3.1.49 \(\int \frac {\cot ^2(x)}{(1+\cot (x))^{5/2}} \, dx\) [49]

Optimal. Leaf size=143 \[ \frac {1}{4} \sqrt {-1+\sqrt {2}} \text {ArcTan}\left (\frac {3-2 \sqrt {2}+\left (1-\sqrt {2}\right ) \cot (x)}{\sqrt {2 \left (-7+5 \sqrt {2}\right )} \sqrt {1+\cot (x)}}\right )+\frac {1}{4} \sqrt {1+\sqrt {2}} \tanh ^{-1}\left (\frac {3+2 \sqrt {2}+\left (1+\sqrt {2}\right ) \cot (x)}{\sqrt {2 \left (7+5 \sqrt {2}\right )} \sqrt {1+\cot (x)}}\right )+\frac {1}{3 (1+\cot (x))^{3/2}}-\frac {1}{\sqrt {1+\cot (x)}} \]

[Out]

1/3/(1+cot(x))^(3/2)-1/(1+cot(x))^(1/2)+1/4*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)+1/4*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 = 0.17, antiderivative size = 143, 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 = {3623, 3610, 12, 3617, 3616, 209, 213} \begin {gather*} \frac {1}{4} \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 {1}{\sqrt {\cot (x)+1}}+\frac {1}{3 (\cot (x)+1)^{3/2}}+\frac {1}{4} \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])^(5/2),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]])]
)/4 + (Sqrt[1 + Sqrt[2]]*ArcTanh[(3 + 2*Sqrt[2] + (1 + Sqrt[2])*Cot[x])/(Sqrt[2*(7 + 5*Sqrt[2])]*Sqrt[1 + Cot[
x]])])/4 + 1/(3*(1 + Cot[x])^(3/2)) - 1/Sqrt[1 + Cot[x]]

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 3610

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]

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 3623

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]

Rubi steps

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

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Mathematica [C] Result contains complex when optimal does not.
time = 0.43, size = 75, normalized size = 0.52 \begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {1+\cot (x)}}{\sqrt {1-i}}\right )}{2 \sqrt {1-i}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {1+\cot (x)}}{\sqrt {1+i}}\right )}{2 \sqrt {1+i}}+\frac {-2-3 \cot (x)}{3 (1+\cot (x))^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cot[x]^2/(1 + Cot[x])^(5/2),x]

[Out]

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

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Maple [A]
time = 0.24, size = 197, normalized size = 1.38

method	result	size
derivativedivides	\(\frac {\sqrt {2}\, \left (\frac {\sqrt {2+2 \sqrt {2}}\, \ln \left (1+\cot \left (x \right )+\sqrt {2}+\sqrt {1+\cot \left (x \right )}\, \sqrt {2+2 \sqrt {2}}\right )}{2}+\frac {2 \left (1-\sqrt {2}\right ) \arctan \left (\frac {2 \sqrt {1+\cot \left (x \right )}+\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{\sqrt {-2+2 \sqrt {2}}}\right )}{8}+\frac {\sqrt {2}\, \left (-\frac {\sqrt {2+2 \sqrt {2}}\, \ln \left (1+\cot \left (x \right )+\sqrt {2}-\sqrt {1+\cot \left (x \right )}\, \sqrt {2+2 \sqrt {2}}\right )}{2}+\frac {2 \left (1-\sqrt {2}\right ) \arctan \left (\frac {2 \sqrt {1+\cot \left (x \right )}-\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{\sqrt {-2+2 \sqrt {2}}}\right )}{8}+\frac {1}{3 \left (1+\cot \left (x \right )\right )^{\frac {3}{2}}}-\frac {1}{\sqrt {1+\cot \left (x \right )}}\)	\(197\)
default	\(\frac {\sqrt {2}\, \left (\frac {\sqrt {2+2 \sqrt {2}}\, \ln \left (1+\cot \left (x \right )+\sqrt {2}+\sqrt {1+\cot \left (x \right )}\, \sqrt {2+2 \sqrt {2}}\right )}{2}+\frac {2 \left (1-\sqrt {2}\right ) \arctan \left (\frac {2 \sqrt {1+\cot \left (x \right )}+\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{\sqrt {-2+2 \sqrt {2}}}\right )}{8}+\frac {\sqrt {2}\, \left (-\frac {\sqrt {2+2 \sqrt {2}}\, \ln \left (1+\cot \left (x \right )+\sqrt {2}-\sqrt {1+\cot \left (x \right )}\, \sqrt {2+2 \sqrt {2}}\right )}{2}+\frac {2 \left (1-\sqrt {2}\right ) \arctan \left (\frac {2 \sqrt {1+\cot \left (x \right )}-\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{\sqrt {-2+2 \sqrt {2}}}\right )}{8}+\frac {1}{3 \left (1+\cot \left (x \right )\right )^{\frac {3}{2}}}-\frac {1}{\sqrt {1+\cot \left (x \right )}}\)	\(197\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(x)^2/(1+cot(x))^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/8*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)))+1/8*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)))+1/3/(1+cot(x))^(3/2)-1/(1+cot(x))^(1/
2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)^2/(1+cot(x))^(5/2),x, algorithm="maxima")

