Integrand size = 14, antiderivative size = 63 \[ \int \frac {e^{3 i \arctan (a x)}}{x^2} \, dx=-\frac {\sqrt {1+a^2 x^2}}{x}-\frac {4 a \sqrt {1+a^2 x^2}}{i+a x}-3 i a \text {arctanh}\left (\sqrt {1+a^2 x^2}\right ) \]
[Out]
Time = 0.41 (sec) , antiderivative size = 63, 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, 270, 272, 65, 214, 665} \[ \int \frac {e^{3 i \arctan (a x)}}{x^2} \, dx=-3 i a \text {arctanh}\left (\sqrt {a^2 x^2+1}\right )-\frac {4 a \sqrt {a^2 x^2+1}}{a x+i}-\frac {\sqrt {a^2 x^2+1}}{x} \]
[In]
[Out]
Rule 65
Rule 214
Rule 270
Rule 272
Rule 665
Rule 5168
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {(1+i a x)^2}{x^2 (1-i a x) \sqrt {1+a^2 x^2}} \, dx \\ & = \int \left (\frac {1}{x^2 \sqrt {1+a^2 x^2}}+\frac {3 i a}{x \sqrt {1+a^2 x^2}}-\frac {4 i a^2}{(i+a x) \sqrt {1+a^2 x^2}}\right ) \, dx \\ & = (3 i a) \int \frac {1}{x \sqrt {1+a^2 x^2}} \, dx-\left (4 i a^2\right ) \int \frac {1}{(i+a x) \sqrt {1+a^2 x^2}} \, dx+\int \frac {1}{x^2 \sqrt {1+a^2 x^2}} \, dx \\ & = -\frac {\sqrt {1+a^2 x^2}}{x}-\frac {4 a \sqrt {1+a^2 x^2}}{i+a x}+\frac {1}{2} (3 i a) \text {Subst}\left (\int \frac {1}{x \sqrt {1+a^2 x}} \, dx,x,x^2\right ) \\ & = -\frac {\sqrt {1+a^2 x^2}}{x}-\frac {4 a \sqrt {1+a^2 x^2}}{i+a x}+\frac {(3 i) \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} \\ & = -\frac {\sqrt {1+a^2 x^2}}{x}-\frac {4 a \sqrt {1+a^2 x^2}}{i+a x}-3 i a \text {arctanh}\left (\sqrt {1+a^2 x^2}\right ) \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.97 \[ \int \frac {e^{3 i \arctan (a x)}}{x^2} \, dx=\sqrt {1+a^2 x^2} \left (-\frac {1}{x}-\frac {4 a}{i+a x}\right )+3 i a \log (x)-3 i a \log \left (1+\sqrt {1+a^2 x^2}\right ) \]
[In]
[Out]
Time = 0.34 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.27
method | result | size |
default | \(-\frac {1}{x \sqrt {a^{2} x^{2}+1}}-\frac {5 a^{2} x}{\sqrt {a^{2} x^{2}+1}}+\frac {i a}{\sqrt {a^{2} x^{2}+1}}+3 i a \left (\frac {1}{\sqrt {a^{2} x^{2}+1}}-\operatorname {arctanh}\left (\frac {1}{\sqrt {a^{2} x^{2}+1}}\right )\right )\) | \(80\) |
risch | \(-\frac {\sqrt {a^{2} x^{2}+1}}{x}+i a \left (-3 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {a^{2} x^{2}+1}}\right )+\frac {4 i \sqrt {\left (x +\frac {i}{a}\right )^{2} a^{2}-2 i a \left (x +\frac {i}{a}\right )}}{a \left (x +\frac {i}{a}\right )}\right )\) | \(82\) |
meijerg | \(-\frac {2 a^{2} x^{2}+1}{x \sqrt {a^{2} x^{2}+1}}+\frac {3 i a \left (-\sqrt {\pi }+\frac {\sqrt {\pi }}{\sqrt {a^{2} x^{2}+1}}-\sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {a^{2} x^{2}+1}}{2}\right )+\frac {\left (2-2 \ln \left (2\right )+2 \ln \left (x \right )+\ln \left (a^{2}\right )\right ) \sqrt {\pi }}{2}\right )}{\sqrt {\pi }}-\frac {3 a^{2} x}{\sqrt {a^{2} x^{2}+1}}-\frac {i a \left (\sqrt {\pi }-\frac {\sqrt {\pi }}{\sqrt {a^{2} x^{2}+1}}\right )}{\sqrt {\pi }}\) | \(140\) |
[In]
[Out]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 109 vs. \(2 (53) = 106\).
Time = 0.26 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.73 \[ \int \frac {e^{3 i \arctan (a x)}}{x^2} \, dx=-\frac {5 \, a^{2} x^{2} + 5 i \, a x + 3 \, {\left (i \, a^{2} x^{2} - a x\right )} \log \left (-a x + \sqrt {a^{2} x^{2} + 1} + 1\right ) + 3 \, {\left (-i \, a^{2} x^{2} + a x\right )} \log \left (-a x + \sqrt {a^{2} x^{2} + 1} - 1\right ) + \sqrt {a^{2} x^{2} + 1} {\left (5 \, a x + i\right )}}{a x^{2} + i \, x} \]
[In]
[Out]
\[ \int \frac {e^{3 i \arctan (a x)}}{x^2} \, dx=- i \left (\int \frac {i}{a^{2} x^{4} \sqrt {a^{2} x^{2} + 1} + x^{2} \sqrt {a^{2} x^{2} + 1}}\, dx + \int \left (- \frac {3 a x}{a^{2} x^{4} \sqrt {a^{2} x^{2} + 1} + x^{2} \sqrt {a^{2} x^{2} + 1}}\right )\, dx + \int \frac {a^{3} x^{3}}{a^{2} x^{4} \sqrt {a^{2} x^{2} + 1} + x^{2} \sqrt {a^{2} x^{2} + 1}}\, dx + \int \left (- \frac {3 i a^{2} x^{2}}{a^{2} x^{4} \sqrt {a^{2} x^{2} + 1} + x^{2} \sqrt {a^{2} x^{2} + 1}}\right )\, dx\right ) \]
[In]
[Out]
none
Time = 0.21 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.95 \[ \int \frac {e^{3 i \arctan (a x)}}{x^2} \, dx=-\frac {5 \, a^{2} x}{\sqrt {a^{2} x^{2} + 1}} - 3 i \, a \operatorname {arsinh}\left (\frac {1}{a {\left | x \right |}}\right ) + \frac {4 i \, a}{\sqrt {a^{2} x^{2} + 1}} - \frac {1}{\sqrt {a^{2} x^{2} + 1} x} \]
[In]
[Out]
\[ \int \frac {e^{3 i \arctan (a x)}}{x^2} \, dx=\int { \frac {{\left (i \, a x + 1\right )}^{3}}{{\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x^{2}} \,d x } \]
[In]
[Out]
Time = 0.06 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.19 \[ \int \frac {e^{3 i \arctan (a x)}}{x^2} \, dx=-a\,\mathrm {atanh}\left (\sqrt {a^2\,x^2+1}\right )\,3{}\mathrm {i}-\frac {\sqrt {a^2\,x^2+1}}{x}-\frac {4\,a^2\,\sqrt {a^2\,x^2+1}}{\left (x\,\sqrt {a^2}+\frac {\sqrt {a^2}\,1{}\mathrm {i}}{a}\right )\,\sqrt {a^2}} \]
[In]
[Out]