Optimal. Leaf size=490 \[ -\frac {2 d \sqrt {1-a^2 x^2} \cos ^{-1}(a x) \log \left (c x^n\right )}{a}-\frac {4 e x \log \left (c x^n\right )}{9 a^2}-\frac {2 e x^2 \sqrt {1-a^2 x^2} \cos ^{-1}(a x) \log \left (c x^n\right )}{9 a}+\frac {4}{9} n x \left (\frac {2 e}{a^2}+9 d\right )+\frac {2 d n \sqrt {1-a^2 x^2} \cos ^{-1}(a x)}{a}+\frac {2 e n x^2 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)}{27 a}+\frac {2 e n x}{27 a^2}-\frac {4 e \sqrt {1-a^2 x^2} \cos ^{-1}(a x) \log \left (c x^n\right )}{9 a^3}-\frac {2 i n \left (9 a^2 d+2 e\right ) \text {Li}_2\left (-i e^{i \cos ^{-1}(a x)}\right )}{9 a^3}+\frac {2 i n \left (9 a^2 d+2 e\right ) \text {Li}_2\left (i e^{i \cos ^{-1}(a x)}\right )}{9 a^3}+\frac {2 n \sqrt {1-a^2 x^2} \left (9 a^2 d+2 e\right ) \cos ^{-1}(a x)}{9 a^3}+\frac {4 i n \left (9 a^2 d+2 e\right ) \cos ^{-1}(a x) \tan ^{-1}\left (e^{i \cos ^{-1}(a x)}\right )}{9 a^3}-\frac {2 e n \left (1-a^2 x^2\right )^{3/2} \cos ^{-1}(a x)}{27 a^3}+\frac {4 e n \sqrt {1-a^2 x^2} \cos ^{-1}(a x)}{27 a^3}+d x \cos ^{-1}(a x)^2 \log \left (c x^n\right )+\frac {1}{3} e x^3 \cos ^{-1}(a x)^2 \log \left (c x^n\right )-d n x \cos ^{-1}(a x)^2-\frac {1}{9} e n x^3 \cos ^{-1}(a x)^2-2 d x \log \left (c x^n\right )-\frac {2}{27} e x^3 \log \left (c x^n\right )+2 d n x+\frac {2}{27} e n x^3 \]
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Rubi [A] time = 0.70, antiderivative size = 490, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 14, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {4668, 4620, 4678, 8, 4628, 4708, 30, 2387, 6, 4698, 4710, 4181, 2279, 2391} \[ -\frac {2 i n \left (9 a^2 d+2 e\right ) \text {PolyLog}\left (2,-i e^{i \cos ^{-1}(a x)}\right )}{9 a^3}+\frac {2 i n \left (9 a^2 d+2 e\right ) \text {PolyLog}\left (2,i e^{i \cos ^{-1}(a x)}\right )}{9 a^3}-\frac {2 d \sqrt {1-a^2 x^2} \cos ^{-1}(a x) \log \left (c x^n\right )}{a}-\frac {4 e x \log \left (c x^n\right )}{9 a^2}-\frac {2 e x^2 \sqrt {1-a^2 x^2} \cos ^{-1}(a x) \log \left (c x^n\right )}{9 a}-\frac {4 e \sqrt {1-a^2 x^2} \cos ^{-1}(a x) \log \left (c x^n\right )}{9 a^3}+\frac {2 n \sqrt {1-a^2 x^2} \left (9 a^2 d+2 e\right ) \cos ^{-1}(a x)}{9 a^3}+\frac {4}{9} n x \left (\frac {2 e}{a^2}+9 d\right )+\frac {4 i n \left (9 a^2 d+2 e\right ) \cos ^{-1}(a x) \tan ^{-1}\left (e^{i \cos ^{-1}(a x)}\right )}{9 a^3}+\frac {2 d n \sqrt {1-a^2 x^2} \cos ^{-1}(a x)}{a}+\frac {2 e n x^2 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)}{27 a}-\frac {2 e n \left (1-a^2 x^2\right )^{3/2} \cos ^{-1}(a x)}{27 a^3}+\frac {4 e n \sqrt {1-a^2 x^2} \cos ^{-1}(a x)}{27 a^3}+\frac {2 e n x}{27 a^2}+d x \cos ^{-1}(a x)^2 \log \left (c x^n\right )+\frac {1}{3} e x^3 \cos ^{-1}(a x)^2 \log \left (c x^n\right )-d n x \cos ^{-1}(a x)^2-\frac {1}{9} e n x^3 \cos ^{-1}(a x)^2-2 d x \log \left (c x^n\right )-\frac {2}{27} e x^3 \log \left (c x^n\right )+2 d n x+\frac {2}{27} e n x^3 \]
Antiderivative was successfully verified.
