Integrand size = 16, antiderivative size = 76 \[ \int \frac {e^{2 i \arctan (a+b x)}}{x^3} \, dx=-\frac {i-a}{2 (i+a) x^2}+\frac {2 i b}{(i+a)^2 x}-\frac {2 b^2 \log (x)}{(1-i a)^3}+\frac {2 b^2 \log (i+a+b x)}{(1-i a)^3} \]
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Time = 0.04 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {5203, 78} \[ \int \frac {e^{2 i \arctan (a+b x)}}{x^3} \, dx=-\frac {2 b^2 \log (x)}{(1-i a)^3}+\frac {2 b^2 \log (a+b x+i)}{(1-i a)^3}+\frac {2 i b}{(a+i)^2 x}-\frac {-a+i}{2 (a+i) x^2} \]
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Rule 78
Rule 5203
Rubi steps \begin{align*} \text {integral}& = \int \frac {1+i a+i b x}{x^3 (1-i a-i b x)} \, dx \\ & = \int \left (\frac {i-a}{(i+a) x^3}-\frac {2 i b}{(i+a)^2 x^2}+\frac {2 i b^2}{(i+a)^3 x}-\frac {2 i b^3}{(i+a)^3 (i+a+b x)}\right ) \, dx \\ & = -\frac {i-a}{2 (i+a) x^2}+\frac {2 i b}{(i+a)^2 x}-\frac {2 b^2 \log (x)}{(1-i a)^3}+\frac {2 b^2 \log (i+a+b x)}{(1-i a)^3} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.83 \[ \int \frac {e^{2 i \arctan (a+b x)}}{x^3} \, dx=\frac {(i+a) \left (1+a^2+4 i b x\right )+4 i b^2 x^2 \log (x)-4 i b^2 x^2 \log (i+a+b x)}{2 (i+a)^3 x^2} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 200 vs. \(2 (65 ) = 130\).
Time = 0.34 (sec) , antiderivative size = 201, normalized size of antiderivative = 2.64
method | result | size |
parallelrisch | \(\frac {4 i a^{4} b x +12 i \ln \left (b x +a +i\right ) x^{2} a \,b^{2}-1+4 i \ln \left (x \right ) x^{2} a^{3} b^{2}+12 \ln \left (x \right ) x^{2} a^{2} b^{2}-4 i \ln \left (b x +a +i\right ) x^{2} a^{3} b^{2}-12 \ln \left (b x +a +i\right ) x^{2} a^{2} b^{2}-4 i a^{3}-12 i \ln \left (x \right ) x^{2} a \,b^{2}+a^{6}-4 \ln \left (x \right ) x^{2} b^{2}+4 \ln \left (b x +a +i\right ) x^{2} b^{2}+8 a^{3} b x -2 i a^{5}+a^{4}-2 i a +8 a b x -4 b x i-a^{2}}{2 \left (a^{4}+2 a^{2}+1\right ) \left (a^{2}+1\right ) x^{2}}\) | \(201\) |
default | \(-\frac {-a^{2}+2 i a +1}{2 \left (a^{2}+1\right ) x^{2}}+\frac {2 b \left (i a^{2}+2 a -i\right )}{\left (a^{2}+1\right )^{2} x}+\frac {2 b^{2} \left (i a^{3}+3 a^{2}-3 i a -1\right ) \ln \left (x \right )}{\left (a^{2}+1\right )^{3}}-\frac {2 b^{3} \left (\frac {\left (i a^{3} b +3 a^{2} b -3 i a b -b \right ) \ln \left (b^{2} x^{2}+2 a b x +a^{2}+1\right )}{2 b^{2}}+\frac {\left (i a^{4}-6 i a^{2}+4 a^{3}+i-4 a -\frac {\left (i a^{3} b +3 a^{2} b -3 i a b -b \right ) a}{b}\right ) \arctan \left (\frac {2 b^{2} x +2 a b}{2 b}\right )}{b}\right )}{\left (a^{2}+1\right )^{3}}\) | \(211\) |
risch | \(\frac {\frac {2 i b x}{a^{2}+2 i a -1}+\frac {a -i}{2 i+2 a}}{x^{2}}+\frac {b^{2} \ln \left (4 a^{8} b^{2} x^{2}+8 a^{9} b x +4 a^{10}+16 a^{6} b^{2} x^{2}+32 a^{7} b x +20 a^{8}+24 a^{4} b^{2} x^{2}+48 a^{5} b x +40 a^{6}+16 a^{2} b^{2} x^{2}+32 a^{3} b x +40 a^{4}+4 b^{2} x^{2}+8 a b x +20 a^{2}+4\right )}{i a^{3}-3 a^{2}-3 i a +1}-\frac {2 i b^{2} \arctan \left (\frac {\left (-2 a^{4} b -4 a^{2} b -2 b \right ) x -2 a^{5}-4 a^{3}-2 a}{-2 a^{4}-4 a^{2}-2}\right )}{i a^{3}-3 a^{2}-3 i a +1}-\frac {2 b^{2} \ln \left (\left (2 a^{4} b +4 a^{2} b +2 b \right ) x \right )}{i a^{3}-3 a^{2}-3 i a +1}\) | \(287\) |
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Time = 0.27 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.91 \[ \int \frac {e^{2 i \arctan (a+b x)}}{x^3} \, dx=\frac {4 i \, b^{2} x^{2} \log \left (x\right ) - 4 i \, b^{2} x^{2} \log \left (\frac {b x + a + i}{b}\right ) + a^{3} - 4 \, {\left (-i \, a + 1\right )} b x + i \, a^{2} + a + i}{2 \, {\left (a^{3} + 3 i \, a^{2} - 3 \, a - i\right )} x^{2}} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 228 vs. \(2 (60) = 120\).
