Optimal. Leaf size=203 \[ \frac {287 i a^3 \sqrt [4]{1-i a x}}{24 \sqrt [4]{1+i a x}}+\frac {55}{8} i a^3 \tan ^{-1}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-\frac {55}{8} i a^3 \tanh ^{-1}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )+\frac {61 a^2 \sqrt [4]{1-i a x}}{24 x \sqrt [4]{1+i a x}}-\frac {\sqrt [4]{1-i a x}}{3 x^3 \sqrt [4]{1+i a x}}+\frac {13 i a \sqrt [4]{1-i a x}}{12 x^2 \sqrt [4]{1+i a x}} \]
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Rubi [A] time = 0.08, 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 = {5062, 98, 151, 155, 12, 93, 298, 203, 206} \[ \frac {287 i a^3 \sqrt [4]{1-i a x}}{24 \sqrt [4]{1+i a x}}+\frac {61 a^2 \sqrt [4]{1-i a x}}{24 x \sqrt [4]{1+i a x}}+\frac {55}{8} i a^3 \tan ^{-1}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-\frac {55}{8} i a^3 \tanh ^{-1}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )+\frac {13 i a \sqrt [4]{1-i a x}}{12 x^2 \sqrt [4]{1+i a x}}-\frac {\sqrt [4]{1-i a x}}{3 x^3 \sqrt [4]{1+i a x}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 93
Rule 98
Rule 151
Rule 155
Rule 203
Rule 206
Rule 298
Rule 5062
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)^{3/4} (1+i a x)^{5/4}} \, dx\\ &=-\frac {\sqrt [4]{1-i a x}}{3 x^3 \sqrt [4]{1+i a x}}+\frac {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)^{3/4} (1+i a x)^{5/4}} \, dx\\ &=-\frac {\sqrt [4]{1-i a x}}{3 x^3 \sqrt [4]{1+i a x}}+\frac {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)^{3/4} (1+i a x)^{5/4}} \, 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 (1-i a x)^{3/4} \sqrt [4]{1+i a x}} \, 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 (1-i a x)^{3/4} \sqrt [4]{1+i a x}} \, 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 ) \operatorname {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 )\\ &=\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 ) \operatorname {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 ) \operatorname {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] time = 0.03, size = 93, normalized size = 0.46 \[ \frac {\sqrt [4]{1-i a x} \left (-330 i a^3 x^3 \, _2F_1\left (\frac {1}{4},1;\frac {5}{4};\frac {a x+i}{i-a x}\right )+287 i a^3 x^3+61 a^2 x^2+26 i a x-8\right )}{24 x^3 \sqrt [4]{1+i a x}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.75, size = 246, normalized size = 1.21 \[ \frac {{\left (574 \, a^{3} x^{3} - 122 i \, a^{2} x^{2} + 52 \, a x + 16 i\right )} \sqrt {a^{2} x^{2} + 1} \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} - 165 \, {\left (i \, a^{4} x^{4} + a^{3} x^{3}\right )} \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} + 1\right ) - {\left (165 \, a^{4} x^{4} - 165 i \, a^{3} x^{3}\right )} \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} + i\right ) + {\left (165 \, a^{4} x^{4} - 165 i \, a^{3} x^{3}\right )} \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} - i\right ) - 165 \, {\left (-i \, a^{4} x^{4} - a^{3} x^{3}\right )} \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} - 1\right )}{48 \, a x^{4} - 48 i \, x^{3}} \]
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 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.17, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )^{\frac {5}{2}} x^{4}}\, dx \]
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 \[ \int \frac {1}{x^{4} \left (\frac {i \, a x + 1}{\sqrt {a^{2} x^{2} + 1}}\right )^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {1}{x^4\,{\left (\frac {1+a\,x\,1{}\mathrm {i}}{\sqrt {a^2\,x^2+1}}\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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