Integrand size = 16, antiderivative size = 134 \[ \int \frac {e^{-3 i \arctan (a+b x)}}{x} \, dx=\frac {4 \sqrt {1-i a-i b x}}{(1+i a) \sqrt {1+i a+i b x}}+i \text {arcsinh}(a+b x)-\frac {2 (i+a)^{3/2} \text {arctanh}\left (\frac {\sqrt {i+a} \sqrt {1+i a+i b x}}{\sqrt {i-a} \sqrt {1-i a-i b x}}\right )}{(i-a)^{3/2}} \]
[Out]
Time = 0.07 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5203, 100, 163, 55, 633, 221, 95, 214} \[ \int \frac {e^{-3 i \arctan (a+b x)}}{x} \, dx=i \text {arcsinh}(a+b x)-\frac {2 (a+i)^{3/2} \text {arctanh}\left (\frac {\sqrt {a+i} \sqrt {i a+i b x+1}}{\sqrt {-a+i} \sqrt {-i a-i b x+1}}\right )}{(-a+i)^{3/2}}+\frac {4 \sqrt {-i a-i b x+1}}{(1+i a) \sqrt {i a+i b x+1}} \]
[In]
[Out]
Rule 55
Rule 95
Rule 100
Rule 163
Rule 214
Rule 221
Rule 633
Rule 5203
Rubi steps \begin{align*} \text {integral}& = \int \frac {(1-i a-i b x)^{3/2}}{x (1+i a+i b x)^{3/2}} \, dx \\ & = \frac {4 \sqrt {1-i a-i b x}}{(1+i a) \sqrt {1+i a+i b x}}+\frac {2 \int \frac {-\frac {1}{2} i (i+a)^2 b-\frac {1}{2} (1+i a) b^2 x}{x \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}} \, dx}{(i-a) b} \\ & = \frac {4 \sqrt {1-i a-i b x}}{(1+i a) \sqrt {1+i a+i b x}}-\frac {(i+a)^2 \int \frac {1}{x \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}} \, dx}{1+i a}+(i b) \int \frac {1}{\sqrt {1-i a-i b x} \sqrt {1+i a+i b x}} \, dx \\ & = \frac {4 \sqrt {1-i a-i b x}}{(1+i a) \sqrt {1+i a+i b x}}-\frac {\left (2 (i+a)^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 )}{1+i a}+(i b) \int \frac {1}{\sqrt {(1-i a) (1+i a)+2 a b x+b^2 x^2}} \, dx \\ & = \frac {4 \sqrt {1-i a-i b x}}{(1+i a) \sqrt {1+i a+i b x}}-\frac {2 (i+a)^{3/2} \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 )}{(i-a)^{3/2}}+\frac {i \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} \\ & = \frac {4 \sqrt {1-i a-i b x}}{(1+i a) \sqrt {1+i a+i b x}}+i \text {arcsinh}(a+b x)-\frac {2 (i+a)^{3/2} \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 )}{(i-a)^{3/2}} \\ \end{align*}
Time = 0.48 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.41 \[ \int \frac {e^{-3 i \arctan (a+b x)}}{x} \, dx=\frac {2 (-1)^{3/4} \sqrt {-i b} \text {arcsinh}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {b} \sqrt {-i (i+a+b x)}}{\sqrt {-i b}}\right )}{\sqrt {b}}+\frac {2 \left (-\frac {2 \sqrt {1+a^2+2 a b x+b^2 x^2}}{-i+a+b x}+\frac {\sqrt {-1+i a} (i+a) \text {arctanh}\left (\frac {\sqrt {-1-i a} \sqrt {-i (i+a+b x)}}{\sqrt {-1+i a} \sqrt {1+i a+i b x}}\right )}{\sqrt {-1-i a}}\right )}{-i+a} \]
[In]
[Out]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1066 vs. \(2 (104 ) = 208\).
