Optimal. Leaf size=337 \[ \frac{x^2 (1-i a x)^{7/4} \sqrt [4]{1+i a x}}{4 a^2}-\frac{(1-i a x)^{7/4} \sqrt [4]{1+i a x} (11-4 i a x)}{32 a^4}-\frac{41 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{64 a^4}+\frac{123 \log \left (\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}-\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+1\right )}{128 \sqrt{2} a^4}-\frac{123 \log \left (\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}+\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+1\right )}{128 \sqrt{2} a^4}-\frac{123 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{64 \sqrt{2} a^4}+\frac{123 \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{64 \sqrt{2} a^4} \]
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Rubi [A] time = 0.21698, antiderivative size = 337, 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, 100, 147, 50, 63, 331, 297, 1162, 617, 204, 1165, 628} \[ \frac{x^2 (1-i a x)^{7/4} \sqrt [4]{1+i a x}}{4 a^2}-\frac{(1-i a x)^{7/4} \sqrt [4]{1+i a x} (11-4 i a x)}{32 a^4}-\frac{41 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{64 a^4}+\frac{123 \log \left (\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}-\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+1\right )}{128 \sqrt{2} a^4}-\frac{123 \log \left (\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}+\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+1\right )}{128 \sqrt{2} a^4}-\frac{123 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{64 \sqrt{2} a^4}+\frac{123 \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{64 \sqrt{2} a^4} \]
Antiderivative was successfully verified.
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Rule 5062
Rule 100
Rule 147
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^3 \, dx &=\int \frac{x^3 (1-i a x)^{3/4}}{(1+i a x)^{3/4}} \, dx\\ &=\frac{x^2 (1-i a x)^{7/4} \sqrt [4]{1+i a x}}{4 a^2}+\frac{\int \frac{x (1-i a x)^{3/4} \left (-2+\frac{3 i a x}{2}\right )}{(1+i a x)^{3/4}} \, dx}{4 a^2}\\ &=\frac{x^2 (1-i a x)^{7/4} \sqrt [4]{1+i a x}}{4 a^2}-\frac{(1-i a x)^{7/4} \sqrt [4]{1+i a x} (11-4 i a x)}{32 a^4}-\frac{(41 i) \int \frac{(1-i a x)^{3/4}}{(1+i a x)^{3/4}} \, dx}{64 a^3}\\ &=-\frac{41 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{64 a^4}+\frac{x^2 (1-i a x)^{7/4} \sqrt [4]{1+i a x}}{4 a^2}-\frac{(1-i a x)^{7/4} \sqrt [4]{1+i a x} (11-4 i a x)}{32 a^4}-\frac{(123 i) \int \frac{1}{\sqrt [4]{1-i a x} (1+i a x)^{3/4}} \, dx}{128 a^3}\\ &=-\frac{41 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{64 a^4}+\frac{x^2 (1-i a x)^{7/4} \sqrt [4]{1+i a x}}{4 a^2}-\frac{(1-i a x)^{7/4} \sqrt [4]{1+i a x} (11-4 i a x)}{32 a^4}+\frac{123 \operatorname{Subst}\left (\int \frac{x^2}{\left (2-x^4\right )^{3/4}} \, dx,x,\sqrt [4]{1-i a x}\right )}{32 a^4}\\ &=-\frac{41 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{64 a^4}+\frac{x^2 (1-i a x)^{7/4} \sqrt [4]{1+i a x}}{4 a^2}-\frac{(1-i a x)^{7/4} \sqrt [4]{1+i a x} (11-4 i a x)}{32 a^4}+\frac{123 \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 )}{32 a^4}\\ &=-\frac{41 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{64 a^4}+\frac{x^2 (1-i a x)^{7/4} \sqrt [4]{1+i a x}}{4 a^2}-\frac{(1-i a x)^{7/4} \sqrt [4]{1+i a x} (11-4 i a x)}{32 a^4}-\frac{123 \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 )}{64 a^4}+\frac{123 \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 )}{64 a^4}\\ &=-\frac{41 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{64 