Optimal. Leaf size=132 \[ \frac {9}{4} a^2 \tan ^{-1}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )+\frac {9}{4} a^2 \tanh ^{-1}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-\frac {(1-i a x)^{7/4} \sqrt [4]{1+i a x}}{2 x^2}+\frac {3 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{4 x} \]
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Rubi [A] time = 0.04, antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {5062, 96, 94, 93, 212, 206, 203} \[ \frac {9}{4} a^2 \tan ^{-1}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )+\frac {9}{4} a^2 \tanh ^{-1}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-\frac {(1-i a x)^{7/4} \sqrt [4]{1+i a x}}{2 x^2}+\frac {3 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{4 x} \]
Antiderivative was successfully verified.
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Rule 93
Rule 94
Rule 96
Rule 203
Rule 206
Rule 212
Rule 5062
Rubi steps
\begin {align*} \int \frac {e^{-\frac {3}{2} i \tan ^{-1}(a x)}}{x^3} \, dx &=\int \frac {(1-i a x)^{3/4}}{x^3 (1+i a x)^{3/4}} \, dx\\ &=-\frac {(1-i a x)^{7/4} \sqrt [4]{1+i a x}}{2 x^2}-\frac {1}{4} (3 i a) \int \frac {(1-i a x)^{3/4}}{x^2 (1+i a x)^{3/4}} \, dx\\ &=\frac {3 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{4 x}-\frac {(1-i a x)^{7/4} \sqrt [4]{1+i a x}}{2 x^2}-\frac {1}{8} \left (9 a^2\right ) \int \frac {1}{x \sqrt [4]{1-i a x} (1+i a x)^{3/4}} \, dx\\ &=\frac {3 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{4 x}-\frac {(1-i a x)^{7/4} \sqrt [4]{1+i a x}}{2 x^2}-\frac {1}{2} \left (9 a^2\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 {3 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{4 x}-\frac {(1-i a x)^{7/4} \sqrt [4]{1+i a x}}{2 x^2}+\frac {1}{4} \left (9 a^2\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}{4} \left (9 a^2\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 {3 i a (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{4 x}-\frac {(1-i a x)^{7/4} \sqrt [4]{1+i a x}}{2 x^2}+\frac {9}{4} a^2 \tan ^{-1}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )+\frac {9}{4} a^2 \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 = 81, normalized size = 0.61 \[ \frac {(1-i a x)^{3/4} \left (6 a^2 x^2 \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {a x+i}{i-a x}\right )-5 a^2 x^2+3 i a x-2\right )}{4 x^2 (1+i a x)^{3/4}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.42, size = 175, normalized size = 1.33 \[ \frac {9 \, a^{2} x^{2} \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} + 1\right ) + 9 i \, a^{2} x^{2} \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} + i\right ) - 9 i \, a^{2} x^{2} \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} - i\right ) - 9 \, a^{2} x^{2} \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} - 1\right ) + {\left (10 \, a^{2} x^{2} + 14 i \, a x - 4\right )} \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}}{8 \, x^{2}} \]
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 = 0.19, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )^{\frac {3}{2}} x^{3}}\, dx \]
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 \[ \int \frac {1}{x^{3} \left (\frac {i \, a x + 1}{\sqrt {a^{2} x^{2} + 1}}\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{x^3\,{\left (\frac {1+a\,x\,1{}\mathrm {i}}{\sqrt {a^2\,x^2+1}}\right )}^{3/2}} \,d x \]
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 \[ \int \frac {1}{x^{3} \left (\frac {i \left (a x - i\right )}{\sqrt {a^{2} x^{2} + 1}}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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