Optimal. Leaf size=66 \[ -\frac {e n p x^{-1+n} \, _2F_1\left (1,-\frac {1-n}{n};2-\frac {1}{n};-\frac {e x^n}{d}\right )}{d (1-n)}-\frac {\log \left (c \left (d+e x^n\right )^p\right )}{x} \]
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Rubi [A]
time = 0.02, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2505, 371}
\begin {gather*} -\frac {\log \left (c \left (d+e x^n\right )^p\right )}{x}-\frac {e n p x^{n-1} \, _2F_1\left (1,-\frac {1-n}{n};2-\frac {1}{n};-\frac {e x^n}{d}\right )}{d (1-n)} \end {gather*}
Antiderivative was successfully verified.
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Rule 371
Rule 2505
Rubi steps
\begin {align*} \int \frac {\log \left (c \left (d+e x^n\right )^p\right )}{x^2} \, dx &=-\frac {\log \left (c \left (d+e x^n\right )^p\right )}{x}+(e n p) \int \frac {x^{-2+n}}{d+e x^n} \, dx\\ &=-\frac {e n p x^{-1+n} \, _2F_1\left (1,-\frac {1-n}{n};2-\frac {1}{n};-\frac {e x^n}{d}\right )}{d (1-n)}-\frac {\log \left (c \left (d+e x^n\right )^p\right )}{x}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 59, normalized size = 0.89 \begin {gather*} \frac {\frac {e n p x^n \, _2F_1\left (1,\frac {-1+n}{n};2-\frac {1}{n};-\frac {e x^n}{d}\right )}{d (-1+n)}-\log \left (c \left (d+e x^n\right )^p\right )}{x} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {\ln \left (c \left (d +e \,x^{n}\right )^{p}\right )}{x^{2}}\, dx\]
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 [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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Sympy [C] Result contains complex when optimal does not.
time = 4.30, size = 46, normalized size = 0.70 \begin {gather*} - \frac {\log {\left (c \left (d + e x^{n}\right )^{p} \right )}}{x} + \frac {p \Phi \left (\frac {d x^{- n} e^{i \pi }}{e}, 1, \frac {1}{n}\right ) \Gamma \left (- \frac {1}{n}\right )}{n x \Gamma \left (1 - \frac {1}{n}\right )} \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 {\ln \left (c\,{\left (d+e\,x^n\right )}^p\right )}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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