Integrand size = 21, antiderivative size = 236 \[ \int \sec ^{10}(c+d x) (a+b \sin (c+d x))^8 \, dx=\frac {128 a b \left (a^2-b^2\right )^3 \sec (c+d x)}{315 d}+\frac {64 a \left (a^2-b^2\right )^2 \sec ^3(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{315 d}+\frac {16 a \left (a^2-b^2\right ) \sec ^5(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^4}{105 d}+\frac {\sec ^9(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{9 d}+\frac {\sec ^7(c+d x) (a+b \sin (c+d x))^6 \left (a b+\left (8 a^2-7 b^2\right ) \sin (c+d x)\right )}{63 d}+\frac {128 a^2 \left (a^2-b^2\right )^3 \tan (c+d x)}{315 d} \] Output:
128/315*a*b*(a^2-b^2)^3*sec(d*x+c)/d+64/315*a*(a^2-b^2)^2*sec(d*x+c)^3*(b+ a*sin(d*x+c))*(a+b*sin(d*x+c))^2/d+16/105*a*(a^2-b^2)*sec(d*x+c)^5*(b+a*si n(d*x+c))*(a+b*sin(d*x+c))^4/d+1/9*sec(d*x+c)^9*(b+a*sin(d*x+c))*(a+b*sin( d*x+c))^7/d+1/63*sec(d*x+c)^7*(a+b*sin(d*x+c))^6*(a*b+(8*a^2-7*b^2)*sin(d* x+c))/d+128/315*a^2*(a^2-b^2)^3*tan(d*x+c)/d
Time = 4.80 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.33 \[ \int \sec ^{10}(c+d x) (a+b \sin (c+d x))^8 \, dx=\frac {\cos (c+d x) \left (-\sec ^{10}(c+d x) (a+b \sin (c+d x))^9+\frac {a \left (35 (a+b \sin (c+d x))^8+8 (a-b) (1-\sin (c+d x)) \left (5 (a+b \sin (c+d x))^7+(a-b) (1-\sin (c+d x)) \left (7 (a+b \sin (c+d x))^6+2 (a-b) (1-\sin (c+d x)) \left (7 (a+b \sin (c+d x))^5+(a-b) (1-\sin (c+d x)) \left (35 (a+b \sin (c+d x))^4-4 (a-b) (1-\sin (c+d x)) \left (5 (a+b \sin (c+d x))^3+(a+b) (1+\sin (c+d x)) \left (7 a^2+6 a b+2 b^2+6 \left (a^2+3 a b+b^2\right ) \sin (c+d x)+\left (2 a^2+6 a b+7 b^2\right ) \sin ^2(c+d x)\right )\right )\right )\right )\right )\right )\right )}{35 (1-\sin (c+d x))^5 (1+\sin (c+d x))^4}\right )}{9 (a-b) d} \] Input:
Integrate[Sec[c + d*x]^10*(a + b*Sin[c + d*x])^8,x]
Output:
(Cos[c + d*x]*(-(Sec[c + d*x]^10*(a + b*Sin[c + d*x])^9) + (a*(35*(a + b*S in[c + d*x])^8 + 8*(a - b)*(1 - Sin[c + d*x])*(5*(a + b*Sin[c + d*x])^7 + (a - b)*(1 - Sin[c + d*x])*(7*(a + b*Sin[c + d*x])^6 + 2*(a - b)*(1 - Sin[ c + d*x])*(7*(a + b*Sin[c + d*x])^5 + (a - b)*(1 - Sin[c + d*x])*(35*(a + b*Sin[c + d*x])^4 - 4*(a - b)*(1 - Sin[c + d*x])*(5*(a + b*Sin[c + d*x])^3 + (a + b)*(1 + Sin[c + d*x])*(7*a^2 + 6*a*b + 2*b^2 + 6*(a^2 + 3*a*b + b^ 2)*Sin[c + d*x] + (2*a^2 + 6*a*b + 7*b^2)*Sin[c + d*x]^2))))))))/(35*(1 - Sin[c + d*x])^5*(1 + Sin[c + d*x])^4)))/(9*(a - b)*d)
Time = 1.16 (sec) , antiderivative size = 231, normalized size of antiderivative = 0.98, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.810, Rules used = {3042, 3170, 25, 3042, 3340, 27, 3042, 3170, 27, 3042, 3170, 27, 3042, 3148, 3042, 4254, 24}
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.
