\(\int \cot ^6(c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^2 \, dx\) [1255]

Optimal result

Integrand size = 29, antiderivative size = 198 \[ \int \cot ^6(c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {3 a b \text {arctanh}(\cos (c+d x))}{128 d}-\frac {\left (a^2+b^2\right ) \cot ^7(c+d x)}{7 d}-\frac {\left (2 a^2+b^2\right ) \cot ^9(c+d x)}{9 d}-\frac {a^2 \cot ^{11}(c+d x)}{11 d}+\frac {3 a b \cot (c+d x) \csc (c+d x)}{128 d}+\frac {a b \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac {a b \cot (c+d x) \csc ^5(c+d x)}{16 d}+\frac {a b \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}-\frac {a b \cot ^5(c+d x) \csc ^5(c+d x)}{5 d} \]

[Out]

3/128*a*b*arctanh(cos(d*x+c))/d-1/7*(a^2+b^2)*cot(d*x+c)^7/d-1/9*(2*a^2+b^2)*cot(d*x+c)^9/d-1/11*a^2*cot(d*x+c
)^11/d+3/128*a*b*cot(d*x+c)*csc(d*x+c)/d+1/64*a*b*cot(d*x+c)*csc(d*x+c)^3/d-1/16*a*b*cot(d*x+c)*csc(d*x+c)^5/d
+1/8*a*b*cot(d*x+c)^3*csc(d*x+c)^5/d-1/5*a*b*cot(d*x+c)^5*csc(d*x+c)^5/d

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {2990, 2691, 3853, 3855, 459} \[ \int \cot ^6(c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^2 \, dx=-\frac {\left (2 a^2+b^2\right ) \cot ^9(c+d x)}{9 d}-\frac {\left (a^2+b^2\right ) \cot ^7(c+d x)}{7 d}-\frac {a^2 \cot ^{11}(c+d x)}{11 d}+\frac {3 a b \text {arctanh}(\cos (c+d x))}{128 d}-\frac {a b \cot ^5(c+d x) \csc ^5(c+d x)}{5 d}+\frac {a b \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}-\frac {a b \cot (c+d x) \csc ^5(c+d x)}{16 d}+\frac {a b \cot (c+d x) \csc ^3(c+d x)}{64 d}+\frac {3 a b \cot (c+d x) \csc (c+d x)}{128 d} \]

[In]

Int[Cot[c + d*x]^6*Csc[c + d*x]^6*(a + b*Sin[c + d*x])^2,x]

[Out]

(3*a*b*ArcTanh[Cos[c + d*x]])/(128*d) - ((a^2 + b^2)*Cot[c + d*x]^7)/(7*d) - ((2*a^2 + b^2)*Cot[c + d*x]^9)/(9
*d) - (a^2*Cot[c + d*x]^11)/(11*d) + (3*a*b*Cot[c + d*x]*Csc[c + d*x])/(128*d) + (a*b*Cot[c + d*x]*Csc[c + d*x
]^3)/(64*d) - (a*b*Cot[c + d*x]*Csc[c + d*x]^5)/(16*d) + (a*b*Cot[c + d*x]^3*Csc[c + d*x]^5)/(8*d) - (a*b*Cot[
c + d*x]^5*Csc[c + d*x]^5)/(5*d)

Rule 459

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[ExpandI
ntegrand[(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] &
& IGtQ[p, 0] && IGtQ[q, 0]

Rule 2691

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b*(a*Sec[e +
f*x])^m*((b*Tan[e + f*x])^(n - 1)/(f*(m + n - 1))), x] - Dist[b^2*((n - 1)/(m + n - 1)), Int[(a*Sec[e + f*x])
^m*(b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] && NeQ[m + n - 1, 0] && Integers
Q[2*m, 2*n]

