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

Optimal. Leaf size=221 \[ -\frac {2}{3} (\cot (x)+1)^{3/2}-2 \sqrt {\cot (x)+1}-\frac {\log \left (\cot (x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\cot (x)+1}+\sqrt {2}+1\right )}{2 \sqrt {1+\sqrt {2}}}+\frac {\log \left (\cot (x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\cot (x)+1}+\sqrt {2}+1\right )}{2 \sqrt {1+\sqrt {2}}}-\sqrt {1+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {\cot (x)+1}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )+\sqrt {1+\sqrt {2}} \tan ^{-1}\left (\frac {2 \sqrt {\cot (x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right ) \]

[Out]

-2/3*(1+cot(x))^(3/2)-2*(1+cot(x))^(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))^(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))^(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^(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^(1/2))^(1/2)

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Rubi [A]  time = 0.21, antiderivative size = 221, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 9, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.818, Rules used = {3528, 12, 3485, 708, 1094, 634, 618, 204, 628} \[ -\frac {2}{3} (\cot (x)+1)^{3/2}-2 \sqrt {\cot (x)+1}-\frac {\log \left (\cot (x)-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\cot (x)+1}+\sqrt {2}+1\right )}{2 \sqrt {1+\sqrt {2}}}+\frac {\log \left (\cot (x)+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {\cot (x)+1}+\sqrt {2}+1\right )}{2 \sqrt {1+\sqrt {2}}}-\sqrt {1+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {\cot (x)+1}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )+\sqrt {1+\sqrt {2}} \tan ^{-1}\left (\frac {2 \sqrt {\cot (x)+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Cot[x]*(1 + Cot[x])^(3/2),x]

[Out]

-(Sqrt[1 + Sqrt[2]]*ArcTan[(Sqrt[2*(1 + Sqrt[2])] - 2*Sqrt[1 + Cot[x]])/Sqrt[2*(-1 + Sqrt[2])]]) + Sqrt[1 + Sq
rt[2]]*ArcTan[(Sqrt[2*(1 + Sqrt[2])] + 2*Sqrt[1 + Cot[x]])/Sqrt[2*(-1 + Sqrt[2])]] - 2*Sqrt[1 + Cot[x]] - (2*(
1 + Cot[x])^(3/2))/3 - Log[1 + Sqrt[2] + Cot[x] - Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 + Cot[x]]]/(2*Sqrt[1 + Sqrt[2]]
) + Log[1 + Sqrt[2] + Cot[x] + Sqrt[2*(1 + Sqrt[2])]*Sqrt[1 + Cot[x]]]/(2*Sqrt[1 + Sqrt[2]])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[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 634

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

Rule 708

Int[1/(Sqrt[(d_) + (e_.)*(x_)]*((a_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2*e, Subst[Int[1/(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 1094

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(-1), x_Symbol] :> With[{q = Rt[a/c, 2]}, With[{r = Rt[2*q - b/c, 2]}
, Dist[1/(2*c*q*r), Int[(r - x)/(q - r*x + x^2), x], x] + Dist[1/(2*c*q*r), Int[(r + x)/(q + r*x + x^2), x], x
]]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && NegQ[b^2 - 4*a*c]

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 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]

Rubi steps

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

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Mathematica [C]  time = 0.26, size = 98, normalized size = 0.44 \[ \frac {\sin (x) \left (-\frac {2}{3} (\cot (x)+1)^{3/2} (\cot (x)+4) (\sin (x)+\cos (x))+(1+i) \sin (x) (\cot (x)+1)^2 \left (\sqrt {1+i} \tanh ^{-1}\left (\frac {\sqrt {\cot (x)+1}}{\sqrt {1+i}}\right )-i \sqrt {1-i} \tanh ^{-1}\left (\frac {\sqrt {\cot (x)+1}}{\sqrt {1-i}}\right )\right )\right )}{(\sin (x)+\cos (x))^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sin[x]*((1 + I)*((-I)*Sqrt[1 - I]*ArcTanh[Sqrt[1 + Cot[x]]/Sqrt[1 - I]] + Sqrt[1 + I]*ArcTanh[Sqrt[1 + Cot[x]
]/Sqrt[1 + I]])*(1 + Cot[x])^2*Sin[x] - (2*(1 + Cot[x])^(3/2)*(4 + Cot[x])*(Cos[x] + Sin[x]))/3))/(Cos[x] + Si
n[x])^2

