Integrand size = 30, antiderivative size = 495 \[ \int \frac {1}{(e \cos (c+d x))^{3/2} \sqrt {a+i a \tan (c+d x)}} \, dx=-\frac {i \sqrt {2} \sqrt {a} \arctan \left (1-\frac {\sqrt {2} \sqrt {e \cos (c+d x)} \sqrt {a-i a \tan (c+d x)}}{\sqrt {a} \sqrt {e}}\right ) \sec (c+d x)}{d e^{3/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {i \sqrt {2} \sqrt {a} \arctan \left (1+\frac {\sqrt {2} \sqrt {e \cos (c+d x)} \sqrt {a-i a \tan (c+d x)}}{\sqrt {a} \sqrt {e}}\right ) \sec (c+d x)}{d e^{3/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {i \sqrt {a} \log \left (a \sqrt {e}-\sqrt {2} \sqrt {a} \sqrt {e \cos (c+d x)} \sqrt {a-i a \tan (c+d x)}+\sqrt {e} \cos (c+d x) (a-i a \tan (c+d x))\right ) \sec (c+d x)}{\sqrt {2} d e^{3/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}-\frac {i \sqrt {a} \log \left (a \sqrt {e}+\sqrt {2} \sqrt {a} \sqrt {e \cos (c+d x)} \sqrt {a-i a \tan (c+d x)}+\sqrt {e} \cos (c+d x) (a-i a \tan (c+d x))\right ) \sec (c+d x)}{\sqrt {2} d e^{3/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}} \]
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Time = 0.47 (sec) , antiderivative size = 495, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {3595, 3594, 303, 1176, 631, 210, 1179, 642} \[ \int \frac {1}{(e \cos (c+d x))^{3/2} \sqrt {a+i a \tan (c+d x)}} \, dx=-\frac {i \sqrt {2} \sqrt {a} \sec (c+d x) \arctan \left (1-\frac {\sqrt {2} \sqrt {a-i a \tan (c+d x)} \sqrt {e \cos (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{d e^{3/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {i \sqrt {2} \sqrt {a} \sec (c+d x) \arctan \left (1+\frac {\sqrt {2} \sqrt {a-i a \tan (c+d x)} \sqrt {e \cos (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{d e^{3/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {i \sqrt {a} \sec (c+d x) \log \left (-\sqrt {2} \sqrt {a} \sqrt {a-i a \tan (c+d x)} \sqrt {e \cos (c+d x)}+\sqrt {e} \cos (c+d x) (a-i a \tan (c+d x))+a \sqrt {e}\right )}{\sqrt {2} d e^{3/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}-\frac {i \sqrt {a} \sec (c+d x) \log \left (\sqrt {2} \sqrt {a} \sqrt {a-i a \tan (c+d x)} \sqrt {e \cos (c+d x)}+\sqrt {e} \cos (c+d x) (a-i a \tan (c+d x))+a \sqrt {e}\right )}{\sqrt {2} d e^{3/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}} \]
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Rule 210
Rule 303
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 3594
Rule 3595
Rubi steps \begin{align*} \text {integral}& = \frac {\sec (c+d x) \int \frac {\sqrt {a-i a \tan (c+d x)}}{\sqrt {e \cos (c+d x)}} \, dx}{e \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}} \\ & = \frac {(4 i a \sec (c+d x)) \text {Subst}\left (\int \frac {x^2}{a^2 e^2+x^4} \, dx,x,\sqrt {e \cos (c+d x)} \sqrt {a-i a \tan (c+d x)}\right )}{d e \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}} \\ & = -\frac {(2 i a \sec (c+d x)) \text {Subst}\left (\int \frac {a e-x^2}{a^2 e^2+x^4} \, dx,x,\sqrt {e \cos (c+d x)} \sqrt {a-i a \tan (c+d x)}\right )}{d e \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {(2 i a \sec (c+d x)) \text {Subst}\left (\int \frac {a e+x^2}{a^2 e^2+x^4} \, dx,x,\sqrt {e \cos (c+d x)} \sqrt {a-i a \tan (c+d x)}\right )}{d e \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}} \\ & = \frac {\left (i \sqrt {a} \sec (c+d x)\right ) \text {Subst}\left (\int \frac {\sqrt {2} \sqrt {a} \sqrt {e}+2 x}{-a e-\sqrt {2} \sqrt {a} \sqrt {e} x-x^2} \, dx,x,\sqrt {e \cos (c+d x)} \sqrt {a-i a \tan (c+d x)}\right )}{\sqrt {2} d e^{3/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {\left (i \sqrt {a} \sec (c+d x)\right ) \text {Subst}\left (\int \frac {\sqrt {2} \sqrt {a} \sqrt {e}-2 x}{-a e+\sqrt {2} \sqrt {a} \sqrt {e} x-x^2} \, dx,x,\sqrt {e \cos (c+d x)} \sqrt {a-i a \tan (c+d x)}\right )}{\sqrt {2} d e^{3/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {(i a \sec (c+d x)) \text {Subst}\left (\int \frac {1}{a e-\sqrt {2} \sqrt {a} \sqrt {e} x+x^2} \, dx,x,\sqrt {e \cos (c+d x)} \sqrt {a-i a \tan (c+d x)}\right )}{d e \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {(i a \sec (c+d x)) \text {Subst}\left (\int \frac {1}{a e+\sqrt {2} \sqrt {a} \sqrt {e} x+x^2} \, dx,x,\sqrt {e \cos (c+d x)} \sqrt {a-i a \tan (c+d x)}\right )}{d e \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}} \\ & = \frac {i \sqrt {a} \log \left (a \sqrt {e}-\sqrt {2} \sqrt {a} \sqrt {e \cos (c+d x)} \sqrt {a-i a \tan (c+d x)}+\sqrt {e} \cos (c+d x) (a-i a \tan (c+d x))\right ) \sec (c+d x)}{\sqrt {2} d e^{3/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}-\frac {i \sqrt {a} \log \left (a \sqrt {e}+\sqrt {2} \sqrt {a} \sqrt {e \cos (c+d x)} \sqrt {a-i a \tan (c+d x)}+\sqrt {e} \cos (c+d x) (a-i a \tan (c+d x))\right ) \sec (c+d x)}{\sqrt {2} d e^{3/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {\left (i \sqrt {2} \sqrt {a} \sec (c+d x)\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt {e \cos (c+d x)} \sqrt {a-i a \tan (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{d e^{3/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}-\frac {\left (i \sqrt {2} \sqrt {a} \sec (c+d x)\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt {e \cos (c+d x)} \sqrt {a-i a \tan (c+d x)}}{\sqrt {a} \sqrt {e}}\right )}{d e^{3/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}} \\ & = -\frac {i \sqrt {2} \sqrt {a} \arctan \left (1-\frac {\sqrt {2} \sqrt {e \cos (c+d x)} \sqrt {a-i a \tan (c+d x)}}{\sqrt {a} \sqrt {e}}\right ) \sec (c+d x)}{d e^{3/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {i \sqrt {2} \sqrt {a} \arctan \left (1+\frac {\sqrt {2} \sqrt {e \cos (c+d x)} \sqrt {a-i a \tan (c+d x)}}{\sqrt {a} \sqrt {e}}\right ) \sec (c+d x)}{d e^{3/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {i \sqrt {a} \log \left (a \sqrt {e}-\sqrt {2} \sqrt {a} \sqrt {e \cos (c+d x)} \sqrt {a-i a \tan (c+d x)}+\sqrt {e} \cos (c+d x) (a-i a \tan (c+d x))\right ) \sec (c+d x)}{\sqrt {2} d e^{3/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}-\frac {i \sqrt {a} \log \left (a \sqrt {e}+\sqrt {2} \sqrt {a} \sqrt {e \cos (c+d x)} \sqrt {a-i a \tan (c+d x)}+\sqrt {e} \cos (c+d x) (a-i a \tan (c+d x))\right ) \sec (c+d x)}{\sqrt {2} d e^{3/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}} \\ \end{align*}
Time = 2.59 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.42 \[ \int \frac {1}{(e \cos (c+d x))^{3/2} \sqrt {a+i a \tan (c+d x)}} \, dx=\frac {i e^{\frac {1}{2} i (c+d x)} \left (2 \arctan \left (1-\sqrt {2} e^{\frac {1}{2} i (c+d x)}\right )-2 \arctan \left (1+\sqrt {2} e^{\frac {1}{2} i (c+d x)}\right )+\log \left (1-\sqrt {2} e^{\frac {1}{2} i (c+d x)}+e^{i (c+d x)}\right )-\log \left (1+\sqrt {2} e^{\frac {1}{2} i (c+d x)}+e^{i (c+d x)}\right )\right )}{\sqrt {2} d e \sqrt {\frac {a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt {e e^{-i (c+d x)} \left (1+e^{2 i (c+d x)}\right )}} \]
