Optimal. Leaf size=102 \[ \frac {11 \sinh ^{-1}(a x)}{2 a^3}-\frac {i (1-i a x)^3}{a^3 \sqrt {a^2 x^2+1}}-\frac {i (3-i a x)^2 \sqrt {a^2 x^2+1}}{3 a^3}-\frac {(3 a x+28 i) \sqrt {a^2 x^2+1}}{6 a^3} \]
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Rubi [A] time = 0.57, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {5060, 1633, 1593, 12, 852, 1635, 1654, 780, 215} \[ -\frac {i (1-i a x)^3}{a^3 \sqrt {a^2 x^2+1}}-\frac {i (3-i a x)^2 \sqrt {a^2 x^2+1}}{3 a^3}-\frac {(3 a x+28 i) \sqrt {a^2 x^2+1}}{6 a^3}+\frac {11 \sinh ^{-1}(a x)}{2 a^3} \]
Antiderivative was successfully verified.
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Rule 12
Rule 215
Rule 780
Rule 852
Rule 1593
Rule 1633
Rule 1635
Rule 1654
Rule 5060
Rubi steps
\begin {align*} \int e^{-3 i \tan ^{-1}(a x)} x^2 \, dx &=\int \frac {x^2 (1-i a x)^2}{(1+i a x) \sqrt {1+a^2 x^2}} \, dx\\ &=(i a) \int \frac {\sqrt {1+a^2 x^2} \left (-\frac {i x^2}{a}-x^3\right )}{(1+i a x)^2} \, dx\\ &=(i a) \int \frac {\left (-\frac {i}{a}-x\right ) x^2 \sqrt {1+a^2 x^2}}{(1+i a x)^2} \, dx\\ &=a^2 \int \frac {x^2 \left (1+a^2 x^2\right )^{3/2}}{a^2 (1+i a x)^3} \, dx\\ &=\int \frac {x^2 \left (1+a^2 x^2\right )^{3/2}}{(1+i a x)^3} \, dx\\ &=\int \frac {x^2 (1-i a x)^3}{\left (1+a^2 x^2\right )^{3/2}} \, dx\\ &=-\frac {i (1-i a x)^3}{a^3 \sqrt {1+a^2 x^2}}-\int \frac {\left (-\frac {3}{a^2}+\frac {i x}{a}\right ) (1-i a x)^2}{\sqrt {1+a^2 x^2}} \, dx\\ &=-\frac {i (1-i a x)^3}{a^3 \sqrt {1+a^2 x^2}}-\frac {i (3-i a x)^2 \sqrt {1+a^2 x^2}}{3 a^3}+\frac {1}{3} \int \frac {\left (-\frac {3}{a^2}+\frac {i x}{a}\right ) (-5+3 i a x)}{\sqrt {1+a^2 x^2}} \, dx\\ &=-\frac {i (1-i a x)^3}{a^3 \sqrt {1+a^2 x^2}}-\frac {i (3-i a x)^2 \sqrt {1+a^2 x^2}}{3 a^3}-\frac {(28 i+3 a x) \sqrt {1+a^2 x^2}}{6 a^3}+\frac {11 \int \frac {1}{\sqrt {1+a^2 x^2}} \, dx}{2 a^2}\\ &=-\frac {i (1-i a x)^3}{a^3 \sqrt {1+a^2 x^2}}-\frac {i (3-i a x)^2 \sqrt {1+a^2 x^2}}{3 a^3}-\frac {(28 i+3 a x) \sqrt {1+a^2 x^2}}{6 a^3}+\frac {11 \sinh ^{-1}(a x)}{2 a^3}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 63, normalized size = 0.62 \[ \frac {33 \sinh ^{-1}(a x)+\frac {\sqrt {a^2 x^2+1} \left (2 i a^3 x^3-7 a^2 x^2-19 i a x-52\right )}{a x-i}}{6 a^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 81, normalized size = 0.79 \[ -\frac {24 \, a x + {\left (33 \, a x - 33 i\right )} \log \left (-a x + \sqrt {a^{2} x^{2} + 1}\right ) - {\left (2 i \, a^{3} x^{3} - 7 \, a^{2} x^{2} - 19 i \, a x - 52\right )} \sqrt {a^{2} x^{2} + 1} - 24 i}{6 \, a^{4} x - 6 i \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {undef} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.17, size = 224, normalized size = 2.20 \[ \frac {4 i \left (\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )\right )^{\frac {5}{2}}}{a^{5} \left (x -\frac {i}{a}\right )^{2}}-\frac {11 i \left (\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )\right )^{\frac {3}{2}}}{3 a^{3}}+\frac {11 \sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}\, x}{2 a^{2}}+\frac {11 \ln \left (\frac {i a +\left (x -\frac {i}{a}\right ) a^{2}}{\sqrt {a^{2}}}+\sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}\right )}{2 a^{2} \sqrt {a^{2}}}+\frac {\left (\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )\right )^{\frac {5}{2}}}{a^{6} \left (x -\frac {i}{a}\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.44, size = 181, normalized size = 1.77 \[ -\frac {i \, {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{a^{5} x^{2} - 2 i \, a^{4} x - a^{3}} - \frac {i \, {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{i \, a^{4} x + a^{3}} - \frac {6 i \, \sqrt {a^{2} x^{2} + 1}}{i \, a^{4} x + a^{3}} - \frac {\sqrt {-a^{2} x^{2} + 4 i \, a x + 3} x}{2 \, a^{2}} + \frac {i \, {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{3 \, a^{3}} + \frac {\arcsin \left (i \, a x + 2\right )}{2 \, a^{3}} + \frac {6 \, \operatorname {arsinh}\left (a x\right )}{a^{3}} - \frac {3 i \, \sqrt {a^{2} x^{2} + 1}}{a^{3}} + \frac {i \, \sqrt {-a^{2} x^{2} + 4 i \, a x + 3}}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 115, normalized size = 1.13 \[ \frac {11\,\mathrm {asinh}\left (x\,\sqrt {a^2}\right )}{2\,a^2\,\sqrt {a^2}}-\frac {\sqrt {a^2\,x^2+1}\,\left (\frac {3\,x\,\sqrt {a^2}}{2\,a^2}+\frac {a\,14{}\mathrm {i}}{3\,{\left (a^2\right )}^{3/2}}-\frac {a^3\,x^2\,1{}\mathrm {i}}{3\,{\left (a^2\right )}^{3/2}}\right )}{\sqrt {a^2}}+\frac {4\,\sqrt {a^2\,x^2+1}}{a^2\,\left (-x\,\sqrt {a^2}+\frac {\sqrt {a^2}\,1{}\mathrm {i}}{a}\right )\,\sqrt {a^2}} \]
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 \[ i \left (\int \frac {x^{2} \sqrt {a^{2} x^{2} + 1}}{a^{3} x^{3} - 3 i a^{2} x^{2} - 3 a x + i}\, dx + \int \frac {a^{2} x^{4} \sqrt {a^{2} x^{2} + 1}}{a^{3} x^{3} - 3 i a^{2} x^{2} - 3 a x + i}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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