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

Optimal. Leaf size=226 \[ -\frac {1}{\sqrt {\cot (x)+1}}-\frac {\log \left (\cot (x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\cot (x)+1}+\sqrt {2}+1\right )}{4 \sqrt {2 \left (1+\sqrt {2}\right )}}+\frac {\log \left (\cot (x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\cot (x)+1}+\sqrt {2}+1\right )}{4 \sqrt {2 \left (1+\sqrt {2}\right )}}+\frac {1}{2} \sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {\cot (x)+1}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )-\frac {1}{2} \sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \tan ^{-1}\left (\frac {2 \sqrt {\cot (x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right ) \]

[Out]

-1/(1+cot(x))^(1/2)+1/4*arctan((-2*(1+cot(x))^(1/2)+(2+2*2^(1/2))^(1/2))/(-2+2*2^(1/2))^(1/2))*(2+2*2^(1/2))^(
1/2)-1/4*arctan((2*(1+cot(x))^(1/2)+(2+2*2^(1/2))^(1/2))/(-2+2*2^(1/2))^(1/2))*(2+2*2^(1/2))^(1/2)-1/4*ln(1+co
t(x)+2^(1/2)-(1+cot(x))^(1/2)*(2+2*2^(1/2))^(1/2))/(2+2*2^(1/2))^(1/2)+1/4*ln(1+cot(x)+2^(1/2)+(1+cot(x))^(1/2
)*(2+2*2^(1/2))^(1/2))/(2+2*2^(1/2))^(1/2)

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Rubi [A]  time = 0.19, antiderivative size = 226, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.909, Rules used = {3529, 21, 3485, 700, 1127, 1161, 618, 204, 1164, 628} \[ -\frac {1}{\sqrt {\cot (x)+1}}-\frac {\log \left (\cot (x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\cot (x)+1}+\sqrt {2}+1\right )}{4 \sqrt {2 \left (1+\sqrt {2}\right )}}+\frac {\log \left (\cot (x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\cot (x)+1}+\sqrt {2}+1\right )}{4 \sqrt {2 \left (1+\sqrt {2}\right )}}+\frac {1}{2} \sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {\cot (x)+1}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )-\frac {1}{2} \sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \tan ^{-1}\left (\frac {2 \sqrt {\cot (x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Cot[x]/(1 + Cot[x])^(3/2),x]

[Out]

(Sqrt[(1 + Sqrt[2])/2]*ArcTan[(Sqrt[2*(1 + Sqrt[2])] - 2*Sqrt[1 + Cot[x]])/Sqrt[2*(-1 + Sqrt[2])]])/2 - (Sqrt[
(1 + Sqrt[2])/2]*ArcTan[(Sqrt[2*(1 + Sqrt[2])] + 2*Sqrt[1 + Cot[x]])/Sqrt[2*(-1 + Sqrt[2])]])/2 - 1/Sqrt[1 + C
ot[x]] - Log[1 + Sqrt[2] + Cot[x] - Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 + Cot[x]]]/(4*Sqrt[2*(1 + Sqrt[2])]) + Log[1
+ Sqrt[2] + Cot[x] + Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 + Cot[x]]]/(4*Sqrt[2*(1 + Sqrt[2])])

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

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 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 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]

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 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 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 3529

