Integrand size = 19, antiderivative size = 385 \[ \int \frac {\sec (c+d x)}{(a+b \sin (c+d x))^8} \, dx=-\frac {\log (1-\sin (c+d x))}{2 (a+b)^8 d}+\frac {\log (1+\sin (c+d x))}{2 (a-b)^8 d}-\frac {8 a b \left (a^2+b^2\right ) \left (a^4+6 a^2 b^2+b^4\right ) \log (a+b \sin (c+d x))}{\left (a^2-b^2\right )^8 d}+\frac {b}{7 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^7}+\frac {a b}{3 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^6}+\frac {b \left (3 a^2+b^2\right )}{5 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^5}+\frac {a b \left (a^2+b^2\right )}{\left (a^2-b^2\right )^4 d (a+b \sin (c+d x))^4}+\frac {b \left (5 a^4+10 a^2 b^2+b^4\right )}{3 \left (a^2-b^2\right )^5 d (a+b \sin (c+d x))^3}+\frac {a b \left (3 a^2+b^2\right ) \left (a^2+3 b^2\right )}{\left (a^2-b^2\right )^6 d (a+b \sin (c+d x))^2}+\frac {b \left (7 a^6+35 a^4 b^2+21 a^2 b^4+b^6\right )}{\left (a^2-b^2\right )^7 d (a+b \sin (c+d x))} \]
-1/2*ln(1-sin(d*x+c))/(a+b)^8/d+1/2*ln(1+sin(d*x+c))/(a-b)^8/d-8*a*b*(a^2+ b^2)*(a^4+6*a^2*b^2+b^4)*ln(a+b*sin(d*x+c))/(a^2-b^2)^8/d+1/7*b/(a^2-b^2)/ d/(a+b*sin(d*x+c))^7+1/3*a*b/(a^2-b^2)^2/d/(a+b*sin(d*x+c))^6+1/5*b*(3*a^2 +b^2)/(a^2-b^2)^3/d/(a+b*sin(d*x+c))^5+a*b*(a^2+b^2)/(a^2-b^2)^4/d/(a+b*si n(d*x+c))^4+1/3*b*(5*a^4+10*a^2*b^2+b^4)/(a^2-b^2)^5/d/(a+b*sin(d*x+c))^3+ a*b*(3*a^4+10*a^2*b^2+3*b^4)/(a^2-b^2)^6/d/(a+b*sin(d*x+c))^2+b*(7*a^6+35* a^4*b^2+21*a^2*b^4+b^6)/(a^2-b^2)^7/d/(a+b*sin(d*x+c))
Time = 1.67 (sec) , antiderivative size = 365, normalized size of antiderivative = 0.95 \[ \int \frac {\sec (c+d x)}{(a+b \sin (c+d x))^8} \, dx=\frac {b \left (-\frac {\log (1-\sin (c+d x))}{2 b (a+b)^8}+\frac {\log (1+\sin (c+d x))}{2 (a-b)^8 b}-\frac {8 a \left (a^2+b^2\right ) \left (a^4+6 a^2 b^2+b^4\right ) \log (a+b \sin (c+d x))}{(a-b)^8 (a+b)^8}+\frac {1}{7 \left (a^2-b^2\right ) (a+b \sin (c+d x))^7}+\frac {a}{3 (a-b)^2 (a+b)^2 (a+b \sin (c+d x))^6}+\frac {3 a^2+b^2}{5 (a-b)^3 (a+b)^3 (a+b \sin (c+d x))^5}+\frac {a \left (a^2+b^2\right )}{(a-b)^4 (a+b)^4 (a+b \sin (c+d x))^4}+\frac {5 a^4+10 a^2 b^2+b^4}{3 (a-b)^5 (a+b)^5 (a+b \sin (c+d x))^3}+\frac {a \left (3 a^2+b^2\right ) \left (a^2+3 b^2\right )}{(a-b)^6 (a+b)^6 (a+b \sin (c+d x))^2}+\frac {7 a^6+35 a^4 b^2+21 a^2 b^4+b^6}{(a-b)^7 (a+b)^7 (a+b \sin (c+d x))}\right )}{d} \]
(b*(-1/2*Log[1 - Sin[c + d*x]]/(b*(a + b)^8) + Log[1 + Sin[c + d*x]]/(2*(a - b)^8*b) - (8*a*(a^2 + b^2)*(a^4 + 6*a^2*b^2 + b^4)*Log[a + b*Sin[c + d* x]])/((a - b)^8*(a + b)^8) + 1/(7*(a^2 - b^2)*(a + b*Sin[c + d*x])^7) + a/ (3*(a - b)^2*(a + b)^2*(a + b*Sin[c + d*x])^6) + (3*a^2 + b^2)/(5*(a - b)^ 3*(a + b)^3*(a + b*Sin[c + d*x])^5) + (a*(a^2 + b^2))/((a - b)^4*(a + b)^4 *(a + b*Sin[c + d*x])^4) + (5*a^4 + 10*a^2*b^2 + b^4)/(3*(a - b)^5*(a + b) ^5*(a + b*Sin[c + d*x])^3) + (a*(3*a^2 + b^2)*(a^2 + 3*b^2))/((a - b)^6*(a + b)^6*(a + b*Sin[c + d*x])^2) + (7*a^6 + 35*a^4*b^2 + 21*a^2*b^4 + b^6)/ ((a - b)^7*(a + b)^7*(a + b*Sin[c + d*x]))))/d
Time = 0.79 (sec) , antiderivative size = 383, normalized size of antiderivative = 0.99, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {3042, 3147, 477, 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.
