Optimal. Leaf size=482 \[ 2 d n x+\frac {2 e n x}{27 a^2}+\frac {4}{9} \left (9 d+\frac {2 e}{a^2}\right ) n x+\frac {2}{27} e n x^3-\frac {2 d n \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{a}-\frac {4 e n \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{27 a^3}-\frac {2 \left (9 a^2 d+2 e\right ) n \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{9 a^3}-\frac {2 e n x^2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{27 a}+\frac {2 e n \left (1-a^2 x^2\right )^{3/2} \sin ^{-1}(a x)}{27 a^3}-d n x \sin ^{-1}(a x)^2-\frac {1}{9} e n x^3 \sin ^{-1}(a x)^2+\frac {4 \left (9 a^2 d+2 e\right ) n \sin ^{-1}(a x) \tanh ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )}{9 a^3}-2 d x \log \left (c x^n\right )-\frac {4 e x \log \left (c x^n\right )}{9 a^2}-\frac {2}{27} e x^3 \log \left (c x^n\right )+\frac {2 d \sqrt {1-a^2 x^2} \sin ^{-1}(a x) \log \left (c x^n\right )}{a}+\frac {4 e \sqrt {1-a^2 x^2} \sin ^{-1}(a x) \log \left (c x^n\right )}{9 a^3}+\frac {2 e x^2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x) \log \left (c x^n\right )}{9 a}+d x \sin ^{-1}(a x)^2 \log \left (c x^n\right )+\frac {1}{3} e x^3 \sin ^{-1}(a x)^2 \log \left (c x^n\right )-\frac {2 i \left (9 a^2 d+2 e\right ) n \text {Li}_2\left (-e^{i \sin ^{-1}(a x)}\right )}{9 a^3}+\frac {2 i \left (9 a^2 d+2 e\right ) n \text {Li}_2\left (e^{i \sin ^{-1}(a x)}\right )}{9 a^3} \]
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Rubi [A]
time = 0.45, antiderivative size = 482, 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 = {4757, 4715, 4767, 8, 4723, 4795, 30, 2434, 6, 4783, 4803, 4268, 2317, 2438}
\begin {gather*} -\frac {2 i n \left (9 a^2 d+2 e\right ) \text {PolyLog}\left (2,-e^{i \text {ArcSin}(a x)}\right )}{9 a^3}+\frac {2 i n \left (9 a^2 d+2 e\right ) \text {PolyLog}\left (2,e^{i \text {ArcSin}(a x)}\right )}{9 a^3}+\frac {2 d \sqrt {1-a^2 x^2} \text {ArcSin}(a x) \log \left (c x^n\right )}{a}+\frac {2 e x^2 \sqrt {1-a^2 x^2} \text {ArcSin}(a x) \log \left (c x^n\right )}{9 a}-\frac {2 d n \sqrt {1-a^2 x^2} \text {ArcSin}(a x)}{a}-\frac {2 e n x^2 \sqrt {1-a^2 x^2} \text {ArcSin}(a x)}{27 a}-\frac {4 e x \log \left (c x^n\right )}{9 a^2}+\frac {4}{9} n x \left (\frac {2 e}{a^2}+9 d\right )+\frac {2 e n x}{27 a^2}+\frac {4 e \sqrt {1-a^2 x^2} \text {ArcSin}(a x) \log \left (c x^n\right )}{9 a^3}-\frac {2 n \sqrt {1-a^2 x^2} \text {ArcSin}(a x) \left (9 a^2 d+2 e\right )}{9 a^3}+\frac {4 n \text {ArcSin}(a x) \left (9 a^2 d+2 e\right ) \tanh ^{-1}\left (e^{i \text {ArcSin}(a x)}\right )}{9 a^3}+\frac {2 e n \left (1-a^2 x^2\right )^{3/2} \text {ArcSin}(a x)}{27 a^3}-\frac {4 e n \sqrt {1-a^2 x^2} \text {ArcSin}(a x)}{27 a^3}+d x \text {ArcSin}(a x)^2 \log \left (c x^n\right )+\frac {1}{3} e x^3 \text {ArcSin}(a x)^2 \log \left (c x^n\right )-d n x \text {ArcSin}(a x)^2-\frac {1}{9} e n x^3 \text {ArcSin}(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 \end {gather*}
Antiderivative was successfully verified.
