Optimal. Leaf size=117 \[ \frac{4 a^3 \sqrt{a^2 x^2+1}}{a x+i}+\frac{14 a^2 \sqrt{a^2 x^2+1}}{3 x}-\frac{3 i a \sqrt{a^2 x^2+1}}{2 x^2}-\frac{\sqrt{a^2 x^2+1}}{3 x^3}+\frac{11}{2} i a^3 \tanh ^{-1}\left (\sqrt{a^2 x^2+1}\right ) \]
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Rubi [A] time = 0.61597, antiderivative size = 117, 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\\ &=(3 i a) \int \frac{1}{x^3 \sqrt{1+a^2 x^2}} \, dx-\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 = 0.0754431, size = 89, normalized size = 0.76 \[ \frac{1}{6} \left (\frac{\sqrt{a^2 x^2+1} \left (52 a^3 x^3+19 i a^2 x^2+7 a x-2 i\right )}{x^3 (a x+i)}+33 i a^3 \log \left (\sqrt{a^2 x^2+1}+1\right )-33 i a^3 \log (x)\right ) \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.07, size = 141, normalized size = 1.2 \begin{align*} 3\,ia \left ( -{\frac{1}{2\,{x}^{2}}{\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}}-{\frac{3\,{a}^{2}}{2} \left ({\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}-{\it Artanh} \left ({\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) \right ) } \right ) -{\frac{13\,{a}^{2}}{3} \left ( -{\frac{1}{x}{\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}}-2\,{\frac{{a}^{2}x}{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) }-{\frac{1}{3\,{x}^{3}}{\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}}-i{a}^{3} \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 = 1.05337, size = 138, normalized size = 1.18 \begin{align*} \frac{26 \, a^{4} x}{3 \, \sqrt{a^{2} x^{2} + 1}} + \frac{11}{2} i \, a^{3} \operatorname{arsinh}\left (\frac{1}{\sqrt{a^{2}}{\left | x \right |}}\right ) - \frac{11 i \, a^{3}}{2 \, \sqrt{a^{2} x^{2} + 1}} + \frac{13 \, a^{2}}{3 \, \sqrt{a^{2} x^{2} + 1} x} - \frac{3 i \, a}{2 \, \sqrt{a^{2} x^{2} + 1} x^{2}} - \frac{1}{3 \, \sqrt{a^{2} x^{2} + 1} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.73378, size = 316, normalized size = 2.7 \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (i a x + 1\right )^{3}}{x^{4} \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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