Integrand size = 21, antiderivative size = 527 \[ \int \frac {\sec ^3(c+d x)}{(a+b \sin (c+d x))^8} \, dx=-\frac {(a+9 b) \log (1-\sin (c+d x))}{4 (a+b)^9 d}+\frac {(a-9 b) \log (1+\sin (c+d x))}{4 (a-b)^9 d}+\frac {8 a b^3 \left (15 a^6+63 a^4 b^2+45 a^2 b^4+5 b^6\right ) \log (a+b \sin (c+d x))}{\left (a^2-b^2\right )^9 d}-\frac {b \left (7 a^2+9 b^2\right )}{14 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^7}-\frac {\sec ^2(c+d x) (b-a \sin (c+d x))}{2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^7}-\frac {a b \left (3 a^2+13 b^2\right )}{6 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^6}-\frac {b \left (5 a^4+50 a^2 b^2+9 b^4\right )}{10 \left (a^2-b^2\right )^4 d (a+b \sin (c+d x))^5}-\frac {a b \left (a^4+20 a^2 b^2+11 b^4\right )}{2 \left (a^2-b^2\right )^5 d (a+b \sin (c+d x))^4}-\frac {b \left (3 a^6+115 a^4 b^2+129 a^2 b^4+9 b^6\right )}{6 \left (a^2-b^2\right )^6 d (a+b \sin (c+d x))^3}-\frac {a b \left (a^6+77 a^4 b^2+147 a^2 b^4+31 b^6\right )}{2 \left (a^2-b^2\right )^7 d (a+b \sin (c+d x))^2}-\frac {b \left (a^8+196 a^6 b^2+574 a^4 b^4+244 a^2 b^6+9 b^8\right )}{2 \left (a^2-b^2\right )^8 d (a+b \sin (c+d x))} \]
-1/4*(a+9*b)*ln(1-sin(d*x+c))/(a+b)^9/d+1/4*(a-9*b)*ln(1+sin(d*x+c))/(a-b) ^9/d+8*a*b^3*(15*a^6+63*a^4*b^2+45*a^2*b^4+5*b^6)*ln(a+b*sin(d*x+c))/(a^2- b^2)^9/d-1/14*b*(7*a^2+9*b^2)/(a^2-b^2)^2/d/(a+b*sin(d*x+c))^7-1/2*sec(d*x +c)^2*(b-a*sin(d*x+c))/(a^2-b^2)/d/(a+b*sin(d*x+c))^7-1/6*a*b*(3*a^2+13*b^ 2)/(a^2-b^2)^3/d/(a+b*sin(d*x+c))^6-1/10*b*(5*a^4+50*a^2*b^2+9*b^4)/(a^2-b ^2)^4/d/(a+b*sin(d*x+c))^5-1/2*a*b*(a^4+20*a^2*b^2+11*b^4)/(a^2-b^2)^5/d/( a+b*sin(d*x+c))^4-1/6*b*(3*a^6+115*a^4*b^2+129*a^2*b^4+9*b^6)/(a^2-b^2)^6/ d/(a+b*sin(d*x+c))^3-1/2*a*b*(a^6+77*a^4*b^2+147*a^2*b^4+31*b^6)/(a^2-b^2) ^7/d/(a+b*sin(d*x+c))^2-1/2*b*(a^8+196*a^6*b^2+574*a^4*b^4+244*a^2*b^6+9*b ^8)/(a^2-b^2)^8/d/(a+b*sin(d*x+c))
Time = 6.47 (sec) , antiderivative size = 770, normalized size of antiderivative = 1.46 \[ \int \frac {\sec ^3(c+d x)}{(a+b \sin (c+d x))^8} \, dx=\frac {b^3 \left (\frac {\sec ^2(c+d x) \left (b^2-a b \sin (c+d x)\right )}{2 b^4 \left (-a^2+b^2\right ) (a+b \sin (c+d x))^7}-\frac {8 a \left (-\frac {\log (1-\sin (c+d x))}{2 b (a+b)^7}+\frac {\log (1+\sin (c+d x))}{2 (a-b)^7 b}-\frac {\left (7 a^6+35 a^4 b^2+21 a^2 b^4+b^6\right ) \log (a+b \sin (c+d x))}{(a-b)^7 (a+b)^7}+\frac {1}{6 \left (a^2-b^2\right ) (a+b \sin (c+d x))^6}+\frac {2 a}{5 (a-b)^2 (a+b)^2 (a+b \sin (c+d x))^5}+\frac {3 a^2+b^2}{4 (a-b)^3 (a+b)^3 (a+b \sin (c+d x))^4}+\frac {4 a \left (a^2+b^2\right )}{3 (a-b)^4 (a+b)^4 (a+b \sin (c+d x))^3}+\frac {5 a^4+10 a^2 b^2+b^4}{2 (a-b)^5 (a+b)^5 (a+b \sin (c+d x))^2}+\frac {2 a \left (3 a^2+b^2\right ) \left (a^2+3 b^2\right )}{(a-b)^6 (a+b)^6 (a+b \sin (c+d x))}\right )+\left (-7 a^2-9 b^2\right ) \left (-\frac {\log (1-\sin (c+d x))}{2 b (a+b)^8}+\frac {\log (1+\sin (c+d x))}{2 (a-b)^8 b}-\frac {8 a \left (a^2+b^2\right ) \left (a^4+6 a^2 b^2+b^4\right ) \log (a+b \sin (c+d x))}{(a-b)^8 (a+b)^8}+\frac {1}{7 \left (a^2-b^2\right ) (a+b \sin (c+d x))^7}+\frac {a}{3 (a-b)^2 (a+b)^2 (a+b \sin (c+d x))^6}+\frac {3 a^2+b^2}{5 (a-b)^3 (a+b)^3 (a+b \sin (c+d x))^5}+\frac {a \left (a^2+b^2\right )}{(a-b)^4 (a+b)^4 (a+b \sin (c+d x))^4}+\frac {5 a^4+10 a^2 b^2+b^4}{3 (a-b)^5 (a+b)^5 (a+b \sin (c+d x))^3}+\frac {a \left (3 a^2+b^2\right ) \left (a^2+3 b^2\right )}{(a-b)^6 (a+b)^6 (a+b \sin (c+d x))^2}+\frac {7 a^6+35 a^4 b^2+21 a^2 b^4+b^6}{(a-b)^7 (a+b)^7 (a+b \sin (c+d x))}\right )}{2 b^2 \left (-a^2+b^2\right )}\right )}{d} \]
(b^3*((Sec[c + d*x]^2*(b^2 - a*b*Sin[c + d*x]))/(2*b^4*(-a^2 + b^2)*(a + b *Sin[c + d*x])^7) - (8*a*(-1/2*Log[1 - Sin[c + d*x]]/(b*(a + b)^7) + Log[1 + Sin[c + d*x]]/(2*(a - b)^7*b) - ((7*a^6 + 35*a^4*b^2 + 21*a^2*b^4 + b^6 )*Log[a + b*Sin[c + d*x]])/((a - b)^7*(a + b)^7) + 1/(6*(a^2 - b^2)*(a + b *Sin[c + d*x])^6) + (2*a)/(5*(a - b)^2*(a + b)^2*(a + b*Sin[c + d*x])^5) + (3*a^2 + b^2)/(4*(a - b)^3*(a + b)^3*(a + b*Sin[c + d*x])^4) + (4*a*(a^2 + b^2))/(3*(a - b)^4*(a + b)^4*(a + b*Sin[c + d*x])^3) + (5*a^4 + 10*a^2*b ^2 + b^4)/(2*(a - b)^5*(a + b)^5*(a + b*Sin[c + d*x])^2) + (2*a*(3*a^2 + b ^2)*(a^2 + 3*b^2))/((a - b)^6*(a + b)^6*(a + b*Sin[c + d*x]))) + (-7*a^2 - 9*b^2)*(-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]))))/(2*b^2*(-a^2 + b^2))))/d
Time = 1.00 (sec) , antiderivative size = 459, normalized size of antiderivative = 0.87, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, 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.
