Optimal. Leaf size=33 \[ \frac{\log \left (\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{n (b c-a d)} \]
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Rubi [A] time = 0.0749734, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.029, Rules used = {2504} \[ \frac{\log \left (\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{n (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 2504
Rubi steps
\begin{align*} \int \frac{1}{(a+b x) (c+d x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )} \, dx &=\frac{\log \left (\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d) n}\\ \end{align*}
Mathematica [A] time = 0.069878, size = 34, normalized size = 1.03 \[ -\frac{\log \left (\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{n (a d-b c)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.815, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( bx+a \right ) \left ( dx+c \right ) } \left ( \ln \left ( e \left ({\frac{bx+a}{dx+c}} \right ) ^{n} \right ) \right ) ^{-1}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.74459, size = 50, normalized size = 1.52 \begin{align*} \frac{\log \left (-\log \left ({\left (b x + a\right )}^{n}\right ) + \log \left ({\left (d x + c\right )}^{n}\right ) - \log \left (e\right )\right )}{b c n - a d n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.486093, size = 78, normalized size = 2.36 \begin{align*} \frac{\log \left (n \log \left (\frac{b x + a}{d x + c}\right ) + \log \left (e\right )\right )}{{\left (b c - a d\right )} n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17881, size = 43, normalized size = 1.3 \begin{align*} \frac{\log \left (n \log \left (\frac{b x + a}{d x + c}\right ) + 1\right )}{b c n - a d n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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