Optimal. Leaf size=448 \[ \frac {b^2 c}{6 e (b d-a e) (b c d+e-a c e) (d+e x)}+\frac {b^3 c^2 \log (1-a c-b c x)}{6 e (b d-a e) (b c d+e-a c e)^2}+\frac {b^3 c \log (1-a c-b c x)}{3 e (b d-a e)^2 (b c d+e-a c e)}-\frac {b \log (1-a c-b c x)}{6 e (b d-a e) (d+e x)^2}-\frac {b^2 \log (1-a c-b c x)}{3 e (b d-a e)^2 (d+e x)}-\frac {b^3 c^2 \log (d+e x)}{6 e (b d-a e) (b c d+e-a c e)^2}-\frac {b^3 c \log (d+e x)}{3 e (b d-a e)^2 (b c d+e-a c e)}+\frac {b^3 \log (1-a c-b c x) \log \left (\frac {b c (d+e x)}{b c d+e-a c e}\right )}{3 e (b d-a e)^3}+\frac {b^3 \text {PolyLog}(2,c (a+b x))}{3 e (b d-a e)^3}-\frac {\text {PolyLog}(2,c (a+b x))}{3 e (d+e x)^3}+\frac {b^3 \text {PolyLog}\left (2,\frac {e (1-a c-b c x)}{b c d+e-a c e}\right )}{3 e (b d-a e)^3} \]
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Rubi [A]
time = 0.31, antiderivative size = 448, normalized size of antiderivative = 1.00, number of steps
used = 15, number of rules used = 9, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.529, Rules used = {6733, 2465,
2440, 2438, 2442, 46, 36, 31, 2441} \begin {gather*} \frac {b^3 c^2 \log (-a c-b c x+1)}{6 e (b d-a e) (-a c e+b c d+e)^2}-\frac {b^3 c^2 \log (d+e x)}{6 e (b d-a e) (-a c e+b c d+e)^2}+\frac {b^3 \text {Li}_2(c (a+b x))}{3 e (b d-a e)^3}+\frac {b^3 \text {Li}_2\left (\frac {e (-a c-b x c+1)}{b c d-a c e+e}\right )}{3 e (b d-a e)^3}+\frac {b^3 c \log (-a c-b c x+1)}{3 e (b d-a e)^2 (-a c e+b c d+e)}-\frac {b^3 c \log (d+e x)}{3 e (b d-a e)^2 (-a c e+b c d+e)}+\frac {b^3 \log (-a c-b c x+1) \log \left (\frac {b c (d+e x)}{-a c e+b c d+e}\right )}{3 e (b d-a e)^3}+\frac {b^2 c}{6 e (d+e x) (b d-a e) (-a c e+b c d+e)}-\frac {b^2 \log (-a c-b c x+1)}{3 e (d+e x) (b d-a e)^2}-\frac {\text {Li}_2(c (a+b x))}{3 e (d+e x)^3}-\frac {b \log (-a c-b c x+1)}{6 e (d+e x)^2 (b d-a e)} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 36
Rule 46
Rule 2438
Rule 2440
Rule 2441
Rule 2442
Rule 2465
Rule 6733
Rubi steps
\begin {align*} \int \frac {\text {Li}_2(c (a+b x))}{(d+e x)^4} \, dx &=-\frac {\text {Li}_2(c (a+b x))}{3 e (d+e x)^3}-\frac {b \int \frac {\log (1-a c-b c x)}{(a+b x) (d+e x)^3} \, dx}{3 e}\\ &=-\frac {\text {Li}_2(c (a+b x))}{3 e (d+e x)^3}-\frac {b \int \left (\frac {b^3 \log (1-a c-b c x)}{(b d-a e)^3 (a+b x)}-\frac {e \log (1-a c-b c x)}{(b d-a e) (d+e x)^3}-\frac {b e \log (1-a c-b c x)}{(b d-a e)^2 (d+e x)^2}-\frac {b^2 e \log (1-a c-b c x)}{(b d-a e)^3 (d+e x)}\right ) \, dx}{3 e}\\ &=-\frac {\text {Li}_2(c (a+b x))}{3 e (d+e x)^3}+\frac {b^3 \int \frac {\log (1-a c-b c x)}{d+e x} \, dx}{3 (b d-a e)^3}-\frac {b^4 \int \frac {\log (1-a c-b c x)}{a+b x} \, dx}{3 e (b d-a e)^3}+\frac {b^2 \int \frac {\log (1-a c-b c x)}{(d+e x)^2} \, dx}{3 (b d-a e)^2}+\frac {b \int \frac {\log (1-a c-b c x)}{(d+e x)^3} \, dx}{3 (b d-a e)}\\ &=-\frac {b \log (1-a c-b c x)}{6 e (b d-a e) (d+e x)^2}-\frac {b^2 \log (1-a c-b c x)}{3 e (b d-a e)^2 (d+e x)}+\frac {b^3 \log (1-a c-b c x) \log \left (\frac {b c (d+e x)}{b c d+e-a c e}\right )}{3 e (b d-a e)^3}-\frac {\text {Li}_2(c (a+b x))}{3 e (d+e x)^3}-\frac {b^3 \text {Subst}\left (\int \frac {\log (1-c x)}{x} \, dx,x,a+b