Integrand size = 43, antiderivative size = 273 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^2 (c i+d i x)^2} \, dx=-\frac {B d^2 n (a+b x)}{(b c-a d)^3 g^2 i^2 (c+d x)}-\frac {b^2 B n (c+d x)}{(b c-a d)^3 g^2 i^2 (a+b x)}+\frac {d^2 (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^3 g^2 i^2 (c+d x)}-\frac {b^2 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^3 g^2 i^2 (a+b x)}-\frac {2 b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (\frac {a+b x}{c+d x}\right )}{(b c-a d)^3 g^2 i^2}+\frac {b B d n \log ^2\left (\frac {a+b x}{c+d x}\right )}{(b c-a d)^3 g^2 i^2} \]
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Time = 0.14 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.093, Rules used = {2561, 45, 2372, 2338} \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^2 (c i+d i x)^2} \, dx=-\frac {b^2 (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{g^2 i^2 (a+b x) (b c-a d)^3}+\frac {d^2 (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{g^2 i^2 (c+d x) (b c-a d)^3}-\frac {2 b d \log \left (\frac {a+b x}{c+d x}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{g^2 i^2 (b c-a d)^3}-\frac {b^2 B n (c+d x)}{g^2 i^2 (a+b x) (b c-a d)^3}-\frac {B d^2 n (a+b x)}{g^2 i^2 (c+d x) (b c-a d)^3}+\frac {b B d n \log ^2\left (\frac {a+b x}{c+d x}\right )}{g^2 i^2 (b c-a d)^3} \]
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Rule 45
Rule 2338
Rule 2372
Rule 2561
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {(b-d x)^2 \left (A+B \log \left (e x^n\right )\right )}{x^2} \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d)^3 g^2 i^2} \\ & = \frac {d^2 (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^3 g^2 i^2 (c+d x)}-\frac {b^2 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^3 g^2 i^2 (a+b x)}-\frac {2 b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (\frac {a+b x}{c+d x}\right )}{(b c-a d)^3 g^2 i^2}-\frac {(B n) \text {Subst}\left (\int \left (d^2-\frac {b^2}{x^2}-\frac {2 b d \log (x)}{x}\right ) \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d)^3 g^2 i^2} \\ & = -\frac {B d^2 n (a+b x)}{(b c-a d)^3 g^2 i^2 (c+d x)}-\frac {b^2 B n (c+d x)}{(b c-a d)^3 g^2 i^2 (a+b x)}+\frac {d^2 (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^3 g^2 i^2 (c+d x)}-\frac {b^2 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^3 g^2 i^2 (a+b x)}-\frac {2 b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (\frac {a+b x}{c+d x}\right )}{(b c-a d)^3 g^2 i^2}+\frac {(2 b B d n) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d)^3 g^2 i^2} \\ & = -\frac {B d^2 n (a+b x)}{(b c-a d)^3 g^2 i^2 (c+d x)}-\frac {b^2 B n (c+d x)}{(b c-a d)^3 g^2 i^2 (a+b x)}+\frac {d^2 (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^3 g^2 i^2 (c+d x)}-\frac {b^2 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^3 g^2 i^2 (a+b x)}-\frac {2 b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (\frac {a+b x}{c+d x}\right )}{(b c-a d)^3 g^2 i^2}+\frac {b B d n \log ^2\left (\frac {a+b x}{c+d x}\right )}{(b c-a d)^3 g^2 i^2} \\ \end{align*}
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.25 (sec) , antiderivative size = 342, normalized size of antiderivative = 1.25 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^2 (c i+d i x)^2} \, dx=\frac {-\frac {b^2 B c n}{a+b x}+\frac {a b B d n}{a+b x}+\frac {b B c d n}{c+d x}-\frac {a B d^2 n}{c+d x}-\frac {b (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{a+b x}+\frac {d (-b c+a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{c+d x}-2 b d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+2 b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)+b B d n \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )\right )-b B d n \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )}{(b c-a d)^3 g^2 i^2} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(665\) vs. \(2(273)=546\).
