Integrand size = 32, antiderivative size = 273 \[ \int \frac {(a+b \log (c (e+f x)))^2}{(d e+d f x) (h+i x)^2} \, dx=-\frac {i (e+f x) (a+b \log (c (e+f x)))^2}{d (f h-e i)^2 (h+i x)}+\frac {2 b f (a+b \log (c (e+f x))) \log \left (\frac {f (h+i x)}{f h-e i}\right )}{d (f h-e i)^2}-\frac {f (a+b \log (c (e+f x)))^2 \log \left (1+\frac {f h-e i}{i (e+f x)}\right )}{d (f h-e i)^2}+\frac {2 b f (a+b \log (c (e+f x))) \operatorname {PolyLog}\left (2,-\frac {f h-e i}{i (e+f x)}\right )}{d (f h-e i)^2}+\frac {2 b^2 f \operatorname {PolyLog}\left (2,-\frac {i (e+f x)}{f h-e i}\right )}{d (f h-e i)^2}+\frac {2 b^2 f \operatorname {PolyLog}\left (3,-\frac {f h-e i}{i (e+f x)}\right )}{d (f h-e i)^2} \]
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Time = 0.38 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.281, Rules used = {2458, 12, 2389, 2379, 2421, 6724, 2355, 2354, 2438} \[ \int \frac {(a+b \log (c (e+f x)))^2}{(d e+d f x) (h+i x)^2} \, dx=\frac {2 b f \operatorname {PolyLog}\left (2,-\frac {f h-e i}{i (e+f x)}\right ) (a+b \log (c (e+f x)))}{d (f h-e i)^2}+\frac {2 b f \log \left (\frac {f (h+i x)}{f h-e i}\right ) (a+b \log (c (e+f x)))}{d (f h-e i)^2}-\frac {i (e+f x) (a+b \log (c (e+f x)))^2}{d (h+i x) (f h-e i)^2}-\frac {f \log \left (\frac {f h-e i}{i (e+f x)}+1\right ) (a+b \log (c (e+f x)))^2}{d (f h-e i)^2}+\frac {2 b^2 f \operatorname {PolyLog}\left (2,-\frac {i (e+f x)}{f h-e i}\right )}{d (f h-e i)^2}+\frac {2 b^2 f \operatorname {PolyLog}\left (3,-\frac {f h-e i}{i (e+f x)}\right )}{d (f h-e i)^2} \]
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Rule 12
Rule 2354
Rule 2355
Rule 2379
Rule 2389
Rule 2421
Rule 2438
Rule 2458
Rule 6724
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {(a+b \log (c x))^2}{d x \left (\frac {f h-e i}{f}+\frac {i x}{f}\right )^2} \, dx,x,e+f x\right )}{f} \\ & = \frac {\text {Subst}\left (\int \frac {(a+b \log (c x))^2}{x \left (\frac {f h-e i}{f}+\frac {i x}{f}\right )^2} \, dx,x,e+f x\right )}{d f} \\ & = \frac {\text {Subst}\left (\int \frac {(a+b \log (c x))^2}{x \left (\frac {f h-e i}{f}+\frac {i x}{f}\right )} \, dx,x,e+f x\right )}{d (f h-e i)}-\frac {i \text {Subst}\left (\int \frac {(a+b \log (c x))^2}{\left (\frac {f h-e i}{f}+\frac {i x}{f}\right )^2} \, dx,x,e+f x\right )}{d f (f h-e i)} \\ & = -\frac {i (e+f x) (a+b \log (c (e+f x)))^2}{d (f h-e i)^2 (h+i x)}-\frac {f (a+b \log (c (e+f x)))^2 \log \left (1+\frac {f h-e i}{i (e+f x)}\right )}{d (f h-e i)^2}+\frac {(2 b f) \text {Subst}\left (\int \frac {\log \left (1+\frac {f h-e i}{i x}\right ) (a+b \log (c x))}{x} \, dx,x,e+f x\right )}{d (f h-e i)^2}+\frac {(2 b i) \text {Subst}\left (\int \frac {a+b \log (c x)}{\frac {f h-e i}{f}+\frac {i x}{f}} \, dx,x,e+f x\right )}{d (f h-e i)^2} \\ & = -\frac {i (e+f x) (a+b \log (c (e+f x)))^2}{d (f h-e i)^2 (h+i x)}+\frac {2 b f (a+b \log (c (e+f x))) \log \left (\frac {f (h+i x)}{f h-e i}\right )}{d (f h-e i)^2}-\frac {f (a+b \log (c (e+f x)))^2 \log \left (1+\frac {f h-e i}{i (e+f x)}\right )}{d (f h-e i)^2}+\frac {2 b f (a+b \log (c (e+f x))) \text {Li}_2\left (-\frac {f h-e i}{i (e+f x)}\right )}{d (f h-e i)^2}-\frac {\left (2 b^2 f\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {i x}{f h-e i}\right )}{x} \, dx,x,e+f x\right )}{d (f h-e i)^2}-\frac {\left (2 b^2 f\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {f h-e i}{i x}\right )}{x} \, dx,x,e+f x\right )}{d (f h-e i)^2} \\ & = -\frac {i (e+f x) (a+b \log (c (e+f x)))^2}{d (f h-e i)^2 (h+i x)}+\frac {2 b f (a+b \log (c (e+f x))) \log \left (\frac {f (h+i x)}{f h-e i}\right )}{d (f h-e i)^2}-\frac {f (a+b \log (c (e+f x)))^2 \log \left (1+\frac {f h-e i}{i (e+f x)}\right )}{d (f h-e i)^2}+\frac {2 b f (a+b \log (c (e+f x))) \text {Li}_2\left (-\frac {f h-e i}{i (e+f x)}\right )}{d (f h-e i)^2}+\frac {2 b^2 f \text {Li}_2\left (-\frac {i (e+f x)}{f h-e i}\right )}{d (f h-e i)^2}+\frac {2 b^2 f \text {Li}_3\left (-\frac {f h-e i}{i (e+f x)}\right )}{d (f h-e i)^2} \\ \end{align*}
Time = 0.40 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.32 \[ \int \frac {(a+b \log (c (e+f x)))^2}{(d e+d f x) (h+i x)^2} \, dx=\frac {3 a^2 (f h-e i)+3 a^2 f (h+i x) \log (e+f x)-3 a^2 f (h+i x) \log (h+i x)+3 a b \left (-2 f (h+i x) \log (e+f x)+2 (f h-e i) \log (c (e+f x))+f (h+i x) \log ^2(c (e+f x))+2 f (h+i x) \log (h+i x)-2 f (h+i x) \left (\log (c (e+f x)) \log \left (\frac {f (h+i x)}{f h-e i}\right )+\operatorname {PolyLog}\left (2,\frac {i (e+f x)}{-f h+e i}\right )\right )\right )+b^2 \left (\log (c (e+f x)) \left (f (h+i x) \log ^2(c (e+f x))+6 f (h+i x) \log \left (\frac {f (h+i x)}{f h-e i}\right )-3 \log (c (e+f x)) \left (i (e+f x)+f (h+i x) \log \left (\frac {f (h+i x)}{f h-e i}\right )\right )\right )-6 f (h+i x) (-1+\log (c (e+f x))) \operatorname {PolyLog}\left (2,\frac {i (e+f x)}{-f h+e i}\right )+6 f (h+i x) \operatorname {PolyLog}\left (3,\frac {i (e+f x)}{-f h+e i}\right )\right )}{3 d (f h-e i)^2 (h+i x)} \]
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\[\int \frac {\left (a +b \ln \left (c \left (f x +e \right )\right )\right )^{2}}{\left (d f x +d e \right ) \left (i x +h \right )^{2}}d x\]
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\[ \int \frac {(a+b \log (c (e+f x)))^2}{(d e+d f x) (h+i x)^2} \, dx=\int { \frac {{\left (b \log \left ({\left (f x + e\right )} c\right ) + a\right )}^{2}}{{\left (d f x + d e\right )} {\left (i x + h\right )}^{2}} \,d x } \]
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\[ \int \frac {(a+b \log (c (e+f x)))^2}{(d e+d f x) (h+i x)^2} \, dx=\frac {\int \frac {a^{2}}{e h^{2} + 2 e h i x + e i^{2} x^{2} + f h^{2} x + 2 f h i x^{2} + f i^{2} x^{3}}\, dx + \int \frac {b^{2} \log {\left (c e + c f x \right )}^{2}}{e h^{2} + 2 e h i x + e i^{2} x^{2} + f h^{2} x + 2 f h i x^{2} + f i^{2} x^{3}}\, dx + \int \frac {2 a b \log {\left (c e + c f x \right )}}{e h^{2} + 2 e h i x + e i^{2} x^{2} + f h^{2} x + 2 f h i x^{2} + f i^{2} x^{3}}\, dx}{d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 622 vs. \(2 (271) = 542\).
