Integrand size = 12, antiderivative size = 94 \[ \int e^{3 i \arctan (a+b x)} \, dx=-\frac {3 i \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{b}-\frac {2 i (1+i a+i b x)^{3/2}}{b \sqrt {1-i a-i b x}}-\frac {3 \text {arcsinh}(a+b x)}{b} \]
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Time = 0.03 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5201, 49, 52, 55, 633, 221} \[ \int e^{3 i \arctan (a+b x)} \, dx=-\frac {3 \text {arcsinh}(a+b x)}{b}-\frac {2 i (i a+i b x+1)^{3/2}}{b \sqrt {-i a-i b x+1}}-\frac {3 i \sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}{b} \]
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Rule 49
Rule 52
Rule 55
Rule 221
Rule 633
Rule 5201
Rubi steps \begin{align*} \text {integral}& = \int \frac {(1+i a+i b x)^{3/2}}{(1-i a-i b x)^{3/2}} \, dx \\ & = -\frac {2 i (1+i a+i b x)^{3/2}}{b \sqrt {1-i a-i b x}}-3 \int \frac {\sqrt {1+i a+i b x}}{\sqrt {1-i a-i b x}} \, dx \\ & = -\frac {3 i \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{b}-\frac {2 i (1+i a+i b x)^{3/2}}{b \sqrt {1-i a-i b x}}-3 \int \frac {1}{\sqrt {1-i a-i b x} \sqrt {1+i a+i b x}} \, dx \\ & = -\frac {3 i \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{b}-\frac {2 i (1+i a+i b x)^{3/2}}{b \sqrt {1-i a-i b x}}-3 \int \frac {1}{\sqrt {(1-i a) (1+i a)+2 a b x+b^2 x^2}} \, dx \\ & = -\frac {3 i \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{b}-\frac {2 i (1+i a+i b x)^{3/2}}{b \sqrt {1-i a-i b x}}-\frac {3 \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{4 b^2}}} \, dx,x,2 a b+2 b^2 x\right )}{2 b^2} \\ & = -\frac {3 i \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{b}-\frac {2 i (1+i a+i b x)^{3/2}}{b \sqrt {1-i a-i b x}}-\frac {3 \text {arcsinh}(a+b x)}{b} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.48 \[ \int e^{3 i \arctan (a+b x)} \, dx=\frac {\sqrt {1+(a+b x)^2} \left (-i+\frac {4}{i+a+b x}\right )}{b}-\frac {3 \text {arcsinh}(a+b x)}{b} \]
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Time = 0.81 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.28
method | result | size |
risch | \(-\frac {i \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{b}-\frac {3 \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{\sqrt {b^{2}}}+\frac {4 \sqrt {\left (x +\frac {i+a}{b}\right )^{2} b^{2}-2 i b \left (x +\frac {i+a}{b}\right )}}{b^{2} \left (x +\frac {i+a}{b}\right )}\) | \(120\) |
default | \(-i b^{3} \left (\frac {x^{2}}{b^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {3 a \left (-\frac {x}{b^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {a \left (-\frac {1}{b^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {2 a \left (2 b^{2} x +2 a b \right )}{b \left (4 b^{2} \left (a^{2}+1\right )-4 a^{2} b^{2}\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}\right )}{b}+\frac {\ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{b^{2} \sqrt {b^{2}}}\right )}{b}-\frac {2 \left (a^{2}+1\right ) \left (-\frac {1}{b^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {2 a \left (2 b^{2} x +2 a b \right )}{b \left (4 b^{2} \left (a^{2}+1\right )-4 a^{2} b^{2}\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}\right )}{b^{2}}\right )+\frac {2 \left (i a +1\right )^{3} \left (2 b^{2} x +2 a b \right )}{\left (4 b^{2} \left (a^{2}+1\right )-4 a^{2} b^{2}\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-3 \left (i a +1\right ) b^{2} \left (-\frac {x}{b^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {a \left (-\frac {1}{b^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {2 a \left (2 b^{2} x +2 a b \right )}{b \left (4 b^{2} \left (a^{2}+1\right )-4 a^{2} b^{2}\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}\right )}{b}+\frac {\ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{b^{2} \sqrt {b^{2}}}\right )+3 i \left (i a +1\right )^{2} b \left (-\frac {1}{b^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {2 a \left (2 b^{2} x +2 a b \right )}{b \left (4 b^{2} \left (a^{2}+1\right )-4 a^{2} b^{2}\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}\right )\) | \(617\) |
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Time = 0.28 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.05 \[ \int e^{3 i \arctan (a+b x)} \, dx=\frac {{\left (-i \, a + 8\right )} b x - i \, a^{2} + 6 \, {\left (b x + a + i\right )} \log \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) - 2 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (i \, b x + i \, a - 5\right )} + 9 \, a + 8 i}{2 \, {\left (b^{2} x + {\left (a + i\right )} b\right )}} \]
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\[ \int e^{3 i \arctan (a+b x)} \, dx=- i \left (\int \frac {i}{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}{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}}{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}}{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 b 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 {b^{3} 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 {3 i b^{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 \frac {3 a b^{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}}\, dx + \int \frac {3 a^{2} b 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 {6 i a b 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\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. 736 vs. \(2 (66) = 132\).
