3.58 \(\int \csc ^2(a+b x) \csc ^6(2 a+2 b x) \, dx\)
Optimal. Leaf size=102 \[ \frac {\tan ^5(a+b x)}{320 b}+\frac {\tan ^3(a+b x)}{32 b}+\frac {15 \tan (a+b x)}{64 b}-\frac {\cot ^7(a+b x)}{448 b}-\frac {3 \cot ^5(a+b x)}{160 b}-\frac {5 \cot ^3(a+b x)}{64 b}-\frac {5 \cot (a+b x)}{16 b} \]
[Out]
-5/16*cot(b*x+a)/b-5/64*cot(b*x+a)^3/b-3/160*cot(b*x+a)^5/b-1/448*cot(b*x+a)^7/b+15/64*tan(b*x+a)/b+1/32*tan(b
*x+a)^3/b+1/320*tan(b*x+a)^5/b
________________________________________________________________________________________
Rubi [A] time = 0.08, antiderivative size = 102, normalized size of antiderivative = 1.00,
number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used =
{4288, 2620, 270} \[ \frac {\tan ^5(a+b x)}{320 b}+\frac {\tan ^3(a+b x)}{32 b}+\frac {15 \tan (a+b x)}{64 b}-\frac {\cot ^7(a+b x)}{448 b}-\frac {3 \cot ^5(a+b x)}{160 b}-\frac {5 \cot ^3(a+b x)}{64 b}-\frac {5 \cot (a+b x)}{16 b} \]
Antiderivative was successfully verified.
[In]
Int[Csc[a + b*x]^2*Csc[2*a + 2*b*x]^6,x]
[Out]
(-5*Cot[a + b*x])/(16*b) - (5*Cot[a + b*x]^3)/(64*b) - (3*Cot[a + b*x]^5)/(160*b) - Cot[a + b*x]^7/(448*b) + (
15*Tan[a + b*x])/(64*b) + Tan[a + b*x]^3/(32*b) + Tan[a + b*x]^5/(320*b)
Rule 270
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]
Rule 2620
Int[csc[(e_.) + (f_.)*(x_)]^(m_.)*sec[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(1 + x^2)^((
m + n)/2 - 1)/x^m, x], x, Tan[e + f*x]], x] /; FreeQ[{e, f}, x] && IntegersQ[m, n, (m + n)/2]
Rule 4288
Int[((f_.)*sin[(a_.) + (b_.)*(x_)])^(n_.)*sin[(c_.) + (d_.)*(x_)]^(p_.), x_Symbol] :> Dist[2^p/f^p, Int[Cos[a
+ b*x]^p*(f*Sin[a + b*x])^(n + p), x], x] /; FreeQ[{a, b, c, d, f, n}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2]
&& IntegerQ[p]
Rubi steps
\begin {align*} \int \csc ^2(a+b x) \csc ^6(2 a+2 b x) \, dx &=\frac {1}{64} \int \csc ^8(a+b x) \sec ^6(a+b x) \, dx\\ &=\frac {\operatorname {Subst}\left (\int \frac {\left (1+x^2\right )^6}{x^8} \, dx,x,\tan (a+b x)\right )}{64 b}\\ &=\frac {\operatorname {Subst}\left (\int \left (15+\frac {1}{x^8}+\frac {6}{x^6}+\frac {15}{x^4}+\frac {20}{x^2}+6 x^2+x^4\right ) \, dx,x,\tan (a+b x)\right )}{64 b}\\ &=-\frac {5 \cot (a+b x)}{16 b}-\frac {5 \cot ^3(a+b x)}{64 b}-\frac {3 \cot ^5(a+b x)}{160 b}-\frac {\cot ^7(a+b x)}{448 b}+\frac {15 \tan (a+b x)}{64 b}+\frac {\tan ^3(a+b x)}{32 b}+\frac {\tan ^5(a+b x)}{320 b}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.07, size = 132, normalized size = 1.29 \[ \frac {33 \tan (a+b x)}{160 b}-\frac {281 \cot (a+b x)}{1120 b}-\frac {\cot (a+b x) \csc ^6(a+b x)}{448 b}-\frac {27 \cot (a+b x) \csc ^4(a+b x)}{2240 b}-\frac {53 \cot (a+b x) \csc ^2(a+b x)}{1120 b}+\frac {\tan (a+b x) \sec ^4(a+b x)}{320 b}+\frac {\tan (a+b x) \sec ^2(a+b x)}{40 b} \]
Antiderivative was successfully verified.
[In]
Integrate[Csc[a + b*x]^2*Csc[2*a + 2*b*x]^6,x]
[Out]
(-281*Cot[a + b*x])/(1120*b) - (53*Cot[a + b*x]*Csc[a + b*x]^2)/(1120*b) - (27*Cot[a + b*x]*Csc[a + b*x]^4)/(2
240*b) - (Cot[a + b*x]*Csc[a + b*x]^6)/(448*b) + (33*Tan[a + b*x])/(160*b) + (Sec[a + b*x]^2*Tan[a + b*x])/(40
*b) + (Sec[a + b*x]^4*Tan[a + b*x])/(320*b)
________________________________________________________________________________________
fricas [A] time = 0.51, size = 118, normalized size = 1.16 \[ -\frac {1024 \, \cos \left (b x + a\right )^{12} - 3584 \, \cos \left (b x + a\right )^{10} + 4480 \, \cos \left (b x + a\right )^{8} - 2240 \, \cos \left (b x + a\right )^{6} + 280 \, \cos \left (b x + a\right )^{4} + 28 \, \cos \left (b x + a\right )^{2} + 7}{2240 \, {\left (b \cos \left (b x + a\right )^{11} - 3 \, b \cos \left (b x + a\right )^{9} + 3 \, b \cos \left (b x + a\right )^{7} - b \cos \left (b x + a\right )^{5}\right )} \sin \left (b x + a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csc(b*x+a)^2*csc(2*b*x+2*a)^6,x, algorithm="fricas")
[Out]
-1/2240*(1024*cos(b*x + a)^12 - 3584*cos(b*x + a)^10 + 4480*cos(b*x + a)^8 - 2240*cos(b*x + a)^6 + 280*cos(b*x
+ a)^4 + 28*cos(b*x + a)^2 + 7)/((b*cos(b*x + a)^11 - 3*b*cos(b*x + a)^9 + 3*b*cos(b*x + a)^7 - b*cos(b*x + a
)^5)*sin(b*x + a))
