Optimal. Leaf size=433 \[ -\frac {\text {Li}_3\left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{b c-a d}-\frac {\text {Li}_2\left (\frac {b (c+d x)}{d (a+b x)}\right ) \log \left (\frac {(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right )}{b c-a d}+\frac {\text {Li}_2\left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \log \left (\frac {(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right )}{b c-a d}-\frac {\log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac {(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right )}{2 (b c-a d)}-\frac {\log \left (\frac {b (e+f x)}{b e-a f}\right ) \log ^2\left (\frac {(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right )}{2 (b c-a d)}+\frac {\log \left (1-\frac {(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right ) \log ^2\left (\frac {(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right )}{2 (b c-a d)}+\frac {\text {Li}_3\left (\frac {b (c+d x)}{d (a+b x)}\right )}{b c-a d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.59, antiderivative size = 445, normalized size of antiderivative = 1.03, number of steps used = 8, number of rules used = 6, integrand size = 65, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.092, Rules used = {2507, 2489, 2488, 2506, 6610, 2503} \[ -\frac {\text {PolyLog}\left (3,1-\frac {(e+f x) (b c-a d)}{(c+d x) (b e-a f)}\right )}{b c-a d}+\frac {\text {PolyLog}\left (2,1-\frac {b c-a d}{b (c+d x)}\right ) \log \left (\frac {(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right )}{b c-a d}-\frac {\log \left (\frac {(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right ) \text {PolyLog}\left (2,1-\frac {(e+f x) (b c-a d)}{(c+d x) (b e-a f)}\right )}{b c-a d}+\frac {\text {PolyLog}\left (3,1-\frac {b c-a d}{b (c+d x)}\right )}{b c-a d}-\frac {\log \left (\frac {b c-a d}{b (c+d x)}\right ) \log ^2\left (\frac {(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right )}{2 (b c-a d)}-\frac {\log \left (\frac {b (e+f x)}{b e-a f}\right ) \log ^2\left (\frac {(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right )}{2 (b c-a d)}+\frac {\log \left (\frac {(e+f x) (b c-a d)}{(c+d x) (b e-a f)}\right ) \log ^2\left (\frac {(c+d x) (b e-a f)}{(a+b x) (d e-c f)}\right )}{2 (b c-a d)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2488
Rule 2489
Rule 2503
Rule 2506
Rule 2507
Rule 6610
Rubi steps
\begin {align*} \int \frac {\log \left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \log \left (\frac {b (e+f x)}{b e-a f}\right )}{(a+b x) (c+d x)} \, dx &=-\frac {\log ^2\left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \log \left (\frac {b (e+f x)}{b e-a f}\right )}{2 (b c-a d)}+\frac {f \int \frac {\log ^2\left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{e+f x} \, dx}{2 (b c-a d)}\\ &=-\frac {\log ^2\left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \log \left (\frac {b (e+f x)}{b e-a f}\right )}{2 (b c-a d)}+\frac {d \int \frac {\log ^2\left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{c+d x} \, dx}{2 (b c-a d)}-\frac {(d e-c f) \int \frac {\log ^2\left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{(c+d x) (e+f x)} \, dx}{2 (b c-a d)}\\ &=-\frac {\log \left (\frac {b c-a d}{b (c+d x)}\right ) \log ^2\left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{2 (b c-a d)}-\frac {\log ^2\left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \log \left (\frac {b (e+f x)}{b e-a f}\right )}{2 (b c-a d)}+\frac {\log ^2\left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \log \left (\frac {(b c-a d) (e+f x)}{(b e-a f) (c+d x)}\right )}{2 (b c-a d)}-\int \frac {\log \left (-\frac {-b c+a d}{b (c+d x)}\right ) \log \left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{(a+b x) (c+d x)} \, dx+\int \frac {\log \left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \log \left (-\frac {(-b c+a d) (e+f x)}{(b e-a f) (c+d x)}\right )}{(a+b x) (c+d x)} \, dx\\ &=-\frac {\log \left (\frac {b c-a d}{b (c+d x)}\right ) \log ^2\left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{2 (b c-a d)}-\frac {\log ^2\left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \log \left (\frac {b (e+f x)}{b e-a f}\right )}{2 (b c-a d)}+\frac {\log ^2\left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \log \left (\frac {(b c-a d) (e+f x)}{(b e-a f) (c+d x)}\right )}{2 (b c-a d)}+\frac {\log \left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \text {Li}_2\left (1-\frac {b c-a d}{b (c+d x)}\right )}{b c-a d}-\frac {\log \left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \text {Li}_2\left (1-\frac {(b c-a d) (e+f x)}{(b e-a f) (c+d x)}\right )}{b c-a d}+\int \frac {\text {Li}_2\left (1+\frac {-b c+a d}{b (c+d x)}\right )}{(a+b x) (c+d x)} \, dx-\int \frac {\text {Li}_2\left (1+\frac {(-b c+a d) (e+f x)}{(b e-a f) (c+d x)}\right )}{(a+b x) (c+d x)} \, dx\\ &=-\frac {\log \left (\frac {b c-a d}{b (c+d x)}\right ) \log ^2\left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{2 (b c-a d)}-\frac {\log ^2\left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \log \left (\frac {b (e+f x)}{b e-a f}\right )}{2 (b c-a d)}+\frac {\log ^2\left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \log \left (\frac {(b c-a d) (e+f x)}{(b e-a f) (c+d x)}\right )}{2 (b c-a d)}+\frac {\log \left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \text {Li}_2\left (1-\frac {b c-a d}{b (c+d x)}\right )}{b c-a d}-\frac {\log \left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \text {Li}_2\left (1-\frac {(b c-a d) (e+f x)}{(b e-a f) (c+d x)}\right )}{b c-a d}+\frac {\text {Li}_3\left (1-\frac {b c-a d}{b (c+d x)}\right )}{b c-a d}-\frac {\text {Li}_3\left (1-\frac {(b c-a d) (e+f x)}{(b e-a f) (c+d x)}\right )}{b c-a d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 0.82, size = 908, normalized size = 2.10 \[ \frac {\log \left (\frac {b (e+f x)}{b e-a f}\right ) \log ^2(c+d x)-\log \left (\frac {d (e+f x)}{d e-c f}\right ) \log ^2(c+d x)-2 \log (a+b x) \log \left (\frac {b (e+f x)}{b e-a f}\right ) \log (c+d x)-2 \log \left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \log \left (\frac {b (e+f x)}{b e-a f}\right ) \log (c+d x)+2 \log (a+b x) \log \left (\frac {d (e+f x)}{d e-c f}\right ) \log (c+d x)+2 \log \left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \log \left (\frac {d (e+f x)}{d e-c f}\right ) \log (c+d x)-\log \left (\frac {a d-b c}{d (a+b x)}\right ) \log ^2\left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )-\log ^2\left (\frac {f (c+d x)}{c f-d e}\right ) \log \left (\frac {b (e+f x)}{b e-a f}\right )-\log ^2\left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \log \left (\frac {b (e+f x)}{b e-a f}\right )+2 \log (a+b x) \log \left (\frac {f (c+d x)}{c f-d e}\right ) \log \left (\frac {b (e+f x)}{b e-a f}\right )+2 \log \left (\frac {f (c+d x)}{c f-d e}\right ) \log \left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \log \left (\frac {b (e+f x)}{b e-a f}\right )+\log ^2\left (\frac {f (c+d x)}{c f-d e}\right ) \log \left (\frac {d (e+f x)}{d e-c f}\right )-2 \log (a+b x) \log \left (\frac {f (c+d x)}{c f-d e}\right ) \log \left (\frac {d (e+f x)}{d e-c f}\right )-2 \log \left (\frac {f (c+d x)}{c f-d e}\right ) \log \left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \log \left (\frac {d (e+f x)}{d e-c f}\right )+\log ^2\left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \log \left (\frac {(a d-b c) (e+f x)}{(d e-c f) (a+b x)}\right )-2 \log \left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \text {Li}_2\left (\frac {b (c+d x)}{d (a+b x)}\right )+2 \log \left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right ) \text {Li}_2\left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )+2 \text {Li}_3\left (\frac {b (c+d x)}{d (a+b x)}\right )-2 \text {Li}_3\left (\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}\right )}{2 b c-2 a d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.60, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\log \left (\frac {b c e - a c f + {\left (b d e - a d f\right )} x}{a d e - a c f + {\left (b d e - b c f\right )} x}\right ) \log \left (\frac {b f x + b e}{b e - a f}\right )}{b d x^{2} + a c + {\left (b c + a d\right )} x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log \left (\frac {{\left (f x + e\right )} b}{b e - a f}\right ) \log \left (\frac {{\left (b e - a f\right )} {\left (d x + c\right )}}{{\left (d e - c f\right )} {\left (b x + a\right )}}\right )}{{\left (b x + a\right )} {\left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 3.09, size = 0, normalized size = 0.00 \[ \int \frac {\ln \left (\frac {\left (f x +e \right ) b}{-a f +b e}\right ) \ln \left (\frac {\left (-a f +b e \right ) \left (d x +c \right )}{\left (-c f +d e \right ) \left (b x +a \right )}\right )}{\left (b x +a \right ) \left (d x +c \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\ln \left (-\frac {b\,\left (e+f\,x\right )}{a\,f-b\,e}\right )\,\ln \left (\frac {\left (a\,f-b\,e\right )\,\left (c+d\,x\right )}{\left (c\,f-d\,e\right )\,\left (a+b\,x\right )}\right )}{\left (a+b\,x\right )\,\left (c+d\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________