Integrand size = 16, antiderivative size = 339 \[ \int \frac {e^{-3 i \arctan (a+b x)}}{x^4} \, dx=-\frac {\left (52-51 i a-2 a^2\right ) b^3 \sqrt {1-i a-i b x}}{6 (i-a)^4 (i+a) \sqrt {1+i a+i b x}}-\frac {(i+a) \sqrt {1-i a-i b x}}{3 (i-a) x^3 \sqrt {1+i a+i b x}}-\frac {7 i b \sqrt {1-i a-i b x}}{6 (i-a)^2 x^2 \sqrt {1+i a+i b x}}+\frac {(19-16 i a) b^2 \sqrt {1-i a-i b x}}{6 (i-a)^3 (i+a) x \sqrt {1+i a+i b x}}+\frac {\left (11 i+18 a-6 i a^2\right ) b^3 \text {arctanh}\left (\frac {\sqrt {i+a} \sqrt {1+i a+i b x}}{\sqrt {i-a} \sqrt {1-i a-i b x}}\right )}{(i-a)^{9/2} (i+a)^{3/2}} \]
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Time = 0.20 (sec) , antiderivative size = 339, 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, 100, 156, 157, 12, 95, 214} \[ \int \frac {e^{-3 i \arctan (a+b x)}}{x^4} \, dx=\frac {\left (-6 i a^2+18 a+11 i\right ) b^3 \text {arctanh}\left (\frac {\sqrt {a+i} \sqrt {i a+i b x+1}}{\sqrt {-a+i} \sqrt {-i a-i b x+1}}\right )}{(-a+i)^{9/2} (a+i)^{3/2}}-\frac {\left (-2 a^2-51 i a+52\right ) b^3 \sqrt {-i a-i b x+1}}{6 (-a+i)^4 (a+i) \sqrt {i a+i b x+1}}+\frac {(19-16 i a) b^2 \sqrt {-i a-i b x+1}}{6 (-a+i)^3 (a+i) x \sqrt {i a+i b x+1}}-\frac {(a+i) \sqrt {-i a-i b x+1}}{3 (-a+i) x^3 \sqrt {i a+i b x+1}}-\frac {7 i b \sqrt {-i a-i b x+1}}{6 (-a+i)^2 x^2 \sqrt {i a+i b x+1}} \]
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Rule 12
Rule 95
Rule 100
Rule 156
Rule 157
Rule 214
Rule 5203
Rubi steps \begin{align*} \text {integral}& = \int \frac {(1-i a-i b x)^{3/2}}{x^4 (1+i a+i b x)^{3/2}} \, dx \\ & = -\frac {(i+a) \sqrt {1-i a-i b x}}{3 (i-a) x^3 \sqrt {1+i a+i b x}}-\frac {\int \frac {7 (i+a) b+6 b^2 x}{x^3 \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}} \, dx}{3 (1+i a)} \\ & = -\frac {(i+a) \sqrt {1-i a-i b x}}{3 (i-a) x^3 \sqrt {1+i a+i b x}}-\frac {7 i b \sqrt {1-i a-i b x}}{6 (i-a)^2 x^2 \sqrt {1+i a+i b x}}+\frac {\int \frac {-\left (\left (19-35 i a-16 a^2\right ) b^2\right )+14 (i+a) b^3 x}{x^2 \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}} \, dx}{6 (1+i a) \left (1+a^2\right )} \\ & = -\frac {(i+a) \sqrt {1-i a-i b x}}{3 (i-a) x^3 \sqrt {1+i a+i b x}}-\frac {7 i b \sqrt {1-i a-i b x}}{6 (i-a)^2 x^2 \sqrt {1+i a+i b x}}+\frac {(19 i+16 a) b^2 \sqrt {1-i a-i b x}}{6 (1+i a)^3 (i+a) x \sqrt {1+i a+i b x}}-\frac {\int \frac {-3 (i+a) \left (11-18 i a-6 a^2\right ) b^3-\left (19-35 i a-16 a^2\right ) b^4 x}{x \sqrt {1-i a-i b x} (1+i a+i b x)^{3/2}} \, dx}{6 (1+i a) \left (1+a^2\right )^2} \\ & = -\frac {\left (52-51 i a-2 a^2\right ) b^3 \sqrt {1-i a-i b x}}{6 (i-a)^4 (i+a) \sqrt {1+i a+i b x}}-\frac {(i+a) \sqrt {1-i a-i b x}}{3 (i-a) x^3 \sqrt {1+i a+i b x}}-\frac {7 i b \sqrt {1-i a-i b x}}{6 (i-a)^2 x^2 \sqrt {1+i a+i b x}}+\frac {(19 i+16 a) b^2 \sqrt {1-i a-i b x}}{6 (1+i a)^3 (i+a) x \sqrt {1+i a+i b x}}-\frac {i \int \frac {3 \left (11-29 i a-24 a^2+6 i a^3\right ) b^4}{x \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}} \, dx}{6 (i-a)^4 (i+a)^2 b} \\ & = -\frac {\left (52-51 i a-2 a^2\right ) b^3 \sqrt {1-i a-i b x}}{6 (i-a)^4 (i+a) \sqrt {1+i a+i b x}}-\frac {(i+a) \sqrt {1-i a-i b x}}{3 (i-a) x^3 \sqrt {1+i a+i b x}}-\frac {7 i b \sqrt {1-i a-i b x}}{6 (i-a)^2 x^2 \sqrt {1+i a+i b x}}+\frac {(19 i+16 a) b^2 \sqrt {1-i a-i b x}}{6 (1+i a)^3 (i+a) x \sqrt {1+i a+i b x}}-\frac {\left (\left (11-18 i a-6 a^2\right ) b^3\right ) \int \frac {1}{x \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}} \, dx}{2 (i-a)^4 (i+a)} \\ & = -\frac {\left (52-51 i a-2 a^2\right ) b^3 \sqrt {1-i a-i b x}}{6 (i-a)^4 (i+a) \sqrt {1+i a+i b x}}-\frac {(i+a) \sqrt {1-i a-i b x}}{3 (i-a) x^3 \sqrt {1+i a+i b x}}-\frac {7 i b \sqrt {1-i a-i b x}}{6 (i-a)^2 x^2 \sqrt {1+i a+i b x}}+\frac {(19 i+16 a) b^2 \sqrt {1-i a-i b x}}{6 (1+i a)^3 (i+a) x \sqrt {1+i a+i b x}}-\frac {\left (\left (11-18 i a-6 a^2\right ) b^3\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)^4 (i+a)} \\ & = -\frac {\left (52-51 i a-2 a^2\right ) b^3 \sqrt {1-i a-i b x}}{6 (i-a)^4 (i+a) \sqrt {1+i a+i b x}}-\frac {(i+a) \sqrt {1-i a-i b x}}{3 (i-a) x^3 \sqrt {1+i a+i b x}}-\frac {7 i b \sqrt {1-i a-i b x}}{6 (i-a)^2 x^2 \sqrt {1+i a+i b x}}+\frac {(19 i+16 a) b^2 \sqrt {1-i a-i b x}}{6 (1+i a)^3 (i+a) x \sqrt {1+i a+i b x}}+\frac {\left (11 i+18 a-6 i a^2\right ) b^3 \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 )}{(i-a)^{9/2} (i+a)^{3/2}} \\ \end{align*}
Time = 0.32 (sec) , antiderivative size = 275, normalized size of antiderivative = 0.81 \[ \int \frac {e^{-3 i \arctan (a+b x)}}{x^4} \, dx=-\frac {-2 (-1-i a)^{7/2} (1-i a) (-i (i+a+b x))^{5/2}-(-1-i a)^{5/2} (3 i+4 a) b x (-i (i+a+b x))^{5/2}+i \left (-11+18 i a+6 a^2\right ) b^2 x^2 \left (-i \sqrt {-1-i a} \sqrt {-i (i+a+b x)} \left (1+a^2+5 i b x+a b x\right )-6 \sqrt {-1+i a} b x \sqrt {1+i a+i b x} \text {arctanh}\left (\frac {\sqrt {-1-i a} \sqrt {-i (i+a+b x)}}{\sqrt {-1+i a} \sqrt {1+i a+i b x}}\right )\right )}{6 (-1-i a)^{5/2} \left (1+a^2\right )^2 x^3 \sqrt {1+i a+i b x}} \]
