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]

(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]]

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Rubi [A]  time = 0.193398, antiderivative size = 139, normalized size of antiderivative = 1., 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 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]

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 \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*}

Mathematica [C]  time = 0.127489, 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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Maple [B]  time = 0.048, size = 249, normalized size = 1.8 \begin{align*}{\frac{\sqrt{2+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{1}{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+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}}}{8}\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+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}{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{1+\cot \left ( x \right ) }}} \end{align*}

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*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))^(1/2)*arc
tan((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))^(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)-1/8*(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))^(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*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/(1+cot(x))^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cot \left (x\right )^{2}}{{\left (\cot \left (x\right ) + 1\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

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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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))^(3/2),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 )}}{\left (\cot{\left (x \right )} + 1\right )^{\frac{3}{2}}}\, dx \end{align*}

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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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cot \left (x\right )^{2}}{{\left (\cot \left (x\right ) + 1\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

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)