Integrand size = 12, antiderivative size = 299 \[ \int e^{\frac {5}{2} i \arctan (a x)} \, dx=-\frac {5 i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{a}-\frac {4 i (1+i a x)^{5/4}}{a \sqrt [4]{1-i a x}}+\frac {5 i \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{\sqrt {2} a}-\frac {5 i \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{\sqrt {2} a}-\frac {5 i \log \left (1+\frac {\sqrt {1-i a x}}{\sqrt {1+i a x}}-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{2 \sqrt {2} a}+\frac {5 i \log \left (1+\frac {\sqrt {1-i a x}}{\sqrt {1+i a x}}+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{2 \sqrt {2} a} \]
[Out]
Time = 0.13 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.917, Rules used = {5169, 49, 52, 65, 338, 303, 1176, 631, 210, 1179, 642} \[ \int e^{\frac {5}{2} i \arctan (a x)} \, dx=\frac {5 i \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{\sqrt {2} a}-\frac {5 i \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{\sqrt {2} a}-\frac {4 i (1+i a x)^{5/4}}{a \sqrt [4]{1-i a x}}-\frac {5 i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{a}-\frac {5 i \log \left (\frac {\sqrt {1-i a x}}{\sqrt {1+i a x}}-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+1\right )}{2 \sqrt {2} a}+\frac {5 i \log \left (\frac {\sqrt {1-i a x}}{\sqrt {1+i a x}}+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+1\right )}{2 \sqrt {2} a} \]
[In]
[Out]
Rule 49
Rule 52
Rule 65
Rule 210
Rule 303
Rule 338
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 5169
Rubi steps \begin{align*} \text {integral}& = \int \frac {(1+i a x)^{5/4}}{(1-i a x)^{5/4}} \, dx \\ & = -\frac {4 i (1+i a x)^{5/4}}{a \sqrt [4]{1-i a x}}-5 \int \frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}} \, dx \\ & = -\frac {5 i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{a}-\frac {4 i (1+i a x)^{5/4}}{a \sqrt [4]{1-i a x}}-\frac {5}{2} \int \frac {1}{\sqrt [4]{1-i a x} (1+i a x)^{3/4}} \, dx \\ & = -\frac {5 i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{a}-\frac {4 i (1+i a x)^{5/4}}{a \sqrt [4]{1-i a x}}-\frac {(10 i) \text {Subst}\left (\int \frac {x^2}{\left (2-x^4\right )^{3/4}} \, dx,x,\sqrt [4]{1-i a x}\right )}{a} \\ & = -\frac {5 i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{a}-\frac {4 i (1+i a x)^{5/4}}{a \sqrt [4]{1-i a x}}-\frac {(10 i) \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 )}{a} \\ & = -\frac {5 i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{a}-\frac {4 i (1+i a x)^{5/4}}{a \sqrt [4]{1-i a x}}+\frac {(5 i) \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{a}-\frac {(5 i) \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{a} \\ & = -\frac {5 i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{a}-\frac {4 i (1+i a x)^{5/4}}{a \sqrt [4]{1-i a x}}-\frac {(5 i) \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{2 a}-\frac {(5 i) \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{2 a}-\frac {(5 i) \text {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{2 \sqrt {2} a}-\frac {(5 i) \text {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{2 \sqrt {2} a} \\ & = -\frac {5 i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{a}-\frac {4 i (1+i a x)^{5/4}}{a \sqrt [4]{1-i a x}}-\frac {5 i \log \left (1+\frac {\sqrt {1-i a x}}{\sqrt {1+i a x}}-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{2 \sqrt {2} a}+\frac {5 i \log \left (1+\frac {\sqrt {1-i a x}}{\sqrt {1+i a x}}+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{2 \sqrt {2} a}-\frac {(5 i) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{\sqrt {2} a}+\frac {(5 i) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{\sqrt {2} a} \\ & = -\frac {5 i (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{a}-\frac {4 i (1+i a x)^{5/4}}{a \sqrt [4]{1-i a x}}+\frac {5 i \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{\sqrt {2} a}-\frac {5 i \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{\sqrt {2} a}-\frac {5 i \log \left (1+\frac {\sqrt {1-i a x}}{\sqrt {1+i a x}}-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{2 \sqrt {2} a}+\frac {5 i \log \left (1+\frac {\sqrt {1-i a x}}{\sqrt {1+i a x}}+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{2 \sqrt {2} a} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.03 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.14 \[ \int e^{\frac {5}{2} i \arctan (a x)} \, dx=-\frac {8 i e^{\frac {9}{2} i \arctan (a x)} \operatorname {Hypergeometric2F1}\left (2,\frac {9}{4},\frac {13}{4},-e^{2 i \arctan (a x)}\right )}{9 a} \]
[In]
[Out]
\[\int {\left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}^{\frac {5}{2}}d x\]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.70 \[ \int e^{\frac {5}{2} i \arctan (a x)} \, dx=-\frac {a \sqrt {\frac {25 i}{a^{2}}} \log \left (\frac {1}{5} i \, a \sqrt {\frac {25 i}{a^{2}}} + \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}\right ) - a \sqrt {\frac {25 i}{a^{2}}} \log \left (-\frac {1}{5} i \, a \sqrt {\frac {25 i}{a^{2}}} + \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}\right ) + a \sqrt {-\frac {25 i}{a^{2}}} \log \left (\frac {1}{5} i \, a \sqrt {-\frac {25 i}{a^{2}}} + \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}\right ) - a \sqrt {-\frac {25 i}{a^{2}}} \log \left (-\frac {1}{5} i \, a \sqrt {-\frac {25 i}{a^{2}}} + \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}\right ) + 2 \, {\left (a x + 9 i\right )} \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}}{2 \, a} \]
[In]
[Out]
\[ \int e^{\frac {5}{2} i \arctan (a x)} \, dx=\int \left (\frac {i a x + 1}{\sqrt {a^{2} x^{2} + 1}}\right )^{\frac {5}{2}}\, dx \]
[In]
[Out]
\[ \int e^{\frac {5}{2} i \arctan (a x)} \, dx=\int { \left (\frac {i \, a x + 1}{\sqrt {a^{2} x^{2} + 1}}\right )^{\frac {5}{2}} \,d x } \]
[In]
[Out]
Exception generated. \[ \int e^{\frac {5}{2} i \arctan (a x)} \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
Timed out. \[ \int e^{\frac {5}{2} i \arctan (a x)} \, dx=\int {\left (\frac {1+a\,x\,1{}\mathrm {i}}{\sqrt {a^2\,x^2+1}}\right )}^{5/2} \,d x \]
[In]
[Out]