Optimal. Leaf size=240 \[ -\frac {31}{128} i a^5 \tan ^{-1}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-\frac {31}{128} i a^5 \tanh ^{-1}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-\frac {611 a^4 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{1920 x}+\frac {269 i a^3 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{960 x^2}+\frac {11 a^2 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{48 x^3}-\frac {(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{5 x^5}-\frac {9 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{40 x^4} \]
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Rubi [A] time = 0.10, antiderivative size = 240, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5062, 99, 151, 12, 93, 212, 206, 203} \[ \frac {269 i a^3 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{960 x^2}+\frac {11 a^2 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{48 x^3}-\frac {611 a^4 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{1920 x}-\frac {31}{128} i a^5 \tan ^{-1}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-\frac {31}{128} i a^5 \tanh ^{-1}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-\frac {9 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{40 x^4}-\frac {(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{5 x^5} \]
Antiderivative was successfully verified.
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Rule 12
Rule 93
Rule 99
Rule 151
Rule 203
Rule 206
Rule 212
Rule 5062
Rubi steps
\begin {align*} \int \frac {e^{\frac {1}{2} i \tan ^{-1}(a x)}}{x^6} \, dx &=\int \frac {\sqrt [4]{1+i a x}}{x^6 \sqrt [4]{1-i a x}} \, dx\\ &=-\frac {(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{5 x^5}+\frac {1}{5} \int \frac {\frac {9 i a}{2}-4 a^2 x}{x^5 \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}}{5 x^5}-\frac {9 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{40 x^4}-\frac {1}{20} \int \frac {\frac {55 a^2}{4}+\frac {27}{2} i a^3 x}{x^4 \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}}{5 x^5}-\frac {9 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{40 x^4}+\frac {11 a^2 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{48 x^3}+\frac {1}{60} \int \frac {-\frac {269 i a^3}{8}+\frac {55 a^4 x}{2}}{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}}{5 x^5}-\frac {9 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{40 x^4}+\frac {11 a^2 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{48 x^3}+\frac {269 i a^3 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{960 x^2}-\frac {1}{120} \int \frac {-\frac {611 a^4}{16}-\frac {269}{8} i a^5 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}}{5 x^5}-\frac {9 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{40 x^4}+\frac {11 a^2 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{48 x^3}+\frac {269 i a^3 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{960 x^2}-\frac {611 a^4 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{1920 x}+\frac {1}{120} \int \frac {465 i a^5}{32 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}}{5 x^5}-\frac {9 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{40 x^4}+\frac {11 a^2 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{48 x^3}+\frac {269 i a^3 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{960 x^2}-\frac {611 a^4 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{1920 x}+\frac {1}{256} \left (31 i a^5\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}}{5 x^5}-\frac {9 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{40 x^4}+\frac {11 a^2 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{48 x^3}+\frac {269 i a^3 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{960 x^2}-\frac {611 a^4 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{1920 x}+\frac {1}{64} \left (31 i a^5\right ) \operatorname {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}}{5 x^5}-\frac {9 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{40 x^4}+\frac {11 a^2 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{48 x^3}+\frac {269 i a^3 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{960 x^2}-\frac {611 a^4 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{1920 x}-\frac {1}{128} \left (31 i a^5\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}{128} \left (31 i a^5\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-i a x)^{3/4} \sqrt [4]{1+i a x}}{5 x^5}-\frac {9 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{40 x^4}+\frac {11 a^2 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{48 x^3}+\frac {269 i a^3 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{960 x^2}-\frac {611 a^4 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{1920 x}-\frac {31}{128} i a^5 \tan ^{-1}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-\frac {31}{128} i a^5 \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 = 111, normalized size = 0.46 \[ \frac {(1-i a x)^{3/4} \left (-310 i a^5 x^5 \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {a x+i}{i-a x}\right )-611 i a^5 x^5-1149 a^4 x^4+978 i a^3 x^3+872 a^2 x^2-816 i a x-384\right )}{1920 x^5 (1+i a x)^{3/4}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.45, size = 199, normalized size = 0.83 \[ \frac {-465 i \, a^{5} x^{5} \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} + 1\right ) + 465 \, a^{5} x^{5} \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} + i\right ) - 465 \, a^{5} x^{5} \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} - i\right ) + 465 i \, a^{5} x^{5} \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} - 1\right ) + {\left (1222 i \, a^{5} x^{5} - 146 \, a^{4} x^{4} + 196 i \, a^{3} x^{3} + 16 \, a^{2} x^{2} - 96 i \, a x - 768\right )} \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}}{3840 \, x^{5}} \]
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 = 0.16, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}}}{x^{6}}\, dx \]
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 \[ \int \frac {\sqrt {\frac {i \, a x + 1}{\sqrt {a^{2} x^{2} + 1}}}}{x^{6}}\,{d x} \]
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 \[ \int \frac {\sqrt {\frac {1+a\,x\,1{}\mathrm {i}}{\sqrt {a^2\,x^2+1}}}}{x^6} \,d x \]
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 \[ \int \frac {\sqrt {\frac {i \left (a x - i\right )}{\sqrt {a^{2} x^{2} + 1}}}}{x^{6}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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