Integrand size = 32, antiderivative size = 142 \[ \int \frac {(a+b \log (c (e+f x)))^2}{(d e+d f x) (h+i x)} \, dx=-\frac {(a+b \log (c (e+f x)))^2 \log \left (1+\frac {f h-e i}{i (e+f x)}\right )}{d (f h-e i)}+\frac {2 b (a+b \log (c (e+f x))) \operatorname {PolyLog}\left (2,-\frac {f h-e i}{i (e+f x)}\right )}{d (f h-e i)}+\frac {2 b^2 \operatorname {PolyLog}\left (3,-\frac {f h-e i}{i (e+f x)}\right )}{d (f h-e i)} \]
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Time = 0.20 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {2458, 12, 2379, 2421, 6724} \[ \int \frac {(a+b \log (c (e+f x)))^2}{(d e+d f x) (h+i x)} \, dx=\frac {2 b \operatorname {PolyLog}\left (2,-\frac {f h-e i}{i (e+f x)}\right ) (a+b \log (c (e+f x)))}{d (f h-e i)}-\frac {\log \left (\frac {f h-e i}{i (e+f x)}+1\right ) (a+b \log (c (e+f x)))^2}{d (f h-e i)}+\frac {2 b^2 \operatorname {PolyLog}\left (3,-\frac {f h-e i}{i (e+f x)}\right )}{d (f h-e i)} \]
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Rule 12
Rule 2379
Rule 2421
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 )} \, 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 )} \, dx,x,e+f x\right )}{d f} \\ & = -\frac {(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)}+\frac {(2 b) \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)} \\ & = -\frac {(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)}+\frac {2 b (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)}-\frac {\left (2 b^2\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)} \\ & = -\frac {(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)}+\frac {2 b (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)}+\frac {2 b^2 \text {Li}_3\left (-\frac {f h-e i}{i (e+f x)}\right )}{d (f h-e i)} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.33 \[ \int \frac {(a+b \log (c (e+f x)))^2}{(d e+d f x) (h+i x)} \, dx=\frac {3 a^2 \log (e+f x)+3 a b \log ^2(c (e+f x))+b^2 \log ^3(c (e+f x))-3 a^2 \log (h+i x)-6 a b \log (c (e+f x)) \log \left (\frac {f (h+i x)}{f h-e i}\right )-3 b^2 \log ^2(c (e+f x)) \log \left (\frac {f (h+i x)}{f h-e i}\right )-6 b (a+b \log (c (e+f x))) \operatorname {PolyLog}\left (2,\frac {i (e+f x)}{-f h+e i}\right )+6 b^2 \operatorname {PolyLog}\left (3,\frac {i (e+f x)}{-f h+e i}\right )}{3 d (f h-e i)} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(333\) vs. \(2(140)=280\).
Time = 0.94 (sec) , antiderivative size = 334, normalized size of antiderivative = 2.35
method | result | size |
parts | \(\frac {a^{2} \left (\frac {\ln \left (i x +h \right )}{e i -f h}-\frac {\ln \left (f x +e \right )}{e i -f h}\right )}{d}+\frac {b^{2} c \left (-\frac {\ln \left (c f x +c e \right )^{3}}{3 c \left (e i -f h \right )}+\frac {\ln \left (c f x +c e \right )^{2} \ln \left (1+\frac {i \left (c f x +c e \right )}{-c e i +h c f}\right )+2 \ln \left (c f x +c e \right ) \operatorname {Li}_{2}\left (-\frac {i \left (c f x +c e \right )}{-c e i +h c f}\right )-2 \,\operatorname {Li}_{3}\left (-\frac {i \left (c f x +c e \right )}{-c e i +h c f}\right )}{c \left (e i -f h \right )}\right )}{d}-\frac {a b \ln \left (c f x +c e \right )^{2}}{d \left (e i -f h \right )}+\frac {2 a b \operatorname {dilog}\left (\frac {-c e i +h c f +i \left (c f x +c e \right )}{-c e i +h c f}\right )}{d \left (e i -f h \right )}+\frac {2 a b \ln \left (c f x +c e \right ) \ln \left (\frac {-c e i +h c f +i \left (c f x +c e \right )}{-c e i +h c f}\right )}{d \left (e i -f h \right )}\) | \(334\) |
