Integrand size = 29, antiderivative size = 151 \[ \int \cot ^6(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {5 a b \text {arctanh}(\cos (c+d x))}{64 d}-\frac {\left (a^2+b^2\right ) \cot ^7(c+d x)}{7 d}-\frac {a^2 \cot ^9(c+d x)}{9 d}+\frac {5 a b \cot (c+d x) \csc (c+d x)}{64 d}-\frac {5 a b \cot (c+d x) \csc ^3(c+d x)}{32 d}+\frac {5 a b \cot ^3(c+d x) \csc ^3(c+d x)}{24 d}-\frac {a b \cot ^5(c+d x) \csc ^3(c+d x)}{4 d} \] Output:
5/64*a*b*arctanh(cos(d*x+c))/d-1/7*(a^2+b^2)*cot(d*x+c)^7/d-1/9*a^2*cot(d* x+c)^9/d+5/64*a*b*cot(d*x+c)*csc(d*x+c)/d-5/32*a*b*cot(d*x+c)*csc(d*x+c)^3 /d+5/24*a*b*cot(d*x+c)^3*csc(d*x+c)^3/d-1/4*a*b*cot(d*x+c)^5*csc(d*x+c)^3/ d
Time = 2.66 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.35 \[ \int \cot ^6(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2 \, dx=-\frac {-40320 a b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+40320 a b \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+\csc ^9(c+d x) \left (4032 \left (8 a^2+b^2\right ) \cos (c+d x)+18816 a^2 \cos (3 (c+d x))+5760 a^2 \cos (5 (c+d x))-2304 b^2 \cos (5 (c+d x))+576 a^2 \cos (7 (c+d x))-1440 b^2 \cos (7 (c+d x))-64 a^2 \cos (9 (c+d x))-288 b^2 \cos (9 (c+d x))+18270 a b \sin (2 (c+d x))+10458 a b \sin (4 (c+d x))+8022 a b \sin (6 (c+d x))+315 a b \sin (8 (c+d x))\right )}{516096 d} \] Input:
Integrate[Cot[c + d*x]^6*Csc[c + d*x]^4*(a + b*Sin[c + d*x])^2,x]
Output:
-1/516096*(-40320*a*b*Log[Cos[(c + d*x)/2]] + 40320*a*b*Log[Sin[(c + d*x)/ 2]] + Csc[c + d*x]^9*(4032*(8*a^2 + b^2)*Cos[c + d*x] + 18816*a^2*Cos[3*(c + d*x)] + 5760*a^2*Cos[5*(c + d*x)] - 2304*b^2*Cos[5*(c + d*x)] + 576*a^2 *Cos[7*(c + d*x)] - 1440*b^2*Cos[7*(c + d*x)] - 64*a^2*Cos[9*(c + d*x)] - 288*b^2*Cos[9*(c + d*x)] + 18270*a*b*Sin[2*(c + d*x)] + 10458*a*b*Sin[4*(c + d*x)] + 8022*a*b*Sin[6*(c + d*x)] + 315*a*b*Sin[8*(c + d*x)]))/d
Time = 1.38 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.06, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.517, Rules used = {3042, 3390, 3042, 3091, 3042, 3091, 3042, 3091, 3042, 4255, 3042, 4257, 4889, 244, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \cot ^6(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^6 (a+b \sin (c+d x))^2}{\sin (c+d x)^{10}}dx\) |
| \(\Big \downarrow \) 3390 |
| \(\displaystyle \int \cot ^6(c+d x) \csc ^4(c+d x) \left (a^2+b^2 \sin ^2(c+d x)\right )dx+2 a b \int \cot ^6(c+d x) \csc ^3(c+d x)dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {a^2+b^2 \sin (c+d x)^2}{\sin (c+d x)^4 \tan (c+d x)^6}dx+2 a b \int \sec \left (c+d x-\frac {\pi }{2}\right )^3 \tan \left (c+d x-\frac {\pi }{2}\right )^6dx\) |
| \(\Big \downarrow \) 3091 |
| \(\displaystyle \int \frac {a^2+b^2 \sin (c+d x)^2}{\sin (c+d x)^4 \tan (c+d x)^6}dx+2 a b \left (-\frac {5}{8} \int \cot ^4(c+d x) \csc ^3(c+d x)dx-\frac {\cot ^5(c+d x) \csc ^3(c+d x)}{8 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {a^2+b^2 \sin (c+d x)^2}{\sin (c+d x)^4 \tan (c+d x)^6}dx+2 a b \left (-\frac {5}{8} \int \sec \left (c+d x-\frac {\pi }{2}\right )^3 \tan \left (c+d x-\frac {\pi }{2}\right )^4dx-\frac {\cot ^5(c+d x) \csc ^3(c+d x)}{8 d}\right )\) |