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Time = 12.92 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.33
method | result | size |
default | \(\frac {\left (-\frac {1}{2}-\frac {i}{2}\right ) \left (i \operatorname {arctanh}\left (\frac {\cos \left (d x +c \right )+\sin \left (d x +c \right )+1}{2 \left (\cos \left (d x +c \right )+1\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}}\right )-\operatorname {arctanh}\left (\frac {-\cos \left (d x +c \right )+\sin \left (d x +c \right )-1}{2 \left (\cos \left (d x +c \right )+1\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}}\right )\right ) \left (\cos \left (d x +c \right )+1+i \sin \left (d x +c \right )\right )}{d \left (\cos \left (d x +c \right )+1\right ) \sqrt {a \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, e \sqrt {e \cos \left (d x +c \right )}}\) | \(161\) |
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Time = 0.27 (sec) , antiderivative size = 341, normalized size of antiderivative = 0.69 \[ \int \frac {1}{(e \cos (c+d x))^{3/2} \sqrt {a+i a \tan (c+d x)}} \, dx=-\frac {1}{2} \, \sqrt {\frac {4 i}{a d^{2} e^{3}}} \log \left (\frac {1}{2} \, a d e^{2} \sqrt {\frac {4 i}{a d^{2} e^{3}}} + \sqrt {2} \sqrt {\frac {1}{2}} \sqrt {e e^{\left (2 i \, d x + 2 i \, c\right )} + e} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )}\right ) + \frac {1}{2} \, \sqrt {\frac {4 i}{a d^{2} e^{3}}} \log \left (-\frac {1}{2} \, a d e^{2} \sqrt {\frac {4 i}{a d^{2} e^{3}}} + \sqrt {2} \sqrt {\frac {1}{2}} \sqrt {e e^{\left (2 i \, d x + 2 i \, c\right )} + e} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )}\right ) + \frac {1}{2} \, \sqrt {-\frac {4 i}{a d^{2} e^{3}}} \log \left (\frac {1}{2} \, a d e^{2} \sqrt {-\frac {4 i}{a d^{2} e^{3}}} + \sqrt {2} \sqrt {\frac {1}{2}} \sqrt {e e^{\left (2 i \, d x + 2 i \, c\right )} + e} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )}\right ) - \frac {1}{2} \, \sqrt {-\frac {4 i}{a d^{2} e^{3}}} \log \left (-\frac {1}{2} \, a d e^{2} \sqrt {-\frac {4 i}{a d^{2} e^{3}}} + \sqrt {2} \sqrt {\frac {1}{2}} \sqrt {e e^{\left (2 i \, d x + 2 i \, c\right )} + e} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )}\right ) \]
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\[ \int \frac {1}{(e \cos (c+d x))^{3/2} \sqrt {a+i a \tan (c+d x)}} \, dx=\int \frac {1}{\left (e \cos {\left (c + d x \right )}\right )^{\frac {3}{2}} \sqrt {i a \left (\tan {\left (c + d x \right )} - i\right )}}\, dx \]
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Time = 0.49 (sec) , antiderivative size = 714, normalized size of antiderivative = 1.44 \[ \int \frac {1}{(e \cos (c+d x))^{3/2} \sqrt {a+i a \tan (c+d x)}} \, dx=\text {Too large to display} \]
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\[ \int \frac {1}{(e \cos (c+d x))^{3/2} \sqrt {a+i a \tan (c+d x)}} \, dx=\int { \frac {1}{\left (e \cos \left (d x + c\right )\right )^{\frac {3}{2}} \sqrt {i \, a \tan \left (d x + c\right ) + a}} \,d x } \]
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Timed out. \[ \int \frac {1}{(e \cos (c+d x))^{3/2} \sqrt {a+i a \tan (c+d x)}} \, dx=\int \frac {1}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{3/2}\,\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}} \,d x \]
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