Optimal. Leaf size=55 \[ \frac {b p \left (d x^n\right )^{2/n} \text {Ei}\left (-\frac {2 \log \left (d x^n\right )}{n}\right )}{2 x^2}-\frac {a+b \log \left (c \log ^p\left (d x^n\right )\right )}{2 x^2} \]
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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 = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2602, 2347,
2209} \begin {gather*} \frac {b p \left (d x^n\right )^{2/n} \text {Ei}\left (-\frac {2 \log \left (d x^n\right )}{n}\right )}{2 x^2}-\frac {a+b \log \left (c \log ^p\left (d x^n\right )\right )}{2 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 2209
Rule 2347
Rule 2602
Rubi steps
\begin {align*} \int \frac {a+b \log \left (c \log ^p\left (d x^n\right )\right )}{x^3} \, dx &=-\frac {a+b \log \left (c \log ^p\left (d x^n\right )\right )}{2 x^2}+\frac {1}{2} (b n p) \int \frac {1}{x^3 \log \left (d x^n\right )} \, dx\\ &=-\frac {a+b \log \left (c \log ^p\left (d x^n\right )\right )}{2 x^2}+\frac {\left (b p \left (d x^n\right )^{2/n}\right ) \text {Subst}\left (\int \frac {e^{-\frac {2 x}{n}}}{x} \, dx,x,\log \left (d x^n\right )\right )}{2 x^2}\\ &=\frac {b p \left (d x^n\right )^{2/n} \text {Ei}\left (-\frac {2 \log \left (d x^n\right )}{n}\right )}{2 x^2}-\frac {a+b \log \left (c \log ^p\left (d x^n\right )\right )}{2 x^2}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 49, normalized size = 0.89 \begin {gather*} -\frac {a-b p \left (d x^n\right )^{2/n} \text {Ei}\left (-\frac {2 \log \left (d x^n\right )}{n}\right )+b \log \left (c \log ^p\left (d x^n\right )\right )}{2 x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {a +b \ln \left (c \ln \left (d \,x^{n}\right )^{p}\right )}{x^{3}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 53, normalized size = 0.96 \begin {gather*} \frac {b d^{\frac {2}{n}} p x^{2} \operatorname {log\_integral}\left (\frac {1}{d^{\frac {2}{n}} x^{2}}\right ) - b p \log \left (n \log \left (x\right ) + \log \left (d\right )\right ) - b \log \left (c\right ) - a}{2 \, x^{2}} \end {gather*}
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 \begin {gather*} \int \frac {a + b \log {\left (c \log {\left (d x^{n} \right )}^{p} \right )}}{x^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {a+b\,\ln \left (c\,{\ln \left (d\,x^n\right )}^p\right )}{x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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