Integrand size = 16, antiderivative size = 229 \[ \int e^{-3 i \arctan (a+b x)} x^2 \, dx=\frac {i (i-a)^2 (1-i a-i b x)^{5/2}}{b^3 \sqrt {1+i a+i b x}}-\frac {\left (11 i-18 a-6 i a^2\right ) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{2 b^3}-\frac {\left (11 i-18 a-6 i a^2\right ) (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{6 b^3}-\frac {i (1-i a-i b x)^{5/2} \sqrt {1+i a+i b x}}{3 b^3}+\frac {\left (11+18 i a-6 a^2\right ) \text {arcsinh}(a+b x)}{2 b^3} \]
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Time = 0.12 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {5203, 91, 81, 52, 55, 633, 221} \[ \int e^{-3 i \arctan (a+b x)} x^2 \, dx=\frac {\left (-6 a^2+18 i a+11\right ) \text {arcsinh}(a+b x)}{2 b^3}-\frac {\left (-6 i a^2-18 a+11 i\right ) \sqrt {i a+i b x+1} (-i a-i b x+1)^{3/2}}{6 b^3}-\frac {\left (-6 i a^2-18 a+11 i\right ) \sqrt {i a+i b x+1} \sqrt {-i a-i b x+1}}{2 b^3}-\frac {i \sqrt {i a+i b x+1} (-i a-i b x+1)^{5/2}}{3 b^3}+\frac {i (-a+i)^2 (-i a-i b x+1)^{5/2}}{b^3 \sqrt {i a+i b x+1}} \]
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Rule 52
Rule 55
Rule 81
Rule 91
Rule 221
Rule 633
Rule 5203
Rubi steps \begin{align*} \text {integral}& = \int \frac {x^2 (1-i a-i b x)^{3/2}}{(1+i a+i b x)^{3/2}} \, dx \\ & = \frac {i (i-a)^2 (1-i a-i b x)^{5/2}}{b^3 \sqrt {1+i a+i b x}}+\frac {i \int \frac {(1-i a-i b x)^{3/2} \left (-((i-a) (3+2 i a) b)-b^2 x\right )}{\sqrt {1+i a+i b x}} \, dx}{b^3} \\ & = \frac {i (i-a)^2 (1-i a-i b x)^{5/2}}{b^3 \sqrt {1+i a+i b x}}-\frac {i (1-i a-i b x)^{5/2} \sqrt {1+i a+i b x}}{3 b^3}+\frac {\left (11+18 i a-6 a^2\right ) \int \frac {(1-i a-i b x)^{3/2}}{\sqrt {1+i a+i b x}} \, dx}{3 b^2} \\ & = \frac {i (i-a)^2 (1-i a-i b x)^{5/2}}{b^3 \sqrt {1+i a+i b x}}+\frac {\left (18 a-i \left (11-6 a^2\right )\right ) (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{6 b^3}-\frac {i (1-i a-i b x)^{5/2} \sqrt {1+i a+i b x}}{3 b^3}+\frac {\left (11+18 i a-6 a^2\right ) \int \frac {\sqrt {1-i a-i b x}}{\sqrt {1+i a+i b x}} \, dx}{2 b^2} \\ & = \frac {i (i-a)^2 (1-i a-i b x)^{5/2}}{b^3 \sqrt {1+i a+i b x}}+\frac {\left (18 a-i \left (11-6 a^2\right )\right ) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{2 b^3}+\frac {\left (18 a-i \left (11-6 a^2\right )\right ) (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{6 b^3}-\frac {i (1-i a-i b x)^{5/2} \sqrt {1+i a+i b x}}{3 b^3}+\frac {\left (11+18 i a-6 a^2\right ) \int \frac {1}{\sqrt {1-i a-i b x} \sqrt {1+i a+i b x}} \, dx}{2 b^2} \\ & = \frac {i (i-a)^2 (1-i a-i b x)^{5/2}}{b^3 \sqrt {1+i a+i b x}}+\frac {\left (18 a-i \left (11-6 a^2\right )\right ) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{2 b^3}+\frac {\left (18 a-i \left (11-6 a^2\right )\right ) (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{6 b^3}-\frac {i (1-i a-i b x)^{5/2} \sqrt {1+i a+i b x}}{3 b^3}+\frac {\left (11+18 i a-6 a^2\right ) \int \frac {1}{\sqrt {(1-i a) (1+i a)+2 a b x+b^2 x^2}} \, dx}{2 b^2} \\ & = \frac {i (i-a)^2 (1-i a-i b x)^{5/2}}{b^3 \sqrt {1+i a+i b x}}+\frac {\left (18 a-i \left (11-6 a^2\right )\right ) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{2 b^3}+\frac {\left (18 a-i \left (11-6 a^2\right )\right ) (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{6 b^3}-\frac {i (1-i a-i b x)^{5/2} \sqrt {1+i a+i b x}}{3 b^3}+\frac {\left (11+18 i a-6 a^2\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 )}{4 b^4} \\ & = \frac {i (i-a)^2 (1-i a-i b x)^{5/2}}{b^3 \sqrt {1+i a+i b x}}+\frac {\left (18 a-i \left (11-6 a^2\right )\right ) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{2 b^3}+\frac {\left (18 a-i \left (11-6 a^2\right )\right ) (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{6 b^3}-\frac {i (1-i a-i b x)^{5/2} \sqrt {1+i a+i b x}}{3 b^3}+\frac {\left (11+18 i a-6 a^2\right ) \text {arcsinh}(a+b x)}{2 b^3} \\ \end{align*}