________________________________________________________________________________________
giac [B] time = 12.22, size = 7477, normalized size = 73.30 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csc(b*x+a)^2*csc(2*b*x+2*a)^6,x, algorithm="giac")
[Out]
-1/71680*(7*(6480*tan(b*x + 4*a)^4*tan(1/2*a)^56 - 2160*tan(b*x + 4*a)^3*tan(1/2*a)^57 + 360*tan(b*x + 4*a)^2*
tan(1/2*a)^58 - 30*tan(b*x + 4*a)*tan(1/2*a)^59 + tan(1/2*a)^60 + 963360*tan(b*x + 4*a)^4*tan(1/2*a)^54 - 4514
40*tan(b*x + 4*a)^3*tan(1/2*a)^55 + 84000*tan(b*x + 4*a)^2*tan(1/2*a)^56 - 6650*tan(b*x + 4*a)*tan(1/2*a)^57 +
210*tan(1/2*a)^58 + 76410000*tan(b*x + 4*a)^4*tan(1/2*a)^52 - 46604880*tan(b*x + 4*a)^3*tan(1/2*a)^53 + 11606
800*tan(b*x + 4*a)^2*tan(1/2*a)^54 - 1351380*tan(b*x + 4*a)*tan(1/2*a)^55 + 61875*tan(1/2*a)^56 - 1737755520*t
an(b*x + 4*a)^4*tan(1/2*a)^50 + 1779239600*tan(b*x + 4*a)^3*tan(1/2*a)^51 - 594557280*tan(b*x + 4*a)^2*tan(1/2
*a)^52 + 84179700*tan(b*x + 4*a)*tan(1/2*a)^53 - 4366140*tan(1/2*a)^54 + 14743616480*tan(b*x + 4*a)^4*tan(1/2*
a)^48 - 21984264480*tan(b*x + 4*a)^3*tan(1/2*a)^49 + 10388004600*tan(b*x + 4*a)^2*tan(1/2*a)^50 - 1902997490*t
an(b*x + 4*a)*tan(1/2*a)^51 + 120469005*tan(1/2*a)^52 - 61028652480*tan(b*x + 4*a)^4*tan(1/2*a)^46 + 130174560
480*tan(b*x + 4*a)^3*tan(1/2*a)^47 - 86273730880*tan(b*x + 4*a)^2*tan(1/2*a)^48 + 21462983130*tan(b*x + 4*a)*t
an(1/2*a)^49 - 1737306714*tan(1/2*a)^50 + 103180381920*tan(b*x + 4*a)^4*tan(1/2*a)^44 - 371517010400*tan(b*x +
4*a)^3*tan(1/2*a)^45 + 365221080480*tan(b*x + 4*a)^2*tan(1/2*a)^46 - 127981989480*tan(b*x + 4*a)*tan(1/2*a)^4
7 + 14001839215*tan(1/2*a)^48 + 111160976000*tan(b*x + 4*a)^4*tan(1/2*a)^42 + 211586603040*tan(b*x + 4*a)^3*ta
n(1/2*a)^43 - 637111944000*tan(b*x + 4*a)^2*tan(1/2*a)^44 + 376048838120*tan(b*x + 4*a)*tan(1/2*a)^45 - 606087
49080*tan(1/2*a)^46 - 724773023760*tan(b*x + 4*a)^4*tan(1/2*a)^40 + 1597018455600*tan(b*x + 4*a)^3*tan(1/2*a)^
41 - 663483964760*tan(b*x + 4*a)^2*tan(1/2*a)^42 - 227449079550*tan(b*x + 4*a)*tan(1/2*a)^43 + 109533462525*ta
n(1/2*a)^44 + 663892404960*tan(b*x + 4*a)^4*tan(1/2*a)^38 - 3633046154960*tan(b*x + 4*a)^3*tan(1/2*a)^39 + 444
2053139040*tan(b*x + 4*a)^2*tan(1/2*a)^40 - 1628950998330*tan(b*x + 4*a)*tan(1/2*a)^41 + 110076988450*tan(1/2*
a)^42 + 1382243479600*tan(b*x + 4*a)^4*tan(1/2*a)^36 - 662161010160*tan(b*x + 4*a)^3*tan(1/2*a)^37 - 394725379
9440*tan(b*x + 4*a)^2*tan(1/2*a)^38 + 3687432890100*tan(b*x + 4*a)*tan(1/2*a)^39 - 757631165865*tan(1/2*a)^40
- 2679939290880*tan(b*x + 4*a)^4*tan(1/2*a)^34 + 9953265569040*tan(b*x + 4*a)^3*tan(1/2*a)^35 - 8576804580640*
tan(b*x + 4*a)^2*tan(1/2*a)^36 + 813320615340*tan(b*x + 4*a)*tan(1/2*a)^37 + 647844828540*tan(1/2*a)^38 - 7553
38211520*tan(b*x + 4*a)^4*tan(1/2*a)^32 - 5830705497280*tan(b*x + 4*a)^3*tan(1/2*a)^33 + 15767445241080*tan(b*
x + 4*a)^2*tan(1/2*a)^34 - 9992067869250*tan(b*x + 4*a)*tan(1/2*a)^35 + 1474839074345*tan(1/2*a)^36 + 39755680
27520*tan(b*x + 4*a)^4*tan(1/2*a)^30 - 11953155299520*tan(b*x + 4*a)^3*tan(1/2*a)^31 + 4861415944320*tan(b*x +
4*a)^2*tan(1/2*a)^32 + 5463157048330*tan(b*x + 4*a)*tan(1/2*a)^33 - 2566729120410*tan(1/2*a)^34 - 75533821152
0*tan(b*x + 4*a)^4*tan(1/2*a)^28 + 11953155299520*tan(b*x + 4*a)^3*tan(1/2*a)^29 - 23059287629120*tan(b*x + 4*
a)^2*tan(1/2*a)^30 + 11707093056720*tan(b*x + 4*a)*tan(1/2*a)^31 - 840595305645*tan(1/2*a)^32 - 2679939290880*
tan(b*x + 4*a)^4*tan(1/2*a)^26 + 5830705497280*tan(b*x + 4*a)^3*tan(1/2*a)^27 + 4861415944320*tan(b*x + 4*a)^2
*tan(1/2*a)^28 - 11707093056720*tan(b*x + 4*a)*tan(1/2*a)^29 + 3742852321200*tan(1/2*a)^30 + 1382243479600*tan
(b*x + 4*a)^4*tan(1/2*a)^24 - 9953265569040*tan(b*x + 4*a)^3*tan(1/2*a)^25 + 15767445241080*tan(b*x + 4*a)^2*t
an(1/2*a)^26 - 5463157048330*tan(b*x + 4*a)*tan(1/2*a)^27 - 840595305645*tan(1/2*a)^28 + 663892404960*tan(b*x
+ 4*a)^4*tan(1/2*a)^22 + 662161010160*tan(b*x + 4*a)^3*tan(1/2*a)^23 - 8576804580640*tan(b*x + 4*a)^2*tan(1/2*
a)^24 + 9992067869250*tan(b*x + 4*a)*tan(1/2*a)^25 - 2566729120410*tan(1/2*a)^26 - 724773023760*tan(b*x + 4*a)
^4*tan(1/2*a)^20 + 3633046154960*tan(b*x + 4*a)^3*tan(1/2*a)^21 - 3947253799440*tan(b*x + 4*a)^2*tan(1/2*a)^22