| \(\displaystyle \int \frac {\sec ^3(c+d x)}{(a+b \sin (c+d x))^8} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\cos (c+d x)^3 (a+b \sin (c+d x))^8}dx\) |
| \(\Big \downarrow \) 3147 |
| \(\displaystyle \frac {b^3 \int \frac {1}{(a+b \sin (c+d x))^8 \left (b^2-b^2 \sin ^2(c+d x)\right )^2}d(b \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 477 |
| \(\displaystyle \frac {\int \left (\frac {8 a \left (15 a^6+63 b^2 a^4+45 b^4 a^2+5 b^6\right ) b^4}{\left (a^2-b^2\right )^9 (a+b \sin (c+d x))}+\frac {4 \left (21 a^6+63 b^2 a^4+27 b^4 a^2+b^6\right ) b^4}{\left (a^2-b^2\right )^8 (a+b \sin (c+d x))^2}+\frac {8 a \left (7 a^4+14 b^2 a^2+3 b^4\right ) b^4}{\left (a^2-b^2\right )^7 (a+b \sin (c+d x))^3}+\frac {\left (35 a^4+42 b^2 a^2+3 b^4\right ) b^4}{\left (a^2-b^2\right )^6 (a+b \sin (c+d x))^4}+\frac {4 a \left (5 a^2+3 b^2\right ) b^4}{\left (a^2-b^2\right )^5 (a+b \sin (c+d x))^5}+\frac {2 \left (5 a^2+b^2\right ) b^4}{\left (a^2-b^2\right )^4 (a+b \sin (c+d x))^6}+\frac {4 a b^4}{\left (a^2-b^2\right )^3 (a+b \sin (c+d x))^7}+\frac {b^4}{\left (a^2-b^2\right )^2 (a+b \sin (c+d x))^8}+\frac {b^2}{4 (a+b)^8 (b-b \sin (c+d x))^2}+\frac {b^2}{4 (a-b)^8 (\sin (c+d x) b+b)^2}+\frac {(a+9 b) b}{4 (a+b)^9 (b-b \sin (c+d x))}+\frac {(a-9 b) b}{4 (a-b)^9 (\sin (c+d x) b+b)}\right )d(b \sin (c+d x))}{b d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {a b^4 \left (5 a^2+3 b^2\right )}{\left (a^2-b^2\right )^5 (a+b \sin (c+d x))^4}-\frac {2 b^4 \left (5 a^2+b^2\right )}{5 \left (a^2-b^2\right )^4 (a+b \sin (c+d x))^5}-\frac {2 a b^4}{3 \left (a^2-b^2\right )^3 (a+b \sin (c+d x))^6}-\frac {b^4}{7 \left (a^2-b^2\right )^2 (a+b \sin (c+d x))^7}-\frac {4 a b^4 \left (7 a^4+14 a^2 b^2+3 b^4\right )}{\left (a^2-b^2\right )^7 (a+b \sin (c+d x))^2}-\frac {b^4 \left (35 a^4+42 a^2 b^2+3 b^4\right )}{3 \left (a^2-b^2\right )^6 (a+b \sin (c+d x))^3}-\frac {4 b^4 \left (21 a^6+63 a^4 b^2+27 a^2 b^4+b^6\right )}{\left (a^2-b^2\right )^8 (a+b \sin (c+d x))}+\frac {8 a b^4 \left (15 a^6+63 a^4 b^2+45 a^2 b^4+5 b^6\right ) \log (a+b \sin (c+d x))}{\left (a^2-b^2\right )^9}+\frac {b^2}{4 (a+b)^8 (b-b \sin (c+d x))}-\frac {b^2}{4 (a-b)^8 (b \sin (c+d x)+b)}-\frac {b (a+9 b) \log (b-b \sin (c+d x))}{4 (a+b)^9}+\frac {b (a-9 b) \log (b \sin (c+d x)+b)}{4 (a-b)^9}}{b d}\) |
