Optimal. Leaf size=133 \[ -\frac {b n}{e m x}-\frac {b e^{\frac {d}{e m}} n \left (f x^m\right )^{\frac {1}{m}} \text {Ei}\left (-\frac {d+e \log \left (f x^m\right )}{e m}\right ) \left (d+e \log \left (f x^m\right )\right )}{e^2 m^2 x}+\frac {e^{\frac {d}{e m}} \left (f x^m\right )^{\frac {1}{m}} \text {Ei}\left (-\frac {d+e \log \left (f x^m\right )}{e m}\right ) \left (a+b \log \left (c x^n\right )\right )}{e m x} \]
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Rubi [A]
time = 0.12, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2347, 2209,
2413, 12, 15, 6617} \begin {gather*} \frac {e^{\frac {d}{e m}} \left (f x^m\right )^{\frac {1}{m}} \left (a+b \log \left (c x^n\right )\right ) \text {Ei}\left (-\frac {d+e \log \left (f x^m\right )}{e m}\right )}{e m x}-\frac {b n e^{\frac {d}{e m}} \left (f x^m\right )^{\frac {1}{m}} \left (d+e \log \left (f x^m\right )\right ) \text {Ei}\left (-\frac {d+e \log \left (f x^m\right )}{e m}\right )}{e^2 m^2 x}-\frac {b n}{e m x} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 15
Rule 2209
Rule 2347
Rule 2413
Rule 6617
Rubi steps
\begin {align*} \int \frac {a+b \log \left (c x^n\right )}{x^2 \left (d+e \log \left (f x^m\right )\right )} \, dx &=\frac {e^{\frac {d}{e m}} \left (f x^m\right )^{\frac {1}{m}} \text {Ei}\left (-\frac {d+e \log \left (f x^m\right )}{e m}\right ) \left (a+b \log \left (c x^n\right )\right )}{e m x}-(b n) \int \frac {e^{\frac {d}{e m}} \left (f x^m\right )^{\frac {1}{m}} \text {Ei}\left (-\frac {d+e \log \left (f x^m\right )}{e m}\right )}{e m x^2} \, dx\\ &=\frac {e^{\frac {d}{e m}} \left (f x^m\right )^{\frac {1}{m}} \text {Ei}\left (-\frac {d+e \log \left (f x^m\right )}{e m}\right ) \left (a+b \log \left (c x^n\right )\right )}{e m x}-\frac {\left (b e^{\frac {d}{e m}} n\right ) \int \frac {\left (f x^m\right )^{\frac {1}{m}} \text {Ei}\left (-\frac {d+e \log \left (f x^m\right )}{e m}\right )}{x^2} \, dx}{e m}\\ &=\frac {e^{\frac {d}{e m}} \left (f x^m\right )^{\frac {1}{m}} \text {Ei}\left (-\frac {d+e \log \left (f x^m\right )}{e m}\right ) \left (a+b \log \left (c x^n\right )\right )}{e m x}-\frac {\left (b e^{\frac {d}{e m}} n \left (f x^m\right )^{\frac {1}{m}}\right ) \int \frac {\text {Ei}\left (-\frac {d+e \log \left (f x^m\right )}{e m}\right )}{x} \, dx}{e m x}\\ &=\frac {e^{\frac {d}{e m}} \left (f x^m\right )^{\frac {1}{m}} \text {Ei}\left (-\frac {d+e \log \left (f x^m\right )}{e m}\right ) \left (a+b \log \left (c x^n\right )\right )}{e m x}-\frac {\left (b e^{\frac {d}{e m}} n \left (f x^m\right )^{\frac {1}{m}}\right ) \text {Subst}\left (\int \text {Ei}\left (-\frac {d+e x}{e m}\right ) \, dx,x,\log \left (f x^m\right )\right )}{e m^2 x}\\ &=\frac {e^{\frac {d}{e m}} \left (f x^m\right )^{\frac {1}{m}} \text {Ei}\left (-\frac {d+e \log \left (f x^m\right )}{e m}\right ) \left (a+b \log \left (c x^n\right )\right )}{e m x}+\frac {\left (b e^{\frac {d}{e m}} n \left (f x^m\right )^{\frac {1}{m}}\right ) \text {Subst}\left (\int \text {Ei}(x) \, dx,x,-\frac {d}{e m}-\frac {\log \left (f x^m\right )}{m}\right )}{e m x}\\ &=-\frac {b n}{e m x}-\frac {b e^{\frac {d}{e m}} n \left (f x^m\right )^{\frac {1}{m}} \text {Ei}\left (-\frac {d}{e m}-\frac {\log \left (f x^m\right )}{m}\right ) \left (\frac {d}{e m}+\frac {\log \left (f x^m\right )}{m}\right )}{e m x}+\frac {e^{\frac {d}{e m}} \left (f x^m\right )^{\frac {1}{m}} \text {Ei}\left (-\frac {d+e \log \left (f x^m\right )}{e m}\right ) \left (a+b \log \left (c x^n\right )\right )}{e m x}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 87, normalized size = 0.65 \begin {gather*} \frac {-b e m n+e^{\frac {d}{e m}} \left (f x^m\right )^{\frac {1}{m}} \text {Ei}\left (-\frac {d+e \log \left (f x^m\right )}{e m}\right ) \left (a e m-b d n-b e n \log \left (f x^m\right )+b e m \log \left (c x^n\right )\right )}{e^2 m^2 x} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.41, size = 2296, normalized size = 17.26
method | result | size |
risch | \(\text {Expression too large to display}\) | \(2296\) |
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 = 0.41, size = 83, normalized size = 0.62 \begin {gather*} -\frac {{\left (b m n e - {\left (b m x e \log \left (c\right ) - b n x e \log \left (f\right ) - b d n x + a m x e\right )} e^{\left (\frac {{\left (e \log \left (f\right ) + d\right )} e^{\left (-1\right )}}{m}\right )} \operatorname {log\_integral}\left (\frac {e^{\left (-\frac {{\left (e \log \left (f\right ) + d\right )} e^{\left (-1\right )}}{m}\right )}}{x}\right )\right )} e^{\left (-2\right )}}{m^{2} x} \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 x^{n} \right )}}{x^{2} \left (d + e \log {\left (f x^{m} \right )}\right )}\, 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 = -0.01 \begin {gather*} \int \frac {a+b\,\ln \left (c\,x^n\right )}{x^2\,\left (d+e\,\ln \left (f\,x^m\right )\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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