- 813320615340*tan(b*x + 4*a)*tan(1/2*a)^23 + 1474839074345*tan(1/2*a)^24 + 111160976000*tan(b*x + 4*a)^4*tan
(1/2*a)^18 - 1597018455600*tan(b*x + 4*a)^3*tan(1/2*a)^19 + 4442053139040*tan(b*x + 4*a)^2*tan(1/2*a)^20 - 368
7432890100*tan(b*x + 4*a)*tan(1/2*a)^21 + 647844828540*tan(1/2*a)^22 + 103180381920*tan(b*x + 4*a)^4*tan(1/2*a
)^16 - 211586603040*tan(b*x + 4*a)^3*tan(1/2*a)^17 - 663483964760*tan(b*x + 4*a)^2*tan(1/2*a)^18 + 16289509983
30*tan(b*x + 4*a)*tan(1/2*a)^19 - 757631165865*tan(1/2*a)^20 - 61028652480*tan(b*x + 4*a)^4*tan(1/2*a)^14 + 37
1517010400*tan(b*x + 4*a)^3*tan(1/2*a)^15 - 637111944000*tan(b*x + 4*a)^2*tan(1/2*a)^16 + 227449079550*tan(b*x
+ 4*a)*tan(1/2*a)^17 + 110076988450*tan(1/2*a)^18 + 14743616480*tan(b*x + 4*a)^4*tan(1/2*a)^12 - 130174560480
*tan(b*x + 4*a)^3*tan(1/2*a)^13 + 365221080480*tan(b*x + 4*a)^2*tan(1/2*a)^14 - 376048838120*tan(b*x + 4*a)*ta
n(1/2*a)^15 + 109533462525*tan(1/2*a)^16 - 1737755520*tan(b*x + 4*a)^4*tan(1/2*a)^10 + 21984264480*tan(b*x + 4
*a)^3*tan(1/2*a)^11 - 86273730880*tan(b*x + 4*a)^2*tan(1/2*a)^12 + 127981989480*tan(b*x + 4*a)*tan(1/2*a)^13 -
60608749080*tan(1/2*a)^14 + 76410000*tan(b*x + 4*a)^4*tan(1/2*a)^8 - 1779239600*tan(b*x + 4*a)^3*tan(1/2*a)^9
+ 10388004600*tan(b*x + 4*a)^2*tan(1/2*a)^10 - 21462983130*tan(b*x + 4*a)*tan(1/2*a)^11 + 14001839215*tan(1/2
*a)^12 + 963360*tan(b*x + 4*a)^4*tan(1/2*a)^6 + 46604880*tan(b*x + 4*a)^3*tan(1/2*a)^7 - 594557280*tan(b*x + 4
*a)^2*tan(1/2*a)^8 + 1902997490*tan(b*x + 4*a)*tan(1/2*a)^9 - 1737306714*tan(1/2*a)^10 + 6480*tan(b*x + 4*a)^4
*tan(1/2*a)^4 + 451440*tan(b*x + 4*a)^3*tan(1/2*a)^5 + 11606800*tan(b*x + 4*a)^2*tan(1/2*a)^6 - 84179700*tan(b
*x + 4*a)*tan(1/2*a)^7 + 120469005*tan(1/2*a)^8 + 2160*tan(b*x + 4*a)^3*tan(1/2*a)^3 + 84000*tan(b*x + 4*a)^2*
tan(1/2*a)^4 + 1351380*tan(b*x + 4*a)*tan(1/2*a)^5 - 4366140*tan(1/2*a)^6 + 360*tan(b*x + 4*a)^2*tan(1/2*a)^2
+ 6650*tan(b*x + 4*a)*tan(1/2*a)^3 + 61875*tan(1/2*a)^4 + 30*tan(b*x + 4*a)*tan(1/2*a) + 210*tan(1/2*a)^2 + 1)
/((243*tan(1/2*a)^25 - 4050*tan(1/2*a)^23 + 28215*tan(1/2*a)^21 - 106200*tan(1/2*a)^19 + 233430*tan(1/2*a)^17
- 304300*tan(1/2*a)^15 + 233430*tan(1/2*a)^13 - 106200*tan(1/2*a)^11 + 28215*tan(1/2*a)^9 - 4050*tan(1/2*a)^7
+ 243*tan(1/2*a)^5)*(6*tan(b*x + 4*a)*tan(1/2*a)^5 - tan(1/2*a)^6 - 20*tan(b*x + 4*a)*tan(1/2*a)^3 + 15*tan(1/
2*a)^4 + 6*tan(b*x + 4*a)*tan(1/2*a) - 15*tan(1/2*a)^2 + 1)^5) + 32*(700*tan(b*x + 4*a)^6*tan(1/2*a)^84 - 1029
00*tan(b*x + 4*a)^6*tan(1/2*a)^82 + 22050*tan(b*x + 4*a)^5*tan(1/2*a)^83 + 175*tan(b*x + 4*a)^4*tan(1/2*a)^84
+ 7200060*tan(b*x + 4*a)^6*tan(1/2*a)^80 - 3103170*tan(b*x + 4*a)^5*tan(1/2*a)^81 + 280770*tan(b*x + 4*a)^4*ta
n(1/2*a)^82 + 2940*tan(b*x + 4*a)^3*tan(1/2*a)^83 + 42*tan(b*x + 4*a)^2*tan(1/2*a)^84 - 316457680*tan(b*x + 4*
a)^6*tan(1/2*a)^78 + 206229240*tan(b*x + 4*a)^5*tan(1/2*a)^79 - 38821965*tan(b*x + 4*a)^4*tan(1/2*a)^80 + 1904
140*tan(b*x + 4*a)^3*tan(1/2*a)^81 + 17640*tan(b*x + 4*a)^2*tan(1/2*a)^82 + 294*tan(b*x + 4*a)*tan(1/2*a)^83 +
5*tan(1/2*a)^84 + 9709765800*tan(b*x + 4*a)^6*tan(1/2*a)^76 - 8530152120*tan(b*x + 4*a)^5*tan(1/2*a)^77 + 244
9598200*tan(b*x + 4*a)^4*tan(1/2*a)^78 - 259255080*tan(b*x + 4*a)^3*tan(1/2*a)^79 + 7360290*tan(b*x + 4*a)^2*t
an(1/2*a)^80 + 47530*tan(b*x + 4*a)*tan(1/2*a)^81 + 462*tan(1/2*a)^82 - 218895118680*tan(b*x + 4*a)^6*tan(1/2*
a)^74 + 243907782300*tan(b*x + 4*a)^5*tan(1/2*a)^75 - 94820555130*tan(b*x + 4*a)^4*tan(1/2*a)^76 + 15442018200
*tan(b*x + 4*a)^3*tan(1/2*a)^77 - 976159968*tan(b*x + 4*a)^2*tan(1/2*a)^78 + 15463224*tan(b*x + 4*a)*tan(1/2*a
)^79 + 47985*tan(1/2*a)^80 + 3722263347880*tan(b*x + 4*a)^6*tan(1/2*a)^72 - 5072116579740*tan(b*x + 4*a)^5*tan
(1/2*a)^73 + 2508017550780*tan(b*x + 4*a)^4*tan(1/2*a)^74 - 555145935120*tan(b*x + 4*a)^3*tan(1/2*a)^75 + 5441
6410020*tan(b*x + 4*a)^2*tan(1/2*a)^76 - 1964255832*tan(b*x + 4*a)*tan(1/2*a)^77 + 13827464*tan(1/2*a)^78 - 48
306853319760*tan(b*x + 4*a)^6*tan(1/2*a)^70 + 78691575627000*tan(b*x + 4*a)^5*tan(1/2*a)^71 - 47699227086490*t
an(b*x + 4*a)^4*tan(1/2*a)^72 + 13463793136080*tan(b*x + 4*a)^3*tan(1/2*a)^73 - 1800097458768*tan(b*x + 4*a)^2
