Integrand size = 25, antiderivative size = 96 \[ \int \frac {e^{4 i \arctan (a x)}}{\sqrt {c+a^2 c x^2}} \, dx=-\frac {2 i c (1+i a x)^3}{3 a \left (c+a^2 c x^2\right )^{3/2}}+\frac {2 i (1+i a x)}{a \sqrt {c+a^2 c x^2}}+\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{a \sqrt {c}} \]
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Time = 0.06 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5183, 683, 667, 223, 212} \[ \int \frac {e^{4 i \arctan (a x)}}{\sqrt {c+a^2 c x^2}} \, dx=\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a \sqrt {c}}-\frac {2 i c (1+i a x)^3}{3 a \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 i (1+i a x)}{a \sqrt {a^2 c x^2+c}} \]
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Rule 212
Rule 223
Rule 667
Rule 683
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 )^{5/2}} \, dx \\ & = -\frac {2 i c (1+i a x)^3}{3 a \left (c+a^2 c x^2\right )^{3/2}}-c \int \frac {(1+i a x)^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx \\ & = -\frac {2 i c (1+i a x)^3}{3 a \left (c+a^2 c x^2\right )^{3/2}}+\frac {2 i (1+i a x)}{a \sqrt {c+a^2 c x^2}}+\int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx \\ & = -\frac {2 i c (1+i a x)^3}{3 a \left (c+a^2 c x^2\right )^{3/2}}+\frac {2 i (1+i a x)}{a \sqrt {c+a^2 c x^2}}+\text {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right ) \\ & = -\frac {2 i c (1+i a x)^3}{3 a \left (c+a^2 c x^2\right )^{3/2}}+\frac {2 i (1+i a x)}{a \sqrt {c+a^2 c x^2}}+\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{a \sqrt {c}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.02 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.74 \[ \int \frac {e^{4 i \arctan (a x)}}{\sqrt {c+a^2 c x^2}} \, dx=-\frac {4 i \sqrt {2+2 a^2 x^2} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-\frac {3}{2},-\frac {1}{2},\frac {1}{2} (1-i a x)\right )}{3 a (1-i a x)^{3/2} \sqrt {c+a^2 c x^2}} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 525 vs. \(2 (80 ) = 160\).
Time = 0.45 (sec) , antiderivative size = 526, normalized size of antiderivative = 5.48
method | result | size |
default | \(\frac {\ln \left (\frac {a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}+c}\right )}{\sqrt {a^{2} c}}+\frac {2 \left (i \sqrt {-a^{2}}-a \right ) \left (\frac {\sqrt {{\left (x +\frac {\sqrt {-a^{2}}}{a^{2}}\right )}^{2} a^{2} c -2 c \sqrt {-a^{2}}\, \left (x +\frac {\sqrt {-a^{2}}}{a^{2}}\right )}}{3 c \sqrt {-a^{2}}\, {\left (x +\frac {\sqrt {-a^{2}}}{a^{2}}\right )}^{2}}-\frac {\sqrt {{\left (x +\frac {\sqrt {-a^{2}}}{a^{2}}\right )}^{2} a^{2} c -2 c \sqrt {-a^{2}}\, \left (x +\frac {\sqrt {-a^{2}}}{a^{2}}\right )}}{3 c \left (x +\frac {\sqrt {-a^{2}}}{a^{2}}\right )}\right )}{a^{3}}-\frac {2 \left (i \sqrt {-a^{2}}+a \right ) \left (-\frac {\sqrt {{\left (x -\frac {\sqrt {-a^{2}}}{a^{2}}\right )}^{2} a^{2} c +2 c \sqrt {-a^{2}}\, \left (x -\frac {\sqrt {-a^{2}}}{a^{2}}\right )}}{3 c \sqrt {-a^{2}}\, {\left (x -\frac {\sqrt {-a^{2}}}{a^{2}}\right )}^{2}}-\frac {\sqrt {{\left (x -\frac {\sqrt {-a^{2}}}{a^{2}}\right )}^{2} a^{2} c +2 c \sqrt {-a^{2}}\, \left (x -\frac {\sqrt {-a^{2}}}{a^{2}}\right )}}{3 c \left (x -\frac {\sqrt {-a^{2}}}{a^{2}}\right )}\right )}{a^{3}}-\frac {2 \left (i \sqrt {-a^{2}}+a \right ) \sqrt {{\left (x -\frac {\sqrt {-a^{2}}}{a^{2}}\right )}^{2} a^{2} c +2 c \sqrt {-a^{2}}\, \left (x -\frac {\sqrt {-a^{2}}}{a^{2}}\right )}}{a^{3} c \left (x -\frac {\sqrt {-a^{2}}}{a^{2}}\right )}+\frac {2 \left (i \sqrt {-a^{2}}-a \right ) \sqrt {{\left (x +\frac {\sqrt {-a^{2}}}{a^{2}}\right )}^{2} a^{2} c -2 c \sqrt {-a^{2}}\, \left (x +\frac {\sqrt {-a^{2}}}{a^{2}}\right )}}{a^{3} c \left (x +\frac {\sqrt {-a^{2}}}{a^{2}}\right )}\) | \(526\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 186 vs. \(2 (75) = 150\).
