Optimal. Leaf size=123 \[ \frac {3}{2} b^2 \text {Li}_3\left (e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )-\frac {3}{2} i b \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac {i \left (a+b \sin ^{-1}(c x)\right )^4}{4 b}+\log \left (1-e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )^3+\frac {3}{4} i b^3 \text {Li}_4\left (e^{2 i \sin ^{-1}(c x)}\right ) \]
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Rubi [A] time = 0.15, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4625, 3717, 2190, 2531, 6609, 2282, 6589} \[ \frac {3}{2} b^2 \text {PolyLog}\left (3,e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )-\frac {3}{2} i b \text {PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac {3}{4} i b^3 \text {PolyLog}\left (4,e^{2 i \sin ^{-1}(c x)}\right )-\frac {i \left (a+b \sin ^{-1}(c x)\right )^4}{4 b}+\log \left (1-e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )^3 \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2282
Rule 2531
Rule 3717
Rule 4625
Rule 6589
Rule 6609
Rubi steps
\begin {align*} \int \frac {\left (a+b \sin ^{-1}(c x)\right )^3}{x} \, dx &=\operatorname {Subst}\left (\int (a+b x)^3 \cot (x) \, dx,x,\sin ^{-1}(c x)\right )\\ &=-\frac {i \left (a+b \sin ^{-1}(c x)\right )^4}{4 b}-2 i \operatorname {Subst}\left (\int \frac {e^{2 i x} (a+b x)^3}{1-e^{2 i x}} \, dx,x,\sin ^{-1}(c x)\right )\\ &=-\frac {i \left (a+b \sin ^{-1}(c x)\right )^4}{4 b}+\left (a+b \sin ^{-1}(c x)\right )^3 \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-(3 b) \operatorname {Subst}\left (\int (a+b x)^2 \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )\\ &=-\frac {i \left (a+b \sin ^{-1}(c x)\right )^4}{4 b}+\left (a+b \sin ^{-1}(c x)\right )^3 \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-\frac {3}{2} i b \left (a+b \sin ^{-1}(c x)\right )^2 \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )+\left (3 i b^2\right ) \operatorname {Subst}\left (\int (a+b x) \text {Li}_2\left (e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )\\ &=-\frac {i \left (a+b \sin ^{-1}(c x)\right )^4}{4 b}+\left (a+b \sin ^{-1}(c x)\right )^3 \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-\frac {3}{2} i b \left (a+b \sin ^{-1}(c x)\right )^2 \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )+\frac {3}{2} b^2 \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_3\left (e^{2 i \sin ^{-1}(c x)}\right )-\frac {1}{2} \left (3 b^3\right ) \operatorname {Subst}\left (\int \text {Li}_3\left (e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )\\ &=-\frac {i \left (a+b \sin ^{-1}(c x)\right )^4}{4 b}+\left (a+b \sin ^{-1}(c x)\right )^3 \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-\frac {3}{2} i b \left (a+b \sin ^{-1}(c x)\right )^2 \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )+\frac {3}{2} b^2 \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_3\left (e^{2 i \sin ^{-1}(c x)}\right )+\frac {1}{4} \left (3 i b^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )\\ &=-\frac {i \left (a+b \sin ^{-1}(c x)\right )^4}{4 b}+\left (a+b \sin ^{-1}(c x)\right )^3 \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-\frac {3}{2} i b \left (a+b \sin ^{-1}(c x)\right )^2 \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )+\frac {3}{2} b^2 \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_3\left (e^{2 i \sin ^{-1}(c x)}\right )+\frac {3}{4} i b^3 \text {Li}_4\left (e^{2 i \sin ^{-1}(c x)}\right )\\ \end {align*}
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Mathematica [A] time = 0.23, size = 244, normalized size = 1.98 \[ a^3 \log (c x)+3 a^2 b \left (\sin ^{-1}(c x) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )-\frac {1}{2} i \left (\sin ^{-1}(c x)^2+\text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )\right )\right )+\frac {1}{8} a b^2 \left (24 i \sin ^{-1}(c x) \text {Li}_2\left (e^{-2 i \sin ^{-1}(c x)}\right )+12 \text {Li}_3\left (e^{-2 i \sin ^{-1}(c x)}\right )+8 i \sin ^{-1}(c x)^3+24 \sin ^{-1}(c x)^2 \log \left (1-e^{-2 i \sin ^{-1}(c x)}\right )-i \pi ^3\right )-\frac {1}{64} i b^3 \left (-96 \sin ^{-1}(c x)^2 \text {Li}_2\left (e^{-2 i \sin ^{-1}(c x)}\right )+96 i \sin ^{-1}(c x) \text {Li}_3\left (e^{-2 i \sin ^{-1}(c x)}\right )+48 \text {Li}_4\left (e^{-2 i \sin ^{-1}(c x)}\right )-16 \sin ^{-1}(c x)^4+64 i \sin ^{-1}(c x)^3 \log \left (1-e^{-2 i \sin ^{-1}(c x)}\right )+\pi ^4\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{3} \arcsin \left (c x\right )^{3} + 3 \, a b^{2} \arcsin \left (c x\right )^{2} + 3 \, a^{2} b \arcsin \left (c x\right ) + a^{3}}{x}, x\right ) \]
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 \[ \int \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{3}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 592, normalized size = 4.81 \[ a^{3} \ln \left (c x \right )-6 i a \,b^{2} \arcsin \left (c x \right ) \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )+b^{3} \arcsin \left (c x \right )^{3} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-3 i a^{2} b \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+6 b^{3} \arcsin \left (c x \right ) \polylog \left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-3 i b^{3} \arcsin \left (c x \right )^{2} \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+b^{3} \arcsin \left (c x \right )^{3} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+6 i b^{3} \polylog \left (4, i c x +\sqrt {-c^{2} x^{2}+1}\right )+6 b^{3} \arcsin \left (c x \right ) \polylog \left (3, i c x +\sqrt {-c^{2} x^{2}+1}\right )-3 i b^{3} \arcsin \left (c x \right )^{2} \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )-\frac {i b^{3} \arcsin \left (c x \right )^{4}}{4}-i a \,b^{2} \arcsin \left (c x \right )^{3}+3 a \,b^{2} \arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-6 i a \,b^{2} \arcsin \left (c x \right ) \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+3 a \,b^{2} \arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+6 a \,b^{2} \polylog \left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+6 a \,b^{2} \polylog \left (3, i c x +\sqrt {-c^{2} x^{2}+1}\right )-3 i a^{2} b \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )+3 a^{2} b \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+3 a^{2} b \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+6 i b^{3} \polylog \left (4, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-\frac {3 i a^{2} b \arcsin \left (c x \right )^{2}}{2} \]
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 \[ a^{3} \log \relax (x) + \int \frac {b^{3} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )^{3} + 3 \, a b^{2} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )^{2} + 3 \, a^{2} b \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^3}{x} \,d x \]
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 \[ \int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{3}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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