| \(\displaystyle \int \sec ^{10}(c+d x) (a+b \sin (c+d x))^8 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+b \sin (c+d x))^8}{\cos (c+d x)^{10}}dx\) |
| \(\Big \downarrow \) 3170 |
| \(\displaystyle \frac {\sec ^9(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{9 d}-\frac {1}{9} \int -\sec ^8(c+d x) (a+b \sin (c+d x))^6 \left (8 a^2+b \sin (c+d x) a-7 b^2\right )dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{9} \int \sec ^8(c+d x) (a+b \sin (c+d x))^6 \left (8 a^2+b \sin (c+d x) a-7 b^2\right )dx+\frac {\sec ^9(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \int \frac {(a+b \sin (c+d x))^6 \left (8 a^2+b \sin (c+d x) a-7 b^2\right )}{\cos (c+d x)^8}dx+\frac {\sec ^9(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{9 d}\) |
| \(\Big \downarrow \) 3340 |
| \(\displaystyle \frac {1}{9} \left (\frac {\sec ^7(c+d x) (a+b \sin (c+d x))^6 \left (\left (8 a^2-7 b^2\right ) \sin (c+d x)+a b\right )}{7 d}-\frac {1}{7} \int -48 a \left (a^2-b^2\right ) \sec ^6(c+d x) (a+b \sin (c+d x))^5dx\right )+\frac {\sec ^9(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{9 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \left (\frac {48}{7} a \left (a^2-b^2\right ) \int \sec ^6(c+d x) (a+b \sin (c+d x))^5dx+\frac {\sec ^7(c+d x) (a+b \sin (c+d x))^6 \left (\left (8 a^2-7 b^2\right ) \sin (c+d x)+a b\right )}{7 d}\right )+\frac {\sec ^9(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {48}{7} a \left (a^2-b^2\right ) \int \frac {(a+b \sin (c+d x))^5}{\cos (c+d x)^6}dx+\frac {\sec ^7(c+d x) (a+b \sin (c+d x))^6 \left (\left (8 a^2-7 b^2\right ) \sin (c+d x)+a b\right )}{7 d}\right )+\frac {\sec ^9(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{9 d}\) |
| \(\Big \downarrow \) 3170 |
| \(\displaystyle \frac {1}{9} \left (\frac {48}{7} a \left (a^2-b^2\right ) \left (\frac {\sec ^5(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^4}{5 d}-\frac {1}{5} \int -4 \left (a^2-b^2\right ) \sec ^4(c+d x) (a+b \sin (c+d x))^3dx\right )+\frac {\sec ^7(c+d x) (a+b \sin (c+d x))^6 \left (\left (8 a^2-7 b^2\right ) \sin (c+d x)+a b\right )}{7 d}\right )+\frac {\sec ^9(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{9 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \left (\frac {48}{7} a \left (a^2-b^2\right ) \left (\frac {4}{5} \left (a^2-b^2\right ) \int \sec ^4(c+d x) (a+b \sin (c+d x))^3dx+\frac {\sec ^5(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^4}{5 d}\right )+\frac {\sec ^7(c+d x) (a+b \sin (c+d x))^6 \left (\left (8 a^2-7 b^2\right ) \sin (c+d x)+a b\right )}{7 d}\right )+\frac {\sec ^9(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {48}{7} a \left (a^2-b^2\right ) \left (\frac {4}{5} \left (a^2-b^2\right ) \int \frac {(a+b \sin (c+d x))^3}{\cos (c+d x)^4}dx+\frac {\sec ^5(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^4}{5 d}\right )+\frac {\sec ^7(c+d x) (a+b \sin (c+d x))^6 \left (\left (8 a^2-7 b^2\right ) \sin (c+d x)+a b\right )}{7 d}\right )+\frac {\sec ^9(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{9 d}\) |
| \(\Big \downarrow \) 3170 |
| \(\displaystyle \frac {1}{9} \left (\frac {48}{7} a \left (a^2-b^2\right ) \left (\frac {4}{5} \left (a^2-b^2\right ) \left (\frac {\sec ^3(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^2}{3 d}-\frac {1}{3} \int -2 \left (a^2-b^2\right ) \sec ^2(c+d x) (a+b \sin (c+d x))dx\right )+\frac {\sec ^5(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^4}{5 d}\right )+\frac {\sec ^7(c+d x) (a+b \sin (c+d x))^6 \left (\left (8 a^2-7 b^2\right ) \sin (c+d x)+a b\right )}{7 d}\right )+\frac {\sec ^9(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{9 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \left (\frac {48}{7} a \left (a^2-b^2\right ) \left (\frac {4}{5} \left (a^2-b^2\right ) \left (\frac {2}{3} \left (a^2-b^2\right ) \int \sec ^2(c+d x) (a+b \sin (c+d x))dx+\frac {\sec ^3(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^2}{3 d}\right )+\frac {\sec ^5(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^4}{5 d}\right )+\frac {\sec ^7(c+d x) (a+b \sin (c+d x))^6 \left (\left (8 a^2-7 b^2\right ) \sin (c+d x)+a b\right )}{7 d}\right )+\frac {\sec ^9(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {48}{7} a \left (a^2-b^2\right ) \left (\frac {4}{5} \left (a^2-b^2\right ) \left (\frac {2}{3} \left (a^2-b^2\right ) \int \frac {a+b \sin (c+d x)}{\cos (c+d x)^2}dx+\frac {\sec ^3(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^2}{3 d}\right )+\frac {\sec ^5(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^4}{5 d}\right )+\frac {\sec ^7(c+d x) (a+b \sin (c+d x))^6 \left (\left (8 a^2-7 b^2\right ) \sin (c+d x)+a b\right )}{7 d}\right )+\frac {\sec ^9(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{9 d}\) |
