Integrand size = 16, antiderivative size = 373 \[ \int e^{\frac {5}{2} i \arctan (a x)} x^3 \, dx=\frac {475 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{64 a^4}-\frac {4 i x^3 (1+i a x)^{5/4}}{a \sqrt [4]{1-i a x}}-\frac {17 x^2 (1-i a x)^{3/4} (1+i a x)^{5/4}}{4 a^2}-\frac {i (521 i-452 a x) (1-i a x)^{3/4} (1+i a x)^{5/4}}{96 a^4}-\frac {475 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{64 \sqrt {2} a^4}+\frac {475 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{64 \sqrt {2} a^4}+\frac {475 \log \left (1+\frac {\sqrt {1-i a x}}{\sqrt {1+i a x}}-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{128 \sqrt {2} a^4}-\frac {475 \log \left (1+\frac {\sqrt {1-i a x}}{\sqrt {1+i a x}}+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{128 \sqrt {2} a^4} \]
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Time = 0.18 (sec) , antiderivative size = 373, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.812, Rules used = {5170, 99, 158, 152, 52, 65, 338, 303, 1176, 631, 210, 1179, 642} \[ \int e^{\frac {5}{2} i \arctan (a x)} x^3 \, dx=-\frac {475 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{64 \sqrt {2} a^4}+\frac {475 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{64 \sqrt {2} a^4}-\frac {i (-452 a x+521 i) (1-i a x)^{3/4} (1+i a x)^{5/4}}{96 a^4}+\frac {475 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{64 a^4}+\frac {475 \log \left (\frac {\sqrt {1-i a x}}{\sqrt {1+i a x}}-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+1\right )}{128 \sqrt {2} a^4}-\frac {475 \log \left (\frac {\sqrt {1-i a x}}{\sqrt {1+i a x}}+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+1\right )}{128 \sqrt {2} a^4}-\frac {17 x^2 (1-i a x)^{3/4} (1+i a x)^{5/4}}{4 a^2}-\frac {4 i x^3 (1+i a x)^{5/4}}{a \sqrt [4]{1-i a x}} \]
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Rule 52
Rule 65
Rule 99
Rule 152
Rule 158
Rule 210
Rule 303
Rule 338
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 5170
Rubi steps \begin{align*} \text {integral}& = \int \frac {x^3 (1+i a x)^{5/4}}{(1-i a x)^{5/4}} \, dx \\ & = -\frac {4 i x^3 (1+i a x)^{5/4}}{a \sqrt [4]{1-i a x}}+\frac {(4 i) \int \frac {x^2 \sqrt [4]{1+i a x} \left (3+\frac {17 i a x}{4}\right )}{\sqrt [4]{1-i a x}} \, dx}{a} \\ & = -\frac {4 i x^3 (1+i a x)^{5/4}}{a \sqrt [4]{1-i a x}}-\frac {17 x^2 (1-i a x)^{3/4} (1+i a x)^{5/4}}{4 a^2}+\frac {i \int \frac {x \sqrt [4]{1+i a x} \left (-\frac {17 i a}{2}+\frac {113 a^2 x}{8}\right )}{\sqrt [4]{1-i a x}} \, dx}{a^3} \\ & = -\frac {4 i x^3 (1+i a x)^{5/4}}{a \sqrt [4]{1-i a x}}-\frac {17 x^2 (1-i a x)^{3/4} (1+i a x)^{5/4}}{4 a^2}-\frac {i (521 i-452 a x) (1-i a x)^{3/4} (1+i a x)^{5/4}}{96 a^4}-\frac {(475 i) \int \frac {\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}} \, dx}{64 a^3} \\ & = \frac {475 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{64 a^4}-\frac {4 i x^3 (1+i a x)^{5/4}}{a \sqrt [4]{1-i a x}}-\frac {17 x^2 (1-i a x)^{3/4} (1+i a x)^{5/4}}{4 a^2}-\frac {i (521 i-452 a x) (1-i a x)^{3/4} (1+i a x)^{5/4}}{96 a^4}-\frac {(475 i) \int \frac {1}{\sqrt [4]{1-i a x} (1+i a x)^{3/4}} \, dx}{128 a^3} \\ & = \frac {475 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{64 a^4}-\frac {4 i x^3 (1+i a x)^{5/4}}{a \sqrt [4]{1-i a x}}-\frac {17 x^2 (1-i a x)^{3/4} (1+i a x)^{5/4}}{4 a^2}-\frac {i (521 i-452 a x) (1-i a x)^{3/4} (1+i a x)^{5/4}}{96 a^4}+\frac {475 \text {Subst}\left (\int \frac {x^2}{\left (2-x^4\right )^{3/4}} \, dx,x,\sqrt [4]{1-i a x}\right )}{32 a^4} \\ & = \frac {475 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{64 a^4}-\frac {4 i x^3 (1+i a x)^{5/4}}{a \sqrt [4]{1-i a x}}-\frac {17 x^2 (1-i a x)^{3/4} (1+i a x)^{5/4}}{4 a^2}-\frac {i (521 i-452 a x) (1-i a x)^{3/4} (1+i a x)^{5/4}}{96 a^4}+\frac {475 \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 )}{32 a^4} \\ & = \frac {475 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{64 a^4}-\frac {4 i x^3 (1+i a x)^{5/4}}{a \sqrt [4]{1-i a x}}-\frac {17 x^2 (1-i a x)^{3/4} (1+i a x)^{5/4}}{4 a^2}-\frac {i (521 i-452 a x) (1-i a x)^{3/4} (1+i a x)^{5/4}}{96 a^4}-\frac {475 \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 )}{64 a^4}+\frac {475 \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 )}{64 