Integrand size = 25, antiderivative size = 69 \[ \int \frac {e^{4 i \arctan (a x)}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=-\frac {i c (1+i a x)^4}{3 a \left (c+a^2 c x^2\right )^{5/2}}+\frac {i c (1+i a x)^5}{15 a \left (c+a^2 c x^2\right )^{5/2}} \]
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Time = 0.05 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {5183, 673, 665} \[ \int \frac {e^{4 i \arctan (a x)}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\frac {i c (1+i a x)^5}{15 a \left (a^2 c x^2+c\right )^{5/2}}-\frac {i c (1+i a x)^4}{3 a \left (a^2 c x^2+c\right )^{5/2}} \]
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Rule 665
Rule 673
Rule 5183
Rubi steps \begin{align*} \text {integral}& = c^2 \int \frac {(1+i a x)^4}{\left (c+a^2 c x^2\right )^{7/2}} \, dx \\ & = -\frac {i c (1+i a x)^4}{3 a \left (c+a^2 c x^2\right )^{5/2}}-\frac {1}{3} c^2 \int \frac {(1+i a x)^5}{\left (c+a^2 c x^2\right )^{7/2}} \, dx \\ & = -\frac {i c (1+i a x)^4}{3 a \left (c+a^2 c x^2\right )^{5/2}}+\frac {i c (1+i a x)^5}{15 a \left (c+a^2 c x^2\right )^{5/2}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.12 \[ \int \frac {e^{4 i \arctan (a x)}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\frac {(1+i a x)^{3/2} (4 i+a x) \sqrt {1+a^2 x^2}}{15 a c \sqrt {1-i a x} (i+a x)^2 \sqrt {c+a^2 c x^2}} \]
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Time = 0.38 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.83
method | result | size |
gosper | \(\frac {\left (-a x +i\right ) \left (a x +i\right ) \left (a x +4 i\right ) \left (i a x +1\right )^{4}}{15 a \left (a^{2} x^{2}+1\right )^{2} \left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}\) | \(57\) |
trager | \(\frac {\left (-a^{5} x^{5}-10 a^{3} x^{3}+20 i x^{2} a^{2}+15 a x -4 i\right ) \sqrt {a^{2} c \,x^{2}+c}}{15 c^{2} \left (a^{2} x^{2}+1\right )^{3} a}\) | \(64\) |
default | \(\frac {x}{c \sqrt {a^{2} c \,x^{2}+c}}+\frac {2 \left (i \sqrt {-a^{2}}-a \right ) \left (\frac {1}{5 c \sqrt {-a^{2}}\, {\left (x +\frac {\sqrt {-a^{2}}}{a^{2}}\right )}^{2} \sqrt {{\left (x +\frac {\sqrt {-a^{2}}}{a^{2}}\right )}^{2} a^{2} c -2 c \sqrt {-a^{2}}\, \left (x +\frac {\sqrt {-a^{2}}}{a^{2}}\right )}}+\frac {3 a^{2} \left (\frac {1}{3 c \sqrt {-a^{2}}\, \left (x +\frac {\sqrt {-a^{2}}}{a^{2}}\right ) \sqrt {{\left (x +\frac {\sqrt {-a^{2}}}{a^{2}}\right )}^{2} a^{2} c -2 c \sqrt {-a^{2}}\, \left (x +\frac {\sqrt {-a^{2}}}{a^{2}}\right )}}+\frac {2 \left (x +\frac {\sqrt {-a^{2}}}{a^{2}}\right ) a^{2} c -2 c \sqrt {-a^{2}}}{3 c^{2} \sqrt {-a^{2}}\, \sqrt {{\left (x +\frac {\sqrt {-a^{2}}}{a^{2}}\right )}^{2} a^{2} c -2 c \sqrt {-a^{2}}\, \left (x +\frac {\sqrt {-a^{2}}}{a^{2}}\right )}}\right )}{5 \sqrt {-a^{2}}}\right )}{a^{3}}-\frac {2 \left (i \sqrt {-a^{2}}+a \right ) \left (-\frac {1}{5 c \sqrt {-a^{2}}\, {\left (x -\frac {\sqrt {-a^{2}}}{a^{2}}\right )}^{2} \sqrt {{\left (x -\frac {\sqrt {-a^{2}}}{a^{2}}\right )}^{2} a^{2} c +2 c \sqrt {-a^{2}}\, \left (x -\frac {\sqrt {-a^{2}}}{a^{2}}\right )}}-\frac {3 a^{2} \left (-\frac {1}{3 c \sqrt {-a^{2}}\, \left (x -\frac {\sqrt {-a^{2}}}{a^{2}}\right ) \sqrt {{\left (x -\frac {\sqrt {-a^{2}}}{a^{2}}\right )}^{2} a^{2} c +2 c \sqrt {-a^{2}}\, \left (x -\frac {\sqrt {-a^{2}}}{a^{2}}\right )}}-\frac {2 \left (x -\frac {\sqrt {-a^{2}}}{a^{2}}\right ) a^{2} c +2 c \sqrt {-a^{2}}}{3 