Optimal. Leaf size=203 \[ -\frac {287 i a^3 \sqrt [4]{1+i a x}}{24 \sqrt [4]{1-i a x}}-\frac {\sqrt [4]{1+i a x}}{3 x^3 \sqrt [4]{1-i a x}}-\frac {13 i a \sqrt [4]{1+i a x}}{12 x^2 \sqrt [4]{1-i a x}}+\frac {61 a^2 \sqrt [4]{1+i a x}}{24 x \sqrt [4]{1-i a x}}+\frac {55}{8} i a^3 \text {ArcTan}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )+\frac {55}{8} i a^3 \tanh ^{-1}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right ) \]
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Rubi [A]
time = 0.06, antiderivative size = 203, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 9, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.562, Rules used = {5170, 100,
156, 160, 12, 95, 218, 212, 209} \begin {gather*} \frac {55}{8} i a^3 \text {ArcTan}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-\frac {287 i a^3 \sqrt [4]{1+i a x}}{24 \sqrt [4]{1-i a x}}+\frac {55}{8} i a^3 \tanh ^{-1}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )+\frac {61 a^2 \sqrt [4]{1+i a x}}{24 x \sqrt [4]{1-i a x}}-\frac {\sqrt [4]{1+i a x}}{3 x^3 \sqrt [4]{1-i a x}}-\frac {13 i a \sqrt [4]{1+i a x}}{12 x^2 \sqrt [4]{1-i a x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 95
Rule 100
Rule 156
Rule 160
Rule 209
Rule 212
Rule 218
Rule 5170
Rubi steps
\begin {align*} \int \frac {e^{\frac {5}{2} i \tan ^{-1}(a x)}}{x^4} \, dx &=\int \frac {(1+i a x)^{5/4}}{x^4 (1-i a x)^{5/4}} \, dx\\ &=-\frac {\sqrt [4]{1+i a x}}{3 x^3 \sqrt [4]{1-i a x}}-\frac {1}{3} \int \frac {-\frac {13 i a}{2}+6 a^2 x}{x^3 (1-i a x)^{5/4} (1+i a x)^{3/4}} \, dx\\ &=-\frac {\sqrt [4]{1+i a x}}{3 x^3 \sqrt [4]{1-i a x}}-\frac {13 i a \sqrt [4]{1+i a x}}{12 x^2 \sqrt [4]{1-i a x}}+\frac {1}{6} \int \frac {-\frac {61 a^2}{4}-13 i a^3 x}{x^2 (1-i a x)^{5/4} (1+i a x)^{3/4}} \, dx\\ &=-\frac {\sqrt [4]{1+i a x}}{3 x^3 \sqrt [4]{1-i a x}}-\frac {13 i a \sqrt [4]{1+i a x}}{12 x^2 \sqrt [4]{1-i a x}}+\frac {61 a^2 \sqrt [4]{1+i a x}}{24 x \sqrt [4]{1-i a x}}-\frac {1}{6} \int \frac {\frac {165 i a^3}{8}-\frac {61 a^4 x}{4}}{x (1-i a x)^{5/4} (1+i a x)^{3/4}} \, dx\\ &=-\frac {287 i a^3 \sqrt [4]{1+i a x}}{24 \sqrt [4]{1-i a x}}-\frac {\sqrt [4]{1+i a x}}{3 x^3 \sqrt [4]{1-i a x}}-\frac {13 i a \sqrt [4]{1+i a x}}{12 x^2 \sqrt [4]{1-i a x}}+\frac {61 a^2 \sqrt [4]{1+i a x}}{24 x \sqrt [4]{1-i a x}}-\frac {i \int \frac {165 a^4}{16 x \sqrt [4]{1-i a x} (1+i a x)^{3/4}} \, dx}{3 a}\\ &=-\frac {287 i a^3 \sqrt [4]{1+i a x}}{24 \sqrt [4]{1-i a x}}-\frac {\sqrt [4]{1+i a x}}{3 x^3 \sqrt [4]{1-i a x}}-\frac {13 i a \sqrt [4]{1+i a x}}{12 x^2 \sqrt [4]{1-i a x}}+\frac {61 a^2 \sqrt [4]{1+i a x}}{24 x \sqrt [4]{1-i a x}}-\frac {1}{16} \left (55 i a^3\right ) \int \frac {1}{x \sqrt [4]{1-i a x} (1+i a x)^{3/4}} \, dx\\ &=-\frac {287 i a^3 \sqrt [4]{1+i a x}}{24 \sqrt [4]{1-i a x}}-\frac {\sqrt [4]{1+i a x}}{3 x^3 \sqrt [4]{1-i a x}}-\frac {13 i a \sqrt [4]{1+i a x}}{12 x^2 \sqrt [4]{1-i a x}}+\frac {61 a^2 \sqrt [4]{1+i a x}}{24 x \sqrt [4]{1-i a x}}-\frac {1}{4} \left (55 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 {287 i a^3 \sqrt [4]{1+i a x}}{24 \sqrt [4]{1-i a x}}-\frac {\sqrt [4]{1+i a x}}{3 x^3 \sqrt [4]{1-i a x}}-\frac {13 i a \sqrt [4]{1+i a x}}{12 x^2 \sqrt [4]{1-i a x}}+\frac {61 a^2 \sqrt [4]{1+i a x}}{24 x \sqrt [4]{1-i a x}}+\frac {1}{8} \left (55 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 (55 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 {287 i a^3 \sqrt [4]{1+i a x}}{24 \sqrt [4]{1-i a x}}-\frac {\sqrt [4]{1+i a x}}{3 x^3 \sqrt [4]{1-i a x}}-\frac {13 i a \sqrt [4]{1+i a x}}{12 x^2 \sqrt [4]{1-i a x}}+\frac {61 a^2 \sqrt [4]{1+i a x}}{24 x \sqrt [4]{1-i a x}}+\frac {55}{8} i a^3 \tan ^{-1}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )+\frac {55}{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 = 106, normalized size = 0.52 \begin {gather*} \frac {-8-34 i a x+87 a^2 x^2-226 i a^3 x^3+287 a^4 x^4+110 a^3 x^3 (i+a x) \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {i+a x}{i-a x}\right )}{24 x^3 \sqrt [4]{1-i a x} (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 \[\int \frac {\left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )^{\frac {5}{2}}}{x^{4}}\, dx\]
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 = 1.93, size = 184, normalized size = 0.91 \begin {gather*} \frac {165 i \, a^{3} x^{3} \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} + 1\right ) - 165 \, a^{3} x^{3} \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} + i\right ) + 165 \, a^{3} x^{3} \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} - i\right ) - 165 i \, a^{3} x^{3} \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} - 1\right ) - 2 \, {\left (287 i \, a^{3} x^{3} - 61 \, a^{2} x^{2} + 26 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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 {{\left (\frac {1+a\,x\,1{}\mathrm {i}}{\sqrt {a^2\,x^2+1}}\right )}^{5/2}}{x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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