Optimal. Leaf size=108 \[ -\frac {437 \sin (x)}{512 \left (1-2 \sin ^2(x)\right )}+\frac {203 \sin (x)}{768 \left (1-2 \sin ^2(x)\right )^2}-\frac {17 \sin (x)}{192 \left (1-2 \sin ^2(x)\right )^3}+\frac {\sin (x)}{32 \left (1-2 \sin ^2(x)\right )^4}-\frac {523}{256} \tanh ^{-1}(\sin (x))+\frac {1483 \tanh ^{-1}\left (\sqrt {2} \sin (x)\right )}{512 \sqrt {2}}-\frac {1}{128} \tan (x) \sec ^3(x)-\frac {43}{256} \tan (x) \sec (x) \]
[Out]
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Rubi [B] time = 1.12, antiderivative size = 786, normalized size of antiderivative = 7.28, number of steps used = 45, number of rules used = 7, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.778, Rules used = {12, 2073, 207, 638, 614, 618, 206} \[ \frac {451 \left (\tan \left (\frac {x}{2}\right )+1\right )}{512 \left (-\tan ^2\left (\frac {x}{2}\right )-2 \tan \left (\frac {x}{2}\right )+1\right )}-\frac {15 \tan \left (\frac {x}{2}\right )+89}{64 \left (-\tan ^2\left (\frac {x}{2}\right )-2 \tan \left (\frac {x}{2}\right )+1\right )}+\frac {89-15 \tan \left (\frac {x}{2}\right )}{64 \left (-\tan ^2\left (\frac {x}{2}\right )+2 \tan \left (\frac {x}{2}\right )+1\right )}-\frac {451 \left (1-\tan \left (\frac {x}{2}\right )\right )}{512 \left (-\tan ^2\left (\frac {x}{2}\right )+2 \tan \left (\frac {x}{2}\right )+1\right )}-\frac {1-43 \tan \left (\frac {x}{2}\right )}{32 \left (-\tan ^2\left (\frac {x}{2}\right )-2 \tan \left (\frac {x}{2}\right )+1\right )^2}-\frac {65 \left (\tan \left (\frac {x}{2}\right )+1\right )}{384 \left (-\tan ^2\left (\frac {x}{2}\right )-2 \tan \left (\frac {x}{2}\right )+1\right )^2}+\frac {65 \left (1-\tan \left (\frac {x}{2}\right )\right )}{384 \left (-\tan ^2\left (\frac {x}{2}\right )+2 \tan \left (\frac {x}{2}\right )+1\right )^2}+\frac {43 \tan \left (\frac {x}{2}\right )+1}{32 \left (-\tan ^2\left (\frac {x}{2}\right )+2 \tan \left (\frac {x}{2}\right )+1\right )^2}+\frac {119 \left (\tan \left (\frac {x}{2}\right )+1\right )}{48 \left (-\tan ^2\left (\frac {x}{2}\right )-2 \tan \left (\frac {x}{2}\right )+1\right )^3}-\frac {11 \left (3 \tan \left (\frac {x}{2}\right )+1\right )}{12 \left (-\tan ^2\left (\frac {x}{2}\right )-2 \tan \left (\frac {x}{2}\right )+1\right )^3}+\frac {11 \left (1-3 \tan \left (\frac {x}{2}\right )\right )}{12 \left (-\tan ^2\left (\frac {x}{2}\right )+2 \tan \left (\frac {x}{2}\right )+1\right )^3}-\frac {119 \left (1-\tan \left (\frac {x}{2}\right )\right )}{48 \left (-\tan ^2\left (\frac {x}{2}\right )+2 \tan \left (\frac {x}{2}\right )+1\right )^3}-\frac {7-17 \tan \left (\frac {x}{2}\right )}{4 \left (-\tan ^2\left (\frac {x}{2}\right )-2 \tan \left (\frac {x}{2}\right )+1\right )^4}+\frac {17 \tan \left (\frac {x}{2}\right )+7}{4 \left (-\tan ^2\left (\frac {x}{2}\right )+2 \tan \left (\frac {x}{2}\right )+1\right )^4}+\frac {45}{256 \left (1-\tan \left (\frac {x}{2}\right )\right )}-\frac {45}{256 \left (\tan \left (\frac {x}{2}\right )+1\right )}-\frac {47}{256 \left (1-\tan \left (\frac {x}{2}\right )\right )^2}+\frac {47}{256 \left (\tan \left (\frac {x}{2}\right )+1\right )^2}+\frac {1}{64 \left (1-\tan \left (\frac {x}{2}\right )\right )^3}-\frac {1}{64 \left (\tan \left (\frac {x}{2}\right )+1\right )^3}-\frac {1}{128 \left (1-\tan \left (\frac {x}{2}\right )\right )^4}+\frac {1}{128 \left (\tan \left (\frac {x}{2}\right )+1\right )^4}-\frac {523}{256} \tanh ^{-1}(\sin (x))-\frac {1483 \log \left (-\sqrt {2} \sin (x)-\sin (x)+\sqrt {2} \cos (x)+\cos (x)+\sqrt {2}+2\right )}{2048 \sqrt {2}}-\frac {1483 \log \left (-\sqrt {2} \sin (x)+\sin (x)-\sqrt {2} \cos (x)+\cos (x)-\sqrt {2}+2\right )}{2048 \sqrt {2}}+\frac {1483 \log \left (\sqrt {2} \sin (x)-\sin (x)-\sqrt {2} \cos (x)+\cos (x)-\sqrt {2}+2\right )}{2048 \sqrt {2}}+\frac {1483 \log \left (\sqrt {2} \sin (x)+\sin (x)+\sqrt {2} \cos (x)+\cos (x)+\sqrt {2}+2\right )}{2048 \sqrt {2}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
Rule 12
Rule 206
Rule 207
Rule 614
Rule 618
Rule 638
Rule 2073
Rubi steps
