Optimal. Leaf size=52 \[ \frac{\sinh ^{-1}(a+b x)}{b}-\frac{i \sqrt{-i a-i b x+1} \sqrt{i a+i b x+1}}{b} \]
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Rubi [A] time = 0.0337881, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {5093, 50, 53, 619, 215} \[ \frac{\sinh ^{-1}(a+b x)}{b}-\frac{i \sqrt{-i a-i b x+1} \sqrt{i a+i b x+1}}{b} \]
Antiderivative was successfully verified.
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Rule 5093
Rule 50
Rule 53
Rule 619
Rule 215
Rubi steps
\begin{align*} \int e^{-i \tan ^{-1}(a+b x)} \, dx &=\int \frac{\sqrt{1-i a-i b x}}{\sqrt{1+i a+i b x}} \, dx\\ &=-\frac{i \sqrt{1-i a-i b x} \sqrt{1+i a+i b x}}{b}+\int \frac{1}{\sqrt{1-i a-i b x} \sqrt{1+i a+i b x}} \, dx\\ &=-\frac{i \sqrt{1-i a-i b x} \sqrt{1+i a+i b x}}{b}+\int \frac{1}{\sqrt{(1-i a) (1+i a)+2 a b x+b^2 x^2}} \, dx\\ &=-\frac{i \sqrt{1-i a-i b x} \sqrt{1+i a+i b x}}{b}+\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{4 b^2}}} \, dx,x,2 a b+2 b^2 x\right )}{2 b^2}\\ &=-\frac{i \sqrt{1-i a-i b x} \sqrt{1+i a+i b x}}{b}+\frac{\sinh ^{-1}(a+b x)}{b}\\ \end{align*}
Mathematica [A] time = 0.0193172, size = 28, normalized size = 0.54 \[ \frac{\sinh ^{-1}(a+b x)-i \sqrt{(a+b x)^2+1}}{b} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.054, size = 122, normalized size = 2.4 \begin{align*}{\frac{-i}{b}\sqrt{ \left ( x-{\frac{i-a}{b}} \right ) ^{2}{b}^{2}+2\,i \left ( x-{\frac{i-a}{b}} \right ) b}}+{\ln \left ({ \left ( ib+ \left ( x-{\frac{i-a}{b}} \right ){b}^{2} \right ){\frac{1}{\sqrt{{b}^{2}}}}}+\sqrt{ \left ( x-{\frac{i-a}{b}} \right ) ^{2}{b}^{2}+2\,i \left ( x-{\frac{i-a}{b}} \right ) b} \right ){\frac{1}{\sqrt{{b}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.49885, size = 47, normalized size = 0.9 \begin{align*} \frac{\operatorname{arsinh}\left (b x + a\right )}{b} - \frac{i \, \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.17001, size = 146, normalized size = 2.81 \begin{align*} \frac{-i \, a - 2 i \, \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1} - 2 \, \log \left (-b x - a + \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}{2 \, b} \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{\sqrt{a^{2} + 2 a b x + b^{2} x^{2} + 1}}{i a + i b x + 1}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12331, size = 70, normalized size = 1.35 \begin{align*} -\frac{\sqrt{{\left (b x + a\right )}^{2} + 1} i}{b} - \frac{\log \left (-a b -{\left (x{\left | b \right |} - \sqrt{{\left (b x + a\right )}^{2} + 1}\right )}{\left | b \right |}\right )}{{\left | b \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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