| \(\displaystyle \int \frac {\sec (c+d x)}{(a+b \sin (c+d x))^8} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\cos (c+d x) (a+b \sin (c+d x))^8}dx\) |
| \(\Big \downarrow \) 3147 |
| \(\displaystyle \frac {b \int \frac {1}{(a+b \sin (c+d x))^8 \left (b^2-b^2 \sin ^2(c+d x)\right )}d(b \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 477 |
| \(\displaystyle \frac {\int \left (-\frac {8 a \left (a^2+b^2\right ) \left (a^4+6 b^2 a^2+b^4\right ) b^2}{\left (a^2-b^2\right )^8 (a+b \sin (c+d x))}-\frac {\left (7 a^6+35 b^2 a^4+21 b^4 a^2+b^6\right ) b^2}{\left (a^2-b^2\right )^7 (a+b \sin (c+d x))^2}-\frac {2 a \left (3 a^2+b^2\right ) \left (a^2+3 b^2\right ) b^2}{\left (a^2-b^2\right )^6 (a+b \sin (c+d x))^3}-\frac {\left (5 a^4+10 b^2 a^2+b^4\right ) b^2}{\left (a^2-b^2\right )^5 (a+b \sin (c+d x))^4}-\frac {4 a \left (a^2+b^2\right ) b^2}{\left (a^2-b^2\right )^4 (a+b \sin (c+d x))^5}-\frac {\left (3 a^2+b^2\right ) b^2}{\left (a^2-b^2\right )^3 (a+b \sin (c+d x))^6}-\frac {2 a b^2}{\left (a^2-b^2\right )^2 (a+b \sin (c+d x))^7}-\frac {b^2}{\left (a^2-b^2\right ) (a+b \sin (c+d x))^8}+\frac {b}{2 (a+b)^8 (b-b \sin (c+d x))}+\frac {b}{2 (a-b)^8 (\sin (c+d x) b+b)}\right )d(b \sin (c+d x))}{b d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {a b^2 \left (3 a^2+b^2\right ) \left (a^2+3 b^2\right )}{\left (a^2-b^2\right )^6 (a+b \sin (c+d x))^2}+\frac {a b^2 \left (a^2+b^2\right )}{\left (a^2-b^2\right )^4 (a+b \sin (c+d x))^4}+\frac {b^2 \left (3 a^2+b^2\right )}{5 \left (a^2-b^2\right )^3 (a+b \sin (c+d x))^5}+\frac {a b^2}{3 \left (a^2-b^2\right )^2 (a+b \sin (c+d x))^6}+\frac {b^2}{7 \left (a^2-b^2\right ) (a+b \sin (c+d x))^7}+\frac {b^2 \left (5 a^4+10 a^2 b^2+b^4\right )}{3 \left (a^2-b^2\right )^5 (a+b \sin (c+d x))^3}-\frac {8 a b^2 \left (a^2+b^2\right ) \left (a^4+6 a^2 b^2+b^4\right ) \log (a+b \sin (c+d x))}{\left (a^2-b^2\right )^8}+\frac {b^2 \left (7 a^6+35 a^4 b^2+21 a^2 b^4+b^6\right )}{\left (a^2-b^2\right )^7 (a+b \sin (c+d x))}-\frac {b \log (b-b \sin (c+d x))}{2 (a+b)^8}+\frac {b \log (b \sin (c+d x)+b)}{2 (a-b)^8}}{b d}\) |
