Integrand size = 14, antiderivative size = 689 \[ \int e^{\frac {1}{4} i \arctan (a x)} x \, dx=\frac {(1-i a x)^{7/8} \sqrt [8]{1+i a x}}{8 a^2}+\frac {(1-i a x)^{7/8} (1+i a x)^{9/8}}{2 a^2}-\frac {\sqrt {2+\sqrt {2}} \arctan \left (\frac {\sqrt {2-\sqrt {2}}-\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{\sqrt {2+\sqrt {2}}}\right )}{32 a^2}-\frac {\sqrt {2-\sqrt {2}} \arctan \left (\frac {\sqrt {2+\sqrt {2}}-\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{\sqrt {2-\sqrt {2}}}\right )}{32 a^2}+\frac {\sqrt {2+\sqrt {2}} \arctan \left (\frac {\sqrt {2-\sqrt {2}}+\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{\sqrt {2+\sqrt {2}}}\right )}{32 a^2}+\frac {\sqrt {2-\sqrt {2}} \arctan \left (\frac {\sqrt {2+\sqrt {2}}+\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{\sqrt {2-\sqrt {2}}}\right )}{32 a^2}+\frac {\sqrt {2-\sqrt {2}} \log \left (1+\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{64 a^2}-\frac {\sqrt {2-\sqrt {2}} \log \left (1+\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{64 a^2}+\frac {\sqrt {2+\sqrt {2}} \log \left (1+\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{64 a^2}-\frac {\sqrt {2+\sqrt {2}} \log \left (1+\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{64 a^2} \]
[Out]
Time = 0.36 (sec) , antiderivative size = 689, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {5170, 81, 52, 65, 338, 305, 1136, 1183, 648, 632, 210, 642} \[ \int e^{\frac {1}{4} i \arctan (a x)} x \, dx=-\frac {\sqrt {2+\sqrt {2}} \arctan \left (\frac {\sqrt {2-\sqrt {2}}-\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{\sqrt {2+\sqrt {2}}}\right )}{32 a^2}-\frac {\sqrt {2-\sqrt {2}} \arctan \left (\frac {\sqrt {2+\sqrt {2}}-\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{\sqrt {2-\sqrt {2}}}\right )}{32 a^2}+\frac {\sqrt {2+\sqrt {2}} \arctan \left (\frac {\sqrt {2-\sqrt {2}}+\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{\sqrt {2+\sqrt {2}}}\right )}{32 a^2}+\frac {\sqrt {2-\sqrt {2}} \arctan \left (\frac {\sqrt {2+\sqrt {2}}+\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{\sqrt {2-\sqrt {2}}}\right )}{32 a^2}+\frac {(1-i a x)^{7/8} (1+i a x)^{9/8}}{2 a^2}+\frac {(1-i a x)^{7/8} \sqrt [8]{1+i a x}}{8 a^2}+\frac {\sqrt {2-\sqrt {2}} \log \left (\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}+1\right )}{64 a^2}-\frac {\sqrt {2-\sqrt {2}} \log \left (\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}+1\right )}{64 a^2}+\frac {\sqrt {2+\sqrt {2}} \log \left (\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}+1\right )}{64 a^2}-\frac {\sqrt {2+\sqrt {2}} \log \left (\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}+1\right )}{64 a^2} \]
[In]
[Out]
Rule 52
Rule 65
Rule 81
Rule 210
Rule 305
Rule 338
Rule 632
Rule 642
Rule 648
Rule 1136
Rule 1183
Rule 5170
