3.1.43 \(\int \cot ^2(x) (1+\cot (x))^{3/2} \, dx\) [43]

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

[Out]

-2/5*(1+cot(x))^(5/2)+2*(1+cot(x))^(1/2)-arctan((3+cot(x)*(1-2^(1/2))-2*2^(1/2))/(1+cot(x))^(1/2)/(-14+10*2^(1
/2))^(1/2))*(2^(1/2)-1)^(1/2)-arctanh((3+2*2^(1/2)+cot(x)*(1+2^(1/2)))/(1+cot(x))^(1/2)/(14+10*2^(1/2))^(1/2))
*(1+2^(1/2))^(1/2)

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Rubi [A]
time = 0.20, antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {3624, 3563, 12, 3617, 3616, 209, 213} \begin {gather*} -\sqrt {\sqrt {2}-1} \text {ArcTan}\left (\frac {\left (1-\sqrt {2}\right ) \cot (x)-2 \sqrt {2}+3}{\sqrt {2 \left (5 \sqrt {2}-7\right )} \sqrt {\cot (x)+1}}\right )-\frac {2}{5} (\cot (x)+1)^{5/2}+2 \sqrt {\cot (x)+1}-\sqrt {1+\sqrt {2}} \tanh ^{-1}\left (\frac {\left (1+\sqrt {2}\right ) \cot (x)+2 \sqrt {2}+3}{\sqrt {2 \left (7+5 \sqrt {2}\right )} \sqrt {\cot (x)+1}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cot[x]^2*(1 + Cot[x])^(3/2),x]

[Out]

-(Sqrt[-1 + Sqrt[2]]*ArcTan[(3 - 2*Sqrt[2] + (1 - Sqrt[2])*Cot[x])/(Sqrt[2*(-7 + 5*Sqrt[2])]*Sqrt[1 + Cot[x]])
]) - Sqrt[1 + Sqrt[2]]*ArcTanh[(3 + 2*Sqrt[2] + (1 + Sqrt[2])*Cot[x])/(Sqrt[2*(7 + 5*Sqrt[2])]*Sqrt[1 + Cot[x]
])] + 2*Sqrt[1 + Cot[x]] - (2*(1 + Cot[x])^(5/2))/5

Rule 12

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

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 213

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(-1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])]
, x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rule 3563

Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((a + b*Tan[c + d*x])^(n - 1)/(d*(n - 1))
), x] + Int[(a^2 - b^2 + 2*a*b*Tan[c + d*x])*(a + b*Tan[c + d*x])^(n - 2), x] /; FreeQ[{a, b, c, d}, x] && NeQ
[a^2 + b^2, 0] && GtQ[n, 1]

