Integrand size = 21, antiderivative size = 473 \[ \int \frac {\sin ^6(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\frac {\left (5 a^6-90 a^4 b^2+200 a^2 b^4-112 b^6\right ) x}{16 a^8}-\frac {2 (a-b)^{3/2} b (a+b)^{3/2} \left (2 a^2-7 b^2\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^8 d}+\frac {b \left (61 a^4-170 a^2 b^2+105 b^4\right ) \sin (c+d x)}{15 a^7 d}-\frac {\left (27 a^4-86 a^2 b^2+56 b^4\right ) \cos (c+d x) \sin (c+d x)}{16 a^6 d}+\frac {\left (15 a^4-52 a^2 b^2+35 b^4\right ) \cos ^2(c+d x) \sin (c+d x)}{15 a^5 b d}-\frac {\left (16 a^4-61 a^2 b^2+42 b^4\right ) \cos ^3(c+d x) \sin (c+d x)}{24 a^4 b^2 d}-\frac {\cos ^3(c+d x) \sin (c+d x)}{3 b d (b+a \cos (c+d x))}+\frac {a \cos ^4(c+d x) \sin (c+d x)}{6 b^2 d (b+a \cos (c+d x))}+\frac {\left (5 a^4-20 a^2 b^2+14 b^4\right ) \cos ^4(c+d x) \sin (c+d x)}{10 a^3 b^2 d (b+a \cos (c+d x))}+\frac {7 b \cos ^5(c+d x) \sin (c+d x)}{30 a^2 d (b+a \cos (c+d x))}-\frac {\cos ^6(c+d x) \sin (c+d x)}{6 a d (b+a \cos (c+d x))} \]
[Out]
Time = 2.02 (sec) , antiderivative size = 473, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {3957, 2975, 3126, 3128, 3102, 2814, 2738, 214} \[ \int \frac {\sin ^6(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\frac {7 b \sin (c+d x) \cos ^5(c+d x)}{30 a^2 d (a \cos (c+d x)+b)}-\frac {2 b (a-b)^{3/2} (a+b)^{3/2} \left (2 a^2-7 b^2\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^8 d}-\frac {\left (16 a^4-61 a^2 b^2+42 b^4\right ) \sin (c+d x) \cos ^3(c+d x)}{24 a^4 b^2 d}+\frac {b \left (61 a^4-170 a^2 b^2+105 b^4\right ) \sin (c+d x)}{15 a^7 d}-\frac {\left (27 a^4-86 a^2 b^2+56 b^4\right ) \sin (c+d x) \cos (c+d x)}{16 a^6 d}+\frac {\left (15 a^4-52 a^2 b^2+35 b^4\right ) \sin (c+d x) \cos ^2(c+d x)}{15 a^5 b d}+\frac {\left (5 a^4-20 a^2 b^2+14 b^4\right ) \sin (c+d x) \cos ^4(c+d x)}{10 a^3 b^2 d (a \cos (c+d x)+b)}+\frac {x \left (5 a^6-90 a^4 b^2+200 a^2 b^4-112 b^6\right )}{16 a^8}+\frac {a \sin (c+d x) \cos ^4(c+d x)}{6 b^2 d (a \cos (c+d x)+b)}-\frac {\sin (c+d x) \cos ^6(c+d x)}{6 a d (a \cos (c+d x)+b)}-\frac {\sin (c+d x) \cos ^3(c+d x)}{3 b d (a \cos (c+d x)+b)} \]
[In]
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Rule 214
Rule 2738
Rule 2814
Rule 2975
Rule 3102
Rule 3126
Rule 3128
Rule 3957
