∫ cot3 (x)_____∘ a + b cot2(x) dx [44]

Optimal. Leaf size=52 \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \cot ^2(x)}}{\sqrt {a-b}}\right )}{\sqrt {a-b}}-\frac {\sqrt {a+b \cot ^2(x)}}{b} \]

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Rubi [A]
time = 0.07, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {3751, 457, 81, 65, 214} \begin {gather*} -\frac {\sqrt {a+b \cot ^2(x)}}{b}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+b \cot ^2(x)}}{\sqrt {a-b}}\right )}{\sqrt {a-b}} \end {gather*}

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub

st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &

& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(c + d*x)^

(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(

d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,

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

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int

[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&

Int[((d_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol]

:> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[c*(ff/f), Subst[Int[(d*ff*(x/c))^m*((a + b*(ff*x)^n)^p/(c^2

+ ff^2*x^2)), x], x, c*(Tan[e + f*x]/ff)], x]] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && (IGtQ[p, 0] || EqQ

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

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Mathematica [A]
time = 0.16, size = 52, normalized size = 1.00 \begin {gather*} -\frac {\frac {b \tanh ^{-1}\left (\frac {\sqrt {a+b \cot ^2(x)}}{\sqrt {a-b}}\right )}{\sqrt {a-b}}+\sqrt {a+b \cot ^2(x)}}{b} \end {gather*}

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method	result	size

derivativedivides	\(-\frac {\sqrt {a +b \left (\cot ^{2}\left (x \right )\right )}}{b}+\frac {\arctan \left (\frac {\sqrt {a +b \left (\cot ^{2}\left (x \right )\right )}}{\sqrt {-a +b}}\right )}{\sqrt {-a +b}}\)	\(44\)
default	\(-\frac {\sqrt {a +b \left (\cot ^{2}\left (x \right )\right )}}{b}+\frac {\arctan \left (\frac {\sqrt {a +b \left (\cot ^{2}\left (x \right )\right )}}{\sqrt {-a +b}}\right )}{\sqrt {-a +b}}\)	\(44\)

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a

ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a-4*b>0)', see `assume?` for

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 118 vs. \(2 (44) = 88\).
time = 3.66, size = 284, normalized size = 5.46 \begin {gather*} \left [\frac {\sqrt {a - b} b \log \left (-2 \, {\left (a^{2} - 2 \, a b + b^{2}\right )} \cos \left (2 \, x\right )^{2} - 2 \, a^{2} + b^{2} + 2 \, {\left ({\left (a - b\right )} \cos \left (2 \, x\right )^{2} - {\left (2 \, a - b\right )} \cos \left (2 \, x\right ) + a\right )} \sqrt {a - b} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}} + 4 \, {\left (a^{2} - a b\right )} \cos \left (2 \, x\right )\right ) - 4 \, {\left (a - b\right )} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}}}{4 \, {\left (a b - b^{2}\right )}}, -\frac {\sqrt {-a + b} b \arctan \left (-\frac {\sqrt {-a + b} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}} {\left (\cos \left (2 \, x\right ) - 1\right )}}{{\left (a - b\right )} \cos \left (2 \, x\right ) - a}\right ) + 2 \, {\left (a - b\right )} \sqrt {\frac {{\left (a - b\right )} \cos \left (2 \, x\right ) - a - b}{\cos \left (2 \, x\right ) - 1}}}{2 \, {\left (a b - b^{2}\right )}}\right ] \end {gather*}

[1/4*(sqrt(a - b)*b*log(-2*(a^2 - 2*a*b + b^2)*cos(2*x)^2 - 2*a^2 + b^2 + 2*((a - b)*cos(2*x)^2 - (2*a - b)*co

s(2*x) + a)*sqrt(a - b)*sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1)) + 4*(a^2 - a*b)*cos(2*x)) - 4*(a - b)*

sqrt(((a - b)*cos(2*x) - a - b)/(cos(2*x) - 1)))/(a*b - b^2), -1/2*(sqrt(-a + b)*b*arctan(-sqrt(-a + b)*sqrt((

(a - b)*cos(2*x) - a - b)/(cos(2*x) - 1))*(cos(2*x) - 1)/((a - b)*cos(2*x) - a)) + 2*(a - b)*sqrt(((a - b)*cos

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\cot ^{3}{\left (x \right )}}{\sqrt {a + b \cot ^{2}{\left (x \right )}}}\, dx \end {gather*}

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 96 vs. \(2 (44) = 88\).
time = 0.48, size = 96, normalized size = 1.85 \begin {gather*} \frac {\frac {\log \left ({\left (\sqrt {a - b} \sin \left (x\right ) - \sqrt {a \sin \left (x\right )^{2} - b \sin \left (x\right )^{2} + b}\right )}^{2}\right )}{\sqrt {a - b}} + \frac {4 \, \sqrt {a - b}}{{\left (\sqrt {a - b} \sin \left (x\right ) - \sqrt {a \sin \left (x\right )^{2} - b \sin \left (x\right )^{2} + b}\right )}^{2} - b}}{2 \, \mathrm {sgn}\left (\sin \left (x\right )\right )} \end {gather*}

1/2*(log((sqrt(a - b)*sin(x) - sqrt(a*sin(x)^2 - b*sin(x)^2 + b))^2)/sqrt(a - b) + 4*sqrt(a - b)/((sqrt(a - b)

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Mupad [B]
time = 1.21, size = 44, normalized size = 0.85 \begin {gather*} -\frac {\sqrt {b\,{\mathrm {cot}\left (x\right )}^2+a}}{b}-\frac {\mathrm {atanh}\left (\frac {\sqrt {b\,{\mathrm {cot}\left (x\right )}^2+a}}{\sqrt {a-b}}\right )}{\sqrt {a-b}} \end {gather*}

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3.1.44 \(\int \frac {\cot ^3(x)}{\sqrt {a+b \cot ^2(x)}} \, dx\) [44]