risch | \(\frac {a^{2} \ln \left (i x +h \right )}{d \left (e i -f h \right )}-\frac {a^{2} \ln \left (f x +e \right )}{d \left (e i -f h \right )}-\frac {b^{2} \ln \left (c f x +c e \right )^{3}}{3 d \left (e i -f h \right )}+\frac {b^{2} \ln \left (c f x +c e \right )^{2} \ln \left (1+\frac {i \left (c f x +c e \right )}{-c e i +h c f}\right )}{d \left (e i -f h \right )}+\frac {2 b^{2} \ln \left (c f x +c e \right ) \operatorname {Li}_{2}\left (-\frac {i \left (c f x +c e \right )}{-c e i +h c f}\right )}{d \left (e i -f h \right )}-\frac {2 b^{2} \operatorname {Li}_{3}\left (-\frac {i \left (c f x +c e \right )}{-c e i +h c f}\right )}{d \left (e i -f h \right )}-\frac {a b \ln \left (c f x +c e \right )^{2}}{d \left (e i -f h \right )}+\frac {2 a b \operatorname {dilog}\left (\frac {-c e i +h c f +i \left (c f x +c e \right )}{-c e i +h c f}\right )}{d \left (e i -f h \right )}+\frac {2 a b \ln \left (c f x +c e \right ) \ln \left (\frac {-c e i +h c f +i \left (c f x +c e \right )}{-c e i +h c f}\right )}{d \left (e i -f h \right )}\) | \(365\) |
derivativedivides | \(\frac {-\frac {c f \,a^{2} \ln \left (c f x +c e \right )}{d \left (e i -f h \right )}+\frac {c f \,a^{2} \ln \left (c e i -h c f -i \left (c f x +c e \right )\right )}{d \left (e i -f h \right )}-\frac {c^{2} f \,b^{2} \left (\frac {\ln \left (c f x +c e \right )^{3}}{3 c \left (e i -f h \right )}-\frac {\ln \left (c f x +c e \right )^{2} \ln \left (1+\frac {i \left (c f x +c e \right )}{-c e i +h c f}\right )+2 \ln \left (c f x +c e \right ) \operatorname {Li}_{2}\left (-\frac {i \left (c f x +c e \right )}{-c e i +h c f}\right )-2 \,\operatorname {Li}_{3}\left (-\frac {i \left (c f x +c e \right )}{-c e i +h c f}\right )}{c \left (e i -f h \right )}\right )}{d}-\frac {2 c^{2} f a b \left (\frac {\ln \left (c f x +c e \right )^{2}}{2 c \left (e i -f h \right )}-\frac {i \left (\frac {\operatorname {dilog}\left (\frac {-c e i +h c f +i \left (c f x +c e \right )}{-c e i +h c f}\right )}{i}+\frac {\ln \left (c f x +c e \right ) \ln \left (\frac {-c e i +h c f +i \left (c f x +c e \right )}{-c e i +h c f}\right )}{i}\right )}{c \left (e i -f h \right )}\right )}{d}}{c f}\) | \(374\) |
default | \(\frac {-\frac {c f \,a^{2} \ln \left (c f x +c e \right )}{d \left (e i -f h \right )}+\frac {c f \,a^{2} \ln \left (c e i -h c f -i \left (c f x +c e \right )\right )}{d \left (e i -f h \right )}-\frac {c^{2} f \,b^{2} \left (\frac {\ln \left (c f x +c e \right )^{3}}{3 c \left (e i -f h \right )}-\frac {\ln \left (c f x +c e \right )^{2} \ln \left (1+\frac {i \left (c f x +c e \right )}{-c e i +h c f}\right )+2 \ln \left (c f x +c e \right ) \operatorname {Li}_{2}\left (-\frac {i \left (c f x +c e \right )}{-c e i +h c f}\right )-2 \,\operatorname {Li}_{3}\left (-\frac {i \left (c f x +c e \right )}{-c e i +h c f}\right )}{c \left (e i -f h \right )}\right )}{d}-\frac {2 c^{2} f a b \left (\frac {\ln \left (c f x +c e \right )^{2}}{2 c \left (e i -f h \right )}-\frac {i \left (\frac {\operatorname {dilog}\left (\frac {-c e i +h c f +i \left (c f x +c e \right )}{-c e i +h c f}\right )}{i}+\frac {\ln \left (c f x +c e \right ) \ln \left (\frac {-c e i +h c f +i \left (c f x +c e \right )}{-c e i +h c f}\right )}{i}\right )}{c \left (e i -f h \right )}\right )}{d}}{c f}\) | \(374\) |
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\[ \int \frac {(a+b \log (c (e+f x)))^2}{(d e+d f x) (h+i x)} \, dx=\int { \frac {{\left (b \log \left ({\left (f x + e\right )} c\right ) + a\right )}^{2}}{{\left (d f x + d e\right )} {\left (i x + h\right )}} \,d x } \]
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\[ \int \frac {(a+b \log (c (e+f x)))^2}{(d e+d f x) (h+i x)} \, dx=\frac {\int \frac {a^{2}}{e h + e i x + f h x + f i x^{2}}\, dx + \int \frac {b^{2} \log {\left (c e + c f x \right )}^{2}}{e h + e i x + f h x + f i x^{2}}\, dx + \int \frac {2 a b \log {\left (c e + c f x \right )}}{e h + e i x + f h x + f i x^{2}}\, dx}{d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 331 vs. \(2 (141) = 282\).
Time = 0.27 (sec) , antiderivative size = 331, normalized size of antiderivative = 2.33 \[ \int \frac {(a+b \log (c (e+f x)))^2}{(d e+d f x) (h+i x)} \, dx=a^{2} {\left (\frac {\log \left (f x + e\right )}{d f h - d e i} - \frac {\log \left (i x + h\right )}{d f h - d e i}\right )} - \frac {{\left (\log \left (f x + e\right )^{2} \log \left (\frac {f i x + e i}{f h - e i} + 1\right ) + 2 \, {\rm Li}_2\left (-\frac {f i x + e i}{f h - e i}\right ) \log \left (f x + e\right ) - 2 \, {\rm Li}_{3}(-\frac {f i x + e i}{f h - e i})\right )} b^{2}}{{\left (f h - e i\right )} d} - \frac {2 \, {\left (b^{2} \log \left (c\right ) + a b\right )} {\left (\log \left (f x + e\right ) \log \left (\frac {f i x + e i}{f h - e i} + 1\right ) + {\rm Li}_2\left (-\frac {f i x + e i}{f h - e i}\right )\right )}}{{\left (f h - e i\right )} d} - \frac {{\left (b^{2} \log \left (c\right )^{2} + 2 \, a b \log \left (c\right )\right )} \log \left (i x + h\right )}{{\left (f h - e i\right )} d} + \frac {b^{2} \log \left (f x + e\right )^{3} + 3 \, {\left (b^{2} \log \left (c\right ) + a b\right )} \log \left (f x + e\right )^{2} + 3 \, {\left (b^{2} \log \left (c\right )^{2} + 2 \, a b \log \left (c\right )\right )} \log \left (f x + e\right )}{3 \, {\left (f h - e i\right )} d} \]
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\[ \int \frac {(a+b \log (c (e+f x)))^2}{(d e+d f x) (h+i x)} \, dx=\int { \frac {{\left (b \log \left ({\left (f x + e\right )} c\right ) + a\right )}^{2}}{{\left (d f x + d e\right )} {\left (i x + h\right )}} \,d x } \]
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Timed out. \[ \int \frac {(a+b \log (c (e+f x)))^2}{(d e+d f x) (h+i x)} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,\left (e+f\,x\right )\right )\right )}^2}{\left (h+i\,x\right )\,\left (d\,e+d\,f\,x\right )} \,d x \]
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