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Rule 6
Rule 8
Rule 30
Rule 2317
Rule 2434
Rule 2438
Rule 4268
Rule 4715
Rule 4723
Rule 4757
Rule 4767
Rule 4783
Rule 4795
Rule 4803
Rubi steps
\begin {align*} \int \left (d+e x^2\right ) \sin ^{-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} \sin ^{-1}(a x) \log \left (c x^n\right )}{a}+\frac {4 e \sqrt {1-a^2 x^2} \sin ^{-1}(a x) \log \left (c x^n\right )}{9 a^3}+\frac {2 e x^2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x) \log \left (c x^n\right )}{9 a}+d x \sin ^{-1}(a x)^2 \log \left (c x^n\right )+\frac {1}{3} e x^3 \sin ^{-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} \sin ^{-1}(a x)}{a x}+\frac {4 e \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{9 a^3 x}+\frac {2 e x \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{9 a}+d \sin ^{-1}(a x)^2+\frac {1}{3} e x^2 \sin ^{-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} \sin ^{-1}(a x) \log \left (c x^n\right )}{a}+\frac {4 e \sqrt {1-a^2 x^2} \sin ^{-1}(a x) \log \left (c x^n\right )}{9 a^3}+\frac {2 e x^2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x) \log \left (c x^n\right )}{9 a}+d x \sin ^{-1}(a x)^2 \log \left (c x^n\right )+\frac {1}{3} e x^3 \sin ^{-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} \sin ^{-1}(a x)}{x}+\frac {2 e x \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{9 a}+d \sin ^{-1}(a x)^2+\frac {1}{3} e x^2 \sin ^{-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} \sin ^{-1}(a x) \log \left (c x^n\right )}{a}+\frac {4 e \sqrt {1-a^2 x^2} \sin ^{-1}(a x) \log \left (c x^n\right )}{9 a^3}+\frac {2 e x^2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x) \log \left (c x^n\right )}{9 a}+d x \sin ^{-1}(a x)^2 \log \left (c x^n\right )+\frac {1}{3} e x^3 \sin ^{-1}(a x)^2 \log \left (c x^n\right )-(d n) \int \sin ^{-1}(a x)^2 \, dx-\frac {1}{3} (e n) \int x^2 \sin ^{-1}(a x)^2 \, dx-\frac {(2 e n) \int x \sqrt {1-a^2 x^2} \sin ^{-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} \sin ^{-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} \sin ^{-1}(a x)}{9 a^3}+\frac {2 e n \left (1-a^2 x^2\right )^{3/2} \sin ^{-1}(a x)}{27 a^3}-d n x \sin ^{-1}(a x)^2-\frac {1}{9} e n x^3 \sin ^{-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} \sin ^{-1}(a x) \log \left (c x^n\right )}{a}+\frac {4 e \sqrt {1-a^2 x^2} \sin ^{-1}(a x) \log \left (c x^n\right )}{9 a^3}+\frac {2 e x^2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x) \log \left (c x^n\right )}{9 a}+d x \sin ^{-1}(a x)^2 \log \left (c x^n\right )+\frac {1}{3} e x^3 \sin ^{-1}(a x)^2 \log \left (c x^n\right )+(2 a d n) \int \frac {x \sin ^{-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 \sin ^{-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 {\sin ^{-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} \sin ^{-1}(a x)}{a}-\frac {2 \left (9 a^2 d+2 e\right ) n \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{9 a^3}-\frac {2 e n x^2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{27 a}+\frac {2 e n \left (1-a^2 x^2\right )^{3/2} \sin ^{-1}(a x)}{27 a^3}-d n x \sin ^{-1}(a x)^2-\frac {1}{9} e n x^3 \sin ^{-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} \sin ^{-1}(a x) \log \left (c x^n\right )}{a}+\frac {4 e \sqrt {1-a^2 x^2} \sin ^{-1}(a x) \log \left (c x^n\right )}{9 a^3}+\frac {2 e x^2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x) \log \left (c x^n\right )}{9 a}+d x \sin ^{-1}(a x)^2 \log \left (c x^n\right )+\frac {1}{3} e x^3 \sin ^{-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 \sin ^{-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 ) \text {Subst}\left (\int x \csc (x) \, dx,x,\sin ^{-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} \sin ^{-1}(a x)}{a}-\frac {4 e n \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{27 a^3}-\frac {2 \left (9 a^2 d+2 e\right ) n \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{9 a^3}-\frac {2 e n x^2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{27 a}+\frac {2 e n \left (1-a^2 x^2\right )^{3/2} \sin ^{-1}(a x)}{27 a^3}-d n x \sin ^{-1}(a x)^2-\frac {1}{9} e n x^3 \sin ^{-1}(a x)^2+\frac {4 \left (9 a^2 d+2 e\right ) n \sin ^{-1}(a x) \tanh ^{-1}\left (e^{i \sin ^{-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} \sin ^{-1}(a x) \log \left (c x^n\right )}{a}+\frac {4 e \sqrt {1-a^2 x^2} \sin ^{-1}(a x) \log \left (c x^n\right )}{9 a^3}+\frac {2 e x^2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x) \log \left (c x^n\right )}{9 a}+d x \sin ^{-1}(a x)^2 \log \left (c x^n\right )+\frac {1}{3} e x^3 \sin ^{-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 ) \text {Subst}\left (\int \log \left (1-e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{9 a^3}-\frac {\left (2 \left (9 a^2 d+2 e\right ) n\right ) \text {Subst}\left (\int \log \left (1+e^{i x}\right ) \, dx,x,\sin ^{-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} \sin ^{-1}(a x)}{a}-\frac {4 e n \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{27 a^3}-\frac {2 \left (9 a^2 d+2 e\right ) n \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{9 a^3}-\frac {2 e n x^2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{27 a}+\frac {2 e n \left (1-a^2 x^2\right )^{3/2} \sin ^{-1}(a x)}{27 a^3}-d n x \sin ^{-1}(a x)^2-\frac {1}{9} e n x^3 \sin ^{-1}(a x)^2+\frac {4 \left (9 a^2 d+2 e\right ) n \sin ^{-1}(a x) \tanh ^{-1}\left (e^{i \sin ^{-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} \sin ^{-1}(a x) \log \left (c x^n\right )}{a}+\frac {4 e \sqrt {1-a^2 x^2} \sin ^{-1}(a x) \log \left (c x^n\right )}{9 a^3}+\frac {2 e x^2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x) \log \left (c x^n\right )}{9 a}+d x \sin ^{-1}(a x)^2 \log \left (c x^n\right )+\frac {1}{3} e x^3 \sin ^{-1}(a x)^2 \log \left (c x^n\right )-\frac {\left (2 i \left (9 a^2 d+2 e\right ) n\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{i \sin ^{-1}(a x)}\right )}{9 a^3}+\frac {\left (2 i \left (9 a^2 d+2 e\right ) n\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i \sin ^{-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} \sin ^{-1}(a x)}{a}-\frac {4 e n \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{27 a^3}-\frac {2 \left (9 a^2 d+2 e\right ) n \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{9 a^3}-\frac {2 e n x^2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)}{27 a}+\frac {2 e n \left (1-a^2 x^2\right )^{3/2} \sin ^{-1}(a x)}{27 a^3}-d n x \sin ^{-1}(a x)^2-\frac {1}{9} e n x^3 \sin ^{-1}(a x)^2+\frac {4 \left (9 a^2 d+2 e\right ) n \sin ^{-1}(a x) \tanh ^{-1}\left (e^{i \sin ^{-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} \sin ^{-1}(a x) \log \left (c x^n\right )}{a}+\frac {4 e \sqrt {1-a^2 x^2} \sin ^{-1}(a x) \log \left (c x^n\right )}{9 a^3}+\frac {2 e x^2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x) \log \left (c x^n\right )}{9 a}+d x \sin ^{-1}(a x)^2 \log \left (c x^n\right )+\frac {1}{3} e x^3 \sin ^{-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 (-e^{i \sin ^{-1}(a x)}\right )}{9 a^3}+\frac {2 i \left (9 a^2 d+2 e\right ) n \text {Li}_2\left (e^{i \sin ^{-1}(a x)}\right )}{9 a^3}\\ \end {align*}
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Mathematica [A]
time = 0.60, size = 456, normalized size = 0.95 \begin {gather*} \frac {162 a^3 d n x+26 a e n x+2 a^3 e n x^3-108 a^2 d n \sqrt {1-a^2 x^2} \sin ^{-1}(a x)-14 e n \sqrt {1-a^2 x^2} \sin ^{-1}(a x)-4 a^2 e n x^2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)-27 a^3 d n x \sin ^{-1}(a x)^2-3 a^3 e n x^3 \sin ^{-1}(a x)^2-54 a^2 d n \sin ^{-1}(a x) \log \left (1-e^{i \sin ^{-1}(a x)}\right )-12 e n \sin ^{-1}(a x) \log \left (1-e^{i \sin ^{-1}(a x)}\right )+54 a^2 d n \sin ^{-1}(a x) \log \left (1+e^{i \sin ^{-1}(a x)}\right )+12 e n \sin ^{-1}(a x) \log \left (1+e^{i \sin ^{-1}(a x)}\right )-54 a^3 d x \log \left (c x^n\right )-12 a e x \log \left (c x^n\right )-2 a^3 e x^3 \log \left (c x^n\right )+54 a^2 d \sqrt {1-a^2 x^2} \sin ^{-1}(a x) \log \left (c x^n\right )+12 e \sqrt {1-a^2 x^2} \sin ^{-1}(a x) \log \left (c x^n\right )+6 a^2 e x^2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x) \log \left (c x^n\right )+27 a^3 d x \sin ^{-1}(a x)^2 \log \left (c x^n\right )+9 a^3 e x^3 \sin ^{-1}(a x)^2 \log \left (c x^n\right )-6 i \left (9 a^2 d+2 e\right ) n \text {Li}_2\left (-e^{i \sin ^{-1}(a x)}\right )+6 i \left (9 a^2 d+2 e\right ) n \text {Li}_2\left (e^{i \sin ^{-1}(a x)}\right )}{27 a^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 3.98, size = 0, normalized size = 0.00 \[\int \left (e \,x^{2}+d \right ) \arcsin \left (a x \right )^{2} \ln \left (c \,x^{n}\right )\, dx\]
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 \begin {gather*} \text {Failed to integrate} \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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (d + e x^{2}\right ) \log {\left (c x^{n} \right )} \operatorname {asin}^{2}{\left (a x \right )}\, dx \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 \ln \left (c\,x^n\right )\,{\mathrm {asin}\left (a\,x\right )}^2\,\left (e\,x^2+d\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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