Integrand size = 16, antiderivative size = 324 \[ \int e^{-3 i \arctan (a+b x)} x^4 \, dx=\frac {2 i x^4 (1-i a-i b x)^{3/2}}{b \sqrt {1+i a+i b x}}+\frac {3 \left (19 i-68 a-88 i a^2+48 a^3+8 i a^4\right ) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{8 b^5}-\frac {3 (17 i-16 a) x^2 (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{20 b^3}-\frac {11 x^3 (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{5 b^2}+\frac {i (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x} \left (163+458 i a-422 a^2-112 i a^3-2 \left (61 i-118 a-52 i a^2\right ) b x\right )}{40 b^5}-\frac {3 \left (19+68 i a-88 a^2-48 i a^3+8 a^4\right ) \text {arcsinh}(a+b x)}{8 b^5} \]
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Time = 0.20 (sec) , antiderivative size = 324, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5203, 99, 158, 152, 52, 55, 633, 221} \[ \int e^{-3 i \arctan (a+b x)} x^4 \, dx=\frac {i (-i a-i b x+1)^{3/2} \sqrt {i a+i b x+1} \left (-112 i a^3-2 \left (-52 i a^2-118 a+61 i\right ) b x-422 a^2+458 i a+163\right )}{40 b^5}-\frac {3 \left (8 a^4-48 i a^3-88 a^2+68 i a+19\right ) \text {arcsinh}(a+b x)}{8 b^5}+\frac {3 \left (8 i a^4+48 a^3-88 i a^2-68 a+19 i\right ) \sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}{8 b^5}-\frac {3 (-16 a+17 i) x^2 (-i a-i b x+1)^{3/2} \sqrt {i a+i b x+1}}{20 b^3}-\frac {11 x^3 (-i a-i b x+1)^{3/2} \sqrt {i a+i b x+1}}{5 b^2}+\frac {2 i x^4 (-i a-i b x+1)^{3/2}}{b \sqrt {i a+i b x+1}} \]
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Rule 52
Rule 55
Rule 99
Rule 152
Rule 158
Rule 221
Rule 633
Rule 5203
Rubi steps \begin{align*} \text {integral}& = \int \frac {x^4 (1-i a-i b x)^{3/2}}{(1+i a+i b x)^{3/2}} \, dx \\ & = \frac {2 i x^4 (1-i a-i b x)^{3/2}}{b \sqrt {1+i a+i b x}}-\frac {(2 i) \int \frac {x^3 \sqrt {1-i a-i b x} \left (4 (1-i a)-\frac {11 i b x}{2}\right )}{\sqrt {1+i a+i b x}} \, dx}{b} \\ & = \frac {2 i x^4 (1-i a-i b x)^{3/2}}{b \sqrt {1+i a+i b x}}-\frac {11 x^3 (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{5 b^2}-\frac {(2 i) \int \frac {x^2 \sqrt {1-i a-i b x} \left (\frac {33}{2} (1+i a) (i+a) b+\frac {3}{2} (17+16 i a) b^2 x\right )}{\sqrt {1+i a+i b x}} \, dx}{5 b^3} \\ & = \frac {2 i x^4 (1-i a-i b x)^{3/2}}{b \sqrt {1+i a+i b x}}-\frac {3 (17 i-16 a) x^2 (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{20 b^3}-\frac {11 x^3 (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{5 b^2}-\frac {i \int \frac {x \sqrt {1-i a-i b x} \left (3 (17 i-16 a) (i-a) (1-i a) b^2-\frac {3}{2} \left (118 a-i \left (61-52 a^2\right )\right ) b^3 x\right )}{\sqrt {1+i a+i b x}} \, dx}{10 b^5} \\ & = \frac {2 i x^4 (1-i a-i b x)^{3/2}}{b \sqrt {1+i a+i b x}}-\frac {3 (17 i-16 a) x^2 (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{20 b^3}-\frac {11 x^3 (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{5 b^2}+\frac {i (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x} \left (163+458 i a-422 a^2-112 i a^3-2 \left (61 i-118 a-52 i a^2\right ) b x\right )}{40 b^5}-\frac {\left (3 \left (19+68 i a-88 a^2-48 i a^3+8 a^4\right )\right ) \int \frac {\sqrt {1-i a-i b x}}{\sqrt {1+i a+i b x}} \, dx}{8 b^4} \\ & = \frac {2 i x^4 (1-i a-i b x)^{3/2}}{b \sqrt {1+i a+i b x}}+\frac {3 \left (19 i-68 a-88 i a^2+48 a^3+8 i a^4\right ) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{8 b^5}-\frac {3 (17 i-16 a) x^2 (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{20 b^3}-\frac {11 x^3 (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{5 b^2}+\frac {i (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x} \left (163+458 