| \(\Big \downarrow \) 3148 |
| \(\displaystyle \frac {1}{9} \left (\frac {48}{7} a \left (a^2-b^2\right ) \left (\frac {4}{5} \left (a^2-b^2\right ) \left (\frac {2}{3} \left (a^2-b^2\right ) \left (a \int \sec ^2(c+d x)dx+\frac {b \sec (c+d x)}{d}\right )+\frac {\sec ^3(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^2}{3 d}\right )+\frac {\sec ^5(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^4}{5 d}\right )+\frac {\sec ^7(c+d x) (a+b \sin (c+d x))^6 \left (\left (8 a^2-7 b^2\right ) \sin (c+d x)+a b\right )}{7 d}\right )+\frac {\sec ^9(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{9} \left (\frac {48}{7} a \left (a^2-b^2\right ) \left (\frac {4}{5} \left (a^2-b^2\right ) \left (\frac {2}{3} \left (a^2-b^2\right ) \left (a \int \csc \left (c+d x+\frac {\pi }{2}\right )^2dx+\frac {b \sec (c+d x)}{d}\right )+\frac {\sec ^3(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^2}{3 d}\right )+\frac {\sec ^5(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^4}{5 d}\right )+\frac {\sec ^7(c+d x) (a+b \sin (c+d x))^6 \left (\left (8 a^2-7 b^2\right ) \sin (c+d x)+a b\right )}{7 d}\right )+\frac {\sec ^9(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{9 d}\) |
| \(\Big \downarrow \) 4254 |
| \(\displaystyle \frac {1}{9} \left (\frac {48}{7} a \left (a^2-b^2\right ) \left (\frac {4}{5} \left (a^2-b^2\right ) \left (\frac {2}{3} \left (a^2-b^2\right ) \left (\frac {b \sec (c+d x)}{d}-\frac {a \int 1d(-\tan (c+d x))}{d}\right )+\frac {\sec ^3(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^2}{3 d}\right )+\frac {\sec ^5(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^4}{5 d}\right )+\frac {\sec ^7(c+d x) (a+b \sin (c+d x))^6 \left (\left (8 a^2-7 b^2\right ) \sin (c+d x)+a b\right )}{7 d}\right )+\frac {\sec ^9(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{9 d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {1}{9} \left (\frac {\sec ^7(c+d x) (a+b \sin (c+d x))^6 \left (\left (8 a^2-7 b^2\right ) \sin (c+d x)+a b\right )}{7 d}+\frac {48}{7} a \left (a^2-b^2\right ) \left (\frac {4}{5} \left (a^2-b^2\right ) \left (\frac {2}{3} \left (a^2-b^2\right ) \left (\frac {a \tan (c+d x)}{d}+\frac {b \sec (c+d x)}{d}\right )+\frac {\sec ^3(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^2}{3 d}\right )+\frac {\sec ^5(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^4}{5 d}\right )\right )+\frac {\sec ^9(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{9 d}\) |
Input:
Int[Sec[c + d*x]^10*(a + b*Sin[c + d*x])^8,x]
Output:
(Sec[c + d*x]^9*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x])^7)/(9*d) + ((Sec [c + d*x]^7*(a + b*Sin[c + d*x])^6*(a*b + (8*a^2 - 7*b^2)*Sin[c + d*x]))/( 7*d) + (48*a*(a^2 - b^2)*((Sec[c + d*x]^5*(b + a*Sin[c + d*x])*(a + b*Sin[ c + d*x])^4)/(5*d) + (4*(a^2 - b^2)*((Sec[c + d*x]^3*(b + a*Sin[c + d*x])* (a + b*Sin[c + d*x])^2)/(3*d) + (2*(a^2 - b^2)*((b*Sec[c + d*x])/d + (a*Ta n[c + d*x])/d))/3))/5))/7)/9
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[(-b)*((g*Cos[e + f*x])^(p + 1)/(f*g*(p + 1))), x] + Simp[a Int[(g*Cos[e + f*x])^p, x], x] /; FreeQ[{a, b, e, f, g, p}, x] && (IntegerQ[2*p] || NeQ[a^2 - b^2, 0])
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[(-(g*Cos[e + f*x])^(p + 1))*(a + b*Sin[e + f*x ])^(m - 1)*((b + a*Sin[e + f*x])/(f*g*(p + 1))), x] + Simp[1/(g^2*(p + 1)) Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^(m - 2)*(b^2*(m - 1) + a^2*(p + 2) + a*b*(m + p + 1)*Sin[e + f*x]), x], x] /; FreeQ[{a, b, e, f, g }, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 1] && LtQ[p, -1] && (IntegersQ[2*m, 2* p] || IntegerQ[m])
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-(g* Cos[e + f*x])^(p + 1))*(a + b*Sin[e + f*x])^m*((d + c*Sin[e + f*x])/(f*g*(p + 1))), x] + Simp[1/(g^2*(p + 1)) Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Si n[e + f*x])^(m - 1)*Simp[a*c*(p + 2) + b*d*m + b*c*(m + p + 2)*Sin[e + f*x] , x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[a^2 - b^2, 0] && GtQ [m, 0] && LtQ[p, -1] && IntegerQ[2*m] && !(EqQ[m, 1] && NeQ[c^2 - d^2, 0] && SimplerQ[c + d*x, a + b*x])
Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp andIntegrand[(1 + x^2)^(n/2 - 1), x], x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(586\) vs. \(2(224)=448\).
Time = 3.77 (sec) , antiderivative size = 587, normalized size of antiderivative = 2.49
| method | result | size |
| parallelrisch | \(\frac {-630 a^{8} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{17}-5040 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{16} a^{7} b +\left (1680 a^{8}-23520 a^{6} b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{15}-70560 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14} a^{5} b^{3}+\left (-9576 a^{8}-28224 a^{6} b^{2}-141120 a^{4} b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}-47040 a^{3} \left (a^{4}+\frac {5}{2} b^{2} a^{2}+4 b^{4}\right ) b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}+\left (10224 a^{8}-159264 a^{6} b^{2}-241920 a^{4} b^{4}-161280 b^{6} a^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}+\left (-352800 a^{5} b^{3}-282240 a^{3} b^{5}-80640 a \,b^{7}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+\left (-21316 a^{8}-79744 a^{6} b^{2}-488320 a^{4} b^{4}-179200 b^{6} a^{2}-17920 b^{8}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}-70560 a \left (a^{6}+3 a^{4} b^{2}+\frac {28}{5} a^{2} b^{4}+\frac {24}{35} b^{6}\right ) b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+\left (10224 a^{8}-159264 a^{6} b^{2}-241920 a^{4} b^{4}-161280 b^{6} a^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}+\left (-211680 a^{5} b^{3}-112896 a^{3} b^{5}-21504 a \,b^{7}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\left (-9576 a^{8}-28224 a^{6} b^{2}-141120 a^{4} b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+\left (-20160 a^{7} b -30240 a^{5} b^{3}-32256 a^{3} b^{5}+9216 a \,b^{7}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (1680 a^{8}-23520 a^{6} b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (-10080 a^{5} b^{3}+8064 a^{3} b^{5}-2304 a \,b^{7}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-630 a^{8} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-560 a^{7} b +1120 a^{5} b^{3}-896 a^{3} b^{5}+256 a \,b^{7}}{315 d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{9}}\) | \(587\) |
| derivativedivides | \(\frac {-a^{8} \left (-\frac {128}{315}-\frac {\sec \left (d x +c \right )^{8}}{9}-\frac {8 \sec \left (d x +c \right )^{6}}{63}-\frac {16 \sec \left (d x +c \right )^{4}}{105}-\frac {64 \sec \left (d x +c \right )^{2}}{315}\right ) \tan \left (d x +c \right )+\frac {8 a^{7} b}{9 \cos \left (d x +c \right )^{9}}+28 a^{6} b^{2} \left (\frac {\sin \left (d x +c \right )^{3}}{9 \cos \left (d x +c \right )^{9}}+\frac {2 \sin \left (d x +c \right )^{3}}{21 \cos \left (d x +c \right )^{7}}+\frac {8 \sin \left (d x +c \right )^{3}}{105 \cos \left (d x +c \right )^{5}}+\frac {16 \sin \left (d x +c \right )^{3}}{315 \cos \left (d x +c \right )^{3}}\right )+56 a^{5} b^{3} \left (\frac {\sin \left (d x +c \right )^{4}}{9 \cos \left (d x +c \right )^{9}}+\frac {5 \sin \left (d x +c \right )^{4}}{63 \cos \left (d x +c \right )^{7}}+\frac {\sin \left (d x +c \right )^{4}}{21 \cos \left (d x +c \right )^{5}}+\frac {\sin \left (d x +c \right )^{4}}{63 \cos \left (d x +c \right )^{3}}-\frac {\sin \left (d x +c \right )^{4}}{63 \cos \left (d x +c \right )}-\frac {\left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )}{63}\right )+70 a^{4} b^{4} \left (\frac {\sin \left (d x +c \right )^{5}}{9 \cos \left (d x +c \right )^{9}}+\frac {4 \sin \left (d x +c \right )^{5}}{63 \cos \left (d x +c \right )^{7}}+\frac {8 \sin \left (d x +c \right )^{5}}{315 \cos \left (d x +c \right )^{5}}\right )+56 a^{3} b^{5} \left (\frac {\sin \left (d x +c \right )^{6}}{9 \cos \left (d x +c \right )^{9}}+\frac {\sin \left (d x +c \right )^{6}}{21 \cos \left (d x +c \right )^{7}}+\frac {\sin \left (d x +c \right )^{6}}{105 \cos \left (d x +c \right )^{5}}-\frac {\sin \left (d x +c \right )^{6}}{315 \cos \left (d x +c \right )^{3}}+\frac {\sin \left (d x +c \right )^{6}}{105 \cos \left (d x +c \right )}+\frac {\left (\frac {8}{3}+\sin \left (d x +c \right )^{4}+\frac {4 \sin \left (d x +c \right )^{2}}{3}\right ) \cos \left (d x +c \right )}{105}\right )+28 b^{6} a^{2} \left (\frac {\sin \left (d x +c \right )^{7}}{9 \cos \left (d x +c \right )^{9}}+\frac {2 \sin \left (d x +c \right )^{7}}{63 \cos \left (d x +c \right )^{7}}\right )+8 a \,b^{7} \left (\frac {\sin \left (d x +c \right )^{8}}{9 \cos \left (d x +c \right )^{9}}+\frac {\sin \left (d x +c \right )^{8}}{63 \cos \left (d x +c \right )^{7}}-\frac {\sin \left (d x +c \right )^{8}}{315 \cos \left (d x +c \right )^{5}}+\frac {\sin \left (d x +c \right )^{8}}{315 \cos \left (d x +c \right )^{3}}-\frac {\sin \left (d x +c \right )^{8}}{63 \cos \left (d x +c \right )}-\frac {\left (\frac {16}{5}+\sin \left (d x +c \right )^{6}+\frac {6 \sin \left (d x +c \right )^{4}}{5}+\frac {8 \sin \left (d x +c \right )^{2}}{5}\right ) \cos \left (d x +c \right )}{63}\right )+\frac {b^{8} \sin \left (d x +c \right )^{9}}{9 \cos \left (d x +c \right )^{9}}}{d}\) | \(662\) |