Rule 2990

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*
(x_)])^2, x_Symbol] :> Dist[2*a*(b/d), Int[(g*Cos[e + f*x])^p*(d*Sin[e + f*x])^(n + 1), x], x] + Int[(g*Cos[e
+ f*x])^p*(d*Sin[e + f*x])^n*(a^2 + b^2*Sin[e + f*x]^2), x] /; FreeQ[{a, b, d, e, f, g, n, p}, x] && NeQ[a^2 -
b^2, 0]

Rule 3853

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Csc[c + d*x])^(n - 1)/(d*(n
- 1))), x] + Dist[b^2*((n - 2)/(n - 1)), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n,
1] && IntegerQ[2*n]

Rule 3855

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps \begin{align*} \text {integral}& = (2 a b) \int \cot ^6(c+d x) \csc ^5(c+d x) \, dx+\int \cot ^6(c+d x) \csc ^6(c+d x) \left (a^2+b^2 \sin ^2(c+d x)\right ) \, dx \\ & = -\frac {a b \cot ^5(c+d x) \csc ^5(c+d x)}{5 d}-(a b) \int \cot ^4(c+d x) \csc ^5(c+d x) \, dx+\frac {\text {Subst}\left (\int \frac {\left (1+x^2\right ) \left (a^2+\left (a^2+b^2\right ) x^2\right )}{x^{12}} \, dx,x,\tan (c+d x)\right )}{d} \\ & = \frac {a b \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}-\frac {a b \cot ^5(c+d x) \csc ^5(c+d x)}{5 d}+\frac {1}{8} (3 a b) \int \cot ^2(c+d x) \csc ^5(c+d x) \, dx+\frac {\text {Subst}\left (\int \left (\frac {a^2}{x^{12}}+\frac {2 a^2+b^2}{x^{10}}+\frac {a^2+b^2}{x^8}\right ) \, dx,x,\tan (c+d x)\right )}{d} \\ & = -\frac {\left (a^2+b^2\right ) \cot ^7(c+d x)}{7 d}-\frac {\left (2 a^2+b^2\right ) \cot ^9(c+d x)}{9 d}-\frac {a^2 \cot ^{11}(c+d x)}{11 d}-\frac {a b \cot (c+d x) \csc ^5(c+d x)}{16 d}+\frac {a b \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}-\frac {a b \cot ^5(c+d x) \csc ^5(c+d x)}{5 d}-\frac {1}{16} (a b) \int \csc ^5(c+d x) \, dx \\ & = -\frac {\left (a^2+b^2\right ) \cot ^7(c+d x)}{7 d}-\frac {\left (2 a^2+b^2\right ) \cot ^9(c+d x)}{9 d}-\frac {a^2 \cot ^{11}(c+d x)}{11 d}+\frac {a b \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac {a b \cot (c+d x) \csc ^5(c+d x)}{16 d}+\frac {a b \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}-\frac {a b \cot ^5(c+d x) \csc ^5(c+d x)}{5 d}-\frac {1}{64} (3 a b) \int \csc ^3(c+d x) \, dx \\ & = -\frac {\left (a^2+b^2\right ) \cot ^7(c+d x)}{7 d}-\frac {\left (2 a^2+b^2\right ) \cot ^9(c+d x)}{9 d}-\frac {a^2 \cot ^{11}(c+d x)}{11 d}+\frac {3 a b \cot (c+d x) \csc (c+d x)}{128 d}+\frac {a b \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac {a b \cot (c+d x) \csc ^5(c+d x)}{16 d}+\frac {a b \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}-\frac {a b \cot ^5(c+d x) \csc ^5(c+d x)}{5 d}-\frac {1}{128} (3 a b) \int \csc (c+d x) \, dx \\ & = \frac {3 a b \text {arctanh}(\cos (c+d x))}{128 d}-\frac {\left (a^2+b^2\right ) \cot ^7(c+d x)}{7 d}-\frac {\left (2 a^2+b^2\right ) \cot ^9(c+d x)}{9 d}-\frac {a^2 \cot ^{11}(c+d x)}{11 d}+\frac {3 a b \cot (c+d x) \csc (c+d x)}{128 d}+\frac {a b \cot (c+d x) \csc ^3(c+d x)}{64 d}-\frac {a b \cot (c+d x) \csc ^5(c+d x)}{16 d}+\frac {a b \cot ^3(c+d x) \csc ^5(c+d x)}{8 d}-\frac {a b \cot ^5(c+d x) \csc ^5(c+d x)}{5 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.17 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.26 \[ \int \cot ^6(c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^2 \, dx=-\frac {-5322240 a b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+5322240 a b \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+\csc ^{11}(c+d x) \left (1478400 \left (8 a^2+b^2\right ) \cos (c+d x)+42240 \left (160 a^2-b^2\right ) \cos (3 (c+d x))+1943040 a^2 \cos (5 (c+d x))-865920 b^2 \cos (5 (c+d x))+140800 a^2 \cos (7 (c+d x))-499840 b^2 \cos (7 (c+d x))-28160 a^2 \cos (9 (c+d x))-77440 b^2 \cos (9 (c+d x))+2560 a^2 \cos (11 (c+d x))+7040 b^2 \cos (11 (c+d x))+5828130 a b \sin (2 (c+d x))+4790016 a b \sin (4 (c+d x))+2302839 a b \sin (6 (c+d x))+110880 a b \sin (8 (c+d x))-10395 a b \sin (10 (c+d x))\right )}{227082240 d} \]

