Integrand size = 24, antiderivative size = 41 \[ \int \frac {e^{2 i \arctan (a x)}}{\sqrt {1+a^2 x^2}} \, dx=-\frac {2 i \sqrt {1+i a x}}{a \sqrt {1-i a x}}-\frac {\text {arcsinh}(a x)}{a} \]
[Out]
Time = 0.03 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {5181, 49, 41, 221} \[ \int \frac {e^{2 i \arctan (a x)}}{\sqrt {1+a^2 x^2}} \, dx=-\frac {\text {arcsinh}(a x)}{a}-\frac {2 i \sqrt {1+i a x}}{a \sqrt {1-i a x}} \]
[In]
[Out]
Rule 41
Rule 49
Rule 221
Rule 5181
Rubi steps \begin{align*} \text {integral}& = \int \frac {\sqrt {1+i a x}}{(1-i a x)^{3/2}} \, dx \\ & = -\frac {2 i \sqrt {1+i a x}}{a \sqrt {1-i a x}}-\int \frac {1}{\sqrt {1-i a x} \sqrt {1+i a x}} \, dx \\ & = -\frac {2 i \sqrt {1+i a x}}{a \sqrt {1-i a x}}-\int \frac {1}{\sqrt {1+a^2 x^2}} \, dx \\ & = -\frac {2 i \sqrt {1+i a x}}{a \sqrt {1-i a x}}-\frac {\text {arcsinh}(a x)}{a} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.27 \[ \int \frac {e^{2 i \arctan (a x)}}{\sqrt {1+a^2 x^2}} \, dx=-\frac {2 i \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}+\arcsin \left (\frac {\sqrt {1-i a x}}{\sqrt {2}}\right )\right )}{a} \]
[In]
[Out]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 86 vs. \(2 (34 ) = 68\).
Time = 0.26 (sec) , antiderivative size = 87, normalized size of antiderivative = 2.12
method | result | size |
default | \(\frac {x}{\sqrt {a^{2} x^{2}+1}}-a^{2} \left (-\frac {x}{a^{2} \sqrt {a^{2} x^{2}+1}}+\frac {\ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}+1}\right )}{a^{2} \sqrt {a^{2}}}\right )-\frac {2 i}{a \sqrt {a^{2} x^{2}+1}}\) | \(87\) |
meijerg | \(\frac {x}{\sqrt {a^{2} x^{2}+1}}+\frac {2 i \left (\sqrt {\pi }-\frac {\sqrt {\pi }}{\sqrt {a^{2} x^{2}+1}}\right )}{a \sqrt {\pi }}-\frac {-\frac {\sqrt {\pi }\, x \left (a^{2}\right )^{\frac {3}{2}}}{a^{2} \sqrt {a^{2} x^{2}+1}}+\frac {\sqrt {\pi }\, \left (a^{2}\right )^{\frac {3}{2}} \operatorname {arcsinh}\left (a x \right )}{a^{3}}}{\sqrt {\pi }\, \sqrt {a^{2}}}\) | \(96\) |
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.32 \[ \int \frac {e^{2 i \arctan (a x)}}{\sqrt {1+a^2 x^2}} \, dx=\frac {2 \, a x + {\left (a x + i\right )} \log \left (-a x + \sqrt {a^{2} x^{2} + 1}\right ) + 2 \, \sqrt {a^{2} x^{2} + 1} + 2 i}{a^{2} x + i \, a} \]
[In]
[Out]
\[ \int \frac {e^{2 i \arctan (a x)}}{\sqrt {1+a^2 x^2}} \, dx=- \int \frac {a^{2} x^{2}}{a^{2} x^{2} \sqrt {a^{2} x^{2} + 1} + \sqrt {a^{2} x^{2} + 1}}\, dx - \int \left (- \frac {2 i a x}{a^{2} x^{2} \sqrt {a^{2} x^{2} + 1} + \sqrt {a^{2} x^{2} + 1}}\right )\, dx - \int \left (- \frac {1}{a^{2} x^{2} \sqrt {a^{2} x^{2} + 1} + \sqrt {a^{2} x^{2} + 1}}\right )\, dx \]
[In]
[Out]
none
Time = 0.19 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.98 \[ \int \frac {e^{2 i \arctan (a x)}}{\sqrt {1+a^2 x^2}} \, dx=\frac {2 \, x}{\sqrt {a^{2} x^{2} + 1}} - \frac {\operatorname {arsinh}\left (a x\right )}{a} - \frac {2 i}{\sqrt {a^{2} x^{2} + 1} a} \]
[In]
[Out]
\[ \int \frac {e^{2 i \arctan (a x)}}{\sqrt {1+a^2 x^2}} \, dx=\int { \frac {{\left (i \, a x + 1\right )}^{2}}{{\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}}} \,d x } \]
[In]
[Out]
Time = 0.58 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.34 \[ \int \frac {e^{2 i \arctan (a x)}}{\sqrt {1+a^2 x^2}} \, dx=-\frac {\mathrm {asinh}\left (x\,\sqrt {a^2}\right )}{\sqrt {a^2}}+\frac {2\,\sqrt {a^2\,x^2+1}}{\left (x\,\sqrt {a^2}+\frac {\sqrt {a^2}\,1{}\mathrm {i}}{a}\right )\,\sqrt {a^2}} \]
[In]
[Out]