Integrand size = 21, antiderivative size = 328 \[ \int \frac {\sec ^5(c+d x)}{(a+b \sin (c+d x))^3} \, dx=-\frac {3 \left (a^2+5 a b+8 b^2\right ) \log (1-\sin (c+d x))}{16 (a+b)^5 d}+\frac {3 \left (a^2-5 a b+8 b^2\right ) \log (1+\sin (c+d x))}{16 (a-b)^5 d}-\frac {3 b^5 \left (7 a^2+b^2\right ) \log (a+b \sin (c+d x))}{\left (a^2-b^2\right )^5 d}-\frac {3 b \left (a^4-5 a^2 b^2-4 b^4\right )}{8 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^2}-\frac {\sec ^4(c+d x) (b-a \sin (c+d x))}{4 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^2}-\frac {3 a b \left (a^4-6 a^2 b^2-27 b^4\right )}{8 \left (a^2-b^2\right )^4 d (a+b \sin (c+d x))}+\frac {\sec ^2(c+d x) \left (2 b \left (a^2+3 b^2\right )+a \left (3 a^2-11 b^2\right ) \sin (c+d x)\right )}{8 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^2} \]
-3/16*(a^2+5*a*b+8*b^2)*ln(1-sin(d*x+c))/(a+b)^5/d+3/16*(a^2-5*a*b+8*b^2)* ln(1+sin(d*x+c))/(a-b)^5/d-3*b^5*(7*a^2+b^2)*ln(a+b*sin(d*x+c))/(a^2-b^2)^ 5/d-3/8*b*(a^4-5*a^2*b^2-4*b^4)/(a^2-b^2)^3/d/(a+b*sin(d*x+c))^2-1/4*sec(d *x+c)^4*(b-a*sin(d*x+c))/(a^2-b^2)/d/(a+b*sin(d*x+c))^2-3/8*a*b*(a^4-6*a^2 *b^2-27*b^4)/(a^2-b^2)^4/d/(a+b*sin(d*x+c))+1/8*sec(d*x+c)^2*(2*b*(a^2+3*b ^2)+a*(3*a^2-11*b^2)*sin(d*x+c))/(a^2-b^2)^2/d/(a+b*sin(d*x+c))^2
Time = 2.00 (sec) , antiderivative size = 388, normalized size of antiderivative = 1.18 \[ \int \frac {\sec ^5(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {\frac {2 \sec ^4(c+d x) (b-a \sin (c+d x))}{(a+b \sin (c+d x))^2}+\frac {\sec ^2(c+d x) \left (2 b \left (a^2+3 b^2\right )+a \left (3 a^2-11 b^2\right ) \sin (c+d x)\right )}{\left (-a^2+b^2\right ) (a+b \sin (c+d x))^2}-\frac {b \left (3 \left (a^4-5 a^2 b^2-4 b^4\right ) \left (-\frac {\log (1-\sin (c+d x))}{b (a+b)^3}+\frac {\log (1+\sin (c+d x))}{(a-b)^3 b}-\frac {2 \left (3 a^2+b^2\right ) \log (a+b \sin (c+d x))}{(a-b)^3 (a+b)^3}+\frac {1}{\left (a^2-b^2\right ) (a+b \sin (c+d x))^2}+\frac {4 a}{(a-b)^2 (a+b)^2 (a+b \sin (c+d x))}\right )-3 a \left (3 a^2-11 b^2\right ) \left (-\frac {\log (1-\sin (c+d x))}{2 b (a+b)^2}+\frac {\log (1+\sin (c+d x))}{2 (a-b)^2 b}-\frac {2 a \log (a+b \sin (c+d x))}{(a-b)^2 (a+b)^2}+\frac {1}{\left (a^2-b^2\right ) (a+b \sin (c+d x))}\right )\right )}{-a^2+b^2}}{8 \left (-a^2+b^2\right ) d} \]
((2*Sec[c + d*x]^4*(b - a*Sin[c + d*x]))/(a + b*Sin[c + d*x])^2 + (Sec[c + d*x]^2*(2*b*(a^2 + 3*b^2) + a*(3*a^2 - 11*b^2)*Sin[c + d*x]))/((-a^2 + b^ 2)*(a + b*Sin[c + d*x])^2) - (b*(3*(a^4 - 5*a^2*b^2 - 4*b^4)*(-(Log[1 - Si n[c + d*x]]/(b*(a + b)^3)) + Log[1 + Sin[c + d*x]]/((a - b)^3*b) - (2*(3*a ^2 + b^2)*Log[a + b*Sin[c + d*x]])/((a - b)^3*(a + b)^3) + 1/((a^2 - b^2)* (a + b*Sin[c + d*x])^2) + (4*a)/((a - b)^2*(a + b)^2*(a + b*Sin[c + d*x])) ) - 3*a*(3*a^2 - 11*b^2)*(-1/2*Log[1 - Sin[c + d*x]]/(b*(a + b)^2) + Log[1 + Sin[c + d*x]]/(2*(a - b)^2*b) - (2*a*Log[a + b*Sin[c + d*x]])/((a - b)^ 2*(a + b)^2) + 1/((a^2 - b^2)*(a + b*Sin[c + d*x])))))/(-a^2 + b^2))/(8*(- a^2 + b^2)*d)
Time = 0.63 (sec) , antiderivative size = 286, 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.
