Optimal. Leaf size=22 \[ \frac {x \sin ^{-1}(x)}{\sqrt {1+x^2}}-\frac {1}{2} \sin ^{-1}\left (x^2\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {197, 4755, 281,
222} \begin {gather*} \frac {x \text {ArcSin}(x)}{\sqrt {x^2+1}}-\frac {\text {ArcSin}\left (x^2\right )}{2} \end {gather*}
Antiderivative was successfully verified.
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Rule 197
Rule 222
Rule 281
Rule 4755
Rubi steps
\begin {align*} \int \frac {\sin ^{-1}(x)}{\left (1+x^2\right )^{3/2}} \, dx &=\frac {x \sin ^{-1}(x)}{\sqrt {1+x^2}}-\int \frac {x}{\sqrt {1-x^4}} \, dx\\ &=\frac {x \sin ^{-1}(x)}{\sqrt {1+x^2}}-\frac {1}{2} \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2}} \, dx,x,x^2\right )\\ &=\frac {x \sin ^{-1}(x)}{\sqrt {1+x^2}}-\frac {1}{2} \sin ^{-1}\left (x^2\right )\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 22, normalized size = 1.00 \begin {gather*} \frac {x \sin ^{-1}(x)}{\sqrt {1+x^2}}-\frac {1}{2} \sin ^{-1}\left (x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.08, size = 0, normalized size = 0.00 \[\int \frac {\arcsin \left (x \right )}{\left (x^{2}+1\right )^{\frac {3}{2}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 3.80, size = 18, normalized size = 0.82 \begin {gather*} \frac {x \arcsin \left (x\right )}{\sqrt {x^{2} + 1}} - \frac {1}{2} \, \arcsin \left (x^{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 56 vs.
\(2 (18) = 36\).
time = 0.66, size = 56, normalized size = 2.55 \begin {gather*} \frac {2 \, \sqrt {x^{2} + 1} x \arcsin \left (x\right ) + {\left (x^{2} + 1\right )} \arctan \left (\frac {\sqrt {x^{2} + 1} \sqrt {-x^{2} + 1} x^{2}}{x^{4} - 1}\right )}{2 \, {\left (x^{2} + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 18.16, size = 78, normalized size = 3.55 \begin {gather*} \frac {x \operatorname {asin}{\left (x \right )}}{\sqrt {x^{2} + 1}} + \frac {i {G_{6, 6}^{6, 2}\left (\begin {matrix} \frac {1}{4}, \frac {3}{4} & \frac {1}{2}, \frac {1}{2}, 1, 1 \\0, \frac {1}{4}, \frac {1}{2}, \frac {3}{4}, 1, 0 & \end {matrix} \middle | {\frac {1}{x^{4}}} \right )}}{8 \pi ^{\frac {3}{2}}} - \frac {{G_{6, 6}^{2, 6}\left (\begin {matrix} - \frac {1}{2}, - \frac {1}{4}, 0, \frac {1}{4}, \frac {1}{2}, 1 & \\- \frac {1}{4}, \frac {1}{4} & - \frac {1}{2}, 0, 0, 0 \end {matrix} \middle | {\frac {e^{- 2 i \pi }}{x^{4}}} \right )}}{8 \pi ^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.46, size = 18, normalized size = 0.82 \begin {gather*} \frac {x \arcsin \left (x\right )}{\sqrt {x^{2} + 1}} - \frac {1}{2} \, \arcsin \left (x^{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 = -0.05 \begin {gather*} \int \frac {\mathrm {asin}\left (x\right )}{{\left (x^2+1\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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