Optimal. Leaf size=59 \[ -\frac {2 i (-a+i)^2 \log (-a-b x+i)}{b^3}+\frac {2 (1+i a) x}{b^2}-\frac {i x^2}{b}-\frac {x^3}{3} \]
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Rubi [A] time = 0.05, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {5095, 77} \[ \frac {2 (1+i a) x}{b^2}-\frac {2 i (-a+i)^2 \log (-a-b x+i)}{b^3}-\frac {i x^2}{b}-\frac {x^3}{3} \]
Antiderivative was successfully verified.
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Rule 77
Rule 5095
Rubi steps
\begin {align*} \int e^{-2 i \tan ^{-1}(a+b x)} x^2 \, dx &=\int \frac {x^2 (1-i a-i b x)}{1+i a+i b x} \, dx\\ &=\int \left (\frac {2 i (-i+a)}{b^2}-\frac {2 i x}{b}-x^2-\frac {2 i (-i+a)^2}{b^2 (-i+a+b x)}\right ) \, dx\\ &=\frac {2 (1+i a) x}{b^2}-\frac {i x^2}{b}-\frac {x^3}{3}-\frac {2 i (i-a)^2 \log (i-a-b x)}{b^3}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 55, normalized size = 0.93 \[ \frac {b x \left (6 i a-b^2 x^2-3 i b x+6\right )-6 i (a-i)^2 \log (-a-b x+i)}{3 b^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 53, normalized size = 0.90 \[ -\frac {b^{3} x^{3} + 3 i \, b^{2} x^{2} + 6 \, {\left (-i \, a - 1\right )} b x - {\left (-6 i \, a^{2} - 12 \, a + 6 i\right )} \log \left (\frac {b x + a - i}{b}\right )}{3 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.14, size = 120, normalized size = 2.03 \[ \frac {2 \, {\left (a^{2} i + 2 \, a - i\right )} \log \left (\frac {1}{\sqrt {{\left (b x + a\right )}^{2} + 1} {\left | b \right |}}\right )}{b^{3}} + \frac {{\left (b i x + a i + 1\right )}^{3} {\left (\frac {3 \, {\left (a b - 2 \, b i\right )} i}{{\left (b i x + a i + 1\right )} b} - \frac {3 \, {\left (a^{2} b^{2} - 6 \, a b^{2} i - 5 \, b^{2}\right )} i^{2}}{{\left (b i x + a i + 1\right )}^{2} b^{2}} - 1\right )}}{3 \, b^{3} i^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 143, normalized size = 2.42 \[ -\frac {x^{3}}{3}-\frac {i x^{2}}{b}+\frac {2 i a x}{b^{2}}+\frac {2 x}{b^{2}}+\frac {2 \arctan \left (b x +a \right ) a^{2}}{b^{3}}-\frac {i \ln \left (b^{2} x^{2}+2 a b x +a^{2}+1\right ) a^{2}}{b^{3}}-\frac {2 \arctan \left (b x +a \right )}{b^{3}}-\frac {4 i \arctan \left (b x +a \right ) a}{b^{3}}+\frac {i \ln \left (b^{2} x^{2}+2 a b x +a^{2}+1\right )}{b^{3}}-\frac {2 \ln \left (b^{2} x^{2}+2 a b x +a^{2}+1\right ) a}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 52, normalized size = 0.88 \[ -\frac {b^{2} x^{3} + 3 i \, b x^{2} + 6 \, {\left (-i \, a - 1\right )} x}{3 \, b^{2}} + \frac {{\left (-2 i \, a^{2} - 4 \, a + 2 i\right )} \log \left (i \, b x + i \, a + 1\right )}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.54, size = 90, normalized size = 1.53 \[ -\ln \left (x+\frac {a-\mathrm {i}}{b}\right )\,\left (\frac {4\,a}{b^3}+\frac {\left (2\,a^2-2\right )\,1{}\mathrm {i}}{b^3}\right )+x^2\,\left (\frac {a-\mathrm {i}}{2\,b}-\frac {a+1{}\mathrm {i}}{2\,b}\right )-\frac {x^3}{3}-\frac {x\,\left (\frac {a-\mathrm {i}}{b}-\frac {a+1{}\mathrm {i}}{b}\right )\,\left (a-\mathrm {i}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.36, size = 53, normalized size = 0.90 \[ - \frac {x^{3}}{3} - x \left (- \frac {2 i a}{b^{2}} - \frac {2}{b^{2}}\right ) - \frac {i x^{2}}{b} - \frac {2 i \left (a - i\right )^{2} \log {\left (i a + i b x + 1 \right )}}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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