Optimal. Leaf size=233 \[ \frac {2467 a^4 \sqrt [4]{1-i a x}}{192 \sqrt [4]{1+i a x}}+\frac {475}{64} a^4 \tan ^{-1}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-\frac {475}{64} a^4 \tanh ^{-1}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-\frac {521 i a^3 \sqrt [4]{1-i a x}}{192 x \sqrt [4]{1+i a x}}+\frac {113 a^2 \sqrt [4]{1-i a x}}{96 x^2 \sqrt [4]{1+i a x}}-\frac {\sqrt [4]{1-i a x}}{4 x^4 \sqrt [4]{1+i a x}}+\frac {17 i a \sqrt [4]{1-i a x}}{24 x^3 \sqrt [4]{1+i a x}} \]
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Rubi [A] time = 0.10, antiderivative size = 233, normalized size of antiderivative = 1.00, number of steps used = 11, 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} \[ \frac {113 a^2 \sqrt [4]{1-i a x}}{96 x^2 \sqrt [4]{1+i a x}}+\frac {2467 a^4 \sqrt [4]{1-i a x}}{192 \sqrt [4]{1+i a x}}-\frac {521 i a^3 \sqrt [4]{1-i a x}}{192 x \sqrt [4]{1+i a x}}+\frac {475}{64} a^4 \tan ^{-1}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-\frac {475}{64} a^4 \tanh ^{-1}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )+\frac {17 i a \sqrt [4]{1-i a x}}{24 x^3 \sqrt [4]{1+i a x}}-\frac {\sqrt [4]{1-i a x}}{4 x^4 \sqrt [4]{1+i a x}} \]
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^5} \, dx &=\int \frac {(1-i a x)^{5/4}}{x^5 (1+i a x)^{5/4}} \, dx\\ &=-\frac {\sqrt [4]{1-i a x}}{4 x^4 \sqrt [4]{1+i a x}}-\frac {1}{4} \int \frac {\frac {17 i a}{2}+8 a^2 x}{x^4 (1-i a x)^{3/4} (1+i a x)^{5/4}} \, dx\\ &=-\frac {\sqrt [4]{1-i a x}}{4 x^4 \sqrt [4]{1+i a x}}+\frac {17 i a \sqrt [4]{1-i a x}}{24 x^3 \sqrt [4]{1+i a x}}+\frac {1}{12} \int \frac {-\frac {113 a^2}{4}+\frac {51}{2} i a^3 x}{x^3 (1-i a x)^{3/4} (1+i a x)^{5/4}} \, dx\\ &=-\frac {\sqrt [4]{1-i a x}}{4 x^4 \sqrt [4]{1+i a x}}+\frac {17 i a \sqrt [4]{1-i a x}}{24 x^3 \sqrt [4]{1+i a x}}+\frac {113 a^2 \sqrt [4]{1-i a x}}{96 x^2 \sqrt [4]{1+i a x}}-\frac {1}{24} \int \frac {-\frac {521 i a^3}{8}-\frac {113 a^4 x}{2}}{x^2 (1-i a x)^{3/4} (1+i a x)^{5/4}} \, dx\\ &=-\frac {\sqrt [4]{1-i a x}}{4 x^4 \sqrt [4]{1+i a x}}+\frac {17 i a \sqrt [4]{1-i a x}}{24 x^3 \sqrt [4]{1+i a x}}+\frac {113 a^2 \sqrt [4]{1-i a x}}{96 x^2 \sqrt [4]{1+i a x}}-\frac {521 i a^3 \sqrt [4]{1-i a x}}{192 x \sqrt [4]{1+i a x}}+\frac {1}{24} \int \frac {\frac {1425 a^4}{16}-\frac {521}{8} i a^5 x}{x (1-i a x)^{3/4} (1+i a x)^{5/4}} \, dx\\ &=\frac {2467 a^4 \sqrt [4]{1-i a x}}{192 \sqrt [4]{1+i a x}}-\frac {\sqrt [4]{1-i a x}}{4 x^4 \sqrt [4]{1+i a x}}+\frac {17 i a \sqrt [4]{1-i a x}}{24 x^3 \sqrt [4]{1+i a x}}+\frac {113 a^2 \sqrt [4]{1-i a x}}{96 x^2 \sqrt [4]{1+i a x}}-\frac {521 i a^3 \sqrt [4]{1-i a x}}{192 x \sqrt [4]{1+i a x}}-\frac {i \int \frac {1425 i a^5}{32 x (1-i a x)^{3/4} \sqrt [4]{1+i a x}} \, dx}{12 a}\\ &=\frac {2467 a^4 \sqrt [4]{1-i a x}}{192 \sqrt [4]{1+i a x}}-\frac {\sqrt [4]{1-i a x}}{4 x^4 \sqrt [4]{1+i a x}}+\frac {17 i a \sqrt [4]{1-i a x}}{24 x^3 \sqrt [4]{1+i a x}}+\frac {113 a^2 \sqrt [4]{1-i a x}}{96 x^2 \sqrt [4]{1+i a x}}-\frac {521 i a^3 \sqrt [4]{1-i a x}}{192 x \sqrt [4]{1+i