*tan(1/2*a)^74 + 101416919940*tan(b*x + 4*a)*tan(1/2*a)^75 - 1647303294*tan(1/2*a)^76 + 478852179070860*tan(b*
x + 4*a)^6*tan(1/2*a)^68 - 920259549793080*tan(b*x + 4*a)^5*tan(1/2*a)^69 + 668627834124600*tan(b*x + 4*a)^4*t
an(1/2*a)^70 - 231870649855320*tan(b*x + 4*a)^3*tan(1/2*a)^71 + 39617163658212*tan(b*x + 4*a)^2*tan(1/2*a)^72
- 3053123538948*tan(b*x + 4*a)*tan(1/2*a)^73 + 77871263652*tan(1/2*a)^74 - 3590491658734820*tan(b*x + 4*a)^6*t
an(1/2*a)^66 + 8099233169503770*tan(b*x + 4*a)^5*tan(1/2*a)^67 - 6967640085374745*tan(b*x + 4*a)^4*tan(1/2*a)^
68 + 2903918016468840*tan(b*x + 4*a)^3*tan(1/2*a)^69 - 610729793019360*tan(b*x + 4*a)^2*tan(1/2*a)^70 + 602369
54245272*tan(b*x + 4*a)*tan(1/2*a)^71 - 2105389907166*tan(1/2*a)^72 + 19896259338686220*tan(b*x + 4*a)^6*tan(1
/2*a)^64 - 52908136354185210*tan(b*x + 4*a)^5*tan(1/2*a)^65 + 53736728263219610*tan(b*x + 4*a)^4*tan(1/2*a)^66
- 26624174150101500*tan(b*x + 4*a)^3*tan(1/2*a)^67 + 6746600775679002*tan(b*x + 4*a)^2*tan(1/2*a)^68 - 820187
477390456*tan(b*x + 4*a)*tan(1/2*a)^69 + 36705931718472*tan(1/2*a)^70 - 77771578076512320*tan(b*x + 4*a)^6*tan
(1/2*a)^62 + 248694609613265760*tan(b*x + 4*a)^5*tan(1/2*a)^63 - 300960195956707245*tan(b*x + 4*a)^4*tan(1/2*a
)^64 + 177302439156331860*tan(b*x + 4*a)^3*tan(1/2*a)^65 - 53651600975241528*tan(b*x + 4*a)^2*tan(1/2*a)^66 +
7877252164835550*tan(b*x + 4*a)*tan(1/2*a)^67 - 434705306175315*tan(1/2*a)^68 + 192844994494822880*tan(b*x + 4
*a)^6*tan(1/2*a)^60 - 787509809051626080*tan(b*x + 4*a)^5*tan(1/2*a)^61 + 1173510851306099040*tan(b*x + 4*a)^4
*tan(1/2*a)^62 - 835950959826956960*tan(b*x + 4*a)^3*tan(1/2*a)^63 + 303563975744062530*tan(b*x + 4*a)^2*tan(1
/2*a)^64 - 53507434964136174*tan(b*x + 4*a)*tan(1/2*a)^65 + 3573865030470070*tan(1/2*a)^66 - 20027398869814224
0*tan(b*x + 4*a)^6*tan(1/2*a)^58 + 1398151947986755920*tan(b*x + 4*a)^5*tan(1/2*a)^59 - 2880661846092155160*ta
n(b*x + 4*a)^4*tan(1/2*a)^60 + 2628180105420371040*tan(b*x + 4*a)^3*tan(1/2*a)^61 - 1179839882273931648*tan(b*
x + 4*a)^2*tan(1/2*a)^62 + 252811149558104928*tan(b*x + 4*a)*tan(1/2*a)^63 - 20413710218743023*tan(1/2*a)^64 -
371753255147504160*tan(b*x + 4*a)^6*tan(1/2*a)^56 - 100566683363511120*tan(b*x + 4*a)^5*tan(1/2*a)^57 + 29544
17021031525840*tan(b*x + 4*a)^4*tan(1/2*a)^58 - 4615789529427552000*tan(b*x + 4*a)^3*tan(1/2*a)^59 + 286913927
0141795952*tan(b*x + 4*a)^2*tan(1/2*a)^60 - 789182123619651552*tan(b*x + 4*a)*tan(1/2*a)^61 + 7904790248539766
4*tan(1/2*a)^62 + 1588175918421104320*tan(b*x + 4*a)^6*tan(1/2*a)^54 - 5705288320074446880*tan(b*x + 4*a)^5*ta
n(1/2*a)^55 + 5543116031154667560*tan(b*x + 4*a)^4*tan(1/2*a)^56 + 291827962982908800*tan(b*x + 4*a)^3*tan(1/2
*a)^57 - 2907035119602048960*tan(b*x + 4*a)^2*tan(1/2*a)^58 + 1372067944595797680*tan(b*x + 4*a)*tan(1/2*a)^59
- 190524603741625608*tan(1/2*a)^60 - 1764268417682817480*tan(b*x + 4*a)^6*tan(1/2*a)^52 + 1185150741338248272
0*tan(b*x + 4*a)^5*tan(1/2*a)^53 - 23636877153497649440*tan(b*x + 4*a)^4*tan(1/2*a)^54 + 18852555003710349600*
tan(b*x + 4*a)^3*tan(1/2*a)^55 - 5514689294400576528*tan(b*x + 4*a)^2*tan(1/2*a)^56 - 74535612123099568*tan(b*
x + 4*a)*tan(1/2*a)^57 + 190797055644582576*tan(1/2*a)^58 - 1248453345588419880*tan(b*x + 4*a)^6*tan(1/2*a)^50
- 4107878664125019420*tan(b*x + 4*a)^5*tan(1/2*a)^51 + 26521780108972032990*tan(b*x + 4*a)^4*tan(1/2*a)^52 -
39323126797282860000*tan(b*x + 4*a)^3*tan(1/2*a)^53 + 23453137263033690240*tan(b*x + 4*a)^2*tan(1/2*a)^54 - 56
08964036362402464*tan(b*x + 4*a)*tan(1/2*a)^55 + 366017355229752504*tan(1/2*a)^56 + 5286270479895116600*tan(b*
x + 4*a)^6*tan(1/2*a)^48 - 21060700379637099300*tan(b*x + 4*a)^5*tan(1/2*a)^49 + 18283250157335063460*tan(b*x
+ 4*a)^4*tan(1/2*a)^50 + 14094690574782962200*tan(b*x + 4*a)^3*tan(1/2*a)^51 - 26565501133755260460*tan(b*x +
4*a)^2*tan(1/2*a)^52 + 11739546935802688800*tan(b*x + 4*a)*tan(1/2*a)^53 - 1551520053876868320*tan(1/2*a)^54 -
4019457841895096160*tan(b*x + 4*a)^6*tan(1/2*a)^46 + 33873365085581604240*tan(b*x + 4*a)^5*tan(1/2*a)^47 - 79
355355678133689210*tan(b*x + 4*a)^4*tan(1/2*a)^48 + 69758331625489475640*tan(b*x + 4*a)^3*tan(1/2*a)^49 - 1783
