Optimal. Leaf size=339 \[ \frac{x \sqrt [4]{1+i a x} (1-i a x)^{7/4}}{3 a^2}+\frac{i \sqrt [4]{1+i a x} (1-i a x)^{7/4}}{4 a^3}+\frac{17 i \sqrt [4]{1+i a x} (1-i a x)^{3/4}}{24 a^3}-\frac{17 i \log \left (\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}-\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+1\right )}{16 \sqrt{2} a^3}+\frac{17 i \log \left (\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}+\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+1\right )}{16 \sqrt{2} a^3}+\frac{17 i \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{8 \sqrt{2} a^3}-\frac{17 i \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{8 \sqrt{2} a^3} \]
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Rubi [A] time = 0.219318, antiderivative size = 339, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 12, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.75, Rules used = {5062, 90, 80, 50, 63, 331, 297, 1162, 617, 204, 1165, 628} \[ \frac{x \sqrt [4]{1+i a x} (1-i a x)^{7/4}}{3 a^2}+\frac{i \sqrt [4]{1+i a x} (1-i a x)^{7/4}}{4 a^3}+\frac{17 i \sqrt [4]{1+i a x} (1-i a x)^{3/4}}{24 a^3}-\frac{17 i \log \left (\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}-\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+1\right )}{16 \sqrt{2} a^3}+\frac{17 i \log \left (\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}+\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+1\right )}{16 \sqrt{2} a^3}+\frac{17 i \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{8 \sqrt{2} a^3}-\frac{17 i \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{8 \sqrt{2} a^3} \]
Antiderivative was successfully verified.
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Rule 5062
Rule 90
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{3}{2} i \tan ^{-1}(a x)} x^2 \, dx &=\int \frac{x^2 (1-i a x)^{3/4}}{(1+i a x)^{3/4}} \, dx\\ &=\frac{x (1-i a x)^{7/4} \sqrt [4]{1+i a x}}{3 a^2}+\frac{\int \frac{(1-i a x)^{3/4} \left (-1+\frac{3 i a x}{2}\right )}{(1+i a x)^{3/4}} \, dx}{3 a^2}\\ &=\frac{i (1-i a x)^{7/4} \sqrt [4]{1+i a x}}{4 a^3}+\frac{x (1-i a x)^{7/4} \sqrt [4]{1+i a x}}{3 a^2}-\frac{17 \int \frac{(1-i a x)^{3/4}}{(1+i a x)^{3/4}} \, dx}{24 a^2}\\ &=\frac{17 i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{24 a^3}+\frac{i (1-i a x)^{7/4} \sqrt [4]{1+i a x}}{4 a^3}+\frac{x (1-i a x)^{7/4} \sqrt [4]{1+i a x}}{3 a^2}-\frac{17 \int \frac{1}{\sqrt [4]{1-i a x} (1+i a x)^{3/4}} \, dx}{16 a^2}\\ &=\frac{17 i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{24 a^3}+\frac{i (1-i a x)^{7/4} \sqrt [4]{1+i a x}}{4 a^3}+\frac{x (1-i a x)^{7/4} \sqrt [4]{1+i a x}}{3 a^2}-\frac{(17 i) \operatorname{Subst}\left (\int \frac{x^2}{\left (2-x^4\right )^{3/4}} \, dx,x,\sqrt [4]{1-i a x}\right )}{4 a^3}\\ &=\frac{17 i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{24 a^3}+\frac{i (1-i a x)^{7/4} \sqrt [4]{1+i a x}}{4 a^3}+\frac{x (1-i a x)^{7/4} \sqrt [4]{1+i a x}}{3 a^2}-\frac{(17 i) \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 )}{4 a^3}\\ &=\frac{17 i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{24 a^3}+\frac{i (1-i a x)^{7/4} \sqrt [4]{1+i a x}}{4 a^3}+\frac{x (1-i a x)^{7/4} \sqrt [4]{1+i a x}}{3 a^2}+\frac{(17 i) \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 )}{8 a^3}-\frac{(17 i) \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 )}{8 a^3}\\ &=\frac{17 i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{24 a^3}+\frac{i (1-i a x)^{7/4} \sqrt [4]{1+i a x}}{4 a^3}+\frac{x (1-i a x)^{7/4} \sqrt [4]{1+i a x}}{3 a^2}-\frac{(17 i) \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 )}{16 a^3}-\frac{(17 i) \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 )}{16 a^3}-\frac{(17 i) \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 )}{16 \sqrt{2} a^3}-\frac{(17 i) \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 )}{16 \sqrt{2} a^3}\\ &=\frac{17 i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{24 a^3}+\frac{i (1-i a x)^{7/4} \sqrt [4]{1+i a x}}{4 a^3}+\frac{x (1-i a x)^{7/4} \sqrt [4]{1+i a x}}{3 a^2}-\frac{17 i \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 )}{16 \sqrt{2} a^3}+\frac{17 i \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 )}{16 \sqrt{2} a^3}-\frac{(17 i) \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 )}{8 \sqrt{2} a^3}+\frac{(17 i) \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 )}{8 \sqrt{2} a^3}\\ &=\frac{17 i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{24 a^3}+\frac{i (1-i a x)^{7/4} \sqrt [4]{1+i a x}}{4 a^3}+\frac{x (1-i a x)^{7/4} \sqrt [4]{1+i a x}}{3 a^2}+\frac{17 i \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{8 \sqrt{2} a^3}-\frac{17 i \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{8 \sqrt{2} a^3}-\frac{17 i \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 )}{16 \sqrt{2} a^3}+\frac{17 i \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 )}{16 \sqrt{2} a^3}\\ \end{align*}
Mathematica [C] time = 0.0285041, size = 73, normalized size = 0.22 \[ \frac{(1-i a x)^{7/4} \left (7 \sqrt [4]{1+i a x} (4 a x+3 i)-17 i \sqrt [4]{2} \text{Hypergeometric2F1}\left (\frac{3}{4},\frac{7}{4},\frac{11}{4},\frac{1}{2} (1-i a x)\right )\right )}{84 a^3} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.158, size = 0, normalized size = 0. \begin{align*} \int{{x}^{2} \left ({(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) ^{-{\frac{3}{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^{2}}{\left (\frac{i \, a x + 1}{\sqrt{a^{2} x^{2} + 1}}\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.05278, size = 678, normalized size = 2. \begin{align*} -\frac{12 \, a^{3} \sqrt{\frac{289 i}{64 \, a^{6}}} \log \left (\frac{8}{17} i \, a^{3} \sqrt{\frac{289 i}{64 \, a^{6}}} + \sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}}\right ) - 12 \, a^{3} \sqrt{\frac{289 i}{64 \, a^{6}}} \log \left (-\frac{8}{17} i \, a^{3} \sqrt{\frac{289 i}{64 \, a^{6}}} + \sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}}\right ) + 12 \, a^{3} \sqrt{-\frac{289 i}{64 \, a^{6}}} \log \left (\frac{8}{17} i \, a^{3} \sqrt{-\frac{289 i}{64 \, a^{6}}} + \sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}}\right ) - 12 \, a^{3} \sqrt{-\frac{289 i}{64 \, a^{6}}} \log \left (-\frac{8}{17} i \, a^{3} \sqrt{-\frac{289 i}{64 \, a^{6}}} + \sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}}\right ) +{\left (8 \, a^{3} x^{3} + 22 i \, a^{2} x^{2} - 37 \, a x - 23 i\right )} \sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}}}{24 \, a^{3}} \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^{2}}{\left (\frac{i \, a x + 1}{\sqrt{a^{2} x^{2} + 1}}\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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