Integrand size = 24, antiderivative size = 82 \[ \int \sec ^6(c+d x) (a+i a \tan (c+d x))^5 \, dx=-\frac {i (a+i a \tan (c+d x))^8}{2 a^3 d}+\frac {4 i (a+i a \tan (c+d x))^9}{9 a^4 d}-\frac {i (a+i a \tan (c+d x))^{10}}{10 a^5 d} \]
[Out]
Time = 0.08 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3568, 45} \[ \int \sec ^6(c+d x) (a+i a \tan (c+d x))^5 \, dx=-\frac {i (a+i a \tan (c+d x))^{10}}{10 a^5 d}+\frac {4 i (a+i a \tan (c+d x))^9}{9 a^4 d}-\frac {i (a+i a \tan (c+d x))^8}{2 a^3 d} \]
[In]
[Out]
Rule 45
Rule 3568
Rubi steps \begin{align*} \text {integral}& = -\frac {i \text {Subst}\left (\int (a-x)^2 (a+x)^7 \, dx,x,i a \tan (c+d x)\right )}{a^5 d} \\ & = -\frac {i \text {Subst}\left (\int \left (4 a^2 (a+x)^7-4 a (a+x)^8+(a+x)^9\right ) \, dx,x,i a \tan (c+d x)\right )}{a^5 d} \\ & = -\frac {i (a+i a \tan (c+d x))^8}{2 a^3 d}+\frac {4 i (a+i a \tan (c+d x))^9}{9 a^4 d}-\frac {i (a+i a \tan (c+d x))^{10}}{10 a^5 d} \\ \end{align*}
Time = 0.53 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.77 \[ \int \sec ^6(c+d x) (a+i a \tan (c+d x))^5 \, dx=\frac {a^5 \sec ^{10}(c+d x) (5+23 \cos (2 (c+d x))-22 i \sin (2 (c+d x))) (-i \cos (8 (c+d x))+\sin (8 (c+d x)))}{180 d} \]
[In]
[Out]
Time = 181.82 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.24
method | result | size |
risch | \(\frac {128 i a^{5} \left (120 \,{\mathrm e}^{14 i \left (d x +c \right )}+210 \,{\mathrm e}^{12 i \left (d x +c \right )}+252 \,{\mathrm e}^{10 i \left (d x +c \right )}+210 \,{\mathrm e}^{8 i \left (d x +c \right )}+120 \,{\mathrm e}^{6 i \left (d x +c \right )}+45 \,{\mathrm e}^{4 i \left (d x +c \right )}+10 \,{\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{45 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{10}}\) | \(102\) |
derivativedivides | \(\frac {i a^{5} \left (\frac {\sin ^{6}\left (d x +c \right )}{10 \cos \left (d x +c \right )^{10}}+\frac {\sin ^{6}\left (d x +c \right )}{20 \cos \left (d x +c \right )^{8}}+\frac {\sin ^{6}\left (d x +c \right )}{60 \cos \left (d x +c \right )^{6}}\right )+5 a^{5} \left (\frac {\sin ^{5}\left (d x +c \right )}{9 \cos \left (d x +c \right )^{9}}+\frac {4 \left (\sin ^{5}\left (d x +c \right )\right )}{63 \cos \left (d x +c \right )^{7}}+\frac {8 \left (\sin ^{5}\left (d x +c \right )\right )}{315 \cos \left (d x +c \right )^{5}}\right )-10 i a^{5} \left (\frac {\sin ^{4}\left (d x +c \right )}{8 \cos \left (d x +c \right )^{8}}+\frac {\sin ^{4}\left (d x +c \right )}{12 \cos \left (d x +c \right )^{6}}+\frac {\sin ^{4}\left (d x +c \right )}{24 \cos \left (d x +c \right )^{4}}\right )-10 a^{5} \left (\frac {\sin ^{3}\left (d x +c \right )}{7 \cos \left (d x +c \right )^{7}}+\frac {4 \left (\sin ^{3}\left (d x +c \right )\right )}{35 \cos \left (d x +c \right )^{5}}+\frac {8 \left (\sin ^{3}\left (d x +c \right )\right )}{105 \cos \left (d x +c \right )^{3}}\right )+\frac {5 i a^{5}}{6 \cos \left (d x +c \right )^{6}}-a^{5} \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )}{d}\) | \(295\) |
default | \(\frac {i a^{5} \left (\frac {\sin ^{6}\left (d x +c \right )}{10 \cos \left (d x +c \right )^{10}}+\frac {\sin ^{6}\left (d x +c \right )}{20 \cos \left (d x +c \right )^{8}}+\frac {\sin ^{6}\left (d x +c \right )}{60 \cos \left (d x +c \right )^{6}}\right )+5 a^{5} \left (\frac {\sin ^{5}\left (d x +c \right )}{9 \cos \left (d x +c \right )^{9}}+\frac {4 \left (\sin ^{5}\left (d x +c \right )\right )}{63 \cos \left (d x +c \right )^{7}}+\frac {8 \left (\sin ^{5}\left (d x +c \right )\right )}{315 \cos \left (d x +c \right )^{5}}\right )-10 i a^{5} \left (\frac {\sin ^{4}\left (d x +c \right )}{8 \cos \left (d x +c \right )^{8}}+\frac {\sin ^{4}\left (d x +c \right )}{12 \cos \left (d x +c \right )^{6}}+\frac {\sin ^{4}\left (d x +c \right )}{24 \cos \left (d x +c \right )^{4}}\right )-10 a^{5} \left (\frac {\sin ^{3}\left (d x +c \right )}{7 \cos \left (d x +c \right )^{7}}+\frac {4 \left (\sin ^{3}\left (d x +c \right )\right )}{35 \cos \left (d x +c \right )^{5}}+\frac {8 \left (\sin ^{3}\left (d x +c \right )\right )}{105 \cos \left (d x +c \right )^{3}}\right )+\frac {5 i a^{5}}{6 \cos \left (d x +c \right )^{6}}-a^{5} \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )}{d}\) | \(295\) |
[In]
[Out]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 229 vs. \(2 (64) = 128\).