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Time = 2.11 (sec) , antiderivative size = 392, normalized size of antiderivative = 1.16
method | result | size |
risch | \(-\frac {i \left (2 a^{2} b^{4} x^{4}+27 i a \,b^{4} x^{4}+2 a^{3} b^{3} x^{3}+45 i a^{2} b^{3} x^{3}+9 i b^{2} x^{2} a^{3}-28 x^{4} b^{4}+2 a^{5} b x -9 i x \,a^{4} b -58 a \,b^{3} x^{3}-9 i b^{3} x^{3}+2 a^{6}-26 a^{2} b^{2} x^{2}+9 i a \,b^{2} x^{2}+4 a^{3} b x -18 i a^{2} b x +6 a^{4}-26 b^{2} x^{2}+2 a b x -9 b x i+6 a^{2}+2\right )}{6 x^{3} \left (i+a \right ) \left (a -i\right )^{4} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}-\frac {b^{3} \left (-\frac {\left (6 a^{3}+12 i a^{2}+7 a +11 i\right ) \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right )}{\left (i-a \right ) \sqrt {a^{2}+1}}+\frac {8 \left (a^{2}+1\right ) \sqrt {\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )}}{b \left (i-a \right ) \left (x -\frac {i-a}{b}\right )}\right )}{2 \left (i+a \right ) \left (a^{4}-4 i a^{3}-6 a^{2}+4 i a +1\right )}\) | \(392\) |
default | \(\text {Expression too large to display}\) | \(3637\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 839 vs. \(2 (223) = 446\).
Time = 0.30 (sec) , antiderivative size = 839, normalized size of antiderivative = 2.47 \[ \int \frac {e^{-3 i \arctan (a+b x)}}{x^4} \, dx=\frac {{\left (-2 i \, a^{2} + 51 \, a + 52 i\right )} b^{4} x^{4} + {\left (-2 i \, a^{3} + 49 \, a^{2} + i \, a + 52\right )} b^{3} x^{3} + 3 \, \sqrt {\frac {{\left (36 \, a^{4} + 216 i \, a^{3} - 456 \, a^{2} - 396 i \, a + 121\right )} b^{6}}{a^{12} - 6 i \, a^{11} - 12 \, a^{10} + 2 i \, a^{9} - 27 \, a^{8} + 36 i \, a^{7} + 36 i \, a^{5} + 27 \, a^{4} + 2 i \, a^{3} + 12 \, a^{2} - 6 i \, a - 1}} {\left ({\left (a^{5} - 3 i \, a^{4} - 2 \, a^{3} - 2 i \, a^{2} - 3 \, a + i\right )} b x^{4} + {\left (a^{6} - 4 i \, a^{5} - 5 \, a^{4} - 5 \, a^{2} + 4 i \, a + 1\right )} x^{3}\right )} \log \left (-\frac {{\left (6 \, a^{2} + 18 i \, a - 11\right )} b^{4} x - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (6 \, a^{2} + 18 i \, a - 11\right )} b^{3} + {\left (a^{7} - 3 i \, a^{6} - a^{5} - 5 i \, a^{4} - 5 \, a^{3} - i \, a^{2} - 3 \, a + i\right )} \sqrt {\frac {{\left (36 \, a^{4} + 216 i \, a^{3} - 456 \, a^{2} - 396 i \, a + 121\right )} b^{6}}{a^{12} - 6 i \, a^{11} - 12 \, a^{10} + 2 i \, a^{9} - 27 \, a^{8} + 36 i \, a^{7} + 36 i \, a^{5} + 27 \, a^{4} + 2 i \, a^{3} + 