Rubi steps \begin{align*} \text {integral}& = \int \frac {\cos ^2(c+d x) \sin ^6(c+d x)}{(-b-a \cos (c+d x))^2} \, dx \\ & = -\frac {\cos ^3(c+d x) \sin (c+d x)}{3 b d (b+a \cos (c+d x))}+\frac {a \cos ^4(c+d x) \sin (c+d x)}{6 b^2 d (b+a \cos (c+d x))}+\frac {7 b \cos ^5(c+d x) \sin (c+d x)}{30 a^2 d (b+a \cos (c+d x))}-\frac {\cos ^6(c+d x) \sin (c+d x)}{6 a d (b+a \cos (c+d x))}+\frac {\int \frac {\cos ^4(c+d x) \left (60 \left (3 a^4-10 a^2 b^2+7 b^4\right )+12 a b \left (5 a^2-2 b^2\right ) \cos (c+d x)-12 \left (20 a^4-65 a^2 b^2+42 b^4\right ) \cos ^2(c+d x)\right )}{(-b-a \cos (c+d x))^2} \, dx}{360 a^2 b^2} \\ & = -\frac {\cos ^3(c+d x) \sin (c+d x)}{3 b d (b+a \cos (c+d x))}+\frac {a \cos ^4(c+d x) \sin (c+d x)}{6 b^2 d (b+a \cos (c+d x))}+\frac {\left (5 a^4-20 a^2 b^2+14 b^4\right ) \cos ^4(c+d x) \sin (c+d x)}{10 a^3 b^2 d (b+a \cos (c+d x))}+\frac {7 b \cos ^5(c+d x) \sin (c+d x)}{30 a^2 d (b+a \cos (c+d x))}-\frac {\cos ^6(c+d x) \sin (c+d x)}{6 a d (b+a \cos (c+d x))}-\frac {\int \frac {\cos ^3(c+d x) \left (144 \left (5 a^6-25 a^4 b^2+34 a^2 b^4-14 b^6\right )+12 a b \left (10 a^4-17 a^2 b^2+7 b^4\right ) \cos (c+d x)-60 \left (16 a^6-77 a^4 b^2+103 a^2 b^4-42 b^6\right ) \cos ^2(c+d x)\right )}{-b-a \cos (c+d x)} \, dx}{360 a^3 b^2 \left (a^2-b^2\right )} \\ & = -\frac {\left (16 a^4-61 a^2 b^2+42 b^4\right ) \cos ^3(c+d x) \sin (c+d x)}{24 a^4 b^2 d}-\frac {\cos ^3(c+d x) \sin (c+d x)}{3 b d (b+a \cos (c+d x))}+\frac {a \cos ^4(c+d x) \sin (c+d x)}{6 b^2 d (b+a \cos (c+d x))}+\frac {\left (5 a^4-20 a^2 b^2+14 b^4\right ) \cos ^4(c+d x) \sin (c+d x)}{10 a^3 b^2 d (b+a \cos (c+d x))}+\frac {7 b \cos ^5(c+d x) \sin (c+d x)}{30 a^2 d (b+a \cos (c+d x))}-\frac {\cos ^6(c+d x) \sin (c+d x)}{6 a d (b+a \cos (c+d x))}+\frac {\int \frac {\cos ^2(c+d x) \left (180 b \left (16 a^6-77 a^4 b^2+103 a^2 b^4-42 b^6\right )+36 a b^2 \left (15 a^4-29 a^2 b^2+14 b^4\right ) \cos (c+d x)-288 b \left (15 a^6-67 a^4 b^2+87 a^2 b^4-35 b^6\right ) \cos ^2(c+d x)\right )}{-b-a \cos (c+d x)} \, dx}{1440 a^4 b^2 \left (a^2-b^2\right )} \\ & = \frac {\left (15 a^4-52 a^2 b^2+35 b^4\right ) \cos ^2(c+d x) \sin (c+d x)}{15 a^5 b d}-\frac {\left (16 a^4-61 a^2 b^2+42 b^4\right ) \cos ^3(c+d x) \sin (c+d x)}{24 a^4 b^2 d}-\frac {\cos ^3(c+d x) \sin (c+d x)}{3 b d (b+a \cos (c+d x))}+\frac {a \cos ^4(c+d x) \sin (c+d x)}{6 b^2 d (b+a \cos (c+d x))}+\frac {\left (5 a^4-20 a^2 b^2+14 b^4\right ) \cos ^4(c+d x) \sin (c+d x)}{10 a^3 b^2 d (b+a \cos (c+d x))}+\frac {7 b \cos ^5(c+d x) \sin (c+d x)}{30 a^2 d (b+a \cos (c+d x))}-\frac {\cos ^6(c+d x) \sin (c+d x)}{6 a d (b+a \cos (c+d x))}-\frac {\int \frac {\cos (c+d x) \left (576 b^2 \left (15 a^6-67 a^4 b^2+87 a^2 b^4-35 b^6\right )+36 a