[In]

Integrate[Cot[c + d*x]^6*Csc[c + d*x]^6*(a + b*Sin[c + d*x])^2,x]

[Out]

-1/227082240*(-5322240*a*b*Log[Cos[(c + d*x)/2]] + 5322240*a*b*Log[Sin[(c + d*x)/2]] + Csc[c + d*x]^11*(147840
0*(8*a^2 + b^2)*Cos[c + d*x] + 42240*(160*a^2 - b^2)*Cos[3*(c + d*x)] + 1943040*a^2*Cos[5*(c + d*x)] - 865920*
b^2*Cos[5*(c + d*x)] + 140800*a^2*Cos[7*(c + d*x)] - 499840*b^2*Cos[7*(c + d*x)] - 28160*a^2*Cos[9*(c + d*x)]
- 77440*b^2*Cos[9*(c + d*x)] + 2560*a^2*Cos[11*(c + d*x)] + 7040*b^2*Cos[11*(c + d*x)] + 5828130*a*b*Sin[2*(c
+ d*x)] + 4790016*a*b*Sin[4*(c + d*x)] + 2302839*a*b*Sin[6*(c + d*x)] + 110880*a*b*Sin[8*(c + d*x)] - 10395*a*
b*Sin[10*(c + d*x)]))/d

Maple [A] (verified)