| default | \(\frac {-a^{8} \left (-\frac {128}{315}-\frac {\sec \left (d x +c \right )^{8}}{9}-\frac {8 \sec \left (d x +c \right )^{6}}{63}-\frac {16 \sec \left (d x +c \right )^{4}}{105}-\frac {64 \sec \left (d x +c \right )^{2}}{315}\right ) \tan \left (d x +c \right )+\frac {8 a^{7} b}{9 \cos \left (d x +c \right )^{9}}+28 a^{6} b^{2} \left (\frac {\sin \left (d x +c \right )^{3}}{9 \cos \left (d x +c \right )^{9}}+\frac {2 \sin \left (d x +c \right )^{3}}{21 \cos \left (d x +c \right )^{7}}+\frac {8 \sin \left (d x +c \right )^{3}}{105 \cos \left (d x +c \right )^{5}}+\frac {16 \sin \left (d x +c \right )^{3}}{315 \cos \left (d x +c \right )^{3}}\right )+56 a^{5} b^{3} \left (\frac {\sin \left (d x +c \right )^{4}}{9 \cos \left (d x +c \right )^{9}}+\frac {5 \sin \left (d x +c \right )^{4}}{63 \cos \left (d x +c \right )^{7}}+\frac {\sin \left (d x +c \right )^{4}}{21 \cos \left (d x +c \right )^{5}}+\frac {\sin \left (d x +c \right )^{4}}{63 \cos \left (d x +c \right )^{3}}-\frac {\sin \left (d x +c \right )^{4}}{63 \cos \left (d x +c \right )}-\frac {\left (2+\sin \left (d x +c \right )^{2}\right ) \cos \left (d x +c \right )}{63}\right )+70 a^{4} b^{4} \left (\frac {\sin \left (d x +c \right )^{5}}{9 \cos \left (d x +c \right )^{9}}+\frac {4 \sin \left (d x +c \right )^{5}}{63 \cos \left (d x +c \right )^{7}}+\frac {8 \sin \left (d x +c \right )^{5}}{315 \cos \left (d x +c \right )^{5}}\right )+56 a^{3} b^{5} \left (\frac {\sin \left (d x +c \right )^{6}}{9 \cos \left (d x +c \right )^{9}}+\frac {\sin \left (d x +c \right )^{6}}{21 \cos \left (d x +c \right )^{7}}+\frac {\sin \left (d x +c \right )^{6}}{105 \cos \left (d x +c \right )^{5}}-\frac {\sin \left (d x +c \right )^{6}}{315 \cos \left (d x +c \right )^{3}}+\frac {\sin \left (d x +c \right )^{6}}{105 \cos \left (d x +c \right )}+\frac {\left (\frac {8}{3}+\sin \left (d x +c \right )^{4}+\frac {4 \sin \left (d x +c \right )^{2}}{3}\right ) \cos \left (d x +c \right )}{105}\right )+28 b^{6} a^{2} \left (\frac {\sin \left (d x +c \right )^{7}}{9 \cos \left (d x +c \right )^{9}}+\frac {2 \sin \left (d x +c \right )^{7}}{63 \cos \left (d x +c \right )^{7}}\right )+8 a \,b^{7} \left (\frac {\sin \left (d x +c \right )^{8}}{9 \cos \left (d x +c \right )^{9}}+\frac {\sin \left (d x +c \right )^{8}}{63 \cos \left (d x +c \right )^{7}}-\frac {\sin \left (d x +c \right )^{8}}{315 \cos \left (d x +c \right )^{5}}+\frac {\sin \left (d x +c \right )^{8}}{315 \cos \left (d x +c \right )^{3}}-\frac {\sin \left (d x +c \right )^{8}}{63 \cos \left (d x +c \right )}-\frac {\left (\frac {16}{5}+\sin \left (d x +c \right )^{6}+\frac {6 \sin \left (d x +c \right )^{4}}{5}+\frac {8 \sin \left (d x +c \right )^{2}}{5}\right ) \cos \left (d x +c \right )}{63}\right )+\frac {b^{8} \sin \left (d x +c \right )^{9}}{9 \cos \left (d x +c \right )^{9}}}{d}\) | \(662\) |
| risch | \(\frac {2 i \left (-280 b^{6} a^{2}-448 a^{6} b^{2}+560 a^{4} b^{4}+35 b^{8}+128 a^{8}+96768 i a^{3} b^{5} {\mathrm e}^{11 i \left (d x +c \right )}+3360 i a \,b^{7} {\mathrm e}^{15 i \left (d x +c \right )}+3360 i a \,b^{7} {\mathrm e}^{3 i \left (d x +c \right )}-56448 i a^{3} b^{5} {\mathrm e}^{13 i \left (d x +c \right )}-4032 i a \,b^{7} {\mathrm e}^{13 i \left (d x +c \right )}+161280 i a^{5} b^{3} {\mathrm e}^{7 i \left (d x +c \right )}-71680 i a^{7} b \,{\mathrm e}^{9 i \left (d x +c \right )}+96768 i a^{3} b^{5} {\mathrm e}^{7 i \left (d x +c \right )}+22752 i a \,b^{7} {\mathrm e}^{7 i \left (d x +c \right )}-4032 i a \,b^{7} {\mathrm e}^{5 i \left (d x +c \right )}+22752 i a \,b^{7} {\mathrm e}^{11 i \left (d x +c \right )}-179200 i a^{5} b^{3} {\mathrm e}^{9 i \left (d x +c \right )}-195328 i a^{3} b^{5} {\mathrm e}^{9 i \left (d x +c \right )}-11392 i a \,b^{7} {\mathrm e}^{9 i \left (d x +c \right )}+161280 i a^{5} b^{3} {\mathrm e}^{11 i \left (d x +c \right )}-56448 i a^{3} b^{5} {\mathrm e}^{5 i \left (d x +c \right )}+246960 a^{4} b^{4} {\mathrm e}^{8 i \left (d x +c \right )}+52920 a^{2} b^{6} {\mathrm e}^{8 i \left (d x +c \right )}-16128 a^{6} b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-2520 a^{2} b^{6} {\mathrm e}^{2 i \left (d x +c \right )}-141120 a^{6} b^{2} {\mathrm e}^{10 i \left (d x +c \right )}-176400 a^{4} b^{4} {\mathrm e}^{10 i \left (d x +c \right )}-52920 a^{2} b^{6} {\mathrm e}^{6 i \left (d x +c \right )}-17640 a^{2} b^{6} {\mathrm e}^{14 i \left (d x +c \right )}-88200 a^{2} b^{6} {\mathrm e}^{10 i \left (d x +c \right )}+84672 a^{6} b^{2} {\mathrm e}^{8 i \left (d x +c \right )}-70560 a^{4} b^{4} {\mathrm e}^{6 i \left (d x +c \right )}+29400 a^{2} b^{6} {\mathrm e}^{12 i \left (d x +c \right )}+117600 a^{4} b^{4} {\mathrm e}^{12 i \left (d x +c \right )}+7560 a^{2} b^{6} {\mathrm e}^{4 i \left (d x +c \right )}+5040 a^{4} b^{4} {\mathrm e}^{2 i \left (d x +c \right )}-4032 a^{6} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-37632 a^{6} b^{2} {\mathrm e}^{6 i \left (d x +c \right )}+20160 a^{4} b^{4} {\mathrm e}^{4 i \left (d x +c \right )}+1152 a^{8} {\mathrm e}^{2 i \left (d x +c \right )}+10752 a^{8} {\mathrm e}^{6 i \left (d x +c \right )}+2940 b^{8} {\mathrm e}^{12 i \left (d x +c \right )}+315 b^{8} {\mathrm e}^{16 i \left (d x +c \right )}+4410 b^{8} {\mathrm e}^{8 i \left (d x +c \right )}+1260 b^{8} {\mathrm e}^{4 i \left (d x +c \right )}+4608 a^{8} {\mathrm e}^{4 i \left (d x +c \right )}+16128 \,{\mathrm e}^{8 i \left (d x +c \right )} a^{8}\right )}{315 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{9}}\) | \(745\) |
Input:
int(sec(d*x+c)^10*(a+b*sin(d*x+c))^8,x,method=_RETURNVERBOSE)
Output:
1/315*(-630*a^8*tan(1/2*d*x+1/2*c)^17-5040*tan(1/2*d*x+1/2*c)^16*a^7*b+(16 80*a^8-23520*a^6*b^2)*tan(1/2*d*x+1/2*c)^15-70560*tan(1/2*d*x+1/2*c)^14*a^ 5*b^3+(-9576*a^8-28224*a^6*b^2-141120*a^4*b^4)*tan(1/2*d*x+1/2*c)^13-47040 *a^3*(a^4+5/2*b^2*a^2+4*b^4)*b*tan(1/2*d*x+1/2*c)^12+(10224*a^8-159264*a^6 *b^2-241920*a^4*b^4-161280*a^2*b^6)*tan(1/2*d*x+1/2*c)^11+(-352800*a^5*b^3 -282240*a^3*b^5-80640*a*b^7)*tan(1/2*d*x+1/2*c)^10+(-21316*a^8-79744*a^6*b ^2-488320*a^4*b^4-179200*a^2*b^6-17920*b^8)*tan(1/2*d*x+1/2*c)^9-70560*a*( a^6+3*a^4*b^2+28/5*a^2*b^4+24/35*b^6)*b*tan(1/2*d*x+1/2*c)^8+(10224*a^8-15 9264*a^6*b^2-241920*a^4*b^4-161280*a^2*b^6)*tan(1/2*d*x+1/2*c)^7+(-211680* a^5*b^3-112896*a^3*b^5-21504*a*b^7)*tan(1/2*d*x+1/2*c)^6+(-9576*a^8-28224* a^6*b^2-141120*a^4*b^4)*tan(1/2*d*x+1/2*c)^5+(-20160*a^7*b-30240*a^5*b^3-3 2256*a^3*b^5+9216*a*b^7)*tan(1/2*d*x+1/2*c)^4+(1680*a^8-23520*a^6*b^2)*tan (1/2*d*x+1/2*c)^3+(-10080*a^5*b^3+8064*a^3*b^5-2304*a*b^7)*tan(1/2*d*x+1/2 *c)^2-630*a^8*tan(1/2*d*x+1/2*c)-560*a^7*b+1120*a^5*b^3-896*a^3*b^5+256*a* b^7)/d/(tan(1/2*d*x+1/2*c)^2-1)^9
Time = 0.11 (sec) , antiderivative size = 336, normalized size of antiderivative = 1.42 \[ \int \sec ^{10}(c+d x) (a+b \sin (c+d x))^8 \, dx=-\frac {840 \, a b^{7} \cos \left (d x + c\right )^{6} - 280 \, a^{7} b - 1960 \, a^{5} b^{3} - 1960 \, a^{3} b^{5} - 280 \, a b^{7} - 504 \, {\left (7 \, a^{3} b^{5} + 3 \, a b^{7}\right )} \cos \left (d x + c\right )^{4} + 360 \, {\left (7 \, a^{5} b^{3} + 14 \, a^{3} b^{5} + 3 \, a b^{7}\right )} \cos \left (d x + c\right )^{2} - {\left ({\left (128 \, a^{8} - 448 \, a^{6} b^{2} + 560 \, a^{4} b^{4} - 280 \, a^{2} b^{6} + 35 \, b^{8}\right )} \cos \left (d x + c\right )^{8} + 35 \, a^{8} + 980 \, a^{6} b^{2} + 2450 \, a^{4} b^{4} + 980 \, a^{2} b^{6} + 35 \, b^{8} + 4 \, {\left (16 \, a^{8} - 56 \, a^{6} b^{2} + 70 \, a^{4} b^{4} - 35 \, a^{2} b^{6} - 35 \, b^{8}\right )} \cos \left (d x + c\right )^{6} + 6 \, {\left (8 \, a^{8} - 28 \, a^{6} b^{2} + 35 \, a^{4} b^{4} + 350 \, a^{2} b^{6} + 35 \, b^{8}\right )} \cos \left (d x + c\right )^{4} + 20 \, {\left (2 \, a^{8} - 7 \, a^{6} b^{2} - 175 \, a^{4} b^{4} - 133 \, a^{2} b^{6} - 7 \, b^{8}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{315 \, d \cos \left (d x + c\right )^{9}} \] Input:
integrate(sec(d*x+c)^10*(a+b*sin(d*x+c))^8,x, algorithm="fricas")
Output:
-1/315*(840*a*b^7*cos(d*x + c)^6 - 280*a^7*b - 1960*a^5*b^3 - 1960*a^3*b^5 - 280*a*b^7 - 504*(7*a^3*b^5 + 3*a*b^7)*cos(d*x + c)^4 + 360*(7*a^5*b^3 + 14*a^3*b^5 + 3*a*b^7)*cos(d*x + c)^2 - ((128*a^8 - 448*a^6*b^2 + 560*a^4* b^4 - 280*a^2*b^6 + 35*b^8)*cos(d*x + c)^8 + 35*a^8 + 980*a^6*b^2 + 2450*a ^4*b^4 + 980*a^2*b^6 + 35*b^8 + 4*(16*a^8 - 56*a^6*b^2 + 70*a^4*b^4 - 35*a ^2*b^6 - 35*b^8)*cos(d*x + c)^6 + 6*(8*a^8 - 28*a^6*b^2 + 35*a^4*b^4 + 350 *a^2*b^6 + 35*b^8)*cos(d*x + c)^4 + 20*(2*a^8 - 7*a^6*b^2 - 175*a^4*b^4 - 133*a^2*b^6 - 7*b^8)*cos(d*x + c)^2)*sin(d*x + c))/(d*cos(d*x + c)^9)
Timed out. \[ \int \sec ^{10}(c+d x) (a+b \sin (c+d x))^8 \, dx=\text {Timed out} \] Input:
integrate(sec(d*x+c)**10*(a+b*sin(d*x+c))**8,x)
Output:
Timed out
Time = 0.04 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.33 \[ \int \sec ^{10}(c+d x) (a+b \sin (c+d x))^8 \, dx=\frac {35 \, b^{8} \tan \left (d x + c\right )^{9} + {\left (35 \, \tan \left (d x + c\right )^{9} + 180 \, \tan \left (d x + c\right )^{7} + 378 \, \tan \left (d x + c\right )^{5} + 420 \, \tan \left (d x + c\right )^{3} + 315 \, \tan \left (d x + c\right )\right )} a^{8} + 28 \, {\left (35 \, \tan \left (d x + c\right )^{9} + 135 \, \tan \left (d x + c\right )^{7} + 189 \, \tan \left (d x + c\right )^{5} + 105 \, \tan \left (d x + c\right )^{3}\right )} a^{6} b^{2} + 70 \, {\left (35 \, \tan \left (d x + c\right )^{9} + 90 \, \tan \left (d x + c\right )^{7} + 63 \, \tan \left (d x + c\right )^{5}\right )} a^{4} b^{4} + 140 \, {\left (7 \, \tan \left (d x + c\right )^{9} + 9 \, \tan \left (d x + c\right )^{7}\right )} a^{2} b^{6} - \frac {280 \, {\left (9 \, \cos \left (d x + c\right )^{2} - 7\right )} a^{5} b^{3}}{\cos \left (d x + c\right )^{9}} + \frac {56 \, {\left (63 \, \cos \left (d x + c\right )^{4} - 90 \, \cos \left (d x + c\right )^{2} + 35\right )} a^{3} b^{5}}{\cos \left (d x + c\right )^{9}} - \frac {8 \, {\left (105 \, \cos \left (d x + c\right )^{6} - 189 \, \cos \left (d x + c\right )^{4} + 135 \, \cos \left (d x + c\right )^{2} - 35\right )} a b^{7}}{\cos \left (d x + c\right )^{9}} + \frac {280 \, a^{7} b}{\cos \left (d x + c\right )^{9}}}{315 \, d} \] Input:
integrate(sec(d*x+c)^10*(a+b*sin(d*x+c))^8,x, algorithm="maxima")
Output:
1/315*(35*b^8*tan(d*x + c)^9 + (35*tan(d*x + c)^9 + 180*tan(d*x + c)^7 + 3 78*tan(d*x + c)^5 + 420*tan(d*x + c)^3 + 315*tan(d*x + c))*a^8 + 28*(35*ta n(d*x + c)^9 + 135*tan(d*x + c)^7 + 189*tan(d*x + c)^5 + 105*tan(d*x + c)^ 3)*a^6*b^2 + 70*(35*tan(d*x + c)^9 + 90*tan(d*x + c)^7 + 63*tan(d*x + c)^5 )*a^4*b^4 + 140*(7*tan(d*x + c)^9 + 9*tan(d*x + c)^7)*a^2*b^6 - 280*(9*cos (d*x + c)^2 - 7)*a^5*b^3/cos(d*x + c)^9 + 56*(63*cos(d*x + c)^4 - 90*cos(d *x + c)^2 + 35)*a^3*b^5/cos(d*x + c)^9 - 8*(105*cos(d*x + c)^6 - 189*cos(d *x + c)^4 + 135*cos(d*x + c)^2 - 35)*a*b^7/cos(d*x + c)^9 + 280*a^7*b/cos( d*x + c)^9)/d
Leaf count of result is larger than twice the leaf count of optimal. 892 vs. \(2 (224) = 448\).
Time = 0.24 (sec) , antiderivative size = 892, normalized size of antiderivative = 3.78 \[ \int \sec ^{10}(c+d x) (a+b \sin (c+d x))^8 \, dx=\text {Too large to display} \] Input:
integrate(sec(d*x+c)^10*(a+b*sin(d*x+c))^8,x, algorithm="giac")
Output:
-2/315*(315*a^8*tan(1/2*d*x + 1/2*c)^17 + 2520*a^7*b*tan(1/2*d*x + 1/2*c)^ 16 - 840*a^8*tan(1/2*d*x + 1/2*c)^15 + 11760*a^6*b^2*tan(1/2*d*x + 1/2*c)^ 15 + 35280*a^5*b^3*tan(1/2*d*x + 1/2*c)^14 + 4788*a^8*tan(1/2*d*x + 1/2*c) ^13 + 14112*a^6*b^2*tan(1/2*d*x + 1/2*c)^13 + 70560*a^4*b^4*tan(1/2*d*x + 1/2*c)^13 + 23520*a^7*b*tan(1/2*d*x + 1/2*c)^12 + 58800*a^5*b^3*tan(1/2*d* x + 1/2*c)^12 + 94080*a^3*b^5*tan(1/2*d*x + 1/2*c)^12 - 5112*a^8*tan(1/2*d *x + 1/2*c)^11 + 79632*a^6*b^2*tan(1/2*d*x + 1/2*c)^11 + 120960*a^4*b^4*ta n(1/2*d*x + 1/2*c)^11 + 80640*a^2*b^6*tan(1/2*d*x + 1/2*c)^11 + 176400*a^5 *b^3*tan(1/2*d*x + 1/2*c)^10 + 141120*a^3*b^5*tan(1/2*d*x + 1/2*c)^10 + 40 320*a*b^7*tan(1/2*d*x + 1/2*c)^10 + 10658*a^8*tan(1/2*d*x + 1/2*c)^9 + 398 72*a^6*b^2*tan(1/2*d*x + 1/2*c)^9 + 244160*a^4*b^4*tan(1/2*d*x + 1/2*c)^9 + 89600*a^2*b^6*tan(1/2*d*x + 1/2*c)^9 + 8960*b^8*tan(1/2*d*x + 1/2*c)^9 + 35280*a^7*b*tan(1/2*d*x + 1/2*c)^8 + 105840*a^5*b^3*tan(1/2*d*x + 1/2*c)^ 8 + 197568*a^3*b^5*tan(1/2*d*x + 1/2*c)^8 + 24192*a*b^7*tan(1/2*d*x + 1/2* c)^8 - 5112*a^8*tan(1/2*d*x + 1/2*c)^7 + 79632*a^6*b^2*tan(1/2*d*x + 1/2*c )^7 + 120960*a^4*b^4*tan(1/2*d*x + 1/2*c)^7 + 80640*a^2*b^6*tan(1/2*d*x + 1/2*c)^7 + 105840*a^5*b^3*tan(1/2*d*x + 1/2*c)^6 + 56448*a^3*b^5*tan(1/2*d *x + 1/2*c)^6 + 10752*a*b^7*tan(1/2*d*x + 1/2*c)^6 + 4788*a^8*tan(1/2*d*x + 1/2*c)^5 + 14112*a^6*b^2*tan(1/2*d*x + 1/2*c)^5 + 70560*a^4*b^4*tan(1/2* d*x + 1/2*c)^5 + 10080*a^7*b*tan(1/2*d*x + 1/2*c)^4 + 15120*a^5*b^3*tan...
