Integrand size = 28, antiderivative size = 155 \[ \int \frac {(a+i a \tan (c+d x))^3}{(e \sec (c+d x))^{13/2}} \, dx=\frac {14 a^3 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{39 d e^6 \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}+\frac {14 a^3 \sin (c+d x)}{117 d e^5 (e \sec (c+d x))^{3/2}}-\frac {2 i (a+i a \tan (c+d x))^3}{13 d (e \sec (c+d x))^{13/2}}-\frac {28 i \left (a^3+i a^3 \tan (c+d x)\right )}{117 d e^2 (e \sec (c+d x))^{9/2}} \]
[Out]
Time = 0.19 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {3578, 3577, 3854, 3856, 2719} \[ \int \frac {(a+i a \tan (c+d x))^3}{(e \sec (c+d x))^{13/2}} \, dx=\frac {14 a^3 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{39 d e^6 \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}+\frac {14 a^3 \sin (c+d x)}{117 d e^5 (e \sec (c+d x))^{3/2}}-\frac {28 i \left (a^3+i a^3 \tan (c+d x)\right )}{117 d e^2 (e \sec (c+d x))^{9/2}}-\frac {2 i (a+i a \tan (c+d x))^3}{13 d (e \sec (c+d x))^{13/2}} \]
[In]
[Out]
Rule 2719
Rule 3577
Rule 3578
Rule 3854
Rule 3856
Rubi steps \begin{align*} \text {integral}& = -\frac {2 i (a+i a \tan (c+d x))^3}{13 d (e \sec (c+d x))^{13/2}}+\frac {(7 a) \int \frac {(a+i a \tan (c+d x))^2}{(e \sec (c+d x))^{9/2}} \, dx}{13 e^2} \\ & = -\frac {2 i (a+i a \tan (c+d x))^3}{13 d (e \sec (c+d x))^{13/2}}-\frac {28 i \left (a^3+i a^3 \tan (c+d x)\right )}{117 d e^2 (e \sec (c+d x))^{9/2}}+\frac {\left (35 a^3\right ) \int \frac {1}{(e \sec (c+d x))^{5/2}} \, dx}{117 e^4} \\ & = \frac {14 a^3 \sin (c+d x)}{117 d e^5 (e \sec (c+d x))^{3/2}}-\frac {2 i (a+i a \tan (c+d x))^3}{13 d (e \sec (c+d x))^{13/2}}-\frac {28 i \left (a^3+i a^3 \tan (c+d x)\right )}{117 d e^2 (e \sec (c+d x))^{9/2}}+\frac {\left (7 a^3\right ) \int \frac {1}{\sqrt {e \sec (c+d x)}} \, dx}{39 e^6} \\ & = \frac {14 a^3 \sin (c+d x)}{117 d e^5 (e \sec (c+d x))^{3/2}}-\frac {2 i (a+i a \tan (c+d x))^3}{13 d (e \sec (c+d x))^{13/2}}-\frac {28 i \left (a^3+i a^3 \tan (c+d x)\right )}{117 d e^2 (e \sec (c+d x))^{9/2}}+\frac {\left (7 a^3\right ) \int \sqrt {\cos (c+d x)} \, dx}{39 e^6 \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}} \\ & = \frac {14 a^3 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{39 d e^6 \sqrt {\cos (c+d x)} \sqrt {e \sec (c+d x)}}+\frac {14 a^3 \sin (c+d x)}{117 d e^5 (e \sec (c+d x))^{3/2}}-\frac {2 i (a+i a \tan (c+d x))^3}{13 d (e \sec (c+d x))^{13/2}}-\frac {28 i \left (a^3+i a^3 \tan (c+d x)\right )}{117 d e^2 (e \sec (c+d x))^{9/2}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 4.31 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.94 \[ \int \frac {(a+i a \tan (c+d x))^3}{(e \sec (c+d x))^{13/2}} \, dx=\frac {a^3 \sqrt {e \sec (c+d x)} (-i \cos (3 (c+d x))+\sin (3 (c+d x))) \left (62+8 \cos (2 (c+d x))-54 \cos (4 (c+d x))+56 e^{-2 i (c+d x)} \sqrt {1+e^{2 i (c+d x)}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-e^{2 i (c+d x)}\right )+42 i \sin (2 (c+d x))+63 i \sin (4 (c+d x))\right )}{468 d e^7} \]
[In]
[Out]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 351 vs. \(2 (159 ) = 318\).