(-1/4*(b*(a + 9*b)*Log[b - b*Sin[c + d*x]])/(a + b)^9 + (8*a*b^4*(15*a^6 + 63*a^4*b^2 + 45*a^2*b^4 + 5*b^6)*Log[a + b*Sin[c + d*x]])/(a^2 - b^2)^9 + ((a - 9*b)*b*Log[b + b*Sin[c + d*x]])/(4*(a - b)^9) + b^2/(4*(a + b)^8*(b - b*Sin[c + d*x])) - b^4/(7*(a^2 - b^2)^2*(a + b*Sin[c + d*x])^7) - (2*a* b^4)/(3*(a^2 - b^2)^3*(a + b*Sin[c + d*x])^6) - (2*b^4*(5*a^2 + b^2))/(5*( a^2 - b^2)^4*(a + b*Sin[c + d*x])^5) - (a*b^4*(5*a^2 + 3*b^2))/((a^2 - b^2 )^5*(a + b*Sin[c + d*x])^4) - (b^4*(35*a^4 + 42*a^2*b^2 + 3*b^4))/(3*(a^2 - b^2)^6*(a + b*Sin[c + d*x])^3) - (4*a*b^4*(7*a^4 + 14*a^2*b^2 + 3*b^4))/ ((a^2 - b^2)^7*(a + b*Sin[c + d*x])^2) - (4*b^4*(21*a^6 + 63*a^4*b^2 + 27* a^2*b^4 + b^6))/((a^2 - b^2)^8*(a + b*Sin[c + d*x])) - b^2/(4*(a - b)^8*(b + b*Sin[c + d*x])))/(b*d)
3.5.66.3.1 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 = 39.80 (sec) , antiderivative size = 433, normalized size of antiderivative = 0.82
| method | result | size |
| derivativedivides | \(\frac {-\frac {b^{3}}{7 \left (a +b \right )^{2} \left (a -b \right )^{2} \left (a +b \sin \left (d x +c \right )\right )^{7}}-\frac {b^{3} \left (35 a^{4}+42 a^{2} b^{2}+3 b^{4}\right )}{3 \left (a +b \right )^{6} \left (a -b \right )^{6} \left (a +b \sin \left (d x +c \right )\right )^{3}}-\frac {2 a \,b^{3}}{3 \left (a -b \right )^{3} \left (a +b \right )^{3} \left (a +b \sin \left (d x +c \right )\right )^{6}}-\frac {2 b^{3} \left (5 a^{2}+b^{2}\right )}{5 \left (a +b \right )^{4} \left (a -b \right )^{4} \left (a +b \sin \left (d x +c \right )\right )^{5}}-\frac {4 b^{3} \left (21 a^{6}+63 a^{4} b^{2}+27 a^{2} b^{4}+b^{6}\right )}{\left (a +b \right )^{8} \left (a -b \right )^{8} \left (a +b \sin \left (d x +c \right )\right )}-\frac {b^{3} a \left (5 a^{2}+3 b^{2}\right )}{\left (a +b \right )^{5} \left (a -b \right )^{5} \left (a +b \sin \left (d x +c \right )\right )^{4}}-\frac {4 b^{3} a \left (7 a^{4}+14 a^{2} b^{2}+3 b^{4}\right )}{\left (a -b \right )^{7} \left (a +b \right )^{7} \left (a +b \sin \left (d x +c \right )\right )^{2}}+\frac {8 b^{3} a \left (15 a^{6}+63 a^{4} b^{2}+45 a^{2} b^{4}+5 b^{6}\right ) \ln \left (a +b \sin \left (d x +c \right )\right )}{\left (a -b \right )^{9} \left (a +b \right )^{9}}-\frac {1}{4 \left (a +b \right )^{8} \left (\sin \left (d x +c \right )-1\right )}+\frac {\left (-a -9 b \right ) \ln \left (\sin \left (d x +c \right )-1\right )}{4 \left (a +b \right )^{9}}-\frac {1}{4 \left (a -b \right )^{8} \left (1+\sin \left (d x +c \right )\right )}+\frac {\left (a -9 b \right ) \ln \left (1+\sin \left (d x +c \right )\right )}{4 \left (a -b \right )^{9}}}{d}\) | \(433\) |
