Optimal. Leaf size=92 \[ -\frac {9 \sqrt {1+a^2 x^2}}{2 a^2}-\frac {3 \left (1+a^2 x^2\right )^{3/2}}{2 a^2 (1+i a x)}-\frac {\left (1+a^2 x^2\right )^{5/2}}{a^2 (1+i a x)^3}-\frac {9 i \sinh ^{-1}(a x)}{2 a^2} \]
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Rubi [A]
time = 0.23, 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 = {5168, 1647,
1607, 12, 807, 679, 221} \begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 221
Rule 679
Rule 807
Rule 1607
Rule 1647
Rule 5168
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 = 0.03, size = 60, normalized size = 0.65 \begin {gather*} \sqrt {1+a^2 x^2} \left (-\frac {3}{a^2}+\frac {i x}{2 a}+\frac {4 i}{a^2 (-i+a x)}\right )-\frac {9 i \sinh ^{-1}(a x)}{2 a^2} \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. 462 vs. \(2 (77 ) = 154\).
time = 0.10, size = 463, normalized size = 5.03
method | result | size |
risch | \(\frac {i \left (a x +6 i\right ) \sqrt {a^{2} x^{2}+1}}{2 a^{2}}-\frac {9 i \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}+1}\right )}{2 a \sqrt {a^{2}}}+\frac {4 i \sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}}{a^{3} \left (x -\frac {i}{a}\right )}\) | \(104\) |
default | \(\frac {i \left (-\frac {i \left (\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )\right )^{\frac {5}{2}}}{a \left (x -\frac {i}{a}\right )^{2}}+3 i a \left (\frac {\left (\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )\right )^{\frac {3}{2}}}{3}+i a \left (\frac {\left (2 \left (x -\frac {i}{a}\right ) a^{2}+2 i a \right ) \sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}}{4 a^{2}}+\frac {\ln \left (\frac {i a +\left (x -\frac {i}{a}\right ) a^{2}}{\sqrt {a^{2}}}+\sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}\right )}{2 \sqrt {a^{2}}}\right )\right )\right )}{a^{3}}-\frac {\frac {i \left (\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )\right )^{\frac {5}{2}}}{a \left (x -\frac {i}{a}\right )^{3}}-2 i a \left (-\frac {i \left (\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )\right )^{\frac {5}{2}}}{a \left (x -\frac {i}{a}\right )^{2}}+3 i a \left (\frac {\left (\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )\right )^{\frac {3}{2}}}{3}+i a \left (\frac {\left (2 \left (x -\frac {i}{a}\right ) a^{2}+2 i a \right ) \sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}}{4 a^{2}}+\frac {\ln \left (\frac {i a +\left (x -\frac {i}{a}\right ) a^{2}}{\sqrt {a^{2}}}+\sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}\right )}{2 \sqrt {a^{2}}}\right )\right )\right )}{a^{4}}\) | \(463\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 112, normalized size = 1.22 \begin {gather*} -\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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.56, size = 72, normalized size = 0.78 \begin {gather*} \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 \, {\left (a^{3} x - i \, a^{2}\right )}} \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 {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 ) \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 = 105, normalized size = 1.14 \begin {gather*} -\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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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