Optimal. Leaf size=63 \[ -\frac{4 a \sqrt{a^2 x^2+1}}{a x+i}-\frac{\sqrt{a^2 x^2+1}}{x}-3 i a \tanh ^{-1}\left (\sqrt{a^2 x^2+1}\right ) \]
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Rubi [A] time = 0.559741, antiderivative size = 63, 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, 264, 266, 63, 208, 651} \[ -\frac{4 a \sqrt{a^2 x^2+1}}{a x+i}-\frac{\sqrt{a^2 x^2+1}}{x}-3 i a \tanh ^{-1}\left (\sqrt{a^2 x^2+1}\right ) \]
Antiderivative was successfully verified.
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Rule 5060
Rule 6742
Rule 264
Rule 266
Rule 63
Rule 208
Rule 651
Rubi steps
\begin{align*} \int \frac{e^{3 i \tan ^{-1}(a x)}}{x^2} \, dx &=\int \frac{(1+i a x)^2}{x^2 (1-i a x) \sqrt{1+a^2 x^2}} \, dx\\ &=\int \left (\frac{1}{x^2 \sqrt{1+a^2 x^2}}+\frac{3 i a}{x \sqrt{1+a^2 x^2}}-\frac{4 i a^2}{(i+a x) \sqrt{1+a^2 x^2}}\right ) \, dx\\ &=(3 i a) \int \frac{1}{x \sqrt{1+a^2 x^2}} \, dx-\left (4 i a^2\right ) \int \frac{1}{(i+a x) \sqrt{1+a^2 x^2}} \, dx+\int \frac{1}{x^2 \sqrt{1+a^2 x^2}} \, dx\\ &=-\frac{\sqrt{1+a^2 x^2}}{x}-\frac{4 a \sqrt{1+a^2 x^2}}{i+a x}+\frac{1}{2} (3 i a) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1+a^2 x}} \, dx,x,x^2\right )\\ &=-\frac{\sqrt{1+a^2 x^2}}{x}-\frac{4 a \sqrt{1+a^2 x^2}}{i+a x}+\frac{(3 i) \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}\\ &=-\frac{\sqrt{1+a^2 x^2}}{x}-\frac{4 a \sqrt{1+a^2 x^2}}{i+a x}-3 i a \tanh ^{-1}\left (\sqrt{1+a^2 x^2}\right )\\ \end{align*}
Mathematica [A] time = 0.0444158, size = 61, normalized size = 0.97 \[ \sqrt{a^2 x^2+1} \left (-\frac{1}{x}-\frac{4 a}{a x+i}\right )-3 i a \log \left (\sqrt{a^2 x^2+1}+1\right )+3 i a \log (x) \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.069, size = 80, normalized size = 1.3 \begin{align*}{ia{\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}}-5\,{\frac{{a}^{2}x}{\sqrt{{a}^{2}{x}^{2}+1}}}-{\frac{1}{x}{\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}}+3\,ia \left ({\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}-{\it Artanh} \left ({\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05147, size = 84, normalized size = 1.33 \begin{align*} -\frac{5 \, a^{2} x}{\sqrt{a^{2} x^{2} + 1}} - 3 i \, a \operatorname{arsinh}\left (\frac{1}{\sqrt{a^{2}}{\left | x \right |}}\right ) + \frac{4 i \, a}{\sqrt{a^{2} x^{2} + 1}} - \frac{1}{\sqrt{a^{2} x^{2} + 1} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.79216, size = 247, normalized size = 3.92 \begin{align*} -\frac{5 \, a^{2} x^{2} + 5 i \, a x + 3 \,{\left (i \, a^{2} x^{2} - a x\right )} \log \left (-a x + \sqrt{a^{2} x^{2} + 1} + 1\right ) + 3 \,{\left (-i \, a^{2} x^{2} + a x\right )} \log \left (-a x + \sqrt{a^{2} x^{2} + 1} - 1\right ) + \sqrt{a^{2} x^{2} + 1}{\left (5 \, a x + i\right )}}{a x^{2} + i \, x} \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^{2} \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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