Optimal. Leaf size=2498 \[ \text {result too large to display} \]
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Rubi [A] time = 2.67, antiderivative size = 2498, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 9, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6603, 2438, 2394, 2315, 2437, 2435, 2440, 2391, 6597} \[ \text {result too large to display} \]
Antiderivative was successfully verified.
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Rule 2315
Rule 2391
Rule 2394
Rule 2435
Rule 2437
Rule 2438
Rule 2440
Rule 6597
Rule 6603
Rubi steps
\begin {align*} \int \frac {\left (g+h \log \left (f (d+e x)^n\right )\right ) \text {Li}_2(c (a+b x))}{x^2} \, dx &=-\frac {\left (g+h \log \left (f (d+e x)^n\right )\right ) \text {Li}_2(c (a+b x))}{x}-b \int \left (\frac {\log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )}{a x}-\frac {b \log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )}{a (a+b x)}\right ) \, dx+(e h n) \int \left (\frac {\text {Li}_2(c (a+b x))}{d x}-\frac {e \text {Li}_2(c (a+b x))}{d (d+e x)}\right ) \, dx\\ &=-\frac {\left (g+h \log \left (f (d+e x)^n\right )\right ) \text {Li}_2(c (a+b x))}{x}-\frac {b \int \frac {\log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )}{x} \, dx}{a}+\frac {b^2 \int \frac {\log (1-a c-b c x) \left (g+h \log \left (f (d+e x)^n\right )\right )}{a+b x} \, dx}{a}+\frac {(e h n) \int \frac {\text {Li}_2(c (a+b x))}{x} \, dx}{d}-\frac {\left (e^2 h n\right ) \int \frac {\text {Li}_2(c (a+b x))}{d+e x} \, dx}{d}\\ &=\frac {e h n \log (x) \text {Li}_2(c (a+b x))}{d}-\frac {e h n \log (d+e x) \text {Li}_2(c (a+b x))}{d}-\frac {\left (g+h \log \left (f (d+e x)^n\right )\right ) \text {Li}_2(c (a+b x))}{x}+\frac {b \operatorname {Subst}\left (\int \frac {\log \left (-\frac {-a b c-b (1-a c)}{b}-c x\right ) \left (g+h \log \left (f \left (-\frac {-b d+a e}{b}+\frac {e x}{b}\right )^n\right )\right )}{x} \, dx,x,a+b x\right )}{a}-\frac {(b g) \int \frac {\log (1-a c-b c x)}{x} \, dx}{a}-\frac {(b h) \int \frac {\log (1-a c-b c x) \log \left (f (d+e x)^n\right )}{x} \, dx}{a}+\frac {(b e h n) \int \frac {\log (x) \log (1-a c-b c x)}{a+b x} \, dx}{d}-\frac {(b e h n) \int \frac {\log (1-a c-b c x) \log (d+e x)}{a+b x} \, dx}{d}\\ &=-\frac {b g \log \left (\frac {b c x}{1-a c}\right ) \log (1-a c-b c x)}{a}+\frac {e h n \log (x) \text {Li}_2(c (a+b x))}{d}-\frac {e h n \log (d+e x) \text {Li}_2(c (a+b x))}{d}-\frac {\left (g+h \log \left (f (d+e x)^n\right )\right ) \text {Li}_2(c (a+b x))}{x}+\frac {(b g) \operatorname {Subst}\left (\int \frac {\log \left (-\frac {-a b c-b (1-a c)}{b}-c x\right )}{x} \, dx,x,a+b x\right )}{a}-\frac {\left (b^2 c g\right ) \int \frac {\log \left (-\frac {b c x}{-1+a c}\right )}{1-a c-b c x} \, dx}{a}+\frac {(b h) \operatorname {Subst}\left (\int \frac {\log \left (-\frac {-a b c-b (1-a c)}{b}-c x\right ) \log \left (f \left (-\frac {-b d+a e}{b}+\frac {e x}{b}\right )^n\right )}{x} \, dx,x,a+b x\right )}{a}-\frac {(b h n) \int \frac {\log (1-a c-b c x) \log (d+e x)}{x} \, dx}{a}+\frac {(e h n) \operatorname {Subst}\left (\int \frac {\log \left (-\frac {a}{b}+\frac {x}{b}\right ) \log \left (-\frac {-a b c-b (1-a c)}{b}-c x\right )}{x} \, dx,x,a+b x\right )}{d}-\frac {(e h n) \operatorname {Subst}\left (\int \frac {\log \left (-\frac {-a b c-b (1-a c)}{b}-c x\right ) \log \left (-\frac {-b d+a e}{b}+\frac {e x}{b}\right )}{x} \, dx,x,a+b x\right )}{d}+\frac {\left (b h \left (n \log (d+e x)-\log \left (f (d+e x)^n\right )\right )\right ) \int \frac {\log (1-a c-b c x)}{x} \, dx}{a}\\ &=-\frac {b g \log \left (\frac {b c x}{1-a c}\right ) \log (1-a c-b c x)}{a}-\frac {b h n \log \left (\frac {b c x}{1-a c}\right ) \log (1-a c-b c x) \log (d+e x)}{a}-\frac {b h n \left (\log \left (\frac {b c x}{1-a c}\right )+\log \left (\frac {b c d+e-a c e}{b c (d+e x)}\right )-\log \left (\frac {(b c d+e-a c e) x}{(1-a c) (d+e x)}\right )\right ) \log ^2\left (\frac {(1-a c) (d+e x)}{d (1-a c-b c x)}\right )}{2 a}+\frac {b h n \left (\log \left (\frac {b c x}{1-a c}\right )-\log \left (-\frac {e x}{d}\right )\right ) \left (\log (1-a c-b c x)+\log \left (\frac {(1-a c) (d+e x)}{d (1-a c-b c x)}\right )\right )^2}{2 a}+\frac {b h \log \left (\frac {b c x}{1-a c}\right ) \log (1-a c-b c x) \left (n \log (d+e x)-\log \left (f (d+e x)^n\right )\right )}{a}-\frac {e h n \left (\log (c (a+b x))+\log \left (\frac {b c d+e-a c e}{b c (d+e x)}\right )-\log \left (\frac {(b c d+e-a c e) (a+b x)}{b (d+e x)}\right )\right ) \log ^2\left (\frac {b (d+e x)}{(b d-a e) (1-c (a+b x))}\right )}{2 d}+\frac {e h n \log (x) \log \left (1+\frac {b x}{a}\right ) \log (1-c (a+b x))}{d}-\frac {e h n \log (c (a+b x)) \log (d+e x) \log (1-c (a+b x))}{d}+\frac {e h n \left (\log (c (a+b x))-\log \left (-\frac {e (a+b x)}{b d-a e}\right )\right ) \left (\log \left (\frac {b (d+e x)}{(b d-a e) (1-c (a+b x))}\right )+\log (1-c (a+b x))\right )^2}{2 d}+\frac {e h n \left (\log \left (1+\frac {b x}{a}\right )+\log \left (\frac {1-a c}{1-c (a+b x)}\right )-\log \left (\frac {(1-a c) (a+b x)}{a (1-c (a+b x))}\right )\right ) \log ^2\left (-\frac {a (1-c (a+b x))}{b x}\right )}{2 d}+\frac {e h n \left (\log (c (a+b x))-\log \left (1+\frac {b x}{a}\right )\right ) \left (\log (x)+\log \left (-\frac {a (1-c (a+b x))}{b x}\right )\right )^2}{2 d}+\frac {e h n \left (\log (1-c (a+b x))-\log \left (-\frac {a (1-c (a+b x))}{b x}\right )\right ) \text {Li}_2\left (-\frac {b x}{a}\right )}{d}-\frac {b g \text {Li}_2(c (a+b x))}{a}+\frac {e h n \log (x) \text {Li}_2(c (a+b x))}{d}-\frac {e h n \log (d+e x) \text {Li}_2(c (a+b x))}{d}-\frac {\left (g+h \log \left (f (d+e x)^n\right )\right ) \text {Li}_2(c (a+b x))}{x}-\frac {b g \text {Li}_2\left (1-\frac {b c x}{1-a c}\right )}{a}-\frac {b h n \left (\log (d+e x)-\log \left (\frac {(1-a c) (d+e x)}{d (1-a c-b c x)}\right )\right ) \text {Li}_2\left (1-\frac {b c x}{1-a c}\right )}{a}-\frac {b h n \log \left (\frac {(1-a c) (d+e x)}{d (1-a c-b c x)}\right ) \text {Li}_2\left (\frac {d (1-a c-b c x)}{(1-a c) (d+e x)}\right )}{a}+\frac {b h n \log \left (\frac {(1-a c) (d+e x)}{d (1-a c-b c x)}\right ) \text {Li}_2\left (-\frac {e (1-a c-b c x)}{b c (d+e x)}\right )}{a}-\frac {e h n \left (\log \left (\frac {b (d+e x)}{(b d-a e) (1-c (a+b x))}\right )+\log (1-c (a+b x))\right ) \text {Li}_2\left (\frac {b (d+e x)}{b d-a e}\right )}{d}-\frac {b h n \left (\log (1-a c-b c x)+\log \left (\frac {(1-a c) (d+e x)}{d (1-a c-b c x)}\right )\right ) \text {Li}_2\left (1+\frac {e x}{d}\right )}{a}+\frac {e h n \log \left (-\frac {a (1-c (a+b x))}{b x}\right ) \text {Li}_2\left (-\frac {b x}{a (1-c (a+b x))}\right )}{d}-\frac {e h n \log \left (-\frac {a (1-c (a+b x))}{b x}\right ) \text {Li}_2\left (-\frac {b c x}{1-c (a+b x)}\right )}{d}-\frac {e h n \left (\log (d+e x)-\log \left (\frac {b (d+e x)}{(b d-a e) (1-c (a+b x))}\right )\right ) \text {Li}_2(1-c (a+b x))}{d}+\frac {e h n \left (\log (x)+\log \left (-\frac {a (1-c (a+b x))}{b x}\right )\right ) \text {Li}_2(1-c (a+b x))}{d}+\frac {e h n \log \left (\frac {b (d+e x)}{(b d-a e) (1-c (a+b x))}\right ) \text {Li}_2\left (-\frac {e (1-c (a+b x))}{b c (d+e x)}\right )}{d}-\frac {e h n \log \left (\frac {b (d+e x)}{(b d-a e) (1-c (a+b x))}\right ) \text {Li}_2\left (\frac {(b d-a e) (1-c (a+b x))}{b (d+e x)}\right )}{d}-\frac {e h n \text {Li}_3\left (-\frac {b x}{a}\right )}{d}+\frac {b h n \text {Li}_3\left (1-\frac {b c x}{1-a c}\right )}{a}-\frac {b h n \text {Li}_3\left (\frac {d (1-a c-b c x)}{(1-a c) (d+e x)}\right )}{a}+\frac {b h n \text {Li}_3\left (-\frac {e (1-a c-b c x)}{b c (d+e x)}\right )}{a}+\frac {e h n \text {Li}_3\left (\frac {b (d+e x)}{b d-a e}\right )}{d}+\frac {b h n \text {Li}_3\left (1+\frac {e x}{d}\right )}{a}+\frac {e h n \text {Li}_3\left (-\frac {b x}{a (1-c (a+b x))}\right )}{d}-\frac {e h n \text {Li}_3\left (-\frac {b c x}{1-c (a+b x)}\right )}{d}+\frac {e h n \text {Li}_3\left (-\frac {e (1-c (a+b x))}{b c (d+e x)}\right )}{d}-\frac {e h n \text {Li}_3\left (\frac {(b d-a e) (1-c (a+b x))}{b (d+e x)}\right )}{d}+\frac {(b h n) \operatorname {Subst}\left (\int \frac {\log \left (-\frac {-a b c-b (1-a c)}{b}-c x\right ) \log \left (-\frac {-b d+a e}{b}+\frac {e x}{b}\right )}{x} \, dx,x,a+b x\right )}{a}-\frac {\left (b h \left (n \log (d+e x)-\log \left (f (d+e x)^n\right )\right )\right ) \operatorname {Subst}\left (\int \frac {\log \left (-\frac {-a b c-b (1-a c)}{b}-c x\right )}{x} \, dx,x,a+b x\right )}{a}+\frac {\left (b^2 c h \left (n \log (d+e x)-\log \left (f (d+e x)^n\right )\right )\right ) \int \frac {\log \left (-\frac {b c x}{-1+a c}\right )}{1-a c-b c x} \, dx}{a}\\ &=-\frac {b g \log \left (\frac {b c x}{1-a c}\right ) \log (1-a c-b c x)}{a}-\frac {b h n \log \left (\frac {b c x}{1-a c}\right ) \log (1-a c-b c x) \log (d+e x)}{a}-\frac {b h n \left (\log \left (\frac {b c x}{1-a c}\right )+\log \left (\frac {b c d+e-a c e}{b c (d+e x)}\right )-\log \left (\frac {(b c d+e-a c e) x}{(1-a c) (d+e x)}\right )\right ) \log ^2\left (\frac {(1-a c) (d+e x)}{d (1-a c-b c x)}\right )}{2 a}+\frac {b h n \left (\log \left (\frac {b c x}{1-a c}\right )-\log \left (-\frac {e x}{d}\right )\right ) \left (\log (1-a c-b c x)+\log \left (\frac {(1-a c) (d+e x)}{d (1-a c-b c x)}\right )\right )^2}{2 a}+\frac {b h \log \left (\frac {b c x}{1-a c}\right ) \log (1-a c-b c x) \left (n \log (d+e x)-\log \left (f (d+e x)^n\right )\right )}{a}+\frac {b h n \left (\log (c (a+b x))+\log \left (\frac {b c d+e-a c e}{b c (d+e x)}\right )-\log \left (\frac {(b c d+e-a c e) (a+b x)}{b (d+e x)}\right )\right ) \log ^2\left (\frac {b (d+e x)}{(b d-a e) (1-c (a+b x))}\right )}{2 a}-\frac {e h n \left (\log (c (a+b x))+\log \left (\frac {b c d+e-a c e}{b c (d+e x)}\right )-\log \left (\frac {(b c d+e-a c e) (a+b x)}{b (d+e x)}\right )\right ) \log ^2\left (\frac {b (d+e x)}{(b d-a e) (1-c (a+b x))}\right )}{2 d}+\frac {e h n \log (x) \log \left (1+\frac {b x}{a}\right ) \log (1-c (a+b x))}{d}+\frac {b h n \log (c (a+b x)) \log (d+e x) \log (1-c (a+b x))}{a}-\frac {e h n \log (c (a+b x)) \log (d+e x) \log (1-c (a+b x))}{d}-\frac {b h n \left (\log (c (a+b x))-\log \left (-\frac {e (a+b x)}{b d-a e}\right )\right ) \left (\log \left (\frac {b (d+e x)}{(b d-a e) (1-c (a+b x))}\right )+\log (1-c (a+b x))\right )^2}{2 a}+\frac {e h n \left (\log (c (a+b x))-\log \left (-\frac {e (a+b x)}{b d-a e}\right )\right ) \left (\log \left (\frac {b (d+e x)}{(b d-a e) (1-c (a+b x))}\right )+\log (1-c (a+b x))\right )^2}{2 d}+\frac {e h n \left (\log \left (1+\frac {b x}{a}\right )+\log \left (\frac {1-a c}{1-c (a+b x)}\right )-\log \left (\frac {(1-a c) (a+b x)}{a (1-c (a+b x))}\right )\right ) \log ^2\left (-\frac {a (1-c (a+b x))}{b x}\right )}{2 d}+\frac {e h