Time = 9.42 (sec) , antiderivative size = 666, normalized size of antiderivative = 2.44
method | result | size |
parallelrisch | \(\frac {-2 B x \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a^{4} b \,c^{3} d^{2} n +2 B x \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a^{3} b^{2} c^{4} d n +B \,x^{2} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )^{2} a^{3} b^{2} c^{3} d^{2}-B \,x^{2} a^{4} b \,c^{2} d^{3} n^{2}+2 B \,x^{2} a^{3} b^{2} c^{3} d^{2} n^{2}-B \,x^{2} a^{2} b^{3} c^{4} d \,n^{2}+2 A \,x^{2} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a^{3} b^{2} c^{3} d^{2}+A \,x^{2} a^{4} b \,c^{2} d^{3} n -A \,x^{2} a^{2} b^{3} c^{4} d n +B x \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )^{2} a^{4} b \,c^{3} d^{2}+B x \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )^{2} a^{3} b^{2} c^{4} d +B x \,a^{4} b \,c^{3} d^{2} n^{2}+B x \,a^{3} b^{2} c^{4} d \,n^{2}+2 A x \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a^{4} b \,c^{3} d^{2}+2 A x \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a^{3} b^{2} c^{4} d -A x \,a^{4} b \,c^{3} d^{2} n +A x \,a^{3} b^{2} c^{4} d n -B x \,a^{5} c^{2} d^{3} n^{2}-B x \,a^{2} b^{3} c^{5} n^{2}+A x \,a^{5} c^{2} d^{3} n -A x \,a^{2} b^{3} c^{5} n +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )^{2} a^{4} b \,c^{4} d -B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a^{5} c^{3} d^{2} n +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a^{3} b^{2} c^{5} n +2 A \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a^{4} b \,c^{4} d}{i^{2} g^{2} \left (d x +c \right ) \left (b x +a \right ) \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (a d -c b \right ) c^{3} a^{3} n}\) | \(666\) |
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Time = 0.34 (sec) , antiderivative size = 450, normalized size of antiderivative = 1.65 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^2 (c i+d i x)^2} \, dx=-\frac {A b^{2} c^{2} - A a^{2} d^{2} + {\left (B b^{2} d^{2} n x^{2} + B a b c d n + {\left (B b^{2} c d + B a b d^{2}\right )} n x\right )} \log \left (\frac {b x + a}{d x + c}\right )^{2} + {\left (B b^{2} c^{2} - 2 \, B a b c d + B a^{2} d^{2}\right )} n + 2 \, {\left (A b^{2} c d - A a b d^{2}\right )} x + {\left (B b^{2} c^{2} - B a^{2} d^{2} + 2 \, {\left (B b^{2} c d - B a b d^{2}\right )} x + 2 \, {\left (B b^{2} d^{2} x^{2} + B a b c d + {\left (B b^{2} c d + B a b d^{2}\right )} x\right )} \log \left (\frac {b x + a}{d x + c}\right )\right )} \log \left (e\right ) + {\left (2 \, A b^{2} d^{2} x^{2} + 2 \, A a b c d + {\left (B b^{2} c^{2} - B a^{2} d^{2}\right )} n + 2 \, {\left (A b^{2} c d + A a b d^{2} + {\left (B b^{2} c d - B a b d^{2}\right )} n\right )} x\right )} \log \left (\frac {b x + a}{d x + c}\right )}{{\left (b^{4} c^{3} d - 3 \, a b^{3} c^{2} d^{2} + 3 \, a^{2} b^{2} c d^{3} - a^{3} b d^{4}\right )} g^{2} i^{2} x^{2} + {\left (b^{4} c^{4} - 2 \, a b^{3} c^{3} d + 2 \, a^{3} b c d^{3} - a^{4} d^{4}\right )} g^{2} i^{2} x + {\left (a b^{3} c^{4} - 3 \, a^{2} b^{2} c^{3} d + 3 \, a^{3} b c^{2} d^{2} - a^{4} c d^{3}\right )} g^{2} i^{2}} \]
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Timed out. \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^2 (c i+d i x)^2} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 862 vs. \(2 (273) = 546\).