Time = 0.30 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.94 \[ \int \frac {e^{4 i \arctan (a x)}}{\sqrt {c+a^2 c x^2}} \, dx=\frac {3 \, {\left (a^{3} c x^{2} + 2 i \, a^{2} c x - a c\right )} \sqrt {\frac {1}{a^{2} c}} \log \left (\frac {2 \, {\left (a^{2} c x + \sqrt {a^{2} c x^{2} + c} a^{2} c \sqrt {\frac {1}{a^{2} c}}\right )}}{x}\right ) - 3 \, {\left (a^{3} c x^{2} + 2 i \, a^{2} c x - a c\right )} \sqrt {\frac {1}{a^{2} c}} \log \left (\frac {2 \, {\left (a^{2} c x - \sqrt {a^{2} c x^{2} + c} a^{2} c \sqrt {\frac {1}{a^{2} c}}\right )}}{x}\right ) - 8 \, \sqrt {a^{2} c x^{2} + c} {\left (2 \, a x + i\right )}}{6 \, {\left (a^{3} c x^{2} + 2 i \, a^{2} c x - a c\right )}} \]
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\[ \int \frac {e^{4 i \arctan (a x)}}{\sqrt {c+a^2 c x^2}} \, dx=\int \frac {\left (a x - i\right )^{4}}{\sqrt {c \left (a^{2} x^{2} + 1\right )} \left (a^{2} x^{2} + 1\right )^{2}}\, dx \]
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\[ \int \frac {e^{4 i \arctan (a x)}}{\sqrt {c+a^2 c x^2}} \, dx=\int { \frac {{\left (i \, a x + 1\right )}^{4}}{\sqrt {a^{2} c x^{2} + c} {\left (a^{2} x^{2} + 1\right )}^{2}} \,d x } \]
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none
Time = 0.35 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.38 \[ \int \frac {e^{4 i \arctan (a x)}}{\sqrt {c+a^2 c x^2}} \, dx=-\frac {\log \left ({\left | -\sqrt {a^{2} c} x + \sqrt {a^{2} c x^{2} + c} \right |}\right )}{a \sqrt {c}} - \frac {8 \, {\left (3 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} + c}\right )}^{2} + 3 i \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} + c}\right )} \sqrt {c} - 2 \, c\right )}}{3 \, {\left (i \, \sqrt {a^{2} c} x - i \, \sqrt {a^{2} c x^{2} + c} - \sqrt {c}\right )}^{3} a} \]
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Timed out. \[ \int \frac {e^{4 i \arctan (a x)}}{\sqrt {c+a^2 c x^2}} \, dx=\int \frac {{\left (1+a\,x\,1{}\mathrm {i}\right )}^4}{\sqrt {c\,a^2\,x^2+c}\,{\left (a^2\,x^2+1\right )}^2} \,d x \]
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