Optimal. Leaf size=92 \[ -\frac {\left (a^2 x^2+1\right )^{5/2}}{a^2 (1+i a x)^3}-\frac {3 \left (a^2 x^2+1\right )^{3/2}}{2 a^2 (1+i a x)}-\frac {9 \sqrt {a^2 x^2+1}}{2 a^2}-\frac {9 i \sinh ^{-1}(a x)}{2 a^2} \]
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Rubi [A] time = 0.32, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {5060, 1633, 1593, 12, 793, 665, 215} \[ -\frac {\left (a^2 x^2+1\right )^{5/2}}{a^2 (1+i a x)^3}-\frac {3 \left (a^2 x^2+1\right )^{3/2}}{2 a^2 (1+i a x)}-\frac {9 \sqrt {a^2 x^2+1}}{2 a^2}-\frac {9 i \sinh ^{-1}(a x)}{2 a^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 215
Rule 665
Rule 793
Rule 1593
Rule 1633
Rule 5060
Rubi steps
\begin {align*} \int e^{-3 i \tan ^{-1}(a x)} x \, dx &=\int \frac {x (1-i a x)^2}{(1+i a x) \sqrt {1+a^2 x^2}} \, dx\\ &=(i a) \int \frac {\left (-\frac {i x}{a}-x^2\right ) \sqrt {1+a^2 x^2}}{(1+i a x)^2} \, dx\\ &=(i a) \int \frac {\left (-\frac {i}{a}-x\right ) x \sqrt {1+a^2 x^2}}{(1+i a x)^2} \, dx\\ &=a^2 \int \frac {x \left (1+a^2 x^2\right )^{3/2}}{a^2 (1+i a x)^3} \, dx\\ &=\int \frac {x \left (1+a^2 x^2\right )^{3/2}}{(1+i a x)^3} \, dx\\ &=-\frac {\left (1+a^2 x^2\right )^{5/2}}{a^2 (1+i a x)^3}-\frac {(3 i) \int \frac {\left (1+a^2 x^2\right )^{3/2}}{(1+i a x)^2} \, dx}{a}\\ &=-\frac {3 \left (1+a^2 x^2\right )^{3/2}}{2 a^2 (1+i a x)}-\frac {\left (1+a^2 x^2\right )^{5/2}}{a^2 (1+i a x)^3}-\frac {(9 i) \int \frac {\sqrt {1+a^2 x^2}}{1+i a x} \, dx}{2 a}\\ &=-\frac {9 \sqrt {1+a^2 x^2}}{2 a^2}-\frac {3 \left (1+a^2 x^2\right )^{3/2}}{2 a^2 (1+i a x)}-\frac {\left (1+a^2 x^2\right )^{5/2}}{a^2 (1+i a x)^3}-\frac {(9 i) \int \frac {1}{\sqrt {1+a^2 x^2}} \, dx}{2 a}\\ &=-\frac {9 \sqrt {1+a^2 x^2}}{2 a^2}-\frac {3 \left (1+a^2 x^2\right )^{3/2}}{2 a^2 (1+i a x)}-\frac {\left (1+a^2 x^2\right )^{5/2}}{a^2 (1+i a x)^3}-\frac {9 i \sinh ^{-1}(a x)}{2 a^2}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 60, normalized size = 0.65 \[ \sqrt {a^2 x^2+1} \left (\frac {4 i}{a^2 (a x-i)}-\frac {3}{a^2}+\frac {i x}{2 a}\right )-\frac {9 i \sinh ^{-1}(a x)}{2 a^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 72, normalized size = 0.78 \[ \frac {8 i \, a x - 9 \, {\left (-i \, a x - 1\right )} \log \left (-a x + \sqrt {a^{2} x^{2} + 1}\right ) + \sqrt {a^{2} x^{2} + 1} {\left (i \, a^{2} x^{2} - 5 \, a x + 14 i\right )} + 8}{2 \, a^{3} x - 2 i \, a^{2}} \]
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 = 226, normalized size = 2.46 \[ \frac {3 \left (\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )\right )^{\frac {5}{2}}}{a^{4} \left (x -\frac {i}{a}\right )^{2}}-\frac {3 \left (\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )\right )^{\frac {3}{2}}}{a^{2}}-\frac {9 i \sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}\, x}{2 a}-\frac {9 i \ln \left (\frac {i a +\left (x -\frac {i}{a}\right ) a^{2}}{\sqrt {a^{2}}}+\sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}\right )}{2 a \sqrt {a^{2}}}-\frac {i \left (\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )\right )^{\frac {5}{2}}}{a^{5} \left (x -\frac {i}{a}\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 112, normalized size = 1.22 \[ -\frac {{\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{a^{4} x^{2} - 2 i \, a^{3} x - a^{2}} - \frac {{\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{2 i \, a^{3} x + 2 \, a^{2}} - \frac {6 \, \sqrt {a^{2} x^{2} + 1}}{i \, a^{3} x + a^{2}} - \frac {9 i \, \operatorname {arsinh}\left (a x\right )}{2 \, a^{2}} - \frac {3 \, \sqrt {a^{2} x^{2} + 1}}{2 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.42, size = 105, normalized size = 1.14 \[ -\frac {\sqrt {a^2\,x^2+1}\,\left (\frac {3\,\sqrt {a^2}}{a^2}-\frac {x\,\sqrt {a^2}\,1{}\mathrm {i}}{2\,a}\right )}{\sqrt {a^2}}-\frac {\mathrm {asinh}\left (x\,\sqrt {a^2}\right )\,9{}\mathrm {i}}{2\,a\,\sqrt {a^2}}-\frac {\sqrt {a^2\,x^2+1}\,4{}\mathrm {i}}{a\,\left (-x\,\sqrt {a^2}+\frac {\sqrt {a^2}\,1{}\mathrm {i}}{a}\right )\,\sqrt {a^2}} \]
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 \[ i \left (\int \frac {x \sqrt {a^{2} x^{2} + 1}}{a^{3} x^{3} - 3 i a^{2} x^{2} - 3 a x + i}\, dx + \int \frac {a^{2} x^{3} \sqrt {a^{2} x^{2} + 1}}{a^{3} x^{3} - 3 i a^{2} x^{2} - 3 a x + i}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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