b^3 \left (83 a^4-153 a^2 b^2+70 b^4\right ) \cos (c+d x)-540 b^2 \left (27 a^6-113 a^4 b^2+142 a^2 b^4-56 b^6\right ) \cos ^2(c+d x)\right )}{-b-a \cos (c+d x)} \, dx}{4320 a^5 b^2 \left (a^2-b^2\right )} \\ & = -\frac {\left (27 a^4-86 a^2 b^2+56 b^4\right ) \cos (c+d x) \sin (c+d x)}{16 a^6 d}+\frac {\left (15 a^4-52 a^2 b^2+35 b^4\right ) \cos ^2(c+d x) \sin (c+d x)}{15 a^5 b d}-\frac {\left (16 a^4-61 a^2 b^2+42 b^4\right ) \cos ^3(c+d x) \sin (c+d x)}{24 a^4 b^2 d}-\frac {\cos ^3(c+d x) \sin (c+d x)}{3 b d (b+a \cos (c+d x))}+\frac {a \cos ^4(c+d x) \sin (c+d x)}{6 b^2 d (b+a \cos (c+d x))}+\frac {\left (5 a^4-20 a^2 b^2+14 b^4\right ) \cos ^4(c+d x) \sin (c+d x)}{10 a^3 b^2 d (b+a \cos (c+d x))}+\frac {7 b \cos ^5(c+d x) \sin (c+d x)}{30 a^2 d (b+a \cos (c+d x))}-\frac {\cos ^6(c+d x) \sin (c+d x)}{6 a d (b+a \cos (c+d x))}+\frac {\int \frac {540 b^3 \left (27 a^6-113 a^4 b^2+142 a^2 b^4-56 b^6\right )-36 a b^2 \left (75 a^6-449 a^4 b^2+654 a^2 b^4-280 b^6\right ) \cos (c+d x)-576 b^3 \left (61 a^6-231 a^4 b^2+275 a^2 b^4-105 b^6\right ) \cos ^2(c+d x)}{-b-a \cos (c+d x)} \, dx}{8640 a^6 b^2 \left (a^2-b^2\right )} \\ & = \frac {b \left (61 a^4-170 a^2 b^2+105 b^4\right ) \sin (c+d x)}{15 a^7 d}-\frac {\left (27 a^4-86 a^2 b^2+56 b^4\right ) \cos (c+d x) \sin (c+d x)}{16 a^6 d}+\frac {\left (15 a^4-52 a^2 b^2+35 b^4\right ) \cos ^2(c+d x) \sin (c+d x)}{15 a^5 b d}-\frac {\left (16 a^4-61 a^2 b^2+42 b^4\right ) \cos ^3(c+d x) \sin (c+d x)}{24 a^4 b^2 d}-\frac {\cos ^3(c+d x) \sin (c+d x)}{3 b d (b+a \cos (c+d x))}+\frac {a \cos ^4(c+d x) \sin (c+d x)}{6 b^2 d (b+a \cos (c+d x))}+\frac {\left (5 a^4-20 a^2 b^2+14 b^4\right ) \cos ^4(c+d x) \sin (c+d x)}{10 a^3 b^2 d (b+a \cos (c+d x))}+\frac {7 b \cos ^5(c+d x) \sin (c+d x)}{30 a^2 d (b+a \cos (c+d x))}-\frac {\cos ^6(c+d x) \sin (c+d x)}{6 a d (b+a \cos (c+d x))}-\frac {\int \frac {-540 a b^3 \left (27 a^6-113 a^4 b^2+142 a^2 b^4-56 b^6\right )+540 b^2 \left (5 a^8-95 a^6 b^2+290 a^4 b^4-312 a^2 b^6+112 b^8\right ) \cos (c+d x)}{-b-a \cos (c+d x)} \, dx}{8640 a^7 b^2 \left (a^2-b^2\right )} \\ & = \frac {\left (5 a^6-90 a^4 b^2+200 a^2 b^4-112 b^6\right ) x}{16 a^8}+\frac {b \left (61 a^4-170 a^2 b^2+105 b^4\right ) \sin (c+d x)}{15 a^7 d}-\frac {\left (27 a^4-86 a^2 b^2+56 b^4\right ) \cos (c+d x) \sin (c+d x)}{16 a^6 d}+\frac {\left (15 a^4-52 a^2 b^2+35 b^4\right ) \cos ^2(c+d x) \sin (c+d x)}{15 a^5 b d}-\frac {\left (16 a^4-61 a^2 b^2+42 b^4\right ) \cos ^3(c+d x) \sin (c+d x)}{24 a^4 b^2 