Time = 0.21 (sec) , antiderivative size = 736, normalized size of antiderivative = 7.83 \[ \int e^{3 i \arctan (a+b x)} \, dx=-\frac {6 i \, a^{3} b^{2} x}{{\left (a^{2} b^{2} - {\left (a^{2} + 1\right )} b^{2}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}} + \frac {5 i \, {\left (a^{2} + 1\right )} a b^{2} x}{{\left (a^{2} b^{2} - {\left (a^{2} + 1\right )} b^{2}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}} - \frac {i \, {\left (a^{2} + 1\right )} a^{2} b}{{\left (a^{2} b^{2} - {\left (a^{2} + 1\right )} b^{2}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}} + \frac {6 \, {\left (i \, a b^{2} + b^{2}\right )} a^{2} x}{{\left (a^{2} b^{2} - {\left (a^{2} + 1\right )} b^{2}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}} - \frac {3 \, {\left (i \, a^{2} b + 2 \, a b - i \, b\right )} a b x}{{\left (a^{2} b^{2} - {\left (a^{2} + 1\right )} b^{2}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}} + \frac {{\left (i \, a^{3} + 3 \, a^{2} - 3 i \, a - 1\right )} b^{2} x}{{\left (a^{2} b^{2} - {\left (a^{2} + 1\right )} b^{2}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}} - \frac {i \, b x^{2}}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}} - \frac {3 \, {\left (i \, a^{2} b + 2 \, a b - i \, b\right )} a^{2}}{{\left (a^{2} b^{2} - {\left (a^{2} + 1\right )} b^{2}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}} - \frac {{\left (-i \, a^{3} - 3 \, a^{2} + 3 i \, a + 1\right )} a b}{{\left (a^{2} b^{2} - {\left (a^{2} + 1\right )} b^{2}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}} - \frac {3 \, {\left (i \, a b^{2} + b^{2}\right )} {\left (a^{2} + 1\right )} x}{{\left (a^{2} b^{2} - {\left (a^{2} + 1\right )} b^{2}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}} + \frac {3 i \, a \operatorname {arsinh}\left (\frac {2 \, {\left (b^{2} x + a b\right )}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}}}\right )}{b} + \frac {3 \, {\left (i \, a b^{2} + b^{2}\right )} {\left (a^{2} + 1\right )} a}{{\left (a^{2} b^{2} - {\left (a^{2} + 1\right )} b^{2}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} b} - \frac {2 \, {\left (i \, a^{2} + i\right )}}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} b} - \frac {3 \, {\left (i \, a b^{2} + b^{2}\right )} \operatorname {arsinh}\left (\frac {2 \, {\left (b^{2} x + a b\right )}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}}}\right )}{b^{3}} + \frac {3 \, {\left (i \, a^{2} b + 2 \, a b - i \, b\right )}}{\sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} b^{2}} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 180 vs. \(2 (66) = 132\).
Time = 0.31 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.91 \[ \int e^{3 i \arctan (a+b x)} \, dx=\frac {\log \left (3 \, {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )}^{2} a b + a^{3} b + {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )}^{3} {\left | b \right |} + 3 \, {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} a^{2} {\left | b \right |} + 2 i \, {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )}^{2} b + 2 i \, a^{2} b + 4 \, {\left (i \, x {\left | b \right |} - i \, \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} a {\left | b \right |} - a b - {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} {\left | b \right |}\right )}{{\left | b \right |}} - \frac {i \, \sqrt {{\left (b x + a\right )}^{2} + 1}}{b} \]
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Timed out. \[ \int e^{3 i \arctan (a+b x)} \, dx=\int \frac {{\left (1+a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}\right )}^3}{{\left ({\left (a+b\,x\right )}^2+1\right )}^{3/2}} \,d x \]
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