n \left (\log (c (a+b x))-\log \left (1+\frac {b x}{a}\right )\right ) \left (\log (x)+\log \left (-\frac {a (1-c (a+b x))}{b x}\right )\right )^2}{2 d}+\frac {e h n \left (\log (1-c (a+b x))-\log \left (-\frac {a (1-c (a+b x))}{b x}\right )\right ) \text {Li}_2\left (-\frac {b x}{a}\right )}{d}-\frac {b g \text {Li}_2(c (a+b x))}{a}+\frac {e h n \log (x) \text {Li}_2(c (a+b x))}{d}-\frac {e h n \log (d+e x) \text {Li}_2(c (a+b x))}{d}+\frac {b h \left (n \log (d+e x)-\log \left (f (d+e x)^n\right )\right ) \text {Li}_2(c (a+b x))}{a}-\frac {\left (g+h \log \left (f (d+e x)^n\right )\right ) \text {Li}_2(c (a+b x))}{x}-\frac {b g \text {Li}_2\left (1-\frac {b c x}{1-a c}\right )}{a}-\frac {b h n \left (\log (d+e x)-\log \left (\frac {(1-a c) (d+e x)}{d (1-a c-b c x)}\right )\right ) \text {Li}_2\left (1-\frac {b c x}{1-a c}\right )}{a}+\frac {b h \left (n \log (d+e x)-\log \left (f (d+e x)^n\right )\right ) \text {Li}_2\left (1-\frac {b c x}{1-a c}\right )}{a}-\frac {b h n \log \left (\frac {(1-a c) (d+e x)}{d (1-a c-b c x)}\right ) \text {Li}_2\left (\frac {d (1-a c-b c x)}{(1-a c) (d+e x)}\right )}{a}+\frac {b h n \log \left (\frac {(1-a c) (d+e x)}{d (1-a c-b c x)}\right ) \text {Li}_2\left (-\frac {e (1-a c-b c x)}{b c (d+e x)}\right )}{a}+\frac {b h n \left (\log \left (\frac {b (d+e x)}{(b d-a e) (1-c (a+b x))}\right )+\log (1-c (a+b x))\right ) \text {Li}_2\left (\frac {b (d+e x)}{b d-a e}\right )}{a}-\frac {e h n \left (\log \left (\frac {b (d+e x)}{(b d-a e) (1-c (a+b x))}\right )+\log (1-c (a+b x))\right ) \text {Li}_2\left (\frac {b (d+e x)}{b d-a e}\right )}{d}-\frac {b h n \left (\log (1-a c-b c x)+\log \left (\frac {(1-a c) (d+e x)}{d (1-a c-b c x)}\right )\right ) \text {Li}_2\left (1+\frac {e x}{d}\right )}{a}+\frac {e h n \log \left (-\frac {a (1-c (a+b x))}{b x}\right ) \text {Li}_2\left (-\frac {b x}{a (1-c (a+b x))}\right )}{d}-\frac {e h n \log \left (-\frac {a (1-c (a+b x))}{b x}\right ) \text {Li}_2\left (-\frac {b c x}{1-c (a+b x)}\right )}{d}+\frac {b h n \left (\log (d+e x)-\log \left (\frac {b (d+e x)}{(b d-a e) (1-c (a+b x))}\right )\right ) \text {Li}_2(1-c (a+b x))}{a}-\frac {e h n \left (\log (d+e x)-\log \left (\frac {b (d+e x)}{(b d-a e) (1-c (a+b x))}\right )\right ) \text {Li}_2(1-c (a+b x))}{d}+\frac {e h n \left (\log (x)+\log \left (-\frac {a (1-c (a+b x))}{b x}\right )\right ) \text {Li}_2(1-c (a+b x))}{d}-\frac {b h n \log \left (\frac {b (d+e x)}{(b d-a e) (1-c (a+b x))}\right ) \text {Li}_2\left (-\frac {e (1-c (a+b x))}{b c (d+e x)}\right )}{a}+\frac {e h n \log \left (\frac {b (d+e x)}{(b d-a e) (1-c (a+b x))}\right ) \text {Li}_2\left (-\frac {e (1-c (a+b x))}{b c (d+e x)}\right )}{d}+\frac {b h n \log \left (\frac {b (d+e x)}{(b d-a e) (1-c (a+b x))}\right ) \text {Li}_2\left (\frac {(b d-a e) (1-c (a+b x))}{b (d+e x)}\right )}{a}-\frac {e h n \log \left (\frac {b (d+e x)}{(b d-a e) (1-c (a+b x))}\right ) \text {Li}_2\left (\frac {(b d-a e) (1-c (a+b x))}{b (d+e x)}\right )}{d}-\frac {e h n \text {Li}_3\left (-\frac {b x}{a}\right )}{d}+\frac {b h n \text {Li}_3\left (1-\frac {b c x}{1-a c}\right )}{a}-\frac {b h n \text {Li}_3\left (\frac {d (1-a c-b c x)}{(1-a c) (d+e x)}\right )}{a}+\frac {b h n \text {Li}_3\left (-\frac {e (1-a c-b c x)}{b c (d+e x)}\right )}{a}-\frac {b h n \text {Li}_3\left (\frac {b (d+e x)}{b d-a e}\right )}{a}+\frac {e h n \text {Li}_3\left (\frac {b (d+e x)}{b d-a e}\right )}{d}+\frac {b h n \text {Li}_3\left (1+\frac {e x}{d}\right )}{a}+\frac {e h n \text {Li}_3\left (-\frac {b x}{a (1-c (a+b x))}\right )}{d}-\frac {e h n \text {Li}_3\left (-\frac {b c x}{1-c (a+b x)}\right )}{d}-\frac {b h n \text {Li}_3(1-c (a+b x))}{a}-\frac {b h n \text {Li}_3\left (-\frac {e (1-c (a+b x))}{b c (d+e x)}\right )}{a}+\frac {e h n \text {Li}_3\left (-\frac {e (1-c (a+b x))}{b c (d+e x)}\right )}{d}+\frac {b h n \text {Li}_3\left (\frac {(b d-a e) (1-c (a+b x))}{b (d+e x)}\right )}{a}-\frac {e h n \text {Li}_3\left (\frac {(b d-a e) (1-c (a+b x))}{b (d+e x)}\right )}{d}\\ \end {align*}
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Mathematica [A] time = 7.39, size = 2247, normalized size = 0.90 \[ \text {Result too large to show} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.72, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {h {\rm Li}_2\left (b c x + a c\right ) \log \left ({\left (e x + d\right )}^{n} f\right ) + g {\rm Li}_2\left (b c x + a c\right )}{x^{2}}, x\right ) \]
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 \[ \int \frac {{\left (h \log \left ({\left (e x + d\right )}^{n} f\right ) + g\right )} {\rm Li}_2\left ({\left (b x + a\right )} c\right )}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.29, size = 0, normalized size = 0.00 \[ \int \frac {\left (g +h \ln \left (f \left (e x +d \right )^{n}\right )\right ) \polylog \left (2, c \left (b x +a \right )\right )}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (h \log \left ({\left (e x + d\right )}^{n} f\right ) + g\right )} {\rm Li}_2\left ({\left (b x + a\right )} c\right )}{x^{2}}\,{d x} \]
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 \[ \int \frac {\mathrm {polylog}\left (2,c\,\left (a+b\,x\right )\right )\,\left (g+h\,\ln \left (f\,{\left (d+e\,x\right )}^n\right )\right )}{x^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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