (\frac{1}{2} \, x\right )\right )} - \frac{i \, \log \left (\tan \left (\frac{1}{2} \, x\right )^{2}\right )}{4 \, \mathrm{sgn}\left (-\tan \left (\frac{1}{2} \, x\right )^{3} - \tan \left (\frac{1}{2} \, x\right )\right )} + \frac{2 i \, \tan \left (\frac{1}{2} \, x\right )^{2} + i}{8 \, \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*I*tan(1/2*x)^2/sgn(-tan(1/2*x)^3 - tan(1/2*x)) - 1/4*I*log(tan(1/2*x)^2)/sgn(-tan(1/2*x)^3 - tan(1/2*x))
+ 1/8*(2*I*tan(1/2*x)^2 + I)/(sgn(-tan(1/2*x)^3 - tan(1/2*x))*tan(1/2*x)^2)