c^{2} \sqrt {-a^{2}}\, \sqrt {{\left (x -\frac {\sqrt {-a^{2}}}{a^{2}}\right )}^{2} a^{2} c +2 c \sqrt {-a^{2}}\, \left (x -\frac {\sqrt {-a^{2}}}{a^{2}}\right )}}\right )}{5 \sqrt {-a^{2}}}\right )}{a^{3}}-\frac {2 \left (i \sqrt {-a^{2}}+a \right ) \left (-\frac {1}{3 c \sqrt {-a^{2}}\, \left (x -\frac {\sqrt {-a^{2}}}{a^{2}}\right ) \sqrt {{\left (x -\frac {\sqrt {-a^{2}}}{a^{2}}\right )}^{2} a^{2} c +2 c \sqrt {-a^{2}}\, \left (x -\frac {\sqrt {-a^{2}}}{a^{2}}\right )}}-\frac {2 \left (x -\frac {\sqrt {-a^{2}}}{a^{2}}\right ) a^{2} c +2 c \sqrt {-a^{2}}}{3 c^{2} \sqrt {-a^{2}}\, \sqrt {{\left (x -\frac {\sqrt {-a^{2}}}{a^{2}}\right )}^{2} a^{2} c +2 c \sqrt {-a^{2}}\, \left (x -\frac {\sqrt {-a^{2}}}{a^{2}}\right )}}\right )}{a \sqrt {-a^{2}}}-\frac {2 \left (i \sqrt {-a^{2}}-a \right ) \left (\frac {1}{3 c \sqrt {-a^{2}}\, \left (x +\frac {\sqrt {-a^{2}}}{a^{2}}\right ) \sqrt {{\left (x +\frac {\sqrt {-a^{2}}}{a^{2}}\right )}^{2} a^{2} c -2 c \sqrt {-a^{2}}\, \left (x +\frac {\sqrt {-a^{2}}}{a^{2}}\right )}}+\frac {2 \left (x +\frac {\sqrt {-a^{2}}}{a^{2}}\right ) a^{2} c -2 c \sqrt {-a^{2}}}{3 c^{2} \sqrt {-a^{2}}\, \sqrt {{\left (x +\frac {\sqrt {-a^{2}}}{a^{2}}\right )}^{2} a^{2} c -2 c \sqrt {-a^{2}}\, \left (x +\frac {\sqrt {-a^{2}}}{a^{2}}\right )}}\right )}{a \sqrt {-a^{2}}}\) | \(940\) |
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Time = 0.31 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.96 \[ \int \frac {e^{4 i \arctan (a x)}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=-\frac {\sqrt {a^{2} c x^{2} + c} {\left (a^{2} x^{2} + 3 i \, a x + 4\right )}}{15 \, {\left (a^{4} c^{2} x^{3} + 3 i \, a^{3} c^{2} x^{2} - 3 \, a^{2} c^{2} x - i \, a c^{2}\right )}} \]
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\[ \int \frac {e^{4 i \arctan (a x)}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\int \frac {\left (a x - i\right )^{4}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \left (a^{2} x^{2} + 1\right )^{2}}\, dx \]
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\[ \int \frac {e^{4 i \arctan (a x)}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\int { \frac {{\left (i \, a x + 1\right )}^{4}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} {\left (a^{2} x^{2} + 1\right )}^{2}} \,d x } \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 134 vs. \(2 (53) = 106\).
Time = 0.29 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.94 \[ \int \frac {e^{4 i \arctan (a x)}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\frac {2 \, {\left (15 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} + c}\right )}^{3} \sqrt {c} - 5 i \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} + c}\right )}^{2} c - 5 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} + c}\right )} c^{\frac {3}{2}} - i \, c^{2}\right )}}{15 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} + c} + i \, \sqrt {c}\right )}^{5} a c} \]
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Time = 1.14 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.67 \[ \int \frac {e^{4 i \arctan (a x)}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\frac {\sqrt {c\,\left (a^2\,x^2+1\right )}\,\left (a^2\,x^2\,1{}\mathrm {i}-3\,a\,x+4{}\mathrm {i}\right )}{15\,a\,c^2\,{\left (-1+a\,x\,1{}\mathrm {i}\right )}^3} \]
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