Time = 0.24 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.86 \[ \int e^{-3 i \arctan (a+b x)} x^2 \, dx=\frac {2 i a^4+a^3 (51+2 i b x)+a^2 (-50 i+69 b x)+a \left (51-106 i b x+9 b^2 x^2+2 i b^3 x^3\right )+i \left (-52+33 i b x-26 b^2 x^2+9 i b^3 x^3+2 b^4 x^4\right )}{6 b^3 \sqrt {1+a^2+2 a b x+b^2 x^2}}+\frac {\sqrt [4]{-1} \left (11+18 i a-6 a^2\right ) \sqrt {-i b} \text {arcsinh}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {b} \sqrt {-i (i+a+b x)}}{\sqrt {-i b}}\right )}{b^{7/2}} \]
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Time = 0.91 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.17
method | result | size |
risch | \(\frac {i \left (2 b^{2} x^{2}-2 a b x +9 b x i+2 a^{2}-27 i a -28\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{6 b^{3}}-\frac {-\frac {11 \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{\sqrt {b^{2}}}+\frac {6 a^{2} \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{\sqrt {b^{2}}}-\frac {18 i a \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{\sqrt {b^{2}}}+\frac {i \left (8 i a^{2}+16 a -8 i\right ) \sqrt {\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )}}{b^{2} \left (x -\frac {i-a}{b}\right )}}{2 b^{2}}\) | \(268\) |
default | \(\frac {i \left (\frac {\left (\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )\right )^{\frac {3}{2}}}{3}+i b \left (\frac {\left (2 \left (x -\frac {i-a}{b}\right ) b^{2}+2 i b \right ) \sqrt {\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )}}{4 b^{2}}+\frac {\ln \left (\frac {i b +\left (x -\frac {i-a}{b}\right ) b^{2}}{\sqrt {b^{2}}}+\sqrt {\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )}\right )}{2 \sqrt {b^{2}}}\right )\right )}{b^{3}}+\frac {i \left (i-a \right )^{2} \left (\frac {i \left (\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )\right )^{\frac {5}{2}}}{b \left (x -\frac {i-a}{b}\right )^{3}}-2 i b \left (-\frac {i \left (\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )\right )^{\frac {5}{2}}}{b \left (x -\frac {i-a}{b}\right )^{2}}+3 i b \left (\frac {\left (\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )\right )^{\frac {3}{2}}}{3}+i b \left (\frac {\left (2 \left (x -\frac {i-a}{b}\right ) b^{2}+2 i b \right ) \sqrt {\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )}}{4 b^{2}}+\frac {\ln \left (\frac {i b +\left (x -\frac {i-a}{b}\right ) b^{2}}{\sqrt {b^{2}}}+\sqrt {\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )}\right )}{2 \sqrt {b^{2}}}\right )\right )\right )\right )}{b^{5}}+\frac {2 i \left (i-a \right ) \left (-\frac {i \left (\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )\right )^{\frac {5}{2}}}{b \left (x -\frac {i-a}{b}\right )^{2}}+3 i b \left (\frac {\left (\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )\right )^{\frac {3}{2}}}{3}+i b \left (\frac {\left (2 \left (x -\frac {i-a}{b}\right ) b^{2}+2 i b \right ) \sqrt {\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )}}{4 b^{2}}+\frac {\ln \left (\frac {i b +\left (x -\frac {i-a}{b}\right ) b^{2}}{\sqrt {b^{2}}}+\sqrt {\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )}\right )}{2 \sqrt {b^{2}}}\right )\right )\right )}{b^{4}}\) | \(799\) |
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Time = 0.27 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.76 \[ \int e^{-3 i \arctan (a+b x)} x^2 \, dx=\frac {7 i \, a^{4} + 166 \, a^{3} + {\left (7 i \, a^{3} + 159 \, a^{2} - 249 i \, a - 96\right )} b x - 408 i \, a^{2} + 12 \, {\left (6 \, a^{3} + {\left (6 \, a^{2} - 18 i \, a - 11\right )} b x - 24 i \, a^{2} - 29 \, a + 11 i\right )} \log \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) - 4 \, {\left (-2 i \, b^{3} x^{3} + 7 \, b^{2} x^{2} - 2 i \, a^{3} - {\left (16 \, a - 19 i\right )} b x - 53 \, a^{2} + 103 i \, a + 52\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} - 345 \, a + 96 i}{24 \, {\left (b^{4} x + {\left (a - i\right )} b^{3}\right )}} \]
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Timed out. \[ \int e^{-3 i \arctan (a+b x)} x^2 \, dx=\text {Timed out} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 624 vs. \(2 (155) = 310\).
