Integrand size = 16, antiderivative size = 264 \[ \int \frac {e^{3 i \arctan (a+b x)}}{x^3} \, dx=\frac {3 (3 i-2 a) b^2 \sqrt {1+i a+i b x}}{(1+i a) (i+a)^3 \sqrt {1-i a-i b x}}+\frac {(3 i-2 a) b (1+i a+i b x)^{3/2}}{2 (1+i a) (i+a)^2 x \sqrt {1-i a-i b x}}-\frac {(1+i a+i b x)^{5/2}}{2 \left (1+a^2\right ) x^2 \sqrt {1-i a-i b x}}+\frac {3 (3+2 i a) b^2 \text {arctanh}\left (\frac {\sqrt {i+a} \sqrt {1+i a+i b x}}{\sqrt {i-a} \sqrt {1-i a-i b x}}\right )}{\sqrt {i-a} (i+a)^{7/2}} \]
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Time = 0.11 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {5203, 98, 96, 95, 214} \[ \int \frac {e^{3 i \arctan (a+b x)}}{x^3} \, dx=-\frac {(i a+i b x+1)^{5/2}}{2 \left (a^2+1\right ) x^2 \sqrt {-i a-i b x+1}}+\frac {3 (3+2 i a) b^2 \text {arctanh}\left (\frac {\sqrt {a+i} \sqrt {i a+i b x+1}}{\sqrt {-a+i} \sqrt {-i a-i b x+1}}\right )}{\sqrt {-a+i} (a+i)^{7/2}}+\frac {3 (-2 a+3 i) b^2 \sqrt {i a+i b x+1}}{(1+i a) (a+i)^3 \sqrt {-i a-i b x+1}}+\frac {(-2 a+3 i) b (i a+i b x+1)^{3/2}}{2 (1+i a) (a+i)^2 x \sqrt {-i a-i b x+1}} \]
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Rule 95
Rule 96
Rule 98
Rule 214
Rule 5203
Rubi steps \begin{align*} \text {integral}& = \int \frac {(1+i a+i b x)^{3/2}}{x^3 (1-i a-i b x)^{3/2}} \, dx \\ & = -\frac {(1+i a+i b x)^{5/2}}{2 \left (1+a^2\right ) x^2 \sqrt {1-i a-i b x}}+\frac {((3 i-2 a) b) \int \frac {(1+i a+i b x)^{3/2}}{x^2 (1-i a-i b x)^{3/2}} \, dx}{2 \left (1+a^2\right )} \\ & = -\frac {(3 i-2 a) b (1+i a+i b x)^{3/2}}{2 (1-i a) \left (1+a^2\right ) x \sqrt {1-i a-i b x}}-\frac {(1+i a+i b x)^{5/2}}{2 \left (1+a^2\right ) x^2 \sqrt {1-i a-i b x}}-\frac {\left (3 (3 i-2 a) b^2\right ) \int \frac {\sqrt {1+i a+i b x}}{x (1-i a-i b x)^{3/2}} \, dx}{2 (i+a) \left (1+a^2\right )} \\ & = -\frac {3 (3 i-2 a) b^2 \sqrt {1+i a+i b x}}{(i-a) (1-i a)^3 \sqrt {1-i a-i b x}}-\frac {(3 i-2 a) b (1+i a+i b x)^{3/2}}{2 (1-i a) \left (1+a^2\right ) x \sqrt {1-i a-i b x}}-\frac {(1+i a+i b x)^{5/2}}{2 \left (1+a^2\right ) x^2 \sqrt {1-i a-i b x}}+\frac {\left (3 (3 i-2 a) b^2\right ) \int \frac {1}{x \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}} \, dx}{2 (i+a)^3} \\ & = -\frac {3 (3 i-2 a) b^2 \sqrt {1+i a+i b x}}{(i-a) (1-i a)^3 \sqrt {1-i a-i b x}}-\frac {(3 i-2 a) b (1+i a+i b x)^{3/2}}{2 (1-i a) \left (1+a^2\right ) x \sqrt {1-i a-i b x}}-\frac {(1+i a+i b x)^{5/2}}{2 \left (1+a^2\right ) x^2 \sqrt {1-i a-i b x}}+\frac {\left (3 (3 i-2 a) b^2\right ) \text {Subst}\left (\int \frac {1}{-1-i a-(-1+i a) x^2} \, dx,x,\frac {\sqrt {1+i a+i b x}}{\sqrt {1-i a-i b x}}\right )}{(i+a)^3} \\ & = -\frac {3 (3 i-2 a) b^2 \sqrt {1+i a+i b x}}{(i-a) (1-i a)^3 \sqrt {1-i a-i b x}}-\frac {(3 i-2 a) b (1+i a+i b x)^{3/2}}{2 (1-i a) \left (1+a^2\right ) x \sqrt {1-i a-i b x}}-\frac {(1+i a+i b x)^{5/2}}{2 \left (1+a^2\right ) x^2 \sqrt {1-i a-i b x}}+\frac {3 (3+2 i a) b^2 \text {arctanh}\left (\frac {\sqrt {i+a} \sqrt {1+i a+i b x}}{\sqrt {i-a} \sqrt {1-i a-i b x}}\right )}{\sqrt {i-a} (i+a)^{7/2}} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.73 \[ \int \frac {e^{3 i \arctan (a+b x)}}{x^3} \, dx=\frac {\frac {\sqrt {1+i a+i b x} \left (i+a+i a^2+a^3-5 b x+5 i a b x+14 i b^2 x^2-a b^2 x^2\right )}{x^2 \sqrt {-i (i+a+b x)}}-\frac {6 i \sqrt {-1-i a} (-3 i+2 a) b^2 \text {arctanh}\left (\frac {\sqrt {-1-i a} \sqrt {-i (i+a+b x)}}{\sqrt {-1+i a} \sqrt {1+i a+i b x}}\right )}{\sqrt {-1+i a} (-i+a)}}{2 (i+a)^3} \]
