Optimal. Leaf size=551 \[ -\frac {b e^2 n \log \left (1-\frac {d}{d+e x}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d^2 f}-\frac {b e n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d^2 f x}+\frac {b g n \text {Li}_2\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^2}+\frac {b g n \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{\sqrt {g} d+e \sqrt {-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^2}-\frac {2 b g n \text {Li}_2\left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^2}-\frac {g \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f^2}+\frac {g \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{d \sqrt {g}+e \sqrt {-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f^2}+\frac {g \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f^2}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f x^2}+\frac {b^2 e^2 n^2 \text {Li}_2\left (\frac {d}{d+e x}\right )}{d^2 f}+\frac {b^2 e^2 n^2 \log (x)}{d^2 f}-\frac {b^2 g n^2 \text {Li}_3\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{f^2}-\frac {b^2 g n^2 \text {Li}_3\left (\frac {\sqrt {g} (d+e x)}{\sqrt {g} d+e \sqrt {-f}}\right )}{f^2}+\frac {2 b^2 g n^2 \text {Li}_3\left (\frac {e x}{d}+1\right )}{f^2} \]
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Rubi [A] time = 0.95, antiderivative size = 575, normalized size of antiderivative = 1.04, number of steps used = 25, number of rules used = 14, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.483, Rules used = {2416, 2398, 2411, 2347, 2344, 2301, 2317, 2391, 2314, 31, 2396, 2433, 2374, 6589} \[ \frac {b g n \text {PolyLog}\left (2,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^2}+\frac {b g n \text {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{d \sqrt {g}+e \sqrt {-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^2}-\frac {2 b g n \text {PolyLog}\left (2,\frac {e x}{d}+1\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^2}-\frac {b^2 e^2 n^2 \text {PolyLog}\left (2,\frac {e x}{d}+1\right )}{d^2 f}-\frac {b^2 g n^2 \text {PolyLog}\left (3,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{f^2}-\frac {b^2 g n^2 \text {PolyLog}\left (3,\frac {\sqrt {g} (d+e x)}{d \sqrt {g}+e \sqrt {-f}}\right )}{f^2}+\frac {2 b^2 g n^2 \text {PolyLog}\left (3,\frac {e x}{d}+1\right )}{f^2}+\frac {e^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 d^2 f}-\frac {b e^2 n \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d^2 f}-\frac {b e n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d^2 f x}-\frac {g \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f^2}+\frac {g \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{d \sqrt {g}+e \sqrt {-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f^2}+\frac {g \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f^2}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f x^2}+\frac {b^2 e^2 n^2 \log (x)}{d^2 f} \]
Antiderivative was successfully verified.
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Rule 31
Rule 2301
Rule 2314
Rule 2317
Rule 2344
Rule 2347
Rule 2374
Rule 2391
Rule 2396
Rule 2398
Rule 2411
Rule 2416
Rule 2433
Rule 6589
Rubi steps
\begin {align*} \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^3 \left (f+g x^2\right )} \, dx &=\int \left (\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f x^3}-\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f^2 x}+\frac {g^2 x \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f^2 \left (f+g x^2\right )}\right ) \, dx\\ &=\frac {\int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^3} \, dx}{f}-\frac {g \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x} \, dx}{f^2}+\frac {g^2 \int \frac {x \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f+g x^2} \, dx}{f^2}\\ &=-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f x^2}-\frac {g \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f^2}+\frac {g^2 \int \left (-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 \sqrt {g} \left (\sqrt {-f}-\sqrt {g} x\right )}+\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 \sqrt {g} \left (\sqrt {-f}+\sqrt {g} x\right )}\right ) \, dx}{f^2}+\frac {(b e n) \int \frac {a+b \log \left (c (d+e x)^n\right )}{x^2 (d+e x)} \, dx}{f}+\frac {(2 b e g n) \int \frac {\log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d+e