Integrand size = 14, antiderivative size = 163 \[ \int e^{3 i \arctan (a+b x)} x \, dx=-\frac {3 (3-2 i a) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{2 b^2}-\frac {(3-2 i a) \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{2 b^2}-\frac {(1-i a) (1+i a+i b x)^{5/2}}{b^2 \sqrt {1-i a-i b x}}+\frac {3 (3 i+2 a) \text {arcsinh}(a+b x)}{2 b^2} \]
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Time = 0.08 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {5203, 79, 52, 55, 633, 221} \[ \int e^{3 i \arctan (a+b x)} x \, dx=\frac {3 (2 a+3 i) \text {arcsinh}(a+b x)}{2 b^2}-\frac {(1-i a) (i a+i b x+1)^{5/2}}{b^2 \sqrt {-i a-i b x+1}}-\frac {(3-2 i a) \sqrt {-i a-i b x+1} (i a+i b x+1)^{3/2}}{2 b^2}-\frac {3 (3-2 i a) \sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}{2 b^2} \]
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Rule 52
Rule 55
Rule 79
Rule 221
Rule 633
Rule 5203
Rubi steps \begin{align*} \text {integral}& = \int \frac {x (1+i a+i b x)^{3/2}}{(1-i a-i b x)^{3/2}} \, dx \\ & = -\frac {(1-i a) (1+i a+i b x)^{5/2}}{b^2 \sqrt {1-i a-i b x}}+\frac {(3 i+2 a) \int \frac {(1+i a+i b x)^{3/2}}{\sqrt {1-i a-i b x}} \, dx}{b} \\ & = -\frac {(3-2 i a) \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{2 b^2}-\frac {(1-i a) (1+i a+i b x)^{5/2}}{b^2 \sqrt {1-i a-i b x}}+\frac {(3 (3 i+2 a)) \int \frac {\sqrt {1+i a+i b x}}{\sqrt {1-i a-i b x}} \, dx}{2 b} \\ & = -\frac {3 (3-2 i a) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{2 b^2}-\frac {(3-2 i a) \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{2 b^2}-\frac {(1-i a) (1+i a+i b x)^{5/2}}{b^2 \sqrt {1-i a-i b x}}+\frac {(3 (3 i+2 a)) \int \frac {1}{\sqrt {1-i a-i b x} \sqrt {1+i a+i b x}} \, dx}{2 b} \\ & = -\frac {3 (3-2 i a) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{2 b^2}-\frac {(3-2 i a) \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{2 b^2}-\frac {(1-i a) (1+i a+i b x)^{5/2}}{b^2 \sqrt {1-i a-i b x}}+\frac {(3 (3 i+2 a)) \int \frac {1}{\sqrt {(1-i a) (1+i a)+2 a b x+b^2 x^2}} \, dx}{2 b} \\ & = -\frac {3 (3-2 i a) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{2 b^2}-\frac {(3-2 i a) \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{2 b^2}-\frac {(1-i a) (1+i a+i b x)^{5/2}}{b^2 \sqrt {1-i a-i b x}}+\frac {(3 (3 i+2 a)) \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^3} \\ & = -\frac {3 (3-2 i a) \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{2 b^2}-\frac {(3-2 i a) \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}}{2 b^2}-\frac {(1-i a) (1+i a+i b x)^{5/2}}{b^2 \sqrt {1-i a-i b x}}+\frac {3 (3 i+2 a) \text {arcsinh}(a+b x)}{2 b^2} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.81 \[ \int e^{3 i \arctan (a+b x)} x \, dx=\frac {\sqrt {1+i a+i b x} \left (-14+15 i a+a^2+5 i b x-b^2 x^2\right )}{2 b^2 \sqrt {-i (i+a+b x)}}+\frac {3 \sqrt [4]{-1} (3 i+2 a) \sqrt {-i b} \text {arcsinh}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {b} \sqrt {-i (i+a+b x)}}{\sqrt {-i b}}\right )}{b^{5/2}} \]
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Time = 0.82 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.14
method | result | size |
risch | \(\frac {i \left (-b x +a +6 i\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{2 b^{2}}+\frac {\frac {9 i \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{\sqrt {b^{2}}}+\frac {6 a \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{\sqrt {b^{2}}}-\frac {i \left (-8 i a +8\right ) \sqrt {\left (x +\frac {i+a}{b}\right )^{2} b^{2}-2 i b \left (x +\frac {i+a}{b}\right )}}{b^{2} \left (x +\frac {i+a}{b}\right )}}{2 b}\) | \(186\) |
default | \(\text {Expression too large to display}\) | \(1052\) |
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Time = 0.28 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.83 \[ \int e^{3 i \arctan (a+b x)} x \, dx=\frac {3 i \, a^{3} + {\left (3 i \, a^{2} - 44 \, a - 32 i\right )} b x - 47 \, a^{2} - 12 \, {\left ({\left (2 \, a + 3 i\right )} b x + 2 \, a^{2} + 5 i \, a - 3\right )} \log \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) - 4 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (i \, b^{2} x^{2} - i \, a^{2} + 5 \, b x + 15 \, a + 14 i\right )} - 76 i \, a + 32}{8 \, {\left (b^{3} x + {\left (a + i\right )} b^{2}\right )}} \]
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\[ \int e^{3 i \arctan (a+b x)} x \, dx=- i \left (\int \frac {i x}{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}{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}{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^{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 {b^{3} 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 \left (- \frac {3 i a^{2} x}{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^{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 {3 a b^{2} 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 \frac {3 a^{2} b 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 {6 i a b 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\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. 1108 vs. \(2 (113) = 226\).
Time = 0.22 (sec) , antiderivative size = 1108, normalized size of antiderivative = 6.80 \[ \int e^{3 i \arctan (a+b x)} x \, dx=\text {Too large to display} \]
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Time = 0.30 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.28 \[ \int e^{3 i \arctan (a+b x)} x \, dx=-\frac {1}{2} \, \sqrt {{\left (b x + a\right )}^{2} + 1} {\left (\frac {i \, x}{b} + \frac {-i \, a b^{2} + 6 \, b^{2}}{b^{4}}\right )} - \frac {{\left (2 \, a + 3 i\right )} \log \left (3 \, {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )}^{2} a b + a^{3} b + {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )}^{3} {\left | b \right |} + 3 \, {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} a^{2} {\left | b \right |} + 2 i \, {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )}^{2} b + 2 i \, a^{2} b + 4 \, {\left (i \, x {\left | b \right |} - i \, \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} a {\left | b \right |} - a b - {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} {\left | b \right |}\right )}{2 \, b {\left | b \right |}} \]
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Timed out. \[ \int e^{3 i \arctan (a+b x)} x \, dx=\int \frac {x\,{\left (1+a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}\right )}^3}{{\left ({\left (a+b\,x\right )}^2+1\right )}^{3/2}} \,d x \]
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