3.1.45 \(\int \frac {\cot ^2(x)}{\sqrt {1+\cot (x)}} \, dx\) [45]

Optimal. Leaf size=214 \[ -\frac {1}{2} \sqrt {1+\sqrt {2}} \text {ArcTan}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {1+\cot (x)}}{\sqrt {2 \left (-1+\sqrt {2}\right )}}\right )+\frac {1}{2} \sqrt {1+\sqrt {2}} \text {ArcTan}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}+2 \sqrt {1+\cot (x)}}{\sqrt {2 \left (-1+\sqrt {2}\right )}}\right )-2 \sqrt {1+\cot (x)}-\frac {\log \left (1+\sqrt {2}+\cot (x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+\cot (x)}\right )}{4 \sqrt {1+\sqrt {2}}}+\frac {\log \left (1+\sqrt {2}+\cot (x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+\cot (x)}\right )}{4 \sqrt {1+\sqrt {2}}} \]

[Out]

-2*(1+cot(x))^(1/2)-1/4*ln(1+cot(x)+2^(1/2)-(1+cot(x))^(1/2)*(2+2*2^(1/2))^(1/2))/(1+2^(1/2))^(1/2)+1/4*ln(1+c
ot(x)+2^(1/2)+(1+cot(x))^(1/2)*(2+2*2^(1/2))^(1/2))/(1+2^(1/2))^(1/2)-1/2*arctan((-2*(1+cot(x))^(1/2)+(2+2*2^(
1/2))^(1/2))/(-2+2*2^(1/2))^(1/2))*(1+2^(1/2))^(1/2)+1/2*arctan((2*(1+cot(x))^(1/2)+(2+2*2^(1/2))^(1/2))/(-2+2
*2^(1/2))^(1/2))*(1+2^(1/2))^(1/2)

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Rubi [A]
time = 0.14, antiderivative size = 214, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.615, Rules used = {3624, 3566, 722, 1108, 648, 632, 210, 642} \begin {gather*} -\frac {1}{2} \sqrt {1+\sqrt {2}} \text {ArcTan}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {\cot (x)+1}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )+\frac {1}{2} \sqrt {1+\sqrt {2}} \text {ArcTan}\left (\frac {2 \sqrt {\cot (x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )-2 \sqrt {\cot (x)+1}-\frac {\log \left (\cot (x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\cot (x)+1}+\sqrt {2}+1\right )}{4 \sqrt {1+\sqrt {2}}}+\frac {\log \left (\cot (x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\cot (x)+1}+\sqrt {2}+1\right )}{4 \sqrt {1+\sqrt {2}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cot[x]^2/Sqrt[1 + Cot[x]],x]

[Out]

-1/2*(Sqrt[1 + Sqrt[2]]*ArcTan[(Sqrt[2*(1 + Sqrt[2])] - 2*Sqrt[1 + Cot[x]])/Sqrt[2*(-1 + Sqrt[2])]]) + (Sqrt[1
+ Sqrt[2]]*ArcTan[(Sqrt[2*(1 + Sqrt[2])] + 2*Sqrt[1 + Cot[x]])/Sqrt[2*(-1 + Sqrt[2])]])/2 - 2*Sqrt[1 + Cot[x]
] - Log[1 + Sqrt[2] + Cot[x] - Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 + Cot[x]]]/(4*Sqrt[1 + Sqrt[2]]) + Log[1 + Sqrt[2]
+ Cot[x] + Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 + Cot[x]]]/(4*Sqrt[1 + Sqrt[2]])

Rule 210

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

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 722

Int[1/(Sqrt[(d_) + (e_.)*(x_)]*((a_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2*e, Subst[Int[1/(c*d^2 + a*e^2 - 2*c
*d*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0]

Rule 1108

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(-1), x_Symbol] :> With[{q = Rt[a/c, 2]}, With[{r = Rt[2*q - b/c, 2]}
, Dist[1/(2*c*q*r), Int[(r - x)/(q - r*x + x^2), x], x] + Dist[1/(2*c*q*r), Int[(r + x)/(q + r*x + x^2), x], x
]]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && NegQ[b^2 - 4*a*c]

Rule 3566

Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[b/d, Subst[Int[(a + x)^n/(b^2 + x^2), x], x
, b*Tan[c + d*x]], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[a^2 + b^2, 0]

Rule 3624

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^2, x_Symbol] :> Simp[
d^2*((a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1))), x] + Int[(a + b*Tan[e + f*x])^m*Simp[c^2 - d^2 + 2*c*d*Tan[e
+ f*x], x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] &&  !LeQ[m, -1] &&  !(EqQ[m, 2] && EqQ
[a, 0])