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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 \[ \int {\left (\cot \relax (x) + 1\right )}^{\frac {3}{2}} \cot \relax (x)\,{d x} \]

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) + 1)^(3/2)*cot(x), x)

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maple [B]  time = 0.12, size = 452, normalized size = 2.05 \[ -\frac {2 \left (1+\cot \relax (x )\right )^{\frac {3}{2}}}{3}-2 \sqrt {1+\cot \relax (x )}+\frac {\sqrt {2 \sqrt {2}+2}\, \sqrt {2}\, \ln \left (1+\cot \relax (x )+\sqrt {2}-\sqrt {1+\cot \relax (x )}\, \sqrt {2 \sqrt {2}+2}\right )}{4}-\frac {\sqrt {2 \sqrt {2}+2}\, \ln \left (1+\cot \relax (x )+\sqrt {2}-\sqrt {1+\cot \relax (x )}\, \sqrt {2 \sqrt {2}+2}\right )}{2}+\frac {\sqrt {2}\, \left (2 \sqrt {2}+2\right ) \arctan \left (\frac {2 \sqrt {1+\cot \relax (x )}-\sqrt {2 \sqrt {2}+2}}{\sqrt {-2+2 \sqrt {2}}}\right )}{2 \sqrt {-2+2 \sqrt {2}}}-\frac {\left (2 \sqrt {2}+2\right ) \arctan \left (\frac {2 \sqrt {1+\cot \relax (x )}-\sqrt {2 \sqrt {2}+2}}{\sqrt {-2+2 \sqrt {2}}}\right )}{\sqrt {-2+2 \sqrt {2}}}+\frac {2 \arctan \left (\frac {2 \sqrt {1+\cot \relax (x )}-\sqrt {2 \sqrt {2}+2}}{\sqrt {-2+2 \sqrt {2}}}\right ) \sqrt {2}}{\sqrt {-2+2 \sqrt {2}}}-\frac {\sqrt {2 \sqrt {2}+2}\, \sqrt {2}\, \ln \left (1+\cot \relax (x )+\sqrt {2}+\sqrt {1+\cot \relax (x )}\, \sqrt {2 \sqrt {2}+2}\right )}{4}+\frac {\sqrt {2 \sqrt {2}+2}\, \ln \left (1+\cot \relax (x )+\sqrt {2}+\sqrt {1+\cot \relax (x )}\, \sqrt {2 \sqrt {2}+2}\right )}{2}+\frac {\sqrt {2}\, \left (2 \sqrt {2}+2\right ) \arctan \left (\frac {2 \sqrt {1+\cot \relax (x )}+\sqrt {2 \sqrt {2}+2}}{\sqrt {-2+2 \sqrt {2}}}\right )}{2 \sqrt {-2+2 \sqrt {2}}}-\frac {\left (2 \sqrt {2}+2\right ) \arctan \left (\frac {2 \sqrt {1+\cot \relax (x )}+\sqrt {2 \sqrt {2}+2}}{\sqrt {-2+2 \sqrt {2}}}\right )}{\sqrt {-2+2 \sqrt {2}}}+\frac {2 \arctan \left (\frac {2 \sqrt {1+\cot \relax (x )}+\sqrt {2 \sqrt {2}+2}}{\sqrt {-2+2 \sqrt {2}}}\right ) \sqrt {2}}{\sqrt {-2+2 \sqrt {2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(1+cot(x))^(3/2)-2*(1+cot(x))^(1/2)+1/4*(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))-1/2*(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^(
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))-
(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))+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)-1/4*(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))+1/2*(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^(1/2)*(2*2^(1/2)+2)/(-2+2*2^(1/2))^(1/2)*arctan((2*(1+co
t(x))^(1/2)+(2*2^(1/2)+2)^(1/2))/(-2+2*2^(1/2))^(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))+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)

