Optimal. Leaf size=52 \[ \frac {4 i \sqrt {a^2 x^2+1}}{-a x+i}-\tanh ^{-1}\left (\sqrt {a^2 x^2+1}\right )+i \sinh ^{-1}(a x) \]
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Rubi [A] time = 0.57, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5060, 6742, 215, 266, 63, 208, 651} \[ \frac {4 i \sqrt {a^2 x^2+1}}{-a x+i}-\tanh ^{-1}\left (\sqrt {a^2 x^2+1}\right )+i \sinh ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 215
Rule 266
Rule 651
Rule 5060
Rule 6742
Rubi steps
\begin {align*} \int \frac {e^{-3 i \tan ^{-1}(a x)}}{x} \, dx &=\int \frac {(1-i a x)^2}{x (1+i a x) \sqrt {1+a^2 x^2}} \, dx\\ &=\int \left (\frac {i a}{\sqrt {1+a^2 x^2}}+\frac {1}{x \sqrt {1+a^2 x^2}}-\frac {4 a}{(-i+a x) \sqrt {1+a^2 x^2}}\right ) \, dx\\ &=(i a) \int \frac {1}{\sqrt {1+a^2 x^2}} \, dx-(4 a) \int \frac {1}{(-i+a x) \sqrt {1+a^2 x^2}} \, dx+\int \frac {1}{x \sqrt {1+a^2 x^2}} \, dx\\ &=\frac {4 i \sqrt {1+a^2 x^2}}{i-a x}+i \sinh ^{-1}(a x)+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1+a^2 x}} \, dx,x,x^2\right )\\ &=\frac {4 i \sqrt {1+a^2 x^2}}{i-a x}+i \sinh ^{-1}(a x)+\frac {\operatorname {Subst}\left (\int \frac {1}{-\frac {1}{a^2}+\frac {x^2}{a^2}} \, dx,x,\sqrt {1+a^2 x^2}\right )}{a^2}\\ &=\frac {4 i \sqrt {1+a^2 x^2}}{i-a x}+i \sinh ^{-1}(a x)-\tanh ^{-1}\left (\sqrt {1+a^2 x^2}\right )\\ \end {align*}
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Mathematica [A] time = 0.04, size = 55, normalized size = 1.06 \[ -\frac {4 i \sqrt {a^2 x^2+1}}{a x-i}-\log \left (\sqrt {a^2 x^2+1}+1\right )+i \sinh ^{-1}(a x)+\log (x) \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.44, size = 100, normalized size = 1.92 \[ \frac {-4 i \, a x - {\left (a x - i\right )} \log \left (-a x + \sqrt {a^{2} x^{2} + 1} + 1\right ) + {\left (-i \, a x - 1\right )} \log \left (-a x + \sqrt {a^{2} x^{2} + 1}\right ) + {\left (a x - i\right )} \log \left (-a x + \sqrt {a^{2} x^{2} + 1} - 1\right ) - 4 i \, \sqrt {a^{2} x^{2} + 1} - 4}{a x - i} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {undef} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.17, size = 257, normalized size = 4.94 \[ \frac {\left (a^{2} x^{2}+1\right )^{\frac {3}{2}}}{3}+\sqrt {a^{2} x^{2}+1}-\arctanh \left (\frac {1}{\sqrt {a^{2} x^{2}+1}}\right )+\frac {i \left (\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )\right )^{\frac {5}{2}}}{a^{3} \left (x -\frac {i}{a}\right )^{3}}-\frac {\left (\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )\right )^{\frac {5}{2}}}{a^{2} \left (x -\frac {i}{a}\right )^{2}}+\frac {2 \left (\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )\right )^{\frac {3}{2}}}{3}+i a \sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}\, x +\frac {i a \ln \left (\frac {i a +\left (x -\frac {i}{a}\right ) a^{2}}{\sqrt {a^{2}}}+\sqrt {\left (x -\frac {i}{a}\right )^{2} a^{2}+2 i a \left (x -\frac {i}{a}\right )}\right )}{\sqrt {a^{2}}} \]
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 \[ \int \frac {{\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (i \, a x + 1\right )}^{3} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.43, size = 74, normalized size = 1.42 \[ -\mathrm {atanh}\left (\sqrt {a^2\,x^2+1}\right )+\frac {a\,\mathrm {asinh}\left (x\,\sqrt {a^2}\right )\,1{}\mathrm {i}}{\sqrt {a^2}}+\frac {a\,\sqrt {a^2\,x^2+1}\,4{}\mathrm {i}}{\left (-x\,\sqrt {a^2}+\frac {\sqrt {a^2}\,1{}\mathrm {i}}{a}\right )\,\sqrt {a^2}} \]
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 \[ i \left (\int \frac {\sqrt {a^{2} x^{2} + 1}}{a^{3} x^{4} - 3 i a^{2} x^{3} - 3 a x^{2} + i x}\, dx + \int \frac {a^{2} x^{2} \sqrt {a^{2} x^{2} + 1}}{a^{3} x^{4} - 3 i a^{2} x^{3} - 3 a x^{2} + i x}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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