2092647389129648*tan(b*x + 4*a)^2*tan(1/2*a)^50 - 4344999459760312724*tan(b*x + 4*a)*tan(1/2*a)^51 + 177297403
9101333642*tan(1/2*a)^52 - 3363472645057689360*tan(b*x + 4*a)^6*tan(1/2*a)^44 - 4196346243738904080*tan(b*x +
4*a)^5*tan(1/2*a)^45 + 61023004440064909200*tan(b*x + 4*a)^4*tan(1/2*a)^46 - 113594005549024014000*tan(b*x + 4
*a)^3*tan(1/2*a)^47 + 79397061913802193540*tan(b*x + 4*a)^2*tan(1/2*a)^48 - 20785250550683362380*tan(b*x + 4*a
)*tan(1/2*a)^49 + 1158344161574237820*tan(1/2*a)^50 + 7922839707121600240*tan(b*x + 4*a)^6*tan(1/2*a)^42 - 400
62672006803795160*tan(b*x + 4*a)^5*tan(1/2*a)^43 + 50479097014227689220*tan(b*x + 4*a)^4*tan(1/2*a)^44 + 14711
499739562948560*tan(b*x + 4*a)^3*tan(1/2*a)^45 - 61779677247183389760*tan(b*x + 4*a)^2*tan(1/2*a)^46 + 3428327
2085462710800*tan(b*x + 4*a)*tan(1/2*a)^47 - 5294375878659948990*tan(1/2*a)^48 - 3363472645057689360*tan(b*x +
4*a)^6*tan(1/2*a)^40 + 40062672006803795160*tan(b*x + 4*a)^5*tan(1/2*a)^41 - 119703426365609444120*tan(b*x +
4*a)^4*tan(1/2*a)^42 + 134372289532573058400*tan(b*x + 4*a)^3*tan(1/2*a)^43 - 50496361003895322600*tan(b*x + 4
*a)^2*tan(1/2*a)^44 - 4643613116469393360*tan(b*x + 4*a)*tan(1/2*a)^45 + 4170913759258711920*tan(1/2*a)^46 - 4
019457841895096160*tan(b*x + 4*a)^6*tan(1/2*a)^38 + 4196346243738904080*tan(b*x + 4*a)^5*tan(1/2*a)^39 + 50479
097014227689220*tan(b*x + 4*a)^4*tan(1/2*a)^40 - 134372289532573058400*tan(b*x + 4*a)^3*tan(1/2*a)^41 + 120599
543530513343520*tan(b*x + 4*a)^2*tan(1/2*a)^42 - 40569490418667454440*tan(b*x + 4*a)*tan(1/2*a)^43 + 336676510
8190708140*tan(1/2*a)^44 + 5286270479895116600*tan(b*x + 4*a)^6*tan(1/2*a)^36 - 33873365085581604240*tan(b*x +
4*a)^5*tan(1/2*a)^37 + 61023004440064909200*tan(b*x + 4*a)^4*tan(1/2*a)^38 - 14711499739562948560*tan(b*x + 4
*a)^3*tan(1/2*a)^39 - 50496361003895322600*tan(b*x + 4*a)^2*tan(1/2*a)^40 + 40569490418667454440*tan(b*x + 4*a
)*tan(1/2*a)^41 - 8102375952750405800*tan(1/2*a)^42 - 1248453345588419880*tan(b*x + 4*a)^6*tan(1/2*a)^34 + 210
60700379637099300*tan(b*x + 4*a)^5*tan(1/2*a)^35 - 79355355678133689210*tan(b*x + 4*a)^4*tan(1/2*a)^36 + 11359
4005549024014000*tan(b*x + 4*a)^3*tan(1/2*a)^37 - 61779677247183389760*tan(b*x + 4*a)^2*tan(1/2*a)^38 + 464361
3116469393360*tan(b*x + 4*a)*tan(1/2*a)^39 + 3366765108190708140*tan(1/2*a)^40 - 1764268417682817480*tan(b*x +
4*a)^6*tan(1/2*a)^32 + 4107878664125019420*tan(b*x + 4*a)^5*tan(1/2*a)^33 + 18283250157335063460*tan(b*x + 4*
a)^4*tan(1/2*a)^34 - 69758331625489475640*tan(b*x + 4*a)^3*tan(1/2*a)^35 + 79397061913802193540*tan(b*x + 4*a)
^2*tan(1/2*a)^36 - 34283272085462710800*tan(b*x + 4*a)*tan(1/2*a)^37 + 4170913759258711920*tan(1/2*a)^38 + 158
8175918421104320*tan(b*x + 4*a)^6*tan(1/2*a)^30 - 11851507413382482720*tan(b*x + 4*a)^5*tan(1/2*a)^31 + 265217
80108972032990*tan(b*x + 4*a)^4*tan(1/2*a)^32 - 14094690574782962200*tan(b*x + 4*a)^3*tan(1/2*a)^33 - 17832092
647389129648*tan(b*x + 4*a)^2*tan(1/2*a)^34 + 20785250550683362380*tan(b*x + 4*a)*tan(1/2*a)^35 - 529437587865
9948990*tan(1/2*a)^36 - 371753255147504160*tan(b*x + 4*a)^6*tan(1/2*a)^28 + 5705288320074446880*tan(b*x + 4*a)
^5*tan(1/2*a)^29 - 23636877153497649440*tan(b*x + 4*a)^4*tan(1/2*a)^30 + 39323126797282860000*tan(b*x + 4*a)^3
*tan(1/2*a)^31 - 26565501133755260460*tan(b*x + 4*a)^2*tan(1/2*a)^32 + 4344999459760312724*tan(b*x + 4*a)*tan(
1/2*a)^33 + 1158344161574237820*tan(1/2*a)^34 - 200273988698142240*tan(b*x + 4*a)^6*tan(1/2*a)^26 + 1005666833
63511120*tan(b*x + 4*a)^5*tan(1/2*a)^27 + 5543116031154667560*tan(b*x + 4*a)^4*tan(1/2*a)^28 - 188525550037103
49600*tan(b*x + 4*a)^3*tan(1/2*a)^29 + 23453137263033690240*tan(b*x + 4*a)^2*tan(1/2*a)^30 - 11739546935802688
800*tan(b*x + 4*a)*tan(1/2*a)^31 + 1772974039101333642*tan(1/2*a)^32 + 192844994494822880*tan(b*x + 4*a)^6*tan
(1/2*a)^24 - 1398151947986755920*tan(b*x + 4*a)^5*tan(1/2*a)^25 + 2954417021031525840*tan(b*x + 4*a)^4*tan(1/2
*a)^26 - 291827962982908800*tan(b*x + 4*a)^3*tan(1/2*a)^27 - 5514689294400576528*tan(b*x + 4*a)^2*tan(1/2*a)^2
8 + 5608964036362402464*tan(b*x + 4*a)*tan(1/2*a)^29 - 1551520053876868320*tan(1/2*a)^30 - 77771578076512320*t