(-1/2*(b*Log[b - b*Sin[c + d*x]])/(a + b)^8 - (8*a*b^2*(a^2 + b^2)*(a^4 + 6*a^2*b^2 + b^4)*Log[a + b*Sin[c + d*x]])/(a^2 - b^2)^8 + (b*Log[b + b*Sin [c + d*x]])/(2*(a - b)^8) + b^2/(7*(a^2 - b^2)*(a + b*Sin[c + d*x])^7) + ( a*b^2)/(3*(a^2 - b^2)^2*(a + b*Sin[c + d*x])^6) + (b^2*(3*a^2 + b^2))/(5*( a^2 - b^2)^3*(a + b*Sin[c + d*x])^5) + (a*b^2*(a^2 + b^2))/((a^2 - b^2)^4* (a + b*Sin[c + d*x])^4) + (b^2*(5*a^4 + 10*a^2*b^2 + b^4))/(3*(a^2 - b^2)^ 5*(a + b*Sin[c + d*x])^3) + (a*b^2*(3*a^2 + b^2)*(a^2 + 3*b^2))/((a^2 - b^ 2)^6*(a + b*Sin[c + d*x])^2) + (b^2*(7*a^6 + 35*a^4*b^2 + 21*a^2*b^4 + b^6 ))/((a^2 - b^2)^7*(a + b*Sin[c + d*x])))/(b*d)
3.5.65.3.1 Defintions of rubi rules used
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ a^p Int[ExpandIntegrand[(c + d*x)^n*(1 - Rt[-b/a, 2]*x)^p*(1 + Rt[-b/a, 2 ]*x)^p, x], x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[p, 0] && IntegerQ[n] & & NiceSqrtQ[-b/a] && !FractionalPowerFactorQ[Rt[-b/a, 2]]
Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m _.), x_Symbol] :> Simp[1/(b^p*f) Subst[Int[(a + x)^m*(b^2 - x^2)^((p - 1) /2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && IntegerQ[(p - 1)/2] && NeQ[a^2 - b^2, 0]
Time = 19.49 (sec) , antiderivative size = 356, normalized size of antiderivative = 0.92
| method | result | size |
| derivativedivides | \(\frac {\frac {\ln \left (1+\sin \left (d x +c \right )\right )}{2 \left (a -b \right )^{8}}+\frac {b}{7 \left (a -b \right ) \left (a +b \right ) \left (a +b \sin \left (d x +c \right )\right )^{7}}+\frac {a b}{3 \left (a +b \right )^{2} \left (a -b \right )^{2} \left (a +b \sin \left (d x +c \right )\right )^{6}}+\frac {b \left (3 a^{2}+b^{2}\right )}{5 \left (a -b \right )^{3} \left (a +b \right )^{3} \left (a +b \sin \left (d x +c \right )\right )^{5}}+\frac {b \left (5 a^{4}+10 a^{2} b^{2}+b^{4}\right )}{3 \left (a +b \right )^{5} \left (a -b \right )^{5} \left (a +b \sin \left (d x +c \right )\right )^{3}}+\frac {b \left (7 a^{6}+35 a^{4} b^{2}+21 a^{2} b^{4}+b^{6}\right )}{\left (a -b \right )^{7} \left (a +b \right )^{7} \left (a +b \sin \left (d x +c \right )\right )}+\frac {b a \left (a^{2}+b^{2}\right )}{\left (a +b \right )^{4} \left (a -b \right )^{4} \left (a +b \sin \left (d x +c \right )\right )^{4}}+\frac {b a \left (3 a^{4}+10 a^{2} b^{2}+3 b^{4}\right )}{\left (a +b \right )^{6} \left (a -b \right )^{6} \left (a +b \sin \left (d x +c \right )\right )^{2}}-\frac {8 b a \left (a^{6}+7 a^{4} b^{2}+7 a^{2} b^{4}+b^{6}\right ) \ln \left (a +b \sin \left (d x +c \right )\right )}{\left (a +b \right )^{8} \left (a -b \right )^{8}}-\frac {\ln \left (\sin \left (d x +c \right )-1\right )}{2 \left (a +b \right )^{8}}}{d}\) | \(356\) |