| default | \(\frac {-\frac {b^{3}}{7 \left (a +b \right )^{2} \left (a -b \right )^{2} \left (a +b \sin \left (d x +c \right )\right )^{7}}-\frac {b^{3} \left (35 a^{4}+42 a^{2} b^{2}+3 b^{4}\right )}{3 \left (a +b \right )^{6} \left (a -b \right )^{6} \left (a +b \sin \left (d x +c \right )\right )^{3}}-\frac {2 a \,b^{3}}{3 \left (a -b \right )^{3} \left (a +b \right )^{3} \left (a +b \sin \left (d x +c \right )\right )^{6}}-\frac {2 b^{3} \left (5 a^{2}+b^{2}\right )}{5 \left (a +b \right )^{4} \left (a -b \right )^{4} \left (a +b \sin \left (d x +c \right )\right )^{5}}-\frac {4 b^{3} \left (21 a^{6}+63 a^{4} b^{2}+27 a^{2} b^{4}+b^{6}\right )}{\left (a +b \right )^{8} \left (a -b \right )^{8} \left (a +b \sin \left (d x +c \right )\right )}-\frac {b^{3} a \left (5 a^{2}+3 b^{2}\right )}{\left (a +b \right )^{5} \left (a -b \right )^{5} \left (a +b \sin \left (d x +c \right )\right )^{4}}-\frac {4 b^{3} a \left (7 a^{4}+14 a^{2} b^{2}+3 b^{4}\right )}{\left (a -b \right )^{7} \left (a +b \right )^{7} \left (a +b \sin \left (d x +c \right )\right )^{2}}+\frac {8 b^{3} a \left (15 a^{6}+63 a^{4} b^{2}+45 a^{2} b^{4}+5 b^{6}\right ) \ln \left (a +b \sin \left (d x +c \right )\right )}{\left (a -b \right )^{9} \left (a +b \right )^{9}}-\frac {1}{4 \left (a +b \right )^{8} \left (\sin \left (d x +c \right )-1\right )}+\frac {\left (-a -9 b \right ) \ln \left (\sin \left (d x +c \right )-1\right )}{4 \left (a +b \right )^{9}}-\frac {1}{4 \left (a -b \right )^{8} \left (1+\sin \left (d x +c \right )\right )}+\frac {\left (a -9 b \right ) \ln \left (1+\sin \left (d x +c \right )\right )}{4 \left (a -b \right )^{9}}}{d}\) | \(433\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(2067\) |
| risch | \(\text {Expression too large to display}\) | \(4084\) |
1/d*(-1/7*b^3/(a+b)^2/(a-b)^2/(a+b*sin(d*x+c))^7-1/3*b^3*(35*a^4+42*a^2*b^ 2+3*b^4)/(a+b)^6/(a-b)^6/(a+b*sin(d*x+c))^3-2/3*a*b^3/(a-b)^3/(a+b)^3/(a+b *sin(d*x+c))^6-2/5*b^3*(5*a^2+b^2)/(a+b)^4/(a-b)^4/(a+b*sin(d*x+c))^5-4*b^ 3*(21*a^6+63*a^4*b^2+27*a^2*b^4+b^6)/(a+b)^8/(a-b)^8/(a+b*sin(d*x+c))-b^3* a*(5*a^2+3*b^2)/(a+b)^5/(a-b)^5/(a+b*sin(d*x+c))^4-4*b^3*a*(7*a^4+14*a^2*b ^2+3*b^4)/(a-b)^7/(a+b)^7/(a+b*sin(d*x+c))^2+8*b^3*a*(15*a^6+63*a^4*b^2+45 *a^2*b^4+5*b^6)/(a-b)^9/(a+b)^9*ln(a+b*sin(d*x+c))-1/4/(a+b)^8/(sin(d*x+c) -1)+1/4/(a+b)^9*(-a-9*b)*ln(sin(d*x+c)-1)-1/4/(a-b)^8/(1+sin(d*x+c))+1/4*( a-9*b)/(a-b)^9*ln(1+sin(d*x+c)))
Leaf count of result is larger than twice the leaf count of optimal. 3678 vs. \(2 (508) = 1016\).
