Integrand size = 28, antiderivative size = 116 \[ \int \frac {(e \sec (c+d x))^{7/2}}{(a+i a \tan (c+d x))^3} \, dx=-\frac {6 i e^4}{5 a^3 d \sqrt {e \sec (c+d x)}}-\frac {6 e^4 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 a^3 d \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}+\frac {4 i e^2 (e \sec (c+d x))^{3/2}}{5 a d (a+i a \tan (c+d x))^2} \]
[Out]
Time = 0.17 (sec) , antiderivative size = 116, 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 = {3581, 3582, 3856, 2719} \[ \int \frac {(e \sec (c+d x))^{7/2}}{(a+i a \tan (c+d x))^3} \, dx=-\frac {6 i e^4}{5 a^3 d \sqrt {e \sec (c+d x)}}-\frac {6 e^4 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 a^3 d \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}+\frac {4 i e^2 (e \sec (c+d x))^{3/2}}{5 a d (a+i a \tan (c+d x))^2} \]
[In]
[Out]
Rule 2719
Rule 3581
Rule 3582
Rule 3856
Rubi steps \begin{align*} \text {integral}& = \frac {4 i e^2 (e \sec (c+d x))^{3/2}}{5 a d (a+i a \tan (c+d x))^2}-\frac {\left (3 e^2\right ) \int \frac {(e \sec (c+d x))^{3/2}}{a+i a \tan (c+d x)} \, dx}{5 a^2} \\ & = -\frac {6 i e^4}{5 a^3 d \sqrt {e \sec (c+d x)}}+\frac {4 i e^2 (e \sec (c+d x))^{3/2}}{5 a d (a+i a \tan (c+d x))^2}-\frac {\left (3 e^4\right ) \int \frac {1}{\sqrt {e \sec (c+d x)}} \, dx}{5 a^3} \\ & = -\frac {6 i e^4}{5 a^3 d \sqrt {e \sec (c+d x)}}+\frac {4 i e^2 (e \sec (c+d x))^{3/2}}{5 a d (a+i a \tan (c+d x))^2}-\frac {\left (3 e^4\right ) \int \sqrt {\cos (c+d x)} \, dx}{5 a^3 \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}} \\ & = -\frac {6 i e^4}{5 a^3 d \sqrt {e \sec (c+d x)}}-\frac {6 e^4 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 a^3 d \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}+\frac {4 i e^2 (e \sec (c+d x))^{3/2}}{5 a d (a+i a \tan (c+d x))^2} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 1.64 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.01 \[ \int \frac {(e \sec (c+d x))^{7/2}}{(a+i a \tan (c+d x))^3} \, dx=\frac {2 e e^{-i d x} \left (-2+\frac {6 e^{2 i (c+d x)} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{2},\frac {3}{4},-e^{2 i (c+d x)}\right )}{\sqrt {1+e^{2 i (c+d x)}}}\right ) (e \sec (c+d x))^{5/2} (\cos (c+2 d x)+i \sin (c+2 d x))}{5 a^3 d (-i+\tan (c+d x))^3} \]
[In]
[Out]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 472 vs. \(2 (125 ) = 250\).
Time = 8.08 (sec) , antiderivative size = 473, normalized size of antiderivative = 4.08
method | result | size |
default | \(-\frac {2 i \sqrt {e \sec \left (d x +c \right )}\, \left (4 i \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )-3 \left (\cos ^{2}\left (d x +c \right )\right ) E\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}}+3 \left (\cos ^{2}\left (d x +c \right )\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}}+4 i \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )-4 \left (\cos ^{4}\left (d x +c \right )\right )-6 \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, E\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}}\, \cos \left (d x +c \right )+6 \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 ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \cos \left (d x +c \right )-3 i \cos \left (d x +c \right ) \sin \left (d x +c \right )-4 \left (\cos ^{3}\left (d x +c \right )\right )-3 \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, E\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}}+3 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}}+5 \left (\cos ^{2}\left (d x +c \right )\right )+5 \cos \left (d x +c \right )\right ) e^{3}}{5 a^{3} d \left (\cos \left (d x +c \right )+1\right )}\) | \(473\) |
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.98 \[ \int \frac {(e \sec (c+d x))^{7/2}}{(a+i a \tan (c+d x))^3} \, dx=-\frac {2 \, {\left (3 i \, \sqrt {2} e^{\frac {7}{2}} e^{\left (3 i \, d x + 3 i \, c\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right )\right ) + \sqrt {2} {\left (3 i \, e^{3} e^{\left (4 i \, d x + 4 i \, c\right )} + 2 i \, e^{3} e^{\left (2 i \, d x + 2 i \, c\right )} - i \, e^{3}\right )} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )}\right )} e^{\left (-3 i \, d x - 3 i \, c\right )}}{5 \, a^{3} d} \]
[In]
[Out]
Timed out. \[ \int \frac {(e \sec (c+d x))^{7/2}}{(a+i a \tan (c+d x))^3} \, dx=\text {Timed out} \]
[In]
[Out]
Exception generated. \[ \int \frac {(e \sec (c+d x))^{7/2}}{(a+i a \tan (c+d x))^3} \, dx=\text {Exception raised: RuntimeError} \]
[In]
[Out]
\[ \int \frac {(e \sec (c+d x))^{7/2}}{(a+i a \tan (c+d x))^3} \, dx=\int { \frac {\left (e \sec \left (d x + c\right )\right )^{\frac {7}{2}}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{3}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {(e \sec (c+d x))^{7/2}}{(a+i a \tan (c+d x))^3} \, dx=\int \frac {{\left (\frac {e}{\cos \left (c+d\,x\right )}\right )}^{7/2}}{{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^3} \,d x \]
[In]
[Out]