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Time = 1.18 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.02
method | result | size |
risch | \(\frac {i \left (-a \,b^{3} x^{3}+6 i b^{3} x^{3}-a^{2} b^{2} x^{2}+12 i a \,b^{2} x^{2}+a^{3} b x +6 i a^{2} b x +a^{4}+b^{2} x^{2}+a b x +6 b x i+2 a^{2}+1\right )}{2 x^{2} \left (i+a \right )^{3} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}+\frac {b^{2} \left (-\frac {\left (-6 a^{2}+3 i a -9\right ) \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right )}{\left (i+a \right ) \sqrt {a^{2}+1}}+\frac {8 i \left (i a -1\right ) \sqrt {\left (x +\frac {i+a}{b}\right )^{2} b^{2}-2 i b \left (x +\frac {i+a}{b}\right )}}{b \left (i+a \right ) \left (x +\frac {i+a}{b}\right )}\right )}{2 a^{3}+6 i a^{2}-6 a -2 i}\) | \(269\) |
default | \(-\frac {2 i b^{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 {1}{\left (a^{2}+1\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {2 a b \left (2 b^{2} x +2 a b \right )}{\left (a^{2}+1\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}}-\frac {\ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right )}{\left (a^{2}+1\right )^{\frac {3}{2}}}\right )-3 b \left (i a^{2}+2 a -i\right ) \left (-\frac {1}{\left (a^{2}+1\right ) x \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {3 a b \left (\frac {1}{\left (a^{2}+1\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {2 a b \left (2 b^{2} x +2 a b \right )}{\left (a^{2}+1\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}}-\frac {\ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right )}{\left (a^{2}+1\right )^{\frac {3}{2}}}\right )}{a^{2}+1}-\frac {4 b^{2} \left (2 b^{2} x +2 a b \right )}{\left (a^{2}+1\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}}\right )+\left (-i a^{3}-3 a^{2}+3 i a +1\right ) \left (-\frac {1}{2 \left (a^{2}+1\right ) x^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {5 a b \left (-\frac {1}{\left (a^{2}+1\right ) x \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {3 a b \left (\frac {1}{\left (a^{2}+1\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {2 a b \left (2 b^{2} x +2 a b \right )}{\left (a^{2}+1\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}}-\frac {\ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right )}{\left (a^{2}+1\right )^{\frac {3}{2}}}\right )}{a^{2}+1}-\frac {4 b^{2} \left (2 b^{2} x +2 a b \right )}{\left (a^{2}+1\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}}\right )}{2 \left (a^{2}+1\right )}-\frac {3 b^{2} \left (\frac {1}{\left (a^{2}+1\right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {2 a b \left (2 b^{2} x +2 a b \right )}{\left (a^{2}+1\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}}-\frac {\ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right )}{\left (a^{2}+1\right )^{\frac {3}{2}}}\right )}{2 \left (a^{2}+1\right )}\right )\) | \(946\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 574 vs. \(2 (182) = 364\).
