Integrand size = 16, antiderivative size = 227 \[ \int e^{3 i \arctan (a+b x)} x^2 \, dx=\frac {\left (11 i+18 a-6 i a^2\right ) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{2 b^3}+\frac {\left (11 i+18 a-6 i a^2\right ) \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{6 b^3}-\frac {i (i+a)^2 (1+i a+i b x)^{5/2}}{b^3 \sqrt {1-i a-i b x}}+\frac {i \sqrt {1-i a-i b x} (1+i a+i b x)^{5/2}}{3 b^3}+\frac {\left (11-18 i a-6 a^2\right ) \text {arcsinh}(a+b x)}{2 b^3} \]
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Time = 0.12 (sec) , antiderivative size = 227, 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 ((3-2 i a) (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 \sqrt {1-i a-i b x} (1+i a+i b x)^{5/2}}{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 {\left (11 i+18 a-6 i a^2\right ) \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{6 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 \sqrt {1-i a-i b x} (1+i a+i b x)^{5/2}}{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 {\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 ) \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{6 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 \sqrt {1-i a-i b x} (1+i a+i b x)^{5/2}}{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 {\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 ) \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{6 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 \sqrt {1-i a-i b x} (1+i a+i b x)^{5/2}}{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 {\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 ) \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{6 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 \sqrt {1-i a-i b x} (1+i a+i b x)^{5/2}}{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 {\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 ) \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{6 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 \sqrt {1-i a-i b x} (1+i a+i b x)^{5/2}}{3 b^3}+\frac {\left (11-18 i a-6 a^2\right ) \text {arcsinh}(a+b x)}{2 b^3} \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.70 \[ \int e^{3 i \arctan (a+b x)} x^2 \, dx=\frac {\sqrt {1+i a+i b x} \left (52 i-53 i a^2-2 a^3+19 b x+7 i b^2 x^2-2 b^3 x^3+a (103-16 i b x)\right )}{6 b^3 \sqrt {-i (i+a+b x)}}+\frac {(-1)^{3/4} \left (-11+18 i a+6 a^2\right ) \text {arcsinh}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {b} \sqrt {-i (i+a+b x)}}{\sqrt {-i b}}\right )}{\sqrt {-i b} b^{5/2}} \]
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Time = 1.21 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.14
method | result | size |
risch | \(-\frac {i \left (2 b^{2} x^{2}-2 a b x -9 b x i+2 a^{2}+27 i a -28\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{6 b^{3}}-\frac {-\frac {11 \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{\sqrt {b^{2}}}+\frac {6 a^{2} \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{\sqrt {b^{2}}}+\frac {18 i a \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{\sqrt {b^{2}}}-\frac {i \left (-8 i a^{2}+16 a +8 i\right ) \sqrt {\left (x +\frac {i+a}{b}\right )^{2} b^{2}-2 i b \left (x +\frac {i+a}{b}\right )}}{b^{2} \left (x +\frac {i+a}{b}\right )}}{2 b^{2}}\) | \(259\) |
default | \(\text {Expression too large to display}\) | \(1785\) |
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Time = 0.27 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.77 \[ \int e^{3 i \arctan (a+b x)} x^2 \, dx=\frac {-7 i \, a^{4} + 166 \, a^{3} + {\left (-7 i \, a^{3} + 159 \, a^{2} + 249 i \, a - 96\right )} b x + 408 i \, a^{2} + 12 \, {\left (6 \, a^{3} + {\left (6 \, a^{2} + 18 i \, a - 11\right )} b x + 24 i \, a^{2} - 29 \, a - 11 i\right )} \log \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) - 4 \, {\left (2 i \, b^{3} x^{3} + 7 \, b^{2} x^{2} + 2 i \, a^{3} - {\left (16 \, a + 19 i\right )} b x - 53 \, a^{2} - 103 i \, a + 52\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} - 345 \, a - 96 i}{24 \, {\left (b^{4} x + {\left (a + i\right )} b^{3}\right )}} \]
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\[ \int e^{3 i \arctan (a+b x)} x^2 \, dx=- i \left (\int \frac {i x^{2}}{a^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\, dx + \int \left (- \frac {3 a x^{2}}{a^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\right )\, dx + \int \frac {a^{3} x^{2}}{a^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\, dx + \int \left (- \frac {3 b x^{3}}{a^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\right )\, dx + \int \frac {b^{3} x^{5}}{a^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\, dx + \int \left (- \frac {3 i a^{2} x^{2}}{a^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\right )\, dx + \int \left (- \frac {3 i b^{2} x^{4}}{a^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\right )\, dx + \int \frac {3 a b^{2} x^{4}}{a^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\, dx + \int \frac {3 a^{2} b x^{3}}{a^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\, dx + \int \left (- \frac {6 i a b x^{3}}{a^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\right )\, dx\right ) \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1608 vs. \(2 (155) = 310\).
Time = 0.20 (sec) , antiderivative size = 1608, normalized size of antiderivative = 7.08 \[ \int e^{3 i \arctan (a+b x)} x^2 \, dx=\text {Too large to display} \]
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Time = 0.31 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.07 \[ \int e^{3 i \arctan (a+b x)} x^2 \, dx=-\frac {1}{6} \, \sqrt {{\left (b x + a\right )}^{2} + 1} {\left (x {\left (\frac {2 i \, x}{b} - \frac {2 i \, a b^{6} - 9 \, b^{6}}{b^{8}}\right )} - \frac {-2 i \, a^{2} b^{5} + 27 \, a b^{5} + 28 i \, b^{5}}{b^{8}}\right )} + \frac {{\left (6 \, a^{2} + 18 i \, a - 11\right )} \log \left (3 \, {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )}^{2} a b + a^{3} b + {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )}^{3} {\left | b \right |} + 3 \, {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} a^{2} {\left | b \right |} + 2 i \, {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )}^{2} b + 2 i \, a^{2} b + 4 \, {\left (i \, x {\left | b \right |} - i \, \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} a {\left | b \right |} - a b - {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} {\left | b \right |}\right )}{6 \, b^{2} {\left | b \right |}} \]
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Timed out. \[ \int e^{3 i \arctan (a+b x)} x^2 \, dx=\int \frac {x^2\,{\left (1+a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}\right )}^3}{{\left ({\left (a+b\,x\right )}^2+1\right )}^{3/2}} \,d x \]
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