Optimal. Leaf size=170 \[ -\frac {(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{3 x^3}-\frac {5 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{12 x^2}+\frac {11 a^2 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{24 x}+\frac {3}{8} i a^3 \text {ArcTan}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )+\frac {3}{8} i a^3 \tanh ^{-1}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right ) \]
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Rubi [A]
time = 0.05, antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 8, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5170, 101, 156,
12, 95, 218, 212, 209} \begin {gather*} \frac {3}{8} i a^3 \text {ArcTan}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )+\frac {3}{8} i a^3 \tanh ^{-1}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )+\frac {11 a^2 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{24 x}-\frac {(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{3 x^3}-\frac {5 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{12 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 95
Rule 101
Rule 156
Rule 209
Rule 212
Rule 218
Rule 5170
Rubi steps
\begin {align*} \int \frac {e^{\frac {1}{2} i \tan ^{-1}(a x)}}{x^4} \, dx &=\int \frac {\sqrt [4]{1+i a x}}{x^4 \sqrt [4]{1-i a x}} \, dx\\ &=-\frac {(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{3 x^3}+\frac {1}{3} \int \frac {\frac {5 i a}{2}-2 a^2 x}{x^3 \sqrt [4]{1-i a x} (1+i a x)^{3/4}} \, dx\\ &=-\frac {(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{3 x^3}-\frac {5 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{12 x^2}-\frac {1}{6} \int \frac {\frac {11 a^2}{4}+\frac {5}{2} i a^3 x}{x^2 \sqrt [4]{1-i a x} (1+i a x)^{3/4}} \, dx\\ &=-\frac {(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{3 x^3}-\frac {5 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{12 x^2}+\frac {11 a^2 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{24 x}+\frac {1}{6} \int -\frac {9 i a^3}{8 x \sqrt [4]{1-i a x} (1+i a x)^{3/4}} \, dx\\ &=-\frac {(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{3 x^3}-\frac {5 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{12 x^2}+\frac {11 a^2 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{24 x}-\frac {1}{16} \left (3 i a^3\right ) \int \frac {1}{x \sqrt [4]{1-i a x} (1+i a x)^{3/4}} \, dx\\ &=-\frac {(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{3 x^3}-\frac {5 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{12 x^2}+\frac {11 a^2 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{24 x}-\frac {1}{4} \left (3 i a^3\right ) \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 )\\ &=-\frac {(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{3 x^3}-\frac {5 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{12 x^2}+\frac {11 a^2 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{24 x}+\frac {1}{8} \left (3 i a^3\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )+\frac {1}{8} \left (3 i a^3\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )\\ &=-\frac {(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{3 x^3}-\frac {5 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{12 x^2}+\frac {11 a^2 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{24 x}+\frac {3}{8} i a^3 \tan ^{-1}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )+\frac {3}{8} i a^3 \tanh ^{-1}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.02, size = 93, normalized size = 0.55 \begin {gather*} \frac {(1-i a x)^{3/4} \left (-8-18 i a x+21 a^2 x^2+11 i a^3 x^3+6 i a^3 x^3 \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {i+a x}{i-a x}\right )\right )}{24 x^3 (1+i a x)^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}}}{x^{4}}\, dx\]
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 = 2.70, size = 184, normalized size = 1.08 \begin {gather*} \frac {9 i \, a^{3} x^{3} \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} + 1\right ) - 9 \, a^{3} x^{3} \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} + i\right ) + 9 \, a^{3} x^{3} \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} - i\right ) - 9 i \, a^{3} x^{3} \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} - 1\right ) - 2 \, {\left (11 i \, a^{3} x^{3} - a^{2} x^{2} + 2 i \, a x + 8\right )} \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}}{48 \, x^{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 \frac {\sqrt {\frac {i \left (a x - i\right )}{\sqrt {a^{2} x^{2} + 1}}}}{x^{4}}\, 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 = -0.01 \begin {gather*} \int \frac {\sqrt {\frac {1+a\,x\,1{}\mathrm {i}}{\sqrt {a^2\,x^2+1}}}}{x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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