Time = 3.17 (sec) , antiderivative size = 3678, normalized size of antiderivative = 6.98 \[ \int \frac {\sec ^3(c+d x)}{(a+b \sin (c+d x))^8} \, dx=\text {Too large to display} \]
1/420*(210*a^16*b - 1680*a^14*b^3 + 5880*a^12*b^5 - 11760*a^10*b^7 + 14700 *a^8*b^9 - 11760*a^6*b^11 + 5880*a^4*b^13 - 1680*a^2*b^15 + 210*b^17 - 210 *(a^10*b^7 + 195*a^8*b^9 + 378*a^6*b^11 - 330*a^4*b^13 - 235*a^2*b^15 - 9* b^17)*cos(d*x + c)^8 + 70*(63*a^12*b^5 + 10015*a^10*b^7 + 18468*a^8*b^9 - 14274*a^6*b^11 - 12025*a^4*b^13 - 2157*a^2*b^15 - 90*b^17)*cos(d*x + c)^6 - 14*(525*a^14*b^3 + 59730*a^12*b^5 + 174637*a^10*b^7 + 77130*a^8*b^9 - 19 4265*a^6*b^11 - 106450*a^4*b^13 - 10785*a^2*b^15 - 522*b^17)*cos(d*x + c)^ 4 + 2*(735*a^16*b + 37165*a^14*b^3 + 437199*a^12*b^5 + 836549*a^10*b^7 - 1 11195*a^8*b^9 - 812385*a^6*b^11 - 362915*a^4*b^13 - 23569*a^2*b^15 - 1584* b^17)*cos(d*x + c)^2 + 3360*(7*(15*a^8*b^9 + 63*a^6*b^11 + 45*a^4*b^13 + 5 *a^2*b^15)*cos(d*x + c)^8 - 7*(75*a^10*b^7 + 360*a^8*b^9 + 414*a^6*b^11 + 160*a^4*b^13 + 15*a^2*b^15)*cos(d*x + c)^6 + 7*(45*a^12*b^5 + 339*a^10*b^7 + 810*a^8*b^9 + 654*a^6*b^11 + 185*a^4*b^13 + 15*a^2*b^15)*cos(d*x + c)^4 - (15*a^14*b^3 + 378*a^12*b^5 + 1893*a^10*b^7 + 3260*a^8*b^9 + 2121*a^6*b ^11 + 490*a^4*b^13 + 35*a^2*b^15)*cos(d*x + c)^2 + ((15*a^7*b^10 + 63*a^5* b^12 + 45*a^3*b^14 + 5*a*b^16)*cos(d*x + c)^8 - 3*(105*a^9*b^8 + 456*a^7*b ^10 + 378*a^5*b^12 + 80*a^3*b^14 + 5*a*b^16)*cos(d*x + c)^6 + (525*a^11*b^ 6 + 2835*a^9*b^8 + 4266*a^7*b^10 + 2254*a^5*b^12 + 345*a^3*b^14 + 15*a*b^1 6)*cos(d*x + c)^4 - (105*a^13*b^4 + 966*a^11*b^6 + 2835*a^9*b^8 + 2948*a^7 *b^10 + 1183*a^5*b^12 + 150*a^3*b^14 + 5*a*b^16)*cos(d*x + c)^2)*sin(d*...
Timed out. \[ \int \frac {\sec ^3(c+d x)}{(a+b \sin (c+d x))^8} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 1670 vs. \(2 (508) = 1016\).
Time = 0.27 (sec) , antiderivative size = 1670, normalized size of antiderivative = 3.17 \[ \int \frac {\sec ^3(c+d x)}{(a+b \sin (c+d x))^8} \, dx=\text {Too large to display} \]