Rubi steps

\begin {align*} \int \frac {\cot ^2(x)}{\sqrt {1+\cot (x)}} \, dx &=-2 \sqrt {1+\cot (x)}-\int \frac {1}{\sqrt {1+\cot (x)}} \, dx\\ &=-2 \sqrt {1+\cot (x)}+\text {Subst}\left (\int \frac {1}{\sqrt {1+x} \left (1+x^2\right )} \, dx,x,\cot (x)\right )\\ &=-2 \sqrt {1+\cot (x)}+2 \text {Subst}\left (\int \frac {1}{2-2 x^2+x^4} \, dx,x,\sqrt {1+\cot (x)}\right )\\ &=-2 \sqrt {1+\cot (x)}+\frac {\text {Subst}\left (\int \frac {\sqrt {2 \left (1+\sqrt {2}\right )}-x}{\sqrt {2}-\sqrt {2 \left (1+\sqrt {2}\right )} x+x^2} \, dx,x,\sqrt {1+\cot (x)}\right )}{2 \sqrt {1+\sqrt {2}}}+\frac {\text {Subst}\left (\int \frac {\sqrt {2 \left (1+\sqrt {2}\right )}+x}{\sqrt {2}+\sqrt {2 \left (1+\sqrt {2}\right )} x+x^2} \, dx,x,\sqrt {1+\cot (x)}\right )}{2 \sqrt {1+\sqrt {2}}}\\ &=-2 \sqrt {1+\cot (x)}+\frac {\text {Subst}\left (\int \frac {1}{\sqrt {2}-\sqrt {2 \left (1+\sqrt {2}\right )} x+x^2} \, dx,x,\sqrt {1+\cot (x)}\right )}{2 \sqrt {2}}+\frac {\text {Subst}\left (\int \frac {1}{\sqrt {2}+\sqrt {2 \left (1+\sqrt {2}\right )} x+x^2} \, dx,x,\sqrt {1+\cot (x)}\right )}{2 \sqrt {2}}-\frac {\text {Subst}\left (\int \frac {-\sqrt {2 \left (1+\sqrt {2}\right )}+2 x}{\sqrt {2}-\sqrt {2 \left (1+\sqrt {2}\right )} x+x^2} \, dx,x,\sqrt {1+\cot (x)}\right )}{4 \sqrt {1+\sqrt {2}}}+\frac {\text {Subst}\left (\int \frac {\sqrt {2 \left (1+\sqrt {2}\right )}+2 x}{\sqrt {2}+\sqrt {2 \left (1+\sqrt {2}\right )} x+x^2} \, dx,x,\sqrt {1+\cot (x)}\right )}{4 \sqrt {1+\sqrt {2}}}\\ &=-2 \sqrt {1+\cot (x)}-\frac {\log \left (1+\sqrt {2}+\cot (x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+\cot (x)}\right )}{4 \sqrt {1+\sqrt {2}}}+\frac {\log \left (1+\sqrt {2}+\cot (x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+\cot (x)}\right )}{4 \sqrt {1+\sqrt {2}}}-\frac {\text {Subst}\left (\int \frac {1}{2 \left (1-\sqrt {2}\right )-x^2} \, dx,x,-\sqrt {2 \left (1+\sqrt {2}\right )}+2 \sqrt {1+\cot (x)}\right )}{\sqrt {2}}-\frac {\text {Subst}\left (\int \frac {1}{2 \left (1-\sqrt {2}\right )-x^2} \, dx,x,\sqrt {2 \left (1+\sqrt {2}\right )}+2 \sqrt {1+\cot (x)}\right )}{\sqrt {2}}\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {1+\cot (x)}}{\sqrt {2 \left (-1+\sqrt {2}\right )}}\right )}{2 \sqrt {-1+\sqrt {2}}}+\frac {\tan ^{-1}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}+2 \sqrt {1+\cot (x)}}{\sqrt {2 \left (-1+\sqrt {2}\right )}}\right )}{2 \sqrt {-1+\sqrt {2}}}-2 \sqrt {1+\cot (x)}-\frac {\log \left (1+\sqrt {2}+\cot (x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+\cot (x)}\right )}{4 \sqrt {1+\sqrt {2}}}+\frac {\log \left (1+\sqrt {2}+\cot (x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+\cot (x)}\right )}{4 \sqrt {1+\sqrt {2}}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.19, size = 67, normalized size = 0.31 \begin {gather*} \frac {1}{2} (1-i)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {1+\cot (x)}}{\sqrt {1-i}}\right )+\frac {1}{2} (1+i)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {1+\cot (x)}}{\sqrt {1+i}}\right )-2 \sqrt {1+\cot (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cot[x]^2/Sqrt[1 + Cot[x]],x]

