Integrand size = 23, antiderivative size = 314 \[ \int \frac {\sin ^6(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx=\frac {5 (a+2 b) \left (a^2+16 a b+16 b^2\right ) x}{16 a^6}-\frac {5 \sqrt {b} \sqrt {a+b} (a+4 b) (3 a+4 b) \arctan \left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b}}\right )}{8 a^6 f}-\frac {\left (33 a^2+110 a b+80 b^2\right ) \cos (e+f x) \sin (e+f x)}{48 a^3 f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac {(9 a+10 b) \cos ^3(e+f x) \sin (e+f x)}{24 a^2 f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac {\cos ^3(e+f x) \sin ^3(e+f x)}{6 a f \left (a+b+b \tan ^2(e+f x)\right )^2}-\frac {5 b \left (9 a^2+32 a b+24 b^2\right ) \tan (e+f x)}{48 a^4 f \left (a+b+b \tan ^2(e+f x)\right )^2}-\frac {5 b \left (5 a^2+20 a b+16 b^2\right ) \tan (e+f x)}{16 a^5 f \left (a+b+b \tan ^2(e+f x)\right )} \]
[Out]
Time = 0.57 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {4217, 481, 592, 541, 536, 209, 211} \[ \int \frac {\sin ^6(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx=-\frac {5 \sqrt {b} \sqrt {a+b} (a+4 b) (3 a+4 b) \arctan \left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b}}\right )}{8 a^6 f}+\frac {(9 a+10 b) \sin (e+f x) \cos ^3(e+f x)}{24 a^2 f \left (a+b \tan ^2(e+f x)+b\right )^2}+\frac {5 x (a+2 b) \left (a^2+16 a b+16 b^2\right )}{16 a^6}-\frac {5 b \left (5 a^2+20 a b+16 b^2\right ) \tan (e+f x)}{16 a^5 f \left (a+b \tan ^2(e+f x)+b\right )}-\frac {5 b \left (9 a^2+32 a b+24 b^2\right ) \tan (e+f x)}{48 a^4 f \left (a+b \tan ^2(e+f x)+b\right )^2}-\frac {\left (33 a^2+110 a b+80 b^2\right ) \sin (e+f x) \cos (e+f x)}{48 a^3 f \left (a+b \tan ^2(e+f x)+b\right )^2}+\frac {\sin ^3(e+f x) \cos ^3(e+f x)}{6 a f \left (a+b \tan ^2(e+f x)+b\right )^2} \]
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Rule 209
Rule 211
Rule 481
Rule 536
Rule 541
Rule 592
Rule 4217
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {x^6}{\left (1+x^2\right )^4 \left (a+b+b x^2\right )^3} \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {\cos ^3(e+f x) \sin ^3(e+f x)}{6 a f \left (a+b+b \tan ^2(e+f x)\right )^2}-\frac {\text {Subst}\left (\int \frac {x^2 \left (3 (a+b)+(-6 a-7 b) x^2\right )}{\left (1+x^2\right )^3 \left (a+b+b x^2\right )^3} \, dx,x,\tan (e+f x)\right )}{6 a f} \\ & = \frac {(9 a+10 b) \cos ^3(e+f x) \sin (e+f x)}{24 a^2 f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac {\cos ^3(e+f x) \sin ^3(e+f x)}{6 a f \left (a+b+b \tan ^2(e+f x)\right )^2}-\frac {\text {Subst}\left (\int \frac {(a+b) (9 a+10 b)+\left (-24 a^2-91 a b-70 b^2\right ) x^2}{\left (1+x^2\right )^2 \left (a+b+b x^2\right )^3} \, dx,x,\tan (e+f x)\right )}{24 a^2 f} \\ & = -\frac {\left (33 a^2+110 a b+80 b^2\right ) \cos (e+f x) \sin (e+f x)}{48 a^3 f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac {(9 a+10 b) \cos ^3(e+f x) \sin (e+f x)}{24 a^2 f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac {\cos ^3(e+f x) \sin ^3(e+f x)}{6 a f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac {\text {Subst}\left (\int \frac {5 (a+b) \left (3 a^2+18 a b+16 b^2\right )-5 b \left (33 a^2+110 a b+80 b^2\right ) x^2}{\left (1+x^2\right ) \left (a+b+b x^2\right )^3} \, dx,x,\tan (e+f x)\right )}{48 a^3 f} \\ & = -\frac {\left (33 a^2+110 a b+80 b^2\right ) \cos (e+f x) \sin (e+f x)}{48 a^3 f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac {(9 a+10 b) \cos ^3(e+f x) \sin (e+f x)}{24 a^2 f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac {\cos ^3(e+f x) \sin ^3(e+f x)}{6 a f \left (a+b+b \tan ^2(e+f x)\right )^2}-\frac {5 b \left (9 a^2+32 a b+24 b^2\right ) \tan (e+f x)}{48 a^4 f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac {\text {Subst}\left (\int \frac {60 (a+b)^2 \left (a^2+8 a b+8 b^2\right )-60 b (a+b) \left (9 a^2+32 a b+24 b^2\right ) x^2}{\left (1+x^2\right ) \left (a+b+b x^2\right )^2} \, dx,x,\tan (e+f x)\right )}{192 a^4 (a+b) f} \\ & = -\frac {\left (33 a^2+110 a b+80 b^2\right ) \cos (e+f x) \sin (e+f x)}{48 a^3 f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac {(9 a+10 b) \cos ^3(e+f x) \sin (e+f x)}{24 a^2 f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac {\cos ^3(e+f x) \sin ^3(e+f x)}{6 a f \left (a+b+b \tan ^2(e+f x)\right )^2}-\frac {5 b \left (9 a^2+32 a b+24 b^2\right ) \tan (e+f x)}{48 a^4 f \left (a+b+b \tan ^2(e+f x)\right )^2}-\frac {5 b \left (5 a^2+20 a b+16 b^2\right ) \tan (e+f x)}{16 a^5 f \left (a+b+b \tan ^2(e+f x)\right )}+\frac {\text {Subst}\left (\int \frac {120 (a+b)^3 \left (a^2+12 a b+16 b^2\right )-120 b (a+b)^2 \left (5 a^2+20 a b+16 b^2\right ) x^2}{\left (1+x^2\right ) \left (a+b+b x^2\right )} \, dx,x,\tan (e+f x)\right )}{384 a^5 (a+b)^2 f} \\ & = -\frac {\left (33 a^2+110 a b+80 b^2\right ) \cos (e+f x) \sin (e+f x)}{48 a^3 f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac {(9 a+10 b) \cos ^3(e+f x) \sin (e+f x)}{24 a^2 f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac {\cos ^3(e+f x) \sin ^3(e+f x)}{6 a f \left (a+b+b \tan ^2(e+f x)\right )^2}-\frac {5 b \left (9 a^2+32 a b+24 b^2\right ) \tan (e+f x)}{48 a^4 f \left (a+b+b \tan ^2(e+f x)\right )^2}-\frac {5 b \left (5 a^2+20 a b+16 b^2\right ) \tan (e+f x)}{16 a^5 f \left (a+b+b \tan ^2(e+f x)\right )}-\frac {(5 b (a+b) (a+4 b) (3 a+4 b)) \text {Subst}\left (\int \frac {1}{a+b+b x^2} \, dx,x,\tan (e+f x)\right )}{8 a^6 f}+\frac {\left (5 (a+2 b) \left (a^2+16 a b+16 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (e+f x)\right )}{16 a^6 f} \\ & = \frac {5 (a+2 b) \left (a^2+16 a b+16 b^2\right ) x}{16 a^6}-\frac {5 \sqrt {b} \sqrt {a+b} (a+4 b) (3 a+4 b) \arctan \left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b}}\right )}{8 a^6 f}-\frac {\left (33 a^2+110 a b+80 b^2\right ) \cos (e+f x) \sin (e+f x)}{48 a^3 f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac {(9 a+10 b) \cos ^3(e+f x) \sin (e+f x)}{24 a^2 f \left (a+b+b \tan ^2(e+f x)\right )^2}+\frac {\cos ^3(e+f x) \sin ^3(e+f x)}{6 a f \left (a+b+b \tan ^2(e+f x)\right )^2}-\frac {5 b \left (9 a^2+32 a b+24 b^2\right ) \tan (e+f x)}{48 a^4 f \left (a+b+b \tan ^2(e+f x)\right )^2}-\frac {5 b \left (5 a^2+20 a b+16 b^2\right ) \tan (e+f x)}{16 a^5 f \left (a+b+b \tan ^2(e+f x)\right )} \\ \end{align*}
Result contains complex when optimal does not.