x\right )}{3 e (b d-a e)^3}+\frac {\left (b^4 c\right ) \int \frac {\log \left (-\frac {b c (d+e x)}{-b c d-(1-a c) e}\right )}{1-a c-b c x} \, dx}{3 e (b d-a e)^3}-\frac {\left (b^3 c\right ) \int \frac {1}{(1-a c-b c x) (d+e x)} \, dx}{3 e (b d-a e)^2}-\frac {\left (b^2 c\right ) \int \frac {1}{(1-a c-b c x) (d+e x)^2} \, dx}{6 e (b d-a e)}\\ &=-\frac {b \log (1-a c-b c x)}{6 e (b d-a e) (d+e x)^2}-\frac {b^2 \log (1-a c-b c x)}{3 e (b d-a e)^2 (d+e x)}+\frac {b^3 \log (1-a c-b c x) \log \left (\frac {b c (d+e x)}{b c d+e-a c e}\right )}{3 e (b d-a e)^3}+\frac {b^3 \text {Li}_2(c (a+b x))}{3 e (b d-a e)^3}-\frac {\text {Li}_2(c (a+b x))}{3 e (d+e x)^3}-\frac {b^3 \text {Subst}\left (\int \frac {\log \left (1+\frac {e x}{-b c d-(1-a c) e}\right )}{x} \, dx,x,1-a c-b c x\right )}{3 e (b d-a e)^3}-\frac {\left (b^2 c\right ) \int \left (-\frac {b^2 c^2}{(b c d+e-a c e)^2 (-1+a c+b c x)}+\frac {e}{(b c d+(1-a c) e) (d+e x)^2}+\frac {b c e}{(b c d+(1-a c) e)^2 (d+e x)}\right ) \, dx}{6 e (b d-a e)}-\frac {\left (b^3 c\right ) \int \frac {1}{d+e x} \, dx}{3 (b d-a e)^2 (b c d+e-a c e)}-\frac {\left (b^4 c^2\right ) \int \frac {1}{1-a c-b c x} \, dx}{3 e (b d-a e)^2 (b c d+e-a c e)}\\ &=\frac {b^2 c}{6 e (b d-a e) (b c d+e-a c e) (d+e x)}+\frac {b^3 c^2 \log (1-a c-b c x)}{6 e (b d-a e) (b c d+e-a c e)^2}+\frac {b^3 c \log (1-a c-b c x)}{3 e (b d-a e)^2 (b c d+e-a c e)}-\frac {b \log (1-a c-b c x)}{6 e (b d-a e) (d+e x)^2}-\frac {b^2 \log (1-a c-b c x)}{3 e (b d-a e)^2 (d+e x)}-\frac {b^3 c^2 \log (d+e x)}{6 e (b d-a e) (b c d+e-a c e)^2}-\frac {b^3 c \log (d+e x)}{3 e (b d-a e)^2 (b c d+e-a c e)}+\frac {b^3 \log (1-a c-b c x) \log \left (\frac {b c (d+e x)}{b c d+e-a c e}\right )}{3 e (b d-a e)^3}+\frac {b^3 \text {Li}_2(c (a+b x))}{3 e (b d-a e)^3}-\frac {\text {Li}_2(c (a+b x))}{3 e (d+e x)^3}+\frac {b^3 \text {Li}_2\left (\frac {e (1-a c-b c x)}{b c d+e-a c e}\right )}{3 e (b d-a e)^3}\\ \end {align*}
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Mathematica [A]
time = 0.39, size = 313, normalized size = 0.70 \begin {gather*} \frac {-\frac {2 \text {PolyLog}(2,c (a+b x))}{(d+e x)^3}+\frac {b \left (-\frac {(b d-a e)^2 \log (1-a c-b c x)}{(d+e x)^2}-\frac {2 b (b d-a e) \log (1-a c-b c x)}{d+e x}+\frac {2 b^2 c (b d-a e) (\log (1-a c-b c x)-\log (d+e x))}{b c d+e-a c e}+\frac {b c (b d-a e)^2 (b c d+e-a c e+b c (d+e x) \log (1-a c-b c x)-b c (d+e x) \log (d+e x))}{(b c d+e-a c e)^2 (d+e x)}+2 b^2 \log (1-a c-b c x) \log \left (\frac {b c (d+e x)}{b c d+e-a c e}\right )+2 b^2 \text {PolyLog}(2,c (a+b x))+2 b^2 \text {PolyLog}\left (2,\frac {e (-1+a c+b c x)}{-b c d+(-1+a c) e}\right )\right )}{(b d-a e)^3}}{6 e} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.14, size = 648, normalized size = 1.45
method | result | size |