1/420*(3360*(15*a^7*b^3 + 63*a^5*b^5 + 45*a^3*b^7 + 5*a*b^9)*log(b*sin(d*x + c) + a)/(a^18 - 9*a^16*b^2 + 36*a^14*b^4 - 84*a^12*b^6 + 126*a^10*b^8 - 126*a^8*b^10 + 84*a^6*b^12 - 36*a^4*b^14 + 9*a^2*b^16 - b^18) + 105*(a - 9*b)*log(sin(d*x + c) + 1)/(a^9 - 9*a^8*b + 36*a^7*b^2 - 84*a^6*b^3 + 126* a^5*b^4 - 126*a^4*b^5 + 84*a^3*b^6 - 36*a^2*b^7 + 9*a*b^8 - b^9) - 105*(a + 9*b)*log(sin(d*x + c) - 1)/(a^9 + 9*a^8*b + 36*a^7*b^2 + 84*a^6*b^3 + 12 6*a^5*b^4 + 126*a^4*b^5 + 84*a^3*b^6 + 36*a^2*b^7 + 9*a*b^8 + b^9) - 2*(84 0*a^14*b + 33490*a^12*b^3 + 57724*a^10*b^5 + 16354*a^8*b^7 - 1496*a^6*b^9 + 814*a^4*b^11 - 236*a^2*b^13 + 30*b^15 - 105*(a^8*b^7 + 196*a^6*b^9 + 574 *a^4*b^11 + 244*a^2*b^13 + 9*b^15)*sin(d*x + c)^8 - 105*(7*a^9*b^6 + 1252* a^7*b^8 + 3514*a^5*b^10 + 1348*a^3*b^12 + 23*a*b^14)*sin(d*x + c)^7 - 35*( 63*a^10*b^5 + 10066*a^8*b^7 + 26194*a^6*b^9 + 7384*a^4*b^11 - 681*a^2*b^13 - 18*b^15)*sin(d*x + c)^6 - 35*(105*a^11*b^4 + 14506*a^9*b^6 + 32254*a^7* b^8 + 160*a^5*b^10 - 3951*a^3*b^12 - 66*a*b^14)*sin(d*x + c)^5 - 7*(525*a^ 12*b^3 + 59310*a^10*b^5 + 83812*a^8*b^7 - 98528*a^6*b^9 - 44663*a^4*b^11 - 438*a^2*b^13 - 18*b^15)*sin(d*x + c)^4 - 7*(315*a^13*b^2 + 25930*a^11*b^4 - 20896*a^9*b^6 - 166336*a^7*b^8 - 53641*a^5*b^10 - 386*a^3*b^12 - 26*a*b ^14)*sin(d*x + c)^3 - (735*a^14*b + 30550*a^12*b^3 - 361856*a^10*b^5 - 919 070*a^8*b^7 - 252845*a^6*b^9 - 3050*a^4*b^11 + 310*a^2*b^13 - 54*b^15)*sin (d*x + c)^2 - 7*(15*a^15 - 420*a^13*b^2 - 26140*a^11*b^4 - 52264*a^9*b^...
Leaf count of result is larger than twice the leaf count of optimal. 1327 vs. \(2 (508) = 1016\).
Time = 0.55 (sec) , antiderivative size = 1327, normalized size of antiderivative = 2.52 \[ \int \frac {\sec ^3(c+d x)}{(a+b \sin (c+d x))^8} \, dx=\text {Too large to display} \]
1/420*(3360*(15*a^7*b^4 + 63*a^5*b^6 + 45*a^3*b^8 + 5*a*b^10)*log(abs(b*si n(d*x + c) + a))/(a^18*b - 9*a^16*b^3 + 36*a^14*b^5 - 84*a^12*b^7 + 126*a^ 10*b^9 - 126*a^8*b^11 + 84*a^6*b^13 - 36*a^4*b^15 + 9*a^2*b^17 - b^19) + 1 05*(a - 9*b)*log(abs(sin(d*x + c) + 1))/(a^9 - 9*a^8*b + 36*a^7*b^2 - 84*a ^6*b^3 + 126*a^5*b^4 - 126*a^4*b^5 + 84*a^3*b^6 - 36*a^2*b^7 + 9*a*b^8 - b ^9) - 105*(a + 9*b)*log(abs(sin(d*x + c) - 1))/(a^9 + 9*a^8*b + 36*a^7*b^2 + 84*a^6*b^3 + 126*a^5*b^4 + 126*a^4*b^5 + 84*a^3*b^6 + 36*a^2*b^7 + 9*a* b^8 + b^9) + 210*(120*a^7*b^3*sin(d*x + c)^2 + 504*a^5*b^5*sin(d*x + c)^2 + 360*a^3*b^7*sin(d*x + c)^2 + 40*a*b^9*sin(d*x + c)^2 - a^10*sin(d*x + c) - 27*a^8*b^2*sin(d*x + c) - 42*a^6*b^4*sin(d*x + c) + 42*a^4*b^6*sin(d*x + c) + 27*a^2*b^8*sin(d*x + c) + b^10*sin(d*x + c) + 8*a^9*b - 72*a^7*b^3 - 504*a^5*b^5 - 408*a^3*b^7 - 48*a*b^9)/((a^18 - 9*a^16*b^2 + 36*a^14*b^4 - 84*a^12*b^6 + 126*a^10*b^8 - 126*a^8*b^10 + 84*a^6*b^12 - 36*a^4*b^14 + 9*a^2*b^16 - b^18)*(sin(d*x + c)^2 - 1)) - 4*(32670*a^7*b^10*sin(d*x + c)^ 7 + 137214*a^5*b^12*sin(d*x + c)^7 + 98010*a^3*b^14*sin(d*x + c)^7 + 10890 *a*b^16*sin(d*x + c)^7 + 237510*a^8*b^9*sin(d*x + c)^6 + 978138*a^6*b^11*s in(d*x + c)^6 + 670950*a^4*b^13*sin(d*x + c)^6 + 65310*a^2*b^15*sin(d*x + c)^6 - 420*b^17*sin(d*x + c)^6 + 741930*a^9*b^8*sin(d*x + c)^5 + 2987334*a ^7*b^10*sin(d*x + c)^5 + 1959930*a^5*b^12*sin(d*x + c)^5 + 166530*a^3*b^14 *sin(d*x + c)^5 - 1260*a*b^16*sin(d*x + c)^5 + 1291675*a^10*b^7*sin(d*x...
Time = 9.61 (sec) , antiderivative size = 1443, normalized size of antiderivative = 2.74 \[ \int \frac {\sec ^3(c+d x)}{(a+b \sin (c+d x))^8} \, dx=\text {Too large to display} \]
((sin(c + d*x)^7*(23*a*b^14 + 1348*a^3*b^12 + 3514*a^5*b^10 + 1252*a^7*b^8 + 7*a^9*b^6))/(2*(a^16 + b^16 - 8*a^2*b^14 + 28*a^4*b^12 - 56*a^6*b^10 + 70*a^8*b^8 - 56*a^10*b^6 + 28*a^12*b^4 - 8*a^14*b^2)) - (420*a^14*b + 15*b ^15 - 118*a^2*b^13 + 407*a^4*b^11 - 748*a^6*b^9 + 8177*a^8*b^7 + 28862*a^1 0*b^5 + 16745*a^12*b^3)/(105*(a^2 - b^2)*(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*(7384*a^4*b^11 - 681*a^2*b^13 - 18*b^15 + 26194*a^6*b^9 + 10066*a^8* b^7 + 63*a^10*b^5))/(6*(a^16 + b^16 - 8*a^2*b^14 + 28*a^4*b^12 - 56*a^6*b^ 10 + 70*a^8*b^8 - 56*a^10*b^6 + 28*a^12*b^4 - 8*a^14*b^2)) + (sin(c + d*x) ^8*(9*b^15 + 244*a^2*b^13 + 574*a^4*b^11 + 196*a^6*b^9 + a^8*b^7))/(2*(a^1 6 + b^16 - 8*a^2*b^14 + 28*a^4*b^12 - 56*a^6*b^10 + 70*a^8*b^8 - 56*a^10*b ^6 + 28*a^12*b^4 - 8*a^14*b^2)) + (sin(c + d*x)^5*(160*a^5*b^10 - 3951*a^3 *b^12 - 66*a*b^14 + 32254*a^7*b^8 + 14506*a^9*b^6 + 105*a^11*b^4))/(6*(a^1 6 + b^16 - 8*a^2*b^14 + 28*a^4*b^12 - 56*a^6*b^10 + 70*a^8*b^8 - 56*a^10*b ^6 + 28*a^12*b^4 - 8*a^14*b^2)) + (sin(c + d*x)^4*(18*b^13 + 456*a^2*b^11 + 45119*a^4*b^9 + 143647*a^6*b^7 + 59835*a^8*b^5 + 525*a^10*b^3))/(30*(a^1 4 - 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)^2*(54*b^15 - 735*a^14*b - 310*a^2*b^13 + 3050*a^4*b^11 + 252845*a^6*b^9 + 919070*a^8*b^7 + 361856*a^10*b^5 - 30550* a^12*b^3))/(210*(a^2 - b^2)*(a^14 - b^14 + 7*a^2*b^12 - 21*a^4*b^10 + 3...