x} \, dx}{f^2}\\ &=-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f x^2}-\frac {g \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f^2}-\frac {g^{3/2} \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\sqrt {-f}-\sqrt {g} x} \, dx}{2 f^2}+\frac {g^{3/2} \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\sqrt {-f}+\sqrt {g} x} \, dx}{2 f^2}+\frac {(b n) \operatorname {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^2} \, dx,x,d+e x\right )}{f}+\frac {(2 b g n) \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (-\frac {e \left (-\frac {d}{e}+\frac {x}{e}\right )}{d}\right )}{x} \, dx,x,d+e x\right )}{f^2}\\ &=-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f x^2}-\frac {g \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f^2}+\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 f^2}+\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 f^2}-\frac {2 b g n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (1+\frac {e x}{d}\right )}{f^2}+\frac {(b n) \operatorname {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{\left (-\frac {d}{e}+\frac {x}{e}\right )^2} \, dx,x,d+e x\right )}{d f}-\frac {(b e n) \operatorname {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x \left (-\frac {d}{e}+\frac {x}{e}\right )} \, dx,x,d+e x\right )}{d f}-\frac {(b e g n) \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{d+e x} \, dx}{f^2}-\frac {(b e g n) \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{d+e x} \, dx}{f^2}+\frac {\left (2 b^2 g n^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (\frac {x}{d}\right )}{x} \, dx,x,d+e x\right )}{f^2}\\ &=-\frac {b e n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d^2 f x}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f x^2}-\frac {g \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f^2}+\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 f^2}+\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 f^2}-\frac {2 b g n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (1+\frac {e x}{d}\right )}{f^2}+\frac {2 b^2 g n^2 \text {Li}_3\left (1+\frac {e x}{d}\right )}{f^2}-\frac {(b e n) \operatorname {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{-\frac {d}{e}+\frac {x}{e}} \, dx,x,d+e x\right )}{d^2 f}+\frac {\left (b e^2 n\right ) \operatorname {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x} \, dx,x,d+e x\right )}{d^2 f}-\frac {(b g n) \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (\frac {e \left (\frac {e \sqrt {-f}+d \sqrt {g}}{e}-\frac {\sqrt {g} x}{e}\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{x} \, dx,x,d+e x\right )}{f^2}-\frac {(b g n) \operatorname {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (\frac {e \left (\frac {e \sqrt {-f}-d \sqrt {g}}{e}+\frac {\sqrt {g} x}{e}\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{x} \, dx,x,d+e x\right )}{f^2}+\frac {\left (b^2 e n^2\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x}{e}} \, dx,x,d+e x\right )}{d^2 f}\\ &=\frac {b^2 e^2 n^2 \log (x)}{d^2 f}-\frac {b e n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d^2 f x}-\frac {b e^2 n \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d^2 f}+\frac {e^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 d^2 f}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f x^2}-\frac {g \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f^2}+\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 f^2}+\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 f^2}+\frac {b g n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{f^2}+\frac {b g n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{f^2}-\frac {2 b g n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (1+\frac {e x}{d}\right )}{f^2}+\frac {2 b^2 g n^2 \text {Li}_3\left (1+\frac {e x}{d}\right )}{f^2}+\frac {\left (b^2 e^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {x}{d}\right )}{x} \, dx,x,d+e x\right )}{d^2 f}-\frac {\left (b^2 g n^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {\sqrt {g} x}{e \sqrt {-f}-d \sqrt {g}}\right )}{x} \, dx,x,d+e x\right )}{f^2}-\frac {\left (b^2 g n^2\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (\frac {\sqrt {g} x}{e \sqrt {-f}+d \sqrt {g}}\right )}{x} \, dx,x,d+e x\right )}{f^2}\\ &=\frac {b^2 e^2 n^2 \log (x)}{d^2 f}-\frac {b e n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d^2 