Time = 16.54 (sec) , antiderivative size = 659, normalized size of antiderivative = 2.79 \[ \int \sec ^{10}(c+d x) (a+b \sin (c+d x))^8 \, dx =\text {Too large to display} \] Input:
int((a + b*sin(c + d*x))^8/cos(c + d*x)^10,x)
Output:
(a - b)^8/(2*d*(tan(c/2 + (d*x)/2) + 1)^8) - (a + b)^8/(9*d*(tan(c/2 + (d* x)/2) - 1)^9) - (a + b)^8/(2*d*(tan(c/2 + (d*x)/2) - 1)^8) - (a - b)^8/(9* d*(tan(c/2 + (d*x)/2) + 1)^9) - ((a + b)^7*(37*a + 21*b))/(28*d*(tan(c/2 + (d*x)/2) - 1)^7) - ((a + b)^7*(55*a + 7*b))/(24*d*(tan(c/2 + (d*x)/2) - 1 )^6) + ((a - b)^5*(65*a*b^2 + 191*a^2*b + 187*a^3 + 5*b^3))/(128*d*(tan(c/ 2 + (d*x)/2) + 1)^2) + ((a - b)^5*(67*a*b^2 - 67*a^2*b - 463*a^3 + 15*b^3) )/(192*d*(tan(c/2 + (d*x)/2) + 1)^3) + ((a - b)^6*(18*a*b + 95*a^2 - b^2)) /(32*d*(tan(c/2 + (d*x)/2) + 1)^4) + ((a - b)^6*(114*a*b - 241*a^2 + 15*b^ 2))/(80*d*(tan(c/2 + (d*x)/2) + 1)^5) - ((a - b)^7*(37*a - 21*b))/(28*d*(t an(c/2 + (d*x)/2) + 1)^7) + ((a - b)^7*(55*a - 7*b))/(24*d*(tan(c/2 + (d*x )/2) + 1)^6) + ((a + b)^6*(18*a*b - 95*a^2 + b^2))/(32*d*(tan(c/2 + (d*x)/ 2) - 1)^4) - ((a + b)^5*(65*a*b^2 - 191*a^2*b + 187*a^3 - 5*b^3))/(128*d*( tan(c/2 + (d*x)/2) - 1)^2) + ((a + b)^5*(67*a*b^2 + 67*a^2*b - 463*a^3 - 1 5*b^3))/(192*d*(tan(c/2 + (d*x)/2) - 1)^3) - ((a + b)^6*(114*a*b + 241*a^2 - 15*b^2))/(80*d*(tan(c/2 + (d*x)/2) - 1)^5) - (a*(a + b)^4*(20*a*b^2 - 2 9*a^2*b + 16*a^3 - 5*b^3))/(16*d*(tan(c/2 + (d*x)/2) - 1)) - (a*(a - b)^4* (20*a*b^2 + 29*a^2*b + 16*a^3 + 5*b^3))/(16*d*(tan(c/2 + (d*x)/2) + 1))
Time = 0.19 (sec) , antiderivative size = 782, normalized size of antiderivative = 3.31 \[ \int \sec ^{10}(c+d x) (a+b \sin (c+d x))^8 \, dx =\text {Too large to display} \] Input:
int(sec(d*x+c)^10*(a+b*sin(d*x+c))^8,x)
Output:
( - 280*cos(c + d*x)*sin(c + d*x)**8*a**7*b + 560*cos(c + d*x)*sin(c + d*x )**8*a**5*b**3 - 448*cos(c + d*x)*sin(c + d*x)**8*a**3*b**5 + 128*cos(c + d*x)*sin(c + d*x)**8*a*b**7 + 1120*cos(c + d*x)*sin(c + d*x)**6*a**7*b - 2 240*cos(c + d*x)*sin(c + d*x)**6*a**5*b**3 + 1792*cos(c + d*x)*sin(c + d*x )**6*a**3*b**5 - 512*cos(c + d*x)*sin(c + d*x)**6*a*b**7 - 1680*cos(c + d* x)*sin(c + d*x)**4*a**7*b + 3360*cos(c + d*x)*sin(c + d*x)**4*a**5*b**3 - 2688*cos(c + d*x)*sin(c + d*x)**4*a**3*b**5 + 768*cos(c + d*x)*sin(c + d*x )**4*a*b**7 + 1120*cos(c + d*x)*sin(c + d*x)**2*a**7*b - 2240*cos(c + d*x) *sin(c + d*x)**2*a**5*b**3 + 1792*cos(c + d*x)*sin(c + d*x)**2*a**3*b**5 - 512*cos(c + d*x)*sin(c + d*x)**2*a*b**7 - 280*cos(c + d*x)*a**7*b + 560*c os(c + d*x)*a**5*b**3 - 448*cos(c + d*x)*a**3*b**5 + 128*cos(c + d*x)*a*b* *7 + 128*sin(c + d*x)**9*a**8 - 448*sin(c + d*x)**9*a**6*b**2 + 560*sin(c + d*x)**9*a**4*b**4 - 280*sin(c + d*x)**9*a**2*b**6 + 35*sin(c + d*x)**9*b **8 - 576*sin(c + d*x)**7*a**8 + 2016*sin(c + d*x)**7*a**6*b**2 - 2520*sin (c + d*x)**7*a**4*b**4 + 1260*sin(c + d*x)**7*a**2*b**6 + 840*sin(c + d*x) **6*a*b**7 + 1008*sin(c + d*x)**5*a**8 - 3528*sin(c + d*x)**5*a**6*b**2 + 4410*sin(c + d*x)**5*a**4*b**4 + 3528*sin(c + d*x)**4*a**3*b**5 - 1008*sin (c + d*x)**4*a*b**7 - 840*sin(c + d*x)**3*a**8 + 2940*sin(c + d*x)**3*a**6 *b**2 + 2520*sin(c + d*x)**2*a**5*b**3 - 2016*sin(c + d*x)**2*a**3*b**5 + 576*sin(c + d*x)**2*a*b**7 + 315*sin(c + d*x)*a**8 + 280*a**7*b - 560*a...