Optimal. Leaf size=45 \[ \frac{\log \left (a+b x^4\right )}{4 (b c-a d)}-\frac{\log \left (c+d x^4\right )}{4 (b c-a d)} \]
[Out]
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Rubi [A] time = 0.093747, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{\log \left (a+b x^4\right )}{4 (b c-a d)}-\frac{\log \left (c+d x^4\right )}{4 (b c-a d)} \]
Antiderivative was successfully verified.
[In] Int[x^3/((a + b*x^4)*(c + d*x^4)),x]
[Out]
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Rubi in Sympy [A] time = 12.7466, size = 36, normalized size = 0.8 \[ - \frac{\log{\left (a + b x^{4} \right )}}{4 \left (a d - b c\right )} + \frac{\log{\left (c + d x^{4} \right )}}{4 \left (a d - b c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3/(b*x**4+a)/(d*x**4+c),x)
[Out]
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Mathematica [A] time = 0.0316985, size = 31, normalized size = 0.69 \[ \frac{\log \left (a+b x^4\right )-\log \left (c+d x^4\right )}{4 b c-4 a d} \]
Antiderivative was successfully verified.
[In] Integrate[x^3/((a + b*x^4)*(c + d*x^4)),x]
[Out]
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Maple [A] time = 0.01, size = 42, normalized size = 0.9 \[{\frac{\ln \left ( d{x}^{4}+c \right ) }{4\,ad-4\,bc}}-{\frac{\ln \left ( b{x}^{4}+a \right ) }{4\,ad-4\,bc}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3/(b*x^4+a)/(d*x^4+c),x)
[Out]
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Maxima [A] time = 1.36802, size = 55, normalized size = 1.22 \[ \frac{\log \left (b x^{4} + a\right )}{4 \,{\left (b c - a d\right )}} - \frac{\log \left (d x^{4} + c\right )}{4 \,{\left (b c - a d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/((b*x^4 + a)*(d*x^4 + c)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.213667, size = 42, normalized size = 0.93 \[ \frac{\log \left (b x^{4} + a\right ) - \log \left (d x^{4} + c\right )}{4 \,{\left (b c - a d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/((b*x^4 + a)*(d*x^4 + c)),x, algorithm="fricas")
[Out]
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Sympy [A] time = 3.01317, size = 138, normalized size = 3.07 \[ \frac{\log{\left (x^{4} + \frac{- \frac{a^{2} d^{2}}{a d - b c} + \frac{2 a b c d}{a d - b c} + a d - \frac{b^{2} c^{2}}{a d - b c} + b c}{2 b d} \right )}}{4 \left (a d - b c\right )} - \frac{\log{\left (x^{4} + \frac{\frac{a^{2} d^{2}}{a d - b c} - \frac{2 a b c d}{a d - b c} + a d + \frac{b^{2} c^{2}}{a d - b c} + b c}{2 b d} \right )}}{4 \left (a d - b c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3/(b*x**4+a)/(d*x**4+c),x)
[Out]
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GIAC/XCAS [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/((b*x^4 + a)*(d*x^4 + c)),x, algorithm="giac")
[Out]