an(b*x + 4*a)^6*tan(1/2*a)^22 + 787509809051626080*tan(b*x + 4*a)^5*tan(1/2*a)^23 - 2880661846092155160*tan(b*
x + 4*a)^4*tan(1/2*a)^24 + 4615789529427552000*tan(b*x + 4*a)^3*tan(1/2*a)^25 - 2907035119602048960*tan(b*x +
4*a)^2*tan(1/2*a)^26 + 74535612123099568*tan(b*x + 4*a)*tan(1/2*a)^27 + 366017355229752504*tan(1/2*a)^28 + 198
96259338686220*tan(b*x + 4*a)^6*tan(1/2*a)^20 - 248694609613265760*tan(b*x + 4*a)^5*tan(1/2*a)^21 + 1173510851
306099040*tan(b*x + 4*a)^4*tan(1/2*a)^22 - 2628180105420371040*tan(b*x + 4*a)^3*tan(1/2*a)^23 + 28691392701417
95952*tan(b*x + 4*a)^2*tan(1/2*a)^24 - 1372067944595797680*tan(b*x + 4*a)*tan(1/2*a)^25 + 190797055644582576*t
an(1/2*a)^26 - 3590491658734820*tan(b*x + 4*a)^6*tan(1/2*a)^18 + 52908136354185210*tan(b*x + 4*a)^5*tan(1/2*a)
^19 - 300960195956707245*tan(b*x + 4*a)^4*tan(1/2*a)^20 + 835950959826956960*tan(b*x + 4*a)^3*tan(1/2*a)^21 -
1179839882273931648*tan(b*x + 4*a)^2*tan(1/2*a)^22 + 789182123619651552*tan(b*x + 4*a)*tan(1/2*a)^23 - 1905246
03741625608*tan(1/2*a)^24 + 478852179070860*tan(b*x + 4*a)^6*tan(1/2*a)^16 - 8099233169503770*tan(b*x + 4*a)^5
*tan(1/2*a)^17 + 53736728263219610*tan(b*x + 4*a)^4*tan(1/2*a)^18 - 177302439156331860*tan(b*x + 4*a)^3*tan(1/
2*a)^19 + 303563975744062530*tan(b*x + 4*a)^2*tan(1/2*a)^20 - 252811149558104928*tan(b*x + 4*a)*tan(1/2*a)^21
+ 79047902485397664*tan(1/2*a)^22 - 48306853319760*tan(b*x + 4*a)^6*tan(1/2*a)^14 + 920259549793080*tan(b*x +
4*a)^5*tan(1/2*a)^15 - 6967640085374745*tan(b*x + 4*a)^4*tan(1/2*a)^16 + 26624174150101500*tan(b*x + 4*a)^3*ta
n(1/2*a)^17 - 53651600975241528*tan(b*x + 4*a)^2*tan(1/2*a)^18 + 53507434964136174*tan(b*x + 4*a)*tan(1/2*a)^1
9 - 20413710218743023*tan(1/2*a)^20 + 3722263347880*tan(b*x + 4*a)^6*tan(1/2*a)^12 - 78691575627000*tan(b*x +
4*a)^5*tan(1/2*a)^13 + 668627834124600*tan(b*x + 4*a)^4*tan(1/2*a)^14 - 2903918016468840*tan(b*x + 4*a)^3*tan(
1/2*a)^15 + 6746600775679002*tan(b*x + 4*a)^2*tan(1/2*a)^16 - 7877252164835550*tan(b*x + 4*a)*tan(1/2*a)^17 +
3573865030470070*tan(1/2*a)^18 - 218895118680*tan(b*x + 4*a)^6*tan(1/2*a)^10 + 5072116579740*tan(b*x + 4*a)^5*
tan(1/2*a)^11 - 47699227086490*tan(b*x + 4*a)^4*tan(1/2*a)^12 + 231870649855320*tan(b*x + 4*a)^3*tan(1/2*a)^13
- 610729793019360*tan(b*x + 4*a)^2*tan(1/2*a)^14 + 820187477390456*tan(b*x + 4*a)*tan(1/2*a)^15 - 43470530617
5315*tan(1/2*a)^16 + 9709765800*tan(b*x + 4*a)^6*tan(1/2*a)^8 - 243907782300*tan(b*x + 4*a)^5*tan(1/2*a)^9 + 2
508017550780*tan(b*x + 4*a)^4*tan(1/2*a)^10 - 13463793136080*tan(b*x + 4*a)^3*tan(1/2*a)^11 + 39617163658212*t
an(b*x + 4*a)^2*tan(1/2*a)^12 - 60236954245272*tan(b*x + 4*a)*tan(1/2*a)^13 + 36705931718472*tan(1/2*a)^14 - 3
16457680*tan(b*x + 4*a)^6*tan(1/2*a)^6 + 8530152120*tan(b*x + 4*a)^5*tan(1/2*a)^7 - 94820555130*tan(b*x + 4*a)
^4*tan(1/2*a)^8 + 555145935120*tan(b*x + 4*a)^3*tan(1/2*a)^9 - 1800097458768*tan(b*x + 4*a)^2*tan(1/2*a)^10 +
3053123538948*tan(b*x + 4*a)*tan(1/2*a)^11 - 2105389907166*tan(1/2*a)^12 + 7200060*tan(b*x + 4*a)^6*tan(1/2*a)
^4 - 206229240*tan(b*x + 4*a)^5*tan(1/2*a)^5 + 2449598200*tan(b*x + 4*a)^4*tan(1/2*a)^6 - 15442018200*tan(b*x
+ 4*a)^3*tan(1/2*a)^7 + 54416410020*tan(b*x + 4*a)^2*tan(1/2*a)^8 - 101416919940*tan(b*x + 4*a)*tan(1/2*a)^9 +
77871263652*tan(1/2*a)^10 - 102900*tan(b*x + 4*a)^6*tan(1/2*a)^2 + 3103170*tan(b*x + 4*a)^5*tan(1/2*a)^3 - 38
821965*tan(b*x + 4*a)^4*tan(1/2*a)^4 + 259255080*tan(b*x + 4*a)^3*tan(1/2*a)^5 - 976159968*tan(b*x + 4*a)^2*ta
n(1/2*a)^6 + 1964255832*tan(b*x + 4*a)*tan(1/2*a)^7 - 1647303294*tan(1/2*a)^8 + 700*tan(b*x + 4*a)^6 - 22050*t
an(b*x + 4*a)^5*tan(1/2*a) + 280770*tan(b*x + 4*a)^4*tan(1/2*a)^2 - 1904140*tan(b*x + 4*a)^3*tan(1/2*a)^3 + 73
60290*tan(b*x + 4*a)^2*tan(1/2*a)^4 - 15463224*tan(b*x + 4*a)*tan(1/2*a)^5 + 13827464*tan(1/2*a)^6 + 175*tan(b
*x + 4*a)^4 - 2940*tan(b*x + 4*a)^3*tan(1/2*a) + 17640*tan(b*x + 4*a)^2*tan(1/2*a)^2 - 47530*tan(b*x + 4*a)*ta
n(1/2*a)^3 + 47985*tan(1/2*a)^4 + 42*tan(b*x + 4*a)^2 - 294*tan(b*x + 4*a)*tan(1/2*a) + 462*tan(1/2*a)^2 + 5)/
((tan(1/2*a)^42 - 105*tan(1/2*a)^40 + 4830*tan(1/2*a)^38 - 127582*tan(1/2*a)^36 + 2131605*tan(1/2*a)^34 - 2341
