Integrand size = 30, antiderivative size = 85 \[ \int (e \cos (c+d x))^{3/2} \sqrt {a+i a \tan (c+d x)} \, dx=\frac {4 i a e \sqrt {e \cos (c+d x)} \sec (c+d x)}{3 d \sqrt {a+i a \tan (c+d x)}}-\frac {2 i (e \cos (c+d x))^{3/2} \sqrt {a+i a \tan (c+d x)}}{3 d} \]
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Time = 0.24 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.01, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {3596, 3578, 3569} \[ \int (e \cos (c+d x))^{3/2} \sqrt {a+i a \tan (c+d x)} \, dx=\frac {4 i a \sec ^2(c+d x) (e \cos (c+d x))^{3/2}}{3 d \sqrt {a+i a \tan (c+d x)}}-\frac {2 i \sqrt {a+i a \tan (c+d x)} (e \cos (c+d x))^{3/2}}{3 d} \]
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Rule 3569
Rule 3578
Rule 3596
Rubi steps \begin{align*} \text {integral}& = \left ((e \cos (c+d x))^{3/2} (e \sec (c+d x))^{3/2}\right ) \int \frac {\sqrt {a+i a \tan (c+d x)}}{(e \sec (c+d x))^{3/2}} \, dx \\ & = -\frac {2 i (e \cos (c+d x))^{3/2} \sqrt {a+i a \tan (c+d x)}}{3 d}+\frac {\left (2 a (e \cos (c+d x))^{3/2} (e \sec (c+d x))^{3/2}\right ) \int \frac {\sqrt {e \sec (c+d x)}}{\sqrt {a+i a \tan (c+d x)}} \, dx}{3 e^2} \\ & = \frac {4 i a (e \cos (c+d x))^{3/2} \sec ^2(c+d x)}{3 d \sqrt {a+i a \tan (c+d x)}}-\frac {2 i (e \cos (c+d x))^{3/2} \sqrt {a+i a \tan (c+d x)}}{3 d} \\ \end{align*}
Time = 1.20 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.66 \[ \int (e \cos (c+d x))^{3/2} \sqrt {a+i a \tan (c+d x)} \, dx=\frac {2 e \sqrt {e \cos (c+d x)} (i \cos (c+d x)+2 \sin (c+d x)) \sqrt {a+i a \tan (c+d x)}}{3 d} \]
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Time = 8.58 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.59
method | result | size |
default | \(\frac {2 \left (i \cos \left (d x +c \right )+2 \sin \left (d x +c \right )\right ) \sqrt {e \cos \left (d x +c \right )}\, \sqrt {a \left (1+i \tan \left (d x +c \right )\right )}\, e}{3 d}\) | \(50\) |
risch | \(-\frac {i e \sqrt {2}\, \sqrt {e \cos \left (d x +c \right )}\, \sqrt {\frac {a \,{\mathrm e}^{2 i \left (d x +c \right )}}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}\, \left (-2 \cos \left (d x +c \right )+4 i \sin \left (d x +c \right )\right )}{3 d}\) | \(65\) |
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Time = 0.25 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.80 \[ \int (e \cos (c+d x))^{3/2} \sqrt {a+i a \tan (c+d x)} \, dx=\frac {\sqrt {2} \sqrt {\frac {1}{2}} \sqrt {e e^{\left (2 i \, d x + 2 i \, c\right )} + e} {\left (-i \, e e^{\left (2 i \, d x + 2 i \, c\right )} + 3 i \, e\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )}}{3 \, d} \]
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Timed out. \[ \int (e \cos (c+d x))^{3/2} \sqrt {a+i a \tan (c+d x)} \, dx=\text {Timed out} \]
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Time = 0.73 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.69 \[ \int (e \cos (c+d x))^{3/2} \sqrt {a+i a \tan (c+d x)} \, dx=\frac {{\left (-i \, e \cos \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 3 i \, e \cos \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + e \sin \left (\frac {3}{2} \, d x + \frac {3}{2} \, c\right ) + 3 \, e \sin \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \sqrt {a} \sqrt {e}}{3 \, d} \]
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\[ \int (e \cos (c+d x))^{3/2} \sqrt {a+i a \tan (c+d x)} \, dx=\int { \left (e \cos \left (d x + c\right )\right )^{\frac {3}{2}} \sqrt {i \, a \tan \left (d x + c\right ) + a} \,d x } \]
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Time = 0.58 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.04 \[ \int (e \cos (c+d x))^{3/2} \sqrt {a+i a \tan (c+d x)} \, dx=\frac {2\,e\,\sqrt {e\,\left (2\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )}\,\left ({\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,2{}\mathrm {i}+2\,\sin \left (c+d\,x\right )-\mathrm {i}\right )\,\sqrt {\frac {a\,\left (2\,{\cos \left (c+d\,x\right )}^2+\sin \left (2\,c+2\,d\,x\right )\,1{}\mathrm {i}\right )}{2\,{\cos \left (c+d\,x\right )}^2}}}{3\,d} \]
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