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

Optimal. Leaf size=223 \[ -\frac{2}{3} (\cot (x)+1)^{3/2}+\frac{\log \left (\cot (x)-\sqrt{2 \left (1+\sqrt{2}\right )} \sqrt{\cot (x)+1}+\sqrt{2}+1\right )}{2 \sqrt{2 \left (1+\sqrt{2}\right )}}-\frac{\log \left (\cot (x)+\sqrt{2 \left (1+\sqrt{2}\right )} \sqrt{\cot (x)+1}+\sqrt{2}+1\right )}{2 \sqrt{2 \left (1+\sqrt{2}\right )}}-\sqrt{\frac{1}{2} \left (1+\sqrt{2}\right )} \tan ^{-1}\left (\frac{\sqrt{2 \left (1+\sqrt{2}\right )}-2 \sqrt{\cot (x)+1}}{\sqrt{2 \left (\sqrt{2}-1\right )}}\right )+\sqrt{\frac{1}{2} \left (1+\sqrt{2}\right )} \tan ^{-1}\left (\frac{2 \sqrt{\cot (x)+1}+\sqrt{2 \left (1+\sqrt{2}\right )}}{\sqrt{2 \left (\sqrt{2}-1\right )}}\right ) \]

[Out]

-(Sqrt[(1 + Sqrt[2])/2]*ArcTan[(Sqrt[2*(1 + Sqrt[2])] - 2*Sqrt[1 + Cot[x]])/Sqrt[2*(-1 + Sqrt[2])]]) + Sqrt[(1
+ Sqrt[2])/2]*ArcTan[(Sqrt[2*(1 + Sqrt[2])] + 2*Sqrt[1 + Cot[x]])/Sqrt[2*(-1 + Sqrt[2])]] - (2*(1 + Cot[x])^(
3/2))/3 + Log[1 + Sqrt[2] + Cot[x] - Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 + Cot[x]]]/(2*Sqrt[2*(1 + Sqrt[2])]) - Log[1
+ Sqrt[2] + Cot[x] + Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 + Cot[x]]]/(2*Sqrt[2*(1 + Sqrt[2])])

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Rubi [A]  time = 0.266433, antiderivative size = 223, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.692, Rules used = {3543, 3485, 700, 1127, 1161, 618, 204, 1164, 628} \[ -\frac{2}{3} (\cot (x)+1)^{3/2}+\frac{\log \left (\cot (x)-\sqrt{2 \left (1+\sqrt{2}\right )} \sqrt{\cot (x)+1}+\sqrt{2}+1\right )}{2 \sqrt{2 \left (1+\sqrt{2}\right )}}-\frac{\log \left (\cot (x)+\sqrt{2 \left (1+\sqrt{2}\right )} \sqrt{\cot (x)+1}+\sqrt{2}+1\right )}{2 \sqrt{2 \left (1+\sqrt{2}\right )}}-\sqrt{\frac{1}{2} \left (1+\sqrt{2}\right )} \tan ^{-1}\left (\frac{\sqrt{2 \left (1+\sqrt{2}\right )}-2 \sqrt{\cot (x)+1}}{\sqrt{2 \left (\sqrt{2}-1\right )}}\right )+\sqrt{\frac{1}{2} \left (1+\sqrt{2}\right )} \tan ^{-1}\left (\frac{2 \sqrt{\cot (x)+1}+\sqrt{2 \left (1+\sqrt{2}\right )}}{\sqrt{2 \left (\sqrt{2}-1\right )}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Cot[x]^2*Sqrt[1 + Cot[x]],x]

[Out]

-(Sqrt[(1 + Sqrt[2])/2]*ArcTan[(Sqrt[2*(1 + Sqrt[2])] - 2*Sqrt[1 + Cot[x]])/Sqrt[2*(-1 + Sqrt[2])]]) + Sqrt[(1
+ Sqrt[2])/2]*ArcTan[(Sqrt[2*(1 + Sqrt[2])] + 2*Sqrt[1 + Cot[x]])/Sqrt[2*(-1 + Sqrt[2])]] - (2*(1 + Cot[x])^(
3/2))/3 + Log[1 + Sqrt[2] + Cot[x] - Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 + Cot[x]]]/(2*Sqrt[2*(1 + Sqrt[2])]) - Log[1
+ Sqrt[2] + Cot[x] + Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 + Cot[x]]]/(2*Sqrt[2*(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 700

