Optimal. Leaf size=337 \[ \frac{x^2 \sqrt [4]{1-i a x} (1+i a x)^{7/4}}{4 a^2}-\frac{\sqrt [4]{1-i a x} (1+i a x)^{7/4} (11+4 i a x)}{32 a^4}-\frac{41 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{64 a^4}+\frac{123 \log \left (\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}-\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+1\right )}{128 \sqrt{2} a^4}-\frac{123 \log \left (\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}+\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+1\right )}{128 \sqrt{2} a^4}+\frac{123 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{64 \sqrt{2} a^4}-\frac{123 \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{64 \sqrt{2} a^4} \]
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Rubi [A] time = 0.214526, 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, 240, 211, 1165, 628, 1162, 617, 204} \[ \frac{x^2 \sqrt [4]{1-i a x} (1+i a x)^{7/4}}{4 a^2}-\frac{\sqrt [4]{1-i a x} (1+i a x)^{7/4} (11+4 i a x)}{32 a^4}-\frac{41 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{64 a^4}+\frac{123 \log \left (\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}-\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+1\right )}{128 \sqrt{2} a^4}-\frac{123 \log \left (\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}+\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+1\right )}{128 \sqrt{2} a^4}+\frac{123 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{64 \sqrt{2} a^4}-\frac{123 \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{64 \sqrt{2} a^4} \]
Antiderivative was successfully verified.
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Rule 5062
Rule 100
Rule 147
Rule 50
Rule 63
Rule 240
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
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 \sqrt [4]{1-i a x} (1+i a x)^{7/4}}{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 \sqrt [4]{1-i a x} (1+i a x)^{7/4}}{4 a^2}-\frac{\sqrt [4]{1-i a x} (1+i a x)^{7/4} (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 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{64 a^4}+\frac{x^2 \sqrt [4]{1-i a x} (1+i a x)^{7/4}}{4 a^2}-\frac{\sqrt [4]{1-i a x} (1+i a x)^{7/4} (11+4 i a x)}{32 a^4}+\frac{(123 i) \int \frac{1}{(1-i a x)^{3/4} \sqrt [4]{1+i a x}} \, dx}{128 a^3}\\ &=-\frac{41 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{64 a^4}+\frac{x^2 \sqrt [4]{1-i a x} (1+i a x)^{7/4}}{4 a^2}-\frac{\sqrt [4]{1-i a x} (1+i a x)^{7/4} (11+4 i a x)}{32 a^4}-\frac{123 \operatorname{Subst}\left (\int \frac{1}{\sqrt [4]{2-x^4}} \, dx,x,\sqrt [4]{1-i a x}\right )}{32 a^4}\\ &=-\frac{41 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{64 a^4}+\frac{x^2 \sqrt [4]{1-i a x} (1+i a x)^{7/4}}{4 a^2}-\frac{\sqrt [4]{1-i a x} (1+i a x)^{7/4} (11+4 i a x)}{32 a^4}-\frac{123 \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 )}{32 a^4}\\ &=-\frac{41 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{64 a^4}+\frac{x^2 \sqrt [4]{1-i a x} (1+i a x)^{7/4}}{4 a^2}-\frac{\sqrt [4]{1-i a x} (1+i a x)^{7/4} (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 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{64 a^4}+\frac{x^2 \sqrt [4]{1-i a x} (1+i a x)^{7/4}}{4 a^2}-\frac{\sqrt [4]{1-i a x} (1+i a x)^{7/4} (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 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{64 a^4}+\frac{x^2 \sqrt [4]{1-i a x} (1+i a x)^{7/4}}{4 a^2}-\frac{\sqrt [4]{1-i a x} (1+i a x)^{7/4} (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 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{64 a^4}+\frac{x^2 \sqrt [4]{1-i a x} (1+i a x)^{7/4}}{4 a^2}-\frac{\sqrt [4]{1-i a x} (1+i a x)^{7/4} (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 = 0.104815, size = 148, normalized size = 0.44 \[ \frac{\sqrt [4]{1-i a x} \left (-24\ 2^{3/4} \text{Hypergeometric2F1}\left (-\frac{11}{4},\frac{1}{4},\frac{5}{4},\frac{1}{2} (1-i a x)\right )+8\ 2^{3/4} \text{Hypergeometric2F1}\left (-\frac{7}{4},\frac{1}{4},\frac{5}{4},\frac{1}{2} (1-i a x)\right )+2\ 2^{3/4} \text{Hypergeometric2F1}\left (-\frac{3}{4},\frac{1}{4},\frac{5}{4},\frac{1}{2} (1-i a x)\right )+i a^3 x^3 (1+i a x)^{3/4}+a^2 x^2 (1+i a x)^{3/4}\right )}{4 a^4} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.175, size = 0, normalized size = 0. \begin{align*} \int \left ({(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) ^{{\frac{3}{2}}}{x}^{3}\, 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^{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 = 1.70543, size = 756, normalized size = 2.24 \begin{align*} \frac{32 \, a^{4} \sqrt{\frac{15129 i}{4096 \, a^{8}}} \log \left (\frac{64}{123} i \, 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} i \, 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} i \, 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} i \, a^{4} \sqrt{-\frac{15129 i}{4096 \, a^{8}}} + \sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}}\right ) +{\left (16 i \, a^{3} x^{3} + 24 \, a^{2} x^{2} - 30 i \, a x - 63\right )} \sqrt{a^{2} x^{2} + 1} \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 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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