[Out]

((1 - I)^(3/2)*ArcTanh[Sqrt[1 + Cot[x]]/Sqrt[1 - I]])/2 + ((1 + I)^(3/2)*ArcTanh[Sqrt[1 + Cot[x]]/Sqrt[1 + I]]
)/2 - 2*Sqrt[1 + Cot[x]]

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Maple [A]
time = 0.26, size = 283, normalized size = 1.32

method	result	size
derivativedivides	\(-2 \sqrt {1+\cot \left (x \right )}-\frac {\left (-\sqrt {2+2 \sqrt {2}}\, \sqrt {2}+2 \sqrt {2+2 \sqrt {2}}\right ) \ln \left (1+\cot \left (x \right )+\sqrt {2}-\sqrt {1+\cot \left (x \right )}\, \sqrt {2+2 \sqrt {2}}\right )}{8}-\frac {\left (-2 \sqrt {2}+\frac {\left (-\sqrt {2+2 \sqrt {2}}\, \sqrt {2}+2 \sqrt {2+2 \sqrt {2}}\right ) \sqrt {2+2 \sqrt {2}}}{2}\right ) \arctan \left (\frac {2 \sqrt {1+\cot \left (x \right )}-\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{2 \sqrt {-2+2 \sqrt {2}}}+\frac {\left (-\sqrt {2+2 \sqrt {2}}\, \sqrt {2}+2 \sqrt {2+2 \sqrt {2}}\right ) \ln \left (1+\cot \left (x \right )+\sqrt {2}+\sqrt {1+\cot \left (x \right )}\, \sqrt {2+2 \sqrt {2}}\right )}{8}+\frac {\left (2 \sqrt {2}-\frac {\left (-\sqrt {2+2 \sqrt {2}}\, \sqrt {2}+2 \sqrt {2+2 \sqrt {2}}\right ) \sqrt {2+2 \sqrt {2}}}{2}\right ) \arctan \left (\frac {2 \sqrt {1+\cot \left (x \right )}+\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{2 \sqrt {-2+2 \sqrt {2}}}\)	\(283\)
default	\(-2 \sqrt {1+\cot \left (x \right )}-\frac {\left (-\sqrt {2+2 \sqrt {2}}\, \sqrt {2}+2 \sqrt {2+2 \sqrt {2}}\right ) \ln \left (1+\cot \left (x \right )+\sqrt {2}-\sqrt {1+\cot \left (x \right )}\, \sqrt {2+2 \sqrt {2}}\right )}{8}-\frac {\left (-2 \sqrt {2}+\frac {\left (-\sqrt {2+2 \sqrt {2}}\, \sqrt {2}+2 \sqrt {2+2 \sqrt {2}}\right ) \sqrt {2+2 \sqrt {2}}}{2}\right ) \arctan \left (\frac {2 \sqrt {1+\cot \left (x \right )}-\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{2 \sqrt {-2+2 \sqrt {2}}}+\frac {\left (-\sqrt {2+2 \sqrt {2}}\, \sqrt {2}+2 \sqrt {2+2 \sqrt {2}}\right ) \ln \left (1+\cot \left (x \right )+\sqrt {2}+\sqrt {1+\cot \left (x \right )}\, \sqrt {2+2 \sqrt {2}}\right )}{8}+\frac {\left (2 \sqrt {2}-\frac {\left (-\sqrt {2+2 \sqrt {2}}\, \sqrt {2}+2 \sqrt {2+2 \sqrt {2}}\right ) \sqrt {2+2 \sqrt {2}}}{2}\right ) \arctan \left (\frac {2 \sqrt {1+\cot \left (x \right )}+\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{2 \sqrt {-2+2 \sqrt {2}}}\)	\(283\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(1+cot(x))^(1/2)-1/8*(-(2+2*2^(1/2))^(1/2)*2^(1/2)+2*(2+2*2^(1/2))^(1/2))*ln(1+cot(x)+2^(1/2)-(1+cot(x))^(1
/2)*(2+2*2^(1/2))^(1/2))-1/2*(-2*2^(1/2)+1/2*(-(2+2*2^(1/2))^(1/2)*2^(1/2)+2*(2+2*2^(1/2))^(1/2))*(2+2*2^(1/2)
)^(1/2))/(-2+2*2^(1/2))^(1/2)*arctan((2*(1+cot(x))^(1/2)-(2+2*2^(1/2))^(1/2))/(-2+2*2^(1/2))^(1/2))+1/8*(-(2+2
*2^(1/2))^(1/2)*2^(1/2)+2*(2+2*2^(1/2))^(1/2))*ln(1+cot(x)+2^(1/2)+(1+cot(x))^(1/2)*(2+2*2^(1/2))^(1/2))+1/2*(
2*2^(1/2)-1/2*(-(2+2*2^(1/2))^(1/2)*2^(1/2)+2*(2+2*2^(1/2))^(1/2))*(2+2*2^(1/2))^(1/2))/(-2+2*2^(1/2))^(1/2)*a
rctan((2*(1+cot(x))^(1/2)+(2+2*2^(1/2))^(1/2))/(-2+2*2^(1/2))^(1/2))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(cot(x)^2/sqrt(cot(x) + 1), x)