Rule 3616

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 3617

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 3624

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

Rubi steps

\begin {align*} \int \cot ^2(x) (1+\cot (x))^{3/2} \, dx &=-\frac {2}{5} (1+\cot (x))^{5/2}-\int (1+\cot (x))^{3/2} \, dx\\ &=2 \sqrt {1+\cot (x)}-\frac {2}{5} (1+\cot (x))^{5/2}-\int \frac {2 \cot (x)}{\sqrt {1+\cot (x)}} \, dx\\ &=2 \sqrt {1+\cot (x)}-\frac {2}{5} (1+\cot (x))^{5/2}-2 \int \frac {\cot (x)}{\sqrt {1+\cot (x)}} \, dx\\ &=2 \sqrt {1+\cot (x)}-\frac {2}{5} (1+\cot (x))^{5/2}-\frac {\int \frac {-1-\left (-1-\sqrt {2}\right ) \cot (x)}{\sqrt {1+\cot (x)}} \, dx}{\sqrt {2}}+\frac {\int \frac {-1-\left (-1+\sqrt {2}\right ) \cot (x)}{\sqrt {1+\cot (x)}} \, dx}{\sqrt {2}}\\ &=2 \sqrt {1+\cot (x)}-\frac {2}{5} (1+\cot (x))^{5/2}-\left (-4+3 \sqrt {2}\right ) \text {Subst}\left (\int \frac {1}{2 \left (-1+\sqrt {2}\right )-4 \left (-1+\sqrt {2}\right )^2+x^2} \, dx,x,\frac {1-2 \left (-1+\sqrt {2}\right )-\left (-1+\sqrt {2}\right ) \cot (x)}{\sqrt {1+\cot (x)}}\right )+\left (4+3 \sqrt {2}\right ) \text {Subst}\left (\int \frac {1}{2 \left (-1-\sqrt {2}\right )-4 \left (-1-\sqrt {2}\right )^2+x^2} \, dx,x,\frac {1-2 \left (-1-\sqrt {2}\right )-\left (-1-\sqrt {2}\right ) \cot (x)}{\sqrt {1+\cot (x)}}\right )\\ &=-\sqrt {-1+\sqrt {2}} \tan ^{-1}\left (\frac {3-2 \sqrt {2}+\left (1-\sqrt {2}\right ) \cot (x)}{\sqrt {2 \left (-7+5 \sqrt {2}\right )} \sqrt {1+\cot (x)}}\right )-\sqrt {1+\sqrt {2}} \tanh ^{-1}\left (\frac {3+2 \sqrt {2}+\left (1+\sqrt {2}\right ) \cot (x)}{\sqrt {2 \left (7+5 \sqrt {2}\right )} \sqrt {1+\cot (x)}}\right )+2 \sqrt {1+\cot (x)}-\frac {2}{5} (1+\cot (x))^{5/2}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.34, size = 96, normalized size = 0.69 \begin {gather*} \frac {\sin (x) \left (-2 \left (\frac {\tanh ^{-1}\left (\frac {\sqrt {1+\cot (x)}}{\sqrt {1-i}}\right )}{\sqrt {1-i}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {1+\cot (x)}}{\sqrt {1+i}}\right )}{\sqrt {1+i}}\right ) (1+\cot (x))^2 \sin (x)-\frac {2}{5} (1+\cot (x))^{5/2} \left (-5+2 \cot (x)+\csc ^2(x)\right ) \sin (x)\right )}{(\cos (x)+\sin (x))^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sin[x]*(-2*(ArcTanh[Sqrt[1 + Cot[x]]/Sqrt[1 - I]]/Sqrt[1 - I] + ArcTanh[Sqrt[1 + Cot[x]]/Sqrt[1 + I]]/Sqrt[1
+ I])*(1 + Cot[x])^2*Sin[x] - (2*(1 + Cot[x])^(5/2)*(-5 + 2*Cot[x] + Csc[x]^2)*Sin[x])/5))/(Cos[x] + Sin[x])^2

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Maple [A]
time = 0.26, size = 197, normalized size = 1.42

method	result	size
derivativedivides	\(-\frac {2 \left (1+\cot \left (x \right )\right )^{\frac {5}{2}}}{5}+2 \sqrt {1+\cot \left (x \right )}-\frac {\sqrt {2}\, \left (-\frac {\sqrt {2+2 \sqrt {2}}\, \ln \left (1+\cot \left (x \right )+\sqrt {2}-\sqrt {1+\cot \left (x \right )}\, \sqrt {2+2 \sqrt {2}}\right )}{2}+\frac {2 \left (1-\sqrt {2}\right ) \arctan \left (\frac {2 \sqrt {1+\cot \left (x \right )}-\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{\sqrt {-2+2 \sqrt {2}}}\right )}{2}-\frac {\sqrt {2}\, \left (\frac {\sqrt {2+2 \sqrt {2}}\, \ln \left (1+\cot \left (x \right )+\sqrt {2}+\sqrt {1+\cot \left (x \right )}\, \sqrt {2+2 \sqrt {2}}\right )}{2}+\frac {2 \left (1-\sqrt {2}\right ) \arctan \left (\frac {2 \sqrt {1+\cot \left (x \right )}+\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{\sqrt {-2+2 \sqrt {2}}}\right )}{2}\)	\(197\)
default	\(-\frac {2 \left (1+\cot \left (x \right )\right )^{\frac {5}{2}}}{5}+2 \sqrt {1+\cot \left (x \right )}-\frac {\sqrt {2}\, \left (-\frac {\sqrt {2+2 \sqrt {2}}\, \ln \left (1+\cot \left (x \right )+\sqrt {2}-\sqrt {1+\cot \left (x \right )}\, \sqrt {2+2 \sqrt {2}}\right )}{2}+\frac {2 \left (1-\sqrt {2}\right ) \arctan \left (\frac {2 \sqrt {1+\cot \left (x \right )}-\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{\sqrt {-2+2 \sqrt {2}}}\right )}{2}-\frac {\sqrt {2}\, \left (\frac {\sqrt {2+2 \sqrt {2}}\, \ln \left (1+\cot \left (x \right )+\sqrt {2}+\sqrt {1+\cot \left (x \right )}\, \sqrt {2+2 \sqrt {2}}\right )}{2}+\frac {2 \left (1-\sqrt {2}\right ) \arctan \left (\frac {2 \sqrt {1+\cot \left (x \right )}+\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{\sqrt {-2+2 \sqrt {2}}}\right )}{2}\)	\(197\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/5*(1+cot(x))^(5/2)+2*(1+cot(x))^(1/2)-1/2*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))+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)))-1/2*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))+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
)))

