Integrand size = 25, antiderivative size = 95 \[ \int \frac {e^{5 i \arctan (a x)}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=-\frac {2 \sqrt {1+a^2 x^2}}{3 a c (i+a x)^3 \sqrt {c+a^2 c x^2}}-\frac {i \sqrt {1+a^2 x^2}}{2 a c (i+a x)^2 \sqrt {c+a^2 c x^2}} \]
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Time = 0.06 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {5184, 5181, 45} \[ \int \frac {e^{5 i \arctan (a x)}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=-\frac {i \sqrt {a^2 x^2+1}}{2 a c (a x+i)^2 \sqrt {a^2 c x^2+c}}-\frac {2 \sqrt {a^2 x^2+1}}{3 a c (a x+i)^3 \sqrt {a^2 c x^2+c}} \]
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Rule 45
Rule 5181
Rule 5184
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1+a^2 x^2} \int \frac {e^{5 i \arctan (a x)}}{\left (1+a^2 x^2\right )^{3/2}} \, dx}{c \sqrt {c+a^2 c x^2}} \\ & = \frac {\sqrt {1+a^2 x^2} \int \frac {1+i a x}{(1-i a x)^4} \, dx}{c \sqrt {c+a^2 c x^2}} \\ & = \frac {\sqrt {1+a^2 x^2} \int \left (\frac {2}{(i+a x)^4}+\frac {i}{(i+a x)^3}\right ) \, dx}{c \sqrt {c+a^2 c x^2}} \\ & = -\frac {2 \sqrt {1+a^2 x^2}}{3 a c (i+a x)^3 \sqrt {c+a^2 c x^2}}-\frac {i \sqrt {1+a^2 x^2}}{2 a c (i+a x)^2 \sqrt {c+a^2 c x^2}} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.59 \[ \int \frac {e^{5 i \arctan (a x)}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=-\frac {i (-i+3 a x) \sqrt {1+a^2 x^2}}{6 a c (i+a x)^3 \sqrt {c+a^2 c x^2}} \]
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Time = 0.26 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.49
method | result | size |
risch | \(\frac {\sqrt {a^{2} x^{2}+1}\, \left (-\frac {i x}{2}-\frac {1}{6 a}\right )}{c \sqrt {c \left (a^{2} x^{2}+1\right )}\, \left (a x +i\right )^{3}}\) | \(47\) |
default | \(-\frac {\sqrt {c \left (a^{2} x^{2}+1\right )}\, \left (3 i a x +1\right )}{6 \sqrt {a^{2} x^{2}+1}\, c^{2} a \left (a x +i\right )^{3}}\) | \(48\) |
gosper | \(-\frac {\left (a x +i\right ) \left (-3 a x +i\right ) \left (i a x +1\right )^{5}}{6 a \left (-a x +i\right ) \left (a^{2} x^{2}+1\right )^{\frac {5}{2}} \left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}\) | \(60\) |
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Time = 0.26 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.06 \[ \int \frac {e^{5 i \arctan (a x)}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\frac {\sqrt {a^{2} c x^{2} + c} {\left (i \, a^{2} x^{3} - 3 \, a x^{2} - 6 i \, x\right )} \sqrt {a^{2} x^{2} + 1}}{6 \, {\left (a^{5} c^{2} x^{5} + 3 i \, a^{4} c^{2} x^{4} - 2 \, a^{3} c^{2} x^{3} + 2 i \, a^{2} c^{2} x^{2} - 3 \, a c^{2} x - i \, c^{2}\right )}} \]
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\[ \int \frac {e^{5 i \arctan (a x)}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=i \left (\int \left (- \frac {i}{a^{6} c x^{6} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + 3 a^{4} c x^{4} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + 3 a^{2} c x^{2} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + c \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c}}\right )\, dx + \int \frac {5 a x}{a^{6} c x^{6} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + 3 a^{4} c x^{4} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + 3 a^{2} c x^{2} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + c \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c}}\, dx + \int \left (- \frac {10 a^{3} x^{3}}{a^{6} c x^{6} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + 3 a^{4} c x^{4} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + 3 a^{2} c x^{2} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + c \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c}}\right )\, dx + \int \frac {a^{5} x^{5}}{a^{6} c x^{6} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + 3 a^{4} c x^{4} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + 3 a^{2} c x^{2} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + c \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c}}\, dx + \int \frac {10 i a^{2} x^{2}}{a^{6} c x^{6} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + 3 a^{4} c x^{4} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + 3 a^{2} c x^{2} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + c \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c}}\, dx + \int \left (- \frac {5 i a^{4} x^{4}}{a^{6} c x^{6} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + 3 a^{4} c x^{4} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + 3 a^{2} c x^{2} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + c \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c}}\right )\, dx\right ) \]
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\[ \int \frac {e^{5 i \arctan (a x)}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\int { \frac {{\left (i \, a x + 1\right )}^{5}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} {\left (a^{2} x^{2} + 1\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {e^{5 i \arctan (a x)}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\int { \frac {{\left (i \, a x + 1\right )}^{5}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} {\left (a^{2} x^{2} + 1\right )}^{\frac {5}{2}}} \,d x } \]
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Time = 1.89 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.51 \[ \int \frac {e^{5 i \arctan (a x)}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=-\frac {\sqrt {c\,\left (a^2\,x^2+1\right )}\,\left (3\,a\,x-\mathrm {i}\right )}{6\,a\,c^2\,\sqrt {a^2\,x^2+1}\,{\left (-1+a\,x\,1{}\mathrm {i}\right )}^3} \]
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