derivativedivides | \(\frac {\frac {c^{4} b^{4} \polylog \left (2, x b c +a c \right )}{3 \left (a e c -b c d -e \left (x b c +a c \right )\right )^{3} e}+\frac {c^{4} b^{4} \left (-\frac {\left (\frac {a c}{2 \left (a e c -b c d -e \right )^{2} \left (a e c -b c d +e \left (-x b c -a c +1\right )-e \right )}-\frac {d c b}{2 \left (a e c -b c d -e \right )^{2} e \left (a e c -b c d +e \left (-x b c -a c +1\right )-e \right )}-\frac {1}{2 \left (a e c -b c d -e \right )^{2} \left (a e c -b c d +e \left (-x b c -a c +1\right )-e \right )}-\frac {\ln \left (a e c -b c d +e \left (-x b c -a c +1\right )-e \right )}{2 \left (a e c -b c d -e \right )^{2} e}+\frac {\ln \left (-x b c -a c +1\right ) \left (2 a e c -2 b c d +e \left (-x b c -a c +1\right )-2 e \right ) \left (-x b c -a c +1\right )}{2 \left (a e c -b c d +e \left (-x b c -a c +1\right )-e \right )^{2} \left (a e c -b c d -e \right )^{2}}\right ) e}{c \left (a e -b d \right )}-\frac {\left (-\frac {\ln \left (a e c -b c d +e \left (-x b c -a c +1\right )-e \right )}{\left (a e c -b c d -e \right ) e}+\frac {\ln \left (-x b c -a c +1\right ) \left (-x b c -a c +1\right )}{\left (a e c -b c d -e \right ) \left (a e c -b c d +e \left (-x b c -a c +1\right )-e \right )}\right ) e}{c^{2} \left (a e -b d \right )^{2}}-\frac {\left (\frac {\dilog \left (\frac {a e c -b c d +e \left (-x b c -a c +1\right )-e}{a e c -b c d -e}\right )}{e}+\frac {\ln \left (-x b c -a c +1\right ) \ln \left (\frac {a e c -b c d +e \left (-x b c -a c +1\right )-e}{a e c -b c d -e}\right )}{e}\right ) e}{c^{3} \left (a e -b d \right )^{3}}-\frac {\dilog \left (-x b c -a c +1\right )}{c^{3} \left (a e -b d \right )^{3}}\right )}{3 e}}{b c}\) | \(648\) |
default | \(\frac {\frac {c^{4} b^{4} \polylog \left (2, x b c +a c \right )}{3 \left (a e c -b c d -e \left (x b c +a c \right )\right )^{3} e}+\frac {c^{4} b^{4} \left (-\frac {\left (\frac {a c}{2 \left (a e c -b c d -e \right )^{2} \left (a e c -b c d +e \left (-x b c -a c +1\right )-e \right )}-\frac {d c b}{2 \left (a e c -b c d -e \right )^{2} e \left (a e c -b c d +e \left (-x b c -a c +1\right )-e \right )}-\frac {1}{2 \left (a e c -b c d -e \right )^{2} \left (a e c -b c d +e \left (-x b c -a c +1\right )-e \right )}-\frac {\ln \left (a e c -b c d +e \left (-x b c -a c +1\right )-e \right )}{2 \left (a e c -b c d -e \right )^{2} e}+\frac {\ln \left (-x b c -a c +1\right ) \left (2 a e c -2 b c d +e \left (-x b c -a c +1\right )-2 e \right ) \left (-x b c -a c +1\right )}{2 \left (a e c -b c d +e \left (-x b c -a c +1\right )-e \right )^{2} \left (a e c -b c d -e \right )^{2}}\right ) e}{c \left (a e -b d \right )}-\frac {\left (-\frac {\ln \left (a e c -b c d +e \left (-x b c -a c +1\right )-e \right )}{\left (a e c -b c d -e \right ) e}+\frac {\ln \left (-x b c -a c +1\right ) \left (-x b c -a c +1\right )}{\left (a e c -b c d -e \right ) \left (a e c -b c d +e \left (-x b c -a c +1\right )-e \right )}\right ) e}{c^{2} \left (a e -b d \right )^{2}}-\frac {\left (\frac {\dilog \left (\frac {a e c -b c d +e \left (-x b c -a c +1\right )-e}{a e c -b c d -e}\right )}{e}+\frac {\ln \left (-x b c -a c +1\right ) \ln \left (\frac {a e c -b c d +e \left (-x b c -a c +1\right )-e}{a e c -b c d -e}\right )}{e}\right ) e}{c^{3} \left (a e -b d \right )^{3}}-\frac {\dilog \left (-x b c -a c +1\right )}{c^{3} \left (a e -b d \right )^{3}}\right )}{3 e}}{b c}\) | \(648\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1519 vs.