| default | \(\frac {\frac {\ln \left (1+\sin \left (d x +c \right )\right )}{2 \left (a -b \right )^{8}}+\frac {b}{7 \left (a -b \right ) \left (a +b \right ) \left (a +b \sin \left (d x +c \right )\right )^{7}}+\frac {a b}{3 \left (a +b \right )^{2} \left (a -b \right )^{2} \left (a +b \sin \left (d x +c \right )\right )^{6}}+\frac {b \left (3 a^{2}+b^{2}\right )}{5 \left (a -b \right )^{3} \left (a +b \right )^{3} \left (a +b \sin \left (d x +c \right )\right )^{5}}+\frac {b \left (5 a^{4}+10 a^{2} b^{2}+b^{4}\right )}{3 \left (a +b \right )^{5} \left (a -b \right )^{5} \left (a +b \sin \left (d x +c \right )\right )^{3}}+\frac {b \left (7 a^{6}+35 a^{4} b^{2}+21 a^{2} b^{4}+b^{6}\right )}{\left (a -b \right )^{7} \left (a +b \right )^{7} \left (a +b \sin \left (d x +c \right )\right )}+\frac {b a \left (a^{2}+b^{2}\right )}{\left (a +b \right )^{4} \left (a -b \right )^{4} \left (a +b \sin \left (d x +c \right )\right )^{4}}+\frac {b a \left (3 a^{4}+10 a^{2} b^{2}+3 b^{4}\right )}{\left (a +b \right )^{6} \left (a -b \right )^{6} \left (a +b \sin \left (d x +c \right )\right )^{2}}-\frac {8 b a \left (a^{6}+7 a^{4} b^{2}+7 a^{2} b^{4}+b^{6}\right ) \ln \left (a +b \sin \left (d x +c \right )\right )}{\left (a +b \right )^{8} \left (a -b \right )^{8}}-\frac {\ln \left (\sin \left (d x +c \right )-1\right )}{2 \left (a +b \right )^{8}}}{d}\) | \(356\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(1529\) |
| norman | \(\text {Expression too large to display}\) | \(2136\) |
| risch | \(\text {Expression too large to display}\) | \(2620\) |
1/d*(1/2/(a-b)^8*ln(1+sin(d*x+c))+1/7*b/(a-b)/(a+b)/(a+b*sin(d*x+c))^7+1/3 *a*b/(a+b)^2/(a-b)^2/(a+b*sin(d*x+c))^6+1/5*b*(3*a^2+b^2)/(a-b)^3/(a+b)^3/ (a+b*sin(d*x+c))^5+1/3*b*(5*a^4+10*a^2*b^2+b^4)/(a+b)^5/(a-b)^5/(a+b*sin(d *x+c))^3+b*(7*a^6+35*a^4*b^2+21*a^2*b^4+b^6)/(a-b)^7/(a+b)^7/(a+b*sin(d*x+ c))+b*a*(a^2+b^2)/(a+b)^4/(a-b)^4/(a+b*sin(d*x+c))^4+b*a*(3*a^4+10*a^2*b^2 +3*b^4)/(a+b)^6/(a-b)^6/(a+b*sin(d*x+c))^2-8*b*a*(a^6+7*a^4*b^2+7*a^2*b^4+ b^6)/(a+b)^8/(a-b)^8*ln(a+b*sin(d*x+c))-1/2/(a+b)^8*ln(sin(d*x+c)-1))
Leaf count of result is larger than twice the leaf count of optimal. 3165 vs. \(2 (374) = 748\).