| \(\Big \downarrow \) 3091 |
| \(\displaystyle \int \frac {a^2+b^2 \sin (c+d x)^2}{\sin (c+d x)^4 \tan (c+d x)^6}dx+2 a b \left (-\frac {5}{8} \left (-\frac {1}{2} \int \cot ^2(c+d x) \csc ^3(c+d x)dx-\frac {\cot ^3(c+d x) \csc ^3(c+d x)}{6 d}\right )-\frac {\cot ^5(c+d x) \csc ^3(c+d x)}{8 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {a^2+b^2 \sin (c+d x)^2}{\sin (c+d x)^4 \tan (c+d x)^6}dx+2 a b \left (-\frac {5}{8} \left (-\frac {1}{2} \int \sec \left (c+d x-\frac {\pi }{2}\right )^3 \tan \left (c+d x-\frac {\pi }{2}\right )^2dx-\frac {\cot ^3(c+d x) \csc ^3(c+d x)}{6 d}\right )-\frac {\cot ^5(c+d x) \csc ^3(c+d x)}{8 d}\right )\) |
| \(\Big \downarrow \) 3091 |
| \(\displaystyle \int \frac {a^2+b^2 \sin (c+d x)^2}{\sin (c+d x)^4 \tan (c+d x)^6}dx+2 a b \left (-\frac {5}{8} \left (\frac {1}{2} \left (\frac {1}{4} \int \csc ^3(c+d x)dx+\frac {\cot (c+d x) \csc ^3(c+d x)}{4 d}\right )-\frac {\cot ^3(c+d x) \csc ^3(c+d x)}{6 d}\right )-\frac {\cot ^5(c+d x) \csc ^3(c+d x)}{8 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {a^2+b^2 \sin (c+d x)^2}{\sin (c+d x)^4 \tan (c+d x)^6}dx+2 a b \left (-\frac {5}{8} \left (\frac {1}{2} \left (\frac {1}{4} \int \csc (c+d x)^3dx+\frac {\cot (c+d x) \csc ^3(c+d x)}{4 d}\right )-\frac {\cot ^3(c+d x) \csc ^3(c+d x)}{6 d}\right )-\frac {\cot ^5(c+d x) \csc ^3(c+d x)}{8 d}\right )\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle \int \frac {a^2+b^2 \sin (c+d x)^2}{\sin (c+d x)^4 \tan (c+d x)^6}dx+2 a b \left (-\frac {5}{8} \left (\frac {1}{2} \left (\frac {1}{4} \left (\frac {1}{2} \int \csc (c+d x)dx-\frac {\cot (c+d x) \csc (c+d x)}{2 d}\right )+\frac {\cot (c+d x) \csc ^3(c+d x)}{4 d}\right )-\frac {\cot ^3(c+d x) \csc ^3(c+d x)}{6 d}\right )-\frac {\cot ^5(c+d x) \csc ^3(c+d x)}{8 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {a^2+b^2 \sin (c+d x)^2}{\sin (c+d x)^4 \tan (c+d x)^6}dx+2 a b \left (-\frac {5}{8} \left (\frac {1}{2} \left (\frac {1}{4} \left (\frac {1}{2} \int \csc (c+d x)dx-\frac {\cot (c+d x) \csc (c+d x)}{2 d}\right )+\frac {\cot (c+d x) \csc ^3(c+d x)}{4 d}\right )-\frac {\cot ^3(c+d x) \csc ^3(c+d x)}{6 d}\right )-\frac {\cot ^5(c+d x) \csc ^3(c+d x)}{8 d}\right )\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \int \frac {a^2+b^2 \sin (c+d x)^2}{\sin (c+d x)^4 \tan (c+d x)^6}dx+2 a b \left (-\frac {5}{8} \left (\frac {1}{2} \left (\frac {1}{4} \left (-\frac {\text {arctanh}(\cos (c+d x))}{2 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 d}\right )+\frac {\cot (c+d x) \csc ^3(c+d x)}{4 d}\right )-\frac {\cot ^3(c+d x) \csc ^3(c+d x)}{6 d}\right )-\frac {\cot ^5(c+d x) \csc ^3(c+d x)}{8 d}\right )\) |
| \(\Big \downarrow \) 4889 |
| \(\displaystyle \frac {\int \cot ^{10}(c+d x) \left (a^2+\left (a^2+b^2\right ) \tan ^2(c+d x)\right )d\tan (c+d x)}{d}+2 a b \left (-\frac {5}{8} \left (\frac {1}{2} \left (\frac {1}{4} \left (-\frac {\text {arctanh}(\cos (c+d x))}{2 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 d}\right )+\frac {\cot (c+d x) \csc ^3(c+d x)}{4 d}\right )-\frac {\cot ^3(c+d x) \csc ^3(c+d x)}{6 d}\right )-\frac {\cot ^5(c+d x) \csc ^3(c+d x)}{8 d}\right )\) |