\(2 (443) = 886\).
time = 0.33, size = 1519, normalized size = 3.39 \begin {gather*} -\frac {{\left (\log \left (b c x + a c\right ) \log \left (-b c x - a c + 1\right ) + {\rm Li}_2\left (-b c x - a c + 1\right )\right )} b^{3}}{3 \, {\left (b^{3} d^{3} e - 3 \, a b^{2} d^{2} e^{2} + 3 \, a^{2} b d e^{3} - a^{3} e^{4}\right )}} + \frac {{\left (\log \left (-b c x - a c + 1\right ) \log \left (\frac {b c x e + a c e - e}{b c d - a c e + e} + 1\right ) + {\rm Li}_2\left (-\frac {b c x e + a c e - e}{b c d - a c e + e}\right )\right )} b^{3}}{3 \, {\left (b^{3} d^{3} e - 3 \, a b^{2} d^{2} e^{2} + 3 \, a^{2} b d e^{3} - a^{3} e^{4}\right )}} - \frac {{\left (3 \, b^{4} c^{2} d - 3 \, a b^{3} c^{2} e + 2 \, b^{3} c e\right )} \log \left (x e + d\right )}{6 \, {\left (b^{4} c^{2} d^{4} e + a^{4} c^{2} e^{5} - 2 \, a^{3} c e^{5} - 2 \, {\left (2 \, a b^{3} c^{2} e^{2} - b^{3} c e^{2}\right )} d^{3} + {\left (6 \, a^{2} b^{2} c^{2} e^{3} - 6 \, a b^{2} c e^{3} + b^{2} e^{3}\right )} d^{2} + a^{2} e^{5} - 2 \, {\left (2 \, a^{3} b c^{2} e^{4} - 3 \, a^{2} b c e^{4} + a b e^{4}\right )} d\right )}} + \frac {b^{4} c^{2} d^{4} - {\left (2 \, a b^{3} c^{2} e - b^{3} c e\right )} d^{3} + {\left (a^{2} b^{2} c^{2} e^{2} - a b^{2} c e^{2}\right )} d^{2} + {\left (b^{4} c^{2} d^{2} e^{2} + a^{2} b^{2} c^{2} e^{4} - a b^{2} c e^{4} - {\left (2 \, a b^{3} c^{2} e^{3} - b^{3} c e^{3}\right )} d\right )} x^{2} + 2 \, {\left (b^{4} c^{2} d^{3} e - {\left (2 \, a b^{3} c^{2} e^{2} - b^{3} c e^{2}\right )} d^{2} + {\left (a^{2} b^{2} c^{2} e^{3} - a b^{2} c e^{3}\right )} d\right )} x - 2 \, {\left (b^{4} c^{2} d^{4} + a^{4} c^{2} e^{4} - 2 \, a^{3} c e^{4} - 2 \, {\left (2 \, a b^{3} c^{2} e - b^{3} c e\right )} d^{3} + {\left (6 \, a^{2} b^{2} c^{2} e^{2} - 6 \, a b^{2} c e^{2} + b^{2} e^{2}\right )} d^{2} + a^{2} e^{4} - 2 \, {\left (2 \, a^{3} b c^{2} e^{3} - 3 \, a^{2} b c e^{3} + a b e^{3}\right )} d\right )} {\rm Li}_2\left (b c x + a c\right ) + {\left (4 \, {\left (a b^{3} c^{2} e - b^{3} c e\right )} d^{3} + {\left (3 \, b^{4} c^{2} d e^{3} - 3 \, a b^{3} c^{2} e^{4} + 2 \, b^{3} c e^{4}\right )} x^{3} - {\left (5 \, a^{2} b^{2} c^{2} e^{2} - 8 \, a b^{2} c e^{2} + 3 \, b^{2} e^{2}\right )} d^{2} + {\left (7 \, b^{4} c^{2} d^{2} e^{2} - 