a^4}+\frac{x^2 (1-i a x)^{7/4} \sqrt [4]{1+i a x}}{4 a^2}-\frac{(1-i a x)^{7/4} \sqrt [4]{1+i a x} (11-4 i a x)}{32 a^4}+\frac{123 \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 )}{128 a^4}+\frac{123 \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 )}{128 a^4}+\frac{123 \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 )}{128 \sqrt{2} a^4}+\frac{123 \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 )}{128 \sqrt{2} a^4}\\ &=-\frac{41 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{64 a^4}+\frac{x^2 (1-i a x)^{7/4} \sqrt [4]{1+i a x}}{4 a^2}-\frac{(1-i a x)^{7/4} \sqrt [4]{1+i a x} (11-4 i a x)}{32 a^4}+\frac{123 \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 )}{128 \sqrt{2} a^4}-\frac{123 \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 )}{128 \sqrt{2} a^4}+\frac{123 \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 )}{64 \sqrt{2} a^4}-\frac{123 \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 )}{64 \sqrt{2} a^4}\\ &=-\frac{41 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{64 a^4}+\frac{x^2 (1-i a x)^{7/4} \sqrt [4]{1+i a x}}{4 a^2}-\frac{(1-i a x)^{7/4} \sqrt [4]{1+i a x} (11-4 i a x)}{32 a^4}-\frac{123 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{64 \sqrt{2} a^4}+\frac{123 \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{64 \sqrt{2} a^4}+\frac{123 \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 )}{128 \sqrt{2} a^4}-\frac{123 \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 )}{128 \sqrt{2} a^4}\\ \end{align*}
Mathematica [C] time = 0.102481, size = 127, normalized size = 0.38 \[ \frac{(1-i a x)^{7/4} \left (12 \sqrt [4]{2} \text{Hypergeometric2F1}\left (-\frac{5}{4},\frac{7}{4},\frac{11}{4},\frac{1}{2} (1-i a x)\right )-20 \sqrt [4]{2} \text{Hypergeometric2F1}\left (-\frac{1}{4},\frac{7}{4},\frac{11}{4},\frac{1}{2} (1-i a x)\right )+7 \sqrt [4]{2} \text{Hypergeometric2F1}\left (\frac{3}{4},\frac{7}{4},\frac{11}{4},\frac{1}{2} (1-i a x)\right )+7 a^2 x^2 \sqrt [4]{1+i a x}\right )}{28 a^4} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.161, size = 0, normalized size = 0. \begin{align*} \int{{x}^{3} \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^{3}}{\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 = 2.04081, size = 740, normalized size = 2.2 \begin{align*} -\frac{32 \, a^{4} \sqrt{\frac{15129 i}{4096 \, a^{8}}} \log \left (\frac{64}{123} \, a^{4} \sqrt{\frac{15129 i}{4096 \, a^{8}}} + \sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}}\right ) - 32 \, a^{4} \sqrt{\frac{15129 i}{4096 \, a^{8}}} \log \left (-\frac{64}{123} \, a^{4} \sqrt{\frac{15129 i}{4096 \, a^{8}}} + \sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}}\right ) + 32 \, a^{4} \sqrt{-\frac{15129 i}{4096 \, a^{8}}} \log \left (\frac{64}{123} \, a^{4} \sqrt{-\frac{15129 i}{4096 \, a^{8}}} + \sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}}\right ) - 32 \, a^{4} \sqrt{-\frac{15129 i}{4096 \, a^{8}}} \log \left (-\frac{64}{123} \, a^{4} \sqrt{-\frac{15129 i}{4096 \, a^{8}}} + \sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}}\right ) +{\left (16 \, a^{4} x^{4} + 40 i \, a^{3} x^{3} - 54 \, a^{2} x^{2} - 93 i \, a x + 63\right )} \sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}}}{64 \, a^{4}} \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^{3}}{\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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