Time = 0.77 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.25

method	result	size
derivativedivides	\(\frac {a^{2} \left (-\frac {\cos ^{7}\left (d x +c \right )}{11 \sin \left (d x +c \right )^{11}}-\frac {4 \left (\cos ^{7}\left (d x +c \right )\right )}{99 \sin \left (d x +c \right )^{9}}-\frac {8 \left (\cos ^{7}\left (d x +c \right )\right )}{693 \sin \left (d x +c \right )^{7}}\right )+2 a b \left (-\frac {\cos ^{7}\left (d x +c \right )}{10 \sin \left (d x +c \right )^{10}}-\frac {3 \left (\cos ^{7}\left (d x +c \right )\right )}{80 \sin \left (d x +c \right )^{8}}-\frac {\cos ^{7}\left (d x +c \right )}{160 \sin \left (d x +c \right )^{6}}+\frac {\cos ^{7}\left (d x +c \right )}{640 \sin \left (d x +c \right )^{4}}-\frac {3 \left (\cos ^{7}\left (d x +c \right )\right )}{1280 \sin \left (d x +c \right )^{2}}-\frac {3 \left (\cos ^{5}\left (d x +c \right )\right )}{1280}-\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{256}-\frac {3 \cos \left (d x +c \right )}{256}-\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{256}\right )+b^{2} \left (-\frac {\cos ^{7}\left (d x +c \right )}{9 \sin \left (d x +c \right )^{9}}-\frac {2 \left (\cos ^{7}\left (d x +c \right )\right )}{63 \sin \left (d x +c \right )^{7}}\right )}{d}\)	\(247\)
default	\(\frac {a^{2} \left (-\frac {\cos ^{7}\left (d x +c \right )}{11 \sin \left (d x +c \right )^{11}}-\frac {4 \left (\cos ^{7}\left (d x +c \right )\right )}{99 \sin \left (d x +c \right )^{9}}-\frac {8 \left (\cos ^{7}\left (d x +c \right )\right )}{693 \sin \left (d x +c \right )^{7}}\right )+2 a b \left (-\frac {\cos ^{7}\left (d x +c \right )}{10 \sin \left (d x +c \right )^{10}}-\frac {3 \left (\cos ^{7}\left (d x +c \right )\right )}{80 \sin \left (d x +c \right )^{8}}-\frac {\cos ^{7}\left (d x +c \right )}{160 \sin \left (d x +c \right )^{6}}+\frac {\cos ^{7}\left (d x +c \right )}{640 \sin \left (d x +c \right )^{4}}-\frac {3 \left (\cos ^{7}\left (d x +c \right )\right )}{1280 \sin \left (d x +c \right )^{2}}-\frac {3 \left (\cos ^{5}\left (d x +c \right )\right )}{1280}-\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{256}-\frac {3 \cos \left (d x +c \right )}{256}-\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{256}\right )+b^{2} \left (-\frac {\cos ^{7}\left (d x +c \right )}{9 \sin \left (d x +c \right )^{9}}-\frac {2 \left (\cos ^{7}\left (d x +c \right )\right )}{63 \sin \left (d x +c \right )^{7}}\right )}{d}\)	\(247\)
parallelrisch	\(\frac {-166320 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a b -315 a^{2} \left (\cot ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1386 a b \left (\cot ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+385 \left (a^{2}-4 b^{2}\right ) \left (\cot ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3465 a b \left (\cot ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+495 \left (5 a^{2}+12 b^{2}\right ) \left (\cot ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6930 a b \left (\cot ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-3465 a^{2} \left (\cot ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-27720 a b \left (\cot ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2310 \left (-5 a^{2}-16 b^{2}\right ) \left (\cot ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-13860 a b \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6930 \left (5 a^{2}+12 b^{2}\right ) \cot \left (\frac {d x}{2}+\frac {c}{2}\right )+315 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2}+\frac {22 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a b}{5}+\frac {11 \left (-a^{2}+4 b^{2}\right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{9}-11 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a b +\frac {11 \left (-5 a^{2}-12 b^{2}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}-22 a b \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+11 a^{2} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+88 a b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {22 \left (5 a^{2}+16 b^{2}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+44 a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-110 a^{2}-264 b^{2}\right )}{7096320 d}\)	\(418\)
risch	\(-\frac {-5120 i a^{2}-14080 i b^{2}-281600 i a^{2} {\mathrm e}^{4 i \left (d x +c \right )}+112640 i b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+154880 i b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-10395 a b \,{\mathrm e}^{i \left (d x +c \right )}-2280960 i b^{2} {\mathrm e}^{8 i \left (d x +c \right )}+5828130 a b \,{\mathrm e}^{9 i \left (d x +c \right )}-1520640 i a^{2} {\mathrm e}^{6 i \left (d x +c \right )}+2027520 i b^{2} {\mathrm e}^{6 i \left (d x +c \right )}+4790016 a b \,{\mathrm e}^{7 i \left (d x +c \right )}+56320 i a^{2} {\mathrm e}^{2 i \left (d x +c \right )}+110880 a b \,{\mathrm e}^{3 i \left (d x +c \right )}+887040 i b^{2} {\mathrm e}^{18 i \left (d x +c \right )}-110880 a b \,{\mathrm e}^{19 i \left (d x +c \right )}+10395 a b \,{\mathrm e}^{21 i \left (d x +c \right )}-2365440 i a^{2} {\mathrm e}^{16 i \left (d x +c \right )}-7603200 i a^{2} {\mathrm e}^{8 i \left (d x +c \right )}-295680 i b^{2} {\mathrm e}^{16 i \left (d x +c \right )}-10644480 i a^{2} {\mathrm e}^{10 i \left (d x +c \right )}-13009920 i a^{2} {\mathrm e}^{12 i \left (d x +c \right )}-5828130 a b \,{\mathrm e}^{13 i \left (d x +c \right )}-4730880 i b^{2} {\mathrm e}^{12 i \left (d x +c \right )}+2302839 a b \,{\mathrm e}^{5 i \left (d x +c \right )}+1774080 i b^{2} {\mathrm e}^{10 i \left (d x +c \right )}-2302839 a b \,{\mathrm e}^{17 i \left (d x +c \right )}-5913600 i a^{2} {\mathrm e}^{14 i \left (d x +c \right )}+2365440 i b^{2} {\mathrm e}^{14 i \left (d x +c \right )}-4790016 a b \,{\mathrm e}^{15 i \left (d x +c \right )}}{221760 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{11}}-\frac {3 a b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{128 d}+\frac {3 a b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{128 d}\)	\(456\)

