\(\int (1+\cot ^2(x))^{3/2} \, dx\) [8]

Optimal result

Integrand size = 10, antiderivative size = 22 \[ \int \left (1+\cot ^2(x)\right )^{3/2} \, dx=-\frac {1}{2} \text {arcsinh}(\cot (x))-\frac {1}{2} \cot (x) \sqrt {\csc ^2(x)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3738, 4207, 201, 221} \[ \int \left (1+\cot ^2(x)\right )^{3/2} \, dx=-\frac {1}{2} \text {arcsinh}(\cot (x))-\frac {1}{2} \cot (x) \sqrt {\csc ^2(x)} \]

[In]

Int[(1 + Cot[x]^2)^(3/2),x]

[Out]

-1/2*ArcSinh[Cot[x]] - (Cot[x]*Sqrt[Csc[x]^2])/2

Rule 201

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 221

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]

Rule 3738

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 4207

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
]

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(51\) vs. \(2(22)=44\).

Time = 0.13 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.32 \[ \int \left (1+\cot ^2(x)\right )^{3/2} \, dx=\frac {1}{8} \sqrt {\csc ^2(x)} \left (-\csc ^2\left (\frac {x}{2}\right )-4 \log \left (\cos \left (\frac {x}{2}\right )\right )+4 \log \left (\sin \left (\frac {x}{2}\right )\right )+\sec ^2\left (\frac {x}{2}\right )\right ) \sin (x) \]

[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

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86

method	result	size
derivativedivides	\(-\frac {\cot \left (x \right ) \sqrt {\cot \left (x \right )^{2}+1}}{2}-\frac {\operatorname {arcsinh}\left (\cot \left (x \right )\right )}{2}\)	\(19\)
default	\(-\frac {\cot \left (x \right ) \sqrt {\cot \left (x \right )^{2}+1}}{2}-\frac {\operatorname {arcsinh}\left (\cot \left (x \right )\right )}{2}\)	\(19\)
risch	\(-\frac {i \sqrt {-\frac {{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \left ({\mathrm e}^{2 i x}+1\right )}{{\mathrm e}^{2 i x}-1}-\sqrt {-\frac {{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \ln \left ({\mathrm e}^{i x}+1\right ) \sin \left (x \right )+\sqrt {-\frac {{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \ln \left ({\mathrm e}^{i x}-1\right ) \sin \left (x \right )\)	\(98\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 91 vs. \(2 (16) = 32\).

Time = 0.27 (sec) , antiderivative size = 91, normalized size of antiderivative = 4.14 \[ \int \left (1+\cot ^2(x)\right )^{3/2} \, dx=-\frac {2 \, \sqrt {2} \sqrt {-\frac {1}{\cos \left (2 \, x\right ) - 1}} {\left (\cos \left (2 \, x\right ) + 1\right )} + \log \left (\frac {1}{2} \, \sqrt {2} \sqrt {-\frac {1}{\cos \left (2 \, x\right ) - 1}} \sin \left (2 \, x\right ) + 1\right ) \sin \left (2 \, x\right ) - \log \left (-\frac {1}{2} \, \sqrt {2} \sqrt {-\frac {1}{\cos \left (2 \, x\right ) - 1}} \sin \left (2 \, x\right ) + 1\right ) \sin \left (2 \, x\right )}{4 \, \sin \left (2 \, x\right )} \]

[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)

Sympy [F]

\[ \int \left (1+\cot ^2(x)\right )^{3/2} \, dx=\int \left (\cot ^{2}{\left (x \right )} + 1\right )^{\frac {3}{2}}\, dx \]

[In]

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

[Out]

Integral((cot(x)**2 + 1)**(3/2), x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 300 vs. \(2 (16) = 32\).

Time = 0.37 (sec) , antiderivative size = 300, normalized size of antiderivative = 13.64 \[ \int \left (1+\cot ^2(x)\right )^{3/2} \, dx=-\frac {4 \, {\left (\cos \left (3 \, x\right ) + \cos \left (x\right )\right )} \cos \left (4 \, x\right ) - 4 \, {\left (2 \, \cos \left (2 \, x\right ) - 1\right )} \cos \left (3 \, x\right ) - 8 \, \cos \left (2 \, x\right ) \cos \left (x\right ) + {\left (2 \, {\left (2 \, \cos \left (2 \, x\right ) - 1\right )} \cos \left (4 \, x\right ) - \cos \left (4 \, x\right )^{2} - 4 \, \cos \left (2 \, x\right )^{2} - \sin \left (4 \, x\right )^{2} + 4 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) - 4 \, \sin \left (2 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right ) - 1\right )} \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1\right ) - {\left (2 \, {\left (2 \, \cos \left (2 \, x\right ) - 1\right )} \cos \left (4 \, x\right ) - \cos \left (4 \, x\right )^{2} - 4 \, \cos \left (2 \, x\right )^{2} - \sin \left (4 \, x\right )^{2} + 4 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) - 4 \, \sin \left (2 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right ) - 1\right )} \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \cos \left (x\right ) + 1\right ) + 4 \, {\left (\sin \left (3 \, x\right ) + \sin \left (x\right )\right )} \sin \left (4 \, x\right ) - 8 \, \sin \left (3 \, x\right ) \sin \left (2 \, x\right ) - 8 \, \sin \left (2 \, x\right ) \sin \left (x\right ) + 4 \, \cos \left (x\right )}{4 \, {\left (2 \, {\left (2 \, \cos \left (2 \, x\right ) - 1\right )} \cos \left (4 \, x\right ) - \cos \left (4 \, x\right )^{2} - 4 \, \cos \left (2 \, x\right )^{2} - \sin \left (4 \, x\right )^{2} + 4 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) - 4 \, \sin \left (2 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right ) - 1\right )}} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.45 \[ \int \left (1+\cot ^2(x)\right )^{3/2} \, dx=\frac {1}{4} \, {\left (\frac {2 \, \cos \left (x\right )}{\cos \left (x\right )^{2} - 1} - \log \left (\cos \left (x\right ) + 1\right ) + \log \left (-\cos \left (x\right ) + 1\right )\right )} \mathrm {sgn}\left (\sin \left (x\right )\right ) \]

[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))

Mupad [B] (verification not implemented)

Time = 13.18 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82 \[ \int \left (1+\cot ^2(x)\right )^{3/2} \, dx=-\frac {\mathrm {asinh}\left (\mathrm {cot}\left (x\right )\right )}{2}-\frac {\mathrm {cot}\left (x\right )\,\sqrt {{\mathrm {cot}\left (x\right )}^2+1}}{2} \]

[In]

int((cot(x)^2 + 1)^(3/2),x)

[Out]

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