d}-\frac {\cos ^3(c+d x) \sin (c+d x)}{3 b d (b+a \cos (c+d x))}+\frac {a \cos ^4(c+d x) \sin (c+d x)}{6 b^2 d (b+a \cos (c+d x))}+\frac {\left (5 a^4-20 a^2 b^2+14 b^4\right ) \cos ^4(c+d x) \sin (c+d x)}{10 a^3 b^2 d (b+a \cos (c+d x))}+\frac {7 b \cos ^5(c+d x) \sin (c+d x)}{30 a^2 d (b+a \cos (c+d x))}-\frac {\cos ^6(c+d x) \sin (c+d x)}{6 a d (b+a \cos (c+d x))}+\frac {\left (b \left (2 a^2-7 b^2\right ) \left (a^2-b^2\right )^2\right ) \int \frac {1}{-b-a \cos (c+d x)} \, dx}{a^8} \\ & = \frac {\left (5 a^6-90 a^4 b^2+200 a^2 b^4-112 b^6\right ) x}{16 a^8}+\frac {b \left (61 a^4-170 a^2 b^2+105 b^4\right ) \sin (c+d x)}{15 a^7 d}-\frac {\left (27 a^4-86 a^2 b^2+56 b^4\right ) \cos (c+d x) \sin (c+d x)}{16 a^6 d}+\frac {\left (15 a^4-52 a^2 b^2+35 b^4\right ) \cos ^2(c+d x) \sin (c+d x)}{15 a^5 b d}-\frac {\left (16 a^4-61 a^2 b^2+42 b^4\right ) \cos ^3(c+d x) \sin (c+d x)}{24 a^4 b^2 d}-\frac {\cos ^3(c+d x) \sin (c+d x)}{3 b d (b+a \cos (c+d x))}+\frac {a \cos ^4(c+d x) \sin (c+d x)}{6 b^2 d (b+a \cos (c+d x))}+\frac {\left (5 a^4-20 a^2 b^2+14 b^4\right ) \cos ^4(c+d x) \sin (c+d x)}{10 a^3 b^2 d (b+a \cos (c+d x))}+\frac {7 b \cos ^5(c+d x) \sin (c+d x)}{30 a^2 d (b+a \cos (c+d x))}-\frac {\cos ^6(c+d x) \sin (c+d x)}{6 a d (b+a \cos (c+d x))}+\frac {\left (2 b \left (2 a^2-7 b^2\right ) \left (a^2-b^2\right )^2\right ) \text {Subst}\left (\int \frac {1}{-a-b+(a-b) x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{a^8 d} \\ & = \frac {\left (5 a^6-90 a^4 b^2+200 a^2 b^4-112 b^6\right ) x}{16 a^8}-\frac {2 (a-b)^{3/2} b (a+b)^{3/2} \left (2 a^2-7 b^2\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{a^8 d}+\frac {b \left (61 a^4-170 a^2 b^2+105 b^4\right ) \sin (c+d x)}{15 a^7 d}-\frac {\left (27 a^4-86 a^2 b^2+56 b^4\right ) \cos (c+d x) \sin (c+d x)}{16 a^6 d}+\frac {\left (15 a^4-52 a^2 b^2+35 b^4\right ) \cos ^2(c+d x) \sin (c+d x)}{15 a^5 b d}-\frac {\left (16 a^4-61 a^2 b^2+42 b^4\right ) \cos ^3(c+d x) \sin (c+d x)}{24 a^4 b^2 d}-\frac {\cos ^3(c+d x) \sin (c+d x)}{3 b d (b+a \cos (c+d x))}+\frac {a \cos ^4(c+d x) \sin (c+d x)}{6 b^2 d (b+a \cos (c+d x))}+\frac {\left (5 a^4-20 a^2 b^2+14 b^4\right ) \cos ^4(c+d x) \sin (c+d x)}{10 a^3 b^2 d (b+a \cos (c+d x))}+\frac {7 b \cos ^5(c+d x) \sin (c+d x)}{30 a^2 d (b+a \cos (c+d x))}-\frac {\cos ^6(c+d x) \sin (c+d x)}{6 a d (b+a \cos (c+d x))} \\ \end{align*}
