Optimal. Leaf size=90 \[ -\frac{i x^3 \sqrt{a^2 x^2+1}}{4 a}+\frac{x^2 \sqrt{a^2 x^2+1}}{3 a^2}-\frac{(16-9 i a x) \sqrt{a^2 x^2+1}}{24 a^4}-\frac{3 i \sinh ^{-1}(a x)}{8 a^4} \]
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Rubi [A] time = 0.0664229, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {5060, 833, 780, 215} \[ -\frac{i x^3 \sqrt{a^2 x^2+1}}{4 a}+\frac{x^2 \sqrt{a^2 x^2+1}}{3 a^2}-\frac{(16-9 i a x) \sqrt{a^2 x^2+1}}{24 a^4}-\frac{3 i \sinh ^{-1}(a x)}{8 a^4} \]
Antiderivative was successfully verified.
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Rule 5060
Rule 833
Rule 780
Rule 215
Rubi steps
\begin{align*} \int e^{-i \tan ^{-1}(a x)} x^3 \, dx &=\int \frac{x^3 (1-i a x)}{\sqrt{1+a^2 x^2}} \, dx\\ &=-\frac{i x^3 \sqrt{1+a^2 x^2}}{4 a}+\frac{\int \frac{x^2 \left (3 i a+4 a^2 x\right )}{\sqrt{1+a^2 x^2}} \, dx}{4 a^2}\\ &=\frac{x^2 \sqrt{1+a^2 x^2}}{3 a^2}-\frac{i x^3 \sqrt{1+a^2 x^2}}{4 a}+\frac{\int \frac{x \left (-8 a^2+9 i a^3 x\right )}{\sqrt{1+a^2 x^2}} \, dx}{12 a^4}\\ &=\frac{x^2 \sqrt{1+a^2 x^2}}{3 a^2}-\frac{i x^3 \sqrt{1+a^2 x^2}}{4 a}-\frac{(16-9 i a x) \sqrt{1+a^2 x^2}}{24 a^4}-\frac{(3 i) \int \frac{1}{\sqrt{1+a^2 x^2}} \, dx}{8 a^3}\\ &=\frac{x^2 \sqrt{1+a^2 x^2}}{3 a^2}-\frac{i x^3 \sqrt{1+a^2 x^2}}{4 a}-\frac{(16-9 i a x) \sqrt{1+a^2 x^2}}{24 a^4}-\frac{3 i \sinh ^{-1}(a x)}{8 a^4}\\ \end{align*}
Mathematica [A] time = 0.0431309, size = 56, normalized size = 0.62 \[ \frac{\sqrt{a^2 x^2+1} \left (-6 i a^3 x^3+8 a^2 x^2+9 i a x-16\right )-9 i \sinh ^{-1}(a x)}{24 a^4} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.156, size = 187, normalized size = 2.1 \begin{align*}{\frac{-{\frac{i}{4}}x}{{a}^{3}} \left ({a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}+{\frac{{\frac{5\,i}{8}}x}{{a}^{3}}\sqrt{{a}^{2}{x}^{2}+1}}+{\frac{{\frac{5\,i}{8}}}{{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{1}{3\,{a}^{4}} \left ({a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}-{\frac{1}{{a}^{4}}\sqrt{{a}^{2} \left ( x-{\frac{i}{a}} \right ) ^{2}+2\,ia \left ( x-{\frac{i}{a}} \right ) }}-{\frac{i}{{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 [A] time = 1.49593, size = 103, normalized size = 1.14 \begin{align*} -\frac{i \,{\left (a^{2} x^{2} + 1\right )}^{\frac{3}{2}} x}{4 \, a^{3}} + \frac{5 i \, \sqrt{a^{2} x^{2} + 1} x}{8 \, a^{3}} + \frac{{\left (a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}{3 \, a^{4}} - \frac{3 i \, \operatorname{arsinh}\left (a x\right )}{8 \, a^{4}} - \frac{\sqrt{a^{2} x^{2} + 1}}{a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.80407, size = 146, normalized size = 1.62 \begin{align*} \frac{{\left (-6 i \, a^{3} x^{3} + 8 \, a^{2} x^{2} + 9 i \, a x - 16\right )} \sqrt{a^{2} x^{2} + 1} + 9 i \, \log \left (-a x + \sqrt{a^{2} x^{2} + 1}\right )}{24 \, a^{4}} \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{x^{3} \sqrt{a^{2} x^{2} + 1}}{i a x + 1}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11851, size = 99, normalized size = 1.1 \begin{align*} -\frac{1}{24} \, \sqrt{a^{2} x^{2} + 1}{\left ({\left (2 \,{\left (\frac{3 \, i x}{a} - \frac{4}{a^{2}}\right )} x - \frac{9 \, i}{a^{3}}\right )} x + \frac{16}{a^{4}}\right )} + \frac{3 \, i \log \left (-x{\left | a \right |} + \sqrt{a^{2} x^{2} + 1}\right )}{8 \, a^{3}{\left | a \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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