Optimal. Leaf size=93 \[ -\frac{4 i a^2 \sqrt{a^2 x^2+1}}{-a x+i}+\frac{3 i a \sqrt{a^2 x^2+1}}{x}-\frac{\sqrt{a^2 x^2+1}}{2 x^2}+\frac{9}{2} a^2 \tanh ^{-1}\left (\sqrt{a^2 x^2+1}\right ) \]
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Rubi [A] time = 0.593131, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.571, Rules used = {5060, 6742, 266, 51, 63, 208, 264, 651} \[ -\frac{4 i a^2 \sqrt{a^2 x^2+1}}{-a x+i}+\frac{3 i a \sqrt{a^2 x^2+1}}{x}-\frac{\sqrt{a^2 x^2+1}}{2 x^2}+\frac{9}{2} a^2 \tanh ^{-1}\left (\sqrt{a^2 x^2+1}\right ) \]
Antiderivative was successfully verified.
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Rule 5060
Rule 6742
Rule 266
Rule 51
Rule 63
Rule 208
Rule 264
Rule 651
Rubi steps
\begin{align*} \int \frac{e^{-3 i \tan ^{-1}(a x)}}{x^3} \, dx &=\int \frac{(1-i a x)^2}{x^3 (1+i a x) \sqrt{1+a^2 x^2}} \, dx\\ &=\int \left (\frac{1}{x^3 \sqrt{1+a^2 x^2}}-\frac{3 i a}{x^2 \sqrt{1+a^2 x^2}}-\frac{4 a^2}{x \sqrt{1+a^2 x^2}}+\frac{4 a^3}{(-i+a x) \sqrt{1+a^2 x^2}}\right ) \, dx\\ &=-\left ((3 i a) \int \frac{1}{x^2 \sqrt{1+a^2 x^2}} \, dx\right )-\left (4 a^2\right ) \int \frac{1}{x \sqrt{1+a^2 x^2}} \, dx+\left (4 a^3\right ) \int \frac{1}{(-i+a x) \sqrt{1+a^2 x^2}} \, dx+\int \frac{1}{x^3 \sqrt{1+a^2 x^2}} \, dx\\ &=\frac{3 i a \sqrt{1+a^2 x^2}}{x}-\frac{4 i a^2 \sqrt{1+a^2 x^2}}{i-a x}+\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{1+a^2 x}} \, dx,x,x^2\right )-\left (2 a^2\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}}{2 x^2}+\frac{3 i a \sqrt{1+a^2 x^2}}{x}-\frac{4 i a^2 \sqrt{1+a^2 x^2}}{i-a x}-4 \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{1}{4} a^2 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1+a^2 x}} \, dx,x,x^2\right )\\ &=-\frac{\sqrt{1+a^2 x^2}}{2 x^2}+\frac{3 i a \sqrt{1+a^2 x^2}}{x}-\frac{4 i a^2 \sqrt{1+a^2 x^2}}{i-a x}+4 a^2 \tanh ^{-1}\left (\sqrt{1+a^2 x^2}\right )-\frac{1}{2} \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}}{2 x^2}+\frac{3 i a \sqrt{1+a^2 x^2}}{x}-\frac{4 i a^2 \sqrt{1+a^2 x^2}}{i-a x}+\frac{9}{2} a^2 \tanh ^{-1}\left (\sqrt{1+a^2 x^2}\right )\\ \end{align*}
Mathematica [A] time = 0.0740573, size = 79, normalized size = 0.85 \[ \sqrt{a^2 x^2+1} \left (\frac{4 i a^2}{a x-i}+\frac{3 i a}{x}-\frac{1}{2 x^2}\right )+\frac{9}{2} a^2 \log \left (\sqrt{a^2 x^2+1}+1\right )-\frac{9}{2} a^2 \log (x) \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.095, size = 376, normalized size = 4. \begin{align*} -{\frac{1}{2\,{x}^{2}} \left ({a}^{2}{x}^{2}+1 \right ) ^{{\frac{5}{2}}}}-{\frac{3\,{a}^{2}}{2} \left ({a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}-{\frac{9\,{a}^{2}}{2}\sqrt{{a}^{2}{x}^{2}+1}}+{\frac{9\,{a}^{2}}{2}{\it Artanh} \left ({\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) }-{{\frac{9\,i}{2}}{a}^{3}\ln \left ({{a}^{2}x{\frac{1}{\sqrt{{a}^{2}}}}}+\sqrt{{a}^{2}{x}^{2}+1} \right ){\frac{1}{\sqrt{{a}^{2}}}}}+{\frac{3\,ia}{x} \left ({a}^{2}{x}^{2}+1 \right ) ^{{\frac{5}{2}}}}-{\frac{i}{a} \left ({a}^{2} \left ( x-{\frac{i}{a}} \right ) ^{2}+2\,ia \left ( x-{\frac{i}{a}} \right ) \right ) ^{{\frac{5}{2}}} \left ( x-{\frac{i}{a}} \right ) ^{-3}}-{\frac{9\,i}{2}}{a}^{3}x\sqrt{{a}^{2}{x}^{2}+1}+{\frac{9\,i}{2}}{a}^{3}\sqrt{{a}^{2} \left ( x-{\frac{i}{a}} \right ) ^{2}+2\,ia \left ( x-{\frac{i}{a}} \right ) }x-{ \left ({a}^{2} \left ( x-{\frac{i}{a}} \right ) ^{2}+2\,ia \left ( x-{\frac{i}{a}} \right ) \right ) ^{{\frac{5}{2}}} \left ( x-{\frac{i}{a}} \right ) ^{-2}}+3\,{a}^{2} \left ({a}^{2} \left ( x-{\frac{i}{a}} \right ) ^{2}+2\,ia \left ( x-{\frac{i}{a}} \right ) \right ) ^{3/2}-3\,i{a}^{3}x \left ({a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}+{{\frac{9\,i}{2}}{a}^{3}\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}}}}} \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{{\left (a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}{{\left (i \, a x + 1\right )}^{3} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.6789, size = 292, normalized size = 3.14 \begin{align*} \frac{14 i \, a^{3} x^{3} + 14 \, a^{2} x^{2} + 9 \,{\left (a^{3} x^{3} - i \, a^{2} x^{2}\right )} \log \left (-a x + \sqrt{a^{2} x^{2} + 1} + 1\right ) - 9 \,{\left (a^{3} x^{3} - i \, a^{2} x^{2}\right )} \log \left (-a x + \sqrt{a^{2} x^{2} + 1} - 1\right ) + \sqrt{a^{2} x^{2} + 1}{\left (14 i \, a^{2} x^{2} + 5 \, a x + i\right )}}{2 \,{\left (a x^{3} - i \, x^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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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