Optimal. Leaf size=60 \[ \frac {2 i (1-i a x)^2}{a \sqrt {1+a^2 x^2}}+\frac {3 i \sqrt {1+a^2 x^2}}{a}-\frac {3 \sinh ^{-1}(a x)}{a} \]
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Rubi [A]
time = 0.03, 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 = {5167, 867, 683,
655, 221} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 221
Rule 655
Rule 683
Rule 867
Rule 5167
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 = 0.03, size = 42, normalized size = 0.70 \begin {gather*} \frac {\sqrt {1+a^2 x^2} \left (i+\frac {4}{-i+a x}\right )}{a}-\frac {3 \sinh ^{-1}(a x)}{a} \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. 256 vs. \(2 (53 ) = 106\).
time = 0.11, size = 257, normalized size = 4.28
method | result | size |
risch | \(\frac {i \sqrt {a^{2} x^{2}+1}}{a}-\frac {3 \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}+1}\right )}{\sqrt {a^{2}}}+\frac {4 \sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}}{a^{2} \left (x -\frac {i}{a}\right )}\) | \(93\) |
default | \(\frac {i \left (\frac {i \left (\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )\right )^{\frac {5}{2}}}{a \left (x -\frac {i}{a}\right )^{3}}-2 i a \left (-\frac {i \left (\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )\right )^{\frac {5}{2}}}{a \left (x -\frac {i}{a}\right )^{2}}+3 i a \left (\frac {\left (\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )\right )^{\frac {3}{2}}}{3}+i a \left (\frac {\left (2 \left (x -\frac {i}{a}\right ) a^{2}+2 i a \right ) \sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}}{4 a^{2}}+\frac {\ln \left (\frac {i a +\left (x -\frac {i}{a}\right ) a^{2}}{\sqrt {a^{2}}}+\sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}\right )}{2 \sqrt {a^{2}}}\right )\right )\right )\right )}{a^{3}}\) | \(257\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 65, normalized size = 1.08 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.63, size = 60, normalized size = 1.00 \begin {gather*} \frac {4 \, a x + 3 \, {\left (a x - 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} \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 \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 ) \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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