Optimal. Leaf size=45 \[ -\frac {x \sin ^{-1}(x)}{1+\sqrt {1-x^2}}+\frac {1}{2} \sin ^{-1}(x)^2-\log \left (1+\sqrt {1-x^2}\right ) \]
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Rubi [A]
time = 0.08, antiderivative size = 51, normalized size of antiderivative = 1.13, number of steps
used = 9, number of rules used = 11, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.611, Rules used = {6874, 283,
222, 4875, 4723, 272, 65, 212, 4781, 29, 4737} \begin {gather*} \frac {\sqrt {1-x^2} \sin ^{-1}(x)}{x}-\tanh ^{-1}\left (\sqrt {1-x^2}\right )-\log (x)+\frac {1}{2} \sin ^{-1}(x)^2-\frac {\sin ^{-1}(x)}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 65
Rule 212
Rule 222
Rule 272
Rule 283
Rule 4723
Rule 4737
Rule 4781
Rule 4875
Rule 6874
Rubi steps
\begin {align*} \int \frac {\sin ^{-1}(x)}{1+\sqrt {1-x^2}} \, dx &=\int \left (\frac {\sin ^{-1}(x)}{x^2}-\frac {\sqrt {1-x^2} \sin ^{-1}(x)}{x^2}\right ) \, dx\\ &=\int \frac {\sin ^{-1}(x)}{x^2} \, dx-\int \frac {\sqrt {1-x^2} \sin ^{-1}(x)}{x^2} \, dx\\ &=-\frac {\sin ^{-1}(x)}{x}+\frac {\sqrt {1-x^2} \sin ^{-1}(x)}{x}-\int \frac {1}{x} \, dx+\int \frac {1}{x \sqrt {1-x^2}} \, dx+\int \frac {\sin ^{-1}(x)}{\sqrt {1-x^2}} \, dx\\ &=-\frac {\sin ^{-1}(x)}{x}+\frac {\sqrt {1-x^2} \sin ^{-1}(x)}{x}+\frac {1}{2} \sin ^{-1}(x)^2-\log (x)+\frac {1}{2} \text {Subst}\left (\int \frac {1}{\sqrt {1-x} x} \, dx,x,x^2\right )\\ &=-\frac {\sin ^{-1}(x)}{x}+\frac {\sqrt {1-x^2} \sin ^{-1}(x)}{x}+\frac {1}{2} \sin ^{-1}(x)^2-\log (x)-\text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {1-x^2}\right )\\ &=-\frac {\sin ^{-1}(x)}{x}+\frac {\sqrt {1-x^2} \sin ^{-1}(x)}{x}+\frac {1}{2} \sin ^{-1}(x)^2-\tanh ^{-1}\left (\sqrt {1-x^2}\right )-\log (x)\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 44, normalized size = 0.98 \begin {gather*} \frac {\left (-1+\sqrt {1-x^2}\right ) \sin ^{-1}(x)}{x}+\frac {1}{2} \sin ^{-1}(x)^2-\log \left (1+\sqrt {1-x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Mathics [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {cought exception: maximum recursion depth exceeded while calling a Python object} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\arcsin \left (x \right )}{1+\sqrt {-x^{2}+1}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 63, normalized size = 1.40 \begin {gather*} \frac {x \arcsin \left (x\right )^{2} - 2 \, x \log \left (x\right ) - x \log \left (\sqrt {-x^{2} + 1} + 1\right ) + x \log \left (\sqrt {-x^{2} + 1} - 1\right ) + 2 \, \sqrt {-x^{2} + 1} \arcsin \left (x\right ) - 2 \, \arcsin \left (x\right )}{2 \, x} \end {gather*}
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 \begin {gather*} \int \frac {\operatorname {asin}{\left (x \right )}}{\sqrt {1 - x^{2}} + 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.01, size = 60, normalized size = 1.33 \begin {gather*} \frac {\arcsin ^{2}x}{2}-\frac {x \arcsin x}{\sqrt {-x^{2}+1}+1}-\ln \left (4\right )-2 \ln \left (\sqrt {-x^{2}+1}+1\right )+\ln \left (2 \sqrt {-x^{2}+1}+2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\mathrm {asin}\left (x\right )}{\sqrt {1-x^2}+1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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