[Out]

integrate(cot(x)^2/(cot(x) + 1)^(5/2), x)

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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))^(5/2),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\cot ^{2}{\left (x \right )}}{\left (\cot {\left (x \right )} + 1\right )^{\frac {5}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)**2/(1+cot(x))**(5/2),x)

[Out]

Integral(cot(x)**2/(cot(x) + 1)**(5/2), 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))^(5/2),x, algorithm="giac")

[Out]

integrate(cot(x)^2/(cot(x) + 1)^(5/2), x)

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Mupad [B]
time = 0.80, size = 242, normalized size = 1.69 \begin {gather*} \mathrm {atanh}\left (\frac {4\,\sqrt {2}\,\sqrt {\frac {1}{64}-\frac {\sqrt {2}}{64}}\,\sqrt {\mathrm {cot}\left (x\right )+1}}{64\,\sqrt {\frac {1}{64}-\frac {\sqrt {2}}{64}}\,\sqrt {\frac {\sqrt {2}}{64}+\frac {1}{64}}-1}-\frac {4\,\sqrt {2}\,\sqrt {\frac {\sqrt {2}}{64}+\frac {1}{64}}\,\sqrt {\mathrm {cot}\left (x\right )+1}}{64\,\sqrt {\frac {1}{64}-\frac {\sqrt {2}}{64}}\,\sqrt {\frac {\sqrt {2}}{64}+\frac {1}{64}}-1}\right )\,\left (2\,\sqrt {\frac {1}{64}-\frac {\sqrt {2}}{64}}+2\,\sqrt {\frac {\sqrt {2}}{64}+\frac {1}{64}}\right )-\mathrm {atanh}\left (\frac {4\,\sqrt {2}\,\sqrt {\frac {1}{64}-\frac {\sqrt {2}}{64}}\,\sqrt {\mathrm {cot}\left (x\right )+1}}{64\,\sqrt {\frac {1}{64}-\frac {\sqrt {2}}{64}}\,\sqrt {\frac {\sqrt {2}}{64}+\frac {1}{64}}+1}+\frac {4\,\sqrt {2}\,\sqrt {\frac {\sqrt {2}}{64}+\frac {1}{64}}\,\sqrt {\mathrm {cot}\left (x\right )+1}}{64\,\sqrt {\frac {1}{64}-\frac {\sqrt {2}}{64}}\,\sqrt {\frac {\sqrt {2}}{64}+\frac {1}{64}}+1}\right )\,\left (2\,\sqrt {\frac {1}{64}-\frac {\sqrt {2}}{64}}-2\,\sqrt {\frac {\sqrt {2}}{64}+\frac {1}{64}}\right )-\frac {\mathrm {cot}\left (x\right )+\frac {2}{3}}{{\left (\mathrm {cot}\left (x\right )+1\right )}^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(x)^2/(cot(x) + 1)^(5/2),x)

[Out]

atanh((4*2^(1/2)*(1/64 - 2^(1/2)/64)^(1/2)*(cot(x) + 1)^(1/2))/(64*(1/64 - 2^(1/2)/64)^(1/2)*(2^(1/2)/64 + 1/6
4)^(1/2) - 1) - (4*2^(1/2)*(2^(1/2)/64 + 1/64)^(1/2)*(cot(x) + 1)^(1/2))/(64*(1/64 - 2^(1/2)/64)^(1/2)*(2^(1/2
)/64 + 1/64)^(1/2) - 1))*(2*(1/64 - 2^(1/2)/64)^(1/2) + 2*(2^(1/2)/64 + 1/64)^(1/2)) - atanh((4*2^(1/2)*(1/64
- 2^(1/2)/64)^(1/2)*(cot(x) + 1)^(1/2))/(64*(1/64 - 2^(1/2)/64)^(1/2)*(2^(1/2)/64 + 1/64)^(1/2) + 1) + (4*2^(1
/2)*(2^(1/2)/64 + 1/64)^(1/2)*(cot(x) + 1)^(1/2))/(64*(1/64 - 2^(1/2)/64)^(1/2)*(2^(1/2)/64 + 1/64)^(1/2) + 1)
)*(2*(1/64 - 2^(1/2)/64)^(1/2) - 2*(2^(1/2)/64 + 1/64)^(1/2)) - (cot(x) + 2/3)/(cot(x) + 1)^(3/2)

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