Optimal. Leaf size=295 \[ \frac{(1-i a x)^{3/4} (1+i a x)^{5/4}}{2 a^2}+\frac{(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{4 a^2}+\frac{\log \left (\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}-\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+1\right )}{8 \sqrt{2} a^2}-\frac{\log \left (\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}+\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+1\right )}{8 \sqrt{2} a^2}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{4 \sqrt{2} a^2}+\frac{\tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{4 \sqrt{2} a^2} \]
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Rubi [A] time = 0.17861, antiderivative size = 295, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 11, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.786, Rules used = {5062, 80, 50, 63, 331, 297, 1162, 617, 204, 1165, 628} \[ \frac{(1-i a x)^{3/4} (1+i a x)^{5/4}}{2 a^2}+\frac{(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{4 a^2}+\frac{\log \left (\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}-\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+1\right )}{8 \sqrt{2} a^2}-\frac{\log \left (\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}+\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+1\right )}{8 \sqrt{2} a^2}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{4 \sqrt{2} a^2}+\frac{\tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{4 \sqrt{2} a^2} \]
Antiderivative was successfully verified.
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Rule 5062
Rule 80
Rule 50
Rule 63
Rule 331
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int e^{\frac{1}{2} i \tan ^{-1}(a x)} x \, dx &=\int \frac{x \sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}} \, dx\\ &=\frac{(1-i a x)^{3/4} (1+i a x)^{5/4}}{2 a^2}-\frac{i \int \frac{\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}} \, dx}{4 a}\\ &=\frac{(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{4 a^2}+\frac{(1-i a x)^{3/4} (1+i a x)^{5/4}}{2 a^2}-\frac{i \int \frac{1}{\sqrt [4]{1-i a x} (1+i a x)^{3/4}} \, dx}{8 a}\\ &=\frac{(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{4 a^2}+\frac{(1-i a x)^{3/4} (1+i a x)^{5/4}}{2 a^2}+\frac{\operatorname{Subst}\left (\int \frac{x^2}{\left (2-x^4\right )^{3/4}} \, dx,x,\sqrt [4]{1-i a x}\right )}{2 a^2}\\ &=\frac{(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{4 a^2}+\frac{(1-i a x)^{3/4} (1+i a x)^{5/4}}{2 a^2}+\frac{\operatorname{Subst}\left (\int \frac{x^2}{1+x^4} \, dx,x,\frac{\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{2 a^2}\\ &=\frac{(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{4 a^2}+\frac{(1-i a x)^{3/4} (1+i a x)^{5/4}}{2 a^2}-\frac{\operatorname{Subst}\left (\int \frac{1-x^2}{1+x^4} \, dx,x,\frac{\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{4 a^2}+\frac{\operatorname{Subst}\left (\int \frac{1+x^2}{1+x^4} \, dx,x,\frac{\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{4 a^2}\\ &=\frac{(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{4 a^2}+\frac{(1-i a x)^{3/4} (1+i a x)^{5/4}}{2 a^2}+\frac{\operatorname{Subst}\left (\int \frac{1}{1-\sqrt{2} x+x^2} \, dx,x,\frac{\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{8 a^2}+\frac{\operatorname{Subst}\left (\int \frac{1}{1+\sqrt{2} x+x^2} \, dx,x,\frac{\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{8 a^2}+\frac{\operatorname{Subst}\left (\int \frac{\sqrt{2}+2 x}{-1-\sqrt{2} x-x^2} \, dx,x,\frac{\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{8 \sqrt{2} a^2}+\frac{\operatorname{Subst}\left (\int \frac{\sqrt{2}-2 x}{-1+\sqrt{2} x-x^2} \, dx,x,\frac{\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{8 \sqrt{2} a^2}\\ &=\frac{(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{4 a^2}+\frac{(1-i a x)^{3/4} (1+i a x)^{5/4}}{2 a^2}+\frac{\log \left (1+\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}-\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{8 \sqrt{2} a^2}-\frac{\log \left (1+\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}+\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{8 \sqrt{2} a^2}+\frac{\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{4 \sqrt{2} a^2}-\frac{\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{4 \sqrt{2} a^2}\\ &=\frac{(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{4 a^2}+\frac{(1-i a x)^{3/4} (1+i a x)^{5/4}}{2 a^2}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{4 \sqrt{2} a^2}+\frac{\tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{4 \sqrt{2} a^2}+\frac{\log \left (1+\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}-\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{8 \sqrt{2} a^2}-\frac{\log \left (1+\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}+\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{8 \sqrt{2} a^2}\\ \end{align*}
Mathematica [C] time = 0.0206425, size = 63, normalized size = 0.21 \[ \frac{(1-i a x)^{3/4} \left (2 \sqrt [4]{2} \text{Hypergeometric2F1}\left (-\frac{1}{4},\frac{3}{4},\frac{7}{4},\frac{1}{2} (1-i a x)\right )+3 (1+i a x)^{5/4}\right )}{6 a^2} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.13, size = 0, normalized size = 0. \begin{align*} \int \sqrt{{(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}}}x\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \sqrt{\frac{i \, a x + 1}{\sqrt{a^{2} x^{2} + 1}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.68765, size = 597, normalized size = 2.02 \begin{align*} -\frac{2 \, a^{2} \sqrt{\frac{i}{16 \, a^{4}}} \log \left (4 \, a^{2} \sqrt{\frac{i}{16 \, a^{4}}} + \sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}}\right ) - 2 \, a^{2} \sqrt{\frac{i}{16 \, a^{4}}} \log \left (-4 \, a^{2} \sqrt{\frac{i}{16 \, a^{4}}} + \sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}}\right ) + 2 \, a^{2} \sqrt{-\frac{i}{16 \, a^{4}}} \log \left (4 \, a^{2} \sqrt{-\frac{i}{16 \, a^{4}}} + \sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}}\right ) - 2 \, a^{2} \sqrt{-\frac{i}{16 \, a^{4}}} \log \left (-4 \, a^{2} \sqrt{-\frac{i}{16 \, a^{4}}} + \sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}}\right ) -{\left (2 \, a^{2} x^{2} - i \, a x + 3\right )} \sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}}}{4 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \sqrt{\frac{i \, a x + 1}{\sqrt{a^{2} x^{2} + 1}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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