Optimal. Leaf size=57 \[ \frac{\tan ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{a+b x^4}}\right )}{2 a^{5/4}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{a+b x^4}}\right )}{2 a^{5/4}} \]
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Rubi [A] time = 0.0694466, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{\tan ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{a+b x^4}}\right )}{2 a^{5/4}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{a+b x^4}}\right )}{2 a^{5/4}} \]
Antiderivative was successfully verified.
[In] Int[1/((a - (a - b)*x^4)*(a + b*x^4)^(1/4)),x]
[Out]
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Rubi in Sympy [A] time = 9.03974, size = 49, normalized size = 0.86 \[ \frac{\operatorname{atan}{\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{a + b x^{4}}} \right )}}{2 a^{\frac{5}{4}}} + \frac{\operatorname{atanh}{\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{a + b x^{4}}} \right )}}{2 a^{\frac{5}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(a-(a-b)*x**4)/(b*x**4+a)**(1/4),x)
[Out]
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Mathematica [A] time = 0.109993, size = 76, normalized size = 1.33 \[ \frac{-\log \left (1-\frac{\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )+\log \left (\frac{\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}+1\right )+2 \tan ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{a x^4+b}}\right )}{4 a^{5/4}} \]
Warning: Unable to verify antiderivative.
[In] Integrate[1/((a - (a - b)*x^4)*(a + b*x^4)^(1/4)),x]
[Out]
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Maple [F] time = 0.076, size = 0, normalized size = 0. \[ \int{\frac{1}{a- \left ( a-b \right ){x}^{4}}{\frac{1}{\sqrt [4]{b{x}^{4}+a}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(a-(a-b)*x^4)/(b*x^4+a)^(1/4),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ -\int \frac{1}{{\left ({\left (a - b\right )} x^{4} - a\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-1/(((a - b)*x^4 - a)*(b*x^4 + a)^(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(-1/(((a - b)*x^4 - a)*(b*x^4 + a)^(1/4)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \int \frac{1}{a x^{4} \sqrt [4]{a + b x^{4}} - a \sqrt [4]{a + b x^{4}} - b x^{4} \sqrt [4]{a + b x^{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(a-(a-b)*x**4)/(b*x**4+a)**(1/4),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int -\frac{1}{{\left ({\left (a - b\right )} x^{4} - a\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-1/(((a - b)*x^4 - a)*(b*x^4 + a)^(1/4)),x, algorithm="giac")
[Out]