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

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) + 1)^(3/2)*cot(x), x)

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mupad [B]  time = 0.67, size = 254, normalized size = 1.15 \[ -\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {-\frac {\sqrt {2}}{4}-\frac {1}{4}}\,\sqrt {\mathrm {cot}\relax (x)+1}\,64{}\mathrm {i}}{256\,\sqrt {\frac {\sqrt {2}}{4}-\frac {1}{4}}\,\sqrt {-\frac {\sqrt {2}}{4}-\frac {1}{4}}-64}-\frac {\sqrt {2}\,\sqrt {\frac {\sqrt {2}}{4}-\frac {1}{4}}\,\sqrt {\mathrm {cot}\relax (x)+1}\,64{}\mathrm {i}}{256\,\sqrt {\frac {\sqrt {2}}{4}-\frac {1}{4}}\,\sqrt {-\frac {\sqrt {2}}{4}-\frac {1}{4}}-64}\right )\,\left (\sqrt {-\frac {\sqrt {2}}{4}-\frac {1}{4}}\,2{}\mathrm {i}+\sqrt {\frac {\sqrt {2}}{4}-\frac {1}{4}}\,2{}\mathrm {i}\right )+\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {-\frac {\sqrt {2}}{4}-\frac {1}{4}}\,\sqrt {\mathrm {cot}\relax (x)+1}\,64{}\mathrm {i}}{256\,\sqrt {\frac {\sqrt {2}}{4}-\frac {1}{4}}\,\sqrt {-\frac {\sqrt {2}}{4}-\frac {1}{4}}+64}+\frac {\sqrt {2}\,\sqrt {\frac {\sqrt {2}}{4}-\frac {1}{4}}\,\sqrt {\mathrm {cot}\relax (x)+1}\,64{}\mathrm {i}}{256\,\sqrt {\frac {\sqrt {2}}{4}-\frac {1}{4}}\,\sqrt {-\frac {\sqrt {2}}{4}-\frac {1}{4}}+64}\right )\,\left (\sqrt {-\frac {\sqrt {2}}{4}-\frac {1}{4}}\,2{}\mathrm {i}-\sqrt {\frac {\sqrt {2}}{4}-\frac {1}{4}}\,2{}\mathrm {i}\right )-2\,\sqrt {\mathrm {cot}\relax (x)+1}-\frac {2\,{\left (\mathrm {cot}\relax (x)+1\right )}^{3/2}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(x)*(cot(x) + 1)^(3/2),x)

[Out]

atan((2^(1/2)*(- 2^(1/2)/4 - 1/4)^(1/2)*(cot(x) + 1)^(1/2)*64i)/(256*(2^(1/2)/4 - 1/4)^(1/2)*(- 2^(1/2)/4 - 1/
4)^(1/2) + 64) + (2^(1/2)*(2^(1/2)/4 - 1/4)^(1/2)*(cot(x) + 1)^(1/2)*64i)/(256*(2^(1/2)/4 - 1/4)^(1/2)*(- 2^(1
/2)/4 - 1/4)^(1/2) + 64))*((- 2^(1/2)/4 - 1/4)^(1/2)*2i - (2^(1/2)/4 - 1/4)^(1/2)*2i) - atan((2^(1/2)*(- 2^(1/
2)/4 - 1/4)^(1/2)*(cot(x) + 1)^(1/2)*64i)/(256*(2^(1/2)/4 - 1/4)^(1/2)*(- 2^(1/2)/4 - 1/4)^(1/2) - 64) - (2^(1
/2)*(2^(1/2)/4 - 1/4)^(1/2)*(cot(x) + 1)^(1/2)*64i)/(256*(2^(1/2)/4 - 1/4)^(1/2)*(- 2^(1/2)/4 - 1/4)^(1/2) - 6
4))*((- 2^(1/2)/4 - 1/4)^(1/2)*2i + (2^(1/2)/4 - 1/4)^(1/2)*2i) - 2*(cot(x) + 1)^(1/2) - (2*(cot(x) + 1)^(3/2)
)/3

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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