Int[Sqrt[(d_) + (e_.)*(x_)]/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[2*e, Subst[Int[x^2/(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 1127

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

Rule 1161

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e - b/c, 2]},
Dist[e/(2*c), Int[1/Simp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /
; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - a*e^2, 0] && (GtQ[(2*d)/e - b/c, 0] || ( !Lt
Q[(2*d)/e - b/c, 0] && EqQ[d - e*Rt[a/c, 2], 0]))

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 1164

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e - b/c, 2]},
Dist[e/(2*c*q), Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x
- x^2, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - a*e^2, 0] &&  !GtQ[b^2
- 4*a*c, 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 \cot ^2(x) \sqrt{1+\cot (x)} \, dx &=-\frac{2}{3} (1+\cot (x))^{3/2}-\int \sqrt{1+\cot (x)} \, dx\\ &=-\frac{2}{3} (1+\cot (x))^{3/2}+\operatorname{Subst}\left (\int \frac{\sqrt{1+x}}{1+x^2} \, dx,x,\cot (x)\right )\\ &=-\frac{2}{3} (1+\cot (x))^{3/2}+2 \operatorname{Subst}\left (\int \frac{x^2}{2-2 x^2+x^4} \, dx,x,\sqrt{1+\cot (x)}\right )\\ &=-\frac{2}{3} (1+\cot (x))^{3/2}-\operatorname{Subst}\left (\int \frac{\sqrt{2}-x^2}{2-2 x^2+x^4} \, dx,x,\sqrt{1+\cot (x)}\right )+\operatorname{Subst}\left (\int \frac{\sqrt{2}+x^2}{2-2 x^2+x^4} \, dx,x,\sqrt{1+\cot (x)}\right )\\ &=-\frac{2}{3} (1+\cot (x))^{3/2}+\frac{1}{2} \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 )+\frac{1}{2} \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 )+\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 )}{2 \sqrt{2 \left (1+\sqrt{2}\right )}}+\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 )}{2 \sqrt{2 \left (1+\sqrt{2}\right )}}\\ &=-\frac{2}{3} (1+\cot (x))^{3/2}+\frac{\log \left (1+\sqrt{2}+\cot (x)-\sqrt{2 \left (1+\sqrt{2}\right )} \sqrt{1+\cot (x)}\right )}{2 \sqrt{2 \left (1+\sqrt{2}\right )}}-\frac{\log \left (1+\sqrt{2}+\cot (x)+\sqrt{2 \left (1+\sqrt{2}\right )} \sqrt{1+\cot (x)}\right )}{2 \sqrt{2 \left (1+\sqrt{2}\right )}}-\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 )-\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 )\\ &=\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 )}{\sqrt{2 \left (-1+\sqrt{2}\right )}}+\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 )}{\sqrt{2 \left (-1+\sqrt{2}\right )}}-\frac{2}{3} (1+\cot (x))^{3/2}+\frac{\log \left (1+\sqrt{2}+\cot (x)-\sqrt{2 \left (1+\sqrt{2}\right )} \sqrt{1+\cot (x)}\right )}{2 \sqrt{2 \left (1+\sqrt{2}\right )}}-\frac{\log \left (1+\sqrt{2}+\cot (x)+\sqrt{2 \left (1+\sqrt{2}\right )} \sqrt{1+\cot (x)}\right )}{2 \sqrt{2 \left (1+\sqrt{2}\right )}}\\ \end{align*}

Mathematica [C]  time = 0.159616, size = 69, normalized size = 0.31 \[ -\frac{2}{3} (\cot (x)+1)^{3/2}-i \sqrt{1-i} \tanh ^{-1}\left (\frac{\sqrt{\cot (x)+1}}{\sqrt{1-i}}\right )+i \sqrt{1+i} \tanh ^{-1}\left (\frac{\sqrt{\cot (x)+1}}{\sqrt{1+i}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.13, size = 356, normalized size = 1.6 \begin{align*} -{\frac{2}{3} \left ( 1+\cot \left ( x \right ) \right ) ^{{\frac{3}{2}}}}-{\frac{\sqrt{2+2\,\sqrt{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} \left ( 2+2\,\sqrt{2} \right ) }{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{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+2\,\sqrt{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} \left ( 2+2\,\sqrt{2} \right ) }{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{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 ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(1+cot(x))^(3/2)-1/4*(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)
)+1/2*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))+1/4*(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))-1/2*(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))+1/4*(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))+1/2*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))-1/4*(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))-1/2*(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))

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

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