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Rule 6
Rule 8
Rule 30
Rule 2279
Rule 2387
Rule 2391
Rule 4181
Rule 4620
Rule 4628
Rule 4668
Rule 4678
Rule 4698
Rule 4708
Rule 4710
Rubi steps
\begin {align*} \int \left (d+e x^2\right ) \cos ^{-1}(a x)^2 \log \left (c x^n\right ) \, dx &=-2 d x \log \left (c x^n\right )-\frac {4 e x \log \left (c x^n\right )}{9 a^2}-\frac {2}{27} e x^3 \log \left (c x^n\right )-\frac {2 d \sqrt {1-a^2 x^2} \cos ^{-1}(a x) \log \left (c x^n\right )}{a}-\frac {4 e \sqrt {1-a^2 x^2} \cos ^{-1}(a x) \log \left (c x^n\right )}{9 a^3}-\frac {2 e x^2 \sqrt {1-a^2 x^2} \cos ^{-1}(a x) \log \left (c x^n\right )}{9 a}+d x \cos ^{-1}(a x)^2 \log \left (c x^n\right )+\frac {1}{3} e x^3 \cos ^{-1}(a x)^2 \log \left (c x^n\right )-n \int \left (-2 d-\frac {4 e}{9 a^2}-\frac {2 e x^2}{27}-\frac {2 d \sqrt {1-a^2 x^2} \cos ^{-1}(a x)}{a x}-\frac {4 e \sqrt {1-a^2 x^2} \cos ^{-1}(a x)}{9 a^3 x}-\frac {2 e x \sqrt {1-a^2 x^2} \cos ^{-1}(a x)}{9 a}+d \cos ^{-1}(a x)^2+\frac {1}{3} e x^2 \cos ^{-1}(a x)^2\right ) \, dx\\ &=-2 d x \log \left (c x^n\right )-\frac {4 e x \log \left (c x^n\right )}{9 a^2}-\frac {2}{27} e x^3 \log \left (c x^n\right )-\frac {2 d \sqrt {1-a^2 x^2} \cos ^{-1}(a x) \log \left (c x^n\right )}{a}-\frac {4 e \sqrt {1-a^2 x^2} \cos ^{-1}(a x) \log \left (c x^n\right )}{9 a^3}-\frac {2 e x^2 \sqrt {1-a^2 x^2} \cos ^{-1}(a x) \log \left (c x^n\right )}{9 a}+d x \cos ^{-1}(a x)^2 \log \left (c x^n\right )+\frac {1}{3} e x^3 \cos ^{-1}(a x)^2 \log \left (c x^n\right )-n \int \left (-2 d-\frac {4 e}{9 a^2}-\frac {2 e x^2}{27}+\frac {\left (-\frac {2 d}{a}-\frac {4 e}{9 a^3}\right ) \sqrt {1-a^2 x^2} \cos ^{-1}(a x)}{x}-\frac {2 e x \sqrt {1-a^2 x^2} \cos ^{-1}(a x)}{9 a}+d \cos ^{-1}(a x)^2+\frac {1}{3} e x^2 \cos ^{-1}(a x)^2\right ) \, dx\\ &=\frac {2}{9} \left (9 d+\frac {2 e}{a^2}\right ) n x+\frac {2}{81} e n x^3-2 d x \log \left (c x^n\right )-\frac {4 e x \log \left (c x^n\right )}{9 a^2}-\frac {2}{27} e x^3 \log \left (c x^n\right )-\frac {2 d \sqrt {1-a^2 x^2} \cos ^{-1}(a x) \log \left (c x^n\right )}{a}-\frac {4 e \sqrt {1-a^2 x^2} \cos ^{-1}(a x) \log \left (c x^n\right )}{9 a^3}-\frac {2 e x^2 \sqrt {1-a^2 x^2} \cos ^{-1}(a x) \log \left (c x^n\right )}{9 a}+d x \cos ^{-1}(a x)^2 \log \left (c x^n\right )+\frac {1}{3} e x^3 \cos ^{-1}(a x)^2 \log \left (c x^n\right )-(d n) \int \cos ^{-1}(a x)^2 \, dx-\frac {1}{3} (e n) \int x^2 \cos ^{-1}(a x)^2 \, dx+\frac {(2 e n) \int x \sqrt {1-a^2 x^2} \cos ^{-1}(a x) \, dx}{9 a}+\frac {\left (2 \left (9 a^2 d+2 e\right ) n\right ) \int \frac {\sqrt {1-a^2 x^2} \cos ^{-1}(a x)}{x} \, dx}{9 a^3}\\ &=\frac {2}{9} \left (9 d+\frac {2 e}{a^2}\right ) n x+\frac {2}{81} e n x^3+\frac {2 \left (9 a^2 d+2 e\right ) n \sqrt {1-a^2 x^2} \cos ^{-1}(a x)}{9 a^3}-\frac {2 e n \left (1-a^2 x^2\right )^{3/2} \cos ^{-1}(a x)}{27 a^3}-d n x \cos ^{-1}(a x)^2-\frac {1}{9} e n x^3 \cos ^{-1}(a x)^2-2 d x \log \left (c x^n\right )-\frac {4 e x \log \left (c x^n\right )}{9 a^2}-\frac {2}{27} e x^3 \log \left (c x^n\right )-\frac {2 d \sqrt {1-a^2 x^2} \cos ^{-1}(a x) \log \left (c x^n\right )}{a}-\frac {4 e \sqrt {1-a^2 x^2} \cos ^{-1}(a x) \log \left (c x^n\right )}{9 a^3}-\frac {2 e x^2 \sqrt {1-a^2 x^2} \cos ^{-1}(a x) \log \left (c x^n\right )}{9 a}+d x \cos ^{-1}(a x)^2 \log \left (c x^n\right )+\frac {1}{3} e x^3 \cos ^{-1}(a x)^2 \log \left (c x^n\right )-(2 a d n) \int \frac {x \cos ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx-\frac {(2 e n) \int \left (1-a^2 x^2\right ) \, dx}{27 a^2}-\frac {1}{9} (2 a e n) \int \frac {x^3 \cos ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx+\frac {\left (2 \left (9 a^2 d+2 e\right ) n\right ) \int \frac {\cos ^{-1}(a x)}{x \sqrt {1-a^2 x^2}} \, dx}{9 a^3}+\frac {\left (2 \left (9 a^2 d+2 e\right ) n\right ) \int 1 \, dx}{9 a^2}\\ &=-\frac {2 e n x}{27 a^2}+\frac {2 \left (9 a^2 d+2 e\right ) n x}{9 a^2}+\frac {2}{9} \left (9 d+\frac {2 e}{a^2}\right ) n x+\frac {4}{81} e n x^3+\frac {2 d n \sqrt {1-a^2 x^2} \cos ^{-1}(a x)}{a}+\frac {2 \left (9 a^2 d+2 e\right ) n \sqrt {1-a^2 x^2} \cos ^{-1}(a x)}{9 a^3}+\frac {2 e n x^2 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)}{27 a}-\frac {2 e n \left (1-a^2 x^2\right )^{3/2} \cos ^{-1}(a x)}{27 a^3}-d n x \cos ^{-1}(a x)^2-\frac {1}{9} e n x^3 \cos ^{-1}(a x)^2-2 d x \log \left (c x^n\right )-\frac {4 e x \log \left (c x^n\right )}{9 a^2}-\frac {2}{27} e x^3 \log \left (c x^n\right )-\frac {2 d \sqrt {1-a^2 x^2} \cos ^{-1}(a x) \log \left (c x^n\right )}{a}-\frac {4 e \sqrt {1-a^2 x^2} \cos ^{-1}(a x) \log \left (c x^n\right )}{9 a^3}-\frac {2 e x^2 \sqrt {1-a^2 x^2} \cos ^{-1}(a x) \log \left (c x^n\right )}{9 a}+d x \cos ^{-1}(a x)^2 \log \left (c x^n\right )+\frac {1}{3} e x^3 \cos ^{-1}(a x)^2 \log \left (c x^n\right )+(2 d n) \int 1 \, dx+\frac {1}{27} (2 e n) \int x^2 \, dx-\frac {(4 e n) \int \frac {x \cos ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx}{27 a}-\frac {\left (2 \left (9 a^2 d+2 e\right ) n\right ) \operatorname {Subst}\left (\int x \sec (x) \, dx,x,\cos ^{-1}(a x)\right )}{9 a^3}\\ &=2 d n x-\frac {2 e n x}{27 a^2}+\frac {2 \left (9 a^2 d+2 e\right ) n x}{9 