Time = 0.81 (sec) , antiderivative size = 1067, normalized size of antiderivative = 7.96
method | result | size |
default | \(-\frac {i \left (\frac {\left (b^{2} x^{2}+2 a b x +a^{2}+1\right )^{\frac {3}{2}}}{3}+a b \left (\frac {\left (2 b^{2} x +2 a b \right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{4 b^{2}}+\frac {\left (4 b^{2} \left (a^{2}+1\right )-4 a^{2} b^{2}\right ) \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{8 b^{2} \sqrt {b^{2}}}\right )+\left (a^{2}+1\right ) \left (\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}+\frac {a b \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}}}-\sqrt {a^{2}+1}\, \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 )\right )\right )}{\left (i-a \right )^{3}}-\frac {i \left (-\frac {i \left (\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )\right )^{\frac {5}{2}}}{b \left (x -\frac {i-a}{b}\right )^{2}}+3 i b \left (\frac {\left (\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )\right )^{\frac {3}{2}}}{3}+i b \left (\frac {\left (2 \left (x -\frac {i-a}{b}\right ) b^{2}+2 i b \right ) \sqrt {\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )}}{4 b^{2}}+\frac {\ln \left (\frac {i b +\left (x -\frac {i-a}{b}\right ) b^{2}}{\sqrt {b^{2}}}+\sqrt {\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )}\right )}{2 \sqrt {b^{2}}}\right )\right )\right )}{\left (i-a \right )^{2} b}+\frac {i \left (\frac {i \left (\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )\right )^{\frac {5}{2}}}{b \left (x -\frac {i-a}{b}\right )^{3}}-2 i b \left (-\frac {i \left (\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )\right )^{\frac {5}{2}}}{b \left (x -\frac {i-a}{b}\right )^{2}}+3 i b \left (\frac {\left (\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )\right )^{\frac {3}{2}}}{3}+i b \left (\frac {\left (2 \left (x -\frac {i-a}{b}\right ) b^{2}+2 i b \right ) \sqrt {\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )}}{4 b^{2}}+\frac {\ln \left (\frac {i b +\left (x -\frac {i-a}{b}\right ) b^{2}}{\sqrt {b^{2}}}+\sqrt {\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )}\right )}{2 \sqrt {b^{2}}}\right )\right )\right )\right )}{\left (i-a \right ) b^{2}}+\frac {i \left (\frac {\left (\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )\right )^{\frac {3}{2}}}{3}+i b \left (\frac {\left (2 \left (x -\frac {i-a}{b}\right ) b^{2}+2 i b \right ) \sqrt {\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )}}{4 b^{2}}+\frac {\ln \left (\frac {i b +\left (x -\frac {i-a}{b}\right ) b^{2}}{\sqrt {b^{2}}}+\sqrt {\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )}\right )}{2 \sqrt {b^{2}}}\right )\right )}{\left (i-a \right )^{3}}\) | \(1067\) |
[In]
[Out]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 356 vs. \(2 (90) = 180\).
