Optimal. Leaf size=90 \[ \frac{2 a^2 \sqrt{a^2 x^2+1}}{3 x}+\frac{i a \sqrt{a^2 x^2+1}}{2 x^2}-\frac{\sqrt{a^2 x^2+1}}{3 x^3}-\frac{1}{2} i a^3 \tanh ^{-1}\left (\sqrt{a^2 x^2+1}\right ) \]
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Rubi [A] time = 0.0695705, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {5060, 835, 807, 266, 63, 208} \[ \frac{2 a^2 \sqrt{a^2 x^2+1}}{3 x}+\frac{i a \sqrt{a^2 x^2+1}}{2 x^2}-\frac{\sqrt{a^2 x^2+1}}{3 x^3}-\frac{1}{2} i a^3 \tanh ^{-1}\left (\sqrt{a^2 x^2+1}\right ) \]
Antiderivative was successfully verified.
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Rule 5060
Rule 835
Rule 807
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{-i \tan ^{-1}(a x)}}{x^4} \, dx &=\int \frac{1-i a x}{x^4 \sqrt{1+a^2 x^2}} \, dx\\ &=-\frac{\sqrt{1+a^2 x^2}}{3 x^3}-\frac{1}{3} \int \frac{3 i a+2 a^2 x}{x^3 \sqrt{1+a^2 x^2}} \, dx\\ &=-\frac{\sqrt{1+a^2 x^2}}{3 x^3}+\frac{i a \sqrt{1+a^2 x^2}}{2 x^2}+\frac{1}{6} \int \frac{-4 a^2+3 i a^3 x}{x^2 \sqrt{1+a^2 x^2}} \, dx\\ &=-\frac{\sqrt{1+a^2 x^2}}{3 x^3}+\frac{i a \sqrt{1+a^2 x^2}}{2 x^2}+\frac{2 a^2 \sqrt{1+a^2 x^2}}{3 x}+\frac{1}{2} \left (i a^3\right ) \int \frac{1}{x \sqrt{1+a^2 x^2}} \, dx\\ &=-\frac{\sqrt{1+a^2 x^2}}{3 x^3}+\frac{i a \sqrt{1+a^2 x^2}}{2 x^2}+\frac{2 a^2 \sqrt{1+a^2 x^2}}{3 x}+\frac{1}{4} \left (i a^3\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1+a^2 x}} \, dx,x,x^2\right )\\ &=-\frac{\sqrt{1+a^2 x^2}}{3 x^3}+\frac{i a \sqrt{1+a^2 x^2}}{2 x^2}+\frac{2 a^2 \sqrt{1+a^2 x^2}}{3 x}+\frac{1}{2} (i a) \operatorname{Subst}\left (\int \frac{1}{-\frac{1}{a^2}+\frac{x^2}{a^2}} \, dx,x,\sqrt{1+a^2 x^2}\right )\\ &=-\frac{\sqrt{1+a^2 x^2}}{3 x^3}+\frac{i a \sqrt{1+a^2 x^2}}{2 x^2}+\frac{2 a^2 \sqrt{1+a^2 x^2}}{3 x}-\frac{1}{2} i a^3 \tanh ^{-1}\left (\sqrt{1+a^2 x^2}\right )\\ \end{align*}
Mathematica [A] time = 0.0484899, size = 70, normalized size = 0.78 \[ \frac{1}{6} \left (\frac{\sqrt{a^2 x^2+1} \left (4 a^2 x^2+3 i a x-2\right )}{x^3}-3 i a^3 \log \left (\sqrt{a^2 x^2+1}+1\right )+3 i a^3 \log (x)\right ) \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.081, size = 237, normalized size = 2.6 \begin{align*}{\frac{{\frac{i}{2}}a}{{x}^{2}} \left ({a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}-{\frac{i}{2}}{a}^{3}{\it Artanh} \left ({\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) +{\frac{i}{2}}{a}^{3}\sqrt{{a}^{2}{x}^{2}+1}+{\frac{{a}^{2}}{x} \left ({a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}-{a}^{4}x\sqrt{{a}^{2}{x}^{2}+1}-{{a}^{4}\ln \left ({{a}^{2}x{\frac{1}{\sqrt{{a}^{2}}}}}+\sqrt{{a}^{2}{x}^{2}+1} \right ){\frac{1}{\sqrt{{a}^{2}}}}}-i{a}^{3}\sqrt{{a}^{2} \left ( x-{\frac{i}{a}} \right ) ^{2}+2\,ia \left ( x-{\frac{i}{a}} \right ) }+{{a}^{4}\ln \left ({ \left ( ia+{a}^{2} \left ( x-{\frac{i}{a}} \right ) \right ){\frac{1}{\sqrt{{a}^{2}}}}}+\sqrt{{a}^{2} \left ( x-{\frac{i}{a}} \right ) ^{2}+2\,ia \left ( x-{\frac{i}{a}} \right ) } \right ){\frac{1}{\sqrt{{a}^{2}}}}}-{\frac{1}{3\,{x}^{3}} \left ({a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a^{2} x^{2} + 1}}{{\left (i \, a x + 1\right )} x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.72583, size = 221, normalized size = 2.46 \begin{align*} \frac{-3 i \, a^{3} x^{3} \log \left (-a x + \sqrt{a^{2} x^{2} + 1} + 1\right ) + 3 i \, a^{3} x^{3} \log \left (-a x + \sqrt{a^{2} x^{2} + 1} - 1\right ) + 4 \, a^{3} x^{3} +{\left (4 \, a^{2} x^{2} + 3 i \, a x - 2\right )} \sqrt{a^{2} x^{2} + 1}}{6 \, x^{3}} \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{\sqrt{a^{2} x^{2} + 1}}{x^{4} \left (i a x + 1\right )}\, 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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