Optimal. Leaf size=45 \[ \frac {\log \left (a+b x^2\right )}{2 (b c-a d)}-\frac {\log \left (c+d x^2\right )}{2 (b c-a d)} \]
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Rubi [A]
time = 0.02, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {455, 36, 31}
\begin {gather*} \frac {\log \left (a+b x^2\right )}{2 (b c-a d)}-\frac {\log \left (c+d x^2\right )}{2 (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 36
Rule 455
Rubi steps
\begin {align*} \int \frac {x}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{(a+b x) (c+d x)} \, dx,x,x^2\right )\\ &=\frac {b \text {Subst}\left (\int \frac {1}{a+b x} \, dx,x,x^2\right )}{2 (b c-a d)}-\frac {d \text {Subst}\left (\int \frac {1}{c+d x} \, dx,x,x^2\right )}{2 (b c-a d)}\\ &=\frac {\log \left (a+b x^2\right )}{2 (b c-a d)}-\frac {\log \left (c+d x^2\right )}{2 (b c-a d)}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 31, normalized size = 0.69 \begin {gather*} \frac {\log \left (a+b x^2\right )-\log \left (c+d x^2\right )}{2 b c-2 a d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 42, normalized size = 0.93
method | result | size |
default | \(-\frac {\ln \left (b \,x^{2}+a \right )}{2 \left (a d -b c \right )}+\frac {\ln \left (d \,x^{2}+c \right )}{2 a d -2 b c}\) | \(42\) |
norman | \(-\frac {\ln \left (b \,x^{2}+a \right )}{2 \left (a d -b c \right )}+\frac {\ln \left (d \,x^{2}+c \right )}{2 a d -2 b c}\) | \(42\) |
risch | \(\frac {\ln \left (d \,x^{2}+c \right )}{2 a d -2 b c}-\frac {\ln \left (-b \,x^{2}-a \right )}{2 \left (a d -b c \right )}\) | \(45\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.32, size = 41, normalized size = 0.91 \begin {gather*} \frac {\log \left (b x^{2} + a\right )}{2 \, {\left (b c - a d\right )}} - \frac {\log \left (d x^{2} + c\right )}{2 \, {\left (b c - a d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.93, size = 31, normalized size = 0.69 \begin {gather*} \frac {\log \left (b x^{2} + a\right ) - \log \left (d x^{2} + c\right )}{2 \, {\left (b c - a d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 138 vs.
\(2 (36) = 72\).
time = 0.54, size = 138, normalized size = 3.07 \begin {gather*} \frac {\log {\left (x^{2} + \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 )}}{2 \left (a d - b c\right )} - \frac {\log {\left (x^{2} + \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 )}}{2 \left (a d - b c\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.58, size = 51, normalized size = 1.13 \begin {gather*} \frac {b \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, {\left (b^{2} c - a b d\right )}} - \frac {d \log \left ({\left | d x^{2} + c \right |}\right )}{2 \, {\left (b c d - a d^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.12, size = 148, normalized size = 3.29 \begin {gather*} \frac {2\,\mathrm {atanh}\left (\frac {8\,b^2\,d^2\,x^2}{\left (2\,a\,d-2\,b\,c\right )\,\left (\frac {32\,a\,b^2\,c\,d^2}{4\,a^2\,d^2-8\,a\,b\,c\,d+4\,b^2\,c^2}+\frac {16\,a\,b^2\,d^3\,x^2}{4\,a^2\,d^2-8\,a\,b\,c\,d+4\,b^2\,c^2}+\frac {16\,b^3\,c\,d^2\,x^2}{4\,a^2\,d^2-8\,a\,b\,c\,d+4\,b^2\,c^2}\right )}\right )}{2\,a\,d-2\,b\,c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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