i a-422 a^2-112 i a^3-2 \left (61 i-118 a-52 i a^2\right ) b x\right )}{40 b^5}-\frac {\left (3 \left (19+68 i a-88 a^2-48 i a^3+8 a^4\right )\right ) \int \frac {1}{\sqrt {1-i a-i b x} \sqrt {1+i a+i b x}} \, dx}{8 b^4} \\ & = \frac {2 i x^4 (1-i a-i b x)^{3/2}}{b \sqrt {1+i a+i b x}}+\frac {3 \left (19 i-68 a-88 i a^2+48 a^3+8 i a^4\right ) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{8 b^5}-\frac {3 (17 i-16 a) x^2 (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{20 b^3}-\frac {11 x^3 (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{5 b^2}+\frac {i (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x} \left (163+458 i a-422 a^2-112 i a^3-2 \left (61 i-118 a-52 i a^2\right ) b x\right )}{40 b^5}-\frac {\left (3 \left (19+68 i a-88 a^2-48 i a^3+8 a^4\right )\right ) \int \frac {1}{\sqrt {(1-i a) (1+i a)+2 a b x+b^2 x^2}} \, dx}{8 b^4} \\ & = \frac {2 i x^4 (1-i a-i b x)^{3/2}}{b \sqrt {1+i a+i b x}}+\frac {3 \left (19 i-68 a-88 i a^2+48 a^3+8 i a^4\right ) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{8 b^5}-\frac {3 (17 i-16 a) x^2 (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{20 b^3}-\frac {11 x^3 (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{5 b^2}+\frac {i (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x} \left (163+458 i a-422 a^2-112 i a^3-2 \left (61 i-118 a-52 i a^2\right ) b x\right )}{40 b^5}-\frac {\left (3 \left (19+68 i a-88 a^2-48 i a^3+8 a^4\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{4 b^2}}} \, dx,x,2 a b+2 b^2 x\right )}{16 b^6} \\ & = \frac {2 i x^4 (1-i a-i b x)^{3/2}}{b \sqrt {1+i a+i b x}}+\frac {3 \left (19 i-68 a-88 i a^2+48 a^3+8 i a^4\right ) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{8 b^5}-\frac {3 (17 i-16 a) x^2 (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{20 b^3}-\frac {11 x^3 (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{5 b^2}+\frac {i (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x} \left (163+458 i a-422 a^2-112 i a^3-2 \left (61 i-118 a-52 i a^2\right ) b x\right )}{40 b^5}-\frac {3 \left (19+68 i a-88 a^2-48 i a^3+8 a^4\right ) \text {arcsinh}(a+b x)}{8 b^5} \\ \end{align*}
Time = 0.50 (sec) , antiderivative size = 299, normalized size of antiderivative = 0.92 \[ \int e^{-3 i \arctan (a+b x)} x^4 \, dx=\frac {\frac {448 i+8 i a^6+285 b x+224 i b^2 x^2+95 b^3 x^3-56 i b^4 x^4-30 b^5 x^5+8 i b^6 x^6+a^5 (410+8 i b x)+2 a^4 (-638 i+265 b x)+a^3 \left (-905-2004 i b x+60 b^2 x^2\right )-a^2 \left (836 i+2635 b x+356 i b^2 x^2+20 b^3 x^3\right )+a \left (-1315+1468 i b x-515 b^2 x^2+116 i b^3 x^3+10 b^4 x^4+8 i b^5 x^5\right )}{\sqrt {1+a^2+2 a b x+b^2 x^2}}+\frac {30 \sqrt [4]{-1} \left (19 i-68 a-88 i a^2+48 a^3+8 i a^4\right ) \sqrt {b} \text {arcsinh}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {b} \sqrt {-i (i+a+b x)}}{\sqrt {-i b}}\right )}{\sqrt {-i b}}}{40 b^5} \]
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Time = 1.17 (sec) , antiderivative size = 445, normalized size of antiderivative = 1.37
method | result | size |
risch | \(\frac {i \left (8 x^{4} b^{4}-8 a \,b^{3} x^{3}+30 i b^{3} x^{3}+8 a^{2} b^{2} x^{2}-70 i a \,b^{2} x^{2}-8 a^{3} b x +130 i a^{2} b x +8 a^{4}-250 i a^{3}-64 b^{2} x^{2}+252 a b x -125 b x i-804 a^{2}+835 i a +288\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{40 b^{5}}-\frac {\frac {57 \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{\sqrt {b^{2}}}+\frac {24 a^{4} \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{\sqrt {b^{2}}}-\frac {264 a^{2} \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{\sqrt {b^{2}}}-\frac {144 i a^{3} \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{\sqrt {b^{2}}}+\frac {204 i a \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{\sqrt {b^{2}}}+\frac {i \left (32 i a^{4}+128 a^{3}-192 i a^{2}-128 a +32 i\right ) \sqrt {\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )}}{b^{2} \left (x -\frac {i-a}{b}\right )}}{8 b^{4}}\) | \(445\) |
