Integrand size = 16, antiderivative size = 92 \[ \int \frac {e^{\frac {3}{2} i \arctan (a x)}}{x^2} \, dx=-\frac {\sqrt [4]{1-i a x} (1+i a x)^{3/4}}{x}+3 i a \arctan \left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-3 i a \text {arctanh}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right ) \]
[Out]
Time = 0.02 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {5170, 96, 95, 304, 209, 212} \[ \int \frac {e^{\frac {3}{2} i \arctan (a x)}}{x^2} \, dx=3 i a \arctan \left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-3 i a \text {arctanh}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-\frac {\sqrt [4]{1-i a x} (1+i a x)^{3/4}}{x} \]
[In]
[Out]
Rule 95
Rule 96
Rule 209
Rule 212
Rule 304
Rule 5170
Rubi steps \begin{align*} \text {integral}& = \int \frac {(1+i a x)^{3/4}}{x^2 (1-i a x)^{3/4}} \, dx \\ & = -\frac {\sqrt [4]{1-i a x} (1+i a x)^{3/4}}{x}+\frac {1}{2} (3 i a) \int \frac {1}{x (1-i a x)^{3/4} \sqrt [4]{1+i a x}} \, dx \\ & = -\frac {\sqrt [4]{1-i a x} (1+i a x)^{3/4}}{x}+(6 i a) \text {Subst}\left (\int \frac {x^2}{-1+x^4} \, dx,x,\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right ) \\ & = -\frac {\sqrt [4]{1-i a x} (1+i a x)^{3/4}}{x}-(3 i a) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )+(3 i a) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right ) \\ & = -\frac {\sqrt [4]{1-i a x} (1+i a x)^{3/4}}{x}+3 i a \arctan \left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right )-3 i a \text {arctanh}\left (\frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}}\right ) \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.01 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.74 \[ \int \frac {e^{\frac {3}{2} i \arctan (a x)}}{x^2} \, dx=-\frac {i \sqrt [4]{1-i a x} \left (-i+a x+6 a x \operatorname {Hypergeometric2F1}\left (\frac {1}{4},1,\frac {5}{4},\frac {i+a x}{i-a x}\right )\right )}{x \sqrt [4]{1+i a x}} \]
[In]
[Out]
\[\int \frac {{\left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}^{\frac {3}{2}}}{x^{2}}d x\]
[In]
[Out]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 157 vs. \(2 (64) = 128\).
Time = 0.28 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.71 \[ \int \frac {e^{\frac {3}{2} i \arctan (a x)}}{x^2} \, dx=\frac {-3 i \, a x \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} + 1\right ) - 3 \, a x \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} + i\right ) + 3 \, a x \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} - i\right ) + 3 i \, a x \log \left (\sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}} - 1\right ) - 2 \, \sqrt {a^{2} x^{2} + 1} \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}}{2 \, x} \]
[In]
[Out]
\[ \int \frac {e^{\frac {3}{2} i \arctan (a x)}}{x^2} \, dx=\int \frac {\left (\frac {i \left (a x - i\right )}{\sqrt {a^{2} x^{2} + 1}}\right )^{\frac {3}{2}}}{x^{2}}\, dx \]
[In]
[Out]
\[ \int \frac {e^{\frac {3}{2} i \arctan (a x)}}{x^2} \, dx=\int { \frac {\left (\frac {i \, a x + 1}{\sqrt {a^{2} x^{2} + 1}}\right )^{\frac {3}{2}}}{x^{2}} \,d x } \]
[In]
[Out]
Exception generated. \[ \int \frac {e^{\frac {3}{2} i \arctan (a x)}}{x^2} \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
Timed out. \[ \int \frac {e^{\frac {3}{2} i \arctan (a x)}}{x^2} \, dx=\int \frac {{\left (\frac {1+a\,x\,1{}\mathrm {i}}{\sqrt {a^2\,x^2+1}}\right )}^{3/2}}{x^2} \,d x \]
[In]
[Out]