Integrand size = 34, antiderivative size = 130 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{(a g+b g x)^2} \, dx=-\frac {8 B^2 (c+d x)}{(b c-a d) g^2 (a+b x)}-\frac {4 B (c+d x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{(b c-a d) g^2 (a+b x)}-\frac {(c+d x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{(b c-a d) g^2 (a+b x)} \]
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Time = 0.06 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.088, Rules used = {2550, 2342, 2341} \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{(a g+b g x)^2} \, dx=-\frac {4 B (c+d x) \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{g^2 (a+b x) (b c-a d)}-\frac {(c+d x) \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )^2}{g^2 (a+b x) (b c-a d)}-\frac {8 B^2 (c+d x)}{g^2 (a+b x) (b c-a d)} \]
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Rule 2341
Rule 2342
Rule 2550
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (A+B \log \left (e x^2\right )\right )^2}{x^2} \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d) g^2} \\ & = -\frac {(c+d x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{(b c-a d) g^2 (a+b x)}+\frac {(4 B) \text {Subst}\left (\int \frac {A+B \log \left (e x^2\right )}{x^2} \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d) g^2} \\ & = -\frac {8 B^2 (c+d x)}{(b c-a d) g^2 (a+b x)}-\frac {4 B (c+d x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{(b c-a d) g^2 (a+b x)}-\frac {(c+d x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{(b c-a d) g^2 (a+b x)} \\ \end{align*}
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.27 (sec) , antiderivative size = 321, normalized size of antiderivative = 2.47 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{(a g+b g x)^2} \, dx=-\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2+\frac {4 B \left ((b c-a d) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )+d (a+b x) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )-d (a+b x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)+2 B (b c-a d+d (a+b x) \log (a+b x)-d (a+b x) \log (c+d x))-B d (a+b x) \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )\right )+B d (a+b x) \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )\right )}{b c-a d}}{b g^2 (a+b x)} \]
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Time = 0.94 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.45
method | result | size |
norman | \(\frac {\frac {\left (A^{2}+4 B A +8 B^{2}\right ) x}{g a}+\frac {B^{2} c \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )^{2}}{g \left (a d -c b \right )}+\frac {B^{2} d x \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )^{2}}{g \left (a d -c b \right )}+\frac {2 c B \left (A +2 B \right ) \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )}{g \left (a d -c b \right )}+\frac {2 B d \left (A +2 B \right ) x \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )}{g \left (a d -c b \right )}}{g \left (b x +a \right )}\) | \(188\) |
parallelrisch | \(-\frac {2 A^{2} a \,b^{2} d^{2}-2 A^{2} b^{3} c d -2 B^{2} x \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )^{2} b^{3} d^{2}-8 B^{2} x \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right ) b^{3} d^{2}+16 B^{2} a \,b^{2} d^{2}-16 B^{2} b^{3} c d -4 A B x \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right ) b^{3} d^{2}-4 A B \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right ) b^{3} c d +8 A B a \,b^{2} d^{2}-8 A B \,b^{3} c d -2 B^{2} \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )^{2} b^{3} c d -8 B^{2} \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right ) b^{3} c d}{2 g^{2} \left (b x +a \right ) b^{3} d \left (a d -c b \right )}\) | \(264\) |
risch | \(-\frac {A^{2}}{g^{2} \left (b x +a \right ) b}+\frac {\frac {8 B^{2} x}{a g}+\frac {B^{2} c \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )^{2}}{g \left (a d -c b \right )}+\frac {B^{2} d x \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )^{2}}{g \left (a d -c b \right )}+\frac {4 B^{2} c \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )}{g \left (a d -c b \right )}+\frac {4 B^{2} d x \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )}{g \left (a d -c b \right )}}{g \left (b x +a \right )}+\frac {\frac {4 B A x}{a g}+\frac {2 c A B \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )}{g \left (a d -c b \right )}+\frac {2 A B d x \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )}{g \left (a d -c b \right )}}{g \left (b x +a \right )}\) | \(286\) |
