Optimal. Leaf size=339 \[ -\frac {17 i \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{24 a^3}-\frac {i \sqrt [4]{1-i a x} (1+i a x)^{7/4}}{4 a^3}+\frac {x \sqrt [4]{1-i a x} (1+i a x)^{7/4}}{3 a^2}+\frac {17 i \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{8 \sqrt {2} a^3}-\frac {17 i \text {ArcTan}\left (1+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{8 \sqrt {2} a^3}+\frac {17 i \log \left (1+\frac {\sqrt {1-i a x}}{\sqrt {1+i a x}}-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{16 \sqrt {2} a^3}-\frac {17 i \log \left (1+\frac {\sqrt {1-i a x}}{\sqrt {1+i a x}}+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{16 \sqrt {2} a^3} \]
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Rubi [A]
time = 0.15, antiderivative size = 339, normalized size of antiderivative = 1.00, number of steps
used = 15, number of rules used = 12, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {5170, 92, 81,
52, 65, 246, 217, 1179, 642, 1176, 631, 210} \begin {gather*} \frac {17 i \text {ArcTan}\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 \text {ArcTan}\left (1+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{8 \sqrt {2} a^3}-\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 {x \sqrt [4]{1-i a x} (1+i a x)^{7/4}}{3 a^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 81
Rule 92
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^2 \, dx &=\int \frac {x^2 (1+i a x)^{3/4}}{(1-i a x)^{3/4}} \, dx\\ &=\frac {x \sqrt [4]{1-i a x} (1+i a x)^{7/4}}{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 \sqrt [4]{1-i a x} (1+i a x)^{7/4}}{4 a^3}+\frac {x \sqrt [4]{1-i a x} (1+i a x)^{7/4}}{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 \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{24 a^3}-\frac {i \sqrt [4]{1-i a x} (1+i a x)^{7/4}}{4 a^3}+\frac {x \sqrt [4]{1-i a x} (1+i a x)^{7/4}}{3 a^2}-\frac {17 \int \frac {1}{(1-i a x)^{3/4} \sqrt [4]{1+i a x}} \, dx}{16 a^2}\\ &=-\frac {17 i \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{24 a^3}-\frac {i \sqrt [4]{1-i a x} (1+i a x)^{7/4}}{4 a^3}+\frac {x \sqrt [4]{1-i a x} (1+i a x)^{7/4}}{3 a^2}-\frac {(17 i) \text {Subst}\left (\int \frac {1}{\sqrt [4]{2-x^4}} \, dx,x,\sqrt [4]{1-i a x}\right )}{4 a^3}\\ &=-\frac {17 i \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{24 a^3}-\frac {i \sqrt [4]{1-i a x} (1+i a x)^{7/4}}{4 a^3}+\frac {x \sqrt [4]{1-i a x} (1+i a x)^{7/4}}{3 a^2}-\frac {(17 i) \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 )}{4 a^3}\\ &=-\frac {17 i \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{24 a^3}-\frac {i \sqrt [4]{1-i a x} (1+i a x)^{7/4}}{4 a^3}+\frac {x \sqrt [4]{1-i a x} (1+i a x)^{7/4}}{3 a^2}-\frac {(17 i) \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 )}{8 a^3}-\frac {(17 i) \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 )}{8 a^3}\\ &=-\frac {17 i \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{24 a^3}-\frac {i \sqrt [4]{1-i a x} (1+i a x)^{7/4}}{4 a^3}+\frac {x \sqrt [4]{1-i a x} (1+i a x)^{7/4}}{3 a^2}-\frac {(17 i) \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 )}{16 a^3}-\frac {(17 i) \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 )}{16 a^3}+\frac {(17 i) \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 )}{16 \sqrt {2} a^3}+\frac {(17 i) \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 )}{16 \sqrt {2} a^3}\\ &=-\frac {17 i \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{24 a^3}-\frac {i \sqrt [4]{1-i a x} (1+i a x)^{7/4}}{4 a^3}+\frac {x \sqrt [4]{1-i a x} (1+i a x)^{7/4}}{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) \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 )}{8 \sqrt {2} a^3}+\frac {(17 i) \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 )}{8 \sqrt {2} a^3}\\ &=-\frac {17 i \sqrt [4]{1-i a x} (1+i a x)^{3/4}}{24 a^3}-\frac {i \sqrt [4]{1-i a x} (1+i a x)^{7/4}}{4 a^3}+\frac {x \sqrt [4]{1-i a x} (1+i a x)^{7/4}}{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*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.03, size = 82, normalized size = 0.24 \begin {gather*} \frac {\sqrt [4]{1-i a x} \left ((1+i a x)^{3/4} \left (-3 i+7 a x+4 i a^2 x^2\right )-34 i 2^{3/4} \, _2F_1\left (-\frac {3}{4},\frac {1}{4};\frac {5}{4};\frac {1}{2} (1-i a x)\right )\right )}{12 a^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )^{\frac {3}{2}} x^{2}\, dx\]
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 = 2.09, size = 247, normalized size = 0.73 \begin {gather*} -\frac {12 \, a^{3} \sqrt {\frac {289 i}{64 \, a^{6}}} \log \left (\frac {8}{17} \, 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} \, 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} \, 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} \, a^{3} \sqrt {-\frac {289 i}{64 \, a^{6}}} + \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}\right ) - \sqrt {a^{2} x^{2} + 1} {\left (8 i \, a^{2} x^{2} + 14 \, a x - 23 i\right )} \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}}{24 \, a^{3}} \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^{2} \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^2\,{\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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