Optimal. Leaf size=193 \[ i b c^2 d \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )-\frac {d \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{2 x^2}-\frac {b c d \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{x}+\frac {i c^2 d \left (a+b \sin ^{-1}(c x)\right )^3}{3 b}-\frac {1}{2} c^2 d \left (a+b \sin ^{-1}(c x)\right )^2-c^2 d \log \left (1-e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac {1}{2} b^2 c^2 d \text {Li}_3\left (e^{2 i \sin ^{-1}(c x)}\right )+b^2 c^2 d \log (x) \]
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Rubi [A] time = 0.29, antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 10, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4695, 4625, 3717, 2190, 2531, 2282, 6589, 4693, 29, 4641} \[ i b c^2 d \text {PolyLog}\left (2,e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{2} b^2 c^2 d \text {PolyLog}\left (3,e^{2 i \sin ^{-1}(c x)}\right )-\frac {d \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{2 x^2}-\frac {b c d \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{x}+\frac {i c^2 d \left (a+b \sin ^{-1}(c x)\right )^3}{3 b}-\frac {1}{2} c^2 d \left (a+b \sin ^{-1}(c x)\right )^2-c^2 d \log \left (1-e^{2 i \sin ^{-1}(c x)}\right ) \left (a+b \sin ^{-1}(c x)\right )^2+b^2 c^2 d \log (x) \]
Antiderivative was successfully verified.
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Rule 29
Rule 2190
Rule 2282
Rule 2531
Rule 3717
Rule 4625
Rule 4641
Rule 4693
Rule 4695
Rule 6589
Rubi steps
\begin {align*} \int \frac {\left (d-c^2 d x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{x^3} \, dx &=-\frac {d \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{2 x^2}+(b c d) \int \frac {\sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{x^2} \, dx-\left (c^2 d\right ) \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{x} \, dx\\ &=-\frac {b c d \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{x}-\frac {d \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{2 x^2}-\left (c^2 d\right ) \operatorname {Subst}\left (\int (a+b x)^2 \cot (x) \, dx,x,\sin ^{-1}(c x)\right )+\left (b^2 c^2 d\right ) \int \frac {1}{x} \, dx-\left (b c^3 d\right ) \int \frac {a+b \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {b c d \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{x}-\frac {1}{2} c^2 d \left (a+b \sin ^{-1}(c x)\right )^2-\frac {d \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{2 x^2}+\frac {i c^2 d \left (a+b \sin ^{-1}(c x)\right )^3}{3 b}+b^2 c^2 d \log (x)+\left (2 i c^2 d\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 {b c d \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{x}-\frac {1}{2} c^2 d \left (a+b \sin ^{-1}(c x)\right )^2-\frac {d \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{2 x^2}+\frac {i c^2 d \left (a+b \sin ^{-1}(c x)\right )^3}{3 b}-c^2 d \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 \log (x)+\left (2 b c^2 d\right ) \operatorname {Subst}\left (\int (a+b x) \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )\\ &=-\frac {b c d \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{x}-\frac {1}{2} c^2 d \left (a+b \sin ^{-1}(c x)\right )^2-\frac {d \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{2 x^2}+\frac {i c^2 d \left (a+b \sin ^{-1}(c x)\right )^3}{3 b}-c^2 d \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 \log (x)+i b c^2 d \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )-\left (i b^2 c^2 d\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )\\ &=-\frac {b c d \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{x}-\frac {1}{2} c^2 d \left (a+b \sin ^{-1}(c x)\right )^2-\frac {d \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{2 x^2}+\frac {i c^2 d \left (a+b \sin ^{-1}(c x)\right )^3}{3 b}-c^2 d \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 \log (x)+i b c^2 d \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 c^2 d\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )\\ &=-\frac {b c d \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{x}-\frac {1}{2} c^2 d \left (a+b \sin ^{-1}(c x)\right )^2-\frac {d \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{2 x^2}+\frac {i c^2 d \left (a+b \sin ^{-1}(c x)\right )^3}{3 b}-c^2 d \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 \log (x)+i b c^2 d \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 c^2 d \text {Li}_3\left (e^{2 i \sin ^{-1}(c x)}\right )\\ \end {align*}
