3.8 \(\int (1+\cot ^2(x))^{3/2} \, dx\)

Optimal. Leaf size=22 \[ -\frac{1}{2} \cot (x) \sqrt{\csc ^2(x)}-\frac{1}{2} \sinh ^{-1}(\cot (x)) \]

[Out]

-ArcSinh[Cot[x]]/2 - (Cot[x]*Sqrt[Csc[x]^2])/2

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Rubi [A]  time = 0.0184442, antiderivative size = 22, 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 = {3657, 4122, 195, 215} \[ -\frac{1}{2} \cot (x) \sqrt{\csc ^2(x)}-\frac{1}{2} \sinh ^{-1}(\cot (x)) \]

Antiderivative was successfully verified.

[In]

Int[(1 + Cot[x]^2)^(3/2),x]

[Out]

-ArcSinh[Cot[x]]/2 - (Cot[x]*Sqrt[Csc[x]^2])/2

Rule 3657

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

Rule 4122

Int[((b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[(b*ff)
/f, Subst[Int[(b + b*ff^2*x^2)^(p - 1), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{b, e, f, p}, x] &&  !IntegerQ[p
]

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rubi steps

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

Mathematica [B]  time = 0.0931351, size = 51, normalized size = 2.32 \[ \frac{1}{8} \sin (x) \sqrt{\csc ^2(x)} \left (-\csc ^2\left (\frac{x}{2}\right )+\sec ^2\left (\frac{x}{2}\right )+4 \log \left (\sin \left (\frac{x}{2}\right )\right )-4 \log \left (\cos \left (\frac{x}{2}\right )\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + Cot[x]^2)^(3/2),x]

[Out]

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

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Maple [A]  time = 0.031, size = 19, normalized size = 0.9 \begin{align*} -{\frac{\cot \left ( x \right ) }{2}\sqrt{1+ \left ( \cot \left ( x \right ) \right ) ^{2}}}-{\frac{{\it Arcsinh} \left ( \cot \left ( x \right ) \right ) }{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+cot(x)^2)^(3/2),x)

[Out]

-1/2*cot(x)*(1+cot(x)^2)^(1/2)-1/2*arcsinh(cot(x))

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Maxima [B]  time = 1.66282, size = 405, normalized size = 18.41 \begin{align*} -\frac{4 \,{\left (\cos \left (3 \, x\right ) + \cos \left (x\right )\right )} \cos \left (4 \, x\right ) - 4 \,{\left (2 \, \cos \left (2 \, x\right ) - 1\right )} \cos \left (3 \, x\right ) - 8 \, \cos \left (2 \, x\right ) \cos \left (x\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 )} \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \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 )} \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \cos \left (x\right ) + 1\right ) + 4 \,{\left (\sin \left (3 \, x\right ) + \sin \left (x\right )\right )} \sin \left (4 \, x\right ) - 8 \, \sin \left (3 \, x\right ) \sin \left (2 \, x\right ) - 8 \, \sin \left (2 \, x\right ) \sin \left (x\right ) + 4 \, \cos \left (x\right )}{4 \,{\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+cot(x)^2)^(3/2),x, algorithm="maxima")

[Out]

-1/4*(4*(cos(3*x) + cos(x))*cos(4*x) - 4*(2*cos(2*x) - 1)*cos(3*x) - 8*cos(2*x)*cos(x) + (2*(2*cos(2*x) - 1)*c
os(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)*log(co
s(x)^2 + sin(x)^2 + 2*cos(x) + 1) - (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)*log(cos(x)^2 + sin(x)^2 - 2*cos(x) + 1) + 4*(sin(3*x) + sin
(x))*sin(4*x) - 8*sin(3*x)*sin(2*x) - 8*sin(2*x)*sin(x) + 4*cos(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 [B]  time = 1.6213, size = 266, normalized size = 12.09 \begin{align*} -\frac{2 \, \sqrt{2} \sqrt{-\frac{1}{\cos \left (2 \, x\right ) - 1}}{\left (\cos \left (2 \, x\right ) + 1\right )} + \log \left (\frac{1}{2} \, \sqrt{2} \sqrt{-\frac{1}{\cos \left (2 \, x\right ) - 1}} \sin \left (2 \, x\right ) + 1\right ) \sin \left (2 \, x\right ) - \log \left (-\frac{1}{2} \, \sqrt{2} \sqrt{-\frac{1}{\cos \left (2 \, x\right ) - 1}} \sin \left (2 \, x\right ) + 1\right ) \sin \left (2 \, x\right )}{4 \, \sin \left (2 \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(2*sqrt(2)*sqrt(-1/(cos(2*x) - 1))*(cos(2*x) + 1) + log(1/2*sqrt(2)*sqrt(-1/(cos(2*x) - 1))*sin(2*x) + 1)
*sin(2*x) - log(-1/2*sqrt(2)*sqrt(-1/(cos(2*x) - 1))*sin(2*x) + 1)*sin(2*x))/sin(2*x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (\cot ^{2}{\left (x \right )} + 1\right )^{\frac{3}{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+cot(x)**2)**(3/2),x)

[Out]

Integral((cot(x)**2 + 1)**(3/2), x)

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Giac [A]  time = 1.26172, size = 43, normalized size = 1.95 \begin{align*} \frac{1}{4} \,{\left (\frac{2 \, \cos \left (x\right )}{\cos \left (x\right )^{2} - 1} - \log \left (\cos \left (x\right ) + 1\right ) + \log \left (-\cos \left (x\right ) + 1\right )\right )} \mathrm{sgn}\left (\sin \left (x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(2*cos(x)/(cos(x)^2 - 1) - log(cos(x) + 1) + log(-cos(x) + 1))*sgn(sin(x))