3.62 \(\int \frac{\cot (x)}{\sqrt{a+b \cot ^4(x)}} \, dx\)

Optimal. Leaf size=41 \[ \frac{\tanh ^{-1}\left (\frac{a-b \cot ^2(x)}{\sqrt{a+b} \sqrt{a+b \cot ^4(x)}}\right )}{2 \sqrt{a+b}} \]

[Out]

ArcTanh[(a - b*Cot[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Cot[x]^4])]/(2*Sqrt[a + b])

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Rubi [A]  time = 0.0691749, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {3670, 1248, 725, 206} \[ \frac{\tanh ^{-1}\left (\frac{a-b \cot ^2(x)}{\sqrt{a+b} \sqrt{a+b \cot ^4(x)}}\right )}{2 \sqrt{a+b}} \]

Antiderivative was successfully verified.

[In]

Int[Cot[x]/Sqrt[a + b*Cot[x]^4],x]

[Out]

ArcTanh[(a - b*Cot[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Cot[x]^4])]/(2*Sqrt[a + b])

Rule 3670

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
[n, 2] || EqQ[n, 4] || (IntegerQ[p] && RationalQ[n]))

Rule 1248

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[(d + e*x)^q
*(a + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, c, d, e, p, q}, x]

Rule 725

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rule 206

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

Rubi steps

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

Mathematica [A]  time = 0.018883, size = 41, normalized size = 1. \[ \frac{\tanh ^{-1}\left (\frac{a-b \cot ^2(x)}{\sqrt{a+b} \sqrt{a+b \cot ^4(x)}}\right )}{2 \sqrt{a+b}} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[x]/Sqrt[a + b*Cot[x]^4],x]

[Out]

ArcTanh[(a - b*Cot[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Cot[x]^4])]/(2*Sqrt[a + b])

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Maple [A]  time = 0.035, size = 65, normalized size = 1.6 \begin{align*}{\frac{1}{2}\ln \left ({\frac{1}{1+ \left ( \cot \left ( x \right ) \right ) ^{2}} \left ( 2\,a+2\,b-2\, \left ( 1+ \left ( \cot \left ( x \right ) \right ) ^{2} \right ) b+2\,\sqrt{a+b}\sqrt{ \left ( 1+ \left ( \cot \left ( x \right ) \right ) ^{2} \right ) ^{2}b-2\, \left ( 1+ \left ( \cot \left ( x \right ) \right ) ^{2} \right ) b+a+b} \right ) } \right ){\frac{1}{\sqrt{a+b}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(x)/(a+b*cot(x)^4)^(1/2),x)

[Out]

1/2/(a+b)^(1/2)*ln((2*a+2*b-2*(1+cot(x)^2)*b+2*(a+b)^(1/2)*((1+cot(x)^2)^2*b-2*(1+cot(x)^2)*b+a+b)^(1/2))/(1+c
ot(x)^2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cot \left (x\right )}{\sqrt{b \cot \left (x\right )^{4} + a}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)/(a+b*cot(x)^4)^(1/2),x, algorithm="maxima")

[Out]

integrate(cot(x)/sqrt(b*cot(x)^4 + a), x)

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Fricas [B]  time = 3.29503, size = 686, normalized size = 16.73 \begin{align*} \left [\frac{\log \left (\frac{1}{2} \,{\left (a^{2} + 2 \, a b + b^{2}\right )} \cos \left (2 \, x\right )^{2} + \frac{1}{2} \, a^{2} + \frac{1}{2} \, b^{2} + \frac{1}{2} \,{\left ({\left (a + b\right )} \cos \left (2 \, x\right )^{2} - 2 \, a \cos \left (2 \, x\right ) + a - b\right )} \sqrt{a + b} \sqrt{\frac{{\left (a + b\right )} \cos \left (2 \, x\right )^{2} - 2 \,{\left (a - b\right )} \cos \left (2 \, x\right ) + a + b}{\cos \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1}} -{\left (a^{2} - b^{2}\right )} \cos \left (2 \, x\right )\right )}{4 \, \sqrt{a + b}}, -\frac{\sqrt{-a - b} \arctan \left (\frac{{\left ({\left (a + b\right )} \cos \left (2 \, x\right )^{2} - 2 \, a \cos \left (2 \, x\right ) + a - b\right )} \sqrt{-a - b} \sqrt{\frac{{\left (a + b\right )} \cos \left (2 \, x\right )^{2} - 2 \,{\left (a - b\right )} \cos \left (2 \, x\right ) + a + b}{\cos \left (2 \, x\right )^{2} - 2 \, \cos \left (2 \, x\right ) + 1}}}{{\left (a^{2} + 2 \, a b + b^{2}\right )} \cos \left (2 \, x\right )^{2} + a^{2} + 2 \, a b + b^{2} - 2 \,{\left (a^{2} - b^{2}\right )} \cos \left (2 \, x\right )}\right )}{2 \,{\left (a + b\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)/(a+b*cot(x)^4)^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cot{\left (x \right )}}{\sqrt{a + b \cot ^{4}{\left (x \right )}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)/(a+b*cot(x)**4)**(1/2),x)

[Out]

Integral(cot(x)/sqrt(a + b*cot(x)**4), x)

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Giac [A]  time = 1.51784, size = 78, normalized size = 1.9 \begin{align*} \frac{\log \left ({\left | -{\left (\sqrt{a + b} \cos \left (x\right )^{2} - \sqrt{a \cos \left (x\right )^{4} + b \cos \left (x\right )^{4} - 2 \, a \cos \left (x\right )^{2} + a}\right )}{\left (a + b\right )} + \sqrt{a + b} a \right |}\right )}{2 \, \sqrt{a + b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(x)/(a+b*cot(x)^4)^(1/2),x, algorithm="giac")

[Out]

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