3005*tan(1/2*a)^32 + 170737896*tan(1/2*a)^30 - 822580200*tan(1/2*a)^28 + 2607894450*tan(1/2*a)^26 - 5534841410
*tan(1/2*a)^24 + 8018138100*tan(1/2*a)^22 - 8018138100*tan(1/2*a)^20 + 5534841410*tan(1/2*a)^18 - 2607894450*t
an(1/2*a)^16 + 822580200*tan(1/2*a)^14 - 170737896*tan(1/2*a)^12 + 23413005*tan(1/2*a)^10 - 2131605*tan(1/2*a)
^8 + 127582*tan(1/2*a)^6 - 4830*tan(1/2*a)^4 + 105*tan(1/2*a)^2 - 1)*(tan(b*x + 4*a)*tan(1/2*a)^6 - 15*tan(b*x
+ 4*a)*tan(1/2*a)^4 + 6*tan(1/2*a)^5 + 15*tan(b*x + 4*a)*tan(1/2*a)^2 - 20*tan(1/2*a)^3 - tan(b*x + 4*a) + 6*
tan(1/2*a))^7))/b
________________________________________________________________________________________
maple [A] time = 1.26, size = 123, normalized size = 1.21 \[ \frac {-\frac {1}{7 \sin \left (b x +a \right )^{7} \cos \left (b x +a \right )^{5}}+\frac {12}{35 \sin \left (b x +a \right )^{5} \cos \left (b x +a \right )^{5}}-\frac {24}{35 \sin \left (b x +a \right )^{5} \cos \left (b x +a \right )^{3}}+\frac {64}{35 \sin \left (b x +a \right )^{3} \cos \left (b x +a \right )^{3}}-\frac {128}{35 \sin \left (b x +a \right )^{3} \cos \left (b x +a \right )}+\frac {512}{35 \sin \left (b x +a \right ) \cos \left (b x +a \right )}-\frac {1024 \cot \left (b x +a \right )}{35}}{64 b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(csc(b*x+a)^2*csc(2*b*x+2*a)^6,x)
[Out]
1/64/b*(-1/7/sin(b*x+a)^7/cos(b*x+a)^5+12/35/sin(b*x+a)^5/cos(b*x+a)^5-24/35/sin(b*x+a)^5/cos(b*x+a)^3+64/35/s
in(b*x+a)^3/cos(b*x+a)^3-128/35/sin(b*x+a)^3/cos(b*x+a)+512/35/sin(b*x+a)/cos(b*x+a)-1024/35*cot(b*x+a))
________________________________________________________________________________________
maxima [B] time = 0.48, size = 2710, normalized size = 26.57 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csc(b*x+a)^2*csc(2*b*x+2*a)^6,x, algorithm="maxima")
[Out]
-32/35*((20*sin(10*b*x + 10*a) - 5*sin(8*b*x + 8*a) - 10*sin(6*b*x + 6*a) + 4*sin(4*b*x + 4*a) + 2*sin(2*b*x +
2*a))*cos(24*b*x + 24*a) - 2*(20*sin(10*b*x + 10*a) - 5*sin(8*b*x + 8*a) - 10*sin(6*b*x + 6*a) + 4*sin(4*b*x
+ 4*a) + 2*sin(2*b*x + 2*a))*cos(22*b*x + 22*a) - 4*(20*sin(10*b*x + 10*a) - 5*sin(8*b*x + 8*a) - 10*sin(6*b*x
+ 6*a) + 4*sin(4*b*x + 4*a) + 2*sin(2*b*x + 2*a))*cos(20*b*x + 20*a) + 10*(20*sin(10*b*x + 10*a) - 5*sin(8*b*
x + 8*a) - 10*sin(6*b*x + 6*a) + 4*sin(4*b*x + 4*a) + 2*sin(2*b*x + 2*a))*cos(18*b*x + 18*a) + 5*(20*sin(10*b*
x + 10*a) - 5*sin(8*b*x + 8*a) - 10*sin(6*b*x + 6*a) + 4*sin(4*b*x + 4*a) + 2*sin(2*b*x + 2*a))*cos(16*b*x + 1
6*a) - 20*(20*sin(10*b*x + 10*a) - 5*sin(8*b*x + 8*a) - 10*sin(6*b*x + 6*a) + 4*sin(4*b*x + 4*a) + 2*sin(2*b*x
+ 2*a))*cos(14*b*x + 14*a) - (20*cos(10*b*x + 10*a) - 5*cos(8*b*x + 8*a) - 10*cos(6*b*x + 6*a) + 4*cos(4*b*x
+ 4*a) + 2*cos(2*b*x + 2*a) - 1)*sin(24*b*x + 24*a) + 2*(20*cos(10*b*x + 10*a) - 5*cos(8*b*x + 8*a) - 10*cos(6
*b*x + 6*a) + 4*cos(4*b*x + 4*a) + 2*cos(2*b*x + 2*a) - 1)*sin(22*b*x + 22*a) + 4*(20*cos(10*b*x + 10*a) - 5*c
os(8*b*x + 8*a) - 10*cos(6*b*x + 6*a) + 4*cos(4*b*x + 4*a) + 2*cos(2*b*x + 2*a) - 1)*sin(20*b*x + 20*a) - 10*(
20*cos(10*b*x + 10*a) - 5*cos(8*b*x + 8*a) - 10*cos(6*b*x + 6*a) + 4*cos(4*b*x + 4*a) + 2*cos(2*b*x + 2*a) - 1
)*sin(18*b*x + 18*a) - 5*(20*cos(10*b*x + 10*a) - 5*cos(8*b*x + 8*a) - 10*cos(6*b*x + 6*a) + 4*cos(4*b*x + 4*a
) + 2*cos(2*b*x + 2*a) - 1)*sin(16*b*x + 16*a) + 20*(20*cos(10*b*x + 10*a) - 5*cos(8*b*x + 8*a) - 10*cos(6*b*x
+ 6*a) + 4*cos(4*b*x + 4*a) + 2*cos(2*b*x + 2*a) - 1)*sin(14*b*x + 14*a))/(b*cos(24*b*x + 24*a)^2 + 4*b*cos(2
2*b*x + 22*a)^2 + 16*b*cos(20*b*x + 20*a)^2 + 100*b*cos(18*b*x + 18*a)^2 + 25*b*cos(16*b*x + 16*a)^2 + 400*b*c
os(14*b*x + 14*a)^2 + 400*b*cos(10*b*x + 10*a)^2 + 25*b*cos(8*b*x + 8*a)^2 + 100*b*cos(6*b*x + 6*a)^2 + 16*b*c
os(4*b*x + 4*a)^2 + 4*b*cos(2*b*x + 2*a)^2 + b*sin(24*b*x + 24*a)^2 + 4*b*sin(22*b*x + 22*a)^2 + 16*b*sin(20*b
*x + 20*a)^2 + 100*b*sin(18*b*x + 18*a)^2 + 25*b*sin(16*b*x + 16*a)^2 + 400*b*sin(14*b*x + 14*a)^2 + 400*b*sin
(10*b*x + 10*a)^2 + 25*b*sin(8*b*x + 8*a)^2 + 100*b*sin(6*b*x + 6*a)^2 + 16*b*sin(4*b*x + 4*a)^2 + 16*b*sin(4*