a^2}+\frac {2}{9} \left (9 d+\frac {2 e}{a^2}\right ) n x+\frac {2}{27} e n x^3+\frac {2 d n \sqrt {1-a^2 x^2} \cos ^{-1}(a x)}{a}+\frac {4 e n \sqrt {1-a^2 x^2} \cos ^{-1}(a x)}{27 a^3}+\frac {2 \left (9 a^2 d+2 e\right ) n \sqrt {1-a^2 x^2} \cos ^{-1}(a x)}{9 a^3}+\frac {2 e n x^2 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)}{27 a}-\frac {2 e n \left (1-a^2 x^2\right )^{3/2} \cos ^{-1}(a x)}{27 a^3}-d n x \cos ^{-1}(a x)^2-\frac {1}{9} e n x^3 \cos ^{-1}(a x)^2+\frac {4 i \left (9 a^2 d+2 e\right ) n \cos ^{-1}(a x) \tan ^{-1}\left (e^{i \cos ^{-1}(a x)}\right )}{9 a^3}-2 d x \log \left (c x^n\right )-\frac {4 e x \log \left (c x^n\right )}{9 a^2}-\frac {2}{27} e x^3 \log \left (c x^n\right )-\frac {2 d \sqrt {1-a^2 x^2} \cos ^{-1}(a x) \log \left (c x^n\right )}{a}-\frac {4 e \sqrt {1-a^2 x^2} \cos ^{-1}(a x) \log \left (c x^n\right )}{9 a^3}-\frac {2 e x^2 \sqrt {1-a^2 x^2} \cos ^{-1}(a x) \log \left (c x^n\right )}{9 a}+d x \cos ^{-1}(a x)^2 \log \left (c x^n\right )+\frac {1}{3} e x^3 \cos ^{-1}(a x)^2 \log \left (c x^n\right )+\frac {(4 e n) \int 1 \, dx}{27 a^2}+\frac {\left (2 \left (9 a^2 d+2 e\right ) n\right ) \operatorname {Subst}\left (\int \log \left (1-i e^{i x}\right ) \, dx,x,\cos ^{-1}(a x)\right )}{9 a^3}-\frac {\left (2 \left (9 a^2 d+2 e\right ) n\right ) \operatorname {Subst}\left (\int \log \left (1+i e^{i x}\right ) \, dx,x,\cos ^{-1}(a x)\right )}{9 a^3}\\ &=2 d n x+\frac {2 e n x}{27 a^2}+\frac {2 \left (9 a^2 d+2 e\right ) n x}{9 a^2}+\frac {2}{9} \left (9 d+\frac {2 e}{a^2}\right ) n x+\frac {2}{27} e n x^3+\frac {2 d n \sqrt {1-a^2 x^2} \cos ^{-1}(a x)}{a}+\frac {4 e n \sqrt {1-a^2 x^2} \cos ^{-1}(a x)}{27 a^3}+\frac {2 \left (9 a^2 d+2 e\right ) n \sqrt {1-a^2 x^2} \cos ^{-1}(a x)}{9 a^3}+\frac {2 e n x^2 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)}{27 a}-\frac {2 e n \left (1-a^2 x^2\right )^{3/2} \cos ^{-1}(a x)}{27 a^3}-d n x \cos ^{-1}(a x)^2-\frac {1}{9} e n x^3 \cos ^{-1}(a x)^2+\frac {4 i \left (9 a^2 d+2 e\right ) n \cos ^{-1}(a x) \tan ^{-1}\left (e^{i \cos ^{-1}(a x)}\right )}{9 a^3}-2 d x \log \left (c x^n\right )-\frac {4 e x \log \left (c x^n\right )}{9 a^2}-\frac {2}{27} e x^3 \log \left (c x^n\right )-\frac {2 d \sqrt {1-a^2 x^2} \cos ^{-1}(a x) \log \left (c x^n\right )}{a}-\frac {4 e \sqrt {1-a^2 x^2} \cos ^{-1}(a x) \log \left (c x^n\right )}{9 a^3}-\frac {2 e x^2 \sqrt {1-a^2 x^2} \cos ^{-1}(a x) \log \left (c x^n\right )}{9 a}+d x \cos ^{-1}(a x)^2 \log \left (c x^n\right )+\frac {1}{3} e x^3 \cos ^{-1}(a x)^2 \log \left (c x^n\right )-\frac {\left (2 i \left (9 a^2 