Time = 6.32 (sec) , antiderivative size = 402, normalized size of antiderivative = 0.85 \[ \int \frac {\sin ^6(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\frac {3840 b \left (2 a^2-7 b^2\right ) \left (a^2-b^2\right )^{3/2} \text {arctanh}\left (\frac {(-a+b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )+\frac {600 a^6 b c-10800 a^4 b^3 c+24000 a^2 b^5 c-13440 b^7 c+600 a^6 b d x-10800 a^4 b^3 d x+24000 a^2 b^5 d x-13440 b^7 d x+120 a \left (5 a^6-90 a^4 b^2+200 a^2 b^4-112 b^6\right ) (c+d x) \cos (c+d x)-15 a \left (15 a^6-576 a^4 b^2+1488 a^2 b^4-896 b^6\right ) \sin (c+d x)+1910 a^6 b \sin (2 (c+d x))-5440 a^4 b^3 \sin (2 (c+d x))+3360 a^2 b^5 \sin (2 (c+d x))-180 a^7 \sin (3 (c+d x))+790 a^5 b^2 \sin (3 (c+d x))-560 a^3 b^4 \sin (3 (c+d x))-166 a^6 b \sin (4 (c+d x))+140 a^4 b^3 \sin (4 (c+d x))+40 a^7 \sin (5 (c+d x))-42 a^5 b^2 \sin (5 (c+d x))+14 a^6 b \sin (6 (c+d x))-5 a^7 \sin (7 (c+d x))}{b+a \cos (c+d x)}}{1920 a^8 d} \]
[In]
[Out]
Time = 2.71 (sec) , antiderivative size = 509, normalized size of antiderivative = 1.08
method | result | size |
derivativedivides | \(\frac {\frac {\frac {2 \left (\left (\frac {5}{16} a^{6}+2 a^{5} b -\frac {21}{8} a^{4} b^{2}-8 a^{3} b^{3}+\frac {5}{2} a^{2} b^{4}+6 a \,b^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}+\left (\frac {38}{3} a^{5} b -\frac {87}{8} a^{4} b^{2}+\frac {15}{2} a^{2} b^{4}+30 a \,b^{5}+\frac {85}{48} a^{6}-\frac {136}{3} a^{3} b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}+\left (\frac {172}{5} a^{5} b -\frac {33}{4} a^{4} b^{2}-96 a^{3} b^{3}+5 a^{2} b^{4}+60 a \,b^{5}+\frac {33}{8} a^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}+\left (-\frac {33}{8} a^{6}+\frac {33}{4} a^{4} b^{2}-5 a^{2} b^{4}+\frac {172}{5} a^{5} b -96 a^{3} b^{3}+60 a \,b^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+\left (\frac {38}{3} a^{5} b +\frac {87}{8} a^{4} b^{2}-\frac {136}{3} a^{3} b^{3}-\frac {15}{2} a^{2} b^{4}+30 a \,b^{5}-\frac {85}{48} a^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (2 a^{5} b -8 a^{3} b^{3}+6 a \,b^{5}-\frac {5}{16} a^{6}+\frac {21}{8} a^{4} b^{2}-\frac {5}{2} a^{2} b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{6}}+\frac {\left (5 a^{6}-90 a^{4} b^{2}+200 a^{2} b^{4}-112 b^{6}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}}{a^{8}}+\frac {2 \left (a -b \right )^{2} \left (a +b \right )^{2} b \left (-\frac {a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b}-\frac {\left (2 a^{2}-7 b^{2}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{a^{8}}}{d}\) | \(509\) |
