Optimal. Leaf size=124 \[ -\frac{\left (\sqrt{a}-\sqrt{b}\right ) \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{3/4}}+\frac{\left (\sqrt{a}+\sqrt{b}\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{3/4}}-\frac{\log \left (a-b x^4\right )}{4 b}+\frac{\tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{2 \sqrt{a} \sqrt{b}} \]
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Rubi [A] time = 0.242426, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35 \[ -\frac{\left (\sqrt{a}-\sqrt{b}\right ) \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{3/4}}+\frac{\left (\sqrt{a}+\sqrt{b}\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{3/4}}-\frac{\log \left (a-b x^4\right )}{4 b}+\frac{\tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{2 \sqrt{a} \sqrt{b}} \]
Antiderivative was successfully verified.
[In] Int[(1 + x + x^2 + x^3)/(a - b*x^4),x]
[Out]
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Rubi in Sympy [A] time = 30.3221, size = 109, normalized size = 0.88 \[ - \frac{\log{\left (a - b x^{4} \right )}}{4 b} + \frac{\operatorname{atanh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{2 \sqrt{a} \sqrt{b}} - \frac{\left (\sqrt{a} - \sqrt{b}\right ) \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{2 a^{\frac{3}{4}} b^{\frac{3}{4}}} + \frac{\left (\sqrt{a} + \sqrt{b}\right ) \operatorname{atanh}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}}{2 a^{\frac{3}{4}} b^{\frac{3}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((x**3+x**2+x+1)/(-b*x**4+a),x)
[Out]
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Mathematica [A] time = 0.0919052, size = 203, normalized size = 1.64 \[ -\frac{\left (a^{3/4}+\sqrt{a} \sqrt [4]{b}+\sqrt [4]{a} \sqrt{b}\right ) \log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right )}{4 a b^{3/4}}-\frac{\left (-a^{3/4}+\sqrt{a} \sqrt [4]{b}-\sqrt [4]{a} \sqrt{b}\right ) \log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right )}{4 a b^{3/4}}+\frac{\left (\sqrt [4]{a} \sqrt{b}-a^{3/4}\right ) \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 a b^{3/4}}-\frac{\log \left (a-b x^4\right )}{4 b}+\frac{\log \left (\sqrt{a}+\sqrt{b} x^2\right )}{4 \sqrt{a} \sqrt{b}} \]
Antiderivative was successfully verified.
[In] Integrate[(1 + x + x^2 + x^3)/(a - b*x^4),x]
[Out]
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Maple [B] time = 0.005, size = 171, normalized size = 1.4 \[{\frac{1}{4\,a}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}} \right ) ^{-1}} \right ) }+{\frac{1}{2\,a}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ) }-{\frac{1}{4}\ln \left ({1 \left ( -a+{x}^{2}\sqrt{ab} \right ) \left ( -a-{x}^{2}\sqrt{ab} \right ) ^{-1}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{1}{2\,b}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{1}{4\,b}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-{\frac{\ln \left ( b{x}^{4}-a \right ) }{4\,b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((x^3+x^2+x+1)/(-b*x^4+a),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(-(x^3 + x^2 + x + 1)/(b*x^4 - a),x, algorithm="maxima")
[Out]
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(x^3 + x^2 + x + 1)/(b*x^4 - a),x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.95533, size = 187, normalized size = 1.51 \[ - \operatorname{RootSum}{\left (256 t^{4} a^{3} b^{4} - 256 t^{3} a^{3} b^{3} + t^{2} \left (96 a^{3} b^{2} - 96 a^{2} b^{3}\right ) + t \left (- 16 a^{3} b + 32 a^{2} b^{2} - 16 a b^{3}\right ) + a^{3} - 3 a^{2} b + 3 a b^{2} - b^{3}, \left ( t \mapsto t \log{\left (x + \frac{- 64 t^{3} a^{3} b^{3} + 48 t^{2} a^{3} b^{2} + 16 t^{2} a^{2} b^{3} - 12 t a^{3} b + 16 t a^{2} b^{2} - 4 t a b^{3} + a^{3} - 2 a^{2} b + a b^{2}}{a^{2} b - 2 a b^{2} + b^{3}} \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x**3+x**2+x+1)/(-b*x**4+a),x)
[Out]
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GIAC/XCAS [A] time = 0.217803, size = 392, normalized size = 3.16 \[ -\frac{{\rm ln}\left ({\left | b x^{4} - a \right |}\right )}{4 \, b} + \frac{\sqrt{2}{\left (\left (-a b^{3}\right )^{\frac{1}{4}} b^{2} - \sqrt{2} \sqrt{-a b^{3}} b + \left (-a b^{3}\right )^{\frac{3}{4}}\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (-\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (-\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{4 \, a b^{3}} + \frac{\sqrt{2}{\left (\left (-a b^{3}\right )^{\frac{1}{4}} b^{2} + \sqrt{2} \sqrt{-a b^{3}} b + \left (-a b^{3}\right )^{\frac{3}{4}}\right )} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (-\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (-\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{4 \, a b^{3}} + \frac{\sqrt{2}{\left (\left (-a b^{3}\right )^{\frac{1}{4}} b^{2} - \left (-a b^{3}\right )^{\frac{3}{4}}\right )}{\rm ln}\left (x^{2} + \sqrt{2} x \left (-\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{-\frac{a}{b}}\right )}{8 \, a b^{3}} - \frac{\sqrt{2}{\left (\left (-a b^{3}\right )^{\frac{1}{4}} b^{2} - \left (-a b^{3}\right )^{\frac{3}{4}}\right )}{\rm ln}\left (x^{2} - \sqrt{2} x \left (-\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{-\frac{a}{b}}\right )}{8 \, a b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(x^3 + x^2 + x + 1)/(b*x^4 - a),x, algorithm="giac")
[Out]