Optimal. Leaf size=202 \[ -\frac {11}{64} a^4 \tan ^{-1}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-\frac {11}{64} a^4 \tanh ^{-1}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )+\frac {83 i a^3 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{192 x}+\frac {29 a^2 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{96 x^2}-\frac {(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{4 x^4}-\frac {7 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{24 x^3} \]
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Rubi [A] time = 0.08, antiderivative size = 202, normalized size of antiderivative = 1.00, number of steps used = 10, 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} \[ \frac {29 a^2 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{96 x^2}+\frac {83 i a^3 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{192 x}-\frac {11}{64} a^4 \tan ^{-1}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-\frac {11}{64} a^4 \tanh ^{-1}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-\frac {7 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{24 x^3}-\frac {(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{4 x^4} \]
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^5} \, dx &=\int \frac {\sqrt [4]{1+i a x}}{x^5 \sqrt [4]{1-i a x}} \, dx\\ &=-\frac {(1-i a x)^{3/4} \sqrt [4]{1+i a x}}{4 x^4}+\frac {1}{4} \int \frac {\frac {7 i a}{2}-3 a^2 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}}{4 x^4}-\frac {7 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{24 x^3}-\frac {1}{12} \int \frac {\frac {29 a^2}{4}+7 i a^3 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}}{4 x^4}-\frac {7 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{24 x^3}+\frac {29 a^2 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{96 x^2}+\frac {1}{24} \int \frac {-\frac {83 i a^3}{8}+\frac {29 a^4 x}{4}}{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}}{4 x^4}-\frac {7 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{24 x^3}+\frac {29 a^2 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{96 x^2}+\frac {83 i a^3 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{192 x}-\frac {1}{24} \int -\frac {33 a^4}{16 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}}{4 x^4}-\frac {7 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{24 x^3}+\frac {29 a^2 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{96 x^2}+\frac {83 i a^3 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{192 x}+\frac {1}{128} \left (11 a^4\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}}{4 x^4}-\frac {7 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{24 x^3}+\frac {29 a^2 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{96 x^2}+\frac {83 i a^3 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{192 x}+\frac {1}{32} \left (11 a^4\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}}{4 x^4}-\frac {7 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{24 x^3}+\frac {29 a^2 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{96 x^2}+\frac {83 i a^3 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{192 x}-\frac {1}{64} \left (11 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 (11 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-i a x)^{3/4} \sqrt [4]{1+i a x}}{4 x^4}-\frac {7 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{24 x^3}+\frac {29 a^2 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{96 x^2}+\frac {83 i a^3 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{192 x}-\frac {11}{64} a^4 \tan ^{-1}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-\frac {11}{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 = 0.03, size = 99, normalized size = 0.49 \[ -\frac {(1-i a x)^{3/4} \left (22 a^4 x^4 \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {a x+i}{i-a x}\right )+83 a^4 x^4-141 i a^3 x^3-114 a^2 x^2+104 i a x+48\right )}{192 x^4 (1+i a x)^{3/4}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.43, size = 192, normalized size = 0.95 \[ -\frac {33 \, a^{4} x^{4} \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} + 1\right ) + 33 i \, a^{4} x^{4} \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} + i\right ) - 33 i \, a^{4} x^{4} \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} - i\right ) - 33 \, a^{4} x^{4} \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} - 1\right ) - {\left (166 \, a^{4} x^{4} + 50 i \, a^{3} x^{3} + 4 \, a^{2} x^{2} - 16 i \, a x - 96\right )} \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}}{384 \, x^{4}} \]
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 = 0.16, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}}}{x^{5}}\, dx \]
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 \[ \int \frac {\sqrt {\frac {i \, a x + 1}{\sqrt {a^{2} x^{2} + 1}}}}{x^{5}}\,{d x} \]
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 \[ \int \frac {\sqrt {\frac {1+a\,x\,1{}\mathrm {i}}{\sqrt {a^2\,x^2+1}}}}{x^5} \,d x \]
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 \[ \int \frac {\sqrt {\frac {i \left (a x - i\right )}{\sqrt {a^{2} x^{2} + 1}}}}{x^{5}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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