Optimal. Leaf size=72 \[ \frac {(a+b x) \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{-1/n} \text {Ei}\left (\frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{n}\right )}{n (c+d x) (b c-a d)} \]
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Rubi [A] time = 0.03, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {2493} \[ \frac {(a+b x) \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{-1/n} \text {Ei}\left (\frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{n}\right )}{n (c+d x) (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 2493
Rubi steps
\begin {align*} \int \frac {1}{(c+d x)^2 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx &=\frac {(a+b x) \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{-1/n} \text {Ei}\left (\frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{n}\right )}{(b c-a d) n (c+d x)}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 72, normalized size = 1.00 \[ \frac {(a+b x) \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{-1/n} \text {Ei}\left (\frac {\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{n}\right )}{n (c+d x) (b c-a d)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.85, size = 40, normalized size = 0.56 \[ \frac {\operatorname {log\_integral}\left (\frac {{\left (b x + a\right )} e^{\left (\frac {1}{n}\right )}}{d x + c}\right )}{{\left (b c - a d\right )} e^{\left (\frac {1}{n}\right )} n} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.31, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (d x +c \right )^{2} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )}\, dx \]
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 \[ \int \frac {1}{{\left (d x + c\right )}^{2} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )}\,{d x} \]
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 \[ \int \frac {1}{\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\,{\left (c+d\,x\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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