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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]