Optimal. Leaf size=295 \[ \frac {3 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{4 a^2}+\frac {\sqrt [4]{1-i a x} (1+i a x)^{7/4}}{2 a^2}-\frac {9 \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{4 \sqrt {2} a^2}+\frac {9 \text {ArcTan}\left (1+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{4 \sqrt {2} a^2}-\frac {9 \log \left (1+\frac {\sqrt {1-i a x}}{\sqrt {1+i a x}}-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{8 \sqrt {2} a^2}+\frac {9 \log \left (1+\frac {\sqrt {1-i a x}}{\sqrt {1+i a x}}+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{8 \sqrt {2} a^2} \]
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Rubi [A]
time = 0.13, antiderivative size = 295, normalized size of antiderivative = 1.00, number of steps
used = 14, number of rules used = 11, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.786, Rules used = {5170, 81, 52,
65, 246, 217, 1179, 642, 1176, 631, 210} \begin {gather*} -\frac {9 \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{4 \sqrt {2} a^2}+\frac {9 \text {ArcTan}\left (1+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{4 \sqrt {2} a^2}+\frac {\sqrt [4]{1-i a x} (1+i a x)^{7/4}}{2 a^2}+\frac {3 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{4 a^2}-\frac {9 \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 {9 \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 81
Rule 210
Rule 217
Rule 246
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 5170
Rubi steps
\begin {align*} \int e^{\frac {3}{2} i \tan ^{-1}(a x)} x \, dx &=\int \frac {x (1+i a x)^{3/4}}{(1-i a x)^{3/4}} \, dx\\ &=\frac {\sqrt [4]{1-i a x} (1+i a x)^{7/4}}{2 a^2}-\frac {(3 i) \int \frac {(1+i a x)^{3/4}}{(1-i a x)^{3/4}} \, dx}{4 a}\\ &=\frac {3 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{4 a^2}+\frac {\sqrt [4]{1-i a x} (1+i a x)^{7/4}}{2 a^2}-\frac {(9 i) \int \frac {1}{(1-i a x)^{3/4} \sqrt [4]{1+i a x}} \, dx}{8 a}\\ &=\frac {3 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{4 a^2}+\frac {\sqrt [4]{1-i a x} (1+i a x)^{7/4}}{2 a^2}+\frac {9 \text {Subst}\left (\int \frac {1}{\sqrt [4]{2-x^4}} \, dx,x,\sqrt [4]{1-i a x}\right )}{2 a^2}\\ &=\frac {3 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{4 a^2}+\frac {\sqrt [4]{1-i a x} (1+i a x)^{7/4}}{2 a^2}+\frac {9 \text {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 {3 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{4 a^2}+\frac {\sqrt [4]{1-i a x} (1+i a x)^{7/4}}{2 a^2}+\frac {9 \text {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 {9 \text {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 {3 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{4 a^2}+\frac {\sqrt [4]{1-i a x} (1+i a x)^{7/4}}{2 a^2}+\frac {9 \text {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 {9 \text {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 {9 \text {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 {9 \text {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 {3 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{4 a^2}+\frac {\sqrt [4]{1-i a x} (1+i a x)^{7/4}}{2 a^2}-\frac {9 \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 {9 \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 {9 \text {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 {9 \text {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 {3 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{4 a^2}+\frac {\sqrt [4]{1-i a x} (1+i a x)^{7/4}}{2 a^2}-\frac {9 \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 {9 \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 {9 \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 {9 \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*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.01, size = 61, normalized size = 0.21 \begin {gather*} \frac {\sqrt [4]{1-i a x} \left ((1+i a x)^{7/4}+6\ 2^{3/4} \, _2F_1\left (-\frac {3}{4},\frac {1}{4};\frac {5}{4};\frac {1}{2} (1-i a x)\right )\right )}{2 a^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )^{\frac {3}{2}} x\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.30, size = 239, normalized size = 0.81 \begin {gather*} -\frac {2 \, a^{2} \sqrt {\frac {81 i}{16 \, a^{4}}} \log \left (\frac {4}{9} i \, a^{2} \sqrt {\frac {81 i}{16 \, a^{4}}} + \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}\right ) - 2 \, a^{2} \sqrt {\frac {81 i}{16 \, a^{4}}} \log \left (-\frac {4}{9} i \, a^{2} \sqrt {\frac {81 i}{16 \, a^{4}}} + \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}\right ) - 2 \, a^{2} \sqrt {-\frac {81 i}{16 \, a^{4}}} \log \left (\frac {4}{9} i \, a^{2} \sqrt {-\frac {81 i}{16 \, a^{4}}} + \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}\right ) + 2 \, a^{2} \sqrt {-\frac {81 i}{16 \, a^{4}}} \log \left (-\frac {4}{9} i \, a^{2} \sqrt {-\frac {81 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 x + 5\right )} \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}}{4 \, a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x \left (\frac {i \left (a x - i\right )}{\sqrt {a^{2} x^{2} + 1}}\right )^{\frac {3}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x\,{\left (\frac {1+a\,x\,1{}\mathrm {i}}{\sqrt {a^2\,x^2+1}}\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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