\begin {align*} \int \frac {1}{(\cos (x)+\cos (3 x))^5} \, dx &=2 \operatorname {Subst}\left (\int \frac {\left (1+x^2\right )^{14}}{32 \left (1-7 x^2+7 x^4-x^6\right )^5} \, dx,x,\tan \left (\frac {x}{2}\right )\right )\\ &=\frac {1}{16} \operatorname {Subst}\left (\int \frac {\left (1+x^2\right )^{14}}{\left (1-7 x^2+7 x^4-x^6\right )^5} \, dx,x,\tan \left (\frac {x}{2}\right )\right )\\ &=\frac {1}{16} \operatorname {Subst}\left (\int \left (\frac {1}{2 (-1+x)^5}+\frac {3}{4 (-1+x)^4}+\frac {47}{8 (-1+x)^3}+\frac {45}{16 (-1+x)^2}-\frac {1}{2 (1+x)^5}+\frac {3}{4 (1+x)^4}-\frac {47}{8 (1+x)^3}+\frac {45}{16 (1+x)^2}+\frac {523}{8 \left (-1+x^2\right )}-\frac {64 (5+12 x)}{\left (-1-2 x+x^2\right )^5}-\frac {176 (2+x)}{\left (-1-2 x+x^2\right )^4}-\frac {4 (21+22 x)}{\left (-1-2 x+x^2\right )^3}+\frac {-52+37 x}{\left (-1-2 x+x^2\right )^2}-\frac {36}{-1-2 x+x^2}+\frac {64 (-5+12 x)}{\left (-1+2 x+x^2\right )^5}+\frac {176 (-2+x)}{\left (-1+2 x+x^2\right )^4}+\frac {4 (-21+22 x)}{\left (-1+2 x+x^2\right )^3}+\frac {-52-37 x}{\left (-1+2 x+x^2\right )^2}-\frac {36}{-1+2 x+x^2}\right ) \, dx,x,\tan \left (\frac {x}{2}\right )\right )\\ &=-\frac {1}{128 \left (1-\tan \left (\frac {x}{2}\right )\right )^4}+\frac {1}{64 \left (1-\tan \left (\frac {x}{2}\right )\right )^3}-\frac {47}{256 \left (1-\tan \left (\frac {x}{2}\right )\right )^2}+\frac {45}{256 \left (1-\tan \left (\frac {x}{2}\right )\right )}+\frac {1}{128 \left (1+\tan \left (\frac {x}{2}\right )\right )^4}-\frac {1}{64 \left (1+\tan \left (\frac {x}{2}\right )\right )^3}+\frac {47}{256 \left (1+\tan \left (\frac {x}{2}\right )\right )^2}-\frac {45}{256 \left (1+\tan \left (\frac {x}{2}\right )\right )}+\frac {1}{16} \operatorname {Subst}\left (\int \frac {-52+37 x}{\left (-1-2 x+x^2\right )^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )+\frac {1}{16} \operatorname {Subst}\left (\int \frac {-52-37 x}{\left (-1+2 x+x^2\right )^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )-\frac {1}{4} \operatorname {Subst}\left (\int \frac {21+22 x}{\left (-1-2 x+x^2\right )^3} \, dx,x,\tan \left (\frac {x}{2}\right )\right )+\frac {1}{4} \operatorname {Subst}\left (\int \frac {-21+22 x}{\left (-1+2 x+x^2\right )^3} \, dx,x,\tan \left (\frac {x}{2}\right )\right )-\frac {9}{4} \operatorname {Subst}\left (\int \frac {1}{-1-2 x+x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )-\frac {9}{4} \operatorname {Subst}\left (\int \frac {1}{-1+2 x+x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )-4 \operatorname {Subst}\left (\int \frac {5+12 x}{\left (-1-2 x+x^2\right )^5} \, dx,x,\tan \left (\frac {x}{2}\right )\right )+4 \operatorname {Subst}\left (\int \frac {-5+12 x}{\left (-1+2 x+x^2\right )^5} \, dx,x,\tan \left (\frac {x}{2}\right )\right )+\frac {523}{128} \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )-11 \operatorname {Subst}\left (\int \frac {2+x}{\left (-1-2 x+x^2\right )^4} \, dx,x,\tan \left (\frac {x}{2}\right )\right )+11 \operatorname {Subst}\left (\int \frac {-2+x}{\left (-1+2 x+x^2\right )^4} \, dx,x,\tan \left (\frac {x}{2}\right )\right )\\ &=-\frac {523}{256} \tanh ^{-1}(\sin (x))-\frac {1}{128 \left (1-\tan \left (\frac {x}{2}\right )\right )^4}+\frac {1}{64 \left (1-\tan \left (\frac {x}{2}\right )\right )^3}-\frac {47}{256 \left (1-\tan \left (\frac {x}{2}\right )\right )^2}+\frac {45}{256 \left (1-\tan \left (\frac {x}{2}\right )\right )}+\frac {1}{128 \left (1+\tan \left (\frac {x}{2}\right )\right )^4}-\frac {1}{64 \left (1+\tan \left (\frac {x}{2}\right )\right )^3}+\frac {47}{256 \left (1+\tan \left (\frac {x}{2}\right )\right )^2}-\frac {45}{256 \left (1+\tan \left (\frac {x}{2}\right )\right )}-\frac {7-17 \tan \left (\frac {x}{2}\right )}{4 \left (1-2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )^4}-\frac {11 \left (1+3 \tan \left (\frac {x}{2}\right )\right )}{12 \left (1-2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )^3}-\frac {1-43 \tan \left (\frac {x}{2}\right )}{32 \left (1-2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )^2}-\frac {89+15 \tan \left (\frac {x}{2}\right )}{64 \left (1-2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )}+\frac {7+17 \tan \left (\frac {x}{2}\right )}{4 \left (1+2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )^4}+\frac {11 \left (1-3 \tan \left (\frac {x}{2}\right )\right )}{12 \left (1+2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )^3}+\frac {1+43 \tan \left (\frac {x}{2}\right )}{32 \left (1+2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )^2}+\frac {89-15 \tan \left (\frac {x}{2}\right )}{64 \left (1+2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )}+\frac {15}{64} \operatorname {Subst}\left (\int \frac {1}{-1-2 x+x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )+\frac {15}{64} \operatorname {Subst}\left (\int \frac {1}{-1+2 x+x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )+\frac {129}{32} \operatorname {Subst}\left (\int \frac {1}{\left (-1-2 x+x^2\right )^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )+\frac {129}{32} \operatorname {Subst}\left (\int \frac {1}{\left (-1+2 x+x^2\right )^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )+\frac {9}{2} \operatorname {Subst}\left (\int \frac {1}{8-x^2} \, dx,x,-2+2 \tan \left (\frac {x}{2}\right )\right )+\frac {9}{2} \operatorname {Subst}\left (\int \frac {1}{8-x^2} \, dx,x,2+2 \tan \left (\frac {x}{2}\right )\right )+\frac {55}{4} \operatorname {Subst}\left (\int \frac {1}{\left (-1-2 x+x^2\right )^3} \, dx,x,\tan \left (\frac {x}{2}\right )\right )+\frac {55}{4} \operatorname {Subst}\left (\int \frac {1}{\left (-1+2 x+x^2\right )^3} \, dx,x,\tan \left (\frac {x}{2}\right )\right )+\frac {119}{4} \operatorname {Subst}\left (\int \frac {1}{\left (-1-2 x+x^2\right )^4} \, dx,x,\tan \left (\frac {x}{2}\right )\right )+\frac {119}{4} \operatorname {Subst}\left (\int \frac {1}{\left (-1+2 x+x^2\right )^4} \, dx,x,\tan \left (\frac {x}{2}\right )\right )\\ &=-\frac {523}{256} \tanh ^{-1}(\sin (x))-\frac {9 \log \left (2+\sqrt {2}+\cos (x)+\sqrt {2} \cos (x)-\sin (x)-\sqrt {2} \sin (x)\right )}{16 \sqrt {2}}-\frac {9 \log \left (2-\sqrt {2}+\cos (x)-\sqrt {2} \cos (x)+\sin (x)-\sqrt {2} \sin (x)\right )}{16 \sqrt {2}}+\frac {9 \log \left (2-\sqrt {2}+\cos (x)-\sqrt {2} \cos (x)-\sin (x)+\sqrt {2} \sin (x)\right )}{16 \sqrt {2}}+\frac {9 \log \left (2+\sqrt {2}+\cos (x)+\sqrt {2} \cos (x)+\sin (x)+\sqrt {2} \sin (x)\right )}{16 \sqrt {2}}-\frac {1}{128 \left (1-\tan \left (\frac {x}{2}\right )\right )^4}+\frac {1}{64 \left (1-\tan \left (\frac {x}{2}\right )\right )^3}-\frac {47}{256 \left (1-\tan \left (\frac {x}{2}\right )\right )^2}+\frac {45}{256 \left (1-\tan \left (\frac {x}{2}\right )\right )}+\frac {1}{128 \left (1+\tan \left (\frac {x}{2}\right )\right )^4}-\frac {1}{64 \left (1+\tan \left (\frac {x}{2}\right )\right )^3}+\frac {47}{256 \left (1+\tan \left (\frac {x}{2}\right )\right )^2}-\frac {45}{256 \left (1+\tan \left (\frac {x}{2}\right )\right )}-\frac {7-17 \tan \left (\frac {x}{2}\right )}{4 \left (1-2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )^4}+\frac {119 \left (1+\tan \left (\frac {x}{2}\right )\right )}{48 \left (1-2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )^3}-\frac {11 \left (1+3 \tan \left (\frac {x}{2}\right )\right )}{12 \left (1-2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )^3}-\frac {1-43 \tan \left (\frac {x}{2}\right )}{32 \left (1-2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )^2}-\frac {55 \left (1+\tan \left (\frac {x}{2}\right )\right )}{32 \left (1-2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )^2}+\frac {129 \left (1+\tan \left (\frac {x}{2}\right )\right )}{128 \left (1-2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )}-\frac {89+15 \tan \left (\frac {x}{2}\right )}{64 \left (1-2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )}+\frac {7+17 \tan \left (\frac {x}{2}\right )}{4 \left (1+2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )^4}+\frac {11 \left (1-3 \tan \left (\frac {x}{2}\right )\right )}{12 \left (1+2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )^3}-\frac {119 \left (1-\tan \left (\frac {x}{2}\right )\right )}{48 \left (1+2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )^3}+\frac {55 \left (1-\tan \left (\frac {x}{2}\right )\right )}{32 \left (1+2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )^2}+\frac {1+43 \tan \left (\frac {x}{2}\right )}{32 \left (1+2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )^2}+\frac {89-15 \tan \left (\frac {x}{2}\right )}{64 \left (1+2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )}-\frac {129 \left (1-\tan \left (\frac {x}{2}\right )\right )}{128 \left (1+2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )}-\frac {15}{32} \operatorname {Subst}\left (\int \frac {1}{8-x^2} \, dx,x,-2+2 \tan \left (\frac {x}{2}\right )\right )-\frac {15}{32} \operatorname {Subst}\left (\int \frac {1}{8-x^2} \, dx,x,2+2 \tan \left (\frac {x}{2}\right )\right )-\frac {129}{128} \operatorname {Subst}\left (\int \frac {1}{-1-2 x+x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )-\frac {129}{128} \operatorname {Subst}\left (\int \frac {1}{-1+2 x+x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )-\frac {165}{32} \operatorname {Subst}\left (\int \frac {1}{\left (-1-2 x+x^2\right )^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )-\frac {165}{32} \operatorname {Subst}\left (\int \frac {1}{\left (-1+2 x+x^2\right )^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )-\frac {595}{48} \operatorname {Subst}\left (\int \frac {1}{\left (-1-2 x+x^2\right )^3} \, dx,x,\tan \left (\frac {x}{2}\right )\right )-\frac {595}{48} \operatorname {Subst}\left (\int \frac {1}{\left (-1+2 x+x^2\right )^3} \, dx,x,\tan \left (\frac {x}{2}\right )\right )\\ &=-\frac {523}{256} \tanh ^{-1}(\sin (x))-\frac {129 \log \left (2+\sqrt {2}+\cos (x)+\sqrt {2} \cos (x)-\sin (x)-\sqrt {2} \sin (x)\right )}{256 \sqrt {2}}-\frac {129 \log \left (2-\sqrt {2}+\cos (x)-\sqrt {2} \cos (x)+\sin (x)-\sqrt {2} \sin (x)\right )}{256 \sqrt {2}}+\frac {129 \log \left (2-\sqrt {2}+\cos (x)-\sqrt {2} \cos (x)-\sin (x)+\sqrt {2} \sin (x)\right )}{256 \sqrt {2}}+\frac {129 \log \left (2+\sqrt {2}+\cos (x)+\sqrt {2} \cos (x)+\sin (x)+\sqrt {2} \sin (x)\right )}{256 \sqrt {2}}-\frac {1}{128 \left (1-\tan \left (\frac {x}{2}\right )\right )^4}+\frac {1}{64 \left (1-\tan \left (\frac {x}{2}\right )\right )^3}-\frac {47}{256 \left (1-\tan \left (\frac {x}{2}\right )\right )^2}+\frac {45}{256 \left (1-\tan \left (\frac {x}{2}\right )\right )}+\frac {1}{128 \left (1+\tan \left (\frac {x}{2}\right )\right )^4}-\frac {1}{64 \left (1+\tan \left (\frac {x}{2}\right )\right )^3}+\frac {47}{256 \left (1+\tan \left (\frac {x}{2}\right )\right )^2}-\frac {45}{256 \left (1+\tan \left (\frac {x}{2}\right )\right )}-\frac {7-17 \tan \left (\frac {x}{2}\right )}{4 \left (1-2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )^4}+\frac {119 \left (1+\tan \left (\frac {x}{2}\right )\right )}{48 \left (1-2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )^3}-\frac {11 \left (1+3 \tan \left (\frac {x}{2}\right )\right )}{12 \left (1-2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )^3}-\frac {1-43 \tan \left (\frac {x}{2}\right )}{32 \left (1-2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )^2}-\frac {65 \left (1+\tan \left (\frac {x}{2}\right )\right )}{384 \left (1-2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )^2}-\frac {9 \left (1+\tan \left (\frac {x}{2}\right )\right )}{32 \left (1-2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )}-\frac {89+15 \tan \left (\frac {x}{2}\right )}{64 \left (1-2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )}+\frac {7+17 \tan \left (\frac {x}{2}\right )}{4 \left (1+2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )^4}+\frac {11 \left (1-3 \tan \left (\frac {x}{2}\right )\right )}{12 \left (1+2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )^3}-\frac {119 \left (1-\tan \left (\frac {x}{2}\right )\right )}{48 \left (1+2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )^3}+\frac {65 \left (1-\tan \left (\frac {x}{2}\right )\right )}{384 \left (1+2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )^2}+\frac {1+43 \tan \left (\frac {x}{2}\right )}{32 \left (1+2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )^2}+\frac {89-15 \tan \left (\frac {x}{2}\right )}{64 \left (1+2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )}+\frac {9 \left (1-\tan \left (\frac {x}{2}\right )\right )}{32 \left (1+2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )}+\frac {165}{128} \operatorname {Subst}\left (\int \frac {1}{-1-2 