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

Rubi steps

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

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Mathematica [C]  time = 0.14, size = 71, normalized size = 0.31 \[ -\frac {1}{\sqrt {\cot (x)+1}}+\frac {1}{2} i \sqrt {1-i} \tanh ^{-1}\left (\frac {\sqrt {\cot (x)+1}}{\sqrt {1-i}}\right )-\frac {1}{2} i \sqrt {1+i} \tanh ^{-1}\left (\frac {\sqrt {\cot (x)+1}}{\sqrt {1+i}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(I/2)*Sqrt[1 - I]*ArcTanh[Sqrt[1 + Cot[x]]/Sqrt[1 - I]] - (I/2)*Sqrt[1 + I]*ArcTanh[Sqrt[1 + Cot[x]]/Sqrt[1 +
I]] - 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)/(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 \[ \int \frac {\cot \relax (x)}{{\left (\cot \relax (x) + 1\right )}^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.13, size = 356, normalized size = 1.58 \[ -\frac {\sqrt {2 \sqrt {2}+2}\, \sqrt {2}\, \ln \left (1+\cot \relax (x )+\sqrt {2}-\sqrt {1+\cot \relax (x )}\, \sqrt {2 \sqrt {2}+2}\right )}{8}-\frac {\sqrt {2}\, \left (2 \sqrt {2}+2\right ) \arctan \left (\frac {2 \sqrt {1+\cot \relax (x )}-\sqrt {2 \sqrt {2}+2}}{\sqrt {-2+2 \sqrt {2}}}\right )}{4 \sqrt {-2+2 \sqrt {2}}}+\frac {\sqrt {2 \sqrt {2}+2}\, \ln \left (1+\cot \relax (x )+\sqrt {2}-\sqrt {1+\cot \relax (x )}\, \sqrt {2 \sqrt {2}+2}\right )}{8}+\frac {\left (2 \sqrt {2}+2\right ) \arctan \left (\frac {2 \sqrt {1+\cot \relax (x )}-\sqrt {2 \sqrt {2}+2}}{\sqrt {-2+2 \sqrt {2}}}\right )}{4 \sqrt {-2+2 \sqrt {2}}}+\frac {\sqrt {2 \sqrt {2}+2}\, \sqrt {2}\, \ln \left (1+\cot \relax (x )+\sqrt {2}+\sqrt {1+\cot \relax (x )}\, \sqrt {2 \sqrt {2}+2}\right )}{8}-\frac {\sqrt {2}\, \left (2 \sqrt {2}+2\right ) \arctan \left (\frac {2 \sqrt {1+\cot \relax (x )}+\sqrt {2 \sqrt {2}+2}}{\sqrt {-2+2 \sqrt {2}}}\right )}{4 \sqrt {-2+2 \sqrt {2}}}-\frac {\sqrt {2 \sqrt {2}+2}\, \ln \left (1+\cot \relax (x )+\sqrt {2}+\sqrt {1+\cot \relax (x )}\, \sqrt {2 \sqrt {2}+2}\right )}{8}+\frac {\left (2 \sqrt {2}+2\right ) \arctan \left (\frac {2 \sqrt {1+\cot \relax (x )}+\sqrt {2 \sqrt {2}+2}}{\sqrt {-2+2 \sqrt {2}}}\right )}{4 \sqrt {-2+2 \sqrt {2}}}-\frac {1}{\sqrt {1+\cot \relax (x )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(2*2^(1/2)+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))-1/4*2^(1/2)*(2*2^(1
/2)+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+cot(x))^(1/2)*(2*2^(1/2)+2)^(1/2))+1/4*(2*2^(1/2)+2)/(-2+2*2^(1/2))^(1/2)*a
rctan((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)*2^(1/2)*ln(1+cot(
x)+2^(1/2)+(1+cot(x))^(1/2)*(2*2^(1/2)+2)^(1/2))-1/4*2^(1/2)*(2*2^(1/2)+2)/(-2+2*2^(1/2))^(1/2)*arctan((2*(1+c
ot(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+cot(x))
^(1/2)*(2*2^(1/2)+2)^(1/2))+1/4*(2*2^(1/2)+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/(1+cot(x))^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.40, size = 121, normalized size = 0.54 \[ -\mathrm {atanh}\left (32\,\sqrt {\mathrm {cot}\relax (x)+1}\,{\left (\sqrt {-\frac {\sqrt {2}}{32}-\frac {1}{32}}+\sqrt {\frac {\sqrt {2}}{32}-\frac {1}{32}}\right )}^3\right )\,\left (2\,\sqrt {-\frac {\sqrt {2}}{32}-\frac {1}{32}}+2\,\sqrt {\frac {\sqrt {2}}{32}-\frac {1}{32}}\right )-\frac {1}{\sqrt {\mathrm {cot}\relax (x)+1}}-\mathrm {atanh}\left (32\,\sqrt {\mathrm {cot}\relax (x)+1}\,{\left (\sqrt {-\frac {\sqrt {2}}{32}-\frac {1}{32}}-\sqrt {\frac {\sqrt {2}}{32}-\frac {1}{32}}\right )}^3\right )\,\left (2\,\sqrt {-\frac {\sqrt {2}}{32}-\frac {1}{32}}-2\,\sqrt {\frac {\sqrt {2}}{32}-\frac {1}{32}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(x)/(cot(x) + 1)^(3/2),x)

[Out]

- atanh(32*(cot(x) + 1)^(1/2)*((- 2^(1/2)/32 - 1/32)^(1/2) + (2^(1/2)/32 - 1/32)^(1/2))^3)*(2*(- 2^(1/2)/32 -
1/32)^(1/2) + 2*(2^(1/2)/32 - 1/32)^(1/2)) - 1/(cot(x) + 1)^(1/2) - atanh(32*(cot(x) + 1)^(1/2)*((- 2^(1/2)/32
- 1/32)^(1/2) - (2^(1/2)/32 - 1/32)^(1/2))^3)*(2*(- 2^(1/2)/32 - 1/32)^(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 \[ \int \frac {\cot {\relax (x )}}{\left (\cot {\relax (x )} + 1\right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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