parts | \(-\frac {A^{2}}{g^{2} \left (b x +a \right ) b}+\frac {\frac {8 B^{2} x}{a g}+\frac {B^{2} c \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )^{2}}{g \left (a d -c b \right )}+\frac {B^{2} d x \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )^{2}}{g \left (a d -c b \right )}+\frac {4 B^{2} c \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )}{g \left (a d -c b \right )}+\frac {4 B^{2} d x \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )}{g \left (a d -c b \right )}}{g \left (b x +a \right )}+\frac {\frac {4 B A x}{a g}+\frac {2 c A B \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )}{g \left (a d -c b \right )}+\frac {2 A B d x \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )}{g \left (a d -c b \right )}}{g \left (b x +a \right )}\) | \(286\) |
derivativedivides | \(-\frac {-\frac {d^{2} A^{2}}{g^{2} \left (\frac {a d -c b}{d x +c}+b \right ) \left (a d -c b \right )}+\frac {\frac {8 d^{2} B^{2}}{b g \left (d x +c \right )}-\frac {4 d^{2} B^{2} \ln \left (\frac {e \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2}}{d^{2}}\right )}{g \left (a d -c b \right )}-\frac {d^{2} B^{2} \ln \left (\frac {e \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2}}{d^{2}}\right )^{2}}{g \left (a d -c b \right )}}{g \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )}+\frac {\frac {4 d^{2} A B}{b g \left (d x +c \right )}-\frac {2 d^{2} A B \ln \left (\frac {e \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2}}{d^{2}}\right )}{g \left (a d -c b \right )}}{g \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )}}{d}\) | \(306\) |
default | \(-\frac {-\frac {d^{2} A^{2}}{g^{2} \left (\frac {a d -c b}{d x +c}+b \right ) \left (a d -c b \right )}+\frac {\frac {8 d^{2} B^{2}}{b g \left (d x +c \right )}-\frac {4 d^{2} B^{2} \ln \left (\frac {e \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2}}{d^{2}}\right )}{g \left (a d -c b \right )}-\frac {d^{2} B^{2} \ln \left (\frac {e \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2}}{d^{2}}\right )^{2}}{g \left (a d -c b \right )}}{g \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )}+\frac {\frac {4 d^{2} A B}{b g \left (d x +c \right )}-\frac {2 d^{2} A B \ln \left (\frac {e \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2}}{d^{2}}\right )}{g \left (a d -c b \right )}}{g \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )}}{d}\) | \(306\) |
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Time = 0.27 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.54 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{(a g+b g x)^2} \, dx=-\frac {{\left (A^{2} + 4 \, A B + 8 \, B^{2}\right )} b c - {\left (A^{2} + 4 \, A B + 8 \, B^{2}\right )} a d + {\left (B^{2} b d x + B^{2} b c\right )} \log \left (\frac {b^{2} e x^{2} + 2 \, a b e x + a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )^{2} + 2 \, {\left ({\left (A B + 2 \, B^{2}\right )} b d x + {\left (A B + 2 \, B^{2}\right )} b c\right )} \log \left (\frac {b^{2} e x^{2} + 2 \, a b e x + a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{{\left (b^{3} c - a b^{2} d\right )} g^{2} x + {\left (a b^{2} c - a^{2} b d\right )} g^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 454 vs. \(2 (112) = 224\).
Time = 1.18 (sec) , antiderivative size = 454, normalized size of antiderivative = 3.49 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{(a g+b g x)^2} \, dx=- \frac {4 B d \left (A + 2 B\right ) \log {\left (x + \frac {4 A B a d^{2} + 4 A B b c d + 8 B^{2} a d^{2} + 8 B^{2} b c d - \frac {4 B a^{2} d^{3} \left (A + 2 B\right )}{a d - b c} + \frac {8 B a b c d^{2} \left (A + 2 B\right )}{a d - b c} - \frac {4 B b^{2} c^{2} d \left (A + 2 B\right )}{a d - b c}}{8 A B b d^{2} + 16 B^{2} b d^{2}} \right )}}{b g^{2} \left (a d - b c\right )} + \frac {4 B d \left (A + 2 B\right ) \log {\left (x + \frac {4 A B a d^{2} + 4 A B b c d + 8 B^{2} a d^{2} + 8 B^{2} b c d + \frac {4 B a^{2} d^{3} \left (A + 2 B\right )}{a d - b c} - \frac {8 B a b c d^{2} \left (A + 2 B\right )}{a d - b c} + \frac {4 B b^{2} c^{2} d \left (A + 2 B\right )}{a d - b c}}{8 A B b d^{2} + 16 B^{2} b d^{2}} \right )}}{b g^{2} \left (a d - b c\right )} + \frac {\left (- 2 A B - 4 B^{2}\right ) \log {\left (\frac {e \left (a + b x\right )^{2}}{\left (c + d x\right )^{2}} \right )}}{a b g^{2} + b^{2} g^{2} x} + \frac {\left (B^{2} c + B^{2} d x\right ) \log {\left (\frac {e \left (a + b x\right )^{2}}{\left (c + d x\right )^{2}} \right )}^{2}}{a^{2} d g^{2} - a b c g^{2} + a b d g^{2} x - b^{2} c g^{2} x} + \frac {- A^{2} - 4 A B - 8 B^{2}}{a b g^{2} + b^{2} g^{2} x} \]
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Leaf count of result is larger than twice the leaf count of optimal. 574 vs. \(2 (130) = 260\).
