Optimal. Leaf size=55 \[ \frac{1}{2} x^2 \left (a+b \log \left (c \log ^p\left (d x^n\right )\right )\right )-\frac{1}{2} b p x^2 \left (d x^n\right )^{-2/n} \text{Ei}\left (\frac{2 \log \left (d x^n\right )}{n}\right ) \]
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Rubi [A] time = 0.0349517, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {2522, 2310, 2178} \[ \frac{1}{2} x^2 \left (a+b \log \left (c \log ^p\left (d x^n\right )\right )\right )-\frac{1}{2} b p x^2 \left (d x^n\right )^{-2/n} \text{Ei}\left (\frac{2 \log \left (d x^n\right )}{n}\right ) \]
Antiderivative was successfully verified.
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Rule 2522
Rule 2310
Rule 2178
Rubi steps
\begin{align*} \int x \left (a+b \log \left (c \log ^p\left (d x^n\right )\right )\right ) \, dx &=\frac{1}{2} x^2 \left (a+b \log \left (c \log ^p\left (d x^n\right )\right )\right )-\frac{1}{2} (b n p) \int \frac{x}{\log \left (d x^n\right )} \, dx\\ &=\frac{1}{2} x^2 \left (a+b \log \left (c \log ^p\left (d x^n\right )\right )\right )-\frac{1}{2} \left (b p x^2 \left (d x^n\right )^{-2/n}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{2 x}{n}}}{x} \, dx,x,\log \left (d x^n\right )\right )\\ &=-\frac{1}{2} b p x^2 \left (d x^n\right )^{-2/n} \text{Ei}\left (\frac{2 \log \left (d x^n\right )}{n}\right )+\frac{1}{2} x^2 \left (a+b \log \left (c \log ^p\left (d x^n\right )\right )\right )\\ \end{align*}
Mathematica [A] time = 0.0567191, size = 49, normalized size = 0.89 \[ \frac{1}{2} x^2 \left (a+b \log \left (c \log ^p\left (d x^n\right )\right )-b p \left (d x^n\right )^{-2/n} \text{Ei}\left (\frac{2 \log \left (d x^n\right )}{n}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.061, size = 0, normalized size = 0. \begin{align*} \int x \left ( a+b\ln \left ( c \left ( \ln \left ( d{x}^{n} \right ) \right ) ^{p} \right ) \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{2} \, a x^{2} - \frac{1}{2} \,{\left (2 \, n p \int \frac{x}{2 \,{\left (\log \left (d\right ) + \log \left (x^{n}\right )\right )}}\,{d x} - x^{2} \log \left (c\right ) - x^{2} \log \left ({\left (\log \left (d\right ) + \log \left (x^{n}\right )\right )}^{p}\right )\right )} b \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.85521, size = 161, normalized size = 2.93 \begin{align*} \frac{b d^{\frac{2}{n}} p x^{2} \log \left (n \log \left (x\right ) + \log \left (d\right )\right ) - b p \logintegral \left (d^{\frac{2}{n}} x^{2}\right ) +{\left (b x^{2} \log \left (c\right ) + a x^{2}\right )} d^{\frac{2}{n}}}{2 \, d^{\frac{2}{n}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \left (a + b \log{\left (c \log{\left (d x^{n} \right )}^{p} \right )}\right )\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.31949, size = 76, normalized size = 1.38 \begin{align*} \frac{1}{2} \, b p x^{2} \log \left (n \log \left (x\right ) + \log \left (d\right )\right ) + \frac{1}{2} \, b x^{2} \log \left (c\right ) + \frac{1}{2} \, a x^{2} - \frac{b p{\rm Ei}\left (\frac{2 \, \log \left (d\right )}{n} + 2 \, \log \left (x\right )\right )}{2 \, d^{\frac{2}{n}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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