Optimal. Leaf size=75 \[ -\frac {\sinh ^{-1}(a x)}{2 a^3}+\frac {x \sqrt {a^2 x^2+1}}{2 a^2}-\frac {i \left (a^2 x^2+1\right )^{3/2}}{3 a^3}+\frac {i \sqrt {a^2 x^2+1}}{a^3} \]
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Rubi [A] time = 0.05, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {5060, 797, 641, 195, 215} \[ -\frac {i \left (a^2 x^2+1\right )^{3/2}}{3 a^3}+\frac {x \sqrt {a^2 x^2+1}}{2 a^2}+\frac {i \sqrt {a^2 x^2+1}}{a^3}-\frac {\sinh ^{-1}(a x)}{2 a^3} \]
Antiderivative was successfully verified.
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Rule 195
Rule 215
Rule 641
Rule 797
Rule 5060
Rubi steps
\begin {align*} \int e^{-i \tan ^{-1}(a x)} x^2 \, dx &=\int \frac {x^2 (1-i a x)}{\sqrt {1+a^2 x^2}} \, dx\\ &=-\frac {\int \frac {1-i a x}{\sqrt {1+a^2 x^2}} \, dx}{a^2}+\frac {\int (1-i a x) \sqrt {1+a^2 x^2} \, dx}{a^2}\\ &=\frac {i \sqrt {1+a^2 x^2}}{a^3}-\frac {i \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 {i \sqrt {1+a^2 x^2}}{a^3}+\frac {x \sqrt {1+a^2 x^2}}{2 a^2}-\frac {i \left (1+a^2 x^2\right )^{3/2}}{3 a^3}-\frac {\sinh ^{-1}(a x)}{a^3}+\frac {\int \frac {1}{\sqrt {1+a^2 x^2}} \, dx}{2 a^2}\\ &=\frac {i \sqrt {1+a^2 x^2}}{a^3}+\frac {x \sqrt {1+a^2 x^2}}{2 a^2}-\frac {i \left (1+a^2 x^2\right )^{3/2}}{3 a^3}-\frac {\sinh ^{-1}(a x)}{2 a^3}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 46, normalized size = 0.61 \[ \frac {-3 \sinh ^{-1}(a x)+\left (-2 i a^2 x^2+3 a x+4 i\right ) \sqrt {a^2 x^2+1}}{6 a^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 51, normalized size = 0.68 \[ \frac {\sqrt {a^{2} x^{2} + 1} {\left (-2 i \, a^{2} x^{2} + 3 \, a x + 4 i\right )} + 3 \, \log \left (-a x + \sqrt {a^{2} x^{2} + 1}\right )}{6 \, a^{3}} \]
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 [B] time = 0.16, size = 168, normalized size = 2.24 \[ -\frac {i \left (a^{2} x^{2}+1\right )^{\frac {3}{2}}}{3 a^{3}}+\frac {x \sqrt {a^{2} x^{2}+1}}{2 a^{2}}+\frac {\ln \left (\frac {x \,a^{2}}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}+1}\right )}{2 a^{2} \sqrt {a^{2}}}+\frac {i \sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}}{a^{3}}-\frac {\ln \left (\frac {i a +\left (x -\frac {i}{a}\right ) a^{2}}{\sqrt {a^{2}}}+\sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}\right )}{a^{2} \sqrt {a^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 59, normalized size = 0.79 \[ \frac {\sqrt {a^{2} x^{2} + 1} x}{2 \, a^{2}} - \frac {i \, {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{3 \, a^{3}} - \frac {\operatorname {arsinh}\left (a x\right )}{2 \, a^{3}} + \frac {i \, \sqrt {a^{2} x^{2} + 1}}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.42, size = 71, normalized size = 0.95 \[ \frac {\sqrt {a^2\,x^2+1}\,\left (\frac {x\,\sqrt {a^2}}{2\,a^2}+\frac {a\,2{}\mathrm {i}}{3\,{\left (a^2\right )}^{3/2}}-\frac {a^3\,x^2\,1{}\mathrm {i}}{3\,{\left (a^2\right )}^{3/2}}\right )}{\sqrt {a^2}}-\frac {\mathrm {asinh}\left (x\,\sqrt {a^2}\right )}{2\,a^2\,\sqrt {a^2}} \]
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 \[ - i \int \frac {x^{2} \sqrt {a^{2} x^{2} + 1}}{a x - i}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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