Optimal. Leaf size=371 \[ 3 i b c^2 d^3 \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )-\frac {3}{4} c^2 d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2-\frac {3}{2} c^2 d^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac {b c d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{x}-\frac {d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{2 x^2}+\frac {i c^2 d^3 \left (a+b \sin ^{-1}(c x)\right )^3}{b}+\frac {3}{32} c^2 d^3 \left (a+b \sin ^{-1}(c x)\right )^2-3 c^2 d^3 \log \left (1-e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac {7}{8} b c^3 d^3 x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )+\frac {3}{16} b c^3 d^3 x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{32} b^2 c^6 d^3 x^4-\frac {21}{32} b^2 c^4 d^3 x^2-\frac {3}{2} b^2 c^2 d^3 \text {Li}_3\left (e^{2 i \sin ^{-1}(c x)}\right )+b^2 c^2 d^3 \log (x) \]
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Rubi [A] time = 0.72, antiderivative size = 371, normalized size of antiderivative = 1.00, number of steps used = 28, number of rules used = 15, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {4695, 4699, 4625, 3717, 2190, 2531, 2282, 6589, 4647, 4641, 30, 4649, 14, 266, 43} \[ 3 i b c^2 d^3 \text {PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )-\frac {3}{2} b^2 c^2 d^3 \text {PolyLog}\left (3,e^{2 i \sin ^{-1}(c x)}\right )-\frac {7}{8} b c^3 d^3 x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )+\frac {3}{16} b c^3 d^3 x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {3}{4} c^2 d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2-\frac {3}{2} c^2 d^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac {b c d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{x}-\frac {d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{2 x^2}+\frac {i c^2 d^3 \left (a+b \sin ^{-1}(c x)\right )^3}{b}+\frac {3}{32} c^2 d^3 \left (a+b \sin ^{-1}(c x)\right )^2-3 c^2 d^3 \log \left (1-e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{32} b^2 c^6 d^3 x^4-\frac {21}{32} b^2 c^4 d^3 x^2+b^2 c^2 d^3 \log (x) \]
Antiderivative was successfully verified.
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Rule 14
Rule 30
Rule 43
Rule 266
Rule 2190
Rule 2282
Rule 2531
Rule 3717
Rule 4625
Rule 4641
Rule 4647
Rule 4649
Rule 4695
Rule 4699
Rule 6589
Rubi steps
\begin {align*} \int \frac {\left (d-c^2 d x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{x^3} \, dx &=-\frac {d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{2 x^2}-\left (3 c^2 d\right ) \int \frac {\left (d-c^2 d x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{x} \, dx+\left (b c d^3\right ) \int \frac {\left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{x^2} \, dx\\ &=-\frac {b c d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{x}-\frac {3}{4} c^2 d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2-\frac {d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{2 x^2}-\left (3 c^2 d^2\right ) \int \frac {\left (d-c^2 d x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{x} \, dx+\left (b^2 c^2 d^3\right ) \int \frac {\left (1-c^2 x^2\right )^2}{x} \, dx+\frac {1}{2} \left (3 b c^3 d^3\right ) \int \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx-\left (5 b c^3 d^3\right ) \int \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx\\ &=-\frac {7}{8} b c^3 d^3 x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {b c d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{x}-\frac {3}{2} c^2 d^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac {3}{4} c^2 d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2-\frac {d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{2 x^2}-\left (3 c^2 d^3\right ) \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{x} \, dx+\frac {1}{2} \left (b^2 c^2 d^3\right ) \operatorname {Subst}\left (\int \frac {\left (1-c^2 x\right )^2}{x} \, dx,x,x^2\right )+\frac {1}{8} \left (9 b c^3 d^3\right ) \int \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx+\left (3 b c^3 d^3\right ) \int \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx-\frac {1}{4} \left (15 b c^3 d^3\right ) \int \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx-\frac {1}{8} \left (3 b^2 c^4 d^3\right ) \int x \left (1-c^2 x^2\right ) \, dx+\frac {1}{4} \left (5 b^2 c^4 d^3\right ) \int x \left (1-c^2 x^2\right ) \, dx\\ &=\frac {3}{16} b c^3 d^3 x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {7}{8} b c^3 d^3 x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {b c d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{x}-\frac {3}{2} c^2 d^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac {3}{4} c^2 d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2-\frac {d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{2 x^2}-\left (3 c^2 d^3\right ) \operatorname {Subst}\left (\int (a+b x)^2 \cot (x) \, dx,x,\sin ^{-1}(c x)\right )+\frac {1}{2} \left (b^2 c^2 d^3\right ) \operatorname {Subst}\left (\int \left (-2 c^2+\frac {1}{x}+c^4 x\right ) \, dx,x,x^2\right )+\frac {1}{16} \left (9 b c^3 d^3\right ) \int \frac {a+b \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx+\frac {1}{2} \left (3 b c^3 d^3\right ) \int \frac {a+b \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx-\frac {1}{8} \left (15 b c^3 d^3\right ) \int \frac {a+b \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx-\frac {1}{8} \left (3 b^2 c^4 d^3\right ) \int \left (x-c^2 x^3\right ) \, dx-\frac {1}{16} \left (9 b^2 c^4 d^3\right ) \int x \, dx+\frac {1}{4} \left (5 b^2 c^4 d^3\right ) \int \left (x-c^2 x^3\right ) \, dx-\frac {1}{2} \left (3 b^2 c^4 d^3\right ) \int x \, dx+\frac {1}{8} \left (15 b^2 c^4 d^3\right ) \int x \, dx\\ &=-\frac {21}{32} b^2 c^4 d^3 x^2+\frac {1}{32} b^2 c^6 d^3 x^4+\frac {3}{16} b c^3 d^3 x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {7}{8} b c^3 d^3 x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {b c d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{x}+\frac {3}{32} c^2 d^3 \left (a+b \sin ^{-1}(c x)\right )^2-\frac {3}{2} c^2 d^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac {3}{4} c^2 d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2-\frac {d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{2 x^2}+\frac {i c^2 d^3 \left (a+b \sin ^{-1}(c x)\right )^3}{b}+b^2 c^2 d^3 \log (x)+\left (6 i c^2 d^3\right ) \operatorname {Subst}\left (\int \frac {e^{2 i x} (a+b x)^2}{1-e^{2 i x}} \, dx,x,\sin ^{-1}(c x)\right )\\ &=-\frac {21}{32} b^2 c^4 d^3 x^2+\frac {1}{32} b^2 c^6 d^3 x^4+\frac {3}{16} b c^3 d^3 x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {7}{8} b c^3 d^3 x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {b c d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{x}+\frac {3}{32} c^2 d^3 \left (a+b \sin ^{-1}(c x)\right )^2-\frac {3}{2} c^2 d^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac {3}{4} c^2 d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2-\frac {d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{2 x^2}+\frac {i c^2 d^3 \left (a+b \sin ^{-1}(c x)\right )^3}{b}-3 c^2 d^3 \left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )+b^2 c^2 d^3 \log (x)+\left (6 b c^2 d^3\right ) \operatorname {Subst}\left (\int (a+b x) \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )\\ &=-\frac {21}{32} b^2 c^4 d^3 x^2+\frac {1}{32} b^2 c^6 d^3 x^4+\frac {3}{16} b c^3 d^3 x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {7}{8} b c^3 d^3 x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {b c d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{x}+\frac {3}{32} c^2 d^3 \left (a+b \sin ^{-1}(c x)\right )^2-\frac {3}{2} c^2 d^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac {3}{4} c^2 d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2-\frac {d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{2 x^2}+\frac {i c^2 d^3 \left (a+b \sin ^{-1}(c x)\right )^3}{b}-3 c^2 d^3 \left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )+b^2 c^2 d^3 \log (x)+3 i b c^2 d^3 \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )-\left (3 i b^2 c^2 d^3\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )\\ &=-\frac {21}{32} b^2 c^4 d^3 x^2+\frac {1}{32} b^2 c^6 d^3 x^4+\frac {3}{16} b c^3 d^3 x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {7}{8} b c^3 d^3 x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {b c d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{x}+\frac {3}{32} c^2 d^3 \left (a+b \sin ^{-1}(c x)\right )^2-\frac {3}{2} c^2 d^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac {3}{4} c^2 d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2-\frac {d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{2 x^2}+\frac {i c^2 d^3 \left (a+b \sin ^{-1}(c x)\right )^3}{b}-3 c^2 d^3 \left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )+b^2 c^2 d^3 \log (x)+3 i b c^2 d^3 \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )-\frac {1}{2} \left (3 b^2 c^2 d^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )\\ &=-\frac {21}{32} b^2 c^4 d^3 x^2+\frac {1}{32} b^2 c^6 d^3 x^4+\frac {3}{16} b c^3 d^3 x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {7}{8} b c^3 d^3 x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {b c d^3 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{x}+\frac {3}{32} c^2 d^3 \left (a+b \sin ^{-1}(c x)\right )^2-\frac {3}{2} c^2 d^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac {3}{4} c^2 d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2-\frac {d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2}{2 x^2}+\frac {i c^2 d^3 \left (a+b \sin ^{-1}(c x)\right )^3}{b}-3 c^2 d^3 \left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )+b^2 c^2 d^3 \log (x)+3 i b c^2 d^3 \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )-\frac {3}{2} b^2 c^2 d^3 \text {Li}_3\left (e^{2 i \sin ^{-1}(c x)}\right )\\ \end {align*}
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Mathematica [A] time = 1.32, size = 494, normalized size = 1.33 \[ \frac {1}{256} d^3 \left (-64 a^2 c^6 x^4+384 a^2 c^4 x^2-768 a^2 c^2 \log (x)-\frac {128 a^2}{x^2}-128 a b c^6 x^4 \sin ^{-1}(c x)+768 a b c^4 x^2 \sin ^{-1}(c x)+768 i a b c^2 \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )-\frac {256 a b c \sqrt {1-c^2 x^2}}{x}+768 i a b c^2 \sin ^{-1}(c x)^2-336 a b c^2 \sin ^{-1}(c x)-1536 a b c^2 \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-32 a b c^5 x^3 \sqrt {1-c^2 x^2}+336 a b c^3 x \sqrt {1-c^2 x^2}-\frac {256 a b \sin ^{-1}(c x)}{x^2}-768 i b^2 c^2 \sin ^{-1}(c x) \text {Li}_2\left (e^{-2 i \sin ^{-1}(c x)}\right )-384 b^2 c^2 \text {Li}_3\left (e^{-2 i \sin ^{-1}(c x)}\right )-\frac {256 b^2 c \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{x}+256 b^2 c^2 \log (c x)-256 i b^2 c^2 \sin ^{-1}(c x)^3+160 b^2 c^2 \sin ^{-1}(c x) \sin \left (2 \sin ^{-1}(c x)\right )+4 b^2 c^2 \sin ^{-1}(c x) \sin \left (4 \sin ^{-1}(c x)\right )-768 b^2 c^2 \sin ^{-1}(c x)^2 \log \left (1-e^{-2 i \sin ^{-1}(c x)}\right )+80 b^2 c^2 \cos \left (2 \sin ^{-1}(c x)\right )-160 b^2 c^2 \sin ^{-1}(c x)^2 \cos \left (2 \sin ^{-1}(c x)\right )+b^2 c^2 \cos \left (4 \sin ^{-1}(c x)\right )-8 b^2 c^2 \sin ^{-1}(c x)^2 \cos \left (4 \sin ^{-1}(c x)\right )+32 i \pi ^3 b^2 c^2-\frac {128 b^2 \sin ^{-1}(c x)^2}{x^2}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {a^{2} c^{6} d^{3} x^{6} - 3 \, a^{2} c^{4} d^{3} x^{4} + 3 \, a^{2} c^{2} d^{3} x^{2} - a^{2} d^{3} + {\left (b^{2} c^{6} d^{3} x^{6} - 3 \, b^{2} c^{4} d^{3} x^{4} + 3 \, b^{2} c^{2} d^{3} x^{2} - b^{2} d^{3}\right )} \arcsin \left (c x\right )^{2} + 2 \, {\left (a b c^{6} d^{3} x^{6} - 3 \, a b c^{4} d^{3} x^{4} + 3 \, a b c^{2} d^{3} x^{2} - a b d^{3}\right )} \arcsin \left (c x\right )}{x^{3}}, x\right ) \]