2 \, a^{2} b^{2} c^{2} e^{4} + 4 \, a b^{2} c e^{4} - 2 \, b^{2} e^{4} - {\left (5 \, a b^{3} c^{2} e^{3} - 2 \, b^{3} c e^{3}\right )} d\right )} x^{2} + {\left (a^{3} b c^{2} e^{3} - 2 \, a^{2} b c e^{3} + a b e^{3}\right )} d + {\left (4 \, b^{4} c^{2} d^{3} e + a^{3} b c^{2} e^{4} - 2 \, a^{2} b c e^{4} + 2 \, {\left (a b^{3} c^{2} e^{2} - 2 \, b^{3} c e^{2}\right )} d^{2} + a b e^{4} - {\left (7 \, a^{2} b^{2} c^{2} e^{3} - 12 \, a b^{2} c e^{3} + 5 \, b^{2} e^{3}\right )} d\right )} x\right )} \log \left (-b c x - a c + 1\right )}{6 \, {\left (b^{4} c^{2} d^{7} e - 2 \, {\left (2 \, a b^{3} c^{2} e^{2} - b^{3} c e^{2}\right )} d^{6} + {\left (6 \, a^{2} b^{2} c^{2} e^{3} - 6 \, a b^{2} c e^{3} + b^{2} e^{3}\right )} d^{5} - 2 \, {\left (2 \, a^{3} b c^{2} e^{4} - 3 \, a^{2} b c e^{4} + a b e^{4}\right )} d^{4} + {\left (a^{4} c^{2} e^{5} - 2 \, a^{3} c e^{5} + a^{2} e^{5}\right )} d^{3} + {\left (b^{4} c^{2} d^{4} e^{4} + a^{4} c^{2} e^{8} - 2 \, a^{3} c e^{8} - 2 \, {\left (2 \, a b^{3} c^{2} e^{5} - b^{3} c e^{5}\right )} d^{3} + {\left (6 \, a^{2} b^{2} c^{2} e^{6} - 6 \, a b^{2} c e^{6} + b^{2} e^{6}\right )} d^{2} + a^{2} e^{8} - 2 \, {\left (2 \, a^{3} b c^{2} e^{7} - 3 \, a^{2} b c e^{7} + a b e^{7}\right )} d\right )} x^{3} + 3 \, {\left (b^{4} c^{2} d^{5} e^{3} - 2 \, {\left (2 \, a b^{3} c^{2} e^{4} - b^{3} c e^{4}\right )} d^{4} + {\left (6 \, a^{2} b^{2} c^{2} e^{5} - 6 \, a b^{2} c e^{5} + b^{2} e^{5}\right )} d^{3} - 2 \, {\left (2 \, a^{3} b c^{2} e^{6} - 3 \, a^{2} b c e^{6} + a b e^{6}\right )} d^{2} + {\left (a^{4} c^{2} e^{7} - 2 \, a^{3} c e^{7} + a^{2} e^{7}\right )} d\right )} x^{2} + 3 \, {\left (b^{4} c^{2} d^{6} e^{2} - 2 \, {\left (2 \, a b^{3} c^{2} e^{3} - b^{3} c e^{3}\right )} d^{5} + {\left (6 \, a^{2} b^{2} c^{2} e^{4} - 6 \, a b^{2} c e^{4} + b^{2} e^{4}\right )} d^{4} - 2 \, {\left (2 \, a^{3} b c^{2} e^{5} - 3 \, a^{2} b c e^{5} + a b e^{5}\right )} d^{3} + {\left (a^{4} c^{2} e^{6} - 2 \, a^{3} c e^{6} + a^{2} e^{6}\right )} d^{2}\right )} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\mathrm {polylog}\left (2,c\,\left (a+b\,x\right )\right )}{{\left (d+e\,x\right )}^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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