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 = 0.03, antiderivative size = 55, normalized size of antiderivative = 1.00, 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 2178
Rule 2310
Rule 2522
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*}
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Mathematica [A] time = 0.06, 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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fricas [A] time = 0.46, size = 70, normalized size = 1.27 \[ \frac {b d^{\frac {2}{n}} p x^{2} \log \left (n \log \relax (x) + \log \relax (d)\right ) - b p \operatorname {log\_integral}\left (d^{\frac {2}{n}} x^{2}\right ) + {\left (b x^{2} \log \relax (c) + a x^{2}\right )} d^{\frac {2}{n}}}{2 \, d^{\frac {2}{n}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 56, normalized size = 1.02 \[ \frac {1}{2} \, b p x^{2} \log \left (n \log \relax (x) + \log \relax (d)\right ) + \frac {1}{2} \, b x^{2} \log \relax (c) + \frac {1}{2} \, a x^{2} - \frac {b p {\rm Ei}\left (\frac {2 \, \log \relax (d)}{n} + 2 \, \log \relax (x)\right )}{2 \, d^{\frac {2}{n}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.46, size = 0, normalized size = 0.00 \[ \int \left (b \ln \left (c \ln \left (d \,x^{n}\right )^{p}\right )+a \right ) x\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{2} \, a x^{2} - \frac {1}{2} \, {\left (2 \, n p \int \frac {x}{2 \, {\left (\log \relax (d) + \log \left (x^{n}\right )\right )}}\,{d x} - x^{2} \log \relax (c) - x^{2} \log \left ({\left (\log \relax (d) + \log \left (x^{n}\right )\right )}^{p}\right )\right )} b \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int x\,\left (a+b\,\ln \left (c\,{\ln \left (d\,x^n\right )}^p\right )\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \left (a + b \log {\left (c \log {\left (d x^{n} \right )}^{p} \right )}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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