3.445 \(\int \frac{\cot (x)}{\sqrt [4]{a^4-b^4 \csc ^2(x)}} \, dx\)

Optimal. Leaf size=54 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt [4]{a^4-b^4 \csc ^2(x)}}{a}\right )}{a}-\frac{\tan ^{-1}\left (\frac{\sqrt [4]{a^4-b^4 \csc ^2(x)}}{a}\right )}{a} \]

[Out]

-(ArcTan[(a^4 - b^4*Csc[x]^2)^(1/4)/a]/a) + ArcTanh[(a^4 - b^4*Csc[x]^2)^(1/4)/a]/a

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Rubi [A]  time = 0.0874378, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {4139, 266, 63, 298, 203, 206} \[ \frac{\tanh ^{-1}\left (\frac{\sqrt [4]{a^4-b^4 \csc ^2(x)}}{a}\right )}{a}-\frac{\tan ^{-1}\left (\frac{\sqrt [4]{a^4-b^4 \csc ^2(x)}}{a}\right )}{a} \]

Antiderivative was successfully verified.

[In]

Int[Cot[x]/(a^4 - b^4*Csc[x]^2)^(1/4),x]

[Out]

-(ArcTan[(a^4 - b^4*Csc[x]^2)^(1/4)/a]/a) + ArcTanh[(a^4 - b^4*Csc[x]^2)^(1/4)/a]/a

Rule 4139

Int[((a_) + (b_.)*((c_.)*sec[(e_.) + (f_.)*(x_)])^(n_))^(p_.)*tan[(e_.) + (f_.)*(x_)]^(m_.), x_Symbol] :> With
[{ff = FreeFactors[Sec[e + f*x], x]}, Dist[1/f, Subst[Int[((-1 + ff^2*x^2)^((m - 1)/2)*(a + b*(c*ff*x)^n)^p)/x
, x], x, Sec[e + f*x]/ff], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && IntegerQ[(m - 1)/2] && (GtQ[m, 0] || EqQ[
n, 2] || EqQ[n, 4] || IGtQ[p, 0] || IntegersQ[2*n, p])

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 63

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,
b, c, d, m, n, x]

Rule 298

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b),
2]]}, Dist[s/(2*b), Int[1/(r + s*x^2), x], x] - Dist[s/(2*b), Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &
&  !GtQ[a/b, 0]

Rule 203

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

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

Mathematica [B]  time = 0.445366, size = 245, normalized size = 4.54 \[ \frac{\sqrt [4]{a^4 \cos (2 x)-a^4+2 b^4} \left (-\log \left (\frac{a^2 \sin (x)}{\sqrt{b^4-a^4 \sin ^2(x)}}-\frac{\sqrt{2} a \sqrt{\sin (x)}}{\sqrt [4]{b^4-a^4 \sin ^2(x)}}+1\right )+\log \left (\frac{a^2 \sin (x)}{\sqrt{b^4-a^4 \sin ^2(x)}}+\frac{\sqrt{2} a \sqrt{\sin (x)}}{\sqrt [4]{b^4-a^4 \sin ^2(x)}}+1\right )-2 \tan ^{-1}\left (1-\frac{\sqrt{2} a \sqrt{\sin (x)}}{\sqrt [4]{b^4-a^4 \sin ^2(x)}}\right )+2 \tan ^{-1}\left (\frac{\sqrt{2} a \sqrt{\sin (x)}}{\sqrt [4]{b^4-a^4 \sin ^2(x)}}+1\right )\right )}{2\ 2^{3/4} a \sqrt{\sin (x)} \sqrt [4]{a^4-b^4 \csc ^2(x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[x]/(a^4 - b^4*Csc[x]^2)^(1/4),x]

[Out]

((-a^4 + 2*b^4 + a^4*Cos[2*x])^(1/4)*(-2*ArcTan[1 - (Sqrt[2]*a*Sqrt[Sin[x]])/(b^4 - a^4*Sin[x]^2)^(1/4)] + 2*A
rcTan[1 + (Sqrt[2]*a*Sqrt[Sin[x]])/(b^4 - a^4*Sin[x]^2)^(1/4)] - Log[1 + (a^2*Sin[x])/Sqrt[b^4 - a^4*Sin[x]^2]
- (Sqrt[2]*a*Sqrt[Sin[x]])/(b^4 - a^4*Sin[x]^2)^(1/4)] + Log[1 + (a^2*Sin[x])/Sqrt[b^4 - a^4*Sin[x]^2] + (Sqr
t[2]*a*Sqrt[Sin[x]])/(b^4 - a^4*Sin[x]^2)^(1/4)]))/(2*2^(3/4)*a*(a^4 - b^4*Csc[x]^2)^(1/4)*Sqrt[Sin[x]])

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Maple [F]  time = 0.103, size = 0, normalized size = 0. \begin{align*} \int{\cot \left ( x \right ){\frac{1}{\sqrt [4]{{a}^{4}-{b}^{4} \left ( \csc \left ( x \right ) \right ) ^{2}}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.42083, size = 100, normalized size = 1.85 \begin{align*} -\frac{\arctan \left (\frac{{\left (a^{4} - \frac{b^{4}}{\sin \left (x\right )^{2}}\right )}^{\frac{1}{4}}}{a}\right )}{a} + \frac{\log \left (a +{\left (a^{4} - \frac{b^{4}}{\sin \left (x\right )^{2}}\right )}^{\frac{1}{4}}\right )}{2 \, a} - \frac{\log \left (-a +{\left (a^{4} - \frac{b^{4}}{\sin \left (x\right )^{2}}\right )}^{\frac{1}{4}}\right )}{2 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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)/(a^4-b^4*csc(x)^2)^(1/4),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cot{\left (x \right )}}{\sqrt [4]{\left (a^{2} - b^{2} \csc{\left (x \right )}\right ) \left (a^{2} + b^{2} \csc{\left (x \right )}\right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(cot(x)/((a**2 - b**2*csc(x))*(a**2 + b**2*csc(x)))**(1/4), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cot \left (x\right )}{{\left (-b^{4} \csc \left (x\right )^{2} + a^{4}\right )}^{\frac{1}{4}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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