Optimal. Leaf size=62 \[ -\frac {b \log \left (a+b x^4\right )}{4 a (b c-a d)}+\frac {d \log \left (c+d x^4\right )}{4 c (b c-a d)}+\frac {\log (x)}{a c} \]
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Rubi [A] time = 0.06, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {446, 72} \[ -\frac {b \log \left (a+b x^4\right )}{4 a (b c-a d)}+\frac {d \log \left (c+d x^4\right )}{4 c (b c-a d)}+\frac {\log (x)}{a c} \]
Antiderivative was successfully verified.
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Rule 72
Rule 446
Rubi steps
\begin {align*} \int \frac {1}{x \left (a+b x^4\right ) \left (c+d x^4\right )} \, dx &=\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{x (a+b x) (c+d x)} \, dx,x,x^4\right )\\ &=\frac {1}{4} \operatorname {Subst}\left (\int \left (\frac {1}{a c x}+\frac {b^2}{a (-b c+a d) (a+b x)}+\frac {d^2}{c (b c-a d) (c+d x)}\right ) \, dx,x,x^4\right )\\ &=\frac {\log (x)}{a c}-\frac {b \log \left (a+b x^4\right )}{4 a (b c-a d)}+\frac {d \log \left (c+d x^4\right )}{4 c (b c-a d)}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 54, normalized size = 0.87 \[ \frac {-b c \log \left (a+b x^4\right )+a d \log \left (c+d x^4\right )-4 a d \log (x)+4 b c \log (x)}{4 a b c^2-4 a^2 c d} \]
Antiderivative was successfully verified.
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fricas [A] time = 2.05, size = 54, normalized size = 0.87 \[ -\frac {b c \log \left (b x^{4} + a\right ) - a d \log \left (d x^{4} + c\right ) - 4 \, {\left (b c - a d\right )} \log \relax (x)}{4 \, {\left (a b c^{2} - a^{2} c d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 73, normalized size = 1.18 \[ -\frac {b^{2} \log \left ({\left | b x^{4} + a \right |}\right )}{4 \, {\left (a b^{2} c - a^{2} b d\right )}} + \frac {d^{2} \log \left ({\left | d x^{4} + c \right |}\right )}{4 \, {\left (b c^{2} d - a c d^{2}\right )}} + \frac {\log \left (x^{4}\right )}{4 \, a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 59, normalized size = 0.95 \[ \frac {b \ln \left (b \,x^{4}+a \right )}{4 \left (a d -b c \right ) a}-\frac {d \ln \left (d \,x^{4}+c \right )}{4 \left (a d -b c \right ) c}+\frac {\ln \relax (x )}{a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.71, size = 61, normalized size = 0.98 \[ -\frac {b \log \left (b x^{4} + a\right )}{4 \, {\left (a b c - a^{2} d\right )}} + \frac {d \log \left (d x^{4} + c\right )}{4 \, {\left (b c^{2} - a c d\right )}} + \frac {\log \left (x^{4}\right )}{4 \, a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.49, size = 58, normalized size = 0.94 \[ \frac {b\,\ln \left (b\,x^4+a\right )}{4\,a^2\,d-4\,a\,b\,c}+\frac {d\,\ln \left (d\,x^4+c\right )}{4\,b\,c^2-4\,a\,c\,d}+\frac {\ln \relax (x)}{a\,c} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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