Optimal. Leaf size=337 \[ -\frac {41 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{64 a^4}+\frac {x^2 (1-i a x)^{7/4} \sqrt [4]{1+i a x}}{4 a^2}-\frac {(1-i a x)^{7/4} \sqrt [4]{1+i a x} (11-4 i a x)}{32 a^4}-\frac {123 \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{64 \sqrt {2} a^4}+\frac {123 \text {ArcTan}\left (1+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{64 \sqrt {2} a^4}+\frac {123 \log \left (1+\frac {\sqrt {1-i a x}}{\sqrt {1+i a x}}-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{128 \sqrt {2} a^4}-\frac {123 \log \left (1+\frac {\sqrt {1-i a x}}{\sqrt {1+i a x}}+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{128 \sqrt {2} a^4} \]
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Rubi [A]
time = 0.16, antiderivative size = 337, 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, 102,
152, 52, 65, 338, 303, 1176, 631, 210, 1179, 642} \begin {gather*} -\frac {123 \text {ArcTan}\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 \text {ArcTan}\left (1+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{64 \sqrt {2} a^4}-\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 {x^2 (1-i a x)^{7/4} \sqrt [4]{1+i a x}}{4 a^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 102
Rule 152
Rule 210
Rule 303
Rule 338
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^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 \text {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 \text {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 \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 )}{64 a^4}+\frac {123 \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 )}{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 \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 )}{128 a^4}+\frac {123 \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 )}{128 a^4}+\frac {123 \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 )}{128 \sqrt {2} a^4}+\frac {123 \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 )}{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 \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 )}{64 \sqrt {2} a^4}-\frac {123 \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 )}{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*}
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Mathematica [A]
time = 0.28, size = 272, normalized size = 0.81 \begin {gather*} \frac {\frac {512 e^{\frac {13}{2} i \text {ArcTan}(a x)}}{\left (1+e^{2 i \text {ArcTan}(a x)}\right )^4}-\frac {1152 e^{\frac {9}{2} i \text {ArcTan}(a x)}}{\left (1+e^{2 i \text {ArcTan}(a x)}\right )^3}+\frac {1008 e^{\frac {5}{2} i \text {ArcTan}(a x)}}{\left (1+e^{2 i \text {ArcTan}(a x)}\right )^2}-\frac {532 e^{\frac {1}{2} i \text {ArcTan}(a x)}}{1+e^{2 i \text {ArcTan}(a x)}}+123 \sqrt [4]{-1} \log \left (e^{-2 i \text {ArcTan}(a x)} \left (\sqrt [4]{-1}-e^{\frac {1}{2} i \text {ArcTan}(a x)}\right )\right )+123 (-1)^{3/4} \log \left (e^{-2 i \text {ArcTan}(a x)} \left ((-1)^{3/4}-e^{\frac {1}{2} i \text {ArcTan}(a x)}\right )\right )-123 \sqrt [4]{-1} \log \left (e^{-2 i \text {ArcTan}(a x)} \left (\sqrt [4]{-1}+e^{\frac {1}{2} i \text {ArcTan}(a x)}\right )\right )-123 (-1)^{3/4} \log \left (e^{-2 i \text {ArcTan}(a x)} \left ((-1)^{3/4}+e^{\frac {1}{2} i \text {ArcTan}(a x)}\right )\right )}{128 a^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {x^{3}}{\left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )^{\frac {3}{2}}}\, dx\]
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 = 2.13, size = 251, normalized size = 0.74 \begin {gather*} -\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 {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 \frac {x^{3}}{\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 \frac {x^3}{{\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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