Optimal. Leaf size=62 \[ \frac {\log (x)}{a c}-\frac {b \log \left (a+b x^2\right )}{2 a (b c-a d)}+\frac {d \log \left (c+d x^2\right )}{2 c (b c-a d)} \]
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Rubi [A]
time = 0.04, 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 = {457, 84}
\begin {gather*} -\frac {b \log \left (a+b x^2\right )}{2 a (b c-a d)}+\frac {d \log \left (c+d x^2\right )}{2 c (b c-a d)}+\frac {\log (x)}{a c} \end {gather*}
Antiderivative was successfully verified.
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Rule 84
Rule 457
Rubi steps
\begin {align*} \int \frac {1}{x \left (a+b x^2\right ) \left (c+d x^2\right )} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{x (a+b x) (c+d x)} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {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^2\right )\\ &=\frac {\log (x)}{a c}-\frac {b \log \left (a+b x^2\right )}{2 a (b c-a d)}+\frac {d \log \left (c+d x^2\right )}{2 c (b c-a d)}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 54, normalized size = 0.87 \begin {gather*} \frac {2 b c \log (x)-2 a d \log (x)-b c \log \left (a+b x^2\right )+a d \log \left (c+d x^2\right )}{2 a b c^2-2 a^2 c d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 59, normalized size = 0.95
method | result | size |
default | \(\frac {b \ln \left (b \,x^{2}+a \right )}{2 a \left (a d -b c \right )}-\frac {d \ln \left (d \,x^{2}+c \right )}{2 c \left (a d -b c \right )}+\frac {\ln \left (x \right )}{a c}\) | \(59\) |
norman | \(\frac {b \ln \left (b \,x^{2}+a \right )}{2 a \left (a d -b c \right )}-\frac {d \ln \left (d \,x^{2}+c \right )}{2 c \left (a d -b c \right )}+\frac {\ln \left (x \right )}{a c}\) | \(59\) |
risch | \(\frac {b \ln \left (b \,x^{2}+a \right )}{2 a \left (a d -b c \right )}-\frac {d \ln \left (d \,x^{2}+c \right )}{2 c \left (a d -b c \right )}+\frac {\ln \left (x \right )}{a c}\) | \(59\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 61, normalized size = 0.98 \begin {gather*} -\frac {b \log \left (b x^{2} + a\right )}{2 \, {\left (a b c - a^{2} d\right )}} + \frac {d \log \left (d x^{2} + c\right )}{2 \, {\left (b c^{2} - a c d\right )}} + \frac {\log \left (x^{2}\right )}{2 \, a c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.24, size = 54, normalized size = 0.87 \begin {gather*} -\frac {b c \log \left (b x^{2} + a\right ) - a d \log \left (d x^{2} + c\right ) - 2 \, {\left (b c - a d\right )} \log \left (x\right )}{2 \, {\left (a b c^{2} - a^{2} c d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.70, size = 73, normalized size = 1.18 \begin {gather*} -\frac {b^{2} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, {\left (a b^{2} c - a^{2} b d\right )}} + \frac {d^{2} \log \left ({\left | d x^{2} + c \right |}\right )}{2 \, {\left (b c^{2} d - a c d^{2}\right )}} + \frac {\log \left (x^{2}\right )}{2 \, a c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.18, size = 58, normalized size = 0.94 \begin {gather*} \frac {b\,\ln \left (b\,x^2+a\right )}{2\,a^2\,d-2\,a\,b\,c}+\frac {d\,\ln \left (d\,x^2+c\right )}{2\,b\,c^2-2\,a\,c\,d}+\frac {\ln \left (x\right )}{a\,c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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