Time = 17.42 (sec) , antiderivative size = 1639, normalized size of antiderivative = 5.22 \[ \int \frac {\sin ^6(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx=\frac {5 (a+2 b+a \cos (2 e+2 f x))^3 \sec ^6(e+f x) \left (\frac {\left (3 a^2+8 a b+8 b^2\right ) \arctan \left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b}}\right )}{(a+b)^{5/2}}-\frac {a \sqrt {b} \left (3 a^2+16 a b+16 b^2+3 a (a+2 b) \cos (2 (e+f x))\right ) \sin (2 (e+f x))}{(a+b)^2 (a+2 b+a \cos (2 (e+f x)))^2}\right )}{65536 b^{5/2} f \left (a+b \sec ^2(e+f x)\right )^3}-\frac {15 (a+2 b+a \cos (2 e+2 f x))^3 \sec ^6(e+f x) \left (-\frac {6 a^2 \arctan \left (\frac {\sec (f x) (\cos (2 e)-i \sin (2 e)) (-((a+2 b) \sin (f x))+a \sin (2 e+f x))}{2 \sqrt {a+b} \sqrt {b (\cos (e)-i \sin (e))^4}}\right ) (\cos (2 e)-i \sin (2 e))}{\sqrt {a+b} \sqrt {b (\cos (e)-i \sin (e))^4}}+\frac {a \sec (2 e) \left (\left (-9 a^4-16 a^3 b+48 a^2 b^2+128 a b^3+64 b^4\right ) \sin (2 f x)+a \left (-3 a^3+2 a^2 b+24 a b^2+16 b^3\right ) \sin (2 (e+2 f x))+\left (3 a^4-64 a^2 b^2-128 a b^3-64 b^4\right ) \sin (4 e+2 f x)\right )+\left (9 a^5+18 a^4 b-64 a^3 b^2-256 a^2 b^3-320 a b^4-128 b^5\right ) \tan (2 e)}{a^2 (a+2 b+a \cos (2 (e+f x)))^2}\right )}{262144 b^2 (a+b)^2 f \left (a+b \sec ^2(e+f x)\right )^3}+\frac {3 (a+2 b+a \cos (2 e+2 f x))^3 \sec ^6(e+f x) \left (-1536 (a+2 b) x-\frac {3 \left (a^6-8 a^5 b+120 a^4 b^2+1280 a^3 b^3+3200 a^2 b^4+3072 a b^5+1024 b^6\right ) \arctan \left (\frac {\sec (f x) (\cos (2 e)-i \sin (2 e)) (-((a+2 b) \sin (f x))+a \sin (2 e+f x))}{2 \sqrt {a+b} \sqrt {b (\cos (e)-i \sin (e))^4}}\right ) (\cos (2 e)-i \sin (2 e))}{b^2 (a+b)^{5/2} f \sqrt {b (\cos (e)-i \sin (e))^4}}+\frac {4 \left (a^4+32 a^3 b+160 a^2 b^2+256 a b^3+128 b^4\right ) \sec (2 e) ((a+2 b) \sin (2 e)-a \sin (2 f x))}{b (a+b) f (a+2 b+a \cos (2 (e+f x)))^2}+\frac {256 a \sin (2 (e+f x))}{f}+\frac {a \left (-3 a^5+26 a^4 b+736 a^3 b^2+2624 a^2 b^3+3200 a b^4+1280 b^5\right ) \sec (2 e) \sin (2 f x)+\left (3 a^6-24 a^5 b-920 a^4 b^2-4864 a^3 b^3-10112 a^2 b^4-9216 a b^5-3072 b^6\right ) \tan (2 e)}{b^2 (a+b)^2 f (a+2 b+a \cos (2 (e+f x)))}\right )}{65536 a^4 \left (a+b \sec ^2(e+f x)\right )^3}-\frac {(a+2 b+a \cos (2 e+2 f x))^3 \sec ^6(e+f x) \left (-6144 \left (7 a^3+54 a^2 b+120 a b^2+80 b^3\right ) x-\frac {3 \left (3 a^8-64 a^7 b+2240 a^6 b^2+53760 a^5 b^3+313600 a^4 b^4+802816 a^3 b^5+1032192 a^2 b^6+655360 a b^7+163840 b^8\right ) \arctan \left (\frac {\sec (f x) (\cos (2 e)-i \sin (2 e)) (-((a+2 b) \sin (f x))+a \sin (2 e+f x))}{2 \sqrt {a+b} \sqrt {b (\cos (e)-i \sin (e))^4}}\right ) (\cos (2 e)-i \sin (2 e))}{b^2 (a+b)^{5/2} f \sqrt {b (\cos (e)-i \sin (e))^4}}+\frac {12 \left (a^6+72 a^5 b+840 a^4 b^2+3584 a^3 b^3+6912 a^2 b^4+6144 a b^5+2048 b^6\right ) \sec (2 e) ((a+2 b) \sin (2 e)-a \sin (2 f x))}{b (a+b) f (a+2 b+a \cos (2 (e+f x)))^2}+\frac {1152 a \left (7 a^2+32 a b+32 b^2\right ) (-i \cos (2 (e+f x))+\sin (2 (e+f x)))}{f}+\frac {1152 a \left (7 a^2+32 a b+32 b^2\right ) (i \cos (2 (e+f x))+\sin (2 (e+f x)))}{f}+\frac {192 a^2 (a+2 b) (-6 i \cos (4 (e+f x))-6 \sin (4 (e+f x)))}{f}+\frac {1152 i a^2 (a+2 b) (\cos (4 (e+f x))+i \sin (4 (e+f x)))}{f}+\frac {256 a^3 \sin (6 (e+f x))}{f}+\frac {3 \left (3 a \left (-a^7+22 a^6 b+1352 a^5 b^2+11312 a^4 b^3+37120 a^3 b^4+57856 a^2 b^5+43008 a b^6+12288 b^7\right ) \sec (2 e) \sin (2 f x)+\left (3 a^8-64 a^7 b-4480 a^6 b^2-45696 a^5 b^3-196928 a^4 b^4-438272 a^3 b^5-528384 a^2 b^6-327680 a b^7-81920 b^8\right ) \tan (2 e)\right )}{b^2 (a+b)^2 f (a+2 b+a \cos (2 (e+f x)))}\right )}{393216 a^6 \left (a+b \sec ^2(e+f x)\right )^3} \]
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Time = 11.70 (sec) , antiderivative size = 247, normalized size of antiderivative = 0.79
method | result | size |
derivativedivides | \(\frac {\frac {\frac {\left (-\frac {27}{8} a^{2} b -3 a \,b^{2}-\frac {11}{16} a^{3}\right ) \tan \left (f x +e \right )^{5}+\left (-6 a^{2} b -6 a \,b^{2}-\frac {5}{6} a^{3}\right ) \tan \left (f x +e \right )^{3}+\left (-\frac {5}{16} a^{3}-\frac {21}{8} a^{2} b -3 a \,b^{2}\right ) \tan \left (f x +e \right )}{\left (1+\tan \left (f x +e \right )^{2}\right )^{3}}+\frac {5 \left (a^{3}+18 a^{2} b +48 a \,b^{2}+32 b^{3}\right ) \arctan \left (\tan \left (f x +e \right )\right )}{16}}{a^{6}}-\frac {\left (a +b \right ) b \left (\frac {\left (\frac {7}{8} a^{2} b +2 a \,b^{2}\right ) \tan \left (f x +e \right )^{3}+\frac {a \left (9 a^{2}+25 a b +16 b^{2}\right ) \tan \left (f x +e \right )}{8}}{\left (a +b +b \tan \left (f x +e \right )^{2}\right )^{2}}+\frac {5 \left (3 a^{2}+16 a b +16 b^{2}\right ) \arctan \left (\frac {b \tan \left (f x +e \right )}{\sqrt {\left (a +b \right ) b}}\right )}{8 \sqrt {\left (a +b \right ) b}}\right )}{a^{6}}}{f}\) | \(247\) |
default | \(\frac {\frac {\frac {\left (-\frac {27}{8} a^{2} b -3 a \,b^{2}-\frac {11}{16} a^{3}\right ) \tan \left (f x +e \right )^{5}+\left (-6 a^{2} b -6 a \,b^{2}-\frac {5}{6} a^{3}\right ) \tan \left (f x +e \right )^{3}+\left (-\frac {5}{16} a^{3}-\frac {21}{8} a^{2} b -3 a \,b^{2}\right ) \tan \left (f x +e \right )}{\left (1+\tan \left (f x +e \right )^{2}\right )^{3}}+\frac {5 \left (a^{3}+18 a^{2} b +48 a \,b^{2}+32 b^{3}\right ) \arctan \left (\tan \left (f x +e \right )\right )}{16}}{a^{6}}-\frac {\left (a +b \right ) b \left (\frac {\left (\frac {7}{8} a^{2} b +2 a \,b^{2}\right ) \tan \left (f x +e \right )^{3}+\frac {a \left (9 a^{2}+25 a b +16 b^{2}\right ) \tan \left (f x +e \right )}{8}}{\left (a +b +b \tan \left (f x +e \right )^{2}\right )^{2}}+\frac {5 \left (3 a^{2}+16 a b +16 b^{2}\right ) \arctan \left (\frac {b \tan \left (f x +e \right )}{\sqrt {\left (a +b \right ) b}}\right )}{8 \sqrt {\left (a +b \right ) b}}\right )}{a^{6}}}{f}\) | \(247\) |
risch | \(\frac {5 x}{16 a^{3}}+\frac {45 x b}{8 a^{4}}+\frac {15 x \,b^{2}}{a^{5}}+\frac {10 x \,b^{3}}{a^{6}}+\frac {15 i {\mathrm e}^{2 i \left (f x +e \right )}}{128 a^{3} f}+\frac {3 i {\mathrm e}^{-4 i \left (f x +e \right )} b}{64 a^{4} f}+\frac {3 i {\mathrm e}^{2 i \left (f x +e \right )} b}{4 a^{4} f}+\frac {3 i {\mathrm e}^{2 i \left (f x +e \right )} b^{2}}{4 a^{5} f}-\frac {i {\mathrm e}^{-6 i \left (f x +e \right )}}{384 a^{3} f}-\frac {3 i {\mathrm e}^{4 i \left (f x +e \right )}}{128 a^{3} f}-\frac {i b \left (9 a^{4} {\mathrm e}^{6 i \left (f x +e \right )}+49 a^{3} b \,{\mathrm e}^{6 i \left (f x +e \right )}+80 a^{2} b^{2} {\mathrm e}^{6 i \left (f x +e \right )}+40 a \,b^{3} {\mathrm e}^{6 i \left (f x +e \right )}+27 a^{4} {\mathrm e}^{4 i \left (f x +e \right )}+153 a^{3} b \,{\mathrm e}^{4 i \left (f x +e \right )}+342 a^{2} b^{2} {\mathrm e}^{4 i \left (f x +e \right )}+360 a \,b^{3} {\mathrm e}^{4 i \left (f x +e \right )}+144 b^{4} {\mathrm e}^{4 i \left (f x +e \right )}+27 \,{\mathrm e}^{2 i \left (f x +e \right )} a^{4}+131 a^{3} b \,{\mathrm e}^{2 i \left (f x +e \right )}+208 a^{2} b^{2} {\mathrm e}^{2 i \left (f x +e \right )}+104 a \,b^{3} {\mathrm e}^{2 i \left (f x +e \right )}+9 a^{4}+27 a^{3} b +18 a^{2} b^{2}\right )}{4 a^{6} f \left (a \,{\mathrm e}^{4 i \left (f x +e \right )}+2 a \,{\mathrm e}^{2 i \left (f x +e \right )}+4 b \,{\mathrm e}^{2 i \left (f x +e \right )}+a \right )^{2}}+\frac {i {\mathrm e}^{6 i \left (f x +e \right )}}{384 a^{3} f}-\frac {3 i {\mathrm e}^{4 i \left (f x +e \right )} b}{64 a^{4} f}-\frac {3 i {\mathrm e}^{-2 i \left (f x +e \right )} b^{2}}{4 a^{5} f}-\frac {3 i {\mathrm e}^{-2 i \left (f x +e \right )} b}{4 a^{4} f}+\frac {3 i {\mathrm e}^{-4 i \left (f x +e \right )}}{128 a^{3} f}-\frac {15 i {\mathrm e}^{-2 i \left (f x +e \right )}}{128 a^{3} f}-\frac {15 \sqrt {-a b -b^{2}}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {2 i \sqrt {-a b -b^{2}}-a -2 b}{a}\right )}{16 f \,a^{4}}-\frac {5 \sqrt {-a b -b^{2}}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {2 i \sqrt {-a b -b^{2}}-a -2 b}{a}\right ) b}{f \,a^{5}}-\frac {5 \sqrt {-a b -b^{2}}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-\frac {2 i \sqrt {-a b -b^{2}}-a -2 b}{a}\right ) b^{2}}{f \,a^{6}}+\frac {15 \sqrt {-a b -b^{2}}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+\frac {2 i \sqrt {-a b -b^{2}}+a +2 b}{a}\right )}{16 f \,a^{4}}+\frac {5 \sqrt {-a b -b^{2}}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+\frac {2 i \sqrt {-a b -b^{2}}+a +2 b}{a}\right ) b}{f \,a^{5}}+\frac {5 \sqrt {-a b -b^{2}}\, \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+\frac {2 i \sqrt {-a b -b^{2}}+a +2 b}{a}\right ) b^{2}}{f \,a^{6}}\) | \(872\) |
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Time = 0.36 (sec) , antiderivative size = 930, normalized size of antiderivative = 2.96 \[ \int \frac {\sin ^6(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx=\left [\frac {30 \, {\left (a^{5} + 18 \, a^{4} b + 48 \, a^{3} b^{2} + 32 \, a^{2} b^{3}\right )} f x \cos \left (f x + e\right )^{4} + 60 \, {\left (a^{4} b + 18 \, a^{3} b^{2} + 48 \, a^{2} b^{3} + 32 \, a b^{4}\right )} f x \cos \left (f x + e\right )^{2} + 30 \, {\left (a^{3} b^{2} + 18 \, a^{2} b^{3} + 48 \, a b^{4} + 32 \, b^{5}\right )} f x + 15 \, {\left ({\left (3 \, a^{4} + 16 \, a^{3} b + 16 \, a^{2} b^{2}\right )} \cos \left (f x + e\right )^{4} + 3 \, a^{2} b^{2} + 16 \, a b^{3} + 16 \, b^{4} + 2 \, {\left (3 \, a^{3} b + 16 \, a^{2} b^{2} + 16 \, a b^{3}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {-a b - b^{2}} \log \left (\frac {{\left (a^{2} + 8 \, a b + 8 \, b^{2}\right )} \cos \left (f x + e\right )^{4} - 2 \, {\left (3 \, a b + 4 \, b^{2}\right )} \cos \left (f x + e\right )^{2} + 4 \, {\left ({\left (a + 2 \, b\right )} \cos \left (f x + e\right )^{3} - b \cos \left (f x + e\right )\right )} \sqrt {-a b - b^{2}} \sin \left (f x + e\right ) + b^{2}}{a^{2} \cos \left (f x + e\right )^{4} + 2 \, a b \cos \left (f x + e\right )^{2} + b^{2}}\right ) - 2 \, {\left (8 \, a^{5} \cos \left (f x + e\right )^{9} - 2 \, {\left (13 \, a^{5} + 10 \, a^{4} b\right )} \cos \left (f x + e\right )^{7} + {\left (33 \, a^{5} + 110 \, a^{4} b + 80 \, a^{3} b^{2}\right )} \cos \left (f x + e\right )^{5} + 20 \, {\left (6 \, a^{4} b + 23 \, a^{3} b^{2} + 18 \, a^{2} b^{3}\right )} \cos \left (f x + e\right )^{3} + 15 \, {\left (5 \, a^{3} b^{2} + 20 \, a^{2} b^{3} + 16 \, a b^{4}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{96 \, {\left (a^{8} f \cos \left (f x + e\right )^{4} + 2 \, a^{7} b f \cos \left (f x + e\right )^{2} + a^{6} b^{2} f\right )}}, \frac {15 \, {\left (a^{5} + 18 \, a^{4} b + 48 \, a^{3} b^{2} + 32 \, a^{2} b^{3}\right )} f x \cos \left (f x + e\right )^{4} + 30 \, {\left (a^{4} b + 18 \, a^{3} b^{2} + 48 \, a^{2} b^{3} + 32 \, a b^{4}\right )} f x \cos \left (f x + e\right )^{2} + 15 \, {\left (a^{3} b^{2} + 18 \, a^{2} b^{3} + 48 \, a b^{4} + 32 \, b^{5}\right )} f x + 15 \, {\left ({\left (3 \, a^{4} + 16 \, a^{3} b + 16 \, a^{2} b^{2}\right )} \cos \left (f x + e\right )^{4} + 3 \, a^{2} b^{2} + 16 \, a b^{3} + 16 \, b^{4} + 2 \, {\left (3 \, a^{3} b + 16 \, a^{2} b^{2} + 16 \, a b^{3}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {a b + b^{2}} \arctan \left (\frac {{\left (a + 2 \, b\right )} \cos \left (f x + e\right )^{2} - b}{2 \, \sqrt {a b + b^{2}} \cos \left (f x + e\right ) \sin \left (f x + e\right )}\right ) - {\left (8 \, a^{5} \cos \left (f x + e\right )^{9} - 2 \, {\left (13 \, a^{5} + 10 \, a^{4} b\right )} \cos \left (f x + e\right )^{7} + {\left (33 \, a^{5} + 110 \, a^{4} b + 80 \, a^{3} b^{2}\right )} \cos \left (f x + e\right )^{5} + 20 \, {\left (6 \, a^{4} b + 23 \, a^{3} b^{2} + 18 \, a^{2} b^{3}\right )} \cos \left (f x + e\right )^{3} + 15 \, {\left (5 \, a^{3} b^{2} + 20 \, a^{2} b^{3} + 16 \, a b^{4}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{48 \, {\left (a^{8} f \cos \left (f x + e\right )^{4} + 2 \, a^{7} b f \cos \left (f x + e\right )^{2} + a^{6} b^{2} f\right )}}\right ] \]
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Timed out. \[ \int \frac {\sin ^6(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx=\text {Timed out} \]
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Time = 0.27 (sec) , antiderivative size = 418, normalized size of antiderivative = 1.33 \[ \int \frac {\sin ^6(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx=-\frac {\frac {15 \, {\left (5 \, a^{2} b^{2} + 20 \, a b^{3} + 16 \, b^{4}\right )} \tan \left (f x + e\right )^{9} + 40 \, {\left (3 \, a^{3} b + 19 \, a^{2} b^{2} + 39 \, a b^{3} + 24 \, b^{4}\right )} \tan \left (f x + e\right )^{7} + {\left (33 \, a^{4} + 470 \, a^{3} b + 1910 \, a^{2} b^{2} + 2880 \, a b^{3} + 1440 \, b^{4}\right )} \tan \left (f x + e\right )^{5} + 40 \, {\left (a^{4} + 14 \, a^{3} b + 46 \, a^{2} b^{2} + 57 \, a b^{3} + 24 \, b^{4}\right )} \tan \left (f x + e\right )^{3} + 15 \, {\left (a^{4} + 14 \, a^{3} b + 41 \, a^{2} b^{2} + 44 \, a b^{3} + 16 \, b^{4}\right )} \tan \left (f x + e\right )}{a^{5} b^{2} \tan \left (f x + e\right )^{10} + {\left (2 \, a^{6} b + 5 \, a^{5} b^{2}\right )} \tan \left (f x + e\right )^{8} + a^{7} + 2 \, a^{6} b + a^{5} b^{2} + {\left (a^{7} + 8 \, a^{6} b + 10 \, a^{5} b^{2}\right )} \tan \left (f x + e\right )^{6} + {\left (3 \, a^{7} + 12 \, a^{6} b + 10 \, a^{5} b^{2}\right )} \tan \left (f x + e\right )^{4} + {\left (3 \, a^{7} + 8 \, a^{6} b + 5 \, a^{5} b^{2}\right )} \tan \left (f x + e\right )^{2}} - \frac {15 \, {\left (a^{3} + 18 \, a^{2} b + 48 \, a b^{2} + 32 \, b^{3}\right )} {\left (f x + e\right )}}{a^{6}} + \frac {30 \, {\left (3 \, a^{3} b + 19 \, a^{2} b^{2} + 32 \, a b^{3} + 16 \, b^{4}\right )} \arctan \left (\frac {b \tan \left (f x + e\right )}{\sqrt {{\left (a + b\right )} b}}\right )}{\sqrt {{\left (a + b\right )} b} a^{6}}}{48 \, f} \]
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Time = 0.49 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.12 \[ \int \frac {\sin ^6(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx=\frac {\frac {15 \, {\left (a^{3} + 18 \, a^{2} b + 48 \, a b^{2} + 32 \, b^{3}\right )} {\left (f x + e\right )}}{a^{6}} - \frac {30 \, {\left (3 \, a^{3} b + 19 \, a^{2} b^{2} + 32 \, a b^{3} + 16 \, b^{4}\right )} {\left (\pi \left \lfloor \frac {f x + e}{\pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (b\right ) + \arctan \left (\frac {b \tan \left (f x + e\right )}{\sqrt {a b + b^{2}}}\right )\right )}}{\sqrt {a b + b^{2}} a^{6}} - \frac {6 \, {\left (7 \, a^{2} b^{2} \tan \left (f x + e\right )^{3} + 23 \, a b^{3} \tan \left (f x + e\right )^{3} + 16 \, b^{4} \tan \left (f x + e\right )^{3} + 9 \, a^{3} b \tan \left (f x + e\right ) + 34 \, a^{2} b^{2} \tan \left (f x + e\right ) + 41 \, a b^{3} \tan \left (f x + e\right ) + 16 \, b^{4} \tan \left (f x + e\right )\right )}}{{\left (b \tan \left (f x + e\right )^{2} + a + b\right )}^{2} a^{5}} - \frac {33 \, a^{2} \tan \left (f x + e\right )^{5} + 162 \, a b \tan \left (f x + e\right )^{5} + 144 \, b^{2} \tan \left (f x + e\right )^{5} + 40 \, a^{2} \tan \left (f x + e\right )^{3} + 288 \, a b \tan \left (f x + e\right )^{3} + 288 \, b^{2} \tan \left (f x + e\right )^{3} + 15 \, a^{2} \tan \left (f x + e\right ) + 126 \, a b \tan \left (f x + e\right ) + 144 \, b^{2} \tan \left (f x + e\right )}{{\left (\tan \left (f x + e\right )^{2} + 1\right )}^{3} a^{5}}}{48 \, f} \]
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Time = 22.33 (sec) , antiderivative size = 2117, normalized size of antiderivative = 6.74 \[ \int \frac {\sin ^6(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^3} \, dx=\text {Too large to display} \]
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