Time = 1.74 (sec) , antiderivative size = 3165, normalized size of antiderivative = 8.22 \[ \int \frac {\sec (c+d x)}{(a+b \sin (c+d x))^8} \, dx=\text {Too large to display} \]
-1/210*(2886*a^14*b + 35728*a^12*b^3 + 113862*a^10*b^5 + 11760*a^8*b^7 - 9 7230*a^6*b^9 - 62496*a^4*b^11 - 4158*a^2*b^13 - 352*b^15 - 210*(7*a^8*b^7 + 28*a^6*b^9 - 14*a^4*b^11 - 20*a^2*b^13 - b^15)*cos(d*x + c)^6 + 70*(365* a^10*b^5 + 1378*a^8*b^7 - 602*a^6*b^9 - 944*a^4*b^11 - 187*a^2*b^13 - 10*b ^15)*cos(d*x + c)^4 - 14*(2229*a^12*b^3 + 10223*a^10*b^5 + 7960*a^8*b^7 - 10490*a^6*b^9 - 8915*a^4*b^11 - 949*a^2*b^13 - 58*b^15)*cos(d*x + c)^2 - 1 680*(a^14*b + 28*a^12*b^3 + 189*a^10*b^5 + 400*a^8*b^7 + 315*a^6*b^9 + 84* a^4*b^11 + 7*a^2*b^13 - 7*(a^8*b^7 + 7*a^6*b^9 + 7*a^4*b^11 + a^2*b^13)*co s(d*x + c)^6 + 7*(5*a^10*b^5 + 38*a^8*b^7 + 56*a^6*b^9 + 26*a^4*b^11 + 3*a ^2*b^13)*cos(d*x + c)^4 - 7*(3*a^12*b^3 + 31*a^10*b^5 + 94*a^8*b^7 + 94*a^ 6*b^9 + 31*a^4*b^11 + 3*a^2*b^13)*cos(d*x + c)^2 + (7*a^13*b^2 + 84*a^11*b ^4 + 315*a^9*b^6 + 400*a^7*b^8 + 189*a^5*b^10 + 28*a^3*b^12 + a*b^14 - (a^ 7*b^8 + 7*a^5*b^10 + 7*a^3*b^12 + a*b^14)*cos(d*x + c)^6 + 3*(7*a^9*b^6 + 50*a^7*b^8 + 56*a^5*b^10 + 14*a^3*b^12 + a*b^14)*cos(d*x + c)^4 - (35*a^11 *b^4 + 287*a^9*b^6 + 542*a^7*b^8 + 350*a^5*b^10 + 63*a^3*b^12 + 3*a*b^14)* cos(d*x + c)^2)*sin(d*x + c))*log(b*sin(d*x + c) + a) + 105*(a^15 + 8*a^14 *b + 49*a^13*b^2 + 224*a^12*b^3 + 693*a^11*b^4 + 1512*a^10*b^5 + 2485*a^9* b^6 + 3200*a^8*b^7 + 3235*a^7*b^8 + 2520*a^6*b^9 + 1491*a^5*b^10 + 672*a^4 *b^11 + 231*a^3*b^12 + 56*a^2*b^13 + 7*a*b^14 - 7*(a^9*b^6 + 8*a^8*b^7 + 2 8*a^7*b^8 + 56*a^6*b^9 + 70*a^5*b^10 + 56*a^4*b^11 + 28*a^3*b^12 + 8*a^...
Timed out. \[ \int \frac {\sec (c+d x)}{(a+b \sin (c+d x))^8} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 1160 vs. \(2 (374) = 748\).
Time = 0.25 (sec) , antiderivative size = 1160, normalized size of antiderivative = 3.01 \[ \int \frac {\sec (c+d x)}{(a+b \sin (c+d x))^8} \, dx=\text {Too large to display} \]
-1/210*(1680*(a^7*b + 7*a^5*b^3 + 7*a^3*b^5 + a*b^7)*log(b*sin(d*x + c) + a)/(a^16 - 8*a^14*b^2 + 28*a^12*b^4 - 56*a^10*b^6 + 70*a^8*b^8 - 56*a^6*b^ 10 + 28*a^4*b^12 - 8*a^2*b^14 + b^16) - 2*(1443*a^12*b + 3704*a^10*b^3 + 1 849*a^8*b^5 - 496*a^6*b^7 + 309*a^4*b^9 - 104*a^2*b^11 + 15*b^13 + 105*(7* a^6*b^7 + 35*a^4*b^9 + 21*a^2*b^11 + b^13)*sin(d*x + c)^6 + 105*(45*a^7*b^ 6 + 217*a^5*b^8 + 119*a^3*b^10 + 3*a*b^12)*sin(d*x + c)^5 + 35*(365*a^8*b^ 5 + 1680*a^6*b^7 + 826*a^4*b^9 + 8*a^2*b^11 + b^13)*sin(d*x + c)^4 + 35*(5 33*a^9*b^4 + 2304*a^7*b^6 + 994*a^5*b^8 + 8*a^3*b^10 + a*b^12)*sin(d*x + c )^3 + 21*(743*a^10*b^3 + 2934*a^8*b^5 + 1099*a^6*b^7 + 29*a^4*b^9 - 6*a^2* b^11 + b^13)*sin(d*x + c)^2 + 7*(1023*a^11*b^2 + 3494*a^9*b^4 + 1219*a^7*b ^6 + 29*a^5*b^8 - 6*a^3*b^10 + a*b^12)*sin(d*x + c))/(a^21 - 7*a^19*b^2 + 21*a^17*b^4 - 35*a^15*b^6 + 35*a^13*b^8 - 21*a^11*b^10 + 7*a^9*b^12 - a^7* b^14 + (a^14*b^7 - 7*a^12*b^9 + 21*a^10*b^11 - 35*a^8*b^13 + 35*a^6*b^15 - 21*a^4*b^17 + 7*a^2*b^19 - b^21)*sin(d*x + c)^7 + 7*(a^15*b^6 - 7*a^13*b^ 8 + 21*a^11*b^10 - 35*a^9*b^12 + 35*a^7*b^14 - 21*a^5*b^16 + 7*a^3*b^18 - a*b^20)*sin(d*x + c)^6 + 21*(a^16*b^5 - 7*a^14*b^7 + 21*a^12*b^9 - 35*a^10 *b^11 + 35*a^8*b^13 - 21*a^6*b^15 + 7*a^4*b^17 - a^2*b^19)*sin(d*x + c)^5 + 35*(a^17*b^4 - 7*a^15*b^6 + 21*a^13*b^8 - 35*a^11*b^10 + 35*a^9*b^12 - 2 1*a^7*b^14 + 7*a^5*b^16 - a^3*b^18)*sin(d*x + c)^4 + 35*(a^18*b^3 - 7*a^16 *b^5 + 21*a^14*b^7 - 35*a^12*b^9 + 35*a^10*b^11 - 21*a^8*b^13 + 7*a^6*b...
Leaf count of result is larger than twice the leaf count of optimal. 1010 vs. \(2 (374) = 748\).
Time = 0.46 (sec) , antiderivative size = 1010, normalized size of antiderivative = 2.62 \[ \int \frac {\sec (c+d x)}{(a+b \sin (c+d x))^8} \, dx=\text {Too large to display} \]
-1/210*(1680*(a^7*b^2 + 7*a^5*b^4 + 7*a^3*b^6 + a*b^8)*log(abs(b*sin(d*x + c) + a))/(a^16*b - 8*a^14*b^3 + 28*a^12*b^5 - 56*a^10*b^7 + 70*a^8*b^9 - 56*a^6*b^11 + 28*a^4*b^13 - 8*a^2*b^15 + b^17) - 105*log(abs(sin(d*x + c) + 1))/(a^8 - 8*a^7*b + 28*a^6*b^2 - 56*a^5*b^3 + 70*a^4*b^4 - 56*a^3*b^5 + 28*a^2*b^6 - 8*a*b^7 + b^8) + 105*log(abs(sin(d*x + c) - 1))/(a^8 + 8*a^7 *b + 28*a^6*b^2 + 56*a^5*b^3 + 70*a^4*b^4 + 56*a^3*b^5 + 28*a^2*b^6 + 8*a* b^7 + b^8) - 2*(2178*a^7*b^8*sin(d*x + c)^7 + 15246*a^5*b^10*sin(d*x + c)^ 7 + 15246*a^3*b^12*sin(d*x + c)^7 + 2178*a*b^14*sin(d*x + c)^7 + 15981*a^8 *b^7*sin(d*x + c)^6 + 109662*a^6*b^9*sin(d*x + c)^6 + 105252*a^4*b^11*sin( d*x + c)^6 + 13146*a^2*b^13*sin(d*x + c)^6 - 105*b^15*sin(d*x + c)^6 + 504 63*a^9*b^6*sin(d*x + c)^5 + 338226*a^7*b^8*sin(d*x + c)^5 + 309876*a^5*b^1 0*sin(d*x + c)^5 + 33558*a^3*b^12*sin(d*x + c)^5 - 315*a*b^14*sin(d*x + c) ^5 + 89005*a^10*b^5*sin(d*x + c)^4 + 579635*a^8*b^7*sin(d*x + c)^4 + 50372 0*a^6*b^9*sin(d*x + c)^4 + 47600*a^4*b^11*sin(d*x + c)^4 - 245*a^2*b^13*si n(d*x + c)^4 - 35*b^15*sin(d*x + c)^4 + 94885*a^11*b^4*sin(d*x + c)^3 + 59 5595*a^9*b^6*sin(d*x + c)^3 + 487760*a^7*b^8*sin(d*x + c)^3 + 41720*a^5*b^ 10*sin(d*x + c)^3 - 245*a^3*b^12*sin(d*x + c)^3 - 35*a*b^14*sin(d*x + c)^3 + 61341*a^12*b^3*sin(d*x + c)^2 + 366177*a^10*b^5*sin(d*x + c)^2 + 281631 *a^8*b^7*sin(d*x + c)^2 + 23268*a^6*b^9*sin(d*x + c)^2 - 735*a^4*b^11*sin( d*x + c)^2 + 147*a^2*b^13*sin(d*x + c)^2 - 21*b^15*sin(d*x + c)^2 + 224...