Rubi steps \begin{align*} \text {integral}& = \int \frac {x \sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}} \, dx \\ & = \frac {(1-i a x)^{7/8} (1+i a x)^{9/8}}{2 a^2}-\frac {i \int \frac {\sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}} \, dx}{8 a} \\ & = \frac {(1-i a x)^{7/8} \sqrt [8]{1+i a x}}{8 a^2}+\frac {(1-i a x)^{7/8} (1+i a x)^{9/8}}{2 a^2}-\frac {i \int \frac {1}{\sqrt [8]{1-i a x} (1+i a x)^{7/8}} \, dx}{32 a} \\ & = \frac {(1-i a x)^{7/8} \sqrt [8]{1+i a x}}{8 a^2}+\frac {(1-i a x)^{7/8} (1+i a x)^{9/8}}{2 a^2}+\frac {\text {Subst}\left (\int \frac {x^6}{\left (2-x^8\right )^{7/8}} \, dx,x,\sqrt [8]{1-i a x}\right )}{4 a^2} \\ & = \frac {(1-i a x)^{7/8} \sqrt [8]{1+i a x}}{8 a^2}+\frac {(1-i a x)^{7/8} (1+i a x)^{9/8}}{2 a^2}+\frac {\text {Subst}\left (\int \frac {x^6}{1+x^8} \, dx,x,\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{4 a^2} \\ & = \frac {(1-i a x)^{7/8} \sqrt [8]{1+i a x}}{8 a^2}+\frac {(1-i a x)^{7/8} (1+i a x)^{9/8}}{2 a^2}+\frac {\text {Subst}\left (\int \frac {x^4}{1-\sqrt {2} x^2+x^4} \, dx,x,\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{8 \sqrt {2} a^2}-\frac {\text {Subst}\left (\int \frac {x^4}{1+\sqrt {2} x^2+x^4} \, dx,x,\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{8 \sqrt {2} a^2} \\ & = \frac {(1-i a x)^{7/8} \sqrt [8]{1+i a x}}{8 a^2}+\frac {(1-i a x)^{7/8} (1+i a x)^{9/8}}{2 a^2}-\frac {\text {Subst}\left (\int \frac {1-\sqrt {2} x^2}{1-\sqrt {2} x^2+x^4} \, dx,x,\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{8 \sqrt {2} a^2}+\frac {\text {Subst}\left (\int \frac {1+\sqrt {2} x^2}{1+\sqrt {2} x^2+x^4} \, dx,x,\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{8 \sqrt {2} a^2} \\ & = \frac {(1-i a x)^{7/8} \sqrt [8]{1+i a x}}{8 a^2}+\frac {(1-i a x)^{7/8} (1+i a x)^{9/8}}{2 a^2}+\frac {\text {Subst}\left (\int \frac {\sqrt {2-\sqrt {2}}-\left (1-\sqrt {2}\right ) x}{1-\sqrt {2-\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{16 \sqrt {2 \left (2-\sqrt {2}\right )} a^2}+\frac {\text {Subst}\left (\int \frac {\sqrt {2-\sqrt {2}}+\left (1-\sqrt {2}\right ) x}{1+\sqrt {2-\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{16 \sqrt {2 \left (2-\sqrt {2}\right )} a^2}-\frac {\text {Subst}\left (\int \frac {\sqrt {2+\sqrt {2}}-\left (1+\sqrt {2}\right ) x}{1-\sqrt {2+\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{16 \sqrt {2 \left (2+\sqrt {2}\right )} a^2}-\frac {\text {Subst}\left (\int \frac {\sqrt {2+\sqrt {2}}+\left (1+\sqrt {2}\right ) x}{1+\sqrt {2+\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{16 \sqrt {2 \left (2+\sqrt {2}\right )} a^2} \\ & = \frac {(1-i a x)^{7/8} \sqrt [8]{1+i a x}}{8 a^2}+\frac {(1-i a x)^{7/8} (1+i a x)^{9/8}}{2 a^2}+\frac {\sqrt {\frac {1}{2} \left (3-2 \sqrt {2}\right )} \text {Subst}\left (\int \frac {1}{1-\sqrt {2+\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{32 a^2}+\frac {\sqrt {\frac {1}{2} \left (3-2 \sqrt {2}\right )} \text {Subst}\left (\int \frac {1}{1+\sqrt {2+\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{32 a^2}+\frac {\sqrt {2-\sqrt {2}} \text {Subst}\left (\int \frac {-\sqrt {2-\sqrt {2}}+2 x}{1-\sqrt {2-\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{64 a^2}-\frac {\sqrt {2-\sqrt {2}} \text {Subst}\left (\int \frac {\sqrt {2-\sqrt {2}}+2 x}{1+\sqrt {2-\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{64 a^2}+\frac {\sqrt {2+\sqrt {2}} \text {Subst}\left (\int \frac {-\sqrt {2+\sqrt {2}}+2 x}{1-\sqrt {2+\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{64 a^2}-\frac {\sqrt {2+\sqrt {2}} \text {Subst}\left (\int \frac {\sqrt {2+\sqrt {2}}+2 x}{1+\sqrt {2+\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{64 a^2}+\frac {\sqrt {\frac {1}{2} \left (3+2 \sqrt {2}\right )} \text {Subst}\left (\int \frac {1}{1-\sqrt {2-\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{32 a^2}+\frac {\sqrt {\frac {1}{2} \left (3+2 \sqrt {2}\right )} \text {Subst}\left (\int \frac {1}{1+\sqrt {2-\sqrt {2}} x+x^2} \, dx,x,\frac {\sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{32 a^2} \\ & = \frac {(1-i a x)^{7/8} \sqrt [8]{1+i a x}}{8 a^2}+\frac {(1-i a x)^{7/8} (1+i a x)^{9/8}}{2 a^2}+\frac {\sqrt {2-\sqrt {2}} \log \left (1+\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{64 a^2}-\frac {\sqrt {2-\sqrt {2}} \log \left (1+\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{64 a^2}+\frac {\sqrt {2+\sqrt {2}} \log \left (1+\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{64 a^2}-\frac {\sqrt {2+\sqrt {2}} \log \left (1+\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{64 a^2}-\frac {\sqrt {\frac {1}{2} \left (3-2 \sqrt {2}\right )} \text {Subst}\left (\int \frac {1}{-2+\sqrt {2}-x^2} \, dx,x,-\sqrt {2+\sqrt {2}}+\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{16 a^2}-\frac {\sqrt {\frac {1}{2} \left (3-2 \sqrt {2}\right )} \text {Subst}\left (\int \frac {1}{-2+\sqrt {2}-x^2} \, dx,x,\sqrt {2+\sqrt {2}}+\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{16 a^2}-\frac {\sqrt {\frac {1}{2} \left (3+2 \sqrt {2}\right )} \text {Subst}\left (\int \frac {1}{-2-\sqrt {2}-x^2} \, dx,x,-\sqrt {2-\sqrt {2}}+\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{16 a^2}-\frac {\sqrt {\frac {1}{2} \left (3+2 \sqrt {2}\right )} \text {Subst}\left (\int \frac {1}{-2-\sqrt {2}-x^2} \, dx,x,\sqrt {2-\sqrt {2}}+\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{16 a^2} \\ & = \frac {(1-i a x)^{7/8} \sqrt [8]{1+i a x}}{8 a^2}+\frac {(1-i a x)^{7/8} (1+i a x)^{9/8}}{2 a^2}-\frac {\sqrt {2+\sqrt {2}} \arctan \left (\frac {\sqrt {2-\sqrt {2}}-\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{\sqrt {2+\sqrt {2}}}\right )}{32 a^2}-\frac {\sqrt {2-\sqrt {2}} \arctan \left (\frac {\sqrt {2+\sqrt {2}}-\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{\sqrt {2-\sqrt {2}}}\right )}{32 a^2}+\frac {\sqrt {2+\sqrt {2}} \arctan \left (\frac {\sqrt {2-\sqrt {2}}+\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{\sqrt {2+\sqrt {2}}}\right )}{32 a^2}+\frac {\sqrt {2-\sqrt {2}} \arctan \left (\frac {\sqrt {2+\sqrt {2}}+\frac {2 \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}}{\sqrt {2-\sqrt {2}}}\right )}{32 a^2}+\frac {\sqrt {2-\sqrt {2}} \log \left (1+\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}-\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{64 a^2}-\frac {\sqrt {2-\sqrt {2}} \log \left (1+\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+\frac {\sqrt {2-\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{64 a^2}+\frac {\sqrt {2+\sqrt {2}} \log \left (1+\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}-\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{64 a^2}-\frac {\sqrt {2+\sqrt {2}} \log \left (1+\frac {\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{1-i a x}}{\sqrt [8]{1+i a x}}\right )}{64 a^2} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.02 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.09 \[ \int e^{\frac {1}{4} i \arctan (a x)} x \, dx=\frac {(1-i a x)^{7/8} \left (7 (1+i a x)^{9/8}+2 \sqrt [8]{2} \operatorname {Hypergeometric2F1}\left (-\frac {1}{8},\frac {7}{8},\frac {15}{8},\frac {1}{2} (1-i a x)\right )\right )}{14 a^2} \]