| \(\Big \downarrow \) 244 |
| \(\displaystyle \frac {\int \left (a^2 \cot ^{10}(c+d x)+\left (a^2+b^2\right ) \cot ^8(c+d x)\right )d\tan (c+d x)}{d}+2 a b \left (-\frac {5}{8} \left (\frac {1}{2} \left (\frac {1}{4} \left (-\frac {\text {arctanh}(\cos (c+d x))}{2 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 d}\right )+\frac {\cot (c+d x) \csc ^3(c+d x)}{4 d}\right )-\frac {\cot ^3(c+d x) \csc ^3(c+d x)}{6 d}\right )-\frac {\cot ^5(c+d x) \csc ^3(c+d x)}{8 d}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {1}{7} \left (a^2+b^2\right ) \cot ^7(c+d x)-\frac {1}{9} a^2 \cot ^9(c+d x)}{d}+2 a b \left (-\frac {5}{8} \left (\frac {1}{2} \left (\frac {1}{4} \left (-\frac {\text {arctanh}(\cos (c+d x))}{2 d}-\frac {\cot (c+d x) \csc (c+d x)}{2 d}\right )+\frac {\cot (c+d x) \csc ^3(c+d x)}{4 d}\right )-\frac {\cot ^3(c+d x) \csc ^3(c+d x)}{6 d}\right )-\frac {\cot ^5(c+d x) \csc ^3(c+d x)}{8 d}\right )\) |
Input:
Int[Cot[c + d*x]^6*Csc[c + d*x]^4*(a + b*Sin[c + d*x])^2,x]
Output:
(-1/7*((a^2 + b^2)*Cot[c + d*x]^7) - (a^2*Cot[c + d*x]^9)/9)/d + 2*a*b*(-1 /8*(Cot[c + d*x]^5*Csc[c + d*x]^3)/d - (5*(-1/6*(Cot[c + d*x]^3*Csc[c + d* x]^3)/d + ((Cot[c + d*x]*Csc[c + d*x]^3)/(4*d) + (-1/2*ArcTanh[Cos[c + d*x ]]/d - (Cot[c + d*x]*Csc[c + d*x])/(2*d))/4)/2))/8)
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p , 0]
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] - Simp[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] && IntegersQ[2*m, 2*n]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n _)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^2, x_Symbol] :> Simp[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]
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] + Simp[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]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Int[u_, x_Symbol] :> With[{v = FunctionOfTrig[u, x]}, With[{d = FreeFactors [Tan[v], x]}, Simp[d/Coefficient[v, x, 1] Subst[Int[SubstFor[1/(1 + d^2*x ^2), Tan[v]/d, u, x], x], x, Tan[v]/d], x]] /; !FalseQ[v] && FunctionOfQ[N onfreeFactors[Tan[v], x], u, x]] /; InverseFunctionFreeQ[u, x] && !MatchQ[ u, (v_.)*((c_.)*tan[w_]^(n_.)*tan[z_]^(n_.))^(p_.) /; FreeQ[{c, p}, x] && I ntegerQ[n] && LinearQ[w, x] && EqQ[z, 2*w]]
Time = 0.51 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.26
| method | result | size |
| derivativedivides | \(\frac {a^{2} \left (-\frac {\cos \left (d x +c \right )^{7}}{9 \sin \left (d x +c \right )^{9}}-\frac {2 \cos \left (d x +c \right )^{7}}{63 \sin \left (d x +c \right )^{7}}\right )+2 a b \left (-\frac {\cos \left (d x +c \right )^{7}}{8 \sin \left (d x +c \right )^{8}}-\frac {\cos \left (d x +c \right )^{7}}{48 \sin \left (d x +c \right )^{6}}+\frac {\cos \left (d x +c \right )^{7}}{192 \sin \left (d x +c \right )^{4}}-\frac {\cos \left (d x +c \right )^{7}}{128 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )^{5}}{128}-\frac {5 \cos \left (d x +c \right )^{3}}{384}-\frac {5 \cos \left (d x +c \right )}{128}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{128}\right )-\frac {b^{2} \cos \left (d x +c \right )^{7}}{7 \sin \left (d x +c \right )^{7}}}{d}\) | \(191\) |