default | \(\text {Expression too large to display}\) | \(1360\) |
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Time = 0.28 (sec) , antiderivative size = 264, normalized size of antiderivative = 0.81 \[ \int e^{-3 i \arctan (a+b x)} x^4 \, dx=\frac {62 i \, a^{6} + 2687 \, a^{5} - 11575 i \, a^{4} - 20350 \, a^{3} + {\left (62 i \, a^{5} + 2625 \, a^{4} - 8950 i \, a^{3} - 11400 \, a^{2} + 6340 i \, a + 1280\right )} b x + 17740 i \, a^{2} + 120 \, {\left (8 \, a^{5} - 56 i \, a^{4} - 136 \, a^{3} + {\left (8 \, a^{4} - 48 i \, a^{3} - 88 \, a^{2} + 68 i \, a + 19\right )} b x + 156 i \, a^{2} + 87 \, a - 19 i\right )} \log \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) - 8 \, {\left (-8 i \, b^{5} x^{5} + 22 \, b^{4} x^{4} - 2 \, {\left (16 \, a - 17 i\right )} b^{3} x^{3} - 8 i \, a^{5} + {\left (52 \, a^{2} - 118 i \, a - 61\right )} b^{2} x^{2} - 418 \, a^{4} + 1694 i \, a^{3} - {\left (112 \, a^{3} - 422 i \, a^{2} - 458 \, a + 163 i\right )} b x + 2599 \, a^{2} - 1763 i \, a - 448\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} + 7620 \, a - 1280 i}{320 \, {\left (b^{6} x + {\left (a - i\right )} b^{5}\right )}} \]
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Timed out. \[ \int e^{-3 i \arctan (a+b x)} x^4 \, dx=\text {Timed out} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1368 vs. \(2 (230) = 460\).
Time = 0.30 (sec) , antiderivative size = 1368, normalized size of antiderivative = 4.22 \[ \int e^{-3 i \arctan (a+b x)} x^4 \, dx=\text {Too large to display} \]
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Time = 0.30 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.03 \[ \int e^{-3 i \arctan (a+b x)} x^4 \, dx=-\frac {1}{40} \, \sqrt {{\left (b x + a\right )}^{2} + 1} {\left ({\left (2 \, {\left (x {\left (-\frac {4 i \, x}{b} - \frac {-4 i \, a b^{17} - 15 \, b^{17}}{b^{19}}\right )} - \frac {4 i \, a^{2} b^{16} + 35 \, a b^{16} - 32 i \, b^{16}}{b^{19}}\right )} x - \frac {-8 i \, a^{3} b^{15} - 130 \, a^{2} b^{15} + 252 i \, a b^{15} + 125 \, b^{15}}{b^{19}}\right )} x - \frac {8 i \, a^{4} b^{14} + 250 \, a^{3} b^{14} - 804 i \, a^{2} b^{14} - 835 \, a b^{14} + 288 i \, b^{14}}{b^{19}}\right )} + \frac {{\left (8 \, a^{4} - 48 i \, a^{3} - 88 \, a^{2} + 68 i \, a + 19\right )} \log \left (3 \, {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )}^{2} a b + a^{3} b + {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )}^{3} {\left | b \right |} + 3 \, {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} a^{2} {\left | b \right |} - 2 i \, {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )}^{2} b - 2 i \, a^{2} b + 4 \, {\left (-i \, x {\left | b \right |} + i \, \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} a {\left | b \right |} - a b - {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} {\left | b \right |}\right )}{8 \, b^{4} {\left | b \right |}} \]
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Timed out. \[ \int e^{-3 i \arctan (a+b x)} x^4 \, dx=\int \frac {x^4\,{\left ({\left (a+b\,x\right )}^2+1\right )}^{3/2}}{{\left (1+a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}\right )}^3} \,d x \]
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