Optimal. Leaf size=93 \[ \frac {g m \text {Int}\left (\frac {1}{(f+g x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )},x\right )}{n (b c-a d)}-\frac {\log \left (h (f+g x)^m\right )}{n (b c-a d) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \]
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Rubi [A] time = 0.13, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\log \left (h (f+g x)^m\right )}{(a+b x) (c+d x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\log \left (h (f+g x)^m\right )}{(a+b x) (c+d x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx &=-\frac {\log \left (h (f+g x)^m\right )}{(b c-a d) n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}+\frac {(g m) \int \frac {1}{(f+g x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx}{(b c-a d) n}\\ \end {align*}
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Mathematica [A] time = 1.33, size = 0, normalized size = 0.00 \[ \int \frac {\log \left (h (f+g x)^m\right )}{(a+b x) (c+d x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 1.99, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\log \left ({\left (g x + f\right )}^{m} h\right )}{{\left (b d x^{2} + a c + {\left (b c + a d\right )} x\right )} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log \left ({\left (g x + f\right )}^{m} h\right )}{{\left (b x + a\right )} {\left (d x + c\right )} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F(-1)] time = 180.00, size = 0, normalized size = 0.00 \[ \int \frac {\ln \left (h \left (g x +f \right )^{m}\right )}{\left (b x +a \right ) \left (d x +c \right ) \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ g m \int \frac {1}{b c f n \log \relax (e) - a d f n \log \relax (e) + {\left (b c g n \log \relax (e) - a d g n \log \relax (e)\right )} x + {\left (b c f n - a d f n + {\left (b c g n - a d g n\right )} x\right )} \log \left ({\left (b x + a\right )}^{n}\right ) - {\left (b c f n - a d f n + {\left (b c g n - a d g n\right )} x\right )} \log \left ({\left (d x + c\right )}^{n}\right )}\,{d x} - \frac {\log \left ({\left (g x + f\right )}^{m}\right ) + \log \relax (h)}{b c n \log \relax (e) - a d n \log \relax (e) + {\left (b c n - a d n\right )} \log \left ({\left (b x + a\right )}^{n}\right ) - {\left (b c n - a d n\right )} \log \left ({\left (d x + c\right )}^{n}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\ln \left (h\,{\left (f+g\,x\right )}^m\right )}{{\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}^2\,\left (a+b\,x\right )\,\left (c+d\,x\right )} \,d x \]
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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