Integrand size = 28, antiderivative size = 106 \[ \int \sqrt {e \sec (c+d x)} (a+i a \tan (c+d x))^2 \, dx=\frac {10 i a^2 \sqrt {e \sec (c+d x)}}{3 d}+\frac {10 a^2 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {e \sec (c+d x)}}{3 d}+\frac {2 i \sqrt {e \sec (c+d x)} \left (a^2+i a^2 \tan (c+d x)\right )}{3 d} \]
[Out]
Time = 0.16 (sec) , antiderivative size = 106, 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 = {3579, 3567, 3856, 2720} \[ \int \sqrt {e \sec (c+d x)} (a+i a \tan (c+d x))^2 \, dx=\frac {10 i a^2 \sqrt {e \sec (c+d x)}}{3 d}+\frac {2 i \left (a^2+i a^2 \tan (c+d x)\right ) \sqrt {e \sec (c+d x)}}{3 d}+\frac {10 a^2 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {e \sec (c+d x)}}{3 d} \]
[In]
[Out]
Rule 2720
Rule 3567
Rule 3579
Rule 3856
Rubi steps \begin{align*} \text {integral}& = \frac {2 i \sqrt {e \sec (c+d x)} \left (a^2+i a^2 \tan (c+d x)\right )}{3 d}+\frac {1}{3} (5 a) \int \sqrt {e \sec (c+d x)} (a+i a \tan (c+d x)) \, dx \\ & = \frac {10 i a^2 \sqrt {e \sec (c+d x)}}{3 d}+\frac {2 i \sqrt {e \sec (c+d x)} \left (a^2+i a^2 \tan (c+d x)\right )}{3 d}+\frac {1}{3} \left (5 a^2\right ) \int \sqrt {e \sec (c+d x)} \, dx \\ & = \frac {10 i a^2 \sqrt {e \sec (c+d x)}}{3 d}+\frac {2 i \sqrt {e \sec (c+d x)} \left (a^2+i a^2 \tan (c+d x)\right )}{3 d}+\frac {1}{3} \left (5 a^2 \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx \\ & = \frac {10 i a^2 \sqrt {e \sec (c+d x)}}{3 d}+\frac {10 a^2 \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {e \sec (c+d x)}}{3 d}+\frac {2 i \sqrt {e \sec (c+d x)} \left (a^2+i a^2 \tan (c+d x)\right )}{3 d} \\ \end{align*}
Time = 1.61 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.63 \[ \int \sqrt {e \sec (c+d x)} (a+i a \tan (c+d x))^2 \, dx=\frac {2 a^2 (e \sec (c+d x))^{3/2} \left (6 i \cos (c+d x)+5 \cos ^{\frac {3}{2}}(c+d x) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )-\sin (c+d x)\right )}{3 d e} \]
[In]
[Out]
Time = 10.73 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.36
method | result | size |
default | \(-\frac {2 a^{2} \sqrt {e \sec \left (d x +c \right )}\, \left (-5 i \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, F\left (i \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right ), i\right ) \cos \left (d x +c \right )-5 i F\left (i \left (-\csc \left (d x +c \right )+\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}-6 i+\tan \left (d x +c \right )\right )}{3 d}\) | \(144\) |
parts | \(-\frac {2 i a^{2} \left (\cos \left (d x +c \right )+1\right ) \sqrt {e \sec \left (d x +c \right )}\, \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, F\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), i\right )}{d}+\frac {4 i a^{2} \sqrt {e \sec \left (d x +c \right )}}{d}-\frac {2 a^{2} \sqrt {e \sec \left (d x +c \right )}\, \left (2 i \cos \left (d x +c \right ) F\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}+2 i F\left (i \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right ), i\right ) \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}+\tan \left (d x +c \right )\right )}{3 d}\) | \(241\) |
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.02 \[ \int \sqrt {e \sec (c+d x)} (a+i a \tan (c+d x))^2 \, dx=-\frac {2 \, {\left (\sqrt {2} {\left (-7 i \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} - 5 i \, a^{2}\right )} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )} + 5 \, \sqrt {2} {\left (i \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + i \, a^{2}\right )} \sqrt {e} {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right )\right )}}{3 \, {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]
[In]
[Out]
\[ \int \sqrt {e \sec (c+d x)} (a+i a \tan (c+d x))^2 \, dx=- a^{2} \left (\int \left (- \sqrt {e \sec {\left (c + d x \right )}}\right )\, dx + \int \sqrt {e \sec {\left (c + d x \right )}} \tan ^{2}{\left (c + d x \right )}\, dx + \int \left (- 2 i \sqrt {e \sec {\left (c + d x \right )}} \tan {\left (c + d x \right )}\right )\, dx\right ) \]
[In]
[Out]
\[ \int \sqrt {e \sec (c+d x)} (a+i a \tan (c+d x))^2 \, dx=\int { \sqrt {e \sec \left (d x + c\right )} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2} \,d x } \]
[In]
[Out]
Exception generated. \[ \int \sqrt {e \sec (c+d x)} (a+i a \tan (c+d x))^2 \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
Timed out. \[ \int \sqrt {e \sec (c+d x)} (a+i a \tan (c+d x))^2 \, dx=\int \sqrt {\frac {e}{\cos \left (c+d\,x\right )}}\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^2 \,d x \]
[In]
[Out]