[In]

int(cos(d*x+c)^6*csc(d*x+c)^12*(a+b*sin(d*x+c))^2,x,method=_RETURNVERBOSE)

[Out]

1/d*(a^2*(-1/11/sin(d*x+c)^11*cos(d*x+c)^7-4/99/sin(d*x+c)^9*cos(d*x+c)^7-8/693/sin(d*x+c)^7*cos(d*x+c)^7)+2*a
*b*(-1/10/sin(d*x+c)^10*cos(d*x+c)^7-3/80/sin(d*x+c)^8*cos(d*x+c)^7-1/160/sin(d*x+c)^6*cos(d*x+c)^7+1/640/sin(
d*x+c)^4*cos(d*x+c)^7-3/1280/sin(d*x+c)^2*cos(d*x+c)^7-3/1280*cos(d*x+c)^5-1/256*cos(d*x+c)^3-3/256*cos(d*x+c)
-3/256*ln(csc(d*x+c)-cot(d*x+c)))+b^2*(-1/9/sin(d*x+c)^9*cos(d*x+c)^7-2/63/sin(d*x+c)^7*cos(d*x+c)^7))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 363 vs. \(2 (180) = 360\).

Time = 0.45 (sec) , antiderivative size = 363, normalized size of antiderivative = 1.83 \[ \int \cot ^6(c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {2560 \, {\left (4 \, a^{2} + 11 \, b^{2}\right )} \cos \left (d x + c\right )^{11} - 14080 \, {\left (4 \, a^{2} + 11 \, b^{2}\right )} \cos \left (d x + c\right )^{9} + 126720 \, {\left (a^{2} + b^{2}\right )} \cos \left (d x + c\right )^{7} + 10395 \, {\left (a b \cos \left (d x + c\right )^{10} - 5 \, a b \cos \left (d x + c\right )^{8} + 10 \, a b \cos \left (d x + c\right )^{6} - 10 \, a b \cos \left (d x + c\right )^{4} + 5 \, a b \cos \left (d x + c\right )^{2} - a b\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 10395 \, {\left (a b \cos \left (d x + c\right )^{10} - 5 \, a b \cos \left (d x + c\right )^{8} + 10 \, a b \cos \left (d x + c\right )^{6} - 10 \, a b \cos \left (d x + c\right )^{4} + 5 \, a b \cos \left (d x + c\right )^{2} - a b\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 1386 \, {\left (15 \, a b \cos \left (d x + c\right )^{9} - 70 \, a b \cos \left (d x + c\right )^{7} - 128 \, a b \cos \left (d x + c\right )^{5} + 70 \, a b \cos \left (d x + c\right )^{3} - 15 \, a b \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{887040 \, {\left (d \cos \left (d x + c\right )^{10} - 5 \, d \cos \left (d x + c\right )^{8} + 10 \, d \cos \left (d x + c\right )^{6} - 10 \, d \cos \left (d x + c\right )^{4} + 5 \, d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \]

