3.1.42 \(\int \cot (x) \sqrt {1+\cot (x)} \, dx\) [42]

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

[Out]

-2*(1+cot(x))^(1/2)+1/2*arctan(1/2*(4+cot(x)*(2-2^(1/2))-3*2^(1/2))/(1+cot(x))^(1/2)/(-7+5*2^(1/2))^(1/2))*(-2
+2*2^(1/2))^(1/2)+1/2*arctanh(1/2*(4+3*2^(1/2)+cot(x)*(2+2^(1/2)))/(1+cot(x))^(1/2)/(7+5*2^(1/2))^(1/2))*(2+2*
2^(1/2))^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.18, antiderivative size = 135, normalized size of antiderivative = 1.00, 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 = {3609, 3617, 3616, 209, 213} \begin {gather*} \sqrt {\frac {1}{2} \left (\sqrt {2}-1\right )} \text {ArcTan}\left (\frac {\left (2-\sqrt {2}\right ) \cot (x)-3 \sqrt {2}+4}{2 \sqrt {5 \sqrt {2}-7} \sqrt {\cot (x)+1}}\right )-2 \sqrt {\cot (x)+1}+\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 ) \end {gather*}

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 209

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

Rule 213

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

Rule 3609

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 3616

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 3617

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])

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 ) \text {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 ) \text {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] Result contains complex when optimal does not.
time = 0.10, size = 61, normalized size = 0.45 \begin {gather*} \sqrt {1-i} \tanh ^{-1}\left (\frac {\sqrt {1+\cot (x)}}{\sqrt {1-i}}\right )+\sqrt {1+i} \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]*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]]

________________________________________________________________________________________

Maple [A]
time = 0.28, size = 174, normalized size = 1.29

method	result	size
derivativedivides	\(-2 \sqrt {1+\cot \left (x \right )}+\frac {\sqrt {2+2 \sqrt {2}}\, \ln \left (1+\cot \left (x \right )+\sqrt {2}+\sqrt {1+\cot \left (x \right )}\, \sqrt {2+2 \sqrt {2}}\right )}{4}-\frac {\left (1-\sqrt {2}\right ) \arctan \left (\frac {2 \sqrt {1+\cot \left (x \right )}+\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{\sqrt {-2+2 \sqrt {2}}}-\frac {\sqrt {2+2 \sqrt {2}}\, \ln \left (1+\cot \left (x \right )+\sqrt {2}-\sqrt {1+\cot \left (x \right )}\, \sqrt {2+2 \sqrt {2}}\right )}{4}+\frac {\left (\sqrt {2}-1\right ) \arctan \left (\frac {2 \sqrt {1+\cot \left (x \right )}-\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{\sqrt {-2+2 \sqrt {2}}}\)	\(174\)
default	\(-2 \sqrt {1+\cot \left (x \right )}+\frac {\sqrt {2+2 \sqrt {2}}\, \ln \left (1+\cot \left (x \right )+\sqrt {2}+\sqrt {1+\cot \left (x \right )}\, \sqrt {2+2 \sqrt {2}}\right )}{4}-\frac {\left (1-\sqrt {2}\right ) \arctan \left (\frac {2 \sqrt {1+\cot \left (x \right )}+\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{\sqrt {-2+2 \sqrt {2}}}-\frac {\sqrt {2+2 \sqrt {2}}\, \ln \left (1+\cot \left (x \right )+\sqrt {2}-\sqrt {1+\cot \left (x \right )}\, \sqrt {2+2 \sqrt {2}}\right )}{4}+\frac {\left (\sqrt {2}-1\right ) \arctan \left (\frac {2 \sqrt {1+\cot \left (x \right )}-\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{\sqrt {-2+2 \sqrt {2}}}\)	\(174\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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^(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))+(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))

________________________________________________________________________________________

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

[Out]

integrate(sqrt(cot(x) + 1)*cot(x), x)

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

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

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)

________________________________________________________________________________________

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

[Out]

integrate(sqrt(cot(x) + 1)*cot(x), x)

________________________________________________________________________________________

Mupad [B]
time = 0.48, size = 210, normalized size = 1.56 \begin {gather*} \mathrm {atanh}\left (\frac {\sqrt {\mathrm {cot}\left (x\right )+1}}{4\,\sqrt {\frac {1}{8}-\frac {\sqrt {2}}{8}}}+\frac {\sqrt {\mathrm {cot}\left (x\right )+1}}{4\,\sqrt {\frac {\sqrt {2}}{8}+\frac {1}{8}}}-\frac {\sqrt {2}\,\sqrt {\mathrm {cot}\left (x\right )+1}}{8\,\sqrt {\frac {1}{8}-\frac {\sqrt {2}}{8}}}+\frac {\sqrt {2}\,\sqrt {\mathrm {cot}\left (x\right )+1}}{8\,\sqrt {\frac {\sqrt {2}}{8}+\frac {1}{8}}}\right )\,\left (2\,\sqrt {\frac {1}{8}-\frac {\sqrt {2}}{8}}+2\,\sqrt {\frac {\sqrt {2}}{8}+\frac {1}{8}}\right )-\mathrm {atanh}\left (\frac {\sqrt {\mathrm {cot}\left (x\right )+1}}{4\,\sqrt {\frac {\sqrt {2}}{8}+\frac {1}{8}}}-\frac {\sqrt {\mathrm {cot}\left (x\right )+1}}{4\,\sqrt {\frac {1}{8}-\frac {\sqrt {2}}{8}}}+\frac {\sqrt {2}\,\sqrt {\mathrm {cot}\left (x\right )+1}}{8\,\sqrt {\frac {1}{8}-\frac {\sqrt {2}}{8}}}+\frac {\sqrt {2}\,\sqrt {\mathrm {cot}\left (x\right )+1}}{8\,\sqrt {\frac {\sqrt {2}}{8}+\frac {1}{8}}}\right )\,\left (2\,\sqrt {\frac {1}{8}-\frac {\sqrt {2}}{8}}-2\,\sqrt {\frac {\sqrt {2}}{8}+\frac {1}{8}}\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)*(cot(x) + 1)^(1/2),x)

[Out]

atanh((cot(x) + 1)^(1/2)/(4*(1/8 - 2^(1/2)/8)^(1/2)) + (cot(x) + 1)^(1/2)/(4*(2^(1/2)/8 + 1/8)^(1/2)) - (2^(1/
2)*(cot(x) + 1)^(1/2))/(8*(1/8 - 2^(1/2)/8)^(1/2)) + (2^(1/2)*(cot(x) + 1)^(1/2))/(8*(2^(1/2)/8 + 1/8)^(1/2)))
*(2*(1/8 - 2^(1/2)/8)^(1/2) + 2*(2^(1/2)/8 + 1/8)^(1/2)) - atanh((cot(x) + 1)^(1/2)/(4*(2^(1/2)/8 + 1/8)^(1/2)
) - (cot(x) + 1)^(1/2)/(4*(1/8 - 2^(1/2)/8)^(1/2)) + (2^(1/2)*(cot(x) + 1)^(1/2))/(8*(1/8 - 2^(1/2)/8)^(1/2))
+ (2^(1/2)*(cot(x) + 1)^(1/2))/(8*(2^(1/2)/8 + 1/8)^(1/2)))*(2*(1/8 - 2^(1/2)/8)^(1/2) - 2*(2^(1/2)/8 + 1/8)^(
1/2)) - 2*(cot(x) + 1)^(1/2)

________________________________________________________________________________________