Optimal. Leaf size=3 \[ \tan ^{-1}\left (\sin ^{-1}(x)\right ) \]
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Rubi [A]
time = 0.03, antiderivative size = 3, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {6828, 209}
\begin {gather*} \text {ArcTan}(\text {ArcSin}(x)) \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 6828
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {1-x^2} \left (1+\sin ^{-1}(x)^2\right )} \, dx &=\text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sin ^{-1}(x)\right )\\ &=\tan ^{-1}\left (\sin ^{-1}(x)\right )\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 3, normalized size = 1.00 \begin {gather*} \tan ^{-1}\left (\sin ^{-1}(x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 4, normalized size = 1.33
method | result | size |
derivativedivides | \(\arctan \left (\arcsin \left (x \right )\right )\) | \(4\) |
default | \(\arctan \left (\arcsin \left (x \right )\right )\) | \(4\) |
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 = 0.75, size = 3, normalized size = 1.00 \begin {gather*} \arctan \left (\arcsin \left (x\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.13, size = 3, normalized size = 1.00 \begin {gather*} \operatorname {atan}{\left (\operatorname {asin}{\left (x \right )} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.44, size = 3, normalized size = 1.00 \begin {gather*} \arctan \left (\arcsin \left (x\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.10, size = 43, normalized size = 14.33 \begin {gather*} \frac {\ln \left (\frac {-1+\mathrm {asin}\left (x\right )\,1{}\mathrm {i}}{\sqrt {1-x^2}}\right )\,1{}\mathrm {i}}{2}-\frac {\ln \left (\frac {1+\mathrm {asin}\left (x\right )\,1{}\mathrm {i}}{\sqrt {1-x^2}}\right )\,1{}\mathrm {i}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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