Time = 0.27 (sec) , antiderivative size = 356, normalized size of antiderivative = 2.66 \[ \int \frac {e^{-3 i \arctan (a+b x)}}{x} \, dx=\frac {{\left ({\left (a - i\right )} b x + a^{2} - 2 i \, a - 1\right )} \sqrt {-\frac {a^{3} + 3 i \, a^{2} - 3 \, a - i}{a^{3} - 3 i \, a^{2} - 3 \, a + i}} \log \left (-\frac {{\left (a + i\right )} b x - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (a + i\right )} - {\left (i \, a^{2} + 2 \, a - i\right )} \sqrt {-\frac {a^{3} + 3 i \, a^{2} - 3 \, a - i}{a^{3} - 3 i \, a^{2} - 3 \, a + i}}}{a + i}\right ) - {\left ({\left (a - i\right )} b x + a^{2} - 2 i \, a - 1\right )} \sqrt {-\frac {a^{3} + 3 i \, a^{2} - 3 \, a - i}{a^{3} - 3 i \, a^{2} - 3 \, a + i}} \log \left (-\frac {{\left (a + i\right )} b x - \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} {\left (a + i\right )} - {\left (-i \, a^{2} - 2 \, a + i\right )} \sqrt {-\frac {a^{3} + 3 i \, a^{2} - 3 \, a - i}{a^{3} - 3 i \, a^{2} - 3 \, a + i}}}{a + i}\right ) - 4 \, b x - {\left ({\left (i \, a + 1\right )} b x + i \, a^{2} + 2 \, a - i\right )} \log \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) - 4 \, a - 4 \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} + 4 i}{{\left (a - i\right )} b x + a^{2} - 2 i \, a - 1} \]
[In]
[Out]
\[ \int \frac {e^{-3 i \arctan (a+b x)}}{x} \, dx=i \left (\int \frac {\sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{a^{3} x + 3 a^{2} b x^{2} - 3 i a^{2} x + 3 a b^{2} x^{3} - 6 i a b x^{2} - 3 a x + b^{3} x^{4} - 3 i b^{2} x^{3} - 3 b x^{2} + i x}\, dx + \int \frac {a^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{a^{3} x + 3 a^{2} b x^{2} - 3 i a^{2} x + 3 a b^{2} x^{3} - 6 i a b x^{2} - 3 a x + b^{3} x^{4} - 3 i b^{2} x^{3} - 3 b x^{2} + i x}\, dx + \int \frac {b^{2} x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{a^{3} x + 3 a^{2} b x^{2} - 3 i a^{2} x + 3 a b^{2} x^{3} - 6 i a b x^{2} - 3 a x + b^{3} x^{4} - 3 i b^{2} x^{3} - 3 b x^{2} + i x}\, dx + \int \frac {2 a b x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{a^{3} x + 3 a^{2} b x^{2} - 3 i a^{2} x + 3 a b^{2} x^{3} - 6 i a b x^{2} - 3 a x + b^{3} x^{4} - 3 i b^{2} x^{3} - 3 b x^{2} + i x}\, dx\right ) \]
[In]
[Out]
\[ \int \frac {e^{-3 i \arctan (a+b x)}}{x} \, dx=\int { \frac {{\left ({\left (b x + a\right )}^{2} + 1\right )}^{\frac {3}{2}}}{{\left (i \, b x + i \, a + 1\right )}^{3} x} \,d x } \]
[In]
[Out]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 252 vs. \(2 (90) = 180\).
Time = 0.41 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.88 \[ \int \frac {e^{-3 i \arctan (a+b x)}}{x} \, dx=-\frac {i \, b \log \left (-3 \, {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )}^{2} a b - a^{3} b - {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )}^{3} {\left | b \right |} - 3 \, {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} a^{2} {\left | b \right |} + 2 i \, {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )}^{2} b + 2 i \, a^{2} b - 4 \, {\left (-i \, x {\left | b \right |} + i \, \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} a {\left | b \right |} + a b + {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} {\left | b \right |}\right )}{3 \, {\left | b \right |}} - \frac {{\left (-i \, a^{2} + 2 \, a + i\right )} \log \left (\frac {{\left | -2 \, x {\left | b \right |} + 2 \, \sqrt {{\left (b x + a\right )}^{2} + 1} - 2 \, \sqrt {a^{2} + 1} \right |}}{{\left | -2 \, x {\left | b \right |} + 2 \, \sqrt {{\left (b x + a\right )}^{2} + 1} + 2 \, \sqrt {a^{2} + 1} \right |}}\right )}{\sqrt {a^{2} + 1} {\left (a - i\right )}} \]
[In]
[Out]
Timed out. \[ \int \frac {e^{-3 i \arctan (a+b x)}}{x} \, dx=\int \frac {{\left ({\left (a+b\,x\right )}^2+1\right )}^{3/2}}{x\,{\left (1+a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}\right )}^3} \,d x \]
[In]
[Out]