Optimal. Leaf size=424 \[ -\frac {3}{2} c^2 d x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {c d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{2 b \sqrt {1-c^2 x^2}}-\frac {i c d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}}+b c d \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{x}+\frac {2 b c d \sqrt {d-c^2 d x^2} \log \left (1-e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}}+\frac {3 b c^3 d x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt {1-c^2 x^2}}-\frac {i b^2 c d \sqrt {d-c^2 d x^2} \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )}{\sqrt {1-c^2 x^2}}+\frac {1}{4} b^2 c^2 d x \sqrt {d-c^2 d x^2}-\frac {5 b^2 c d \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{4 \sqrt {1-c^2 x^2}} \]
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Rubi [A] time = 0.40, antiderivative size = 424, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 13, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.448, Rules used = {4695, 4647, 4641, 4627, 321, 216, 4683, 4625, 3717, 2190, 2279, 2391, 195} \[ -\frac {i b^2 c d \sqrt {d-c^2 d x^2} \text {PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )}{\sqrt {1-c^2 x^2}}+\frac {3 b c^3 d x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt {1-c^2 x^2}}-\frac {3}{2} c^2 d x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {c d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{2 b \sqrt {1-c^2 x^2}}-\frac {i c d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}}+b c d \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{x}+\frac {2 b c d \sqrt {d-c^2 d x^2} \log \left (1-e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}}+\frac {1}{4} b^2 c^2 d x \sqrt {d-c^2 d x^2}-\frac {5 b^2 c d \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{4 \sqrt {1-c^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 195
Rule 216
Rule 321
Rule 2190
Rule 2279
Rule 2391
Rule 3717
Rule 4625
Rule 4627
Rule 4641
Rule 4647
Rule 4683
Rule 4695
Rubi steps
\begin {align*} \int \frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{x^2} \, dx &=-\frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{x}-\left (3 c^2 d\right ) \int \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx+\frac {\left (2 b c d \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{x} \, dx}{\sqrt {1-c^2 x^2}}\\ &=b c d \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {3}{2} c^2 d x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{x}+\frac {\left (2 b c d \sqrt {d-c^2 d x^2}\right ) \int \frac {a+b \sin ^{-1}(c x)}{x} \, dx}{\sqrt {1-c^2 x^2}}-\frac {\left (3 c^2 d \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}} \, dx}{2 \sqrt {1-c^2 x^2}}-\frac {\left (b^2 c^2 d \sqrt {d-c^2 d x^2}\right ) \int \sqrt {1-c^2 x^2} \, dx}{\sqrt {1-c^2 x^2}}+\frac {\left (3 b c^3 d \sqrt {d-c^2 d x^2}\right ) \int x \left (a+b \sin ^{-1}(c x)\right ) \, dx}{\sqrt {1-c^2 x^2}}\\ &=-\frac {1}{2} b^2 c^2 d x \sqrt {d-c^2 d x^2}+\frac {3 b c^3 d x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt {1-c^2 x^2}}+b c d \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {3}{2} c^2 d x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{x}-\frac {c d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{2 b \sqrt {1-c^2 x^2}}+\frac {\left (2 b c d \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int (a+b x) \cot (x) \, dx,x,\sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (b^2 c^2 d \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{2 \sqrt {1-c^2 x^2}}-\frac {\left (3 b^2 c^4 d \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{2 \sqrt {1-c^2 x^2}}\\ &=\frac {1}{4} b^2 c^2 d x \sqrt {d-c^2 d x^2}-\frac {b^2 c d \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{2 \sqrt {1-c^2 x^2}}+\frac {3 b c^3 d x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt {1-c^2 x^2}}+b c d \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {3}{2} c^2 d x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {i c d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{x}-\frac {c d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{2 b \sqrt {1-c^2 x^2}}-\frac {\left (4 i b c d \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {e^{2 i x} (a+b x)}{1-e^{2 i x}} \, dx,x,\sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (3 b^2 c^2 d \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{4 \sqrt {1-c^2 x^2}}\\ &=\frac {1}{4} b^2 c^2 d x \sqrt {d-c^2 d x^2}-\frac {5 b^2 c d \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{4 \sqrt {1-c^2 x^2}}+\frac {3 b c^3 d x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt {1-c^2 x^2}}+b c d \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {3}{2} c^2 d x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {i c d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{x}-\frac {c d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{2 b \sqrt {1-c^2 x^2}}+\frac {2 b c d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {\left (2 b^2 c d \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}}\\ &=\frac {1}{4} b^2 c^2 d x \sqrt {d-c^2 d x^2}-\frac {5 b^2 c d \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{4 \sqrt {1-c^2 x^2}}+\frac {3 b c^3 d x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt {1-c^2 x^2}}+b c d \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {3}{2} c^2 d x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {i c d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{x}-\frac {c d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{2 b \sqrt {1-c^2 x^2}}+\frac {2 b c d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )}{\sqrt {1-c^2 x^2}}+\frac {\left (i b^2 c d \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )}{\sqrt {1-c^2 x^2}}\\ &=\frac {1}{4} b^2 