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Maxima [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (\cot {\left (x \right )} + 1\right )^{\frac {3}{2}} \cot ^{2}{\left (x \right )}\, dx \end {gather*}

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) + 1)**(3/2)*cot(x)**2, x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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) + 1)^(3/2)*cot(x)^2, x)

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Mupad [B]
time = 1.00, size = 254, normalized size = 1.83 \begin {gather*} \mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {\frac {1}{4}-\frac {\sqrt {2}}{4}}\,\sqrt {\mathrm {cot}\left (x\right )+1}\,64{}\mathrm {i}}{256\,\sqrt {\frac {1}{4}-\frac {\sqrt {2}}{4}}\,\sqrt {\frac {\sqrt {2}}{4}+\frac {1}{4}}-64}-\frac {\sqrt {2}\,\sqrt {\frac {\sqrt {2}}{4}+\frac {1}{4}}\,\sqrt {\mathrm {cot}\left (x\right )+1}\,64{}\mathrm {i}}{256\,\sqrt {\frac {1}{4}-\frac {\sqrt {2}}{4}}\,\sqrt {\frac {\sqrt {2}}{4}+\frac {1}{4}}-64}\right )\,\left (\sqrt {\frac {1}{4}-\frac {\sqrt {2}}{4}}\,2{}\mathrm {i}+\sqrt {\frac {\sqrt {2}}{4}+\frac {1}{4}}\,2{}\mathrm {i}\right )-\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {\frac {1}{4}-\frac {\sqrt {2}}{4}}\,\sqrt {\mathrm {cot}\left (x\right )+1}\,64{}\mathrm {i}}{256\,\sqrt {\frac {1}{4}-\frac {\sqrt {2}}{4}}\,\sqrt {\frac {\sqrt {2}}{4}+\frac {1}{4}}+64}+\frac {\sqrt {2}\,\sqrt {\frac {\sqrt {2}}{4}+\frac {1}{4}}\,\sqrt {\mathrm {cot}\left (x\right )+1}\,64{}\mathrm {i}}{256\,\sqrt {\frac {1}{4}-\frac {\sqrt {2}}{4}}\,\sqrt {\frac {\sqrt {2}}{4}+\frac {1}{4}}+64}\right )\,\left (\sqrt {\frac {1}{4}-\frac {\sqrt {2}}{4}}\,2{}\mathrm {i}-\sqrt {\frac {\sqrt {2}}{4}+\frac {1}{4}}\,2{}\mathrm {i}\right )+2\,\sqrt {\mathrm {cot}\left (x\right )+1}-\frac {2\,{\left (\mathrm {cot}\left (x\right )+1\right )}^{5/2}}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(x)^2*(cot(x) + 1)^(3/2),x)

[Out]

atan((2^(1/2)*(1/4 - 2^(1/2)/4)^(1/2)*(cot(x) + 1)^(1/2)*64i)/(256*(1/4 - 2^(1/2)/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*(1/4 - 2^(1/2)/4)^(1/2)*(2^(1/2)/4
+ 1/4)^(1/2) - 64))*((1/4 - 2^(1/2)/4)^(1/2)*2i + (2^(1/2)/4 + 1/4)^(1/2)*2i) - atan((2^(1/2)*(1/4 - 2^(1/2)/4
)^(1/2)*(cot(x) + 1)^(1/2)*64i)/(256*(1/4 - 2^(1/2)/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*(1/4 - 2^(1/2)/4)^(1/2)*(2^(1/2)/4 + 1/4)^(1/2) + 64))*((1/4 - 2^
(1/2)/4)^(1/2)*2i - (2^(1/2)/4 + 1/4)^(1/2)*2i) + 2*(cot(x) + 1)^(1/2) - (2*(cot(x) + 1)^(5/2))/5

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