Optimal. Leaf size=268 \[ -\frac {i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{a}-\frac {3 i \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{\sqrt {2} a}+\frac {3 i \text {ArcTan}\left (1+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{\sqrt {2} a}+\frac {3 i \log \left (1+\frac {\sqrt {1-i a x}}{\sqrt {1+i a x}}-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{2 \sqrt {2} a}-\frac {3 i \log \left (1+\frac {\sqrt {1-i a x}}{\sqrt {1+i a x}}+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{2 \sqrt {2} a} \]
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Rubi [A]
time = 0.11, antiderivative size = 268, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 10, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {5169, 52, 65,
338, 303, 1176, 631, 210, 1179, 642} \begin {gather*} -\frac {3 i \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{\sqrt {2} a}+\frac {3 i \text {ArcTan}\left (1+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{\sqrt {2} a}-\frac {i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{a}+\frac {3 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 )}{2 \sqrt {2} a}-\frac {3 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 )}{2 \sqrt {2} a} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 210
Rule 303
Rule 338
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 5169
Rubi steps
\begin {align*} \int e^{-\frac {3}{2} i \tan ^{-1}(a x)} \, dx &=\int \frac {(1-i a x)^{3/4}}{(1+i a x)^{3/4}} \, dx\\ &=-\frac {i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{a}+\frac {3}{2} \int \frac {1}{\sqrt [4]{1-i a x} (1+i a x)^{3/4}} \, dx\\ &=-\frac {i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{a}+\frac {(6 i) \text {Subst}\left (\int \frac {x^2}{\left (2-x^4\right )^{3/4}} \, dx,x,\sqrt [4]{1-i a x}\right )}{a}\\ &=-\frac {i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{a}+\frac {(6 i) \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 )}{a}\\ &=-\frac {i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{a}-\frac {(3 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 )}{a}+\frac {(3 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 )}{a}\\ &=-\frac {i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{a}+\frac {(3 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 )}{2 a}+\frac {(3 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 )}{2 a}+\frac {(3 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 )}{2 \sqrt {2} a}+\frac {(3 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 )}{2 \sqrt {2} a}\\ &=-\frac {i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{a}+\frac {3 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 )}{2 \sqrt {2} a}-\frac {3 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 )}{2 \sqrt {2} a}+\frac {(3 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 )}{\sqrt {2} a}-\frac {(3 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 )}{\sqrt {2} a}\\ &=-\frac {i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{a}-\frac {3 i \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{\sqrt {2} a}+\frac {3 i \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{\sqrt {2} a}+\frac {3 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 )}{2 \sqrt {2} a}-\frac {3 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 )}{2 \sqrt {2} a}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.04, size = 39, normalized size = 0.15 \begin {gather*} -\frac {8 i e^{\frac {1}{2} i \text {ArcTan}(a x)} \, _2F_1\left (\frac {1}{4},2;\frac {5}{4};-e^{2 i \text {ArcTan}(a x)}\right )}{a} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )^{\frac {3}{2}}}\, dx\]
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 = 2.82, size = 209, normalized size = 0.78 \begin {gather*} \frac {a \sqrt {\frac {9 i}{a^{2}}} \log \left (\frac {1}{3} i \, a \sqrt {\frac {9 i}{a^{2}}} + \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}\right ) - a \sqrt {\frac {9 i}{a^{2}}} \log \left (-\frac {1}{3} i \, a \sqrt {\frac {9 i}{a^{2}}} + \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}\right ) + a \sqrt {-\frac {9 i}{a^{2}}} \log \left (\frac {1}{3} i \, a \sqrt {-\frac {9 i}{a^{2}}} + \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}\right ) - a \sqrt {-\frac {9 i}{a^{2}}} \log \left (-\frac {1}{3} i \, a \sqrt {-\frac {9 i}{a^{2}}} + \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}\right ) - 2 \, {\left (a x + i\right )} \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}}{2 \, a} \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 {1}{\left (\frac {i a x + 1}{\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 {1}{{\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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