Time = 25.77 (sec) , antiderivative size = 352, normalized size of antiderivative = 2.27
method | result | size |
risch | \(-\frac {i \left (9 \,{\mathrm e}^{6 i \left (d x +c \right )}+41 \,{\mathrm e}^{4 i \left (d x +c \right )}+83 \,{\mathrm e}^{2 i \left (d x +c \right )}+219\right ) a^{3} \sqrt {2}}{936 d \,e^{6} \sqrt {\frac {e \,{\mathrm e}^{i \left (d x +c \right )}}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}}-\frac {7 i \left (-\frac {2 \left (e \,{\mathrm e}^{2 i \left (d x +c \right )}+e \right )}{e \sqrt {{\mathrm e}^{i \left (d x +c \right )} \left (e \,{\mathrm e}^{2 i \left (d x +c \right )}+e \right )}}+\frac {i \sqrt {-i \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}\, \sqrt {2}\, \sqrt {i \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}\, \sqrt {i {\mathrm e}^{i \left (d x +c \right )}}\, \left (-2 i E\left (\sqrt {-i \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}, \frac {\sqrt {2}}{2}\right )+i F\left (\sqrt {-i \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}, \frac {\sqrt {2}}{2}\right )\right )}{\sqrt {e \,{\mathrm e}^{3 i \left (d x +c \right )}+e \,{\mathrm e}^{i \left (d x +c \right )}}}\right ) a^{3} \sqrt {2}\, \sqrt {e \,{\mathrm e}^{i \left (d x +c \right )} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}}{39 d \,e^{6} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) \sqrt {\frac {e \,{\mathrm e}^{i \left (d x +c \right )}}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}}\) | \(352\) |
default | \(-\frac {2 i a^{3} \left (36 \left (\cos ^{7}\left (d x +c \right )\right )+36 \left (\cos ^{6}\left (d x +c \right )\right )+7 i \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )-13 \left (\cos ^{5}\left (d x +c \right )\right )+5 i \sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right )+21 \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 )-21 \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 )-13 \left (\cos ^{4}\left (d x +c \right )\right )+36 i \left (\cos ^{6}\left (d x +c \right )\right ) \sin \left (d x +c \right )+42 \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}}-42 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 i \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )+21 \sec \left (d x +c \right ) \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}}-21 \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 i \sin \left (d x +c \right )+36 i \left (\cos ^{5}\left (d x +c \right )\right ) \sin \left (d x +c \right )+7 i \cos \left (d x +c \right ) \sin \left (d x +c \right )\right )}{117 e^{6} d \left (\cos \left (d x +c \right )+1\right ) \sqrt {e \sec \left (d x +c \right )}}\) | \(531\) |
parts | \(\text {Expression too large to display}\) | \(1040\) |
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.94 \[ \int \frac {(a+i a \tan (c+d x))^3}{(e \sec (c+d x))^{13/2}} \, dx=\frac {{\left (336 i \, \sqrt {2} a^{3} \sqrt {e} e^{\left (i \, d x + i \, c\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, e^{\left (i \, d x + i \, c\right )}\right )\right ) + \sqrt {2} {\left (-9 i \, a^{3} e^{\left (8 i \, d x + 8 i \, c\right )} - 50 i \, a^{3} e^{\left (6 i \, d x + 6 i \, c\right )} - 124 i \, a^{3} e^{\left (4 i \, d x + 4 i \, c\right )} + 34 i \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + 117 i \, a^{3}\right )} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac {1}{2} i \, d x + \frac {1}{2} i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}}{936 \, d e^{7}} \]
[In]
[Out]
Timed out. \[ \int \frac {(a+i a \tan (c+d x))^3}{(e \sec (c+d x))^{13/2}} \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int \frac {(a+i a \tan (c+d x))^3}{(e \sec (c+d x))^{13/2}} \, dx=\int { \frac {{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{3}}{\left (e \sec \left (d x + c\right )\right )^{\frac {13}{2}}} \,d x } \]
[In]
[Out]
\[ \int \frac {(a+i a \tan (c+d x))^3}{(e \sec (c+d x))^{13/2}} \, dx=\int { \frac {{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{3}}{\left (e \sec \left (d x + c\right )\right )^{\frac {13}{2}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {(a+i a \tan (c+d x))^3}{(e \sec (c+d x))^{13/2}} \, dx=\int \frac {{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^3}{{\left (\frac {e}{\cos \left (c+d\,x\right )}\right )}^{13/2}} \,d x \]
[In]
[Out]