Optimal. Leaf size=118 \[ -\frac{4 a^3 \sqrt{a^2 x^2+1}}{-a x+i}+\frac{14 a^2 \sqrt{a^2 x^2+1}}{3 x}+\frac{3 i a \sqrt{a^2 x^2+1}}{2 x^2}-\frac{\sqrt{a^2 x^2+1}}{3 x^3}-\frac{11}{2} i a^3 \tanh ^{-1}\left (\sqrt{a^2 x^2+1}\right ) \]
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Rubi [A] time = 0.603894, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 9, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.643, Rules used = {5060, 6742, 271, 264, 266, 51, 63, 208, 651} \[ -\frac{4 a^3 \sqrt{a^2 x^2+1}}{-a x+i}+\frac{14 a^2 \sqrt{a^2 x^2+1}}{3 x}+\frac{3 i a \sqrt{a^2 x^2+1}}{2 x^2}-\frac{\sqrt{a^2 x^2+1}}{3 x^3}-\frac{11}{2} i a^3 \tanh ^{-1}\left (\sqrt{a^2 x^2+1}\right ) \]
Antiderivative was successfully verified.
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Rule 5060
Rule 6742
Rule 271
Rule 264
Rule 266
Rule 51
Rule 63
Rule 208
Rule 651
Rubi steps
\begin{align*} \int \frac{e^{-3 i \tan ^{-1}(a x)}}{x^4} \, dx &=\int \frac{(1-i a x)^2}{x^4 (1+i a x) \sqrt{1+a^2 x^2}} \, dx\\ &=\int \left (\frac{1}{x^4 \sqrt{1+a^2 x^2}}-\frac{3 i a}{x^3 \sqrt{1+a^2 x^2}}-\frac{4 a^2}{x^2 \sqrt{1+a^2 x^2}}+\frac{4 i a^3}{x \sqrt{1+a^2 x^2}}-\frac{4 i a^4}{(-i+a x) \sqrt{1+a^2 x^2}}\right ) \, dx\\ &=-\left ((3 i a) \int \frac{1}{x^3 \sqrt{1+a^2 x^2}} \, dx\right )-\left (4 a^2\right ) \int \frac{1}{x^2 \sqrt{1+a^2 x^2}} \, dx+\left (4 i a^3\right ) \int \frac{1}{x \sqrt{1+a^2 x^2}} \, dx-\left (4 i a^4\right ) \int \frac{1}{(-i+a x) \sqrt{1+a^2 x^2}} \, dx+\int \frac{1}{x^4 \sqrt{1+a^2 x^2}} \, dx\\ &=-\frac{\sqrt{1+a^2 x^2}}{3 x^3}+\frac{4 a^2 \sqrt{1+a^2 x^2}}{x}-\frac{4 a^3 \sqrt{1+a^2 x^2}}{i-a x}-\frac{1}{2} (3 i a) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{1+a^2 x}} \, dx,x,x^2\right )-\frac{1}{3} \left (2 a^2\right ) \int \frac{1}{x^2 \sqrt{1+a^2 x^2}} \, dx+\left (2 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{3 i a \sqrt{1+a^2 x^2}}{2 x^2}+\frac{14 a^2 \sqrt{1+a^2 x^2}}{3 x}-\frac{4 a^3 \sqrt{1+a^2 x^2}}{i-a x}+(4 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{1}{4} \left (3 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{3 i a \sqrt{1+a^2 x^2}}{2 x^2}+\frac{14 a^2 \sqrt{1+a^2 x^2}}{3 x}-\frac{4 a^3 \sqrt{1+a^2 x^2}}{i-a x}-4 i a^3 \tanh ^{-1}\left (\sqrt{1+a^2 x^2}\right )+\frac{1}{2} (3 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{3 i a \sqrt{1+a^2 x^2}}{2 x^2}+\frac{14 a^2 \sqrt{1+a^2 x^2}}{3 x}-\frac{4 a^3 \sqrt{1+a^2 x^2}}{i-a x}-\frac{11}{2} i a^3 \tanh ^{-1}\left (\sqrt{1+a^2 x^2}\right )\\ \end{align*}
Mathematica [A] time = 0.0697248, size = 89, normalized size = 0.75 \[ \frac{1}{6} \left (\frac{\sqrt{a^2 x^2+1} \left (52 a^3 x^3-19 i a^2 x^2+7 a x+2 i\right )}{x^3 (a x-i)}-33 i a^3 \log \left (\sqrt{a^2 x^2+1}+1\right )+33 i a^3 \log (x)\right ) \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.09, size = 392, normalized size = 3.3 \begin{align*} -{\frac{16\,i}{3}}{a}^{3} \left ({a}^{2} \left ( x-{\frac{i}{a}} \right ) ^{2}+2\,ia \left ( x-{\frac{i}{a}} \right ) \right ) ^{{\frac{3}{2}}}-{\frac{11\,i}{2}}{a}^{3}{\it Artanh} \left ({\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) +8\,{a}^{4}\sqrt{{a}^{2} \left ( x-{\frac{i}{a}} \right ) ^{2}+2\,ia \left ( x-{\frac{i}{a}} \right ) }x+8\,{\frac{{a}^{4}}{\sqrt{{a}^{2}}}\ln \left ({\frac{1}{\sqrt{{a}^{2}}} \left ( ia+{a}^{2} \left ( x-{\frac{i}{a}} \right ) \right ) }+\sqrt{{a}^{2} \left ( x-{\frac{i}{a}} \right ) ^{2}+2\,ia \left ( x-{\frac{i}{a}} \right ) } \right ) }-{ \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{11\,i}{2}}{a}^{3}\sqrt{{a}^{2}{x}^{2}+1}+{2\,ia \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}}+{\frac{{\frac{3\,i}{2}}a}{{x}^{2}} \left ({a}^{2}{x}^{2}+1 \right ) ^{{\frac{5}{2}}}}+{\frac{11\,i}{6}}{a}^{3} \left ({a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}+{\frac{16\,{a}^{2}}{3\,x} \left ({a}^{2}{x}^{2}+1 \right ) ^{{\frac{5}{2}}}}-{\frac{16\,{a}^{4}x}{3} \left ({a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}-8\,{a}^{4}x\sqrt{{a}^{2}{x}^{2}+1}-8\,{\frac{{a}^{4}}{\sqrt{{a}^{2}}}\ln \left ({\frac{{a}^{2}x}{\sqrt{{a}^{2}}}}+\sqrt{{a}^{2}{x}^{2}+1} \right ) }-{\frac{1}{3\,{x}^{3}} \left ({a}^{2}{x}^{2}+1 \right ) ^{{\frac{5}{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^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.75583, size = 316, normalized size = 2.68 \begin{align*} \frac{52 \, a^{4} x^{4} - 52 i \, a^{3} x^{3} - 33 \,{\left (i \, a^{4} x^{4} + a^{3} x^{3}\right )} \log \left (-a x + \sqrt{a^{2} x^{2} + 1} + 1\right ) - 33 \,{\left (-i \, a^{4} x^{4} - a^{3} x^{3}\right )} \log \left (-a x + \sqrt{a^{2} x^{2} + 1} - 1\right ) +{\left (52 \, a^{3} x^{3} - 19 i \, a^{2} x^{2} + 7 \, a x + 2 i\right )} \sqrt{a^{2} x^{2} + 1}}{6 \,{\left (a x^{4} - i \, x^{3}\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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