Optimal. Leaf size=230 \[ \frac {2 i b \text {Li}_2\left (\frac {e^{i \sin ^{-1}(a+b x)}}{i a-\sqrt {1-a^2}}\right )}{\sqrt {1-a^2}}-\frac {2 i b \text {Li}_2\left (\frac {e^{i \sin ^{-1}(a+b x)}}{i a+\sqrt {1-a^2}}\right )}{\sqrt {1-a^2}}-\frac {2 b \sin ^{-1}(a+b x) \log \left (1-\frac {e^{i \sin ^{-1}(a+b x)}}{-\sqrt {1-a^2}+i a}\right )}{\sqrt {1-a^2}}+\frac {2 b \sin ^{-1}(a+b x) \log \left (1-\frac {e^{i \sin ^{-1}(a+b x)}}{\sqrt {1-a^2}+i a}\right )}{\sqrt {1-a^2}}-\frac {\sin ^{-1}(a+b x)^2}{x} \]
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Rubi [A] time = 0.45, antiderivative size = 208, normalized size of antiderivative = 0.90, number of steps used = 11, number of rules used = 8, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {4805, 4743, 4773, 3323, 2264, 2190, 2279, 2391} \[ \frac {2 b \text {PolyLog}\left (2,-\frac {i e^{i \sin ^{-1}(a+b x)}}{a-\sqrt {a^2-1}}\right )}{\sqrt {a^2-1}}-\frac {2 b \text {PolyLog}\left (2,-\frac {i e^{i \sin ^{-1}(a+b x)}}{\sqrt {a^2-1}+a}\right )}{\sqrt {a^2-1}}+\frac {2 i b \sin ^{-1}(a+b x) \log \left (1+\frac {i e^{i \sin ^{-1}(a+b x)}}{a-\sqrt {a^2-1}}\right )}{\sqrt {a^2-1}}-\frac {2 i b \sin ^{-1}(a+b x) \log \left (1+\frac {i e^{i \sin ^{-1}(a+b x)}}{\sqrt {a^2-1}+a}\right )}{\sqrt {a^2-1}}-\frac {\sin ^{-1}(a+b x)^2}{x} \]
Warning: Unable to verify antiderivative.
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Rule 2190
Rule 2264
Rule 2279
Rule 2391
Rule 3323
Rule 4743
Rule 4773
Rule 4805
Rubi steps
\begin {align*} \int \frac {\sin ^{-1}(a+b x)^2}{x^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\sin ^{-1}(x)^2}{\left (-\frac {a}{b}+\frac {x}{b}\right )^2} \, dx,x,a+b x\right )}{b}\\ &=-\frac {\sin ^{-1}(a+b x)^2}{x}+2 \operatorname {Subst}\left (\int \frac {\sin ^{-1}(x)}{\left (-\frac {a}{b}+\frac {x}{b}\right ) \sqrt {1-x^2}} \, dx,x,a+b x\right )\\ &=-\frac {\sin ^{-1}(a+b x)^2}{x}+2 \operatorname {Subst}\left (\int \frac {x}{-\frac {a}{b}+\frac {\sin (x)}{b}} \, dx,x,\sin ^{-1}(a+b x)\right )\\ &=-\frac {\sin ^{-1}(a+b x)^2}{x}+4 \operatorname {Subst}\left (\int \frac {e^{i x} x}{\frac {i}{b}-\frac {2 a e^{i x}}{b}-\frac {i e^{2 i x}}{b}} \, dx,x,\sin ^{-1}(a+b x)\right )\\ &=-\frac {\sin ^{-1}(a+b x)^2}{x}-\frac {(4 i) \operatorname {Subst}\left (\int \frac {e^{i x} x}{-\frac {2 a}{b}-\frac {2 \sqrt {-1+a^2}}{b}-\frac {2 i e^{i x}}{b}} \, dx,x,\sin ^{-1}(a+b x)\right )}{\sqrt {-1+a^2}}+\frac {(4 i) \operatorname {Subst}\left (\int \frac {e^{i x} x}{-\frac {2 a}{b}+\frac {2 \sqrt {-1+a^2}}{b}-\frac {2 i e^{i x}}{b}} \, dx,x,\sin ^{-1}(a+b x)\right )}{\sqrt {-1+a^2}}\\ &=-\frac {\sin ^{-1}(a+b x)^2}{x}+\frac {2 i b \sin ^{-1}(a+b x) \log \left (1+\frac {i e^{i \sin ^{-1}(a+b x)}}{a-\sqrt {-1+a^2}}\right )}{\sqrt {-1+a^2}}-\frac {2 i b \sin ^{-1}(a+b x) \log \left (1+\frac {i e^{i \sin ^{-1}(a+b x)}}{a+\sqrt {-1+a^2}}\right )}{\sqrt {-1+a^2}}+\frac {(2 i b) \operatorname {Subst}\left (\int \log \left (1-\frac {2 i e^{i x}}{\left (-\frac {2 a}{b}-\frac {2 \sqrt {-1+a^2}}{b}\right ) b}\right ) \, dx,x,\sin ^{-1}(a+b x)\right )}{\sqrt {-1+a^2}}-\frac {(2 i b) \operatorname {Subst}\left (\int \log \left (1-\frac {2 i e^{i x}}{\left (-\frac {2 a}{b}+\frac {2 \sqrt {-1+a^2}}{b}\right ) b}\right ) \, dx,x,\sin ^{-1}(a+b x)\right )}{\sqrt {-1+a^2}}\\ &=-\frac {\sin ^{-1}(a+b x)^2}{x}+\frac {2 i b \sin ^{-1}(a+b x) \log \left (1+\frac {i e^{i \sin ^{-1}(a+b x)}}{a-\sqrt {-1+a^2}}\right )}{\sqrt {-1+a^2}}-\frac {2 i b \sin ^{-1}(a+b x) \log \left (1+\frac {i e^{i \sin ^{-1}(a+b x)}}{a+\sqrt {-1+a^2}}\right )}{\sqrt {-1+a^2}}+\frac {(2 b) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {2 i x}{\left (-\frac {2 a}{b}-\frac {2 \sqrt {-1+a^2}}{b}\right ) b}\right )}{x} \, dx,x,e^{i \sin ^{-1}(a+b x)}\right )}{\sqrt {-1+a^2}}-\frac {(2 b) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {2 i x}{\left (-\frac {2 a}{b}+\frac {2 \sqrt {-1+a^2}}{b}\right ) b}\right )}{x} \, dx,x,e^{i \sin ^{-1}(a+b x)}\right )}{\sqrt {-1+a^2}}\\ &=-\frac {\sin ^{-1}(a+b x)^2}{x}+\frac {2 i b \sin ^{-1}(a+b x) \log \left (1+\frac {i e^{i \sin ^{-1}(a+b x)}}{a-\sqrt {-1+a^2}}\right )}{\sqrt {-1+a^2}}-\frac {2 i b \sin ^{-1}(a+b x) \log \left (1+\frac {i e^{i \sin ^{-1}(a+b x)}}{a+\sqrt {-1+a^2}}\right )}{\sqrt {-1+a^2}}+\frac {2 b \text {Li}_2\left (-\frac {i e^{i \sin ^{-1}(a+b x)}}{a-\sqrt {-1+a^2}}\right )}{\sqrt {-1+a^2}}-\frac {2 b \text {Li}_2\left (-\frac {i e^{i \sin ^{-1}(a+b x)}}{a+\sqrt {-1+a^2}}\right )}{\sqrt {-1+a^2}}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 208, normalized size = 0.90 \[ \frac {2 b x \text {Li}_2\left (\frac {i e^{i \sin ^{-1}(a+b x)}}{\sqrt {a^2-1}-a}\right )-2 b x \text {Li}_2\left (-\frac {i e^{i \sin ^{-1}(a+b x)}}{a+\sqrt {a^2-1}}\right )-\sqrt {a^2-1} \sin ^{-1}(a+b x)^2+2 i b x \sin ^{-1}(a+b x) \left (\log \left (\frac {-\sqrt {a^2-1}+i e^{i \sin ^{-1}(a+b x)}+a}{a-\sqrt {a^2-1}}\right )-\log \left (\frac {\sqrt {a^2-1}+i e^{i \sin ^{-1}(a+b x)}+a}{\sqrt {a^2-1}+a}\right )\right )}{\sqrt {a^2-1} x} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\arcsin \left (b x + a\right )^{2}}{x^{2}}, x\right ) \]
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 \[ \int \frac {\arcsin \left (b x + a\right )^{2}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.65, size = 333, normalized size = 1.45 \[ -\frac {\arcsin \left (b x +a \right )^{2}}{x}-\frac {2 b \sqrt {-a^{2}+1}\, \arcsin \left (b x +a \right ) \ln \left (\frac {i a +\sqrt {-a^{2}+1}-i \left (b x +a \right )-\sqrt {1-\left (b x +a \right )^{2}}}{i a +\sqrt {-a^{2}+1}}\right )}{a^{2}-1}+\frac {2 b \sqrt {-a^{2}+1}\, \arcsin \left (b x +a \right ) \ln \left (\frac {i a -\sqrt {-a^{2}+1}-i \left (b x +a \right )-\sqrt {1-\left (b x +a \right )^{2}}}{i a -\sqrt {-a^{2}+1}}\right )}{a^{2}-1}+\frac {2 i b \sqrt {-a^{2}+1}\, \dilog \left (\frac {i a +\sqrt {-a^{2}+1}-i \left (b x +a \right )-\sqrt {1-\left (b x +a \right )^{2}}}{i a +\sqrt {-a^{2}+1}}\right )}{a^{2}-1}-\frac {2 i b \sqrt {-a^{2}+1}\, \dilog \left (\frac {i a -\sqrt {-a^{2}+1}-i \left (b x +a \right )-\sqrt {1-\left (b x +a \right )^{2}}}{i a -\sqrt {-a^{2}+1}}\right )}{a^{2}-1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
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 \[ \int \frac {{\mathrm {asin}\left (a+b\,x\right )}^2}{x^2} \,d x \]
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 \[ \int \frac {\operatorname {asin}^{2}{\left (a + b x \right )}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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