3.42 \(\int \cot (x) \sqrt{1+\cot (x)} \, dx\)

Optimal. Leaf size=135 \[ -2 \sqrt{\cot (x)+1}+\sqrt{\frac{1}{2} \left (\sqrt{2}-1\right )} \tan ^{-1}\left (\frac{\left (2-\sqrt{2}\right ) \cot (x)-3 \sqrt{2}+4}{2 \sqrt{5 \sqrt{2}-7} \sqrt{\cot (x)+1}}\right )+\sqrt{\frac{1}{2} \left (1+\sqrt{2}\right )} \tanh ^{-1}\left (\frac{\left (2+\sqrt{2}\right ) \cot (x)+3 \sqrt{2}+4}{2 \sqrt{7+5 \sqrt{2}} \sqrt{\cot (x)+1}}\right ) \]

[Out]

Sqrt[(-1 + Sqrt[2])/2]*ArcTan[(4 - 3*Sqrt[2] + (2 - Sqrt[2])*Cot[x])/(2*Sqrt[-7 + 5*Sqrt[2]]*Sqrt[1 + Cot[x]])
] + Sqrt[(1 + Sqrt[2])/2]*ArcTanh[(4 + 3*Sqrt[2] + (2 + Sqrt[2])*Cot[x])/(2*Sqrt[7 + 5*Sqrt[2]]*Sqrt[1 + Cot[x
]])] - 2*Sqrt[1 + Cot[x]]

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Rubi [A]  time = 0.240291, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.454, Rules used = {3528, 3536, 3535, 203, 207} \[ -2 \sqrt{\cot (x)+1}+\sqrt{\frac{1}{2} \left (\sqrt{2}-1\right )} \tan ^{-1}\left (\frac{\left (2-\sqrt{2}\right ) \cot (x)-3 \sqrt{2}+4}{2 \sqrt{5 \sqrt{2}-7} \sqrt{\cot (x)+1}}\right )+\sqrt{\frac{1}{2} \left (1+\sqrt{2}\right )} \tanh ^{-1}\left (\frac{\left (2+\sqrt{2}\right ) \cot (x)+3 \sqrt{2}+4}{2 \sqrt{7+5 \sqrt{2}} \sqrt{\cot (x)+1}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Cot[x]*Sqrt[1 + Cot[x]],x]

[Out]

Sqrt[(-1 + Sqrt[2])/2]*ArcTan[(4 - 3*Sqrt[2] + (2 - Sqrt[2])*Cot[x])/(2*Sqrt[-7 + 5*Sqrt[2]]*Sqrt[1 + Cot[x]])
] + Sqrt[(1 + Sqrt[2])/2]*ArcTanh[(4 + 3*Sqrt[2] + (2 + Sqrt[2])*Cot[x])/(2*Sqrt[7 + 5*Sqrt[2]]*Sqrt[1 + Cot[x
]])] - 2*Sqrt[1 + Cot[x]]

Rule 3528

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

Rule 3536

Int[((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> With[{q =
Rt[a^2 + b^2, 2]}, Dist[1/(2*q), Int[(a*c + b*d + c*q + (b*c - a*d + d*q)*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*
x]], x], x] - Dist[1/(2*q), Int[(a*c + b*d - c*q + (b*c - a*d - d*q)*Tan[e + f*x])/Sqrt[a + b*Tan[e + f*x]], x
], x]] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && NeQ[2
*a*c*d - b*(c^2 - d^2), 0] && (PerfectSquareQ[a^2 + b^2] || RationalQ[a, b, c, d])

Rule 3535

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

Rule 203

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

Rule 207

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

Rubi steps

\begin{align*} \int \cot (x) \sqrt{1+\cot (x)} \, dx &=-2 \sqrt{1+\cot (x)}-\int \frac{1-\cot (x)}{\sqrt{1+\cot (x)}} \, dx\\ &=-2 \sqrt{1+\cot (x)}+\frac{\int \frac{-\sqrt{2}-\left (-2-\sqrt{2}\right ) \cot (x)}{\sqrt{1+\cot (x)}} \, dx}{2 \sqrt{2}}-\frac{\int \frac{\sqrt{2}-\left (-2+\sqrt{2}\right ) \cot (x)}{\sqrt{1+\cot (x)}} \, dx}{2 \sqrt{2}}\\ &=-2 \sqrt{1+\cot (x)}+\left (-4+3 \sqrt{2}\right ) \operatorname{Subst}\left (\int \frac{1}{-2 \sqrt{2} \left (-2+\sqrt{2}\right )-4 \left (-2+\sqrt{2}\right )^2+x^2} \, dx,x,\frac{-\sqrt{2}-2 \left (-2+\sqrt{2}\right )-\left (-2+\sqrt{2}\right ) \cot (x)}{\sqrt{1+\cot (x)}}\right )-\left (4+3 \sqrt{2}\right ) \operatorname{Subst}\left (\int \frac{1}{2 \sqrt{2} \left (-2-\sqrt{2}\right )-4 \left (-2-\sqrt{2}\right )^2+x^2} \, dx,x,\frac{\sqrt{2}-2 \left (-2-\sqrt{2}\right )-\left (-2-\sqrt{2}\right ) \cot (x)}{\sqrt{1+\cot (x)}}\right )\\ &=\sqrt{\frac{1}{2} \left (-1+\sqrt{2}\right )} \tan ^{-1}\left (\frac{4-3 \sqrt{2}+\left (2-\sqrt{2}\right ) \cot (x)}{2 \sqrt{-7+5 \sqrt{2}} \sqrt{1+\cot (x)}}\right )+\sqrt{\frac{1}{2} \left (1+\sqrt{2}\right )} \tanh ^{-1}\left (\frac{4+3 \sqrt{2}+\left (2+\sqrt{2}\right ) \cot (x)}{2 \sqrt{7+5 \sqrt{2}} \sqrt{1+\cot (x)}}\right )-2 \sqrt{1+\cot (x)}\\ \end{align*}

Mathematica [C]  time = 0.0898445, size = 61, normalized size = 0.45 \[ -2 \sqrt{\cot (x)+1}+\sqrt{1-i} \tanh ^{-1}\left (\frac{\sqrt{\cot (x)+1}}{\sqrt{1-i}}\right )+\sqrt{1+i} \tanh ^{-1}\left (\frac{\sqrt{\cot (x)+1}}{\sqrt{1+i}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[x]*Sqrt[1 + Cot[x]],x]

[Out]

Sqrt[1 - I]*ArcTanh[Sqrt[1 + Cot[x]]/Sqrt[1 - I]] + Sqrt[1 + I]*ArcTanh[Sqrt[1 + Cot[x]]/Sqrt[1 + I]] - 2*Sqrt
[1 + Cot[x]]

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Maple [B]  time = 0.041, size = 249, normalized size = 1.8 \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{1}{\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}}{\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{1}{\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)*(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/(-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)*ar
ctan((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+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)*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/(-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 )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(cot(x) + 1)*cot(x), 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)*(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{\left (x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(cot(x) + 1)*cot(x), 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 )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(cot(x) + 1)*cot(x), x)