Optimal. Leaf size=51 \[ \frac {4 i \sqrt {a^2 x^2+1}}{a x+i}-\tanh ^{-1}\left (\sqrt {a^2 x^2+1}\right )-i \sinh ^{-1}(a x) \]
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Rubi [A] time = 0.66, antiderivative size = 51, 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\\ &=-\left ((i a) \int \frac {1}{\sqrt {1+a^2 x^2}} \, dx\right )-(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 = 1.08 \[ \frac {4 i \sqrt {a^2 x^2+1}}{a x+i}-\log \left (\sqrt {a^2 x^2+1}+1\right )-i \sinh ^{-1}(a x)+\log (x) \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.45, size = 100, normalized size = 1.96 \[ \frac {4 i \, a x - {\left (a x + i\right )} \log \left (-a x + \sqrt {a^{2} x^{2} + 1} + 1\right ) + {\left (i \, a x - 1\right )} \log \left (-a x + \sqrt {a^{2} x^{2} + 1}\right ) + {\left (a x + i\right )} \log \left (-a x + \sqrt {a^{2} x^{2} + 1} - 1\right ) + 4 i \, \sqrt {a^{2} x^{2} + 1} - 4}{a x + i} \]
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 [A] time = 0.14, size = 77, normalized size = 1.51 \[ \frac {4 i a x}{\sqrt {a^{2} x^{2}+1}}-\frac {i a \ln \left (\frac {x \,a^{2}}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}+1}\right )}{\sqrt {a^{2}}}+\frac {4}{\sqrt {a^{2} x^{2}+1}}-\arctanh \left (\frac {1}{\sqrt {a^{2} x^{2}+1}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 46, normalized size = 0.90 \[ \frac {4 i \, a x}{\sqrt {a^{2} x^{2} + 1}} + \frac {4}{\sqrt {a^{2} x^{2} + 1}} - i \, \operatorname {arsinh}\left (a x\right ) - \operatorname {arsinh}\left (\frac {1}{a {\left | x \right |}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.43, size = 73, normalized size = 1.43 \[ -\mathrm {atanh}\left (\sqrt {a^2\,x^2+1}\right )-\frac {a\,\mathrm {asinh}\left (x\,\sqrt {a^2}\right )\,1{}\mathrm {i}}{\sqrt {a^2}}+\frac {a\,\sqrt {a^2\,x^2+1}\,4{}\mathrm {i}}{\left (x\,\sqrt {a^2}+\frac {\sqrt {a^2}\,1{}\mathrm {i}}{a}\right )\,\sqrt {a^2}} \]
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 \[ - i \left (\int \frac {i}{a^{2} x^{3} \sqrt {a^{2} x^{2} + 1} + x \sqrt {a^{2} x^{2} + 1}}\, dx + \int \left (- \frac {3 a x}{a^{2} x^{3} \sqrt {a^{2} x^{2} + 1} + x \sqrt {a^{2} x^{2} + 1}}\right )\, dx + \int \frac {a^{3} x^{3}}{a^{2} x^{3} \sqrt {a^{2} x^{2} + 1} + x \sqrt {a^{2} x^{2} + 1}}\, dx + \int \left (- \frac {3 i a^{2} x^{2}}{a^{2} x^{3} \sqrt {a^{2} x^{2} + 1} + x \sqrt {a^{2} x^{2} + 1}}\right )\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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