b*x + 4*a)*sin(2*b*x + 2*a) + 4*b*sin(2*b*x + 2*a)^2 - 2*(2*b*cos(22*b*x + 22*a) + 4*b*cos(20*b*x + 20*a) - 10
*b*cos(18*b*x + 18*a) - 5*b*cos(16*b*x + 16*a) + 20*b*cos(14*b*x + 14*a) - 20*b*cos(10*b*x + 10*a) + 5*b*cos(8
*b*x + 8*a) + 10*b*cos(6*b*x + 6*a) - 4*b*cos(4*b*x + 4*a) - 2*b*cos(2*b*x + 2*a) + b)*cos(24*b*x + 24*a) + 4*
(4*b*cos(20*b*x + 20*a) - 10*b*cos(18*b*x + 18*a) - 5*b*cos(16*b*x + 16*a) + 20*b*cos(14*b*x + 14*a) - 20*b*co
s(10*b*x + 10*a) + 5*b*cos(8*b*x + 8*a) + 10*b*cos(6*b*x + 6*a) - 4*b*cos(4*b*x + 4*a) - 2*b*cos(2*b*x + 2*a)
+ b)*cos(22*b*x + 22*a) - 8*(10*b*cos(18*b*x + 18*a) + 5*b*cos(16*b*x + 16*a) - 20*b*cos(14*b*x + 14*a) + 20*b
*cos(10*b*x + 10*a) - 5*b*cos(8*b*x + 8*a) - 10*b*cos(6*b*x + 6*a) + 4*b*cos(4*b*x + 4*a) + 2*b*cos(2*b*x + 2*
a) - b)*cos(20*b*x + 20*a) + 20*(5*b*cos(16*b*x + 16*a) - 20*b*cos(14*b*x + 14*a) + 20*b*cos(10*b*x + 10*a) -
5*b*cos(8*b*x + 8*a) - 10*b*cos(6*b*x + 6*a) + 4*b*cos(4*b*x + 4*a) + 2*b*cos(2*b*x + 2*a) - b)*cos(18*b*x + 1
8*a) - 10*(20*b*cos(14*b*x + 14*a) - 20*b*cos(10*b*x + 10*a) + 5*b*cos(8*b*x + 8*a) + 10*b*cos(6*b*x + 6*a) -
4*b*cos(4*b*x + 4*a) - 2*b*cos(2*b*x + 2*a) + b)*cos(16*b*x + 16*a) - 40*(20*b*cos(10*b*x + 10*a) - 5*b*cos(8*
b*x + 8*a) - 10*b*cos(6*b*x + 6*a) + 4*b*cos(4*b*x + 4*a) + 2*b*cos(2*b*x + 2*a) - b)*cos(14*b*x + 14*a) - 40*
(5*b*cos(8*b*x + 8*a) + 10*b*cos(6*b*x + 6*a) - 4*b*cos(4*b*x + 4*a) - 2*b*cos(2*b*x + 2*a) + b)*cos(10*b*x +
10*a) + 10*(10*b*cos(6*b*x + 6*a) - 4*b*cos(4*b*x + 4*a) - 2*b*cos(2*b*x + 2*a) + b)*cos(8*b*x + 8*a) - 20*(4*
b*cos(4*b*x + 4*a) + 2*b*cos(2*b*x + 2*a) - b)*cos(6*b*x + 6*a) + 8*(2*b*cos(2*b*x + 2*a) - b)*cos(4*b*x + 4*a
) - 4*b*cos(2*b*x + 2*a) - 2*(2*b*sin(22*b*x + 22*a) + 4*b*sin(20*b*x + 20*a) - 10*b*sin(18*b*x + 18*a) - 5*b*
sin(16*b*x + 16*a) + 20*b*sin(14*b*x + 14*a) - 20*b*sin(10*b*x + 10*a) + 5*b*sin(8*b*x + 8*a) + 10*b*sin(6*b*x
+ 6*a) - 4*b*sin(4*b*x + 4*a) - 2*b*sin(2*b*x + 2*a))*sin(24*b*x + 24*a) + 4*(4*b*sin(20*b*x + 20*a) - 10*b*s
in(18*b*x + 18*a) - 5*b*sin(16*b*x + 16*a) + 20*b*sin(14*b*x + 14*a) - 20*b*sin(10*b*x + 10*a) + 5*b*sin(8*b*x
+ 8*a) + 10*b*sin(6*b*x + 6*a) - 4*b*sin(4*b*x + 4*a) - 2*b*sin(2*b*x + 2*a))*sin(22*b*x + 22*a) - 8*(10*b*si
n(18*b*x + 18*a) + 5*b*sin(16*b*x + 16*a) - 20*b*sin(14*b*x + 14*a) + 20*b*sin(10*b*x + 10*a) - 5*b*sin(8*b*x
+ 8*a) - 10*b*sin(6*b*x + 6*a) + 4*b*sin(4*b*x + 4*a) + 2*b*sin(2*b*x + 2*a))*sin(20*b*x + 20*a) + 20*(5*b*sin
(16*b*x + 16*a) - 20*b*sin(14*b*x + 14*a) + 20*b*sin(10*b*x + 10*a) - 5*b*sin(8*b*x + 8*a) - 10*b*sin(6*b*x +
6*a) + 4*b*sin(4*b*x + 4*a) + 2*b*sin(2*b*x + 2*a))*sin(18*b*x + 18*a) - 10*(20*b*sin(14*b*x + 14*a) - 20*b*si
n(10*b*x + 10*a) + 5*b*sin(8*b*x + 8*a) + 10*b*sin(6*b*x + 6*a) - 4*b*sin(4*b*x + 4*a) - 2*b*sin(2*b*x + 2*a))
*sin(16*b*x + 16*a) - 40*(20*b*sin(10*b*x + 10*a) - 5*b*sin(8*b*x + 8*a) - 10*b*sin(6*b*x + 6*a) + 4*b*sin(4*b
*x + 4*a) + 2*b*sin(2*b*x + 2*a))*sin(14*b*x + 14*a) - 40*(5*b*sin(8*b*x + 8*a) + 10*b*sin(6*b*x + 6*a) - 4*b*
sin(4*b*x + 4*a) - 2*b*sin(2*b*x + 2*a))*sin(10*b*x + 10*a) + 20*(5*b*sin(6*b*x + 6*a) - 2*b*sin(4*b*x + 4*a)
- b*sin(2*b*x + 2*a))*sin(8*b*x + 8*a) - 40*(2*b*sin(4*b*x + 4*a) + b*sin(2*b*x + 2*a))*sin(6*b*x + 6*a) + b)
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mupad [B] time = 0.28, size = 83, normalized size = 0.81 \[ \frac {15\,\mathrm {tan}\left (a+b\,x\right )}{64\,b}+\frac {{\mathrm {tan}\left (a+b\,x\right )}^3}{32\,b}+\frac {{\mathrm {tan}\left (a+b\,x\right )}^5}{320\,b}-\frac {{\mathrm {cot}\left (a+b\,x\right )}^7\,\left (\frac {5\,{\mathrm {tan}\left (a+b\,x\right )}^6}{16}+\frac {5\,{\mathrm {tan}\left (a+b\,x\right )}^4}{64}+\frac {3\,{\mathrm {tan}\left (a+b\,x\right )}^2}{160}+\frac {1}{448}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(sin(a + b*x)^2*sin(2*a + 2*b*x)^6),x)
[Out]
(15*tan(a + b*x))/(64*b) + tan(a + b*x)^3/(32*b) + tan(a + b*x)^5/(320*b) - (cot(a + b*x)^7*((3*tan(a + b*x)^2
)/160 + (5*tan(a + b*x)^4)/64 + (5*tan(a + b*x)^6)/16 + 1/448))/b
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csc(b*x+a)**2*csc(2*b*x+2*a)**6,x)
[Out]
Timed out
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