c^2 d x \sqrt {d-c^2 d x^2}-\frac {5 b^2 c d \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{4 \sqrt {1-c^2 x^2}}+\frac {3 b c^3 d x^2 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt {1-c^2 x^2}}+b c d \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {3}{2} c^2 d x \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {i c d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{x}-\frac {c d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{2 b \sqrt {1-c^2 x^2}}+\frac {2 b c d \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {i b^2 c d \sqrt {d-c^2 d x^2} \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )}{\sqrt {1-c^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 2.67, size = 396, normalized size = 0.93 \[ \frac {36 a^2 c d^{3/2} x \sqrt {1-c^2 x^2} \tan ^{-1}\left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (c^2 x^2-1\right )}\right )-12 a^2 d \sqrt {1-c^2 x^2} \left (c^2 x^2+2\right ) \sqrt {d-c^2 d x^2}-24 a b d \sqrt {d-c^2 d x^2} \left (2 \sqrt {1-c^2 x^2} \sin ^{-1}(c x)-2 c x \log (c x)+c x \sin ^{-1}(c x)^2\right )-6 a b c d x \sqrt {d-c^2 d x^2} \left (2 \sin ^{-1}(c x) \left (\sin ^{-1}(c x)+\sin \left (2 \sin ^{-1}(c x)\right )\right )+\cos \left (2 \sin ^{-1}(c x)\right )\right )-8 b^2 d \sqrt {d-c^2 d x^2} \left (\sin ^{-1}(c x) \left (3 \sqrt {1-c^2 x^2} \sin ^{-1}(c x)+c x \left (\sin ^{-1}(c x)+3 i\right ) \sin ^{-1}(c x)-6 c x \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )\right )+3 i c x \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )\right )-b^2 c d x \sqrt {d-c^2 d x^2} \left (4 \sin ^{-1}(c x)^3+\left (6 \sin ^{-1}(c x)^2-3\right ) \sin \left (2 \sin ^{-1}(c x)\right )+6 \sin ^{-1}(c x) \cos \left (2 \sin ^{-1}(c x)\right )\right )}{24 x \sqrt {1-c^2 x^2}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (a^{2} c^{2} d x^{2} - a^{2} d + {\left (b^{2} c^{2} d x^{2} - b^{2} d\right )} \arcsin \left (c x\right )^{2} + 2 \, {\left (a b c^{2} d x^{2} - a b d\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}}{x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.47, size = 1148, normalized size = 2.71 \[ \frac {2 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) d}{\left (c^{2} x^{2}-1\right ) x}-\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \,c^{4} \arcsin \left (c x \right )^{2} x^{3}}{2 \left (c^{2} x^{2}-1\right )}-a^{2} c^{2} x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}-\frac {a^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{d x}+\frac {2 i a b \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) d c}{c^{2} x^{2}-1}-\frac {3 a^{2} c^{2} d x \sqrt {-c^{2} d \,x^{2}+d}}{2}-\frac {3 a^{2} c^{2} d^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 \sqrt {c^{2} d}}-\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \,c^{2} \arcsin \left (c x \right )^{2} x}{2 \left (c^{2} x^{2}-1\right )}+\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \,c^{4} x^{3}}{4 c^{2} x^{2}-4}-\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \,c^{2} x}{4 \left (c^{2} x^{2}-1\right )}+\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right )^{2} d}{\left (c^{2} x^{2}-1\right ) x}+\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{3} d c}{2 c^{2} x^{2}-2}+\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d c \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{4 c^{2} x^{2}-4}-\frac {2 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}-1\right ) d c}{c^{2} x^{2}-1}-\frac {2 b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d c \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{c^{2} x^{2}-1}-\frac {2 b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d c \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )}{c^{2} x^{2}-1}+\frac {i b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d c \arcsin \left (c x \right )^{2} \sqrt {-c^{2} x^{2}+1}}{c^{2} x^{2}-1}+\frac {a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d c \sqrt {-c^{2} x^{2}+1}}{4 c^{2} x^{2}-4}+\frac {2 i b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d c \sqrt {-c^{2} x^{2}+1}\, \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )}{c^{2} x^{2}-1}+\frac {2 i b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d c \sqrt {-c^{2} x^{2}+1}\, \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )}{c^{2} x^{2}-1}-\frac {b^{2} \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \,c^{3} \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, x^{2}}{2 \left (c^{2} x^{2}-1\right )}+\frac {3 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{2} d c}{2 \left (c^{2} x^{2}-1\right )}-\frac {a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \,c^{3} \sqrt {-c^{2} x^{2}+1}\, x^{2}}{2 \left (c^{2} x^{2}-1\right )}-\frac {a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \,c^{4} \arcsin \left (c x \right ) x^{3}}{c^{2} x^{2}-1}-\frac {a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \,c^{2} \arcsin \left (c x \right ) x}{c^{2} x^{2}-1} \]
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 \[ -\frac {1}{2} \, {\left (3 \, \sqrt {-c^{2} d x^{2} + d} c^{2} d x + 3 \, c d^{\frac {3}{2}} \arcsin \left (c x\right ) + \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}}{x}\right )} a^{2} - \sqrt {d} \int \frac {{\left ({\left (b^{2} c^{2} d x^{2} - b^{2} d\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )^{2} + 2 \, {\left (a b c^{2} d x^{2} - a b d\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )\right )} \sqrt {c x + 1} \sqrt {-c x + 1}}{x^{2}}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^{3/2}}{x^2} \,d x \]
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 \[ \int \frac {\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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