x+x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )+\frac {165}{128} \operatorname {Subst}\left (\int \frac {1}{-1+2 x+x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )+\frac {129}{64} \operatorname {Subst}\left (\int \frac {1}{8-x^2} \, dx,x,-2+2 \tan \left (\frac {x}{2}\right )\right )+\frac {129}{64} \operatorname {Subst}\left (\int \frac {1}{8-x^2} \, dx,x,2+2 \tan \left (\frac {x}{2}\right )\right )+\frac {595}{128} \operatorname {Subst}\left (\int \frac {1}{\left (-1-2 x+x^2\right )^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )+\frac {595}{128} \operatorname {Subst}\left (\int \frac {1}{\left (-1+2 x+x^2\right )^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )\\ &=-\frac {523}{256} \tanh ^{-1}(\sin (x))-\frac {387 \log \left (2+\sqrt {2}+\cos (x)+\sqrt {2} \cos (x)-\sin (x)-\sqrt {2} \sin (x)\right )}{512 \sqrt {2}}-\frac {387 \log \left (2-\sqrt {2}+\cos (x)-\sqrt {2} \cos (x)+\sin (x)-\sqrt {2} \sin (x)\right )}{512 \sqrt {2}}+\frac {387 \log \left (2-\sqrt {2}+\cos (x)-\sqrt {2} \cos (x)-\sin (x)+\sqrt {2} \sin (x)\right )}{512 \sqrt {2}}+\frac {387 \log \left (2+\sqrt {2}+\cos (x)+\sqrt {2} \cos (x)+\sin (x)+\sqrt {2} \sin (x)\right )}{512 \sqrt {2}}-\frac {1}{128 \left (1-\tan \left (\frac {x}{2}\right )\right )^4}+\frac {1}{64 \left (1-\tan \left (\frac {x}{2}\right )\right )^3}-\frac {47}{256 \left (1-\tan \left (\frac {x}{2}\right )\right )^2}+\frac {45}{256 \left (1-\tan \left (\frac {x}{2}\right )\right )}+\frac {1}{128 \left (1+\tan \left (\frac {x}{2}\right )\right )^4}-\frac {1}{64 \left (1+\tan \left (\frac {x}{2}\right )\right )^3}+\frac {47}{256 \left (1+\tan \left (\frac {x}{2}\right )\right )^2}-\frac {45}{256 \left (1+\tan \left (\frac {x}{2}\right )\right )}-\frac {7-17 \tan \left (\frac {x}{2}\right )}{4 \left (1-2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )^4}+\frac {119 \left (1+\tan \left (\frac {x}{2}\right )\right )}{48 \left (1-2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )^3}-\frac {11 \left (1+3 \tan \left (\frac {x}{2}\right )\right )}{12 \left (1-2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )^3}-\frac {1-43 \tan \left (\frac {x}{2}\right )}{32 \left (1-2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )^2}-\frac {65 \left (1+\tan \left (\frac {x}{2}\right )\right )}{384 \left (1-2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )^2}+\frac {451 \left (1+\tan \left (\frac {x}{2}\right )\right )}{512 \left (1-2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )}-\frac {89+15 \tan \left (\frac {x}{2}\right )}{64 \left (1-2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )}+\frac {7+17 \tan \left (\frac {x}{2}\right )}{4 \left (1+2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )^4}+\frac {11 \left (1-3 \tan \left (\frac {x}{2}\right )\right )}{12 \left (1+2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )^3}-\frac {119 \left (1-\tan \left (\frac {x}{2}\right )\right )}{48 \left (1+2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )^3}+\frac {65 \left (1-\tan \left (\frac {x}{2}\right )\right )}{384 \left (1+2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )^2}+\frac {1+43 \tan \left (\frac {x}{2}\right )}{32 \left (1+2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )^2}+\frac {89-15 \tan \left (\frac {x}{2}\right )}{64 \left (1+2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )}-\frac {451 \left (1-\tan \left (\frac {x}{2}\right )\right )}{512 \left (1+2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )}-\frac {595}{512} \operatorname {Subst}\left (\int \frac {1}{-1-2 x+x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )-\frac {595}{512} \operatorname {Subst}\left (\int \frac {1}{-1+2 x+x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )-\frac {165}{64} \operatorname {Subst}\left (\int \frac {1}{8-x^2} \, dx,x,-2+2 \tan \left (\frac {x}{2}\right )\right )-\frac {165}{64} \operatorname {Subst}\left (\int \frac {1}{8-x^2} \, dx,x,2+2 \tan \left (\frac {x}{2}\right )\right )\\ &=-\frac {523}{256} \tanh ^{-1}(\sin (x))-\frac {111 \log \left (2+\sqrt {2}+\cos (x)+\sqrt {2} \cos (x)-\sin (x)-\sqrt {2} \sin (x)\right )}{256 \sqrt {2}}-\frac {111 \log \left (2-\sqrt {2}+\cos (x)-\sqrt {2} \cos (x)+\sin (x)-\sqrt {2} \sin (x)\right )}{256 \sqrt {2}}+\frac {111 \log \left (2-\sqrt {2}+\cos (x)-\sqrt {2} \cos (x)-\sin (x)+\sqrt {2} \sin (x)\right )}{256 \sqrt {2}}+\frac {111 \log \left (2+\sqrt {2}+\cos (x)+\sqrt {2} \cos (x)+\sin (x)+\sqrt {2} \sin (x)\right )}{256 \sqrt {2}}-\frac {1}{128 \left (1-\tan \left (\frac {x}{2}\right )\right )^4}+\frac {1}{64 \left (1-\tan \left (\frac {x}{2}\right )\right )^3}-\frac {47}{256 \left (1-\tan \left (\frac {x}{2}\right )\right )^2}+\frac {45}{256 \left (1-\tan \left (\frac {x}{2}\right )\right )}+\frac {1}{128 \left (1+\tan \left (\frac {x}{2}\right )\right )^4}-\frac {1}{64 \left (1+\tan \left (\frac {x}{2}\right )\right )^3}+\frac {47}{256 \left (1+\tan \left (\frac {x}{2}\right )\right )^2}-\frac {45}{256 \left (1+\tan \left (\frac {x}{2}\right )\right )}-\frac {7-17 \tan \left (\frac {x}{2}\right )}{4 \left (1-2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )^4}+\frac {119 \left (1+\tan \left (\frac {x}{2}\right )\right )}{48 \left (1-2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )^3}-\frac {11 \left (1+3 \tan \left (\frac {x}{2}\right )\right )}{12 \left (1-2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )^3}-\frac {1-43 \tan \left (\frac {x}{2}\right )}{32 \left (1-2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )^2}-\frac {65 \left (1+\tan \left (\frac {x}{2}\right )\right )}{384 \left (1-2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )^2}+\frac {451 \left (1+\tan \left (\frac {x}{2}\right )\right )}{512 \left (1-2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )}-\frac {89+15 \tan \left (\frac {x}{2}\right )}{64 \left (1-2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )}+\frac {7+17 \tan \left (\frac {x}{2}\right )}{4 \left (1+2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )^4}+\frac {11 \left (1-3 \tan \left (\frac {x}{2}\right )\right )}{12 \left (1+2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )^3}-\frac {119 \left (1-\tan \left (\frac {x}{2}\right )\right )}{48 \left (1+2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )^3}+\frac {65 \left (1-\tan \left (\frac {x}{2}\right )\right )}{384 \left (1+2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )^2}+\frac {1+43 \tan \left (\frac {x}{2}\right )}{32 \left (1+2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )^2}+\frac {89-15 \tan \left (\frac {x}{2}\right )}{64 \left (1+2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )}-\frac {451 \left (1-\tan \left (\frac {x}{2}\right )\right )}{512 \left (1+2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )}+\frac {595}{256} \operatorname {Subst}\left (\int \frac {1}{8-x^2} \, dx,x,-2+2 \tan \left (\frac {x}{2}\right )\right )+\frac {595}{256} \operatorname {Subst}\left (\int \frac {1}{8-x^2} \, dx,x,2+2 \tan \left (\frac {x}{2}\right )\right )\\ &=-\frac {523}{256} \tanh ^{-1}(\sin (x))-\frac {1483 \log \left (2+\sqrt {2}+\cos (x)+\sqrt {2} \cos (x)-\sin (x)-\sqrt {2} \sin (x)\right )}{2048 \sqrt {2}}-\frac {1483 \log \left (2-\sqrt {2}+\cos (x)-\sqrt {2} \cos (x)+\sin (x)-\sqrt {2} \sin (x)\right )}{2048 \sqrt {2}}+\frac {1483 \log \left (2-\sqrt {2}+\cos (x)-\sqrt {2} \cos (x)-\sin (x)+\sqrt {2} \sin (x)\right )}{2048 \sqrt {2}}+\frac {1483 \log \left (2+\sqrt {2}+\cos (x)+\sqrt {2} \cos (x)+\sin (x)+\sqrt {2} \sin (x)\right )}{2048 \sqrt {2}}-\frac {1}{128 \left (1-\tan \left (\frac {x}{2}\right )\right )^4}+\frac {1}{64 \left (1-\tan \left (\frac {x}{2}\right )\right )^3}-\frac {47}{256 \left (1-\tan \left (\frac {x}{2}\right )\right )^2}+\frac {45}{256 \left (1-\tan \left (\frac {x}{2}\right )\right )}+\frac {1}{128 \left (1+\tan \left (\frac {x}{2}\right )\right )^4}-\frac {1}{64 \left (1+\tan \left (\frac {x}{2}\right )\right )^3}+\frac {47}{256 \left (1+\tan \left (\frac {x}{2}\right )\right )^2}-\frac {45}{256 \left (1+\tan \left (\frac {x}{2}\right )\right )}-\frac {7-17 \tan \left (\frac {x}{2}\right )}{4 \left (1-2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )^4}+\frac {119 \left (1+\tan \left (\frac {x}{2}\right )\right )}{48 \left (1-2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )^3}-\frac {11 \left (1+3 \tan \left (\frac {x}{2}\right )\right )}{12 \left (1-2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )^3}-\frac {1-43 \tan \left (\frac {x}{2}\right )}{32 \left (1-2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )^2}-\frac {65 \left (1+\tan \left (\frac {x}{2}\right )\right )}{384 \left (1-2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )^2}+\frac {451 \left (1+\tan \left (\frac {x}{2}\right )\right )}{512 \left (1-2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )}-\frac {89+15 \tan \left (\frac {x}{2}\right )}{64 \left (1-2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )}+\frac {7+17 \tan \left (\frac {x}{2}\right )}{4 \left (1+2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )^4}+\frac {11 \left (1-3 \tan \left (\frac {x}{2}\right )\right )}{12 \left (1+2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )^3}-\frac {119 \left (1-\tan \left (\frac {x}{2}\right )\right )}{48 \left (1+2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )^3}+\frac {65 \left (1-\tan \left (\frac {x}{2}\right )\right )}{384 \left (1+2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )^2}+\frac {1+43 \tan \left (\frac {x}{2}\right )}{32 \left (1+2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )^2}+\frac {89-15 \tan \left (\frac {x}{2}\right )}{64 \left (1+2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )}-\frac {451 \left (1-\tan \left (\frac {x}{2}\right )\right )}{512 \left (1+2 \tan \left (\frac {x}{2}\right )-\tan ^2\left (\frac {x}{2}\right )\right )}\\ \end {align*}
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Mathematica [C] time = 6.36, size = 478, normalized size = 4.43 \[ \frac {1483 \log \left (2 \sin (x)+\sqrt {2}\right )}{1024 \sqrt {2}}+\frac {83 \sin (x)}{512 (\cos (x)-\sin (x))^2}+\frac {\sin (x)}{128 (\cos (x)-\sin (x))^4}-\frac {437}{1024 (\cos (x)-\sin (x))}+\frac {437}{1024 (\sin (x)+\cos (x))}-\frac {43}{512 \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )^2}+\frac {43}{512 \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right )^2}+\frac {83 \sin (x)}{512 (\sin (x)+\cos (x))^2}-\frac {17}{768 (\cos (x)-\sin (x))^3}+\frac {17}{768 (\sin (x)+\cos (x))^3}-\frac {1}{512 \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )^4}+\frac {1}{512 \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right )^4}+\frac {\sin (x)}{128 (\sin (x)+\cos (x))^4}+\frac {523}{256} \log \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )-\frac {523}{256} \log \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right )-\frac {1483 \log \left (-\sqrt {2} \sin (x)-\sqrt {2} \cos (x)+2\right )}{2048 \sqrt {2}}+\frac {\left (\frac {1483}{4096}-\frac {1483 i}{4096}\right ) \left (\sqrt {2}+(-1-i)\right ) \log \left (-\sqrt {2} \sin (x)+\sqrt {2} \cos (x)+2\right )}{\sqrt {2}+(-1+i)}-\frac {1483 i \tan ^{-1}\left (\frac {-\sqrt {2} \sin \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )}{-\sin \left (\frac {x}{2}\right )+\sqrt {2} \cos \left (\frac {x}{2}\right )-\cos \left (\frac {x}{2}\right )}\right )}{1024 \sqrt {2}}+\frac {\left (\frac {1483}{2048}+\frac {1483 i}{2048}\right ) \left (\sqrt {2}+(-1-i)\right ) \tan ^{-1}\left (\frac {-\sqrt {2} \sin \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )}{-\sin \left (\frac {x}{2}\right )+\sqrt {2} \cos \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )}\right )}{\sqrt {2}+(-1+i)} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.56, size = 219, normalized size = 2.03 \[ \frac {4449 \, {\left (16 \, \sqrt {2} \cos \relax (x)^{12} - 32 \, \sqrt {2} \cos \relax (x)^{10} + 24 \, \sqrt {2} \cos \relax (x)^{8} - 8 \, \sqrt {2} \cos \relax (x)^{6} + \sqrt {2} \cos \relax (x)^{4}\right )} \log \left (-\frac {2 \, \cos \relax (x)^{2} - 2 \, \sqrt {2} \sin \relax (x) - 3}{2 \, \cos \relax (x)^{2} - 1}\right ) - 6276 \, {\left (16 \, \cos \relax (x)^{12} - 32 \, \cos \relax (x)^{10} + 24 \, \cos \relax (x)^{8} - 8 \, \cos \relax (x)^{6} + \cos \relax (x)^{4}\right )} \log \left (\sin \relax (x) + 1\right ) + 6276 \, {\left (16 \, \cos \relax (x)^{12} - 32 \, \cos \relax (x)^{10} + 24 \, \cos \relax (x)^{8} - 8 \, \cos \relax (x)^{6} + \cos \relax (x)^{4}\right )} \log \left (-\sin \relax (x) + 1\right ) - 4 \, {\left (14616 \, \cos \relax (x)^{10} - 25420 \, \cos \relax (x)^{8} + 15570 \, \cos \relax (x)^{6} - 3677 \, \cos \relax (x)^{4} + 162 \, \cos \relax (x)^{2} + 12\right )} \sin \relax (x)}{6144 \, {\left (16 \, \cos \relax (x)^{12} - 32 \, \cos \relax (x)^{10} + 24 \, \cos \relax (x)^{8} - 8 \, \cos \relax (x)^{6} + \cos \relax (x)^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 104, normalized size = 0.96 \[ -\frac {1483}{2048} \, \sqrt {2} \log \left (\frac {{\left | -2 \, \sqrt {2} + 4 \, \sin \relax (x) \right |}}{{\left | 2 \, \sqrt {2} + 4 \, \sin \relax (x) \right |}}\right ) + \frac {43 \, \sin \relax (x)^{3} - 45 \, \sin \relax (x)}{256 \, {\left (\sin \relax (x)^{2} - 1\right )}^{2}} + \frac {10488 \, \sin \relax (x)^{7} - 14108 \, \sin \relax (x)^{5} + 6514 \, \sin \relax (x)^{3} - 993 \, \sin \relax (x)}{1536 \, {\left (2 \, \sin \relax (x)^{2} - 1\right )}^{4}} - \frac {523}{512} \, \log \left (\sin \relax (x) + 1\right ) + \frac {523}{512} \, \log \left (-\sin \relax (x) + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 95, normalized size = 0.88 \[ \frac {1483 \sqrt {2}\, \arctanh \left (\sqrt {2}\, \sin \relax (x )\right )}{1024}+\frac {523 \ln \left (\sin \relax (x )-1\right )}{512}-\frac {523 \ln \left (\sin \relax (x )+1\right )}{512}+\frac {1}{512 \left (\sin \relax (x )+1\right )^{2}}+\frac {43}{512 \left (\sin \relax (x )+1\right )}-\frac {4 \left (-\frac {437 \left (\sin ^{7}\relax (x )\right )}{256}+\frac {3527 \left (\sin ^{5}\relax (x )\right )}{1536}-\frac {3257 \left (\sin ^{3}\relax (x )\right )}{3072}+\frac {331 \sin \relax (x )}{2048}\right )}{\left (2 \left (\sin ^{2}\relax (x )\right )-1\right )^{4}}-\frac {1}{512 \left (\sin \relax (x )-1\right )^{2}}+\frac {43}{512 \left (\sin \relax (x )-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.25, size = 307, normalized size = 2.84 \[ -\frac {11492\,\sin \left (3\,x\right )+18218\,\sin \left (5\,x\right )+12230\,\sin \left (7\,x\right )+7466\,\sin \left (9\,x\right )+3654\,\sin \left (11\,x\right )+276144\,\mathrm {atanh}\left (\frac {\sin \left (\frac {x}{2}\right )}{\cos \left (\frac {x}{2}\right )}\right )+4764\,\sin \relax (x)+502080\,\cos \left (2\,x\right )\,\mathrm {atanh}\left (\frac {\sin \left (\frac {x}{2}\right )}{\cos \left (\frac {x}{2}\right )}\right )+389112\,\cos \left (4\,x\right )\,\mathrm {atanh}\left (\frac {\sin \left (\frac {x}{2}\right )}{\cos \left (\frac {x}{2}\right )}\right )+251040\,\cos \left (6\,x\right )\,\mathrm {atanh}\left (\frac {\sin \left (\frac {x}{2}\right )}{\cos \left (\frac {x}{2}\right )}\right )+125520\,\cos \left (8\,x\right )\,\mathrm {atanh}\left (\frac {\sin \left (\frac {x}{2}\right )}{\cos \left (\frac {x}{2}\right )}\right )+50208\,\cos \left (10\,x\right )\,\mathrm {atanh}\left (\frac {\sin \left (\frac {x}{2}\right )}{\cos \left (\frac {x}{2}\right )}\right )+12552\,\cos \left (12\,x\right )\,\mathrm {atanh}\left (\frac {\sin \left (\frac {x}{2}\right )}{\cos \left (\frac {x}{2}\right )}\right )-97878\,\sqrt {2}\,\mathrm {atanh}\left (\sqrt {2}\,\sin \relax (x)\right )-177960\,\sqrt {2}\,\mathrm {atanh}\left (\sqrt {2}\,\sin \relax (x)\right )\,\cos \left (2\,x\right )-137919\,\sqrt {2}\,\mathrm {atanh}\left (\sqrt {2}\,\sin \relax (x)\right )\,\cos \left (4\,x\right )-88980\,\sqrt {2}\,\mathrm {atanh}\left (\sqrt {2}\,\sin \relax (x)\right )\,\cos \left (6\,x\right )-44490\,\sqrt {2}\,\mathrm {atanh}\left (\sqrt {2}\,\sin \relax (x)\right )\,\cos \left (8\,x\right )-17796\,\sqrt {2}\,\mathrm {atanh}\left (\sqrt {2}\,\sin \relax (x)\right )\,\cos \left (10\,x\right )-4449\,\sqrt {2}\,\mathrm {atanh}\left (\sqrt {2}\,\sin \relax (x)\right )\,\cos \left (12\,x\right )}{122880\,\cos \left (2\,x\right )+95232\,\cos \left (4\,x\right )+61440\,\cos \left (6\,x\right )+30720\,\cos \left (8\,x\right )+12288\,\cos \left (10\,x\right )+3072\,\cos \left (12\,x\right )+67584} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (\cos {\relax (x )} + \cos {\left (3 x \right )}\right )^{5}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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