Optimal. Leaf size=324 \[ -\frac{2 (1-i a x)^{9/4}}{a^2 \sqrt [4]{1+i a x}}-\frac{5 (1+i a x)^{3/4} (1-i a x)^{5/4}}{2 a^2}-\frac{25 (1+i a x)^{3/4} \sqrt [4]{1-i a x}}{4 a^2}-\frac{25 \log \left (\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}-\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+1\right )}{8 \sqrt{2} a^2}+\frac{25 \log \left (\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}+\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+1\right )}{8 \sqrt{2} a^2}-\frac{25 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{4 \sqrt{2} a^2}+\frac{25 \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{4 \sqrt{2} a^2} \]
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Rubi [A] time = 0.205937, antiderivative size = 324, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 11, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.786, Rules used = {5062, 78, 50, 63, 240, 211, 1165, 628, 1162, 617, 204} \[ -\frac{2 (1-i a x)^{9/4}}{a^2 \sqrt [4]{1+i a x}}-\frac{5 (1+i a x)^{3/4} (1-i a x)^{5/4}}{2 a^2}-\frac{25 (1+i a x)^{3/4} \sqrt [4]{1-i a x}}{4 a^2}-\frac{25 \log \left (\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}-\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+1\right )}{8 \sqrt{2} a^2}+\frac{25 \log \left (\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}+\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+1\right )}{8 \sqrt{2} a^2}-\frac{25 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{4 \sqrt{2} a^2}+\frac{25 \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{4 \sqrt{2} a^2} \]
Antiderivative was successfully verified.
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Rule 5062
Rule 78
Rule 50
Rule 63
Rule 240
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int e^{-\frac{5}{2} i \tan ^{-1}(a x)} x \, dx &=\int \frac{x (1-i a x)^{5/4}}{(1+i a x)^{5/4}} \, dx\\ &=-\frac{2 (1-i a x)^{9/4}}{a^2 \sqrt [4]{1+i a x}}-\frac{(5 i) \int \frac{(1-i a x)^{5/4}}{\sqrt [4]{1+i a x}} \, dx}{a}\\ &=-\frac{2 (1-i a x)^{9/4}}{a^2 \sqrt [4]{1+i a x}}-\frac{5 (1-i a x)^{5/4} (1+i a x)^{3/4}}{2 a^2}-\frac{(25 i) \int \frac{\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}} \, dx}{4 a}\\ &=-\frac{2 (1-i a x)^{9/4}}{a^2 \sqrt [4]{1+i a x}}-\frac{25 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{4 a^2}-\frac{5 (1-i a x)^{5/4} (1+i a x)^{3/4}}{2 a^2}-\frac{(25 i) \int \frac{1}{(1-i a x)^{3/4} \sqrt [4]{1+i a x}} \, dx}{8 a}\\ &=-\frac{2 (1-i a x)^{9/4}}{a^2 \sqrt [4]{1+i a x}}-\frac{25 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{4 a^2}-\frac{5 (1-i a x)^{5/4} (1+i a x)^{3/4}}{2 a^2}+\frac{25 \operatorname{Subst}\left (\int \frac{1}{\sqrt [4]{2-x^4}} \, dx,x,\sqrt [4]{1-i a x}\right )}{2 a^2}\\ &=-\frac{2 (1-i a x)^{9/4}}{a^2 \sqrt [4]{1+i a x}}-\frac{25 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{4 a^2}-\frac{5 (1-i a x)^{5/4} (1+i a x)^{3/4}}{2 a^2}+\frac{25 \operatorname{Subst}\left (\int \frac{1}{1+x^4} \, dx,x,\frac{\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{2 a^2}\\ &=-\frac{2 (1-i a x)^{9/4}}{a^2 \sqrt [4]{1+i a x}}-\frac{25 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{4 a^2}-\frac{5 (1-i a x)^{5/4} (1+i a x)^{3/4}}{2 a^2}+\frac{25 \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{25 \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{2 (1-i a x)^{9/4}}{a^2 \sqrt [4]{1+i a x}}-\frac{25 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{4 a^2}-\frac{5 (1-i a x)^{5/4} (1+i a x)^{3/4}}{2 a^2}+\frac{25 \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{25 \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{25 \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{25 \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{2 (1-i a x)^{9/4}}{a^2 \sqrt [4]{1+i a x}}-\frac{25 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{4 a^2}-\frac{5 (1-i a x)^{5/4} (1+i a x)^{3/4}}{2 a^2}-\frac{25 \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{25 \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{25 \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{25 \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{2 (1-i a x)^{9/4}}{a^2 \sqrt [4]{1+i a x}}-\frac{25 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{4 a^2}-\frac{5 (1-i a x)^{5/4} (1+i a x)^{3/4}}{2 a^2}-\frac{25 \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{25 \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{25 \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{25 \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 = 0.027655, size = 63, normalized size = 0.19 \[ \frac{2 (1-i a x)^{9/4} \left (5\ 2^{3/4} \text{Hypergeometric2F1}\left (\frac{1}{4},\frac{9}{4},\frac{13}{4},\frac{1}{2} (1-i a x)\right )-\frac{9}{\sqrt [4]{1+i a x}}\right )}{9 a^2} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.173, size = 0, normalized size = 0. \begin{align*} \int{x \left ({(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) ^{-{\frac{5}{2}}}}\, 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 \frac{x}{\left (\frac{i \, a x + 1}{\sqrt{a^{2} x^{2} + 1}}\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.78464, size = 757, normalized size = 2.34 \begin{align*} -\frac{2 \,{\left (a^{3} x - i \, a^{2}\right )} \sqrt{\frac{625 i}{16 \, a^{4}}} \log \left (\frac{4}{25} i \, a^{2} \sqrt{\frac{625 i}{16 \, a^{4}}} + \sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}}\right ) - 2 \,{\left (a^{3} x - i \, a^{2}\right )} \sqrt{\frac{625 i}{16 \, a^{4}}} \log \left (-\frac{4}{25} i \, a^{2} \sqrt{\frac{625 i}{16 \, a^{4}}} + \sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}}\right ) - 2 \,{\left (a^{3} x - i \, a^{2}\right )} \sqrt{-\frac{625 i}{16 \, a^{4}}} \log \left (\frac{4}{25} i \, a^{2} \sqrt{-\frac{625 i}{16 \, a^{4}}} + \sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}}\right ) + 2 \,{\left (a^{3} x - i \, a^{2}\right )} \sqrt{-\frac{625 i}{16 \, a^{4}}} \log \left (-\frac{4}{25} i \, a^{2} \sqrt{-\frac{625 i}{16 \, a^{4}}} + \sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}}\right ) - \sqrt{a^{2} x^{2} + 1}{\left (2 i \, a^{2} x^{2} - 9 \, a x + 43 i\right )} \sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}}}{4 \,{\left (a^{3} x - i \, a^{2}\right )}} \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 \frac{x}{\left (\frac{i \, a x + 1}{\sqrt{a^{2} x^{2} + 1}}\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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