Optimal. Leaf size=271 \[ -2 i \sin ^{-1}(a+b x) \text {Li}_2\left (\frac {e^{i \sin ^{-1}(a+b x)}}{i a-\sqrt {1-a^2}}\right )-2 i \sin ^{-1}(a+b x) \text {Li}_2\left (\frac {e^{i \sin ^{-1}(a+b x)}}{i a+\sqrt {1-a^2}}\right )+2 \text {Li}_3\left (\frac {e^{i \sin ^{-1}(a+b x)}}{i a-\sqrt {1-a^2}}\right )+2 \text {Li}_3\left (\frac {e^{i \sin ^{-1}(a+b x)}}{i a+\sqrt {1-a^2}}\right )+\sin ^{-1}(a+b x)^2 \log \left (1-\frac {e^{i \sin ^{-1}(a+b x)}}{-\sqrt {1-a^2}+i a}\right )+\sin ^{-1}(a+b x)^2 \log \left (1-\frac {e^{i \sin ^{-1}(a+b x)}}{\sqrt {1-a^2}+i a}\right )-\frac {1}{3} i \sin ^{-1}(a+b x)^3 \]
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Rubi [A] time = 0.41, antiderivative size = 271, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {4805, 4741, 4521, 2190, 2531, 2282, 6589} \[ -2 i \sin ^{-1}(a+b x) \text {PolyLog}\left (2,\frac {e^{i \sin ^{-1}(a+b x)}}{-\sqrt {1-a^2}+i a}\right )-2 i \sin ^{-1}(a+b x) \text {PolyLog}\left (2,\frac {e^{i \sin ^{-1}(a+b x)}}{\sqrt {1-a^2}+i a}\right )+2 \text {PolyLog}\left (3,\frac {e^{i \sin ^{-1}(a+b x)}}{-\sqrt {1-a^2}+i a}\right )+2 \text {PolyLog}\left (3,\frac {e^{i \sin ^{-1}(a+b x)}}{\sqrt {1-a^2}+i a}\right )+\sin ^{-1}(a+b x)^2 \log \left (1-\frac {e^{i \sin ^{-1}(a+b x)}}{-\sqrt {1-a^2}+i a}\right )+\sin ^{-1}(a+b x)^2 \log \left (1-\frac {e^{i \sin ^{-1}(a+b x)}}{\sqrt {1-a^2}+i a}\right )-\frac {1}{3} i \sin ^{-1}(a+b x)^3 \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2282
Rule 2531
Rule 4521
Rule 4741
Rule 4805
Rule 6589
Rubi steps
\begin {align*} \int \frac {\sin ^{-1}(a+b x)^2}{x} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\sin ^{-1}(x)^2}{-\frac {a}{b}+\frac {x}{b}} \, dx,x,a+b x\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \frac {x^2 \cos (x)}{-\frac {a}{b}+\frac {\sin (x)}{b}} \, dx,x,\sin ^{-1}(a+b x)\right )}{b}\\ &=-\frac {1}{3} i \sin ^{-1}(a+b x)^3+\frac {i \operatorname {Subst}\left (\int \frac {e^{i x} x^2}{-\frac {i a}{b}-\frac {\sqrt {1-a^2}}{b}+\frac {e^{i x}}{b}} \, dx,x,\sin ^{-1}(a+b x)\right )}{b}+\frac {i \operatorname {Subst}\left (\int \frac {e^{i x} x^2}{-\frac {i a}{b}+\frac {\sqrt {1-a^2}}{b}+\frac {e^{i x}}{b}} \, dx,x,\sin ^{-1}(a+b x)\right )}{b}\\ &=-\frac {1}{3} i \sin ^{-1}(a+b x)^3+\sin ^{-1}(a+b x)^2 \log \left (1-\frac {e^{i \sin ^{-1}(a+b x)}}{i a-\sqrt {1-a^2}}\right )+\sin ^{-1}(a+b x)^2 \log \left (1-\frac {e^{i \sin ^{-1}(a+b x)}}{i a+\sqrt {1-a^2}}\right )-2 \operatorname {Subst}\left (\int x \log \left (1+\frac {e^{i x}}{\left (-\frac {i a}{b}-\frac {\sqrt {1-a^2}}{b}\right ) b}\right ) \, dx,x,\sin ^{-1}(a+b x)\right )-2 \operatorname {Subst}\left (\int x \log \left (1+\frac {e^{i x}}{\left (-\frac {i a}{b}+\frac {\sqrt {1-a^2}}{b}\right ) b}\right ) \, dx,x,\sin ^{-1}(a+b x)\right )\\ &=-\frac {1}{3} i \sin ^{-1}(a+b x)^3+\sin ^{-1}(a+b x)^2 \log \left (1-\frac {e^{i \sin ^{-1}(a+b x)}}{i a-\sqrt {1-a^2}}\right )+\sin ^{-1}(a+b x)^2 \log \left (1-\frac {e^{i \sin ^{-1}(a+b x)}}{i a+\sqrt {1-a^2}}\right )-2 i \sin ^{-1}(a+b x) \text {Li}_2\left (\frac {e^{i \sin ^{-1}(a+b x)}}{i a-\sqrt {1-a^2}}\right )-2 i \sin ^{-1}(a+b x) \text {Li}_2\left (\frac {e^{i \sin ^{-1}(a+b x)}}{i a+\sqrt {1-a^2}}\right )+2 i \operatorname {Subst}\left (\int \text {Li}_2\left (-\frac {e^{i x}}{\left (-\frac {i a}{b}-\frac {\sqrt {1-a^2}}{b}\right ) b}\right ) \, dx,x,\sin ^{-1}(a+b x)\right )+2 i \operatorname {Subst}\left (\int \text {Li}_2\left (-\frac {e^{i x}}{\left (-\frac {i a}{b}+\frac {\sqrt {1-a^2}}{b}\right ) b}\right ) \, dx,x,\sin ^{-1}(a+b x)\right )\\ &=-\frac {1}{3} i \sin ^{-1}(a+b x)^3+\sin ^{-1}(a+b x)^2 \log \left (1-\frac {e^{i \sin ^{-1}(a+b x)}}{i a-\sqrt {1-a^2}}\right )+\sin ^{-1}(a+b x)^2 \log \left (1-\frac {e^{i \sin ^{-1}(a+b x)}}{i a+\sqrt {1-a^2}}\right )-2 i \sin ^{-1}(a+b x) \text {Li}_2\left (\frac {e^{i \sin ^{-1}(a+b x)}}{i a-\sqrt {1-a^2}}\right )-2 i \sin ^{-1}(a+b x) \text {Li}_2\left (\frac {e^{i \sin ^{-1}(a+b x)}}{i a+\sqrt {1-a^2}}\right )+2 \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (\frac {x}{i a-\sqrt {1-a^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(a+b x)}\right )+2 \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (\frac {x}{i a+\sqrt {1-a^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(a+b x)}\right )\\ &=-\frac {1}{3} i \sin ^{-1}(a+b x)^3+\sin ^{-1}(a+b x)^2 \log \left (1-\frac {e^{i \sin ^{-1}(a+b x)}}{i a-\sqrt {1-a^2}}\right )+\sin ^{-1}(a+b x)^2 \log \left (1-\frac {e^{i \sin ^{-1}(a+b x)}}{i a+\sqrt {1-a^2}}\right )-2 i \sin ^{-1}(a+b x) \text {Li}_2\left (\frac {e^{i \sin ^{-1}(a+b x)}}{i a-\sqrt {1-a^2}}\right )-2 i \sin ^{-1}(a+b x) \text {Li}_2\left (\frac {e^{i \sin ^{-1}(a+b x)}}{i a+\sqrt {1-a^2}}\right )+2 \text {Li}_3\left (\frac {e^{i \sin ^{-1}(a+b x)}}{i a-\sqrt {1-a^2}}\right )+2 \text {Li}_3\left (\frac {e^{i \sin ^{-1}(a+b x)}}{i a+\sqrt {1-a^2}}\right )\\ \end {align*}
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Mathematica [A] time = 0.04, size = 309, normalized size = 1.14 \[ -2 i \sin ^{-1}(a+b x) \text {Li}_2\left (-\frac {e^{i \sin ^{-1}(a+b x)}}{\left (-\frac {i a}{b}-\frac {\sqrt {1-a^2}}{b}\right ) b}\right )-2 i \sin ^{-1}(a+b x) \text {Li}_2\left (-\frac {e^{i \sin ^{-1}(a+b x)}}{\left (\frac {\sqrt {1-a^2}}{b}-\frac {i a}{b}\right ) b}\right )+2 \text {Li}_3\left (\frac {e^{i \sin ^{-1}(a+b x)}}{i a-\sqrt {1-a^2}}\right )+2 \text {Li}_3\left (\frac {e^{i \sin ^{-1}(a+b x)}}{i a+\sqrt {1-a^2}}\right )+\sin ^{-1}(a+b x)^2 \log \left (1+\frac {e^{i \sin ^{-1}(a+b x)}}{b \left (-\frac {\sqrt {1-a^2}}{b}-\frac {i a}{b}\right )}\right )+\sin ^{-1}(a+b x)^2 \log \left (1+\frac {e^{i \sin ^{-1}(a+b x)}}{b \left (\frac {\sqrt {1-a^2}}{b}-\frac {i a}{b}\right )}\right )-\frac {1}{3} i \sin ^{-1}(a+b x)^3 \]
Antiderivative was successfully verified.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\arcsin \left (b x + a\right )^{2}}{x}, x\right ) \]
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 \[ \int \frac {\arcsin \left (b x + a\right )^{2}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.70, size = 0, normalized size = 0.00 \[ \int \frac {\arcsin \left (b x +a \right )^{2}}{x}\, dx \]
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 \[ \int \frac {\arcsin \left (b x + a\right )^{2}}{x}\,{d x} \]
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 \[ \int \frac {{\mathrm {asin}\left (a+b\,x\right )}^2}{x} \,d x \]
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 \[ \int \frac {\operatorname {asin}^{2}{\left (a + b x \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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