Time = 0.28 (sec) , antiderivative size = 624, normalized size of antiderivative = 2.72 \[ \int e^{-3 i \arctan (a+b x)} x^2 \, dx=\frac {i \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}} a^{2}}{b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3} - 2 i \, b^{4} x - 2 i \, a b^{3} - b^{3}} + \frac {2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}} a}{b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3} - 2 i \, b^{4} x - 2 i \, a b^{3} - b^{3}} + \frac {2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}} a}{2 i \, b^{4} x + 2 i \, a b^{3} + 2 \, b^{3}} + \frac {6 i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a^{2}}{i \, b^{4} x + i \, a b^{3} + b^{3}} - \frac {i \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}}}{b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3} - 2 i \, b^{4} x - 2 i \, a b^{3} - b^{3}} - \frac {2 i \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}}}{2 i \, b^{4} x + 2 i \, a b^{3} + 2 \, b^{3}} + \frac {12 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a}{i \, b^{4} x + i \, a b^{3} + b^{3}} - \frac {3 \, a^{2} \operatorname {arsinh}\left (b x + a\right )}{b^{3}} - \frac {6 i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{i \, b^{4} x + i \, a b^{3} + b^{3}} - \frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 4 i \, b x + 4 i \, a + 3} x}{2 \, b^{2}} + \frac {9 i \, a \operatorname {arsinh}\left (b x + a\right )}{b^{3}} + \frac {i \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}}}{3 \, b^{3}} + \frac {3 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a}{b^{3}} - \frac {\sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 4 i \, b x + 4 i \, a + 3} a}{2 \, b^{3}} + \frac {\arcsin \left (i \, b x + i \, a + 2\right )}{2 \, b^{3}} + \frac {6 \, \operatorname {arsinh}\left (b x + a\right )}{b^{3}} - \frac {3 i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b^{3}} + \frac {i \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + 4 i \, b x + 4 i \, a + 3}}{b^{3}} \]
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Time = 0.30 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.05 \[ \int e^{-3 i \arctan (a+b x)} x^2 \, dx=-\frac {1}{6} \, \sqrt {{\left (b x + a\right )}^{2} + 1} {\left (x {\left (-\frac {2 i \, x}{b} + \frac {2 i \, a b^{6} + 9 \, b^{6}}{b^{8}}\right )} + \frac {-2 i \, a^{2} b^{5} - 27 \, a b^{5} + 28 i \, b^{5}}{b^{8}}\right )} + \frac {{\left (6 \, a^{2} - 18 i \, a - 11\right )} \log \left (3 \, {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )}^{2} a b + a^{3} b + {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )}^{3} {\left | b \right |} + 3 \, {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} a^{2} {\left | b \right |} - 2 i \, {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )}^{2} b - 2 i \, a^{2} b + 4 \, {\left (-i \, x {\left | b \right |} + i \, \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} a {\left | b \right |} - a b - {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} {\left | b \right |}\right )}{6 \, b^{2} {\left | b \right |}} \]
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Timed out. \[ \int e^{-3 i \arctan (a+b x)} x^2 \, dx=\int \frac {x^2\,{\left ({\left (a+b\,x\right )}^2+1\right )}^{3/2}}{{\left (1+a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}\right )}^3} \,d x \]
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