Time = 0.22 (sec) , antiderivative size = 862, normalized size of antiderivative = 3.16 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^2 (c i+d i x)^2} \, dx=-B {\left (\frac {2 \, b d x + b c + a d}{{\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} g^{2} i^{2} x^{2} + {\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} g^{2} i^{2} x + {\left (a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2}\right )} g^{2} i^{2}} + \frac {2 \, b d \log \left (b x + a\right )}{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} g^{2} i^{2}} - \frac {2 \, b d \log \left (d x + c\right )}{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} g^{2} i^{2}}\right )} \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right ) - \frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2} - {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )} \log \left (b x + a\right )^{2} + 2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )} \log \left (b x + a\right ) \log \left (d x + c\right ) - {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )} \log \left (d x + c\right )^{2}\right )} B n}{a b^{3} c^{4} g^{2} i^{2} - 3 \, a^{2} b^{2} c^{3} d g^{2} i^{2} + 3 \, a^{3} b c^{2} d^{2} g^{2} i^{2} - a^{4} c d^{3} g^{2} i^{2} + {\left (b^{4} c^{3} d g^{2} i^{2} - 3 \, a b^{3} c^{2} d^{2} g^{2} i^{2} + 3 \, a^{2} b^{2} c d^{3} g^{2} i^{2} - a^{3} b d^{4} g^{2} i^{2}\right )} x^{2} + {\left (b^{4} c^{4} g^{2} i^{2} - 2 \, a b^{3} c^{3} d g^{2} i^{2} + 2 \, a^{3} b c d^{3} g^{2} i^{2} - a^{4} d^{4} g^{2} i^{2}\right )} x} - A {\left (\frac {2 \, b d x + b c + a d}{{\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} g^{2} i^{2} x^{2} + {\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} g^{2} i^{2} x + {\left (a b^{2} c^{3} - 2 \, a^{2} b c^{2} d + a^{3} c d^{2}\right )} g^{2} i^{2}} + \frac {2 \, b d \log \left (b x + a\right )}{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} g^{2} i^{2}} - \frac {2 \, b d \log \left (d x + c\right )}{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} g^{2} i^{2}}\right )} \]
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Time = 106.59 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.35 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^2 (c i+d i x)^2} \, dx=-{\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )}^{2} {\left (\frac {{\left (d x + c\right )} B n \log \left (\frac {b x + a}{d x + c}\right )}{{\left (b x + a\right )} g^{2} i^{2}} + \frac {{\left (B n + B \log \left (e\right ) + A\right )} {\left (d x + c\right )}}{{\left (b x + a\right )} g^{2} i^{2}}\right )} \]
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Time = 2.06 (sec) , antiderivative size = 432, normalized size of antiderivative = 1.58 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^2 (c i+d i x)^2} \, dx=\frac {B\,b\,d\,{\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}^2}{g^2\,i^2\,n\,{\left (a\,d-b\,c\right )}^3}-\frac {A\,b\,c}{g^2\,i^2\,{\left (a\,d-b\,c\right )}^2\,\left (a+b\,x\right )\,\left (c+d\,x\right )}-\frac {A\,a\,d}{g^2\,i^2\,{\left (a\,d-b\,c\right )}^2\,\left (a+b\,x\right )\,\left (c+d\,x\right )}+\frac {B\,a\,d\,n}{g^2\,i^2\,{\left (a\,d-b\,c\right )}^2\,\left (a+b\,x\right )\,\left (c+d\,x\right )}-\frac {B\,b\,c\,n}{g^2\,i^2\,{\left (a\,d-b\,c\right )}^2\,\left (a+b\,x\right )\,\left (c+d\,x\right )}-\frac {2\,A\,b\,d\,x}{g^2\,i^2\,{\left (a\,d-b\,c\right )}^2\,\left (a+b\,x\right )\,\left (c+d\,x\right )}-\frac {B\,a\,d\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}{g^2\,i^2\,{\left (a\,d-b\,c\right )}^2\,\left (a+b\,x\right )\,\left (c+d\,x\right )}-\frac {B\,b\,c\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}{g^2\,i^2\,{\left (a\,d-b\,c\right )}^2\,\left (a+b\,x\right )\,\left (c+d\,x\right )}-\frac {2\,B\,b\,d\,x\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}{g^2\,i^2\,{\left (a\,d-b\,c\right )}^2\,\left (a+b\,x\right )\,\left (c+d\,x\right )}-\frac {A\,b\,d\,\mathrm {atan}\left (\frac {a\,d\,1{}\mathrm {i}+b\,c\,1{}\mathrm {i}+b\,d\,x\,2{}\mathrm {i}}{a\,d-b\,c}\right )\,4{}\mathrm {i}}{g^2\,i^2\,{\left (a\,d-b\,c\right )}^3} \]
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