a^4} \\ & = \frac {475 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{64 a^4}-\frac {4 i x^3 (1+i a x)^{5/4}}{a \sqrt [4]{1-i a x}}-\frac {17 x^2 (1-i a x)^{3/4} (1+i a x)^{5/4}}{4 a^2}-\frac {i (521 i-452 a x) (1-i a x)^{3/4} (1+i a x)^{5/4}}{96 a^4}+\frac {475 \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 )}{128 a^4}+\frac {475 \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 )}{128 a^4}+\frac {475 \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 )}{128 \sqrt {2} a^4}+\frac {475 \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 )}{128 \sqrt {2} a^4} \\ & = \frac {475 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{64 a^4}-\frac {4 i x^3 (1+i a x)^{5/4}}{a \sqrt [4]{1-i a x}}-\frac {17 x^2 (1-i a x)^{3/4} (1+i a x)^{5/4}}{4 a^2}-\frac {i (521 i-452 a x) (1-i a x)^{3/4} (1+i a x)^{5/4}}{96 a^4}+\frac {475 \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 )}{128 \sqrt {2} a^4}-\frac {475 \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 )}{128 \sqrt {2} a^4}+\frac {475 \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 )}{64 \sqrt {2} a^4}-\frac {475 \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 )}{64 \sqrt {2} a^4} \\ & = \frac {475 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{64 a^4}-\frac {4 i x^3 (1+i a x)^{5/4}}{a \sqrt [4]{1-i a x}}-\frac {17 x^2 (1-i a x)^{3/4} (1+i a x)^{5/4}}{4 a^2}-\frac {i (521 i-452 a x) (1-i a x)^{3/4} (1+i a x)^{5/4}}{96 a^4}-\frac {475 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{64 \sqrt {2} a^4}+\frac {475 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{64 \sqrt {2} a^4}+\frac {475 \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 )}{128 \sqrt {2} a^4}-\frac {475 \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 )}{128 \sqrt {2} a^4} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.03 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.26 \[ \int e^{\frac {5}{2} i \arctan (a x)} x^3 \, dx=\frac {-\sqrt [4]{1+i a x} (-i+a x)^2 \left (59-5 i a x+6 a^2 x^2\right )+380 \sqrt [4]{2} (1-i a x) \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},\frac {3}{4},\frac {7}{4},\frac {1}{2} (1-i a x)\right )}{24 a^4 \sqrt [4]{1-i a x}} \]
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\[\int {\left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}^{\frac {5}{2}} x^{3}d x\]
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none
Time = 0.27 (sec) , antiderivative size = 251, normalized size of antiderivative = 0.67 \[ \int e^{\frac {5}{2} i \arctan (a x)} x^3 \, dx=-\frac {96 \, a^{4} \sqrt {\frac {225625 i}{4096 \, a^{8}}} \log \left (\frac {64}{475} \, a^{4} \sqrt {\frac {225625 i}{4096 \, a^{8}}} + \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}\right ) - 96 \, a^{4} \sqrt {\frac {225625 i}{4096 \, a^{8}}} \log \left (-\frac {64}{475} \, a^{4} \sqrt {\frac {225625 i}{4096 \, a^{8}}} + \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}\right ) + 96 \, a^{4} \sqrt {-\frac {225625 i}{4096 \, a^{8}}} \log \left (\frac {64}{475} \, a^{4} \sqrt {-\frac {225625 i}{4096 \, a^{8}}} + \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}\right ) - 96 \, a^{4} \sqrt {-\frac {225625 i}{4096 \, a^{8}}} \log \left (-\frac {64}{475} \, a^{4} \sqrt {-\frac {225625 i}{4096 \, a^{8}}} + \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}\right ) + {\left (48 \, a^{4} x^{4} - 136 i \, a^{3} x^{3} - 226 \, a^{2} x^{2} + 521 i \, a x - 2467\right )} \sqrt {\frac {i \, \sqrt {a^{2} x^{2} + 1}}{a x + i}}}{192 \, a^{4}} \]
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Timed out. \[ \int e^{\frac {5}{2} i \arctan (a x)} x^3 \, dx=\text {Timed out} \]
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\[ \int e^{\frac {5}{2} i \arctan (a x)} x^3 \, dx=\int { x^{3} \left (\frac {i \, a x + 1}{\sqrt {a^{2} x^{2} + 1}}\right )^{\frac {5}{2}} \,d x } \]
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Exception generated. \[ \int e^{\frac {5}{2} i \arctan (a x)} x^3 \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int e^{\frac {5}{2} i \arctan (a x)} x^3 \, dx=\int x^3\,{\left (\frac {1+a\,x\,1{}\mathrm {i}}{\sqrt {a^2\,x^2+1}}\right )}^{5/2} \,d x \]
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