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]

-(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]])

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Rubi [A]  time = 0.208622, antiderivative size = 221, normalized size of antiderivative = 1., 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 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]

Rule 12

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

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 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 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 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 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 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]

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*}

Mathematica [C]  time = 0.242566, 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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Maple [B]  time = 0.03, size = 452, normalized size = 2.1 \begin{align*} -{\frac{2}{3} \left ( 1+\cot \left ( x \right ) \right ) ^{{\frac{3}{2}}}}-2\,\sqrt{1+\cot \left ( x \right ) }-{\frac{\sqrt{2+2\,\sqrt{2}}\sqrt{2}}{4}\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}\ln \left ( 1+\cot \left ( x \right ) +\sqrt{2}+\sqrt{1+\cot \left ( x \right ) }\sqrt{2+2\,\sqrt{2}} \right ) }+{\frac{ \left ( 2+2\,\sqrt{2} \right ) \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{2+2\,\sqrt{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 ) }+2\,{\frac{\sqrt{2}}{\sqrt{-2+2\,\sqrt{2}}}\arctan \left ({\frac{2\,\sqrt{1+\cot \left ( x \right ) }+\sqrt{2+2\,\sqrt{2}}}{\sqrt{-2+2\,\sqrt{2}}}} \right ) }+{\frac{\sqrt{2+2\,\sqrt{2}}\sqrt{2}}{4}\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}\ln \left ( 1+\cot \left ( x \right ) +\sqrt{2}-\sqrt{1+\cot \left ( x \right ) }\sqrt{2+2\,\sqrt{2}} \right ) }+{\frac{ \left ( 2+2\,\sqrt{2} \right ) \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{2+2\,\sqrt{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 ) }+2\,{\frac{\sqrt{2}}{\sqrt{-2+2\,\sqrt{2}}}\arctan \left ({\frac{2\,\sqrt{1+\cot \left ( x \right ) }-\sqrt{2+2\,\sqrt{2}}}{\sqrt{-2+2\,\sqrt{2}}}} \right ) } \end{align*}

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

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

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

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

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)