Time = 7.28 (sec) , antiderivative size = 937, normalized size of antiderivative = 2.43 \[ \int \frac {\sec (c+d x)}{(a+b \sin (c+d x))^8} \, dx=\frac {\ln \left (a+b\,\sin \left (c+d\,x\right )\right )\,\left (\frac {1}{2\,{\left (a+b\right )}^8}-\frac {1}{2\,{\left (a-b\right )}^8}\right )}{d}+\frac {\frac {1443\,a^{12}\,b+3704\,a^{10}\,b^3+1849\,a^8\,b^5-496\,a^6\,b^7+309\,a^4\,b^9-104\,a^2\,b^{11}+15\,b^{13}}{105\,\left (a^{14}-7\,a^{12}\,b^2+21\,a^{10}\,b^4-35\,a^8\,b^6+35\,a^6\,b^8-21\,a^4\,b^{10}+7\,a^2\,b^{12}-b^{14}\right )}+\frac {\sin \left (c+d\,x\right )\,\left (1023\,a^{11}\,b^2+3494\,a^9\,b^4+1219\,a^7\,b^6+29\,a^5\,b^8-6\,a^3\,b^{10}+a\,b^{12}\right )}{15\,\left (a^{14}-7\,a^{12}\,b^2+21\,a^{10}\,b^4-35\,a^8\,b^6+35\,a^6\,b^8-21\,a^4\,b^{10}+7\,a^2\,b^{12}-b^{14}\right )}+\frac {{\sin \left (c+d\,x\right )}^3\,\left (533\,a^9\,b^4+2304\,a^7\,b^6+994\,a^5\,b^8+8\,a^3\,b^{10}+a\,b^{12}\right )}{3\,\left (a^{14}-7\,a^{12}\,b^2+21\,a^{10}\,b^4-35\,a^8\,b^6+35\,a^6\,b^8-21\,a^4\,b^{10}+7\,a^2\,b^{12}-b^{14}\right )}+\frac {{\sin \left (c+d\,x\right )}^5\,\left (45\,a^7\,b^6+217\,a^5\,b^8+119\,a^3\,b^{10}+3\,a\,b^{12}\right )}{a^{14}-7\,a^{12}\,b^2+21\,a^{10}\,b^4-35\,a^8\,b^6+35\,a^6\,b^8-21\,a^4\,b^{10}+7\,a^2\,b^{12}-b^{14}}+\frac {{\sin \left (c+d\,x\right )}^2\,\left (743\,a^{10}\,b^3+2934\,a^8\,b^5+1099\,a^6\,b^7+29\,a^4\,b^9-6\,a^2\,b^{11}+b^{13}\right )}{5\,\left (a^{14}-7\,a^{12}\,b^2+21\,a^{10}\,b^4-35\,a^8\,b^6+35\,a^6\,b^8-21\,a^4\,b^{10}+7\,a^2\,b^{12}-b^{14}\right )}+\frac {{\sin \left (c+d\,x\right )}^4\,\left (365\,a^8\,b^5+1680\,a^6\,b^7+826\,a^4\,b^9+8\,a^2\,b^{11}+b^{13}\right )}{3\,\left (a^{14}-7\,a^{12}\,b^2+21\,a^{10}\,b^4-35\,a^8\,b^6+35\,a^6\,b^8-21\,a^4\,b^{10}+7\,a^2\,b^{12}-b^{14}\right )}+\frac {{\sin \left (c+d\,x\right )}^6\,\left (7\,a^6\,b^7+35\,a^4\,b^9+21\,a^2\,b^{11}+b^{13}\right )}{a^{14}-7\,a^{12}\,b^2+21\,a^{10}\,b^4-35\,a^8\,b^6+35\,a^6\,b^8-21\,a^4\,b^{10}+7\,a^2\,b^{12}-b^{14}}}{d\,\left (a^7+7\,a^6\,b\,\sin \left (c+d\,x\right )+21\,a^5\,b^2\,{\sin \left (c+d\,x\right )}^2+35\,a^4\,b^3\,{\sin \left (c+d\,x\right )}^3+35\,a^3\,b^4\,{\sin \left (c+d\,x\right )}^4+21\,a^2\,b^5\,{\sin \left (c+d\,x\right )}^5+7\,a\,b^6\,{\sin \left (c+d\,x\right )}^6+b^7\,{\sin \left (c+d\,x\right )}^7\right )}+\frac {\ln \left (\sin \left (c+d\,x\right )+1\right )}{2\,d\,{\left (a-b\right )}^8}-\frac {\ln \left (\sin \left (c+d\,x\right )-1\right )}{2\,d\,{\left (a+b\right )}^8} \]
(log(a + b*sin(c + d*x))*(1/(2*(a + b)^8) - 1/(2*(a - b)^8)))/d + ((1443*a ^12*b + 15*b^13 - 104*a^2*b^11 + 309*a^4*b^9 - 496*a^6*b^7 + 1849*a^8*b^5 + 3704*a^10*b^3)/(105*(a^14 - b^14 + 7*a^2*b^12 - 21*a^4*b^10 + 35*a^6*b^8 - 35*a^8*b^6 + 21*a^10*b^4 - 7*a^12*b^2)) + (sin(c + d*x)*(a*b^12 - 6*a^3 *b^10 + 29*a^5*b^8 + 1219*a^7*b^6 + 3494*a^9*b^4 + 1023*a^11*b^2))/(15*(a^ 14 - b^14 + 7*a^2*b^12 - 21*a^4*b^10 + 35*a^6*b^8 - 35*a^8*b^6 + 21*a^10*b ^4 - 7*a^12*b^2)) + (sin(c + d*x)^3*(a*b^12 + 8*a^3*b^10 + 994*a^5*b^8 + 2 304*a^7*b^6 + 533*a^9*b^4))/(3*(a^14 - b^14 + 7*a^2*b^12 - 21*a^4*b^10 + 3 5*a^6*b^8 - 35*a^8*b^6 + 21*a^10*b^4 - 7*a^12*b^2)) + (sin(c + d*x)^5*(3*a *b^12 + 119*a^3*b^10 + 217*a^5*b^8 + 45*a^7*b^6))/(a^14 - b^14 + 7*a^2*b^1 2 - 21*a^4*b^10 + 35*a^6*b^8 - 35*a^8*b^6 + 21*a^10*b^4 - 7*a^12*b^2) + (s in(c + d*x)^2*(b^13 - 6*a^2*b^11 + 29*a^4*b^9 + 1099*a^6*b^7 + 2934*a^8*b^ 5 + 743*a^10*b^3))/(5*(a^14 - b^14 + 7*a^2*b^12 - 21*a^4*b^10 + 35*a^6*b^8 - 35*a^8*b^6 + 21*a^10*b^4 - 7*a^12*b^2)) + (sin(c + d*x)^4*(b^13 + 8*a^2 *b^11 + 826*a^4*b^9 + 1680*a^6*b^7 + 365*a^8*b^5))/(3*(a^14 - b^14 + 7*a^2 *b^12 - 21*a^4*b^10 + 35*a^6*b^8 - 35*a^8*b^6 + 21*a^10*b^4 - 7*a^12*b^2)) + (sin(c + d*x)^6*(b^13 + 21*a^2*b^11 + 35*a^4*b^9 + 7*a^6*b^7))/(a^14 - b^14 + 7*a^2*b^12 - 21*a^4*b^10 + 35*a^6*b^8 - 35*a^8*b^6 + 21*a^10*b^4 - 7*a^12*b^2))/(d*(a^7 + b^7*sin(c + d*x)^7 + 7*a*b^6*sin(c + d*x)^6 + 21*a^ 5*b^2*sin(c + d*x)^2 + 35*a^4*b^3*sin(c + d*x)^3 + 35*a^3*b^4*sin(c + d...