Optimal. Leaf size=371 \[ -\frac{i (1+i a x)^{3/4} (1-i a x)^{9/4}}{3 a^3}-\frac{2 i (1-i a x)^{9/4}}{a^3 \sqrt [4]{1+i a x}}-\frac{11 i (1+i a x)^{3/4} (1-i a x)^{5/4}}{4 a^3}-\frac{55 i (1+i a x)^{3/4} \sqrt [4]{1-i a x}}{8 a^3}-\frac{55 i \log \left (\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}-\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+1\right )}{16 \sqrt{2} a^3}+\frac{55 i \log \left (\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}+\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+1\right )}{16 \sqrt{2} a^3}-\frac{55 i \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{8 \sqrt{2} a^3}+\frac{55 i \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{8 \sqrt{2} a^3} \]
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Rubi [A] time = 0.241132, antiderivative size = 371, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 12, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.75, Rules used = {5062, 89, 80, 50, 63, 240, 211, 1165, 628, 1162, 617, 204} \[ -\frac{i (1+i a x)^{3/4} (1-i a x)^{9/4}}{3 a^3}-\frac{2 i (1-i a x)^{9/4}}{a^3 \sqrt [4]{1+i a x}}-\frac{11 i (1+i a x)^{3/4} (1-i a x)^{5/4}}{4 a^3}-\frac{55 i (1+i a x)^{3/4} \sqrt [4]{1-i a x}}{8 a^3}-\frac{55 i \log \left (\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}-\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+1\right )}{16 \sqrt{2} a^3}+\frac{55 i \log \left (\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}+\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+1\right )}{16 \sqrt{2} a^3}-\frac{55 i \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{8 \sqrt{2} a^3}+\frac{55 i \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{8 \sqrt{2} a^3} \]
Antiderivative was successfully verified.
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Rule 5062
Rule 89
Rule 80
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^2 \, dx &=\int \frac{x^2 (1-i a x)^{5/4}}{(1+i a x)^{5/4}} \, dx\\ &=-\frac{2 i (1-i a x)^{9/4}}{a^3 \sqrt [4]{1+i a x}}+\frac{(2 i) \int \frac{(1-i a x)^{5/4} \left (-\frac{5 i a}{2}-\frac{a^2 x}{2}\right )}{\sqrt [4]{1+i a x}} \, dx}{a^3}\\ &=-\frac{2 i (1-i a x)^{9/4}}{a^3 \sqrt [4]{1+i a x}}-\frac{i (1-i a x)^{9/4} (1+i a x)^{3/4}}{3 a^3}+\frac{11 \int \frac{(1-i a x)^{5/4}}{\sqrt [4]{1+i a x}} \, dx}{2 a^2}\\ &=-\frac{2 i (1-i a x)^{9/4}}{a^3 \sqrt [4]{1+i a x}}-\frac{11 i (1-i a x)^{5/4} (1+i a x)^{3/4}}{4 a^3}-\frac{i (1-i a x)^{9/4} (1+i a x)^{3/4}}{3 a^3}+\frac{55 \int \frac{\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}} \, dx}{8 a^2}\\ &=-\frac{2 i (1-i a x)^{9/4}}{a^3 \sqrt [4]{1+i a x}}-\frac{55 i \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{8 a^3}-\frac{11 i (1-i a x)^{5/4} (1+i a x)^{3/4}}{4 a^3}-\frac{i (1-i a x)^{9/4} (1+i a x)^{3/4}}{3 a^3}+\frac{55 \int \frac{1}{(1-i a x)^{3/4} \sqrt [4]{1+i a x}} \, dx}{16 a^2}\\ &=-\frac{2 i (1-i a x)^{9/4}}{a^3 \sqrt [4]{1+i a x}}-\frac{55 i \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{8 a^3}-\frac{11 i (1-i a x)^{5/4} (1+i a x)^{3/4}}{4 a^3}-\frac{i (1-i a x)^{9/4} (1+i a x)^{3/4}}{3 a^3}+\frac{(55 i) \operatorname{Subst}\left (\int \frac{1}{\sqrt [4]{2-x^4}} \, dx,x,\sqrt [4]{1-i a x}\right )}{4 a^3}\\ &=-\frac{2 i (1-i a x)^{9/4}}{a^3 \sqrt [4]{1+i a x}}-\frac{55 i \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{8 a^3}-\frac{11 i (1-i a x)^{5/4} (1+i a x)^{3/4}}{4 a^3}-\frac{i (1-i a x)^{9/4} (1+i a x)^{3/4}}{3 a^3}+\frac{(55 i) \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 )}{4 