Optimal. Leaf size=235 \[ \frac {1}{6} d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{2} d^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )-\frac {i d^3 \left (a+b \sin ^{-1}(c x)\right )^2}{2 b}+d^3 \log \left (1-e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{36} b c d^3 x \left (1-c^2 x^2\right )^{5/2}-\frac {7}{72} b c d^3 x \left (1-c^2 x^2\right )^{3/2}-\frac {19}{48} b c d^3 x \sqrt {1-c^2 x^2}-\frac {1}{2} i b d^3 \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )-\frac {19}{48} b d^3 \sin ^{-1}(c x) \]
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Rubi [A] time = 0.28, antiderivative size = 235, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {4683, 4625, 3717, 2190, 2279, 2391, 195, 216} \[ -\frac {1}{2} i b d^3 \text {PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right )+\frac {1}{6} d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} d^3 \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{2} d^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )-\frac {i d^3 \left (a+b \sin ^{-1}(c x)\right )^2}{2 b}+d^3 \log \left (1-e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{36} b c d^3 x \left (1-c^2 x^2\right )^{5/2}-\frac {7}{72} b c d^3 x \left (1-c^2 x^2\right )^{3/2}-\frac {19}{48} b c d^3 x \sqrt {1-c^2 x^2}-\frac {19}{48} b d^3 \sin ^{-1}(c x) \]
Antiderivative was successfully verified.
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Rule 195
Rule 216
Rule 2190
Rule 2279
Rule 2391
Rule 3717
Rule 4625
Rule 4683
Rubi steps
\begin {align*} \int \frac {\left (d-c^2 d x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )}{x} \, dx &=\frac {1}{6} d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )+d \int \frac {\left (d-c^2 d x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{x} \, dx-\frac {1}{6} \left (b c d^3\right ) \int \left (1-c^2 x^2\right )^{5/2} \, dx\\ &=-\frac {1}{36} b c d^3 x \left (1-c^2 x^2\right )^{5/2}+\frac {1}{4} d^3 \left (1-c^2 x^2\right )^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 )+d^2 \int \frac {\left (d-c^2 d x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{x} \, dx-\frac {1}{36} \left (5 b c d^3\right ) \int \left (1-c^2 x^2\right )^{3/2} \, dx-\frac {1}{4} \left (b c d^3\right ) \int \left (1-c^2 x^2\right )^{3/2} \, dx\\ &=-\frac {7}{72} b c d^3 x \left (1-c^2 x^2\right )^{3/2}-\frac {1}{36} b c d^3 x \left (1-c^2 x^2\right )^{5/2}+\frac {1}{2} d^3 \left (1-c^2 x^2\right ) \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 )+\frac {1}{6} d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )+d^3 \int \frac {a+b \sin ^{-1}(c x)}{x} \, dx-\frac {1}{48} \left (5 b c d^3\right ) \int \sqrt {1-c^2 x^2} \, dx-\frac {1}{16} \left (3 b c d^3\right ) \int \sqrt {1-c^2 x^2} \, dx-\frac {1}{2} \left (b c d^3\right ) \int \sqrt {1-c^2 x^2} \, dx\\ &=-\frac {19}{48} b c d^3 x \sqrt {1-c^2 x^2}-\frac {7}{72} b c d^3 x \left (1-c^2 x^2\right )^{3/2}-\frac {1}{36} b c d^3 x \left (1-c^2 x^2\right )^{5/2}+\frac {1}{2} d^3 \left (1-c^2 x^2\right ) \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 )+\frac {1}{6} d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )+d^3 \operatorname {Subst}\left (\int (a+b x) \cot (x) \, dx,x,\sin ^{-1}(c x)\right )-\frac {1}{96} \left (5 b c d^3\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx-\frac {1}{32} \left (3 b c d^3\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx-\frac {1}{4} \left (b c d^3\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {19}{48} b c d^3 x \sqrt {1-c^2 x^2}-\frac {7}{72} b c d^3 x \left (1-c^2 x^2\right )^{3/2}-\frac {1}{36} b c d^3 x \left (1-c^2 x^2\right )^{5/2}-\frac {19}{48} b d^3 \sin ^{-1}(c x)+\frac {1}{2} d^3 \left (1-c^2 x^2\right ) \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 )+\frac {1}{6} d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )-\frac {i d^3 \left (a+b \sin ^{-1}(c x)\right )^2}{2 b}-\left (2 i d^3\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 )\\ &=-\frac {19}{48} b c d^3 x \sqrt {1-c^2 x^2}-\frac {7}{72} b c d^3 x \left (1-c^2 x^2\right )^{3/2}-\frac {1}{36} b c d^3 x \left (1-c^2 x^2\right )^{5/2}-\frac {19}{48} b d^3 \sin ^{-1}(c x)+\frac {1}{2} d^3 \left (1-c^2 x^2\right ) \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 )+\frac {1}{6} d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )-\frac {i d^3 \left (a+b \sin ^{-1}(c x)\right )^2}{2 b}+d^3 \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-\left (b d^3\right ) \operatorname {Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )\\ &=-\frac {19}{48} b c d^3 x \sqrt {1-c^2 x^2}-\frac {7}{72} b c d^3 x \left (1-c^2 x^2\right )^{3/2}-\frac {1}{36} b c d^3 x \left (1-c^2 x^2\right )^{5/2}-\frac {19}{48} b d^3 \sin ^{-1}(c x)+\frac {1}{2} d^3 \left (1-c^2 x^2\right ) \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 )+\frac {1}{6} d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )-\frac {i d^3 \left (a+b \sin ^{-1}(c x)\right )^2}{2 b}+d^3 \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )+\frac {1}{2} \left (i b d^3\right ) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )\\ &=-\frac {19}{48} b c d^3 x \sqrt {1-c^2 x^2}-\frac {7}{72} b c d^3 x \left (1-c^2 x^2\right )^{3/2}-\frac {1}{36} b c d^3 x \left (1-c^2 x^2\right )^{5/2}-\frac {19}{48} b d^3 \sin ^{-1}(c x)+\frac {1}{2} d^3 \left (1-c^2 x^2\right ) \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 )+\frac {1}{6} d^3 \left (1-c^2 x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )-\frac {i d^3 \left (a+b \sin ^{-1}(c x)\right )^2}{2 b}+d^3 \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-\frac {1}{2} i b d^3 \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )\\ \end {align*}
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Mathematica [A] time = 0.23, size = 183, normalized size = 0.78 \[ -\frac {1}{144} d^3 \left (24 a c^6 x^6-108 a c^4 x^4+216 a c^2 x^2-144 a \log (x)+75 b c x \sqrt {1-c^2 x^2}+4 b c^5 x^5 \sqrt {1-c^2 x^2}-22 b c^3 x^3 \sqrt {1-c^2 x^2}+3 b \sin ^{-1}(c x) \left (8 c^6 x^6-36 c^4 x^4+72 c^2 x^2-48 \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-25\right )+72 i b \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )+72 i b \sin ^{-1}(c x)^2\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {a c^{6} d^{3} x^{6} - 3 \, a c^{4} d^{3} x^{4} + 3 \, a c^{2} d^{3} x^{2} - a d^{3} + {\left (b c^{6} d^{3} x^{6} - 3 \, b c^{4} d^{3} x^{4} + 3 \, b c^{2} d^{3} x^{2} - b d^{3}\right )} \arcsin \left (c x\right )}{x}, x\right ) \]
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 \[ \int -\frac {{\left (c^{2} d x^{2} - d\right )}^{3} {\left (b \arcsin \left (c x\right ) + a\right )}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.35, size = 266, normalized size = 1.13 \[ -\frac {d^{3} a \,c^{6} x^{6}}{6}+\frac {3 d^{3} a \,c^{4} x^{4}}{4}-\frac {3 d^{3} a \,c^{2} x^{2}}{2}+d^{3} a \ln \left (c x \right )-\frac {i b \,d^{3} \arcsin \left (c x \right )^{2}}{2}+d^{3} b \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+d^{3} b \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-i d^{3} b \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )-i d^{3} b \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+\frac {d^{3} b \arcsin \left (c x \right ) \cos \left (6 \arcsin \left (c x \right )\right )}{192}-\frac {d^{3} b \sin \left (6 \arcsin \left (c x \right )\right )}{1152}+\frac {d^{3} b \arcsin \left (c x \right ) \cos \left (4 \arcsin \left (c x \right )\right )}{16}-\frac {d^{3} b \sin \left (4 \arcsin \left (c x \right )\right )}{64}+\frac {29 d^{3} b \cos \left (2 \arcsin \left (c x \right )\right ) \arcsin \left (c x \right )}{64}-\frac {29 d^{3} b \sin \left (2 \arcsin \left (c x \right )\right )}{128} \]
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 \[ -\frac {1}{6} \, a c^{6} d^{3} x^{6} + \frac {3}{4} \, a c^{4} d^{3} x^{4} - \frac {3}{2} \, a c^{2} d^{3} x^{2} + a d^{3} \log \relax (x) - \int \frac {{\left (b c^{6} d^{3} x^{6} - 3 \, b c^{4} d^{3} x^{4} + 3 \, b c^{2} d^{3} x^{2} - b d^{3}\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )}{x}\,{d x} \]
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 \[ \int \frac {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^3}{x} \,d x \]
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 \[ - d^{3} \left (\int \left (- \frac {a}{x}\right )\, dx + \int 3 a c^{2} x\, dx + \int \left (- 3 a c^{4} x^{3}\right )\, dx + \int a c^{6} x^{5}\, dx + \int \left (- \frac {b \operatorname {asin}{\left (c x \right )}}{x}\right )\, dx + \int 3 b c^{2} x \operatorname {asin}{\left (c x \right )}\, dx + \int \left (- 3 b c^{4} x^{3} \operatorname {asin}{\left (c x \right )}\right )\, dx + \int b c^{6} x^{5} \operatorname {asin}{\left (c x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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