| default | \(\frac {a^{2} \left (-\frac {\cos \left (d x +c \right )^{7}}{9 \sin \left (d x +c \right )^{9}}-\frac {2 \cos \left (d x +c \right )^{7}}{63 \sin \left (d x +c \right )^{7}}\right )+2 a b \left (-\frac {\cos \left (d x +c \right )^{7}}{8 \sin \left (d x +c \right )^{8}}-\frac {\cos \left (d x +c \right )^{7}}{48 \sin \left (d x +c \right )^{6}}+\frac {\cos \left (d x +c \right )^{7}}{192 \sin \left (d x +c \right )^{4}}-\frac {\cos \left (d x +c \right )^{7}}{128 \sin \left (d x +c \right )^{2}}-\frac {\cos \left (d x +c \right )^{5}}{128}-\frac {5 \cos \left (d x +c \right )^{3}}{384}-\frac {5 \cos \left (d x +c \right )}{128}-\frac {5 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{128}\right )-\frac {b^{2} \cos \left (d x +c \right )^{7}}{7 \sin \left (d x +c \right )^{7}}}{d}\) | \(191\) |
| risch | \(-\frac {8064 i b^{2} {\mathrm e}^{14 i \left (d x +c \right )}+315 a b \,{\mathrm e}^{17 i \left (d x +c \right )}-24192 i b^{2} {\mathrm e}^{12 i \left (d x +c \right )}+13440 i a^{2} {\mathrm e}^{12 i \left (d x +c \right )}+40320 i b^{2} {\mathrm e}^{10 i \left (d x +c \right )}+18270 a b \,{\mathrm e}^{11 i \left (d x +c \right )}+10458 a b \,{\mathrm e}^{13 i \left (d x +c \right )}+3456 i a^{2} {\mathrm e}^{4 i \left (d x +c \right )}-12672 i b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-18270 a b \,{\mathrm e}^{7 i \left (d x +c \right )}+8022 a b \,{\mathrm e}^{15 i \left (d x +c \right )}+40320 i a^{2} {\mathrm e}^{10 i \left (d x +c \right )}+8064 i a^{2} {\mathrm e}^{14 i \left (d x +c \right )}-128 i a^{2}-576 i b^{2}-10458 a b \,{\mathrm e}^{5 i \left (d x +c \right )}+1152 i a^{2} {\mathrm e}^{2 i \left (d x +c \right )}+1152 i b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-315 a b \,{\mathrm e}^{i \left (d x +c \right )}-8022 a b \,{\mathrm e}^{3 i \left (d x +c \right )}-4032 i b^{2} {\mathrm e}^{16 i \left (d x +c \right )}+24192 i a^{2} {\mathrm e}^{8 i \left (d x +c \right )}-32256 i b^{2} {\mathrm e}^{8 i \left (d x +c \right )}+24192 i a^{2} {\mathrm e}^{6 i \left (d x +c \right )}+24192 i b^{2} {\mathrm e}^{6 i \left (d x +c \right )}}{2016 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{9}}-\frac {5 a b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{64 d}+\frac {5 a b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{64 d}\) | \(400\) |
Input:
int(cot(d*x+c)^6*csc(d*x+c)^4*(a+b*sin(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
1/d*(a^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)+2 *a*b*(-1/8/sin(d*x+c)^8*cos(d*x+c)^7-1/48/sin(d*x+c)^6*cos(d*x+c)^7+1/192/ sin(d*x+c)^4*cos(d*x+c)^7-1/128/sin(d*x+c)^2*cos(d*x+c)^7-1/128*cos(d*x+c) ^5-5/384*cos(d*x+c)^3-5/128*cos(d*x+c)-5/128*ln(csc(d*x+c)-cot(d*x+c)))-1/ 7*b^2/sin(d*x+c)^7*cos(d*x+c)^7)
Leaf count of result is larger than twice the leaf count of optimal. 291 vs. \(2 (137) = 274\).
