Optimal. Leaf size=73 \[ \frac {\sqrt {1-a^2 x^2}}{a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}-\frac {\left (1-a^2 x^2\right )^{3/2}}{3 a^3}+\frac {\text {ArcSin}(a x)}{2 a^3} \]
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Rubi [A]
time = 0.04, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6259, 811, 655,
201, 222} \begin {gather*} \frac {\text {ArcSin}(a x)}{2 a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}-\frac {\left (1-a^2 x^2\right )^{3/2}}{3 a^3}+\frac {\sqrt {1-a^2 x^2}}{a^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 201
Rule 222
Rule 655
Rule 811
Rule 6259
Rubi steps
\begin {align*} \int e^{-\tanh ^{-1}(a x)} x^2 \, dx &=\int \frac {x^2 (1-a x)}{\sqrt {1-a^2 x^2}} \, dx\\ &=\frac {\int \frac {1-a x}{\sqrt {1-a^2 x^2}} \, dx}{a^2}-\frac {\int (1-a x) \sqrt {1-a^2 x^2} \, dx}{a^2}\\ &=\frac {\sqrt {1-a^2 x^2}}{a^3}-\frac {\left (1-a^2 x^2\right )^{3/2}}{3 a^3}+\frac {\int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{a^2}-\frac {\int \sqrt {1-a^2 x^2} \, dx}{a^2}\\ &=\frac {\sqrt {1-a^2 x^2}}{a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}-\frac {\left (1-a^2 x^2\right )^{3/2}}{3 a^3}+\frac {\sin ^{-1}(a x)}{a^3}-\frac {\int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{2 a^2}\\ &=\frac {\sqrt {1-a^2 x^2}}{a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}-\frac {\left (1-a^2 x^2\right )^{3/2}}{3 a^3}+\frac {\sin ^{-1}(a x)}{2 a^3}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 43, normalized size = 0.59 \begin {gather*} \frac {\sqrt {1-a^2 x^2} \left (4-3 a x+2 a^2 x^2\right )+3 \text {ArcSin}(a x)}{6 a^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(132\) vs.
\(2(61)=122\).
time = 0.81, size = 133, normalized size = 1.82
method | result | size |
risch | \(-\frac {\left (2 a^{2} x^{2}-3 a x +4\right ) \left (a^{2} x^{2}-1\right )}{6 a^{3} \sqrt {-a^{2} x^{2}+1}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a^{2} \sqrt {a^{2}}}\) | \(72\) |
default | \(-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{3 a^{3}}-\frac {\frac {x \sqrt {-a^{2} x^{2}+1}}{2}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 \sqrt {a^{2}}}}{a^{2}}+\frac {\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}+\frac {a \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{\sqrt {a^{2}}}}{a^{3}}\) | \(133\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.46, size = 61, normalized size = 0.84 \begin {gather*} -\frac {\sqrt {-a^{2} x^{2} + 1} x}{2 \, a^{2}} - \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{3 \, a^{3}} + \frac {\arcsin \left (a x\right )}{2 \, a^{3}} + \frac {\sqrt {-a^{2} x^{2} + 1}}{a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 57, normalized size = 0.78 \begin {gather*} \frac {{\left (2 \, a^{2} x^{2} - 3 \, a x + 4\right )} \sqrt {-a^{2} x^{2} + 1} - 6 \, \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right )}{6 \, a^{3}} \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 {x^{2} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}{a x + 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.83, size = 82, normalized size = 1.12 \begin {gather*} \frac {\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{2\,a^2\,\sqrt {-a^2}}+\frac {\sqrt {1-a^2\,x^2}\,\left (\frac {2\,a}{3\,{\left (-a^2\right )}^{3/2}}-\frac {x\,\sqrt {-a^2}}{2\,a^2}+\frac {a^3\,x^2}{3\,{\left (-a^2\right )}^{3/2}}\right )}{\sqrt {-a^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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