default | \(\frac {\frac {\frac {2 \left (\left (\frac {5}{16} a^{6}+2 a^{5} b -\frac {21}{8} a^{4} b^{2}-8 a^{3} b^{3}+\frac {5}{2} a^{2} b^{4}+6 a \,b^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}+\left (\frac {38}{3} a^{5} b -\frac {87}{8} a^{4} b^{2}+\frac {15}{2} a^{2} b^{4}+30 a \,b^{5}+\frac {85}{48} a^{6}-\frac {136}{3} a^{3} b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}+\left (\frac {172}{5} a^{5} b -\frac {33}{4} a^{4} b^{2}-96 a^{3} b^{3}+5 a^{2} b^{4}+60 a \,b^{5}+\frac {33}{8} a^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}+\left (-\frac {33}{8} a^{6}+\frac {33}{4} a^{4} b^{2}-5 a^{2} b^{4}+\frac {172}{5} a^{5} b -96 a^{3} b^{3}+60 a \,b^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+\left (\frac {38}{3} a^{5} b +\frac {87}{8} a^{4} b^{2}-\frac {136}{3} a^{3} b^{3}-\frac {15}{2} a^{2} b^{4}+30 a \,b^{5}-\frac {85}{48} a^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (2 a^{5} b -8 a^{3} b^{3}+6 a \,b^{5}-\frac {5}{16} a^{6}+\frac {21}{8} a^{4} b^{2}-\frac {5}{2} a^{2} b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{6}}+\frac {\left (5 a^{6}-90 a^{4} b^{2}+200 a^{2} b^{4}-112 b^{6}\right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}}{a^{8}}+\frac {2 \left (a -b \right )^{2} \left (a +b \right )^{2} b \left (-\frac {a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b}-\frac {\left (2 a^{2}-7 b^{2}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{a^{8}}}{d}\) | \(509\) |
risch | \(\frac {2 \sqrt {a^{2}-b^{2}}\, b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i \sqrt {a^{2}-b^{2}}-b}{a}\right )}{d \,a^{4}}-\frac {9 \sqrt {a^{2}-b^{2}}\, b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i \sqrt {a^{2}-b^{2}}-b}{a}\right )}{d \,a^{6}}+\frac {7 \sqrt {a^{2}-b^{2}}\, b^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i \sqrt {a^{2}-b^{2}}-b}{a}\right )}{d \,a^{8}}+\frac {11 i b \,{\mathrm e}^{-i \left (d x +c \right )}}{8 a^{3} d}-\frac {9 i b^{3} {\mathrm e}^{-i \left (d x +c \right )}}{2 a^{5} d}+\frac {3 i b^{5} {\mathrm e}^{-i \left (d x +c \right )}}{a^{7} d}-\frac {11 i b \,{\mathrm e}^{i \left (d x +c \right )}}{8 a^{3} d}+\frac {9 i b^{3} {\mathrm e}^{i \left (d x +c \right )}}{2 a^{5} d}-\frac {3 i b^{5} {\mathrm e}^{i \left (d x +c \right )}}{a^{7} d}+\frac {3 i {\mathrm e}^{-2 i \left (d x +c \right )} b^{2}}{4 a^{4} d}-\frac {5 i {\mathrm e}^{-2 i \left (d x +c \right )} b^{4}}{8 a^{6} d}-\frac {3 i {\mathrm e}^{2 i \left (d x +c \right )} b^{2}}{4 a^{4} d}+\frac {5 i {\mathrm e}^{2 i \left (d x +c \right )} b^{4}}{8 