Time = 0.28 (sec) , antiderivative size = 574, normalized size of antiderivative = 2.17 \[ \int \frac {e^{3 i \arctan (a+b x)}}{x^3} \, dx=\frac {{\left (-i \, a - 14\right )} b^{3} x^{3} + {\left (-i \, a^{2} - 13 \, a - 14 i\right )} b^{2} x^{2} - 3 \, {\left ({\left (a^{3} + 3 i \, a^{2} - 3 \, a - i\right )} b x^{3} + {\left (a^{4} + 4 i \, a^{3} - 6 \, a^{2} - 4 i \, a + 1\right )} x^{2}\right )} \sqrt {\frac {{\left (4 \, a^{2} - 12 i \, a - 9\right )} b^{4}}{a^{8} + 6 i \, a^{7} - 14 \, a^{6} - 14 i \, a^{5} - 14 i \, a^{3} + 14 \, a^{2} + 6 i \, a - 1}} \log \left (-\frac {{\left (2 \, a - 3 i\right )} b^{3} x - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (2 \, a - 3 i\right )} b^{2} + {\left (a^{5} + 3 i \, a^{4} - 2 \, a^{3} + 2 i \, a^{2} - 3 \, a - i\right )} \sqrt {\frac {{\left (4 \, a^{2} - 12 i \, a - 9\right )} b^{4}}{a^{8} + 6 i \, a^{7} - 14 \, a^{6} - 14 i \, a^{5} - 14 i \, a^{3} + 14 \, a^{2} + 6 i \, a - 1}}}{{\left (2 \, a - 3 i\right )} b^{2}}\right ) + 3 \, {\left ({\left (a^{3} + 3 i \, a^{2} - 3 \, a - i\right )} b x^{3} + {\left (a^{4} + 4 i \, a^{3} - 6 \, a^{2} - 4 i \, a + 1\right )} x^{2}\right )} \sqrt {\frac {{\left (4 \, a^{2} - 12 i \, a - 9\right )} b^{4}}{a^{8} + 6 i \, a^{7} - 14 \, a^{6} - 14 i \, a^{5} - 14 i \, a^{3} + 14 \, a^{2} + 6 i \, a - 1}} \log \left (-\frac {{\left (2 \, a - 3 i\right )} b^{3} x - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (2 \, a - 3 i\right )} b^{2} - {\left (a^{5} + 3 i \, a^{4} - 2 \, a^{3} + 2 i \, a^{2} - 3 \, a - i\right )} \sqrt {\frac {{\left (4 \, a^{2} - 12 i \, a - 9\right )} b^{4}}{a^{8} + 6 i \, a^{7} - 14 \, a^{6} - 14 i \, a^{5} - 14 i \, a^{3} + 14 \, a^{2} + 6 i \, a - 1}}}{{\left (2 \, a - 3 i\right )} b^{2}}\right ) + {\left ({\left (-i \, a - 14\right )} b^{2} x^{2} + i \, a^{3} - 5 \, {\left (a + i\right )} b x - a^{2} + i \, a - 1\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{2 \, {\left ({\left (a^{3} + 3 i \, a^{2} - 3 \, a - i\right )} b x^{3} + {\left (a^{4} + 4 i \, a^{3} - 6 \, a^{2} - 4 i \, a + 1\right )} x^{2}\right )}} \]
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\[ \int \frac {e^{3 i \arctan (a+b x)}}{x^3} \, dx=- i \left (\int \frac {i}{a^{2} x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x^{4} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{5} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\, dx + \int \left (- \frac {3 a}{a^{2} x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x^{4} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{5} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\right )\, dx + \int \frac {a^{3}}{a^{2} x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x^{4} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{5} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\, dx + \int \left (- \frac {3 i a^{2}}{a^{2} x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x^{4} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{5} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\right )\, dx + \int \left (- \frac {3 b x}{a^{2} x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x^{4} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{5} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\right )\, dx + \int \frac {b^{3} x^{3}}{a^{2} x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x^{4} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{5} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\, dx + \int \left (- \frac {3 i b^{2} x^{2}}{a^{2} x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x^{4} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{5} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\right )\, dx + \int \frac {3 a b^{2} x^{2}}{a^{2} x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x^{4} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{5} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\, dx + \int \frac {3 a^{2} b x}{a^{2} x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x^{4} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{5} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}\, dx + \int \left (- \frac {6 i a b x}{a^{2} x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + 2 a b x^{4} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + b^{2} x^{5} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} + x^{3} \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. 1536 vs. \(2 (182) = 364\).
Time = 0.21 (sec) , antiderivative size = 1536, normalized size of antiderivative = 5.82 \[ \int \frac {e^{3 i \arctan (a+b x)}}{x^3} \, dx=\text {Too large to display} \]
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\[ \int \frac {e^{3 i \arctan (a+b x)}}{x^3} \, dx=\int { \frac {{\left (i \, b x + i \, a + 1\right )}^{3}}{{\left ({\left (b x + a\right )}^{2} + 1\right )}^{\frac {3}{2}} x^{3}} \,d x } \]
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Timed out. \[ \int \frac {e^{3 i \arctan (a+b x)}}{x^3} \, dx=\int \frac {{\left (1+a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}\right )}^3}{x^3\,{\left ({\left (a+b\,x\right )}^2+1\right )}^{3/2}} \,d x \]
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