Optimal. Leaf size=51 \[ \frac{4 i \sqrt{a^2 x^2+1}}{a x+i}-\tanh ^{-1}\left (\sqrt{a^2 x^2+1}\right )-i \sinh ^{-1}(a x) \]
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Rubi [A] time = 0.663198, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5060, 6742, 215, 266, 63, 208, 651} \[ \frac{4 i \sqrt{a^2 x^2+1}}{a x+i}-\tanh ^{-1}\left (\sqrt{a^2 x^2+1}\right )-i \sinh ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 5060
Rule 6742
Rule 215
Rule 266
Rule 63
Rule 208
Rule 651
Rubi steps
\begin{align*} \int \frac{e^{3 i \tan ^{-1}(a x)}}{x} \, dx &=\int \frac{(1+i a x)^2}{x (1-i a x) \sqrt{1+a^2 x^2}} \, dx\\ &=\int \left (-\frac{i a}{\sqrt{1+a^2 x^2}}+\frac{1}{x \sqrt{1+a^2 x^2}}-\frac{4 a}{(i+a x) \sqrt{1+a^2 x^2}}\right ) \, dx\\ &=-\left ((i a) \int \frac{1}{\sqrt{1+a^2 x^2}} \, dx\right )-(4 a) \int \frac{1}{(i+a x) \sqrt{1+a^2 x^2}} \, dx+\int \frac{1}{x \sqrt{1+a^2 x^2}} \, dx\\ &=\frac{4 i \sqrt{1+a^2 x^2}}{i+a x}-i \sinh ^{-1}(a x)+\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1+a^2 x}} \, dx,x,x^2\right )\\ &=\frac{4 i \sqrt{1+a^2 x^2}}{i+a x}-i \sinh ^{-1}(a x)+\frac{\operatorname{Subst}\left (\int \frac{1}{-\frac{1}{a^2}+\frac{x^2}{a^2}} \, dx,x,\sqrt{1+a^2 x^2}\right )}{a^2}\\ &=\frac{4 i \sqrt{1+a^2 x^2}}{i+a x}-i \sinh ^{-1}(a x)-\tanh ^{-1}\left (\sqrt{1+a^2 x^2}\right )\\ \end{align*}
Mathematica [A] time = 0.0391205, size = 55, normalized size = 1.08 \[ \frac{4 i \sqrt{a^2 x^2+1}}{a x+i}-\log \left (\sqrt{a^2 x^2+1}+1\right )-i \sinh ^{-1}(a x)+\log (x) \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.064, size = 77, normalized size = 1.5 \begin{align*}{4\,iax{\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}}-{ia\ln \left ({{a}^{2}x{\frac{1}{\sqrt{{a}^{2}}}}}+\sqrt{{a}^{2}{x}^{2}+1} \right ){\frac{1}{\sqrt{{a}^{2}}}}}+4\,{\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}-{\it Artanh} \left ({\frac{1}{\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 = 0.999818, size = 82, normalized size = 1.61 \begin{align*} \frac{4 i \, a x}{\sqrt{a^{2} x^{2} + 1}} - \frac{i \, a \operatorname{arsinh}\left (\frac{a^{2} x}{\sqrt{a^{2}}}\right )}{\sqrt{a^{2}}} + \frac{4}{\sqrt{a^{2} x^{2} + 1}} - \operatorname{arsinh}\left (\frac{1}{\sqrt{a^{2}}{\left | x \right |}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.72128, size = 250, normalized size = 4.9 \begin{align*} \frac{4 i \, a x -{\left (a x + i\right )} \log \left (-a x + \sqrt{a^{2} x^{2} + 1} + 1\right ) +{\left (i \, a x - 1\right )} \log \left (-a x + \sqrt{a^{2} x^{2} + 1}\right ) +{\left (a x + i\right )} \log \left (-a x + \sqrt{a^{2} x^{2} + 1} - 1\right ) + 4 i \, \sqrt{a^{2} x^{2} + 1} - 4}{a x + i} \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}}{x \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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