f x}-\frac {b e^2 n \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{d^2 f}+\frac {e^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 d^2 f}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 f x^2}-\frac {g \log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f^2}+\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 f^2}+\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 f^2}+\frac {b g n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{f^2}+\frac {b g n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{f^2}-\frac {b^2 e^2 n^2 \text {Li}_2\left (1+\frac {e x}{d}\right )}{d^2 f}-\frac {2 b g n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (1+\frac {e x}{d}\right )}{f^2}-\frac {b^2 g n^2 \text {Li}_3\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{f^2}-\frac {b^2 g n^2 \text {Li}_3\left (\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{f^2}+\frac {2 b^2 g n^2 \text {Li}_3\left (1+\frac {e x}{d}\right )}{f^2}\\ \end {align*}
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Mathematica [C] time = 0.77, size = 811, normalized size = 1.47 \[ \frac {b^2 \left (d^2 g \left (\log \left (1-\frac {\sqrt {g} (d+e x)}{d \sqrt {g}-i e \sqrt {f}}\right ) \log ^2(d+e x)+2 \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{d \sqrt {g}-i e \sqrt {f}}\right ) \log (d+e x)-2 \text {Li}_3\left (\frac {\sqrt {g} (d+e x)}{d \sqrt {g}-i e \sqrt {f}}\right )\right ) x^2+d^2 g \left (\log \left (1-\frac {\sqrt {g} (d+e x)}{\sqrt {g} d+i e \sqrt {f}}\right ) \log ^2(d+e x)+2 \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{\sqrt {g} d+i e \sqrt {f}}\right ) \log (d+e x)-2 \text {Li}_3\left (\frac {\sqrt {g} (d+e x)}{\sqrt {g} d+i e \sqrt {f}}\right )\right ) x^2-2 d^2 g \left (\log \left (-\frac {e x}{d}\right ) \log ^2(d+e x)+2 \text {Li}_2\left (\frac {e x}{d}+1\right ) \log (d+e x)-2 \text {Li}_3\left (\frac {e x}{d}+1\right )\right ) x^2+f \left (2 e^2 \log (x) x^2-2 e^2 \text {Li}_2\left (\frac {e x}{d}+1\right ) x^2-\log (d+e x) \left (2 e^2 \log \left (-\frac {e x}{d}\right ) x^2+(d+e x) (2 e x+(d-e x) \log (d+e x))\right )\right )\right ) n^2-2 b \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right ) \left (-d^2 g \left (\log (d+e x) \log \left (\frac {e \left (i \sqrt {g} x+\sqrt {f}\right )}{e \sqrt {f}-i d \sqrt {g}}\right )+\text {Li}_2\left (-\frac {i \sqrt {g} (d+e x)}{e \sqrt {f}-i d \sqrt {g}}\right )\right ) x^2-d^2 g \left (\log (d+e x) \log \left (\frac {e \left (\sqrt {f}-i \sqrt {g} x\right )}{i \sqrt {g} d+e \sqrt {f}}\right )+\text {Li}_2\left (\frac {i \sqrt {g} (d+e x)}{i \sqrt {g} d+e \sqrt {f}}\right )\right ) x^2+2 d^2 g \left (\log \left (-\frac {e x}{d}\right ) \log (d+e x)+\text {Li}_2\left (\frac {e x}{d}+1\right )\right ) x^2+f \left (e^2 \log (x) x^2+d e x+\left (d^2-e^2 x^2\right ) \log (d+e x)\right )\right ) n-d^2 f \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2-2 d^2 g x^2 \log (x) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2+d^2 g x^2 \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (g x^2+f\right )}{2 d^2 f^2 x^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} \log \left ({\left (e x + d\right )}^{n} c\right )^{2} + 2 \, a b \log \left ({\left (e x + d\right )}^{n} c\right ) + a^{2}}{g x^{5} + f x^{3}}, x\right ) \]
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 \[ \int \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2}}{{\left (g x^{2} + f\right )} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.25, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \left (e x +d \right )^{n}\right )+a \right )^{2}}{\left (g \,x^{2}+f \right ) x^{3}}\, dx \]
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 \[ \frac {1}{2} \, a^{2} {\left (\frac {g \log \left (g x^{2} + f\right )}{f^{2}} - \frac {2 \, g \log \relax (x)}{f^{2}} - \frac {1}{f x^{2}}\right )} + \int \frac {b^{2} \log \left ({\left (e x + d\right )}^{n}\right )^{2} + b^{2} \log \relax (c)^{2} + 2 \, a b \log \relax (c) + 2 \, {\left (b^{2} \log \relax (c) + a b\right )} \log \left ({\left (e x + d\right )}^{n}\right )}{g x^{5} + f x^{3}}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^2}{x^3\,\left (g\,x^2+f\right )} \,d x \]
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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