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^12*(a+b*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

1/887040*(2560*(4*a^2 + 11*b^2)*cos(d*x + c)^11 - 14080*(4*a^2 + 11*b^2)*cos(d*x + c)^9 + 126720*(a^2 + b^2)*c
os(d*x + c)^7 + 10395*(a*b*cos(d*x + c)^10 - 5*a*b*cos(d*x + c)^8 + 10*a*b*cos(d*x + c)^6 - 10*a*b*cos(d*x + c
)^4 + 5*a*b*cos(d*x + c)^2 - a*b)*log(1/2*cos(d*x + c) + 1/2)*sin(d*x + c) - 10395*(a*b*cos(d*x + c)^10 - 5*a*
b*cos(d*x + c)^8 + 10*a*b*cos(d*x + c)^6 - 10*a*b*cos(d*x + c)^4 + 5*a*b*cos(d*x + c)^2 - a*b)*log(-1/2*cos(d*
x + c) + 1/2)*sin(d*x + c) - 1386*(15*a*b*cos(d*x + c)^9 - 70*a*b*cos(d*x + c)^7 - 128*a*b*cos(d*x + c)^5 + 70
*a*b*cos(d*x + c)^3 - 15*a*b*cos(d*x + c))*sin(d*x + c))/((d*cos(d*x + c)^10 - 5*d*cos(d*x + c)^8 + 10*d*cos(d
*x + c)^6 - 10*d*cos(d*x + c)^4 + 5*d*cos(d*x + c)^2 - d)*sin(d*x + c))

Sympy [F(-1)]

Timed out. \[ \int \cot ^6(c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^2 \, dx=\text {Timed out} \]

[In]

integrate(cos(d*x+c)**6*csc(d*x+c)**12*(a+b*sin(d*x+c))**2,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.99 \[ \int \cot ^6(c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^2 \, dx=-\frac {693 \, a b {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{9} - 70 \, \cos \left (d x + c\right )^{7} - 128 \, \cos \left (d x + c\right )^{5} + 70 \, \cos \left (d x + c\right )^{3} - 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{10} - 5 \, \cos \left (d x + c\right )^{8} + 10 \, \cos \left (d x + c\right )^{6} - 10 \, \cos \left (d x + c\right )^{4} + 5 \, \cos \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac {14080 \, {\left (9 \, \tan \left (d x + c\right )^{2} + 7\right )} b^{2}}{\tan \left (d x + c\right )^{9}} + \frac {1280 \, {\left (99 \, \tan \left (d x + c\right )^{4} + 154 \, \tan \left (d x + c\right )^{2} + 63\right )} a^{2}}{\tan \left (d x + c\right )^{11}}}{887040 \, d} \]

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^12*(a+b*sin(d*x+c))^2,x, algorithm="maxima")

[Out]

-1/887040*(693*a*b*(2*(15*cos(d*x + c)^9 - 70*cos(d*x + c)^7 - 128*cos(d*x + c)^5 + 70*cos(d*x + c)^3 - 15*cos
(d*x + c))/(cos(d*x + c)^10 - 5*cos(d*x + c)^8 + 10*cos(d*x + c)^6 - 10*cos(d*x + c)^4 + 5*cos(d*x + c)^2 - 1)
- 15*log(cos(d*x + c) + 1) + 15*log(cos(d*x + c) - 1)) + 14080*(9*tan(d*x + c)^2 + 7)*b^2/tan(d*x + c)^9 + 12
80*(99*tan(d*x + c)^4 + 154*tan(d*x + c)^2 + 63)*a^2/tan(d*x + c)^11)/d

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 502 vs. \(2 (180) = 360\).

