Integrand size = 30, antiderivative size = 323 \[ \int \sqrt {e \sec (c+d x)} \sqrt {a+i a \tan (c+d x)} \, dx=\frac {i \sqrt {2} \sqrt {a} \sqrt {e} \arctan \left (1-\frac {\sqrt {2} \sqrt {e} \sqrt {a+i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{d}-\frac {i \sqrt {2} \sqrt {a} \sqrt {e} \arctan \left (1+\frac {\sqrt {2} \sqrt {e} \sqrt {a+i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{d}-\frac {i \sqrt {a} \sqrt {e} \log \left (a-\frac {\sqrt {2} \sqrt {a} \sqrt {e} \sqrt {a+i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}+\cos (c+d x) (a+i a \tan (c+d x))\right )}{\sqrt {2} d}+\frac {i \sqrt {a} \sqrt {e} \log \left (a+\frac {\sqrt {2} \sqrt {a} \sqrt {e} \sqrt {a+i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}+\cos (c+d x) (a+i a \tan (c+d x))\right )}{\sqrt {2} d} \]
[Out]
Time = 0.25 (sec) , antiderivative size = 323, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {3576, 303, 1176, 631, 210, 1179, 642} \[ \int \sqrt {e \sec (c+d x)} \sqrt {a+i a \tan (c+d x)} \, dx=\frac {i \sqrt {2} \sqrt {a} \sqrt {e} \arctan \left (1-\frac {\sqrt {2} \sqrt {e} \sqrt {a+i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{d}-\frac {i \sqrt {2} \sqrt {a} \sqrt {e} \arctan \left (1+\frac {\sqrt {2} \sqrt {e} \sqrt {a+i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{d}-\frac {i \sqrt {a} \sqrt {e} \log \left (-\frac {\sqrt {2} \sqrt {a} \sqrt {e} \sqrt {a+i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}+\cos (c+d x) (a+i a \tan (c+d x))+a\right )}{\sqrt {2} d}+\frac {i \sqrt {a} \sqrt {e} \log \left (\frac {\sqrt {2} \sqrt {a} \sqrt {e} \sqrt {a+i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}+\cos (c+d x) (a+i a \tan (c+d x))+a\right )}{\sqrt {2} d} \]
[In]
[Out]
Rule 210
Rule 303
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 3576
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (4 i a e^2\right ) \text {Subst}\left (\int \frac {x^2}{a^2+e^2 x^4} \, dx,x,\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}\right )}{d} \\ & = \frac {(2 i a e) \text {Subst}\left (\int \frac {a-e x^2}{a^2+e^2 x^4} \, dx,x,\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}\right )}{d}-\frac {(2 i a e) \text {Subst}\left (\int \frac {a+e x^2}{a^2+e^2 x^4} \, dx,x,\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}\right )}{d} \\ & = -\frac {(i a) \text {Subst}\left (\int \frac {1}{\frac {a}{e}-\frac {\sqrt {2} \sqrt {a} x}{\sqrt {e}}+x^2} \, dx,x,\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}\right )}{d}-\frac {(i a) \text {Subst}\left (\int \frac {1}{\frac {a}{e}+\frac {\sqrt {2} \sqrt {a} x}{\sqrt {e}}+x^2} \, dx,x,\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}\right )}{d}-\frac {\left (i \sqrt {a} \sqrt {e}\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {a}}{\sqrt {e}}+2 x}{-\frac {a}{e}-\frac {\sqrt {2} \sqrt {a} x}{\sqrt {e}}-x^2} \, dx,x,\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}\right )}{\sqrt {2} d}-\frac {\left (i \sqrt {a} \sqrt {e}\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {a}}{\sqrt {e}}-2 x}{-\frac {a}{e}+\frac {\sqrt {2} \sqrt {a} x}{\sqrt {e}}-x^2} \, dx,x,\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}\right )}{\sqrt {2} d} \\ & = -\frac {i \sqrt {a} \sqrt {e} \log \left (a-\frac {\sqrt {2} \sqrt {a} \sqrt {e} \sqrt {a+i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}+\cos (c+d x) (a+i a \tan (c+d x))\right )}{\sqrt {2} d}+\frac {i \sqrt {a} \sqrt {e} \log \left (a+\frac {\sqrt {2} \sqrt {a} \sqrt {e} \sqrt {a+i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}+\cos (c+d x) (a+i a \tan (c+d x))\right )}{\sqrt {2} d}-\frac {\left (i \sqrt {2} \sqrt {a} \sqrt {e}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt {e} \sqrt {a+i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{d}+\frac {\left (i \sqrt {2} \sqrt {a} \sqrt {e}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt {e} \sqrt {a+i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{d} \\ & = \frac {i \sqrt {2} \sqrt {a} \sqrt {e} \arctan \left (1-\frac {\sqrt {2} \sqrt {e} \sqrt {a+i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{d}-\frac {i \sqrt {2} \sqrt {a} \sqrt {e} \arctan \left (1+\frac {\sqrt {2} \sqrt {e} \sqrt {a+i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{d}-\frac {i \sqrt {a} \sqrt {e} \log \left (a-\frac {\sqrt {2} \sqrt {a} \sqrt {e} \sqrt {a+i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}+\cos (c+d x) (a+i a \tan (c+d x))\right )}{\sqrt {2} d}+\frac {i \sqrt {a} \sqrt {e} \log \left (a+\frac {\sqrt {2} \sqrt {a} \sqrt {e} \sqrt {a+i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}+\cos (c+d x) (a+i a \tan (c+d x))\right )}{\sqrt {2} d} \\ \end{align*}
Time = 1.90 (sec) , antiderivative size = 277, normalized size of antiderivative = 0.86 \[ \int \sqrt {e \sec (c+d x)} \sqrt {a+i a \tan (c+d x)} \, dx=-\frac {2 e \left (\text {arctanh}\left (\frac {\sqrt {1-i \cos (c)+\sin (c)} \sqrt {i-\tan \left (\frac {d x}{2}\right )}}{\sqrt {-1-i \cos (c)-\sin (c)} \sqrt {i+\tan \left (\frac {d x}{2}\right )}}\right ) \sqrt {-1-i \cos (c)-\sin (c)} \sqrt {1+i \cos (c)-\sin (c)}-\text {arctanh}\left (\frac {\sqrt {1+i \cos (c)-\sin (c)} \sqrt {i-\tan \left (\frac {d x}{2}\right )}}{\sqrt {-1+i \cos (c)+\sin (c)} \sqrt {i+\tan \left (\frac {d x}{2}\right )}}\right ) \sqrt {1-i \cos (c)+\sin (c)} \sqrt {-1+i \cos (c)+\sin (c)}\right ) \sqrt {i+\tan \left (\frac {d x}{2}\right )} \sqrt {a+i a \tan (c+d x)}}{d \sqrt {e \sec (c+d x)} \sqrt {1+\cos (2 c)+i \sin (2 c)} \sqrt {i-\tan \left (\frac {d x}{2}\right )}} \]
[In]
[Out]
Time = 11.34 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.49
method | result | size |
default | \(\frac {\left (-1+i\right ) \sqrt {e \sec \left (d x +c \right )}\, \sqrt {a \left (1+i \tan \left (d x +c \right )\right )}\, \left (i \operatorname {arctanh}\left (\frac {-\cos \left (d x +c \right )+\sin \left (d x +c \right )-1}{2 \left (\cos \left (d x +c \right )+1\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}}\right )-\operatorname {arctanh}\left (\frac {\cos \left (d x +c \right )+\sin \left (d x +c \right )+1}{2 \left (\cos \left (d x +c \right )+1\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}}\right )\right ) \cos \left (d x +c \right )}{d \left (-i \cos \left (d x +c \right )+\sin \left (d x +c \right )-i\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}}\) | \(157\) |
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 323, normalized size of antiderivative = 1.00 \[ \int \sqrt {e \sec (c+d x)} \sqrt {a+i a \tan (c+d x)} \, dx=\frac {1}{2} \, \sqrt {\frac {4 i \, a e}{d^{2}}} \log \left (2 \, \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )} + d \sqrt {\frac {4 i \, a e}{d^{2}}}\right ) - \frac {1}{2} \, \sqrt {\frac {4 i \, a e}{d^{2}}} \log \left (2 \, \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )} - d \sqrt {\frac {4 i \, a e}{d^{2}}}\right ) - \frac {1}{2} \, \sqrt {-\frac {4 i \, a e}{d^{2}}} \log \left (2 \, \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )} + d \sqrt {-\frac {4 i \, a e}{d^{2}}}\right ) + \frac {1}{2} \, \sqrt {-\frac {4 i \, a e}{d^{2}}} \log \left (2 \, \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )} - d \sqrt {-\frac {4 i \, a e}{d^{2}}}\right ) \]
[In]
[Out]
\[ \int \sqrt {e \sec (c+d x)} \sqrt {a+i a \tan (c+d x)} \, dx=\int \sqrt {e \sec {\left (c + d x \right )}} \sqrt {i a \left (\tan {\left (c + d x \right )} - i\right )}\, dx \]
[In]
[Out]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1400 vs. \(2 (239) = 478\).
Time = 0.48 (sec) , antiderivative size = 1400, normalized size of antiderivative = 4.33 \[ \int \sqrt {e \sec (c+d x)} \sqrt {a+i a \tan (c+d x)} \, dx=\text {Too large to display} \]
[In]
[Out]
\[ \int \sqrt {e \sec (c+d x)} \sqrt {a+i a \tan (c+d x)} \, dx=\int { \sqrt {e \sec \left (d x + c\right )} \sqrt {i \, a \tan \left (d x + c\right ) + a} \,d x } \]
[In]
[Out]
Timed out. \[ \int \sqrt {e \sec (c+d x)} \sqrt {a+i a \tan (c+d x)} \, dx=\int \sqrt {\frac {e}{\cos \left (c+d\,x\right )}}\,\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}} \,d x \]
[In]
[Out]