Optimal. Leaf size=90 \[ -\frac {3 i \sinh ^{-1}(a x)}{8 a^4}+\frac {x^2 \sqrt {a^2 x^2+1}}{3 a^2}-\frac {i x^3 \sqrt {a^2 x^2+1}}{4 a}-\frac {(16-9 i a x) \sqrt {a^2 x^2+1}}{24 a^4} \]
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Rubi [A] time = 0.07, antiderivative size = 90, normalized size of antiderivative = 1.00, 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 215
Rule 780
Rule 833
Rule 5060
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*}
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Mathematica [A] time = 0.05, 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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fricas [A] time = 0.46, size = 59, normalized size = 0.66 \[ \frac {{\left (-6 i \, a^{3} x^{3} + 8 \, a^{2} x^{2} + 9 i \, a x - 16\right )} \sqrt {a^{2} x^{2} + 1} + 9 i \, \log \left (-a x + \sqrt {a^{2} x^{2} + 1}\right )}{24 \, a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.19, size = 187, normalized size = 2.08 \[ -\frac {i x \left (a^{2} x^{2}+1\right )^{\frac {3}{2}}}{4 a^{3}}+\frac {5 i x \sqrt {a^{2} x^{2}+1}}{8 a^{3}}+\frac {5 i \ln \left (\frac {x \,a^{2}}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}+1}\right )}{8 a^{3} \sqrt {a^{2}}}+\frac {\left (a^{2} x^{2}+1\right )^{\frac {3}{2}}}{3 a^{4}}-\frac {\sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}}{a^{4}}-\frac {i \ln \left (\frac {i a +\left (x -\frac {i}{a}\right ) a^{2}}{\sqrt {a^{2}}}+\sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}\right )}{a^{3} \sqrt {a^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 76, normalized size = 0.84 \[ -\frac {i \, {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x}{4 \, a^{3}} + \frac {5 i \, \sqrt {a^{2} x^{2} + 1} x}{8 \, a^{3}} + \frac {{\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{3 \, a^{4}} - \frac {3 i \, \operatorname {arsinh}\left (a x\right )}{8 \, a^{4}} - \frac {\sqrt {a^{2} x^{2} + 1}}{a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 85, normalized size = 0.94 \[ -\frac {\mathrm {asinh}\left (x\,\sqrt {a^2}\right )\,3{}\mathrm {i}}{8\,a^3\,\sqrt {a^2}}-\frac {\sqrt {a^2\,x^2+1}\,\left (\frac {2}{3\,{\left (a^2\right )}^{3/2}}-\frac {a^2\,x^2}{3\,{\left (a^2\right )}^{3/2}}+\frac {x^3\,{\left (a^2\right )}^{3/2}\,1{}\mathrm {i}}{4\,a^3}-\frac {x\,\sqrt {a^2}\,3{}\mathrm {i}}{8\,a^3}\right )}{\sqrt {a^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - i \int \frac {x^{3} \sqrt {a^{2} x^{2} + 1}}{a x - i}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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