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 = 0.0448873, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, 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 5059
Rule 853
Rule 669
Rule 641
Rule 215
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*}
Mathematica [A] time = 0.0324315, size = 42, normalized size = 0.7 \[ -\frac{3 \sinh ^{-1}(a x)}{a}+\frac{\sqrt{a^2 x^2+1} \left (\frac{4}{a x+i}-i\right )}{a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.054, size = 81, normalized size = 1.4 \begin{align*} 4\,{\frac{x}{\sqrt{{a}^{2}{x}^{2}+1}}}-{ia{x}^{2}{\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}}-{\frac{5\,i}{a}{\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}}-3\,{\frac{1}{\sqrt{{a}^{2}}}\ln \left ({\frac{{a}^{2}x}{\sqrt{{a}^{2}}}}+\sqrt{{a}^{2}{x}^{2}+1} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.99551, size = 89, normalized size = 1.48 \begin{align*} -\frac{i \, a x^{2}}{\sqrt{a^{2} x^{2} + 1}} + \frac{4 \, x}{\sqrt{a^{2} x^{2} + 1}} - \frac{3 \, \operatorname{arsinh}\left (\frac{a^{2} x}{\sqrt{a^{2}}}\right )}{\sqrt{a^{2}}} - \frac{5 i}{\sqrt{a^{2} x^{2} + 1} a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.6227, size = 146, normalized size = 2.43 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (i a x + 1\right )^{3}}{\left (a^{2} x^{2} + 1\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{undef} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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