12 \, a^{2} - 6 i \, a - 1}}}{{\left (6 \, a^{2} + 18 i \, a - 11\right )} b^{3}}\right ) - 3 \, \sqrt {\frac {{\left (36 \, a^{4} + 216 i \, a^{3} - 456 \, a^{2} - 396 i \, a + 121\right )} b^{6}}{a^{12} - 6 i \, a^{11} - 12 \, a^{10} + 2 i \, a^{9} - 27 \, a^{8} + 36 i \, a^{7} + 36 i \, a^{5} + 27 \, a^{4} + 2 i \, a^{3} + 12 \, a^{2} - 6 i \, a - 1}} {\left ({\left (a^{5} - 3 i \, a^{4} - 2 \, a^{3} - 2 i \, a^{2} - 3 \, a + i\right )} b x^{4} + {\left (a^{6} - 4 i \, a^{5} - 5 \, a^{4} - 5 \, a^{2} + 4 i \, a + 1\right )} x^{3}\right )} \log \left (-\frac {{\left (6 \, a^{2} + 18 i \, a - 11\right )} b^{4} x - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (6 \, a^{2} + 18 i \, a - 11\right )} b^{3} - {\left (a^{7} - 3 i \, a^{6} - a^{5} - 5 i \, a^{4} - 5 \, a^{3} - i \, a^{2} - 3 \, a + i\right )} \sqrt {\frac {{\left (36 \, a^{4} + 216 i \, a^{3} - 456 \, a^{2} - 396 i \, a + 121\right )} b^{6}}{a^{12} - 6 i \, a^{11} - 12 \, a^{10} + 2 i \, a^{9} - 27 \, a^{8} + 36 i \, a^{7} + 36 i \, a^{5} + 27 \, a^{4} + 2 i \, a^{3} + 12 \, a^{2} - 6 i \, a - 1}}}{{\left (6 \, a^{2} + 18 i \, a - 11\right )} b^{3}}\right ) + {\left ({\left (-2 i \, a^{2} + 51 \, a + 52 i\right )} b^{3} x^{3} - 2 i \, a^{5} + {\left (16 \, a^{2} + 3 i \, a + 19\right )} b^{2} x^{2} - 2 \, a^{4} - 4 i \, a^{3} - 7 \, {\left (a^{3} - i \, a^{2} + a - i\right )} b x - 4 \, a^{2} - 2 i \, a - 2\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{6 \, {\left ({\left (a^{5} - 3 i \, a^{4} - 2 \, a^{3} - 2 i \, a^{2} - 3 \, a + i\right )} b x^{4} + {\left (a^{6} - 4 i \, a^{5} - 5 \, a^{4} - 5 \, a^{2} + 4 i \, a + 1\right )} x^{3}\right )}} \]
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Timed out. \[ \int \frac {e^{-3 i \arctan (a+b x)}}{x^4} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {e^{-3 i \arctan (a+b x)}}{x^4} \, dx=\text {Exception raised: RuntimeError} \]
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\[ \int \frac {e^{-3 i \arctan (a+b x)}}{x^4} \, dx=\int { \frac {{\left ({\left (b x + a\right )}^{2} + 1\right )}^{\frac {3}{2}}}{{\left (i \, b x + i \, a + 1\right )}^{3} x^{4}} \,d x } \]
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Timed out. \[ \int \frac {e^{-3 i \arctan (a+b x)}}{x^4} \, dx=\int \frac {{\left ({\left (a+b\,x\right )}^2+1\right )}^{3/2}}{x^4\,{\left (1+a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}\right )}^3} \,d x \]
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