Optimal. Leaf size=75 \[ \frac {i \sqrt {1+a^2 x^2}}{a^3}+\frac {x \sqrt {1+a^2 x^2}}{2 a^2}-\frac {i \left (1+a^2 x^2\right )^{3/2}}{3 a^3}-\frac {\sinh ^{-1}(a x)}{2 a^3} \]
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Rubi [A]
time = 0.03, 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 = {5168, 811, 655,
201, 221} \begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 201
Rule 221
Rule 655
Rule 811
Rule 5168
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 = 0.02, size = 46, normalized size = 0.61 \begin {gather*} \frac {\left (4 i+3 a x-2 i a^2 x^2\right ) \sqrt {1+a^2 x^2}-3 \sinh ^{-1}(a x)}{6 a^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 166 vs. \(2 (61 ) = 122\).
time = 0.08, size = 167, normalized size = 2.23
method | result | size |
risch | \(-\frac {i \left (2 a^{2} x^{2}+3 i a x -4\right ) \sqrt {a^{2} x^{2}+1}}{6 a^{3}}-\frac {\ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}+1}\right )}{2 a^{2} \sqrt {a^{2}}}\) | \(67\) |
default | \(-\frac {i \left (a^{2} x^{2}+1\right )^{\frac {3}{2}}}{3 a^{3}}+\frac {\frac {x \sqrt {a^{2} x^{2}+1}}{2}+\frac {\ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}+1}\right )}{2 \sqrt {a^{2}}}}{a^{2}}+\frac {i \left (\sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}+\frac {i a \ln \left (\frac {i a +\left (x -\frac {i}{a}\right ) a^{2}}{\sqrt {a^{2}}}+\sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}\right )}{\sqrt {a^{2}}}\right )}{a^{3}}\) | \(167\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 59, normalized size = 0.79 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.05, size = 51, normalized size = 0.68 \begin {gather*} \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}} \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*} - i \int \frac {x^{2} \sqrt {a^{2} x^{2} + 1}}{a x - i}\, 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 = 0.42, size = 71, normalized size = 0.95 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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