d+2 e\right ) n\right ) \operatorname {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{i \cos ^{-1}(a x)}\right )}{9 a^3}+\frac {\left (2 i \left (9 a^2 d+2 e\right ) n\right ) \operatorname {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{i \cos ^{-1}(a x)}\right )}{9 a^3}\\ &=2 d n x+\frac {2 e n x}{27 a^2}+\frac {2 \left (9 a^2 d+2 e\right ) n x}{9 a^2}+\frac {2}{9} \left (9 d+\frac {2 e}{a^2}\right ) n x+\frac {2}{27} e n x^3+\frac {2 d n \sqrt {1-a^2 x^2} \cos ^{-1}(a x)}{a}+\frac {4 e n \sqrt {1-a^2 x^2} \cos ^{-1}(a x)}{27 a^3}+\frac {2 \left (9 a^2 d+2 e\right ) n \sqrt {1-a^2 x^2} \cos ^{-1}(a x)}{9 a^3}+\frac {2 e n x^2 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)}{27 a}-\frac {2 e n \left (1-a^2 x^2\right )^{3/2} \cos ^{-1}(a x)}{27 a^3}-d n x \cos ^{-1}(a x)^2-\frac {1}{9} e n x^3 \cos ^{-1}(a x)^2+\frac {4 i \left (9 a^2 d+2 e\right ) n \cos ^{-1}(a x) \tan ^{-1}\left (e^{i \cos ^{-1}(a x)}\right )}{9 a^3}-2 d x \log \left (c x^n\right )-\frac {4 e x \log \left (c x^n\right )}{9 a^2}-\frac {2}{27} e x^3 \log \left (c x^n\right )-\frac {2 d \sqrt {1-a^2 x^2} \cos ^{-1}(a x) \log \left (c x^n\right )}{a}-\frac {4 e \sqrt {1-a^2 x^2} \cos ^{-1}(a x) \log \left (c x^n\right )}{9 a^3}-\frac {2 e x^2 \sqrt {1-a^2 x^2} \cos ^{-1}(a x) \log \left (c x^n\right )}{9 a}+d x \cos ^{-1}(a x)^2 \log \left (c x^n\right )+\frac {1}{3} e x^3 \cos ^{-1}(a x)^2 \log \left (c x^n\right )-\frac {2 i \left (9 a^2 d+2 e\right ) n \text {Li}_2\left (-i e^{i \cos ^{-1}(a x)}\right )}{9 a^3}+\frac {2 i \left (9 a^2 d+2 e\right ) n \text {Li}_2\left (i e^{i \cos ^{-1}(a x)}\right )}{9 a^3}\\ \end {align*}
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Mathematica [A] time = 0.82, size = 564, normalized size = 1.15 \[ \frac {d \left (a x \left (\cos ^{-1}(a x)^2-2\right )-2 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)\right ) \left (\log \left (c x^n\right )+n (-\log (x))-n\right )}{a}+\frac {2 d n \left (\sqrt {1-a^2 x^2} \cos ^{-1}(a x)-i \text {Li}_2\left (-i e^{i \cos ^{-1}(a x)}\right )+i \text {Li}_2\left (i e^{i \cos ^{-1}(a x)}\right )+a x-\cos ^{-1}(a x) \log \left (1-i e^{i \cos ^{-1}(a x)}\right )+\cos ^{-1}(a x) \log \left (1+i e^{i \cos ^{-1}(a x)}\right )\right )}{a}+\frac {d n \log (x) \left (-2 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)-2 a x+a x \cos ^{-1}(a x)^2\right )}{a}+\frac {4 e n x}{9 a^2}+\frac {e \left (-6 \cos ^{-1}(a x) \left (9 \sqrt {1-a^2 x^2}+\sin \left (3 \cos ^{-1}(a x)\right )\right )+27 a x \left (\cos ^{-1}(a x)^2-2\right )-\left (2-9 \cos ^{-1}(a x)^2\right ) \cos \left (3 \cos ^{-1}(a x)\right )\right ) \left (3 \left (\log \left (c x^n\right )-n \log (x)\right )-n\right )}{324 a^3}+\frac {4 e n \left (\sqrt {1-a^2 x^2} \cos ^{-1}(a x)-i \text {Li}_2\left (-i e^{i \cos ^{-1}(a x)}\right )+i \text {Li}_2\left (i e^{i \cos ^{-1}(a x)}\right )+a x-\cos ^{-1}(a x) \log \left (1-i e^{i \cos ^{-1}(a x)}\right )+\cos ^{-1}(a x) \log \left (1+i e^{i \cos ^{-1}(a x)}\right )\right )}{9 a^3}+\frac {e n \left (-12 \left (1-a^2 x^2\right )^{3/2} \cos ^{-1}(a x)-9 a x+\cos \left (3 \cos ^{-1}(a x)\right )\right )}{162 a^3}+\frac {e n \log (x) \left (-2 a^3 x^3+9 a^3 x^3 \cos ^{-1}(a x)^2-6 a^2 x^2 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)-12 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)-12 a x\right )}{27 a^3}+2 d n x+\frac {2}{81} e n x^3 \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.70, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (e x^{2} + d\right )} \arccos \left (a x\right )^{2} \log \left (c x^{n}\right ), x\right ) \]
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 \[ \int {\left (e x^{2} + d\right )} \arccos \left (a x\right )^{2} \log \left (c x^{n}\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 11.59, size = 0, normalized size = 0.00 \[ \int \left (e \,x^{2}+d \right ) \arccos \left (a x \right )^{2} \ln \left (c \,x^{n}\right )\, dx \]
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 \[ \frac {1}{3} \, {\left (e x^{3} + 3 \, d x\right )} \arctan \left (\sqrt {a x + 1} \sqrt {-a x + 1}, a x\right )^{2} \log \left (x^{n}\right ) - \frac {1}{9} \, {\left ({\left (e n - 3 \, e \log \relax (c)\right )} x^{3} + 9 \, {\left (d n - d \log \relax (c)\right )} x\right )} \arctan \left (\sqrt {a x + 1} \sqrt {-a x + 1}, a x\right )^{2} - \int \frac {2 \, {\left (3 \, {\left (a e x^{3} + 3 \, a d x\right )} \arctan \left (\sqrt {a x + 1} \sqrt {-a x + 1}, a x\right ) \log \left (x^{n}\right ) - {\left ({\left (a e n - 3 \, a e \log \relax (c)\right )} x^{3} + 9 \, {\left (a d n - a d \log \relax (c)\right )} x\right )} \arctan \left (\sqrt {a x + 1} \sqrt {-a x + 1}, a x\right )\right )} \sqrt {a x + 1} \sqrt {-a x + 1}}{9 \, {\left (a^{2} x^{2} - 1\right )}}\,{d x} \]
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 \[ \int \ln \left (c\,x^n\right )\,{\mathrm {acos}\left (a\,x\right )}^2\,\left (e\,x^2+d\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (d + e x^{2}\right ) \log {\left (c x^{n} \right )} \operatorname {acos}^{2}{\left (a x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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