Time = 0.44 (sec) , antiderivative size = 502, normalized size of antiderivative = 2.54 \[ \int \cot ^6(c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {315 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 1386 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} - 385 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 1540 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 3465 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 2475 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 5940 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 6930 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 3465 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 27720 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 11550 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 36960 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 13860 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 166320 \, a b \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 34650 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 83160 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {502266 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 34650 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 83160 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} - 13860 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 11550 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 36960 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 27720 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 3465 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 6930 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 2475 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 5940 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 3465 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 385 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1540 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1386 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 315 \, a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11}}}{7096320 \, d} \]

[In]

integrate(cos(d*x+c)^6*csc(d*x+c)^12*(a+b*sin(d*x+c))^2,x, algorithm="giac")

[Out]

1/7096320*(315*a^2*tan(1/2*d*x + 1/2*c)^11 + 1386*a*b*tan(1/2*d*x + 1/2*c)^10 - 385*a^2*tan(1/2*d*x + 1/2*c)^9
+ 1540*b^2*tan(1/2*d*x + 1/2*c)^9 - 3465*a*b*tan(1/2*d*x + 1/2*c)^8 - 2475*a^2*tan(1/2*d*x + 1/2*c)^7 - 5940*
b^2*tan(1/2*d*x + 1/2*c)^7 - 6930*a*b*tan(1/2*d*x + 1/2*c)^6 + 3465*a^2*tan(1/2*d*x + 1/2*c)^5 + 27720*a*b*tan
(1/2*d*x + 1/2*c)^4 + 11550*a^2*tan(1/2*d*x + 1/2*c)^3 + 36960*b^2*tan(1/2*d*x + 1/2*c)^3 + 13860*a*b*tan(1/2*
d*x + 1/2*c)^2 - 166320*a*b*log(abs(tan(1/2*d*x + 1/2*c))) - 34650*a^2*tan(1/2*d*x + 1/2*c) - 83160*b^2*tan(1/
2*d*x + 1/2*c) + (502266*a*b*tan(1/2*d*x + 1/2*c)^11 + 34650*a^2*tan(1/2*d*x + 1/2*c)^10 + 83160*b^2*tan(1/2*d
*x + 1/2*c)^10 - 13860*a*b*tan(1/2*d*x + 1/2*c)^9 - 11550*a^2*tan(1/2*d*x + 1/2*c)^8 - 36960*b^2*tan(1/2*d*x +
1/2*c)^8 - 27720*a*b*tan(1/2*d*x + 1/2*c)^7 - 3465*a^2*tan(1/2*d*x + 1/2*c)^6 + 6930*a*b*tan(1/2*d*x + 1/2*c)
^5 + 2475*a^2*tan(1/2*d*x + 1/2*c)^4 + 5940*b^2*tan(1/2*d*x + 1/2*c)^4 + 3465*a*b*tan(1/2*d*x + 1/2*c)^3 + 385
*a^2*tan(1/2*d*x + 1/2*c)^2 - 1540*b^2*tan(1/2*d*x + 1/2*c)^2 - 1386*a*b*tan(1/2*d*x + 1/2*c) - 315*a^2)/tan(1
/2*d*x + 1/2*c)^11)/d

Mupad [B] (verification not implemented)