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 \[ \int -\frac {{\left (c^{2} d x^{2} - d\right )}^{3} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.02, size = 884, normalized size = 2.38 \[ -6 c^{2} d^{3} a b \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+3 i c^{2} d^{3} a b \arcsin \left (c x \right )^{2}+6 i c^{2} d^{3} b^{2} \arcsin \left (c x \right ) \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )+6 i c^{2} d^{3} b^{2} \arcsin \left (c x \right ) \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+6 i c^{2} d^{3} a b \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+6 i c^{2} d^{3} a b \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )-\frac {d^{3} a^{2}}{2 x^{2}}+\frac {c^{2} d^{3} b^{2} \cos \left (4 \arcsin \left (c x \right )\right )}{256}+\frac {3 c^{4} d^{3} a^{2} x^{2}}{2}-\frac {c^{6} d^{3} a^{2} x^{4}}{4}-\frac {d^{3} b^{2} \arcsin \left (c x \right )^{2}}{2 x^{2}}-\frac {5 c^{2} d^{3} b^{2} \arcsin \left (c x \right )^{2}}{8}-3 c^{2} d^{3} a^{2} \ln \left (c x \right )+c^{2} d^{3} b^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 c^{2} d^{3} b^{2} \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )+c^{2} d^{3} b^{2} \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )-6 c^{2} d^{3} b^{2} \polylog \left (3, i c x +\sqrt {-c^{2} x^{2}+1}\right )-6 c^{2} d^{3} b^{2} \polylog \left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-\frac {5 b^{2} c^{4} d^{3} x^{2}}{8}+\frac {c^{2} d^{3} a b \sin \left (4 \arcsin \left (c x \right )\right )}{64}-\frac {5 c^{2} d^{3} a b \arcsin \left (c x \right )}{4}-\frac {c^{2} d^{3} b^{2} \cos \left (4 \arcsin \left (c x \right )\right ) \arcsin \left (c x \right )^{2}}{32}+\frac {c^{2} d^{3} b^{2} \arcsin \left (c x \right ) \sin \left (4 \arcsin \left (c x \right )\right )}{64}-\frac {d^{3} a b \arcsin \left (c x \right )}{x^{2}}+i c^{2} d^{3} a b +i c^{2} d^{3} b^{2} \arcsin \left (c x \right )^{3}+i c^{2} d^{3} b^{2} \arcsin \left (c x \right )-3 c^{2} d^{3} b^{2} \arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-3 c^{2} d^{3} b^{2} \arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+\frac {5 c^{4} d^{3} b^{2} \arcsin \left (c x \right )^{2} x^{2}}{4}+\frac {5 d^{3} b^{2} c^{2}}{16}-\frac {c^{2} d^{3} a b \arcsin \left (c x \right ) \cos \left (4 \arcsin \left (c x \right )\right )}{16}+\frac {5 c^{3} d^{3} b^{2} \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x}{4}-\frac {c \,d^{3} b^{2} \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{x}+\frac {5 c^{3} d^{3} a b \sqrt {-c^{2} x^{2}+1}\, x}{4}+\frac {5 c^{4} d^{3} a b \arcsin \left (c x \right ) x^{2}}{2}-\frac {c \,d^{3} a b \sqrt {-c^{2} x^{2}+1}}{x}-6 c^{2} d^{3} a b \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right ) \]
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 \[ -\frac {1}{4} \, a^{2} c^{6} d^{3} x^{4} + \frac {3}{2} \, a^{2} c^{4} d^{3} x^{2} - 3 \, a^{2} c^{2} d^{3} \log \relax (x) - a b d^{3} {\left (\frac {\sqrt {-c^{2} x^{2} + 1} c}{x} + \frac {\arcsin \left (c x\right )}{x^{2}}\right )} - \frac {a^{2} d^{3}}{2 \, x^{2}} - \int \frac {{\left (b^{2} c^{6} d^{3} x^{6} - 3 \, b^{2} c^{4} d^{3} x^{4} + 3 \, b^{2} c^{2} d^{3} x^{2} - b^{2} d^{3}\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )^{2} + 2 \, {\left (a b c^{6} d^{3} x^{6} - 3 \, a b c^{4} d^{3} x^{4} + 3 \, a b c^{2} d^{3} x^{2}\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )}{x^{3}}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^3}{x^3} \,d x \]
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 \[ - d^{3} \left (\int \left (- \frac {a^{2}}{x^{3}}\right )\, dx + \int \frac {3 a^{2} c^{2}}{x}\, dx + \int \left (- 3 a^{2} c^{4} x\right )\, dx + \int a^{2} c^{6} x^{3}\, dx + \int \left (- \frac {b^{2} \operatorname {asin}^{2}{\left (c x \right )}}{x^{3}}\right )\, dx + \int \left (- \frac {2 a b \operatorname {asin}{\left (c x \right )}}{x^{3}}\right )\, dx + \int \frac {3 b^{2} c^{2} \operatorname {asin}^{2}{\left (c x \right )}}{x}\, dx + \int \left (- 3 b^{2} c^{4} x \operatorname {asin}^{2}{\left (c x \right )}\right )\, dx + \int b^{2} c^{6} x^{3} \operatorname {asin}^{2}{\left (c x \right )}\, dx + \int \frac {6 a b c^{2} \operatorname {asin}{\left (c x \right )}}{x}\, dx + \int \left (- 6 a b c^{4} x \operatorname {asin}{\left (c x \right )}\right )\, dx + \int 2 a b c^{6} x^{3} \operatorname {asin}{\left (c x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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