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 = 0.143375, 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 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt [4]{a^4-b^4 \csc ^2(x)}}{a}\right )}{a}-\frac{\tan ^{-1}\left (\frac{\sqrt [4]{a^4-b^4 \csc ^2(x)}}{a}\right )}{a} \]

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

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(cot(x)/(a**4-b**4*csc(x)**2)**(1/4),x)

[Out]

Timed out

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Mathematica [C]  time = 0.228065, size = 85, normalized size = 1.57 \[ -\frac{\csc ^2(x) \left (a^4 \cos (2 x)-a^4+2 b^4\right ) \, _2F_1\left (\frac{3}{4},1;\frac{7}{4};-\frac{\left (\cos (2 x) a^4-a^4+2 b^4\right ) \csc ^2(x)}{2 a^4}\right )}{3 a^4 \sqrt [4]{a^4-b^4 \csc ^2(x)}} \]

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])*Csc[x]^2*Hypergeometric2F1[3/4, 1, 7/4, -((-a^4
+ 2*b^4 + a^4*Cos[2*x])*Csc[x]^2)/(2*a^4)])/(3*a^4*(a^4 - b^4*Csc[x]^2)^(1/4))

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Maple [F]  time = 0.132, size = 0, normalized size = 0. \[ \int{\cot \left ( x \right ){\frac{1}{\sqrt [4]{{a}^{4}-{b}^{4} \left ( \csc \left ( x \right ) \right ) ^{2}}}}}\, dx \]

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 = 1.49489, size = 100, normalized size = 1.85 \[ -\frac{\arctan \left (\frac{{\left (a^{4} - \frac{b^{4}}{\sin \left (x\right )^{2}}\right )}^{\frac{1}{4}}}{a}\right )}{a} + \frac{\log \left (a +{\left (a^{4} - \frac{b^{4}}{\sin \left (x\right )^{2}}\right )}^{\frac{1}{4}}\right )}{2 \, a} - \frac{\log \left (-a +{\left (a^{4} - \frac{b^{4}}{\sin \left (x\right )^{2}}\right )}^{\frac{1}{4}}\right )}{2 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cot(x)/(-b^4*csc(x)^2 + a^4)^(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. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cot(x)/(-b^4*csc(x)^2 + a^4)^(1/4),x, algorithm="fricas")

[Out]

Timed out

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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