a x}}+\frac {1}{128} \left (475 a^4\right ) \int \frac {1}{x (1-i a x)^{3/4} \sqrt [4]{1+i a x}} \, dx\\ &=\frac {2467 a^4 \sqrt [4]{1-i a x}}{192 \sqrt [4]{1+i a x}}-\frac {\sqrt [4]{1-i a x}}{4 x^4 \sqrt [4]{1+i a x}}+\frac {17 i a \sqrt [4]{1-i a x}}{24 x^3 \sqrt [4]{1+i a x}}+\frac {113 a^2 \sqrt [4]{1-i a x}}{96 x^2 \sqrt [4]{1+i a x}}-\frac {521 i a^3 \sqrt [4]{1-i a x}}{192 x \sqrt [4]{1+i a x}}+\frac {1}{32} \left (475 a^4\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 {2467 a^4 \sqrt [4]{1-i a x}}{192 \sqrt [4]{1+i a x}}-\frac {\sqrt [4]{1-i a x}}{4 x^4 \sqrt [4]{1+i a x}}+\frac {17 i a \sqrt [4]{1-i a x}}{24 x^3 \sqrt [4]{1+i a x}}+\frac {113 a^2 \sqrt [4]{1-i a x}}{96 x^2 \sqrt [4]{1+i a x}}-\frac {521 i a^3 \sqrt [4]{1-i a x}}{192 x \sqrt [4]{1+i a x}}-\frac {1}{64} \left (475 a^4\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}{64} \left (475 a^4\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 {2467 a^4 \sqrt [4]{1-i a x}}{192 \sqrt [4]{1+i a x}}-\frac {\sqrt [4]{1-i a x}}{4 x^4 \sqrt [4]{1+i a x}}+\frac {17 i a \sqrt [4]{1-i a x}}{24 x^3 \sqrt [4]{1+i a x}}+\frac {113 a^2 \sqrt [4]{1-i a x}}{96 x^2 \sqrt [4]{1+i a x}}-\frac {521 i a^3 \sqrt [4]{1-i a x}}{192 x \sqrt [4]{1+i a x}}+\frac {475}{64} a^4 \tan ^{-1}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-\frac {475}{64} a^4 \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 = 0.04, size = 99, normalized size = 0.42 \[ \frac {\sqrt [4]{1-i a x} \left (-2850 a^4 x^4 \, _2F_1\left (\frac {1}{4},1;\frac {5}{4};\frac {a x+i}{i-a x}\right )+2467 a^4 x^4-521 i a^3 x^3+226 a^2 x^2+136 i a x-48\right )}{192 x^4 \sqrt [4]{1+i a x}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.54, size = 252, normalized size = 1.08 \[ \frac {{\left (-4934 i \, a^{4} x^{4} - 1042 \, a^{3} x^{3} - 452 i \, a^{2} x^{2} + 272 \, a x + 96 i\right )} \sqrt {a^{2} x^{2} + 1} \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} - 1425 \, {\left (a^{5} x^{5} - i \, a^{4} x^{4}\right )} \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} + 1\right ) + {\left (1425 i \, a^{5} x^{5} + 1425 \, a^{4} x^{4}\right )} \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} + i\right ) + {\left (-1425 i \, a^{5} x^{5} - 1425 \, a^{4} x^{4}\right )} \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} - i\right ) + 1425 \, {\left (a^{5} x^{5} - i \, a^{4} x^{4}\right )} \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} - 1\right )}{384 \, a x^{5} - 384 i \, x^{4}} \]
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 \[ \int \frac {1}{\left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )^{\frac {5}{2}} x^{5}}\, dx \]
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 \[ \int \frac {1}{x^{5} \left (\frac {i \, a x + 1}{\sqrt {a^{2} x^{2} + 1}}\right )^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {1}{x^5\,{\left (\frac {1+a\,x\,1{}\mathrm {i}}{\sqrt {a^2\,x^2+1}}\right )}^{5/2}} \,d x \]
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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