Time = 16.38 (sec) , antiderivative size = 448, normalized size of antiderivative = 2.26 \[ \int \cot ^6(c+d x) \csc ^6(c+d x) (a+b \sin (c+d x))^2 \, dx=-\frac {\frac {5\,a^2\,\cos \left (c+d\,x\right )}{96}+\frac {5\,b^2\,\cos \left (c+d\,x\right )}{768}+\frac {5\,a^2\,\cos \left (3\,c+3\,d\,x\right )}{168}+\frac {23\,a^2\,\cos \left (5\,c+5\,d\,x\right )}{2688}+\frac {5\,a^2\,\cos \left (7\,c+7\,d\,x\right )}{8064}-\frac {a^2\,\cos \left (9\,c+9\,d\,x\right )}{8064}+\frac {a^2\,\cos \left (11\,c+11\,d\,x\right )}{88704}-\frac {b^2\,\cos \left (3\,c+3\,d\,x\right )}{5376}-\frac {41\,b^2\,\cos \left (5\,c+5\,d\,x\right )}{10752}-\frac {71\,b^2\,\cos \left (7\,c+7\,d\,x\right )}{32256}-\frac {11\,b^2\,\cos \left (9\,c+9\,d\,x\right )}{32256}+\frac {b^2\,\cos \left (11\,c+11\,d\,x\right )}{32256}+\frac {841\,a\,b\,\sin \left (2\,c+2\,d\,x\right )}{32768}+\frac {27\,a\,b\,\sin \left (4\,c+4\,d\,x\right )}{1280}+\frac {3323\,a\,b\,\sin \left (6\,c+6\,d\,x\right )}{327680}+\frac {a\,b\,\sin \left (8\,c+8\,d\,x\right )}{2048}-\frac {3\,a\,b\,\sin \left (10\,c+10\,d\,x\right )}{65536}+\frac {693\,a\,b\,\sin \left (c+d\,x\right )\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{65536}-\frac {495\,a\,b\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,\sin \left (3\,c+3\,d\,x\right )}{65536}+\frac {495\,a\,b\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,\sin \left (5\,c+5\,d\,x\right )}{131072}-\frac {165\,a\,b\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,\sin \left (7\,c+7\,d\,x\right )}{131072}+\frac {33\,a\,b\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,\sin \left (9\,c+9\,d\,x\right )}{131072}-\frac {3\,a\,b\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,\sin \left (11\,c+11\,d\,x\right )}{131072}}{d\,{\sin \left (c+d\,x\right )}^{11}} \]

[In]

int((cos(c + d*x)^6*(a + b*sin(c + d*x))^2)/sin(c + d*x)^12,x)

[Out]

-((5*a^2*cos(c + d*x))/96 + (5*b^2*cos(c + d*x))/768 + (5*a^2*cos(3*c + 3*d*x))/168 + (23*a^2*cos(5*c + 5*d*x)
)/2688 + (5*a^2*cos(7*c + 7*d*x))/8064 - (a^2*cos(9*c + 9*d*x))/8064 + (a^2*cos(11*c + 11*d*x))/88704 - (b^2*c
os(3*c + 3*d*x))/5376 - (41*b^2*cos(5*c + 5*d*x))/10752 - (71*b^2*cos(7*c + 7*d*x))/32256 - (11*b^2*cos(9*c +
9*d*x))/32256 + (b^2*cos(11*c + 11*d*x))/32256 + (841*a*b*sin(2*c + 2*d*x))/32768 + (27*a*b*sin(4*c + 4*d*x))/
1280 + (3323*a*b*sin(6*c + 6*d*x))/327680 + (a*b*sin(8*c + 8*d*x))/2048 - (3*a*b*sin(10*c + 10*d*x))/65536 + (
693*a*b*sin(c + d*x)*log(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/65536 - (495*a*b*log(sin(c/2 + (d*x)/2)/cos(c
/2 + (d*x)/2))*sin(3*c + 3*d*x))/65536 + (495*a*b*log(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2))*sin(5*c + 5*d*x))
/131072 - (165*a*b*log(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2))*sin(7*c + 7*d*x))/131072 + (33*a*b*log(sin(c/2 +
(d*x)/2)/cos(c/2 + (d*x)/2))*sin(9*c + 9*d*x))/131072 - (3*a*b*log(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2))*sin
(11*c + 11*d*x))/131072)/(d*sin(c + d*x)^11)