3.47 \(\int \frac {\cot ^2(x)}{(1+\cot (x))^{3/2}} \, dx\)

Optimal. Leaf size=139 \[ \frac {1}{\sqrt {\cot (x)+1}}+\frac {1}{2} \sqrt {\frac {1}{2} \left (\sqrt {2}-1\right )} \tan ^{-1}\left (\frac {\left (2-\sqrt {2}\right ) \cot (x)-3 \sqrt {2}+4}{2 \sqrt {5 \sqrt {2}-7} \sqrt {\cot (x)+1}}\right )+\frac {1}{2} \sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \tanh ^{-1}\left (\frac {\left (2+\sqrt {2}\right ) \cot (x)+3 \sqrt {2}+4}{2 \sqrt {7+5 \sqrt {2}} \sqrt {\cot (x)+1}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.19, antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {3542, 3536, 3535, 203, 207} \[ \frac {1}{\sqrt {\cot (x)+1}}+\frac {1}{2} \sqrt {\frac {1}{2} \left (\sqrt {2}-1\right )} \tan ^{-1}\left (\frac {\left (2-\sqrt {2}\right ) \cot (x)-3 \sqrt {2}+4}{2 \sqrt {5 \sqrt {2}-7} \sqrt {\cot (x)+1}}\right )+\frac {1}{2} \sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \tanh ^{-1}\left (\frac {\left (2+\sqrt {2}\right ) \cot (x)+3 \sqrt {2}+4}{2 \sqrt {7+5 \sqrt {2}} \sqrt {\cot (x)+1}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Cot[x]^2/(1 + Cot[x])^(3/2),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]]
)])/2 + (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 + 1/Sqrt[1 + Cot[x]]

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])

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 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 3542

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

Rubi steps

\begin {align*} \int \frac {\cot ^2(x)}{(1+\cot (x))^{3/2}} \, dx &=\frac {1}{\sqrt {1+\cot (x)}}+\frac {1}{2} \int \frac {-1+\cot (x)}{\sqrt {1+\cot (x)}} \, dx\\ &=\frac {1}{\sqrt {1+\cot (x)}}+\frac {\int \frac {-\sqrt {2}-\left (2-\sqrt {2}\right ) \cot (x)}{\sqrt {1+\cot (x)}} \, dx}{4 \sqrt {2}}-\frac {\int \frac {\sqrt {2}-\left (2+\sqrt {2}\right ) \cot (x)}{\sqrt {1+\cot (x)}} \, dx}{4 \sqrt {2}}\\ &=\frac {1}{\sqrt {1+\cot (x)}}-\frac {1}{2} \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 )+\frac {1}{2} \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 )\\ &=\frac {1}{2} \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 )+\frac {1}{2} \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 )+\frac {1}{\sqrt {1+\cot (x)}}\\ \end {align*}

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Mathematica [C]  time = 0.13, size = 65, normalized size = 0.47 \[ \frac {1}{\sqrt {\cot (x)+1}}+\frac {1}{2} \sqrt {1-i} \tanh ^{-1}\left (\frac {\sqrt {\cot (x)+1}}{\sqrt {1-i}}\right )+\frac {1}{2} \sqrt {1+i} \tanh ^{-1}\left (\frac {\sqrt {\cot (x)+1}}{\sqrt {1+i}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[x]^2/(1 + Cot[x])^(3/2),x]

[Out]

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

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)^2/(1+cot(x))^(3/2),x, algorithm="fricas")

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cot \relax (x)^{2}}{{\left (\cot \relax (x) + 1\right )}^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)^2/(1+cot(x))^(3/2),x, algorithm="giac")

[Out]

integrate(cot(x)^2/(cot(x) + 1)^(3/2), x)

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maple [B]  time = 0.19, size = 249, normalized size = 1.79 \[ -\frac {\sqrt {2 \sqrt {2}+2}\, \ln \left (1+\cot \relax (x )+\sqrt {2}-\sqrt {1+\cot \relax (x )}\, \sqrt {2 \sqrt {2}+2}\right )}{8}+\frac {\arctan \left (\frac {2 \sqrt {1+\cot \relax (x )}-\sqrt {2 \sqrt {2}+2}}{\sqrt {-2+2 \sqrt {2}}}\right ) \sqrt {2}}{2 \sqrt {-2+2 \sqrt {2}}}-\frac {\arctan \left (\frac {2 \sqrt {1+\cot \relax (x )}-\sqrt {2 \sqrt {2}+2}}{\sqrt {-2+2 \sqrt {2}}}\right )}{2 \sqrt {-2+2 \sqrt {2}}}+\frac {\sqrt {2 \sqrt {2}+2}\, \ln \left (1+\cot \relax (x )+\sqrt {2}+\sqrt {1+\cot \relax (x )}\, \sqrt {2 \sqrt {2}+2}\right )}{8}-\frac {\arctan \left (\frac {2 \sqrt {1+\cot \relax (x )}+\sqrt {2 \sqrt {2}+2}}{\sqrt {-2+2 \sqrt {2}}}\right )}{2 \sqrt {-2+2 \sqrt {2}}}+\frac {\arctan \left (\frac {2 \sqrt {1+\cot \relax (x )}+\sqrt {2 \sqrt {2}+2}}{\sqrt {-2+2 \sqrt {2}}}\right ) \sqrt {2}}{2 \sqrt {-2+2 \sqrt {2}}}+\frac {1}{\sqrt {1+\cot \relax (x )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(x)^2/(1+cot(x))^(3/2),x)

