Optimal. Leaf size=60 \[ \frac {2 i (1-i a x)^2}{a \sqrt {a^2 x^2+1}}+\frac {3 i \sqrt {a^2 x^2+1}}{a}-\frac {3 \sinh ^{-1}(a x)}{a} \]
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Rubi [A] time = 0.04, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5059, 853, 669, 641, 215} \[ \frac {2 i (1-i a x)^2}{a \sqrt {a^2 x^2+1}}+\frac {3 i \sqrt {a^2 x^2+1}}{a}-\frac {3 \sinh ^{-1}(a x)}{a} \]
Antiderivative was successfully verified.
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Rule 215
Rule 641
Rule 669
Rule 853
Rule 5059
Rubi steps
\begin {align*} \int e^{-3 i \tan ^{-1}(a x)} \, dx &=\int \frac {(1-i a x)^2}{(1+i a x) \sqrt {1+a^2 x^2}} \, dx\\ &=\int \frac {(1-i a x)^3}{\left (1+a^2 x^2\right )^{3/2}} \, dx\\ &=\frac {2 i (1-i a x)^2}{a \sqrt {1+a^2 x^2}}-3 \int \frac {1-i a x}{\sqrt {1+a^2 x^2}} \, dx\\ &=\frac {2 i (1-i a x)^2}{a \sqrt {1+a^2 x^2}}+\frac {3 i \sqrt {1+a^2 x^2}}{a}-3 \int \frac {1}{\sqrt {1+a^2 x^2}} \, dx\\ &=\frac {2 i (1-i a x)^2}{a \sqrt {1+a^2 x^2}}+\frac {3 i \sqrt {1+a^2 x^2}}{a}-\frac {3 \sinh ^{-1}(a x)}{a}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 42, normalized size = 0.70 \[ -\frac {3 \sinh ^{-1}(a x)}{a}+\frac {\sqrt {a^2 x^2+1} \left (\frac {4}{a x-i}+i\right )}{a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 60, normalized size = 1.00 \[ \frac {4 \, a x + {\left (3 \, a x - 3 i\right )} \log \left (-a x + \sqrt {a^{2} x^{2} + 1}\right ) + \sqrt {a^{2} x^{2} + 1} {\left (i \, a x + 5\right )} - 4 i}{a^{2} x - i \, a} \]
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 = 0.11, size = 219, normalized size = 3.65 \[ -\frac {\left (\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )\right )^{\frac {5}{2}}}{a^{4} \left (x -\frac {i}{a}\right )^{3}}-\frac {2 i \left (\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )\right )^{\frac {5}{2}}}{a^{3} \left (x -\frac {i}{a}\right )^{2}}+\frac {2 i \left (\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )\right )^{\frac {3}{2}}}{a}-3 \sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}\, x -\frac {3 \ln \left (\frac {i a +\left (x -\frac {i}{a}\right ) a^{2}}{\sqrt {a^{2}}}+\sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}\right )}{\sqrt {a^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 65, normalized size = 1.08 \[ \frac {i \, {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{a^{3} x^{2} - 2 i \, a^{2} x - a} - \frac {3 \, \operatorname {arsinh}\left (a x\right )}{a} + \frac {6 i \, \sqrt {a^{2} x^{2} + 1}}{i \, a^{2} x + a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.42, size = 73, normalized size = 1.22 \[ \frac {\sqrt {a^2\,x^2+1}\,1{}\mathrm {i}}{a}-\frac {3\,\mathrm {asinh}\left (x\,\sqrt {a^2}\right )}{\sqrt {a^2}}-\frac {4\,\sqrt {a^2\,x^2+1}}{\left (-x\,\sqrt {a^2}+\frac {\sqrt {a^2}\,1{}\mathrm {i}}{a}\right )\,\sqrt {a^2}} \]
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 \[ i \left (\int \frac {\sqrt {a^{2} x^{2} + 1}}{a^{3} x^{3} - 3 i a^{2} x^{2} - 3 a x + i}\, dx + \int \frac {a^{2} x^{2} \sqrt {a^{2} x^{2} + 1}}{a^{3} x^{3} - 3 i a^{2} x^{2} - 3 a x + i}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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