Optimal. Leaf size=74 \[ \frac {\sin ^{-1}(a x)}{2 a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}+\frac {\left (1-a^2 x^2\right )^{3/2}}{3 a^3}-\frac {\sqrt {1-a^2 x^2}}{a^3} \]
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Rubi [A] time = 0.05, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6124, 797, 641, 195, 216} \[ \frac {\left (1-a^2 x^2\right )^{3/2}}{3 a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}-\frac {\sqrt {1-a^2 x^2}}{a^3}+\frac {\sin ^{-1}(a x)}{2 a^3} \]
Antiderivative was successfully verified.
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Rule 195
Rule 216
Rule 641
Rule 797
Rule 6124
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 = 0.03, size = 44, normalized size = 0.59 \[ \frac {3 \sin ^{-1}(a x)-\sqrt {1-a^2 x^2} \left (2 a^2 x^2+3 a x+4\right )}{6 a^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.82, size = 57, normalized size = 0.77 \[ -\frac {{\left (2 \, a^{2} x^{2} + 3 \, a x + 4\right )} \sqrt {-a^{2} x^{2} + 1} + 6 \, \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right )}{6 \, a^{3}} \]
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 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 87, normalized size = 1.18 \[ -\frac {x^{2} \sqrt {-a^{2} x^{2}+1}}{3 a}-\frac {2 \sqrt {-a^{2} x^{2}+1}}{3 a^{3}}-\frac {x \sqrt {-a^{2} x^{2}+1}}{2 a^{2}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a^{2} \sqrt {a^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 65, normalized size = 0.88 \[ -\frac {\sqrt {-a^{2} x^{2} + 1} x^{2}}{3 \, a} - \frac {\sqrt {-a^{2} x^{2} + 1} x}{2 \, a^{2}} + \frac {\arcsin \left (a x\right )}{2 \, a^{3}} - \frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{3 \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 82, normalized size = 1.11 \[ \frac {\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{2\,a^2\,\sqrt {-a^2}}+\frac {\sqrt {1-a^2\,x^2}\,\left (\frac {2}{3\,a\,\sqrt {-a^2}}+\frac {a\,x^2}{3\,\sqrt {-a^2}}-\frac {x\,\sqrt {-a^2}}{2\,a^2}\right )}{\sqrt {-a^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.29, size = 133, normalized size = 1.80 \[ a \left (\begin {cases} - \frac {x^{2} \sqrt {- a^{2} x^{2} + 1}}{3 a^{2}} - \frac {2 \sqrt {- a^{2} x^{2} + 1}}{3 a^{4}} & \text {for}\: a \neq 0 \\\frac {x^{4}}{4} & \text {otherwise} \end {cases}\right ) + \begin {cases} - \frac {i x \sqrt {a^{2} x^{2} - 1}}{2 a^{2}} - \frac {i \operatorname {acosh}{\left (a x \right )}}{2 a^{3}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {x^{3}}{2 \sqrt {- a^{2} x^{2} + 1}} - \frac {x}{2 a^{2} \sqrt {- a^{2} x^{2} + 1}} + \frac {\operatorname {asin}{\left (a x \right )}}{2 a^{3}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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