Time = 0.44 (sec) , antiderivative size = 228, normalized size of antiderivative = 3.00 \[ \int \frac {e^{2 i \arctan (a+b x)}}{x^3} \, dx=\frac {2 i b^{2} \log {\left (- \frac {2 a^{4} b^{2}}{\left (a + i\right )^{3}} - \frac {8 i a^{3} b^{2}}{\left (a + i\right )^{3}} + \frac {12 a^{2} b^{2}}{\left (a + i\right )^{3}} + 2 a b^{2} + \frac {8 i a b^{2}}{\left (a + i\right )^{3}} + 4 b^{3} x + 2 i b^{2} - \frac {2 b^{2}}{\left (a + i\right )^{3}} \right )}}{\left (a + i\right )^{3}} - \frac {2 i b^{2} \log {\left (\frac {2 a^{4} b^{2}}{\left (a + i\right )^{3}} + \frac {8 i a^{3} b^{2}}{\left (a + i\right )^{3}} - \frac {12 a^{2} b^{2}}{\left (a + i\right )^{3}} + 2 a b^{2} - \frac {8 i a b^{2}}{\left (a + i\right )^{3}} + 4 b^{3} x + 2 i b^{2} + \frac {2 b^{2}}{\left (a + i\right )^{3}} \right )}}{\left (a + i\right )^{3}} - \frac {- a^{2} - 4 i b x - 1}{x^{2} \cdot \left (2 a^{2} + 4 i a - 2\right )} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 188 vs. \(2 (58) = 116\).
Time = 0.27 (sec) , antiderivative size = 188, normalized size of antiderivative = 2.47 \[ \int \frac {e^{2 i \arctan (a+b x)}}{x^3} \, dx=-\frac {2 \, {\left (a^{3} - 3 i \, a^{2} - 3 \, a + i\right )} b^{2} \arctan \left (\frac {b^{2} x + a b}{b}\right )}{a^{6} + 3 \, a^{4} + 3 \, a^{2} + 1} - \frac {{\left (i \, a^{3} + 3 \, a^{2} - 3 i \, a - 1\right )} b^{2} \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}{a^{6} + 3 \, a^{4} + 3 \, a^{2} + 1} - \frac {2 \, {\left (-i \, a^{3} - 3 \, a^{2} + 3 i \, a + 1\right )} b^{2} \log \left (x\right )}{a^{6} + 3 \, a^{4} + 3 \, a^{2} + 1} + \frac {a^{4} - 2 i \, a^{3} - 4 \, {\left (-i \, a^{2} - 2 \, a + i\right )} b x - 2 i \, a - 1}{2 \, {\left (a^{4} + 2 \, a^{2} + 1\right )} x^{2}} \]
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Time = 0.29 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.17 \[ \int \frac {e^{2 i \arctan (a+b x)}}{x^3} \, dx=\frac {2 \, b^{3} \log \left (b x + a + i\right )}{i \, a^{3} b - 3 \, a^{2} b - 3 i \, a b + b} + \frac {2 \, b^{2} \log \left ({\left | x \right |}\right )}{-i \, a^{3} + 3 \, a^{2} + 3 i \, a - 1} + \frac {a^{3} + i \, a^{2} + 4 i \, {\left (a b + i \, b\right )} x + a + i}{2 \, {\left (a + i\right )}^{3} x^{2}} \]
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Time = 0.79 (sec) , antiderivative size = 154, normalized size of antiderivative = 2.03 \[ \int \frac {e^{2 i \arctan (a+b x)}}{x^3} \, dx=\frac {\frac {a-\mathrm {i}}{2\,\left (a+1{}\mathrm {i}\right )}+\frac {b\,x\,2{}\mathrm {i}}{{\left (a+1{}\mathrm {i}\right )}^2}}{x^2}+\frac {b^2\,\mathrm {atanh}\left (\frac {-a^3-a^2\,3{}\mathrm {i}+3\,a+1{}\mathrm {i}}{{\left (a+1{}\mathrm {i}\right )}^3}+\frac {x\,\left (2\,a^8\,b^2+8\,a^6\,b^2+12\,a^4\,b^2+8\,a^2\,b^2+2\,b^2\right )}{{\left (a+1{}\mathrm {i}\right )}^3\,\left (-b\,a^6+2{}\mathrm {i}\,b\,a^5-b\,a^4+4{}\mathrm {i}\,b\,a^3+b\,a^2+2{}\mathrm {i}\,b\,a+b\right )}\right )\,4{}\mathrm {i}}{{\left (a+1{}\mathrm {i}\right )}^3} \]
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