Integrand size = 16, antiderivative size = 201 \[ \int e^{-i \arctan (a+b x)} x^3 \, dx=-\frac {\left (3+12 i a-12 a^2-8 i a^3\right ) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{8 b^4}+\frac {x^2 (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{4 b^2}-\frac {(1-i a-i b x)^{3/2} \sqrt {1+i a+i b x} \left (7+10 i a-18 a^2-2 (i-6 a) b x\right )}{24 b^4}-\frac {\left (3 i-12 a-12 i a^2+8 a^3\right ) \text {arcsinh}(a+b x)}{8 b^4} \]
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Time = 0.14 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {5203, 102, 152, 52, 55, 633, 221} \[ \int e^{-i \arctan (a+b x)} x^3 \, dx=-\frac {(-i a-i b x+1)^{3/2} \sqrt {i a+i b x+1} \left (-18 a^2-2 (-6 a+i) b x+10 i a+7\right )}{24 b^4}-\frac {\left (8 a^3-12 i a^2-12 a+3 i\right ) \text {arcsinh}(a+b x)}{8 b^4}-\frac {\left (-8 i a^3-12 a^2+12 i a+3\right ) \sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}{8 b^4}+\frac {x^2 (-i a-i b x+1)^{3/2} \sqrt {i a+i b x+1}}{4 b^2} \]
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Rule 52
Rule 55
Rule 102
Rule 152
Rule 221
Rule 633
Rule 5203
Rubi steps \begin{align*} \text {integral}& = \int \frac {x^3 \sqrt {1-i a-i b x}}{\sqrt {1+i a+i b x}} \, dx \\ & = \frac {x^2 (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{4 b^2}+\frac {\int \frac {x \sqrt {1-i a-i b x} \left (-2 \left (1+a^2\right )+(i-6 a) b x\right )}{\sqrt {1+i a+i b x}} \, dx}{4 b^2} \\ & = \frac {x^2 (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{4 b^2}-\frac {(1-i a-i b x)^{3/2} \sqrt {1+i a+i b x} \left (7+10 i a-18 a^2-2 (i-6 a) b x\right )}{24 b^4}-\frac {\left (3 i-12 a-12 i a^2+8 a^3\right ) \int \frac {\sqrt {1-i a-i b x}}{\sqrt {1+i a+i b x}} \, dx}{8 b^3} \\ & = -\frac {\left (3+12 i a-12 a^2-8 i a^3\right ) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{8 b^4}+\frac {x^2 (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{4 b^2}-\frac {(1-i a-i b x)^{3/2} \sqrt {1+i a+i b x} \left (7+10 i a-18 a^2-2 (i-6 a) b x\right )}{24 b^4}-\frac {\left (3 i-12 a-12 i a^2+8 a^3\right ) \int \frac {1}{\sqrt {1-i a-i b x} \sqrt {1+i a+i b x}} \, dx}{8 b^3} \\ & = -\frac {\left (3+12 i a-12 a^2-8 i a^3\right ) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{8 b^4}+\frac {x^2 (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{4 b^2}-\frac {(1-i a-i b x)^{3/2} \sqrt {1+i a+i b x} \left (7+10 i a-18 a^2-2 (i-6 a) b x\right )}{24 b^4}-\frac {\left (3 i-12 a-12 i a^2+8 a^3\right ) \int \frac {1}{\sqrt {(1-i a) (1+i a)+2 a b x+b^2 x^2}} \, dx}{8 b^3} \\ & = -\frac {\left (3+12 i a-12 a^2-8 i a^3\right ) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{8 b^4}+\frac {x^2 (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{4 b^2}-\frac {(1-i a-i b x)^{3/2} \sqrt {1+i a+i b x} \left (7+10 i a-18 a^2-2 (i-6 a) b x\right )}{24 b^4}-\frac {\left (3 i-12 a-12 i a^2+8 a^3\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^5} \\ & = -\frac {\left (3+12 i a-12 a^2-8 i a^3\right ) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{8 b^4}+\frac {x^2 (1-i a-i b x)^{3/2} \sqrt {1+i a+i b x}}{4 b^2}-\frac {(1-i a-i b x)^{3/2} \sqrt {1+i a+i b x} \left (7+10 i a-18 a^2-2 (i-6 a) b x\right )}{24 b^4}-\frac {\left (3 i-12 a-12 i a^2+8 a^3\right ) \text {arcsinh}(a+b x)}{8 b^4} \\ \end{align*}
Time = 0.42 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.00 \[ \int e^{-i \arctan (a+b x)} x^3 \, dx=\frac {\sqrt {1+i a+i b x} \left (-16-38 i a^3+6 a^4+25 i b x+17 b^2 x^2-14 i b^3 x^3-6 b^4 x^4+5 a^2 (1-6 i b x)+i a \left (-23+50 i b x+18 b^2 x^2\right )\right )}{24 b^4 \sqrt {-i (i+a+b x)}}+\frac {(-1)^{3/4} \left (-3-12 i a+12 a^2+8 i a^3\right ) \sqrt {-i b} \text {arcsinh}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {b} \sqrt {-i (i+a+b x)}}{\sqrt {-i b}}\right )}{4 b^{9/2}} \]
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Time = 0.68 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.75
method | result | size |
risch | \(\frac {i \left (-6 b^{3} x^{3}+6 a \,b^{2} x^{2}-8 i x^{2} b^{2}-6 a^{2} b x +20 i a b x +6 a^{3}-44 i a^{2}+9 b x -39 a +16 i\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{24 b^{4}}-\frac {\left (8 a^{3}-12 i a^{2}-12 a +3 i\right ) \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{8 b^{3} \sqrt {b^{2}}}\) | \(150\) |
