Optimal. Leaf size=110 \[ \frac{\sqrt{i a+i b x+1} (-i a-i b x+1)^{3/2}}{2 b^2}+\frac{(1+2 i a) \sqrt{i a+i b x+1} \sqrt{-i a-i b x+1}}{2 b^2}+\frac{(-2 a+i) \sinh ^{-1}(a+b x)}{2 b^2} \]
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Rubi [A] time = 0.0759064, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {5095, 80, 50, 53, 619, 215} \[ \frac{\sqrt{i a+i b x+1} (-i a-i b x+1)^{3/2}}{2 b^2}+\frac{(1+2 i a) \sqrt{i a+i b x+1} \sqrt{-i a-i b x+1}}{2 b^2}+\frac{(-2 a+i) \sinh ^{-1}(a+b x)}{2 b^2} \]
Antiderivative was successfully verified.
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Rule 5095
Rule 80
Rule 50
Rule 53
Rule 619
Rule 215
Rubi steps
\begin{align*} \int e^{-i \tan ^{-1}(a+b x)} x \, dx &=\int \frac{x \sqrt{1-i a-i b x}}{\sqrt{1+i a+i b x}} \, dx\\ &=\frac{(1-i a-i b x)^{3/2} \sqrt{1+i a+i b x}}{2 b^2}+\frac{(i-2 a) \int \frac{\sqrt{1-i a-i b x}}{\sqrt{1+i a+i b x}} \, dx}{2 b}\\ &=\frac{(1+2 i a) \sqrt{1-i a-i b x} \sqrt{1+i a+i b x}}{2 b^2}+\frac{(1-i a-i b x)^{3/2} \sqrt{1+i a+i b x}}{2 b^2}+\frac{(i-2 a) \int \frac{1}{\sqrt{1-i a-i b x} \sqrt{1+i a+i b x}} \, dx}{2 b}\\ &=\frac{(1+2 i a) \sqrt{1-i a-i b x} \sqrt{1+i a+i b x}}{2 b^2}+\frac{(1-i a-i b x)^{3/2} \sqrt{1+i a+i b x}}{2 b^2}+\frac{(i-2 a) \int \frac{1}{\sqrt{(1-i a) (1+i a)+2 a b x+b^2 x^2}} \, dx}{2 b}\\ &=\frac{(1+2 i a) \sqrt{1-i a-i b x} \sqrt{1+i a+i b x}}{2 b^2}+\frac{(1-i a-i b x)^{3/2} \sqrt{1+i a+i b x}}{2 b^2}+\frac{(i-2 a) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{4 b^2}}} \, dx,x,2 a b+2 b^2 x\right )}{4 b^3}\\ &=\frac{(1+2 i a) \sqrt{1-i a-i b x} \sqrt{1+i a+i b x}}{2 b^2}+\frac{(1-i a-i b x)^{3/2} \sqrt{1+i a+i b x}}{2 b^2}+\frac{(i-2 a) \sinh ^{-1}(a+b x)}{2 b^2}\\ \end{align*}
Mathematica [A] time = 0.129312, size = 131, normalized size = 1.19 \[ \frac{\sqrt{i a+i b x+1} \left (a^2-i a-b^2 x^2-3 i b x+2\right )}{2 b^2 \sqrt{-i (a+b x+i)}}+\frac{(-1)^{3/4} (1+2 i a) \sqrt{-i b} \sinh ^{-1}\left (\frac{\left (\frac{1}{2}+\frac{i}{2}\right ) \sqrt{b} \sqrt{-i (a+b x+i)}}{\sqrt{-i b}}\right )}{b^{5/2}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.08, size = 350, normalized size = 3.2 \begin{align*}{\frac{-{\frac{i}{2}}x}{b}\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}+1}}-{\frac{{\frac{i}{2}}a}{{b}^{2}}\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}+1}}-{\frac{{\frac{i}{2}}}{b}\ln \left ({({b}^{2}x+ab){\frac{1}{\sqrt{{b}^{2}}}}}+\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}+1} \right ){\frac{1}{\sqrt{{b}^{2}}}}}+{\frac{ia}{{b}^{2}}\sqrt{ \left ( x-{\frac{i-a}{b}} \right ) ^{2}{b}^{2}+2\,i \left ( x-{\frac{i-a}{b}} \right ) b}}+{\frac{1}{{b}^{2}}\sqrt{ \left ( x-{\frac{i-a}{b}} \right ) ^{2}{b}^{2}+2\,i \left ( x-{\frac{i-a}{b}} \right ) b}}-{\frac{a}{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}}}}}+{\frac{i}{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 = 1.51266, size = 131, normalized size = 1.19 \begin{align*} -\frac{i \, \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1} x}{2 \, b} - \frac{a \operatorname{arsinh}\left (b x + a\right )}{b^{2}} + \frac{i \, \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a}{2 \, b^{2}} + \frac{i \, \operatorname{arsinh}\left (b x + a\right )}{2 \, b^{2}} + \frac{\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.29105, size = 200, normalized size = 1.82 \begin{align*} \frac{3 i \, a^{2} +{\left (8 \, a - 4 i\right )} \log \left (-b x - a + \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) + \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1}{\left (-4 i \, b x + 4 i \, a + 8\right )} + 4 \, a}{8 \, b^{2}} \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 \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 = 1.13039, size = 105, normalized size = 0.95 \begin{align*} -\frac{1}{2} \, \sqrt{{\left (b x + a\right )}^{2} + 1}{\left (\frac{i x}{b} - \frac{a b i + 2 \, b}{b^{3}}\right )} + \frac{{\left (2 \, a - i\right )} \log \left (-a b -{\left (x{\left | b \right |} - \sqrt{{\left (b x + a\right )}^{2} + 1}\right )}{\left | b \right |}\right )}{2 \, b{\left | b \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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