a^3}\\ &=-\frac{2 i (1-i a x)^{9/4}}{a^3 \sqrt [4]{1+i a x}}-\frac{55 i \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{8 a^3}-\frac{11 i (1-i a x)^{5/4} (1+i a x)^{3/4}}{4 a^3}-\frac{i (1-i a x)^{9/4} (1+i a x)^{3/4}}{3 a^3}+\frac{(55 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{(55 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{2 i (1-i a x)^{9/4}}{a^3 \sqrt [4]{1+i a x}}-\frac{55 i \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{8 a^3}-\frac{11 i (1-i a x)^{5/4} (1+i a x)^{3/4}}{4 a^3}-\frac{i (1-i a x)^{9/4} (1+i a x)^{3/4}}{3 a^3}+\frac{(55 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{(55 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{(55 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{(55 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{2 i (1-i a x)^{9/4}}{a^3 \sqrt [4]{1+i a x}}-\frac{55 i \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{8 a^3}-\frac{11 i (1-i a x)^{5/4} (1+i a x)^{3/4}}{4 a^3}-\frac{i (1-i a x)^{9/4} (1+i a x)^{3/4}}{3 a^3}-\frac{55 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{55 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{(55 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{(55 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{2 i (1-i a x)^{9/4}}{a^3 \sqrt [4]{1+i a x}}-\frac{55 i \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{8 a^3}-\frac{11 i (1-i a x)^{5/4} (1+i a x)^{3/4}}{4 a^3}-\frac{i (1-i a x)^{9/4} (1+i a x)^{3/4}}{3 a^3}-\frac{55 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{55 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{55 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{55 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 = 0.0302484, size = 91, normalized size = 0.25 \[ -\frac{\sqrt [4]{1-i a x} (a x+i)^2 \left (11 i 2^{3/4} \sqrt [4]{1+i a x} \text{Hypergeometric2F1}\left (\frac{1}{4},\frac{9}{4},\frac{13}{4},\frac{1}{2} (1-i a x)\right )+3 a x-21 i\right )}{9 a^3 \sqrt [4]{1+i a x}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.173, size = 0, normalized size = 0. \begin{align*} \int{{x}^{2} \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^{2}}{\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 = 1.86518, size = 783, normalized size = 2.11 \begin{align*} \frac{12 \,{\left (a^{4} x - i \, a^{3}\right )} \sqrt{\frac{3025 i}{64 \, a^{6}}} \log \left (\frac{8}{55} \, a^{3} \sqrt{\frac{3025 i}{64 \, a^{6}}} + \sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}}\right ) - 12 \,{\left (a^{4} x - i \, a^{3}\right )} \sqrt{\frac{3025 i}{64 \, a^{6}}} \log \left (-\frac{8}{55} \, a^{3} \sqrt{\frac{3025 i}{64 \, a^{6}}} + \sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}}\right ) - 12 \,{\left (a^{4} x - i \, a^{3}\right )} \sqrt{-\frac{3025 i}{64 \, a^{6}}} \log \left (\frac{8}{55} \, a^{3} \sqrt{-\frac{3025 i}{64 \, a^{6}}} + \sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}}\right ) + 12 \,{\left (a^{4} x - i \, a^{3}\right )} \sqrt{-\frac{3025 i}{64 \, a^{6}}} \log \left (-\frac{8}{55} \, a^{3} \sqrt{-\frac{3025 i}{64 \, a^{6}}} + \sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}}\right ) +{\left (8 i \, a^{3} x^{3} - 26 \, a^{2} x^{2} - 61 i \, a x - 287\right )} \sqrt{a^{2} x^{2} + 1} \sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}}}{24 \,{\left (a^{4} x - i \, a^{3}\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^{2}}{\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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