[In]
[Out]
\[\int {\left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}^{\frac {1}{4}} x d x\]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 428, normalized size of antiderivative = 0.62 \[ \int e^{\frac {1}{4} i \arctan (a x)} x \, dx=-\frac {8 \, a^{2} \left (\frac {i}{1048576 \, a^{8}}\right )^{\frac {1}{4}} \log \left (32 \, a^{2} \left (\frac {i}{1048576 \, a^{8}}\right )^{\frac {1}{4}} + \left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}}\right ) + 8 i \, a^{2} \left (\frac {i}{1048576 \, a^{8}}\right )^{\frac {1}{4}} \log \left (32 i \, a^{2} \left (\frac {i}{1048576 \, a^{8}}\right )^{\frac {1}{4}} + \left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}}\right ) - 8 i \, a^{2} \left (\frac {i}{1048576 \, a^{8}}\right )^{\frac {1}{4}} \log \left (-32 i \, a^{2} \left (\frac {i}{1048576 \, a^{8}}\right )^{\frac {1}{4}} + \left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}}\right ) - 8 \, a^{2} \left (\frac {i}{1048576 \, a^{8}}\right )^{\frac {1}{4}} \log \left (-32 \, a^{2} \left (\frac {i}{1048576 \, a^{8}}\right )^{\frac {1}{4}} + \left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}}\right ) + 8 \, a^{2} \left (-\frac {i}{1048576 \, a^{8}}\right )^{\frac {1}{4}} \log \left (32 \, a^{2} \left (-\frac {i}{1048576 \, a^{8}}\right )^{\frac {1}{4}} + \left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}}\right ) + 8 i \, a^{2} \left (-\frac {i}{1048576 \, a^{8}}\right )^{\frac {1}{4}} \log \left (32 i \, a^{2} \left (-\frac {i}{1048576 \, a^{8}}\right )^{\frac {1}{4}} + \left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}}\right ) - 8 i \, a^{2} \left (-\frac {i}{1048576 \, a^{8}}\right )^{\frac {1}{4}} \log \left (-32 i \, a^{2} \left (-\frac {i}{1048576 \, a^{8}}\right )^{\frac {1}{4}} + \left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}}\right ) - 8 \, a^{2} \left (-\frac {i}{1048576 \, a^{8}}\right )^{\frac {1}{4}} \log \left (-32 \, a^{2} \left (-\frac {i}{1048576 \, a^{8}}\right )^{\frac {1}{4}} + \left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}}\right ) - {\left (4 \, a^{2} x^{2} - i \, a x + 5\right )} \left (\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}\right )^{\frac {1}{4}}}{8 \, a^{2}} \]
[In]
[Out]
\[ \int e^{\frac {1}{4} i \arctan (a x)} x \, dx=\int x \sqrt [4]{\frac {i \left (a x - i\right )}{\sqrt {a^{2} x^{2} + 1}}}\, dx \]
[In]
[Out]
\[ \int e^{\frac {1}{4} i \arctan (a x)} x \, dx=\int { x \left (\frac {i \, a x + 1}{\sqrt {a^{2} x^{2} + 1}}\right )^{\frac {1}{4}} \,d x } \]
[In]
[Out]
Exception generated. \[ \int e^{\frac {1}{4} i \arctan (a x)} x \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
Timed out. \[ \int e^{\frac {1}{4} i \arctan (a x)} x \, dx=\int x\,{\left (\frac {1+a\,x\,1{}\mathrm {i}}{\sqrt {a^2\,x^2+1}}\right )}^{1/4} \,d x \]
[In]
[Out]