Optimal. Leaf size=21 \[ \frac {\log \left (\frac {x^4}{a+b x^4}\right )}{4 a} \]
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Rubi [A]
time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.05, number of steps
used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {272, 36, 29, 31}
\begin {gather*} \frac {\log (x)}{a}-\frac {\log \left (a+b x^4\right )}{4 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 272
Rubi steps
\begin {gather*} \begin {aligned} \text {Integral} &=\frac {1}{4} \text {Subst}\left (\int \frac {1}{x (a+b x)} \, dx,x,x^4\right )\\ &=\frac {\text {Subst}\left (\int \frac {1}{x} \, dx,x,x^4\right )}{4 a}-\frac {b \text {Subst}\left (\int \frac {1}{a+b x} \, dx,x,x^4\right )}{4 a}\\ &=\frac {\log (x)}{a}-\frac {\log \left (a+b x^4\right )}{4 a}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.01, size = 22, normalized size = 1.05 \begin {gather*} \frac {\log (x)}{a}-\frac {\log \left (a+b x^4\right )}{4 a} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 21, normalized size = 1.00
method | result | size |
default | \(-\frac {\ln \left (b \,x^{4}+a \right )}{4 a}+\frac {\ln \left (x \right )}{a}\) | \(21\) |
norman | \(-\frac {\ln \left (b \,x^{4}+a \right )}{4 a}+\frac {\ln \left (x \right )}{a}\) | \(21\) |
risch | \(-\frac {\ln \left (b \,x^{4}+a \right )}{4 a}+\frac {\ln \left (x \right )}{a}\) | \(21\) |
parallelrisch | \(\frac {4 \ln \left (x \right )-\ln \left (b \,x^{4}+a \right )}{4 a}\) | \(21\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.32, size = 23, normalized size = 1.10 \begin {gather*} -\frac {\log \left (b x^{4} + a\right )}{4 \, a} + \frac {\log \left (x^{4}\right )}{4 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.65, size = 18, normalized size = 0.86 \begin {gather*} -\frac {\log \left (b x^{4} + a\right ) - 4 \, \log \left (x\right )}{4 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.11, size = 15, normalized size = 0.71 \begin {gather*} \frac {\log {\left (x \right )}}{a} - \frac {\log {\left (\frac {a}{b} + x^{4} \right )}}{4 a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.45, size = 24, normalized size = 1.14 \begin {gather*} \frac {\log \left (x^{4}\right )}{4 \, a} - \frac {\log \left ({\left | b x^{4} + a \right |}\right )}{4 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.11, size = 18, normalized size = 0.86 \begin {gather*} -\frac {\ln \left (b\,x^4+a\right )-4\,\ln \left (x\right )}{4\,a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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