Optimal. Leaf size=83 \[ -\frac{\left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{2 x^2}-\frac{e r \left (2 a+2 b \log \left (c x^n\right )+b n\right )}{8 x^2}-\frac{b n \left (d+e \log \left (f x^r\right )\right )}{4 x^2}-\frac{b e n r}{8 x^2} \]
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Rubi [A] time = 0.0726216, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {2304, 2366, 12} \[ -\frac{\left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{2 x^2}-\frac{e r \left (2 a+2 b \log \left (c x^n\right )+b n\right )}{8 x^2}-\frac{b n \left (d+e \log \left (f x^r\right )\right )}{4 x^2}-\frac{b e n r}{8 x^2} \]
Antiderivative was successfully verified.
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Rule 2304
Rule 2366
Rule 12
Rubi steps
\begin{align*} \int \frac{\left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{x^3} \, dx &=-\frac{b n \left (d+e \log \left (f x^r\right )\right )}{4 x^2}-\frac{\left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{2 x^2}-(e r) \int \frac{-2 a \left (1+\frac{b n}{2 a}\right )-2 b \log \left (c x^n\right )}{4 x^3} \, dx\\ &=-\frac{b n \left (d+e \log \left (f x^r\right )\right )}{4 x^2}-\frac{\left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{2 x^2}-\frac{1}{4} (e r) \int \frac{-2 a \left (1+\frac{b n}{2 a}\right )-2 b \log \left (c x^n\right )}{x^3} \, dx\\ &=-\frac{b e n r}{8 x^2}-\frac{e r \left (2 a+b n+2 b \log \left (c x^n\right )\right )}{8 x^2}-\frac{b n \left (d+e \log \left (f x^r\right )\right )}{4 x^2}-\frac{\left (a+b \log \left (c x^n\right )\right ) \left (d+e \log \left (f x^r\right )\right )}{2 x^2}\\ \end{align*}
Mathematica [A] time = 0.0696156, size = 64, normalized size = 0.77 \[ -\frac{e (2 a+b n) \log \left (f x^r\right )+2 a d+a e r+b \log \left (c x^n\right ) \left (2 d+2 e \log \left (f x^r\right )+e r\right )+b d n+b e n r}{4 x^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.19, size = 1442, normalized size = 17.4 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.1727, size = 126, normalized size = 1.52 \begin{align*} -\frac{1}{4} \, b e{\left (\frac{r}{x^{2}} + \frac{2 \, \log \left (f x^{r}\right )}{x^{2}}\right )} \log \left (c x^{n}\right ) - \frac{b e n{\left (r + \log \left (f\right ) + \log \left (x^{r}\right )\right )}}{4 \, x^{2}} - \frac{b d n}{4 \, x^{2}} - \frac{a e r}{4 \, x^{2}} - \frac{b d \log \left (c x^{n}\right )}{2 \, x^{2}} - \frac{a e \log \left (f x^{r}\right )}{2 \, x^{2}} - \frac{a d}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.768871, size = 266, normalized size = 3.2 \begin{align*} -\frac{2 \, b e n r \log \left (x\right )^{2} + b d n + 2 \, a d +{\left (b e n + a e\right )} r +{\left (b e r + 2 \, b d\right )} \log \left (c\right ) +{\left (b e n + 2 \, b e \log \left (c\right ) + 2 \, a e\right )} \log \left (f\right ) + 2 \,{\left (b e r \log \left (c\right ) + b e n \log \left (f\right ) + b d n +{\left (b e n + a e\right )} r\right )} \log \left (x\right )}{4 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 11.4372, size = 201, normalized size = 2.42 \begin{align*} - \frac{a d}{2 x^{2}} - \frac{a e r \log{\left (x \right )}}{2 x^{2}} - \frac{a e r}{4 x^{2}} - \frac{a e \log{\left (f \right )}}{2 x^{2}} - \frac{b d n \log{\left (x \right )}}{2 x^{2}} - \frac{b d n}{4 x^{2}} - \frac{b d \log{\left (c \right )}}{2 x^{2}} - \frac{b e n r \log{\left (x \right )}^{2}}{2 x^{2}} - \frac{b e n r \log{\left (x \right )}}{2 x^{2}} - \frac{b e n r}{4 x^{2}} - \frac{b e n \log{\left (f \right )} \log{\left (x \right )}}{2 x^{2}} - \frac{b e n \log{\left (f \right )}}{4 x^{2}} - \frac{b e r \log{\left (c \right )} \log{\left (x \right )}}{2 x^{2}} - \frac{b e r \log{\left (c \right )}}{4 x^{2}} - \frac{b e \log{\left (c \right )} \log{\left (f \right )}}{2 x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.2139, size = 157, normalized size = 1.89 \begin{align*} -\frac{2 \, b n r e \log \left (x\right )^{2} + 2 \, b n r e \log \left (x\right ) + 2 \, b r e \log \left (c\right ) \log \left (x\right ) + 2 \, b n e \log \left (f\right ) \log \left (x\right ) + b n r e + b r e \log \left (c\right ) + b n e \log \left (f\right ) + 2 \, b e \log \left (c\right ) \log \left (f\right ) + 2 \, b d n \log \left (x\right ) + 2 \, a r e \log \left (x\right ) + b d n + a r e + 2 \, b d \log \left (c\right ) + 2 \, a e \log \left (f\right ) + 2 \, a d}{4 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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