3.45 \(\int \frac{\cot ^2(x)}{\sqrt{1+\cot (x)}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.185029, antiderivative size = 214, normalized size of antiderivative = 1., 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 = {3543, 3485, 708, 1094, 634, 618, 204, 628} \[ -2 \sqrt{\cot (x)+1}-\frac{\log \left (\cot (x)-\sqrt{2 \left (1+\sqrt{2}\right )} \sqrt{\cot (x)+1}+\sqrt{2}+1\right )}{4 \sqrt{1+\sqrt{2}}}+\frac{\log \left (\cot (x)+\sqrt{2 \left (1+\sqrt{2}\right )} \sqrt{\cot (x)+1}+\sqrt{2}+1\right )}{4 \sqrt{1+\sqrt{2}}}-\frac{1}{2} \sqrt{1+\sqrt{2}} \tan ^{-1}\left (\frac{\sqrt{2 \left (1+\sqrt{2}\right )}-2 \sqrt{\cot (x)+1}}{\sqrt{2 \left (\sqrt{2}-1\right )}}\right )+\frac{1}{2} \sqrt{1+\sqrt{2}} \tan ^{-1}\left (\frac{2 \sqrt{\cot (x)+1}+\sqrt{2 \left (1+\sqrt{2}\right )}}{\sqrt{2 \left (\sqrt{2}-1\right )}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Cot[x]^2/Sqrt[1 + Cot[x]],x]

[Out]

-(Sqrt[1 + Sqrt[2]]*ArcTan[(Sqrt[2*(1 + Sqrt[2])] - 2*Sqrt[1 + Cot[x]])/Sqrt[2*(-1 + Sqrt[2])]])/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 3543

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])

Rule 3485

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 708

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 1094

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 634

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 618

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 204

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

Rule 628

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]

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)}+\operatorname{Subst}\left (\int \frac{1}{\sqrt{1+x} \left (1+x^2\right )} \, dx,x,\cot (x)\right )\\ &=-2 \sqrt{1+\cot (x)}+2 \operatorname{Subst}\left (\int \frac{1}{2-2 x^2+x^4} \, dx,x,\sqrt{1+\cot (x)}\right )\\ &=-2 \sqrt{1+\cot (x)}+\frac{\operatorname{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{\operatorname{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{\operatorname{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{\operatorname{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{\operatorname{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{\operatorname{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{\operatorname{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{\operatorname{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*}

Mathematica [C]  time = 0.158486, size = 67, normalized size = 0.31 \[ -2 \sqrt{\cot (x)+1}+\frac{1}{2} (1-i)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{\cot (x)+1}}{\sqrt{1-i}}\right )+\frac{1}{2} (1+i)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{\cot (x)+1}}{\sqrt{1+i}}\right ) \]

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 [B]  time = 0.034, size = 442, normalized size = 2.1 \begin{align*} -2\,\sqrt{1+\cot \left ( x \right ) }+{\frac{\sqrt{2+2\,\sqrt{2}}}{4}\ln \left ( 1+\cot \left ( x \right ) +\sqrt{2}+\sqrt{1+\cot \left ( x \right ) }\sqrt{2+2\,\sqrt{2}} \right ) }-{\frac{\sqrt{2+2\,\sqrt{2}}\sqrt{2}}{8}\ln \left ( 1+\cot \left ( x \right ) +\sqrt{2}+\sqrt{1+\cot \left ( x \right ) }\sqrt{2+2\,\sqrt{2}} \right ) }+{\frac{ \left ( 2+2\,\sqrt{2} \right ) \sqrt{2}}{4\,\sqrt{-2+2\,\sqrt{2}}}\arctan \left ({\frac{1}{\sqrt{-2+2\,\sqrt{2}}} \left ( 2\,\sqrt{1+\cot \left ( x \right ) }+\sqrt{2+2\,\sqrt{2}} \right ) } \right ) }-{\frac{2+2\,\sqrt{2}}{2\,\sqrt{-2+2\,\sqrt{2}}}\arctan \left ({\frac{1}{\sqrt{-2+2\,\sqrt{2}}} \left ( 2\,\sqrt{1+\cot \left ( x \right ) }+\sqrt{2+2\,\sqrt{2}} \right ) } \right ) }+{\frac{\sqrt{2}}{\sqrt{-2+2\,\sqrt{2}}}\arctan \left ({\frac{1}{\sqrt{-2+2\,\sqrt{2}}} \left ( 2\,\sqrt{1+\cot \left ( x \right ) }+\sqrt{2+2\,\sqrt{2}} \right ) } \right ) }-{\frac{\sqrt{2+2\,\sqrt{2}}}{4}\ln \left ( 1+\cot \left ( x \right ) +\sqrt{2}-\sqrt{1+\cot \left ( x \right ) }\sqrt{2+2\,\sqrt{2}} \right ) }+{\frac{\sqrt{2+2\,\sqrt{2}}\sqrt{2}}{8}\ln \left ( 1+\cot \left ( x \right ) +\sqrt{2}-\sqrt{1+\cot \left ( x \right ) }\sqrt{2+2\,\sqrt{2}} \right ) }+{\frac{ \left ( 2+2\,\sqrt{2} \right ) \sqrt{2}}{4\,\sqrt{-2+2\,\sqrt{2}}}\arctan \left ({\frac{1}{\sqrt{-2+2\,\sqrt{2}}} \left ( 2\,\sqrt{1+\cot \left ( x \right ) }-\sqrt{2+2\,\sqrt{2}} \right ) } \right ) }-{\frac{2+2\,\sqrt{2}}{2\,\sqrt{-2+2\,\sqrt{2}}}\arctan \left ({\frac{1}{\sqrt{-2+2\,\sqrt{2}}} \left ( 2\,\sqrt{1+\cot \left ( x \right ) }-\sqrt{2+2\,\sqrt{2}} \right ) } \right ) }+{\frac{\sqrt{2}}{\sqrt{-2+2\,\sqrt{2}}}\arctan \left ({\frac{1}{\sqrt{-2+2\,\sqrt{2}}} \left ( 2\,\sqrt{1+\cot \left ( x \right ) }-\sqrt{2+2\,\sqrt{2}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(1+cot(x))^(1/2)+1/4*(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))-1/8*(2+2
*2^(1/2))^(1/2)*2^(1/2)*ln(1+cot(x)+2^(1/2)+(1+cot(x))^(1/2)*(2+2*2^(1/2))^(1/2))+1/4*2^(1/2)*(2+2*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))-1/2*(2+2*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/(-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))*2^(1/2)-1/4*(2+2*2^(1/2))^(1/2)*ln(1+co
t(x)+2^(1/2)-(1+cot(x))^(1/2)*(2+2*2^(1/2))^(1/2))+1/8*(2+2*2^(1/2))^(1/2)*2^(1/2)*ln(1+cot(x)+2^(1/2)-(1+cot(
x))^(1/2)*(2+2*2^(1/2))^(1/2))+1/4*2^(1/2)*(2+2*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))-1/2*(2+2*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/(-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))*2^(1/2)

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

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

[Out]

Timed out

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

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., size = 0, normalized size = 0. \begin{align*} \int \frac{\cot \left (x\right )^{2}}{\sqrt{\cot \left (x\right ) + 1}}\,{d x} \end{align*}

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)