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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(cot(x)**2/sqrt(cot(x) + 1), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(cot(x)^2/sqrt(cot(x) + 1), x)

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Mupad [B]
time = 0.44, size = 238, normalized size = 1.11 \begin {gather*} \mathrm {atanh}\left (\frac {16\,\sqrt {2}\,\sqrt {-\frac {\sqrt {2}}{16}-\frac {1}{16}}\,\sqrt {\mathrm {cot}\left (x\right )+1}}{128\,\sqrt {\frac {\sqrt {2}}{16}-\frac {1}{16}}\,\sqrt {-\frac {\sqrt {2}}{16}-\frac {1}{16}}-8}-\frac {16\,\sqrt {2}\,\sqrt {\frac {\sqrt {2}}{16}-\frac {1}{16}}\,\sqrt {\mathrm {cot}\left (x\right )+1}}{128\,\sqrt {\frac {\sqrt {2}}{16}-\frac {1}{16}}\,\sqrt {-\frac {\sqrt {2}}{16}-\frac {1}{16}}-8}\right )\,\left (2\,\sqrt {-\frac {\sqrt {2}}{16}-\frac {1}{16}}+2\,\sqrt {\frac {\sqrt {2}}{16}-\frac {1}{16}}\right )-\mathrm {atanh}\left (\frac {16\,\sqrt {2}\,\sqrt {-\frac {\sqrt {2}}{16}-\frac {1}{16}}\,\sqrt {\mathrm {cot}\left (x\right )+1}}{128\,\sqrt {\frac {\sqrt {2}}{16}-\frac {1}{16}}\,\sqrt {-\frac {\sqrt {2}}{16}-\frac {1}{16}}+8}+\frac {16\,\sqrt {2}\,\sqrt {\frac {\sqrt {2}}{16}-\frac {1}{16}}\,\sqrt {\mathrm {cot}\left (x\right )+1}}{128\,\sqrt {\frac {\sqrt {2}}{16}-\frac {1}{16}}\,\sqrt {-\frac {\sqrt {2}}{16}-\frac {1}{16}}+8}\right )\,\left (2\,\sqrt {-\frac {\sqrt {2}}{16}-\frac {1}{16}}-2\,\sqrt {\frac {\sqrt {2}}{16}-\frac {1}{16}}\right )-2\,\sqrt {\mathrm {cot}\left (x\right )+1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(x)^2/(cot(x) + 1)^(1/2),x)

[Out]

atanh((16*2^(1/2)*(- 2^(1/2)/16 - 1/16)^(1/2)*(cot(x) + 1)^(1/2))/(128*(2^(1/2)/16 - 1/16)^(1/2)*(- 2^(1/2)/16
- 1/16)^(1/2) - 8) - (16*2^(1/2)*(2^(1/2)/16 - 1/16)^(1/2)*(cot(x) + 1)^(1/2))/(128*(2^(1/2)/16 - 1/16)^(1/2)
*(- 2^(1/2)/16 - 1/16)^(1/2) - 8))*(2*(- 2^(1/2)/16 - 1/16)^(1/2) + 2*(2^(1/2)/16 - 1/16)^(1/2)) - atanh((16*2
^(1/2)*(- 2^(1/2)/16 - 1/16)^(1/2)*(cot(x) + 1)^(1/2))/(128*(2^(1/2)/16 - 1/16)^(1/2)*(- 2^(1/2)/16 - 1/16)^(1
/2) + 8) + (16*2^(1/2)*(2^(1/2)/16 - 1/16)^(1/2)*(cot(x) + 1)^(1/2))/(128*(2^(1/2)/16 - 1/16)^(1/2)*(- 2^(1/2)
/16 - 1/16)^(1/2) + 8))*(2*(- 2^(1/2)/16 - 1/16)^(1/2) - 2*(2^(1/2)/16 - 1/16)^(1/2)) - 2*(cot(x) + 1)^(1/2)

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