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 = 0.04, 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 = 0.03, 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 [A]
time = 0.06, size = 92, normalized size = 1.23
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 | \(i a \left (\frac {x^{2} \sqrt {a^{2} x^{2}+1}}{3 a^{2}}-\frac {2 \sqrt {a^{2} x^{2}+1}}{3 a^{4}}\right )+\frac {x \sqrt {a^{2} x^{2}+1}}{2 a^{2}}-\frac {\ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}+1}\right )}{2 a^{2} \sqrt {a^{2}}}\) | \(92\) |
meijerg | \(\frac {\frac {\sqrt {\pi }\, x \left (a^{2}\right )^{\frac {3}{2}} \sqrt {a^{2} x^{2}+1}}{a^{2}}-\frac {\sqrt {\pi }\, \left (a^{2}\right )^{\frac {3}{2}} \arcsinh \left (a x \right )}{a^{3}}}{2 a^{2} \sqrt {\pi }\, \sqrt {a^{2}}}+\frac {i \left (\frac {4 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (-4 a^{2} x^{2}+8\right ) \sqrt {a^{2} x^{2}+1}}{6}\right )}{2 a^{3} \sqrt {\pi }}\) | \(98\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 62, normalized size = 0.83 \begin {gather*} \frac {i \, \sqrt {a^{2} x^{2} + 1} x^{2}}{3 \, a} + \frac {\sqrt {a^{2} x^{2} + 1} x}{2 \, a^{2}} - \frac {\operatorname {arsinh}\left (a x\right )}{2 \, a^{3}} - \frac {2 i \, \sqrt {a^{2} x^{2} + 1}}{3 \, a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.94, 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 [A]
time = 2.14, size = 75, normalized size = 1.00 \begin {gather*} i 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 ) + \frac {x \sqrt {a^{2} x^{2} + 1}}{2 a^{2}} - \frac {\operatorname {asinh}{\left (a x \right )}}{2 a^{3}} \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 = 0.41, 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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