Time = 0.11 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.93 \[ \int \cot ^6(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {128 \, {\left (2 \, a^{2} + 9 \, b^{2}\right )} \cos \left (d x + c\right )^{9} - 1152 \, {\left (a^{2} + b^{2}\right )} \cos \left (d x + c\right )^{7} + 315 \, {\left (a b \cos \left (d x + c\right )^{8} - 4 \, a b \cos \left (d x + c\right )^{6} + 6 \, a b \cos \left (d x + c\right )^{4} - 4 \, a b \cos \left (d x + c\right )^{2} + a b\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 315 \, {\left (a b \cos \left (d x + c\right )^{8} - 4 \, a b \cos \left (d x + c\right )^{6} + 6 \, a b \cos \left (d x + c\right )^{4} - 4 \, a b \cos \left (d x + c\right )^{2} + a b\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 42 \, {\left (15 \, a b \cos \left (d x + c\right )^{7} + 73 \, a b \cos \left (d x + c\right )^{5} - 55 \, a b \cos \left (d x + c\right )^{3} + 15 \, a b \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{8064 \, {\left (d \cos \left (d x + c\right )^{8} - 4 \, d \cos \left (d x + c\right )^{6} + 6 \, d \cos \left (d x + c\right )^{4} - 4 \, d \cos \left (d x + c\right )^{2} + d\right )} \sin \left (d x + c\right )} \] Input:
integrate(cot(d*x+c)^6*csc(d*x+c)^4*(a+b*sin(d*x+c))^2,x, algorithm="frica s")
Output:
1/8064*(128*(2*a^2 + 9*b^2)*cos(d*x + c)^9 - 1152*(a^2 + b^2)*cos(d*x + c) ^7 + 315*(a*b*cos(d*x + c)^8 - 4*a*b*cos(d*x + c)^6 + 6*a*b*cos(d*x + c)^4 - 4*a*b*cos(d*x + c)^2 + a*b)*log(1/2*cos(d*x + c) + 1/2)*sin(d*x + c) - 315*(a*b*cos(d*x + c)^8 - 4*a*b*cos(d*x + c)^6 + 6*a*b*cos(d*x + c)^4 - 4* a*b*cos(d*x + c)^2 + a*b)*log(-1/2*cos(d*x + c) + 1/2)*sin(d*x + c) - 42*( 15*a*b*cos(d*x + c)^7 + 73*a*b*cos(d*x + c)^5 - 55*a*b*cos(d*x + c)^3 + 15 *a*b*cos(d*x + c))*sin(d*x + c))/((d*cos(d*x + c)^8 - 4*d*cos(d*x + c)^6 + 6*d*cos(d*x + c)^4 - 4*d*cos(d*x + c)^2 + d)*sin(d*x + c))
Timed out. \[ \int \cot ^6(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2 \, dx=\text {Timed out} \] Input:
integrate(cot(d*x+c)**6*csc(d*x+c)**4*(a+b*sin(d*x+c))**2,x)
Output:
Timed out
Time = 0.04 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.02 \[ \int \cot ^6(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2 \, dx=-\frac {21 \, a b {\left (\frac {2 \, {\left (15 \, \cos \left (d x + c\right )^{7} + 73 \, \cos \left (d x + c\right )^{5} - 55 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{8} - 4 \, \cos \left (d x + c\right )^{6} + 6 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} + 1} - 15 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac {1152 \, b^{2}}{\tan \left (d x + c\right )^{7}} + \frac {128 \, {\left (9 \, \tan \left (d x + c\right )^{2} + 7\right )} a^{2}}{\tan \left (d x + c\right )^{9}}}{8064 \, d} \] Input:
integrate(cot(d*x+c)^6*csc(d*x+c)^4*(a+b*sin(d*x+c))^2,x, algorithm="maxim a")
Output:
-1/8064*(21*a*b*(2*(15*cos(d*x + c)^7 + 73*cos(d*x + c)^5 - 55*cos(d*x + c )^3 + 15*cos(d*x + c))/(cos(d*x + c)^8 - 4*cos(d*x + c)^6 + 6*cos(d*x + c) ^4 - 4*cos(d*x + c)^2 + 1) - 15*log(cos(d*x + c) + 1) + 15*log(cos(d*x + c ) - 1)) + 1152*b^2/tan(d*x + c)^7 + 128*(9*tan(d*x + c)^2 + 7)*a^2/tan(d*x + c)^9)/d
Leaf count of result is larger than twice the leaf count of optimal. 408 vs. \(2 (137) = 274\).