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Mathematica [A] time = 0.39, size = 236, normalized size = 1.22 \[ \frac {1}{2} d \left (-2 a^2 c^2 \log (x)-\frac {a^2}{x^2}+2 i a b c^2 \left (\text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )+\sin ^{-1}(c x) \left (\sin ^{-1}(c x)+2 i \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )\right )\right )-\frac {2 a b \left (c x \sqrt {1-c^2 x^2}+\sin ^{-1}(c x)\right )}{x^2}+\frac {1}{12} i b^2 c^2 \left (-24 \sin ^{-1}(c x) \text {Li}_2\left (e^{-2 i \sin ^{-1}(c x)}\right )+12 i \text {Li}_3\left (e^{-2 i \sin ^{-1}(c x)}\right )-8 \sin ^{-1}(c x)^3+24 i \sin ^{-1}(c x)^2 \log \left (1-e^{-2 i \sin ^{-1}(c x)}\right )+\pi ^3\right )-\frac {b^2 \left (-2 c^2 x^2 \log (c x)+2 c x \sqrt {1-c^2 x^2} \sin ^{-1}(c x)+\sin ^{-1}(c x)^2\right )}{x^2}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.67, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {a^{2} c^{2} d x^{2} - a^{2} d + {\left (b^{2} c^{2} d x^{2} - b^{2} d\right )} \arcsin \left (c x\right )^{2} + 2 \, {\left (a b c^{2} d x^{2} - a b d\right )} \arcsin \left (c x\right )}{x^{3}}, x\right ) \]
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 \[ \int -\frac {{\left (c^{2} d x^{2} - d\right )} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.59, size = 564, normalized size = 2.92 \[ -c^{2} d \,a^{2} \ln \left (c x \right )-\frac {d \,a^{2}}{2 x^{2}}+\frac {i c^{2} d \,b^{2} \arcsin \left (c x \right )^{3}}{3}+2 i c^{2} d \,b^{2} \arcsin \left (c x \right ) \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )-\frac {c d \,b^{2} \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{x}-\frac {d \,b^{2} \arcsin \left (c x \right )^{2}}{2 x^{2}}-c^{2} d \,b^{2} \arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+i c^{2} d a b -2 c^{2} d \,b^{2} \polylog \left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-c^{2} d \,b^{2} \arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+i c^{2} d \,b^{2} \arcsin \left (c x \right )-2 c^{2} d \,b^{2} \polylog \left (3, i c x +\sqrt {-c^{2} x^{2}+1}\right )+c^{2} d \,b^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 c^{2} d \,b^{2} \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )+c^{2} d \,b^{2} \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )+2 i c^{2} d a b \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+2 i c^{2} d a b \polylog \left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )-\frac {c d a b \sqrt {-c^{2} x^{2}+1}}{x}-\frac {d a b \arcsin \left (c x \right )}{x^{2}}-2 c^{2} d a b \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 c^{2} d a b \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+2 i c^{2} d \,b^{2} \arcsin \left (c x \right ) \polylog \left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+i c^{2} d a b \arcsin \left (c x \right )^{2} \]
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 \[ -a^{2} c^{2} d \log \relax (x) - a b d {\left (\frac {\sqrt {-c^{2} x^{2} + 1} c}{x} + \frac {\arcsin \left (c x\right )}{x^{2}}\right )} - \frac {a^{2} d}{2 \, x^{2}} - \int \frac {2 \, a b c^{2} d x^{2} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right ) + {\left (b^{2} c^{2} d x^{2} - b^{2} d\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )^{2}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,\left (d-c^2\,d\,x^2\right )}{x^3} \,d x \]
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 \[ - d \left (\int \left (- \frac {a^{2}}{x^{3}}\right )\, dx + \int \frac {a^{2} c^{2}}{x}\, dx + \int \left (- \frac {b^{2} \operatorname {asin}^{2}{\left (c x \right )}}{x^{3}}\right )\, dx + \int \left (- \frac {2 a b \operatorname {asin}{\left (c x \right )}}{x^{3}}\right )\, dx + \int \frac {b^{2} c^{2} \operatorname {asin}^{2}{\left (c x \right )}}{x}\, dx + \int \frac {2 a b c^{2} \operatorname {asin}{\left (c x \right )}}{x}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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