Optimal. Leaf size=83 \[ \frac{\tan ^{-1}\left (\frac{\sqrt [4]{1-a} x}{\sqrt [4]{a+1}}\right )}{2 \sqrt{a+1} \sqrt [4]{1-a^2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{1-a} x}{\sqrt [4]{a+1}}\right )}{2 \sqrt{a+1} \sqrt [4]{1-a^2}} \]
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Rubi [A] time = 0.141138, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{\tan ^{-1}\left (\frac{\sqrt [4]{1-a} x}{\sqrt [4]{a+1}}\right )}{2 \sqrt{a+1} \sqrt [4]{1-a^2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{1-a} x}{\sqrt [4]{a+1}}\right )}{2 \sqrt{a+1} \sqrt [4]{1-a^2}} \]
Antiderivative was successfully verified.
[In] Int[(1 + a + (-1 + a)*x^4)^(-1),x]
[Out]
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Rubi in Sympy [A] time = 16.7366, size = 63, normalized size = 0.76 \[ \frac{\operatorname{atan}{\left (\frac{x \sqrt [4]{- a + 1}}{\sqrt [4]{a + 1}} \right )}}{2 \sqrt [4]{- a + 1} \left (a + 1\right )^{\frac{3}{4}}} + \frac{\operatorname{atanh}{\left (\frac{x \sqrt [4]{- a + 1}}{\sqrt [4]{a + 1}} \right )}}{2 \sqrt [4]{- a + 1} \left (a + 1\right )^{\frac{3}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(1+a+(-1+a)*x**4),x)
[Out]
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Mathematica [A] time = 0.118108, size = 160, normalized size = 1.93 \[ \frac{-\log \left (\sqrt{a-1} x^2-\sqrt{2} \sqrt [4]{a-1} \sqrt [4]{a+1} x+\sqrt{a+1}\right )+\log \left (\sqrt{a-1} x^2+\sqrt{2} \sqrt [4]{a-1} \sqrt [4]{a+1} x+\sqrt{a+1}\right )-2 \tan ^{-1}\left (1-\sqrt{2} \sqrt [4]{\frac{a-1}{a+1}} x\right )+2 \tan ^{-1}\left (\sqrt{2} \sqrt [4]{\frac{a-1}{a+1}} x+1\right )}{4 \sqrt{2} \sqrt [4]{a-1} (a+1)^{3/4}} \]
Warning: Unable to verify antiderivative.
[In] Integrate[(1 + a + (-1 + a)*x^4)^(-1),x]
[Out]
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Maple [B] time = 0.014, size = 170, normalized size = 2.1 \[{\frac{\sqrt{2}}{8+8\,a}\sqrt [4]{{\frac{1+a}{-1+a}}}\ln \left ({1 \left ({x}^{2}+\sqrt [4]{{\frac{1+a}{-1+a}}}x\sqrt{2}+\sqrt{{\frac{1+a}{-1+a}}} \right ) \left ({x}^{2}-\sqrt [4]{{\frac{1+a}{-1+a}}}x\sqrt{2}+\sqrt{{\frac{1+a}{-1+a}}} \right ) ^{-1}} \right ) }+{\frac{\sqrt{2}}{4+4\,a}\sqrt [4]{{\frac{1+a}{-1+a}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{1+a}{-1+a}}}}}}+1 \right ) }+{\frac{\sqrt{2}}{4+4\,a}\sqrt [4]{{\frac{1+a}{-1+a}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{1+a}{-1+a}}}}}}-1 \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(1+a+(-1+a)*x^4),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((a - 1)*x^4 + a + 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.241454, size = 242, normalized size = 2.92 \[ -\left (-\frac{1}{a^{4} + 2 \, a^{3} - 2 \, a - 1}\right )^{\frac{1}{4}} \arctan \left (\frac{{\left (a + 1\right )} \left (-\frac{1}{a^{4} + 2 \, a^{3} - 2 \, a - 1}\right )^{\frac{1}{4}}}{x + \sqrt{x^{2} +{\left (a^{2} + 2 \, a + 1\right )} \sqrt{-\frac{1}{a^{4} + 2 \, a^{3} - 2 \, a - 1}}}}\right ) + \frac{1}{4} \, \left (-\frac{1}{a^{4} + 2 \, a^{3} - 2 \, a - 1}\right )^{\frac{1}{4}} \log \left ({\left (a + 1\right )} \left (-\frac{1}{a^{4} + 2 \, a^{3} - 2 \, a - 1}\right )^{\frac{1}{4}} + x\right ) - \frac{1}{4} \, \left (-\frac{1}{a^{4} + 2 \, a^{3} - 2 \, a - 1}\right )^{\frac{1}{4}} \log \left (-{\left (a + 1\right )} \left (-\frac{1}{a^{4} + 2 \, a^{3} - 2 \, a - 1}\right )^{\frac{1}{4}} + x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((a - 1)*x^4 + a + 1),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.816406, size = 32, normalized size = 0.39 \[ \operatorname{RootSum}{\left (t^{4} \left (256 a^{4} + 512 a^{3} - 512 a - 256\right ) + 1, \left ( t \mapsto t \log{\left (4 t a + 4 t + x \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(1+a+(-1+a)*x**4),x)
[Out]
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GIAC/XCAS [A] time = 0.217449, size = 360, normalized size = 4.34 \[ \frac{{\left (a^{4} - 2 \, a^{3} + 2 \, a - 1\right )}^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (\frac{a + 1}{a - 1}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a + 1}{a - 1}\right )^{\frac{1}{4}}}\right )}{2 \,{\left (\sqrt{2} a^{2} - \sqrt{2}\right )}} + \frac{{\left (a^{4} - 2 \, a^{3} + 2 \, a - 1\right )}^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (\frac{a + 1}{a - 1}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a + 1}{a - 1}\right )^{\frac{1}{4}}}\right )}{2 \,{\left (\sqrt{2} a^{2} - \sqrt{2}\right )}} + \frac{{\left (a^{4} - 2 \, a^{3} + 2 \, a - 1\right )}^{\frac{1}{4}}{\rm ln}\left (x^{2} + \sqrt{2} x \left (\frac{a + 1}{a - 1}\right )^{\frac{1}{4}} + \sqrt{\frac{a + 1}{a - 1}}\right )}{4 \,{\left (\sqrt{2} a^{2} - \sqrt{2}\right )}} - \frac{{\left (a^{4} - 2 \, a^{3} + 2 \, a - 1\right )}^{\frac{1}{4}}{\rm ln}\left (x^{2} - \sqrt{2} x \left (\frac{a + 1}{a - 1}\right )^{\frac{1}{4}} + \sqrt{\frac{a + 1}{a - 1}}\right )}{4 \,{\left (\sqrt{2} a^{2} - \sqrt{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((a - 1)*x^4 + a + 1),x, algorithm="giac")
[Out]