Time = 0.30 (sec) , antiderivative size = 622, normalized size of antiderivative = 2.28 \[ \int \frac {(a+b \log (c (e+f x)))^2}{(d e+d f x) (h+i x)^2} \, dx=a^{2} {\left (\frac {f \log \left (f x + e\right )}{d f^{2} h^{2} - 2 \, d e f h i + d e^{2} i^{2}} - \frac {f \log \left (i x + h\right )}{d f^{2} h^{2} - 2 \, d e f h i + d e^{2} i^{2}} + \frac {1}{d f h^{2} - d e h i + {\left (d f h i - d e i^{2}\right )} x}\right )} - \frac {{\left (\log \left (f x + e\right )^{2} \log \left (\frac {f i x + e i}{f h - e i} + 1\right ) + 2 \, {\rm Li}_2\left (-\frac {f i x + e i}{f h - e i}\right ) \log \left (f x + e\right ) - 2 \, {\rm Li}_{3}(-\frac {f i x + e i}{f h - e i})\right )} b^{2} f}{{\left (f^{2} h^{2} - 2 \, e f h i + e^{2} i^{2}\right )} d} + \frac {3 \, {\left (f h - e i\right )} b^{2} \log \left (c\right )^{2} + {\left (b^{2} f i x + b^{2} f h\right )} \log \left (f x + e\right )^{3} + 6 \, {\left (f h - e i\right )} a b \log \left (c\right ) + 3 \, {\left (a b f h + {\left (f h \log \left (c\right ) - e i\right )} b^{2} + {\left (a b f i + {\left (f i \log \left (c\right ) - f i\right )} b^{2}\right )} x\right )} \log \left (f x + e\right )^{2} + 3 \, {\left (2 \, {\left (f h \log \left (c\right ) - e i\right )} a b + {\left (f h \log \left (c\right )^{2} - 2 \, e i \log \left (c\right )\right )} b^{2} + {\left (2 \, {\left (f i \log \left (c\right ) - f i\right )} a b + {\left (f i \log \left (c\right )^{2} - 2 \, f i \log \left (c\right )\right )} b^{2}\right )} x\right )} \log \left (f x + e\right )}{3 \, {\left ({\left (f^{2} h^{2} i - 2 \, e f h i^{2} + e^{2} i^{3}\right )} d x + {\left (f^{2} h^{3} - 2 \, e f h^{2} i + e^{2} h i^{2}\right )} d\right )}} - \frac {2 \, {\left ({\left (f \log \left (c\right ) - f\right )} b^{2} + a b f\right )} {\left (\log \left (f x + e\right ) \log \left (\frac {f i x + e i}{f h - e i} + 1\right ) + {\rm Li}_2\left (-\frac {f i x + e i}{f h - e i}\right )\right )}}{{\left (f^{2} h^{2} - 2 \, e f h i + e^{2} i^{2}\right )} d} - \frac {{\left (2 \, {\left (f \log \left (c\right ) - f\right )} a b + {\left (f \log \left (c\right )^{2} - 2 \, f \log \left (c\right )\right )} b^{2}\right )} \log \left (i x + h\right )}{{\left (f^{2} h^{2} - 2 \, e f h i + e^{2} i^{2}\right )} d} \]
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\[ \int \frac {(a+b \log (c (e+f x)))^2}{(d e+d f x) (h+i x)^2} \, dx=\int { \frac {{\left (b \log \left ({\left (f x + e\right )} c\right ) + a\right )}^{2}}{{\left (d f x + d e\right )} {\left (i x + h\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {(a+b \log (c (e+f x)))^2}{(d e+d f x) (h+i x)^2} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,\left (e+f\,x\right )\right )\right )}^2}{{\left (h+i\,x\right )}^2\,\left (d\,e+d\,f\,x\right )} \,d x \]
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