Integrand size = 14, antiderivative size = 51 \[ \int \frac {e^{3 i \arctan (a x)}}{x} \, dx=\frac {4 i \sqrt {1+a^2 x^2}}{i+a x}-i \text {arcsinh}(a x)-\text {arctanh}\left (\sqrt {1+a^2 x^2}\right ) \]
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Time = 0.49 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5168, 6874, 221, 272, 65, 214, 665} \[ \int \frac {e^{3 i \arctan (a x)}}{x} \, dx=-\text {arctanh}\left (\sqrt {a^2 x^2+1}\right )+\frac {4 i \sqrt {a^2 x^2+1}}{a x+i}-i \text {arcsinh}(a x) \]
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Rule 65
Rule 214
Rule 221
Rule 272
Rule 665
Rule 5168
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \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 \text {arcsinh}(a x)+\frac {1}{2} \text {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 \text {arcsinh}(a x)+\frac {\text {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 \text {arcsinh}(a x)-\text {arctanh}\left (\sqrt {1+a^2 x^2}\right ) \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.08 \[ \int \frac {e^{3 i \arctan (a x)}}{x} \, dx=\frac {4 i \sqrt {1+a^2 x^2}}{i+a x}-i \text {arcsinh}(a x)+\log (x)-\log \left (1+\sqrt {1+a^2 x^2}\right ) \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 100 vs. \(2 (44 ) = 88\).
Time = 0.27 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.98
method | result | size |
default | \(\frac {4}{\sqrt {a^{2} x^{2}+1}}-\operatorname {arctanh}\left (\frac {1}{\sqrt {a^{2} x^{2}+1}}\right )+\frac {3 i a x}{\sqrt {a^{2} x^{2}+1}}-i a^{3} \left (-\frac {x}{a^{2} \sqrt {a^{2} x^{2}+1}}+\frac {\ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}+1}\right )}{a^{2} \sqrt {a^{2}}}\right )\) | \(101\) |
meijerg | \(\frac {-\sqrt {\pi }+\frac {\sqrt {\pi }}{\sqrt {a^{2} x^{2}+1}}-\sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {a^{2} x^{2}+1}}{2}\right )+\frac {\left (2-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (a^{2}\right )\right ) \sqrt {\pi }}{2}}{\sqrt {\pi }}+\frac {3 i a x}{\sqrt {a^{2} x^{2}+1}}-\frac {3 \left (\sqrt {\pi }-\frac {\sqrt {\pi }}{\sqrt {a^{2} x^{2}+1}}\right )}{\sqrt {\pi }}-\frac {i a \left (-\frac {\sqrt {\pi }\, x \left (a^{2}\right )^{\frac {3}{2}}}{a^{2} \sqrt {a^{2} x^{2}+1}}+\frac {\sqrt {\pi }\, \left (a^{2}\right )^{\frac {3}{2}} \operatorname {arcsinh}\left (a x \right )}{a^{3}}\right )}{\sqrt {\pi }\, \sqrt {a^{2}}}\) | \(162\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 100 vs. \(2 (41) = 82\).
Time = 0.27 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.96 \[ \int \frac {e^{3 i \arctan (a x)}}{x} \, dx=\frac {4 i \, a x - {\left (a x + i\right )} \log \left (-a x + \sqrt {a^{2} x^{2} + 1} + 1\right ) + {\left (i \, a x - 1\right )} \log \left (-a x + \sqrt {a^{2} x^{2} + 1}\right ) + {\left (a x + i\right )} \log \left (-a x + \sqrt {a^{2} x^{2} + 1} - 1\right ) + 4 i \, \sqrt {a^{2} x^{2} + 1} - 4}{a x + i} \]
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\[ \int \frac {e^{3 i \arctan (a x)}}{x} \, dx=- i \left (\int \frac {i}{a^{2} x^{3} \sqrt {a^{2} x^{2} + 1} + x \sqrt {a^{2} x^{2} + 1}}\, dx + \int \left (- \frac {3 a x}{a^{2} x^{3} \sqrt {a^{2} x^{2} + 1} + x \sqrt {a^{2} x^{2} + 1}}\right )\, dx + \int \frac {a^{3} x^{3}}{a^{2} x^{3} \sqrt {a^{2} x^{2} + 1} + x \sqrt {a^{2} x^{2} + 1}}\, dx + \int \left (- \frac {3 i a^{2} x^{2}}{a^{2} x^{3} \sqrt {a^{2} x^{2} + 1} + x \sqrt {a^{2} x^{2} + 1}}\right )\, dx\right ) \]
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Time = 0.20 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.90 \[ \int \frac {e^{3 i \arctan (a x)}}{x} \, dx=\frac {4 i \, a x}{\sqrt {a^{2} x^{2} + 1}} + \frac {4}{\sqrt {a^{2} x^{2} + 1}} - i \, \operatorname {arsinh}\left (a x\right ) - \operatorname {arsinh}\left (\frac {1}{a {\left | x \right |}}\right ) \]
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\[ \int \frac {e^{3 i \arctan (a x)}}{x} \, dx=\int { \frac {{\left (i \, a x + 1\right )}^{3}}{{\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x} \,d x } \]
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Time = 0.51 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.43 \[ \int \frac {e^{3 i \arctan (a x)}}{x} \, dx=-\mathrm {atanh}\left (\sqrt {a^2\,x^2+1}\right )-\frac {a\,\mathrm {asinh}\left (x\,\sqrt {a^2}\right )\,1{}\mathrm {i}}{\sqrt {a^2}}+\frac {a\,\sqrt {a^2\,x^2+1}\,4{}\mathrm {i}}{\left (x\,\sqrt {a^2}+\frac {\sqrt {a^2}\,1{}\mathrm {i}}{a}\right )\,\sqrt {a^2}} \]
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