Time = 0.24 (sec) , antiderivative size = 229, normalized size of antiderivative = 2.79 \[ \int \sec ^6(c+d x) (a+i a \tan (c+d x))^5 \, dx=-\frac {128 \, {\left (-120 i \, a^{5} e^{\left (14 i \, d x + 14 i \, c\right )} - 210 i \, a^{5} e^{\left (12 i \, d x + 12 i \, c\right )} - 252 i \, a^{5} e^{\left (10 i \, d x + 10 i \, c\right )} - 210 i \, a^{5} e^{\left (8 i \, d x + 8 i \, c\right )} - 120 i \, a^{5} e^{\left (6 i \, d x + 6 i \, c\right )} - 45 i \, a^{5} e^{\left (4 i \, d x + 4 i \, c\right )} - 10 i \, a^{5} e^{\left (2 i \, d x + 2 i \, c\right )} - i \, a^{5}\right )}}{45 \, {\left (d e^{\left (20 i \, d x + 20 i \, c\right )} + 10 \, d e^{\left (18 i \, d x + 18 i \, c\right )} + 45 \, d e^{\left (16 i \, d x + 16 i \, c\right )} + 120 \, d e^{\left (14 i \, d x + 14 i \, c\right )} + 210 \, d e^{\left (12 i \, d x + 12 i \, c\right )} + 252 \, d e^{\left (10 i \, d x + 10 i \, c\right )} + 210 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 120 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 45 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 10 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]
[In]
[Out]
\[ \int \sec ^6(c+d x) (a+i a \tan (c+d x))^5 \, dx=i a^{5} \left (\int \left (- i \sec ^{6}{\left (c + d x \right )}\right )\, dx + \int 5 \tan {\left (c + d x \right )} \sec ^{6}{\left (c + d x \right )}\, dx + \int \left (- 10 \tan ^{3}{\left (c + d x \right )} \sec ^{6}{\left (c + d x \right )}\right )\, dx + \int \tan ^{5}{\left (c + d x \right )} \sec ^{6}{\left (c + d x \right )}\, dx + \int 10 i \tan ^{2}{\left (c + d x \right )} \sec ^{6}{\left (c + d x \right )}\, dx + \int \left (- 5 i \tan ^{4}{\left (c + d x \right )} \sec ^{6}{\left (c + d x \right )}\right )\, dx\right ) \]
[In]
[Out]
none
Time = 0.35 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.32 \[ \int \sec ^6(c+d x) (a+i a \tan (c+d x))^5 \, dx=-\frac {-9 i \, a^{5} \tan \left (d x + c\right )^{10} - 50 \, a^{5} \tan \left (d x + c\right )^{9} + 90 i \, a^{5} \tan \left (d x + c\right )^{8} + 210 i \, a^{5} \tan \left (d x + c\right )^{6} + 252 \, a^{5} \tan \left (d x + c\right )^{5} + 240 \, a^{5} \tan \left (d x + c\right )^{3} - 225 i \, a^{5} \tan \left (d x + c\right )^{2} - 90 \, a^{5} \tan \left (d x + c\right )}{90 \, d} \]
[In]
[Out]
none
Time = 0.81 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.32 \[ \int \sec ^6(c+d x) (a+i a \tan (c+d x))^5 \, dx=-\frac {-9 i \, a^{5} \tan \left (d x + c\right )^{10} - 50 \, a^{5} \tan \left (d x + c\right )^{9} + 90 i \, a^{5} \tan \left (d x + c\right )^{8} + 210 i \, a^{5} \tan \left (d x + c\right )^{6} + 252 \, a^{5} \tan \left (d x + c\right )^{5} + 240 \, a^{5} \tan \left (d x + c\right )^{3} - 225 i \, a^{5} \tan \left (d x + c\right )^{2} - 90 \, a^{5} \tan \left (d x + c\right )}{90 \, d} \]
[In]
[Out]
Time = 4.05 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.84 \[ \int \sec ^6(c+d x) (a+i a \tan (c+d x))^5 \, dx=\frac {a^5\,\sin \left (c+d\,x\right )\,\left (90\,{\cos \left (c+d\,x\right )}^9+{\cos \left (c+d\,x\right )}^8\,\sin \left (c+d\,x\right )\,225{}\mathrm {i}-240\,{\cos \left (c+d\,x\right )}^7\,{\sin \left (c+d\,x\right )}^2-252\,{\cos \left (c+d\,x\right )}^5\,{\sin \left (c+d\,x\right )}^4-{\cos \left (c+d\,x\right )}^4\,{\sin \left (c+d\,x\right )}^5\,210{}\mathrm {i}-{\cos \left (c+d\,x\right )}^2\,{\sin \left (c+d\,x\right )}^7\,90{}\mathrm {i}+50\,\cos \left (c+d\,x\right )\,{\sin \left (c+d\,x\right )}^8+{\sin \left (c+d\,x\right )}^9\,9{}\mathrm {i}\right )}{90\,d\,{\cos \left (c+d\,x\right )}^{10}} \]
[In]
[Out]