default | \(-\frac {i \left (\frac {x \left (b^{2} x^{2}+2 a b x +a^{2}+1\right )^{\frac {3}{2}}}{4 b^{2}}-\frac {5 a \left (\frac {\left (b^{2} x^{2}+2 a b x +a^{2}+1\right )^{\frac {3}{2}}}{3 b^{2}}-\frac {a \left (\frac {\left (2 b^{2} x +2 a b \right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{4 b^{2}}+\frac {\left (4 b^{2} \left (a^{2}+1\right )-4 a^{2} b^{2}\right ) \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{8 b^{2} \sqrt {b^{2}}}\right )}{b}\right )}{4 b}-\frac {\left (a^{2}+1\right ) \left (\frac {\left (2 b^{2} x +2 a b \right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{4 b^{2}}+\frac {\left (4 b^{2} \left (a^{2}+1\right )-4 a^{2} b^{2}\right ) \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{8 b^{2} \sqrt {b^{2}}}\right )}{4 b^{2}}\right )}{b}-\frac {i \left (i-a \right )^{2} \left (\frac {\left (2 b^{2} x +2 a b \right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{4 b^{2}}+\frac {\left (4 b^{2} \left (a^{2}+1\right )-4 a^{2} b^{2}\right ) \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{8 b^{2} \sqrt {b^{2}}}\right )}{b^{3}}-\frac {i \left (i-a \right ) \left (\frac {\left (b^{2} x^{2}+2 a b x +a^{2}+1\right )^{\frac {3}{2}}}{3 b^{2}}-\frac {a \left (\frac {\left (2 b^{2} x +2 a b \right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{4 b^{2}}+\frac {\left (4 b^{2} \left (a^{2}+1\right )-4 a^{2} b^{2}\right ) \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{8 b^{2} \sqrt {b^{2}}}\right )}{b}\right )}{b^{2}}+\frac {\left (i a^{3}+3 a^{2}-3 i a -1\right ) \left (\sqrt {\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )}+\frac {i b \ln \left (\frac {i b +\left (x -\frac {i-a}{b}\right ) b^{2}}{\sqrt {b^{2}}}+\sqrt {\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )}\right )}{\sqrt {b^{2}}}\right )}{b^{4}}\) | \(681\) |
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Time = 0.28 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.69 \[ \int e^{-i \arctan (a+b x)} x^3 \, dx=\frac {45 i \, a^{4} + 224 \, a^{3} - 192 i \, a^{2} + 24 \, {\left (8 \, a^{3} - 12 i \, a^{2} - 12 \, a + 3 i\right )} \log \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) - 8 \, {\left (6 i \, b^{3} x^{3} + 2 \, {\left (-3 i \, a - 4\right )} b^{2} x^{2} - 6 i \, a^{3} + {\left (6 i \, a^{2} + 20 \, a - 9 i\right )} b x - 44 \, a^{2} + 39 i \, a + 16\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} - 72 \, a}{192 \, b^{4}} \]
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\[ \int e^{-i \arctan (a+b x)} x^3 \, dx=- i \int \frac {x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{a + b x - i}\, dx \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 308 vs. \(2 (145) = 290\).
Time = 0.30 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.53 \[ \int e^{-i \arctan (a+b x)} x^3 \, dx=-\frac {3 i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a^{2} x}{2 \, b^{3}} - \frac {a^{3} \operatorname {arsinh}\left (b x + a\right )}{b^{4}} - \frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a^{3}}{2 \, b^{4}} - \frac {i \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}} x}{4 \, b^{3}} - \frac {3 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a x}{2 \, b^{3}} + \frac {3 i \, a^{2} \operatorname {arsinh}\left (b x + a\right )}{2 \, b^{4}} + \frac {3 i \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}} a}{4 \, b^{4}} + \frac {3 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a^{2}}{2 \, b^{4}} + \frac {5 i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} x}{8 \, b^{3}} + \frac {3 \, a \operatorname {arsinh}\left (b x + a\right )}{2 \, b^{4}} + \frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}}}{3 \, b^{4}} - \frac {19 i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a}{8 \, b^{4}} - \frac {3 i \, \operatorname {arsinh}\left (b x + a\right )}{8 \, b^{4}} - \frac {\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b^{4}} \]
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Time = 0.28 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.77 \[ \int e^{-i \arctan (a+b x)} x^3 \, dx=-\frac {1}{24} \, \sqrt {{\left (b x + a\right )}^{2} + 1} {\left ({\left (2 \, x {\left (\frac {3 i \, x}{b} - \frac {3 i \, a b^{5} + 4 \, b^{5}}{b^{7}}\right )} - \frac {-6 i \, a^{2} b^{4} - 20 \, a b^{4} + 9 i \, b^{4}}{b^{7}}\right )} x - \frac {6 i \, a^{3} b^{3} + 44 \, a^{2} b^{3} - 39 i \, a b^{3} - 16 \, b^{3}}{b^{7}}\right )} + \frac {{\left (8 \, a^{3} - 12 i \, a^{2} - 12 \, a + 3 i\right )} \log \left (-a b - {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} {\left | b \right |}\right )}{8 \, b^{3} {\left | b \right |}} \]
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Timed out. \[ \int e^{-i \arctan (a+b x)} x^3 \, dx=\int \frac {x^3\,\sqrt {{\left (a+b\,x\right )}^2+1}}{1+a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}} \,d x \]
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