Time = 0.23 (sec) , antiderivative size = 574, normalized size of antiderivative = 4.42 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{(a g+b g x)^2} \, dx=-4 \, {\left ({\left (\frac {1}{b^{2} g^{2} x + a b g^{2}} + \frac {d \log \left (b x + a\right )}{{\left (b^{2} c - a b d\right )} g^{2}} - \frac {d \log \left (d x + c\right )}{{\left (b^{2} c - a b d\right )} g^{2}}\right )} \log \left (\frac {b^{2} e x^{2}}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {2 \, a b e x}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) - \frac {{\left (b d x + a d\right )} \log \left (b x + a\right )^{2} + {\left (b d x + a d\right )} \log \left (d x + c\right )^{2} - 2 \, b c + 2 \, a d - 2 \, {\left (b d x + a d\right )} \log \left (b x + a\right ) + 2 \, {\left (b d x + a d - {\left (b d x + a d\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )}{a b^{2} c g^{2} - a^{2} b d g^{2} + {\left (b^{3} c g^{2} - a b^{2} d g^{2}\right )} x}\right )} B^{2} - 2 \, A B {\left (\frac {\log \left (\frac {b^{2} e x^{2}}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {2 \, a b e x}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{b^{2} g^{2} x + a b g^{2}} + \frac {2}{b^{2} g^{2} x + a b g^{2}} + \frac {2 \, d \log \left (b x + a\right )}{{\left (b^{2} c - a b d\right )} g^{2}} - \frac {2 \, d \log \left (d x + c\right )}{{\left (b^{2} c - a b d\right )} g^{2}}\right )} - \frac {B^{2} \log \left (\frac {b^{2} e x^{2}}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {2 \, a b e x}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )^{2}}{b^{2} g^{2} x + a b g^{2}} - \frac {A^{2}}{b^{2} g^{2} x + a b g^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 380 vs. \(2 (130) = 260\).
Time = 0.72 (sec) , antiderivative size = 380, normalized size of antiderivative = 2.92 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{(a g+b g x)^2} \, dx=-{\left (\frac {B^{2} d}{b^{2} c g^{2} - a b d g^{2}} + \frac {B^{2}}{{\left (b g x + a g\right )} b g}\right )} \log \left (\frac {b^{2} e}{\frac {b^{2} c^{2} g^{2}}{{\left (b g x + a g\right )}^{2}} - \frac {2 \, a b c d g^{2}}{{\left (b g x + a g\right )}^{2}} + \frac {a^{2} d^{2} g^{2}}{{\left (b g x + a g\right )}^{2}} + \frac {2 \, b c d g}{b g x + a g} - \frac {2 \, a d^{2} g}{b g x + a g} + d^{2}}\right )^{2} + \frac {4 \, {\left (A B d + 2 \, B^{2} d\right )} \log \left (\frac {b c g}{b g x + a g} - \frac {a d g}{b g x + a g} + d\right )}{b^{2} c g^{2} - a b d g^{2}} - \frac {2 \, {\left (A B + 2 \, B^{2}\right )} \log \left (\frac {b^{2} e}{\frac {b^{2} c^{2} g^{2}}{{\left (b g x + a g\right )}^{2}} - \frac {2 \, a b c d g^{2}}{{\left (b g x + a g\right )}^{2}} + \frac {a^{2} d^{2} g^{2}}{{\left (b g x + a g\right )}^{2}} + \frac {2 \, b c d g}{b g x + a g} - \frac {2 \, a d^{2} g}{b g x + a g} + d^{2}}\right )}{{\left (b g x + a g\right )} b g} - \frac {A^{2} + 4 \, A B + 8 \, B^{2}}{{\left (b g x + a g\right )} b g} \]
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Time = 2.82 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.75 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{(a g+b g x)^2} \, dx=-{\ln \left (\frac {e\,{\left (a+b\,x\right )}^2}{{\left (c+d\,x\right )}^2}\right )}^2\,\left (\frac {B^2}{b^2\,g^2\,\left (x+\frac {a}{b}\right )}-\frac {B^2\,d}{b\,g^2\,\left (a\,d-b\,c\right )}\right )-\frac {A^2+4\,A\,B+8\,B^2}{x\,b^2\,g^2+a\,b\,g^2}-\frac {\ln \left (\frac {e\,{\left (a+b\,x\right )}^2}{{\left (c+d\,x\right )}^2}\right )\,\left (\frac {4\,B^2}{b^2\,d\,g^2}+\frac {2\,A\,B}{b^2\,d\,g^2}\right )}{\frac {x}{d}+\frac {a}{b\,d}}-\frac {B\,d\,\mathrm {atan}\left (\frac {\left (2\,b\,d\,x+\frac {c\,b^2\,g^2+a\,d\,b\,g^2}{b\,g^2}\right )\,1{}\mathrm {i}}{a\,d-b\,c}\right )\,\left (A+2\,B\right )\,8{}\mathrm {i}}{b\,g^2\,\left (a\,d-b\,c\right )} \]
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