Optimal. Leaf size=354 \[ -\frac {1}{18} b c d^3 x \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {7}{36} b c d^3 x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {19}{24} b c d^3 x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{6} d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{4} d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{2} d^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-i b d^3 \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )-\frac {i d^3 \left (a+b \sin ^{-1}(c x)\right )^3}{3 b}-\frac {19}{48} d^3 \left (a+b \sin ^{-1}(c x)\right )^2+d^3 \log \left (1-e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac {7}{144} b^2 c^4 d^3 x^4+\frac {71}{144} b^2 c^2 d^3 x^2-\frac {1}{108} b^2 d^3 \left (1-c^2 x^2\right )^3+\frac {1}{2} b^2 d^3 \text {Li}_3\left (e^{2 i \sin ^{-1}(c x)}\right ) \]
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Rubi [A] time = 0.66, antiderivative size = 354, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 13, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.482, Rules used = {4699, 4625, 3717, 2190, 2531, 2282, 6589, 4647, 4641, 30, 4649, 14, 261} \[ -i b d^3 \text {PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{2} b^2 d^3 \text {PolyLog}\left (3,e^{2 i \sin ^{-1}(c x)}\right )-\frac {1}{18} b c d^3 x \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {7}{36} b c d^3 x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {19}{24} b c d^3 x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{6} d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{4} d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{2} d^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac {i d^3 \left (a+b \sin ^{-1}(c x)\right )^3}{3 b}-\frac {19}{48} d^3 \left (a+b \sin ^{-1}(c x)\right )^2+d^3 \log \left (1-e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac {7}{144} b^2 c^4 d^3 x^4+\frac {71}{144} b^2 c^2 d^3 x^2-\frac {1}{108} b^2 d^3 \left (1-c^2 x^2\right )^3 \]
Antiderivative was successfully verified.
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Rule 14
Rule 30
Rule 261
Rule 2190
Rule 2282
Rule 2531
Rule 3717
Rule 4625
Rule 4641
Rule 4647
Rule 4649
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} \, dx &=\frac {1}{6} d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2+d \int \frac {\left (d-c^2 d x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2}{x} \, dx-\frac {1}{3} \left (b c d^3\right ) \int \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx\\ &=-\frac {1}{18} b c d^3 x \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{6} d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2+d^2 \int \frac {\left (d-c^2 d x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{x} \, dx-\frac {1}{18} \left (5 b c d^3\right ) \int \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx-\frac {1}{2} \left (b c d^3\right ) \int \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx+\frac {1}{18} \left (b^2 c^2 d^3\right ) \int x \left (1-c^2 x^2\right )^2 \, dx\\ &=-\frac {1}{108} b^2 d^3 \left (1-c^2 x^2\right )^3-\frac {7}{36} b c d^3 x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{18} b c d^3 x \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{2} d^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{4} d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{6} d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2+d^3 \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{x} \, dx-\frac {1}{24} \left (5 b c 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 c d^3\right ) \int \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx-\left (b c d^3\right ) \int \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx+\frac {1}{72} \left (5 b^2 c^2 d^3\right ) \int x \left (1-c^2 x^2\right ) \, dx+\frac {1}{8} \left (b^2 c^2 d^3\right ) \int x \left (1-c^2 x^2\right ) \, dx\\ &=-\frac {1}{108} b^2 d^3 \left (1-c^2 x^2\right )^3-\frac {19}{24} b c d^3 x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {7}{36} b c d^3 x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{18} b c d^3 x \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{2} d^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{4} d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{6} d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2+d^3 \operatorname {Subst}\left (\int (a+b x)^2 \cot (x) \, dx,x,\sin ^{-1}(c x)\right )-\frac {1}{48} \left (5 b c d^3\right ) \int \frac {a+b \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx-\frac {1}{16} \left (3 b c d^3\right ) \int \frac {a+b \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx-\frac {1}{2} \left (b c d^3\right ) \int \frac {a+b \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx+\frac {1}{72} \left (5 b^2 c^2 d^3\right ) \int \left (x-c^2 x^3\right ) \, dx+\frac {1}{48} \left (5 b^2 c^2 d^3\right ) \int x \, dx+\frac {1}{8} \left (b^2 c^2 d^3\right ) \int \left (x-c^2 x^3\right ) \, dx+\frac {1}{16} \left (3 b^2 c^2 d^3\right ) \int x \, dx+\frac {1}{2} \left (b^2 c^2 d^3\right ) \int x \, dx\\ &=\frac {71}{144} b^2 c^2 d^3 x^2-\frac {7}{144} b^2 c^4 d^3 x^4-\frac {1}{108} b^2 d^3 \left (1-c^2 x^2\right )^3-\frac {19}{24} b c d^3 x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {7}{36} b c d^3 x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{18} b c d^3 x \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {19}{48} d^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{2} d^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{4} d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{6} d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2-\frac {i d^3 \left (a+b \sin ^{-1}(c x)\right )^3}{3 b}-\left (2 i 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 {71}{144} b^2 c^2 d^3 x^2-\frac {7}{144} b^2 c^4 d^3 x^4-\frac {1}{108} b^2 d^3 \left (1-c^2 x^2\right )^3-\frac {19}{24} b c d^3 x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {7}{36} b c d^3 x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{18} b c d^3 x \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {19}{48} d^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{2} d^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{4} d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{6} d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2-\frac {i d^3 \left (a+b \sin ^{-1}(c x)\right )^3}{3 b}+d^3 \left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-\left (2 b 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 {71}{144} b^2 c^2 d^3 x^2-\frac {7}{144} b^2 c^4 d^3 x^4-\frac {1}{108} b^2 d^3 \left (1-c^2 x^2\right )^3-\frac {19}{24} b c d^3 x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {7}{36} b c d^3 x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{18} b c d^3 x \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {19}{48} d^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{2} d^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{4} d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{6} d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2-\frac {i d^3 \left (a+b \sin ^{-1}(c x)\right )^3}{3 b}+d^3 \left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-i b d^3 \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )+\left (i b^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 {71}{144} b^2 c^2 d^3 x^2-\frac {7}{144} b^2 c^4 d^3 x^4-\frac {1}{108} b^2 d^3 \left (1-c^2 x^2\right )^3-\frac {19}{24} b c d^3 x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {7}{36} b c d^3 x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{18} b c d^3 x \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {19}{48} d^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{2} d^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{4} d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{6} d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2-\frac {i d^3 \left (a+b \sin ^{-1}(c x)\right )^3}{3 b}+d^3 \left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-i b 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 (b^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 {71}{144} b^2 c^2 d^3 x^2-\frac {7}{144} b^2 c^4 d^3 x^4-\frac {1}{108} b^2 d^3 \left (1-c^2 x^2\right )^3-\frac {19}{24} b c d^3 x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac {7}{36} b c d^3 x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{18} b c d^3 x \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )-\frac {19}{48} d^3 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{2} d^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{4} d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{6} d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )^2-\frac {i d^3 \left (a+b \sin ^{-1}(c x)\right )^3}{3 b}+d^3 \left (a+b \sin ^{-1}(c x)\right )^2 \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-i b 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} b^2 d^3 \text {Li}_3\left (e^{2 i \sin ^{-1}(c x)}\right )\\ \end {align*}
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Mathematica [A] time = 0.81, size = 448, normalized size = 1.27 \[ \frac {d^3 \left (-576 a^2 c^6 x^6+2592 a^2 c^4 x^4-5184 a^2 c^2 x^2+3456 a^2 \log (c x)-1152 a b c^6 x^6 \sin ^{-1}(c x)+5184 a b c^4 x^4 \sin ^{-1}(c x)-3600 a b c x \sqrt {1-c^2 x^2}-10368 a b c^2 x^2 \sin ^{-1}(c x)-192 a b c^5 x^5 \sqrt {1-c^2 x^2}+1056 a b c^3 x^3 \sqrt {1-c^2 x^2}-3456 i a b \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )-3456 i a b \sin ^{-1}(c x)^2+3600 a b \sin ^{-1}(c x)+6912 a b \sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )+3456 i b^2 \sin ^{-1}(c x) \text {Li}_2\left (e^{-2 i \sin ^{-1}(c x)}\right )+1728 b^2 \text {Li}_3\left (e^{-2 i \sin ^{-1}(c x)}\right )+1152 i b^2 \sin ^{-1}(c x)^3-1566 b^2 \sin ^{-1}(c x) \sin \left (2 \sin ^{-1}(c x)\right )-108 b^2 \sin ^{-1}(c x) \sin \left (4 \sin ^{-1}(c x)\right )-6 b^2 \sin ^{-1}(c x) \sin \left (6 \sin ^{-1}(c x)\right )+3456 b^2 \sin ^{-1}(c x)^2 \log \left (1-e^{-2 i \sin ^{-1}(c x)}\right )-783 b^2 \cos \left (2 \sin ^{-1}(c x)\right )+1566 b^2 \sin ^{-1}(c x)^2 \cos \left (2 \sin ^{-1}(c x)\right )-27 b^2 \cos \left (4 \sin ^{-1}(c x)\right )+216 b^2 \sin ^{-1}(c x)^2 \cos \left (4 \sin ^{-1}(c x)\right )-b^2 \cos \left (6 \sin ^{-1}(c x)\right )+18 b^2 \sin ^{-1}(c x)^2 \cos \left (6 \sin ^{-1}(c x)\right )-144 i \pi ^3 b^2\right )}{3456} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.74, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {a^{2} c^{6} d^{3} x^{6} - 3 \, a^{2} c^{4} d^{3} x^{4} + 3 \, a^{2} c^{2} d^{3} x^{2} - a^{2} d^{3} + {\left (b^{2} c^{6} d^{3} x^{6} - 3 \, b^{2} c^{4} d^{3} x^{4} + 3 \, b^{2} c^{2} d^{3} x^{2} - b^{2} d^{3}\right )} \arcsin \left (c x\right )^{2} + 2 \, {\left (a b c^{6} d^{3} x^{6} - 3 \, a b c^{4} d^{3} x^{4} + 3 \, a b c^{2} d^{3} x^{2} - a b d^{3}\right )} \arcsin \left (c x\right )}{x}, x\right ) \]
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 \[ \int -\frac {{\left (c^{2} d x^{2} - d\right )}^{3} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.58, size = 661, normalized size = 1.87 \[ \frac {d^{3} a b \arcsin \left (c x \right ) \cos \left (6 \arcsin \left (c x \right )\right )}{96}+\frac {d^{3} a b \arcsin \left (c x \right ) \cos \left (4 \arcsin \left (c x \right )\right )}{8}+\frac {29 d^{3} a b \cos \left (2 \arcsin \left (c x \right )\right ) \arcsin \left (c x \right )}{32}+2 d^{3} a b \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 d^{3} a b \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-i d^{3} a b \arcsin \left (c x \right )^{2}-2 i d^{3} b^{2} \arcsin \left (c x \right ) \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 i d^{3} a b \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-2 i d^{3} a b \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 i d^{3} b^{2} \arcsin \left (c x \right ) \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-\frac {d^{3} a b \sin \left (6 \arcsin \left (c x \right )\right )}{576}-\frac {d^{3} a b \sin \left (4 \arcsin \left (c x \right )\right )}{32}-\frac {29 d^{3} a b \sin \left (2 \arcsin \left (c x \right )\right )}{64}+\frac {d^{3} b^{2} \cos \left (6 \arcsin \left (c x \right )\right ) \arcsin \left (c x \right )^{2}}{192}-\frac {d^{3} b^{2} \arcsin \left (c x \right ) \sin \left (6 \arcsin \left (c x \right )\right )}{576}+\frac {d^{3} b^{2} \cos \left (4 \arcsin \left (c x \right )\right ) \arcsin \left (c x \right )^{2}}{16}-\frac {d^{3} b^{2} \arcsin \left (c x \right ) \sin \left (4 \arcsin \left (c x \right )\right )}{32}+\frac {29 d^{3} b^{2} \arcsin \left (c x \right )^{2} \cos \left (2 \arcsin \left (c x \right )\right )}{64}-\frac {29 d^{3} b^{2} \arcsin \left (c x \right ) \sin \left (2 \arcsin \left (c x \right )\right )}{64}-\frac {3 d^{3} a^{2} c^{2} x^{2}}{2}+\frac {3 d^{3} a^{2} c^{4} x^{4}}{4}-\frac {d^{3} a^{2} c^{6} x^{6}}{6}+d^{3} b^{2} \arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+d^{3} b^{2} \arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+2 d^{3} b^{2} \polylog \left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+2 d^{3} b^{2} \polylog \left (3, i c x +\sqrt {-c^{2} x^{2}+1}\right )-\frac {i d^{3} b^{2} \arcsin \left (c x \right )^{3}}{3}+d^{3} a^{2} \ln \left (c x \right )-\frac {d^{3} b^{2} \cos \left (6 \arcsin \left (c x \right )\right )}{3456}-\frac {d^{3} b^{2} \cos \left (4 \arcsin \left (c x \right )\right )}{128}-\frac {29 d^{3} b^{2} \cos \left (2 \arcsin \left (c x \right )\right )}{128} \]
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 \[ -\frac {1}{6} \, a^{2} c^{6} d^{3} x^{6} + \frac {3}{4} \, a^{2} c^{4} d^{3} x^{4} - \frac {3}{2} \, a^{2} c^{2} d^{3} x^{2} + a^{2} d^{3} \log \relax (x) - \int \frac {{\left (b^{2} c^{6} d^{3} x^{6} - 3 \, b^{2} c^{4} d^{3} x^{4} + 3 \, b^{2} c^{2} d^{3} x^{2} - b^{2} d^{3}\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )^{2} + 2 \, {\left (a b c^{6} d^{3} x^{6} - 3 \, a b c^{4} d^{3} x^{4} + 3 \, a b c^{2} d^{3} x^{2} - a b d^{3}\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )}{x}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^3}{x} \,d x \]
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 \[ - d^{3} \left (\int \left (- \frac {a^{2}}{x}\right )\, dx + \int 3 a^{2} c^{2} x\, dx + \int \left (- 3 a^{2} c^{4} x^{3}\right )\, dx + \int a^{2} c^{6} x^{5}\, dx + \int \left (- \frac {b^{2} \operatorname {asin}^{2}{\left (c x \right )}}{x}\right )\, dx + \int \left (- \frac {2 a b \operatorname {asin}{\left (c x \right )}}{x}\right )\, dx + \int 3 b^{2} c^{2} x \operatorname {asin}^{2}{\left (c x \right )}\, dx + \int \left (- 3 b^{2} c^{4} x^{3} \operatorname {asin}^{2}{\left (c x \right )}\right )\, dx + \int b^{2} c^{6} x^{5} \operatorname {asin}^{2}{\left (c x \right )}\, dx + \int 6 a b c^{2} x \operatorname {asin}{\left (c x \right )}\, dx + \int \left (- 6 a b c^{4} x^{3} \operatorname {asin}{\left (c x \right )}\right )\, dx + \int 2 a b c^{6} x^{5} \operatorname {asin}{\left (c x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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