a^{6} d}-\frac {\sin \left (6 d x +6 c \right )}{192 a^{2} d}-\frac {45 x \,b^{2}}{8 a^{4}}+\frac {25 x \,b^{4}}{2 a^{6}}-\frac {7 x \,b^{6}}{a^{8}}+\frac {3 \sin \left (4 d x +4 c \right )}{64 a^{2} d}+\frac {b \sin \left (5 d x +5 c \right )}{40 a^{3} d}-\frac {3 \sin \left (4 d x +4 c \right ) b^{2}}{32 a^{4} d}-\frac {7 b \sin \left (3 d x +3 c \right )}{24 d \,a^{3}}+\frac {b^{3} \sin \left (3 d x +3 c \right )}{3 d \,a^{5}}+\frac {5 x}{16 a^{2}}+\frac {2 i \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) b^{2} \left (b \,{\mathrm e}^{i \left (d x +c \right )}+a \right )}{a^{8} d \left (a \,{\mathrm e}^{2 i \left (d x +c \right )}+2 b \,{\mathrm e}^{i \left (d x +c \right )}+a \right )}-\frac {2 \sqrt {a^{2}-b^{2}}\, b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {b +i \sqrt {a^{2}-b^{2}}}{a}\right )}{d \,a^{4}}+\frac {9 \sqrt {a^{2}-b^{2}}\, b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {b +i \sqrt {a^{2}-b^{2}}}{a}\right )}{d \,a^{6}}-\frac {7 \sqrt {a^{2}-b^{2}}\, b^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {b +i \sqrt {a^{2}-b^{2}}}{a}\right )}{d \,a^{8}}-\frac {15 i {\mathrm e}^{-2 i \left (d x +c \right )}}{128 a^{2} d}+\frac {15 i {\mathrm e}^{2 i \left (d x +c \right )}}{128 a^{2} d}\) | \(777\) |
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Time = 0.34 (sec) , antiderivative size = 793, normalized size of antiderivative = 1.68 \[ \int \frac {\sin ^6(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\left [\frac {15 \, {\left (5 \, a^{7} - 90 \, a^{5} b^{2} + 200 \, a^{3} b^{4} - 112 \, a b^{6}\right )} d x \cos \left (d x + c\right ) + 15 \, {\left (5 \, a^{6} b - 90 \, a^{4} b^{3} + 200 \, a^{2} b^{5} - 112 \, b^{7}\right )} d x + 120 \, {\left (2 \, a^{4} b^{2} - 9 \, a^{2} b^{4} + 7 \, b^{6} + {\left (2 \, a^{5} b - 9 \, a^{3} b^{3} + 7 \, a b^{5}\right )} \cos \left (d x + c\right )\right )} \sqrt {a^{2} - b^{2}} \log \left (\frac {2 \, a b \cos \left (d x + c\right ) - {\left (a^{2} - 2 \, b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, \sqrt {a^{2} - b^{2}} {\left (b \cos \left (d x + c\right ) + a\right )} \sin \left (d x + c\right ) + 2 \, a^{2} - b^{2}}{a^{2} \cos \left (d x + c\right )^{2} + 2 \, a b \cos \left (d x + c\right ) + b^{2}}\right ) - {\left (40 \, a^{7} \cos \left (d x + c\right )^{6} - 56 \, a^{6} b \cos \left (d x + c\right )^{5} - 976 \, a^{5} b^{2} + 2720 \, a^{3} b^{4} - 1680 \, a b^{6} - 2 \, {\left (65 \, a^{7} - 42 \, a^{5} b^{2}\right )} \cos \left (d x + c\right )^{4} + 2 \, {\left (111 \, a^{6} b - 70 \, a^{4} b^{3}\right )} \cos \left (d x + c\right )^{3} + {\left (165 \, a^{7} - 458 \, a^{5} b^{2} + 280 \, a^{3} b^{4}\right )} \cos \left (d x + c\right )^{2} - {\left (571 \, a^{6} b - 1430 \, a^{4} b^{3} + 840 \, a^{2} b^{5}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, {\left (a^{9} d \cos \left (d x + c\right ) + a^{8} b d\right )}}, \frac {15 \, {\left (5 \, a^{7} - 90 \, a^{5} b^{2} + 200 \, a^{3} b^{4} - 112 \, a b^{6}\right )} d x \cos \left (d x + c\right ) + 15 \, {\left (5 \, a^{6} b - 90 \, a^{4} b^{3} + 200 \, a^{2} b^{5} - 112 \, b^{7}\right )} d x - 240 \, {\left (2 \, a^{4} b^{2} - 9 \, a^{2} b^{4} + 7 \, b^{6} + {\left (2 \, a^{5} b - 9 \, a^{3} b^{3} + 7 \, a b^{5}\right )} \cos \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}} \arctan \left (-\frac {\sqrt {-a^{2} + b^{2}} {\left (b \cos \left (d x + c\right ) + a\right )}}{{\left (a^{2} - b^{2}\right )} \sin \left (d x + c\right )}\right ) - {\left (40 \, a^{7} \cos \left (d x + c\right )^{6} - 56 \, a^{6} b \cos \left (d x + c\right )^{5} - 976 \, a^{5} b^{2} + 2720 \, a^{3} b^{4} - 1680 \, a b^{6} - 2 \, {\left (65 \, a^{7} - 42 \, a^{5} b^{2}\right )} \cos \left (d x + c\right )^{4} + 2 \, {\left (111 \, a^{6} b - 70 \, a^{4} b^{3}\right )} \cos \left (d x + c\right )^{3} + {\left (165 \, a^{7} - 458 \, a^{5} b^{2} + 280 \, a^{3} b^{4}\right )} \cos \left (d x + c\right )^{2} - {\left (571 \, a^{6} b - 1430 \, a^{4} b^{3} + 840 \, a^{2} b^{5}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, {\left (a^{9} d \cos \left (d x + c\right ) + a^{8} b d\right )}}\right ] \]
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\[ \int \frac {\sin ^6(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\int \frac {\sin ^{6}{\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{2}}\, dx \]
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Exception generated. \[ \int \frac {\sin ^6(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.34 (sec) , antiderivative size = 870, normalized size of antiderivative = 1.84 \[ \int \frac {\sin ^6(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\text {Too large to display} \]
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Time = 17.51 (sec) , antiderivative size = 3692, normalized size of antiderivative = 7.81 \[ \int \frac {\sin ^6(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\text {Too large to display} \]
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