[Out]

-1/8*(2*2^(1/2)+2)^(1/2)*ln(1+cot(x)+2^(1/2)-(1+cot(x))^(1/2)*(2*2^(1/2)+2)^(1/2))+1/2/(-2+2*2^(1/2))^(1/2)*ar
ctan((2*(1+cot(x))^(1/2)-(2*2^(1/2)+2)^(1/2))/(-2+2*2^(1/2))^(1/2))*2^(1/2)-1/2/(-2+2*2^(1/2))^(1/2)*arctan((2
*(1+cot(x))^(1/2)-(2*2^(1/2)+2)^(1/2))/(-2+2*2^(1/2))^(1/2))+1/8*(2*2^(1/2)+2)^(1/2)*ln(1+cot(x)+2^(1/2)+(1+co
t(x))^(1/2)*(2*2^(1/2)+2)^(1/2))-1/2/(-2+2*2^(1/2))^(1/2)*arctan((2*(1+cot(x))^(1/2)+(2*2^(1/2)+2)^(1/2))/(-2+
2*2^(1/2))^(1/2))+1/2/(-2+2*2^(1/2))^(1/2)*arctan((2*(1+cot(x))^(1/2)+(2*2^(1/2)+2)^(1/2))/(-2+2*2^(1/2))^(1/2
))*2^(1/2)+1/(1+cot(x))^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cot \relax (x)^{2}}{{\left (\cot \relax (x) + 1\right )}^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)^2/(1+cot(x))^(3/2),x, algorithm="maxima")

[Out]

integrate(cot(x)^2/(cot(x) + 1)^(3/2), x)

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mupad [B]  time = 0.50, size = 208, normalized size = 1.50 \[ \frac {1}{\sqrt {\mathrm {cot}\relax (x)+1}}-\mathrm {atanh}\left (\frac {\sqrt {\mathrm {cot}\relax (x)+1}}{8\,\sqrt {\frac {\sqrt {2}}{32}+\frac {1}{32}}}-\frac {\sqrt {\mathrm {cot}\relax (x)+1}}{8\,\sqrt {\frac {1}{32}-\frac {\sqrt {2}}{32}}}+\frac {\sqrt {2}\,\sqrt {\mathrm {cot}\relax (x)+1}}{16\,\sqrt {\frac {1}{32}-\frac {\sqrt {2}}{32}}}+\frac {\sqrt {2}\,\sqrt {\mathrm {cot}\relax (x)+1}}{16\,\sqrt {\frac {\sqrt {2}}{32}+\frac {1}{32}}}\right )\,\left (2\,\sqrt {\frac {1}{32}-\frac {\sqrt {2}}{32}}-2\,\sqrt {\frac {\sqrt {2}}{32}+\frac {1}{32}}\right )+\mathrm {atanh}\left (\frac {\sqrt {\mathrm {cot}\relax (x)+1}}{8\,\sqrt {\frac {1}{32}-\frac {\sqrt {2}}{32}}}+\frac {\sqrt {\mathrm {cot}\relax (x)+1}}{8\,\sqrt {\frac {\sqrt {2}}{32}+\frac {1}{32}}}-\frac {\sqrt {2}\,\sqrt {\mathrm {cot}\relax (x)+1}}{16\,\sqrt {\frac {1}{32}-\frac {\sqrt {2}}{32}}}+\frac {\sqrt {2}\,\sqrt {\mathrm {cot}\relax (x)+1}}{16\,\sqrt {\frac {\sqrt {2}}{32}+\frac {1}{32}}}\right )\,\left (2\,\sqrt {\frac {1}{32}-\frac {\sqrt {2}}{32}}+2\,\sqrt {\frac {\sqrt {2}}{32}+\frac {1}{32}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(x)^2/(cot(x) + 1)^(3/2),x)

[Out]

1/(cot(x) + 1)^(1/2) - atanh((cot(x) + 1)^(1/2)/(8*(2^(1/2)/32 + 1/32)^(1/2)) - (cot(x) + 1)^(1/2)/(8*(1/32 -
2^(1/2)/32)^(1/2)) + (2^(1/2)*(cot(x) + 1)^(1/2))/(16*(1/32 - 2^(1/2)/32)^(1/2)) + (2^(1/2)*(cot(x) + 1)^(1/2)
)/(16*(2^(1/2)/32 + 1/32)^(1/2)))*(2*(1/32 - 2^(1/2)/32)^(1/2) - 2*(2^(1/2)/32 + 1/32)^(1/2)) + atanh((cot(x)
+ 1)^(1/2)/(8*(1/32 - 2^(1/2)/32)^(1/2)) + (cot(x) + 1)^(1/2)/(8*(2^(1/2)/32 + 1/32)^(1/2)) - (2^(1/2)*(cot(x)
+ 1)^(1/2))/(16*(1/32 - 2^(1/2)/32)^(1/2)) + (2^(1/2)*(cot(x) + 1)^(1/2))/(16*(2^(1/2)/32 + 1/32)^(1/2)))*(2*
(1/32 - 2^(1/2)/32)^(1/2) + 2*(2^(1/2)/32 + 1/32)^(1/2))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cot ^{2}{\relax (x )}}{\left (\cot {\relax (x )} + 1\right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)**2/(1+cot(x))**(3/2),x)

[Out]

Integral(cot(x)**2/(cot(x) + 1)**(3/2), x)

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