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 = 0.33, 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\\ &=-\left ((i a) \int \frac {\left (\frac {i x}{a}-x^2\right ) \sqrt {1+a^2 x^2}}{(1-i a x)^2} \, dx\right )\\ &=-\left ((i a) \int \frac {\left (\frac {i}{a}-x\right ) x \sqrt {1+a^2 x^2}}{(1-i a x)^2} \, dx\right )\\ &=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 = 0.07, size = 54, normalized size = 0.59 \[ -\frac {i \left (-9 \sinh ^{-1}(a x)+\frac {\sqrt {a^2 x^2+1} \left (a^2 x^2-5 i a x+14\right )}{a x+i}\right )}{2 a^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, 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 [A] time = 0.16, size = 104, normalized size = 1.13 \[ -\frac {i a \,x^{3}}{2 \sqrt {a^{2} x^{2}+1}}-\frac {9 i x}{2 a \sqrt {a^{2} x^{2}+1}}+\frac {9 i \ln \left (\frac {x \,a^{2}}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}+1}\right )}{2 a \sqrt {a^{2}}}-\frac {3 x^{2}}{\sqrt {a^{2} x^{2}+1}}-\frac {7}{a^{2} \sqrt {a^{2} x^{2}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 76, normalized size = 0.83 \[ -\frac {i \, a x^{3}}{2 \, \sqrt {a^{2} x^{2} + 1}} - \frac {3 \, x^{2}}{\sqrt {a^{2} x^{2} + 1}} - \frac {9 i \, x}{2 \, \sqrt {a^{2} x^{2} + 1} a} + \frac {9 i \, \operatorname {arsinh}\left (a x\right )}{2 \, a^{2}} - \frac {7}{\sqrt {a^{2} x^{2} + 1} a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.43, size = 104, normalized size = 1.13 \[ -\frac {\sqrt {a^2\,x^2+1}\,\left (\frac {3\,\sqrt {a^2}}{a^2}+\frac {x\,\sqrt {a^2}\,1{}\mathrm {i}}{2\,a}\right )}{\sqrt {a^2}}+\frac {\mathrm {asinh}\left (x\,\sqrt {a^2}\right )\,9{}\mathrm {i}}{2\,a\,\sqrt {a^2}}-\frac {\sqrt {a^2\,x^2+1}\,4{}\mathrm {i}}{a\,\left (x\,\sqrt {a^2}+\frac {\sqrt {a^2}\,1{}\mathrm {i}}{a}\right )\,\sqrt {a^2}} \]
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 \[ - i \left (\int \frac {i x}{a^{2} x^{2} \sqrt {a^{2} x^{2} + 1} + \sqrt {a^{2} x^{2} + 1}}\, dx + \int \left (- \frac {3 a x^{2}}{a^{2} x^{2} \sqrt {a^{2} x^{2} + 1} + \sqrt {a^{2} x^{2} + 1}}\right )\, dx + \int \frac {a^{3} x^{4}}{a^{2} x^{2} \sqrt {a^{2} x^{2} + 1} + \sqrt {a^{2} x^{2} + 1}}\, dx + \int \left (- \frac {3 i a^{2} x^{3}}{a^{2} x^{2} \sqrt {a^{2} x^{2} + 1} + \sqrt {a^{2} x^{2} + 1}}\right )\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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