Integrand size = 28, antiderivative size = 114 \[ \int \frac {1}{(e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))} \, dx=\frac {10 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {e \sec (c+d x)}}{21 a d e^2}+\frac {10 \sin (c+d x)}{21 a d e \sqrt {e \sec (c+d x)}}+\frac {2 i}{7 d (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))} \]
[Out]
Time = 0.14 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3583, 3854, 3856, 2720} \[ \int \frac {1}{(e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))} \, dx=\frac {10 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {e \sec (c+d x)}}{21 a d e^2}+\frac {10 \sin (c+d x)}{21 a d e \sqrt {e \sec (c+d x)}}+\frac {2 i}{7 d (a+i a \tan (c+d x)) (e \sec (c+d x))^{3/2}} \]
[In]
[Out]
Rule 2720
Rule 3583
Rule 3854
Rule 3856
Rubi steps \begin{align*} \text {integral}& = \frac {2 i}{7 d (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))}+\frac {5 \int \frac {1}{(e \sec (c+d x))^{3/2}} \, dx}{7 a} \\ & = \frac {10 \sin (c+d x)}{21 a d e \sqrt {e \sec (c+d x)}}+\frac {2 i}{7 d (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))}+\frac {5 \int \sqrt {e \sec (c+d x)} \, dx}{21 a e^2} \\ & = \frac {10 \sin (c+d x)}{21 a d e \sqrt {e \sec (c+d x)}}+\frac {2 i}{7 d (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))}+\frac {\left (5 \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{21 a e^2} \\ & = \frac {10 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {e \sec (c+d x)}}{21 a d e^2}+\frac {10 \sin (c+d x)}{21 a d e \sqrt {e \sec (c+d x)}}+\frac {2 i}{7 d (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))} \\ \end{align*}
Time = 1.61 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))} \, dx=-\frac {\sec ^3(c+d x) \left (-14 \cos (c+d x)+2 \cos (3 (c+d x))+20 i \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) (\cos (c+d x)+i \sin (c+d x))+5 i \sin (c+d x)+5 i \sin (3 (c+d x))\right )}{42 a d (e \sec (c+d x))^{3/2} (-i+\tan (c+d x))} \]
[In]
[Out]
Time = 8.86 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.53
method | result | size |
default | \(\frac {\frac {10 i F\left (i \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}{21}+\frac {2 i \left (\cos ^{3}\left (d x +c \right )\right )}{7}+\frac {10 i \sec \left (d x +c \right ) F\left (i \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}{21}+\frac {2 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{7}+\frac {10 \sin \left (d x +c \right )}{21}}{a d \sqrt {e \sec \left (d x +c \right )}\, e}\) | \(174\) |
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.01 \[ \int \frac {1}{(e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))} \, dx=\frac {{\left (\sqrt {2} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (-7 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 9 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 19 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 3 i\right )} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )} - 40 i \, \sqrt {2} \sqrt {e} e^{\left (4 i \, d x + 4 i \, c\right )} {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right )\right )} e^{\left (-4 i \, d x - 4 i \, c\right )}}{84 \, a d e^{2}} \]
[In]
[Out]
\[ \int \frac {1}{(e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))} \, dx=- \frac {i \int \frac {1}{\left (e \sec {\left (c + d x \right )}\right )^{\frac {3}{2}} \tan {\left (c + d x \right )} - i \left (e \sec {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx}{a} \]
[In]
[Out]
Exception generated. \[ \int \frac {1}{(e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))} \, dx=\text {Exception raised: RuntimeError} \]
[In]
[Out]
\[ \int \frac {1}{(e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))} \, dx=\int { \frac {1}{\left (e \sec \left (d x + c\right )\right )^{\frac {3}{2}} {\left (i \, a \tan \left (d x + c\right ) + a\right )}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {1}{(e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))} \, dx=\int \frac {1}{{\left (\frac {e}{\cos \left (c+d\,x\right )}\right )}^{3/2}\,\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )} \,d x \]
[In]
[Out]