Time = 0.30 (sec) , antiderivative size = 408, normalized size of antiderivative = 2.70 \[ \int \cot ^6(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {14 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 63 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 54 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 72 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 336 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 504 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 504 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 336 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1512 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1008 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 5040 \, a b \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 756 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2520 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {14258 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 756 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 2520 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 1008 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 336 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 1512 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 504 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 504 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 336 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 54 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 72 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 63 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 14 \, a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9}}}{64512 \, d} \] Input:
integrate(cot(d*x+c)^6*csc(d*x+c)^4*(a+b*sin(d*x+c))^2,x, algorithm="giac" )
Output:
1/64512*(14*a^2*tan(1/2*d*x + 1/2*c)^9 + 63*a*b*tan(1/2*d*x + 1/2*c)^8 - 5 4*a^2*tan(1/2*d*x + 1/2*c)^7 + 72*b^2*tan(1/2*d*x + 1/2*c)^7 - 336*a*b*tan (1/2*d*x + 1/2*c)^6 - 504*b^2*tan(1/2*d*x + 1/2*c)^5 + 504*a*b*tan(1/2*d*x + 1/2*c)^4 + 336*a^2*tan(1/2*d*x + 1/2*c)^3 + 1512*b^2*tan(1/2*d*x + 1/2* c)^3 + 1008*a*b*tan(1/2*d*x + 1/2*c)^2 - 5040*a*b*log(abs(tan(1/2*d*x + 1/ 2*c))) - 756*a^2*tan(1/2*d*x + 1/2*c) - 2520*b^2*tan(1/2*d*x + 1/2*c) + (1 4258*a*b*tan(1/2*d*x + 1/2*c)^9 + 756*a^2*tan(1/2*d*x + 1/2*c)^8 + 2520*b^ 2*tan(1/2*d*x + 1/2*c)^8 - 1008*a*b*tan(1/2*d*x + 1/2*c)^7 - 336*a^2*tan(1 /2*d*x + 1/2*c)^6 - 1512*b^2*tan(1/2*d*x + 1/2*c)^6 - 504*a*b*tan(1/2*d*x + 1/2*c)^5 + 504*b^2*tan(1/2*d*x + 1/2*c)^4 + 336*a*b*tan(1/2*d*x + 1/2*c) ^3 + 54*a^2*tan(1/2*d*x + 1/2*c)^2 - 72*b^2*tan(1/2*d*x + 1/2*c)^2 - 63*a* b*tan(1/2*d*x + 1/2*c) - 14*a^2)/tan(1/2*d*x + 1/2*c)^9)/d
Time = 22.78 (sec) , antiderivative size = 373, normalized size of antiderivative = 2.47 \[ \int \cot ^6(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{4608\,d}-\frac {b^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{128\,d}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {3\,a^2}{256}+\frac {5\,b^2}{128}\right )}{d}-\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {8\,a^2}{3}+12\,b^2\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {3\,a^2}{7}-\frac {4\,b^2}{7}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (6\,a^2+20\,b^2\right )-4\,b^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\frac {a^2}{9}-\frac {8\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3}+4\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+8\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\frac {a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2}\right )}{512\,d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {a^2}{192}+\frac {3\,b^2}{128}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (\frac {3\,a^2}{3584}-\frac {b^2}{896}\right )}{d}+\frac {a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{64\,d}+\frac {a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{128\,d}-\frac {a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{192\,d}+\frac {a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{1024\,d}-\frac {5\,a\,b\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{64\,d} \] Input:
int((cot(c + d*x)^6*(a + b*sin(c + d*x))^2)/sin(c + d*x)^4,x)
Output:
(a^2*tan(c/2 + (d*x)/2)^9)/(4608*d) - (b^2*tan(c/2 + (d*x)/2)^5)/(128*d) - (tan(c/2 + (d*x)/2)*((3*a^2)/256 + (5*b^2)/128))/d - (cot(c/2 + (d*x)/2)^ 9*(tan(c/2 + (d*x)/2)^6*((8*a^2)/3 + 12*b^2) - tan(c/2 + (d*x)/2)^2*((3*a^ 2)/7 - (4*b^2)/7) - tan(c/2 + (d*x)/2)^8*(6*a^2 + 20*b^2) - 4*b^2*tan(c/2 + (d*x)/2)^4 + a^2/9 - (8*a*b*tan(c/2 + (d*x)/2)^3)/3 + 4*a*b*tan(c/2 + (d *x)/2)^5 + 8*a*b*tan(c/2 + (d*x)/2)^7 + (a*b*tan(c/2 + (d*x)/2))/2))/(512* d) + (tan(c/2 + (d*x)/2)^3*(a^2/192 + (3*b^2)/128))/d - (tan(c/2 + (d*x)/2 )^7*((3*a^2)/3584 - b^2/896))/d + (a*b*tan(c/2 + (d*x)/2)^2)/(64*d) + (a*b *tan(c/2 + (d*x)/2)^4)/(128*d) - (a*b*tan(c/2 + (d*x)/2)^6)/(192*d) + (a*b *tan(c/2 + (d*x)/2)^8)/(1024*d) - (5*a*b*log(tan(c/2 + (d*x)/2)))/(64*d)
Time = 0.18 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.78 \[ \int \cot ^6(c+d x) \csc ^4(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {128 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{8} a^{2}+576 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{8} b^{2}+315 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{7} a b +64 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6} a^{2}-1728 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6} b^{2}-2478 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5} a b -960 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} a^{2}+1728 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} b^{2}+2856 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} a b +1216 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} a^{2}-576 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} b^{2}-1008 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a b -448 \cos \left (d x +c \right ) a^{2}-315 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sin \left (d x +c \right )^{9} a b}{4032 \sin \left (d x +c \right )^{9} d} \] Input:
int(cot(d*x+c)^6*csc(d*x+c)^4*(a+b*sin(d*x+c))^2,x)
Output:
(128*cos(c + d*x)*sin(c + d*x)**8*a**2 + 576*cos(c + d*x)*sin(c + d*x)**8* b**2 + 315*cos(c + d*x)*sin(c + d*x)**7*a*b + 64*cos(c + d*x)*sin(c + d*x) **6*a**2 - 1728*cos(c + d*x)*sin(c + d*x)**6*b**2 - 2478*cos(c + d*x)*sin( c + d*x)**5*a*b - 960*cos(c + d*x)*sin(c + d*x)**4*a**2 + 1728*cos(c + d*x )*sin(c + d*x)**4*b**2 + 2856*cos(c + d*x)*sin(c + d*x)**3*a*b + 1216*cos( c + d*x)*sin(c + d*x)**2*a**2 - 576*cos(c + d*x)*sin(c + d*x)**2*b**2 - 10 08*cos(c + d*x)*sin(c + d*x)*a*b - 448*cos(c + d*x)*a**2 - 315*log(tan((c + d*x)/2))*sin(c + d*x)**9*a*b)/(4032*sin(c + d*x)**9*d)