| \(\displaystyle \int \frac {\sec ^5(c+d x)}{(a+b \sin (c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\cos (c+d x)^5 (a+b \sin (c+d x))^3}dx\) |
| \(\Big \downarrow \) 3147 |
| \(\displaystyle \frac {b^5 \int \frac {1}{(a+b \sin (c+d x))^3 \left (b^2-b^2 \sin ^2(c+d x)\right )^3}d(b \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 477 |
| \(\displaystyle \frac {\int \left (-\frac {3 \left (7 a^2+b^2\right ) b^6}{\left (a^2-b^2\right )^5 (a+b \sin (c+d x))}-\frac {6 a b^6}{\left (a^2-b^2\right )^4 (a+b \sin (c+d x))^2}-\frac {b^6}{\left (a^2-b^2\right )^3 (a+b \sin (c+d x))^3}+\frac {b^3}{8 (a+b)^3 (b-b \sin (c+d x))^3}+\frac {b^3}{8 (a-b)^3 (\sin (c+d x) b+b)^3}+\frac {3 (a+3 b) b^2}{16 (a+b)^4 (b-b \sin (c+d x))^2}+\frac {3 (a-3 b) b^2}{16 (a-b)^4 (\sin (c+d x) b+b)^2}+\frac {3 \left (a^2+5 b a+8 b^2\right ) b}{16 (a+b)^5 (b-b \sin (c+d x))}+\frac {3 \left (a^2-5 b a+8 b^2\right ) b}{16 (a-b)^5 (\sin (c+d x) b+b)}\right )d(b \sin (c+d x))}{b d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {3 b \left (a^2+5 a b+8 b^2\right ) \log (b-b \sin (c+d x))}{16 (a+b)^5}+\frac {3 b \left (a^2-5 a b+8 b^2\right ) \log (b \sin (c+d x)+b)}{16 (a-b)^5}+\frac {6 a b^6}{\left (a^2-b^2\right )^4 (a+b \sin (c+d x))}+\frac {b^6}{2 \left (a^2-b^2\right )^3 (a+b \sin (c+d x))^2}-\frac {3 b^6 \left (7 a^2+b^2\right ) \log (a+b \sin (c+d x))}{\left (a^2-b^2\right )^5}+\frac {b^3}{16 (a+b)^3 (b-b \sin (c+d x))^2}-\frac {b^3}{16 (a-b)^3 (b \sin (c+d x)+b)^2}+\frac {3 b^2 (a+3 b)}{16 (a+b)^4 (b-b \sin (c+d x))}-\frac {3 b^2 (a-3 b)}{16 (a-b)^4 (b \sin (c+d x)+b)}}{b d}\) |
((-3*b*(a^2 + 5*a*b + 8*b^2)*Log[b - b*Sin[c + d*x]])/(16*(a + b)^5) - (3* b^6*(7*a^2 + b^2)*Log[a + b*Sin[c + d*x]])/(a^2 - b^2)^5 + (3*b*(a^2 - 5*a *b + 8*b^2)*Log[b + b*Sin[c + d*x]])/(16*(a - b)^5) + b^3/(16*(a + b)^3*(b - b*Sin[c + d*x])^2) + (3*b^2*(a + 3*b))/(16*(a + b)^4*(b - b*Sin[c + d*x ])) + b^6/(2*(a^2 - b^2)^3*(a + b*Sin[c + d*x])^2) + (6*a*b^6)/((a^2 - b^2 )^4*(a + b*Sin[c + d*x])) - b^3/(16*(a - b)^3*(b + b*Sin[c + d*x])^2) - (3 *(a - 3*b)*b^2)/(16*(a - b)^4*(b + b*Sin[c + d*x])))/(b*d)
3.5.55.3.1 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 = 6.66 (sec) , antiderivative size = 252, normalized size of antiderivative = 0.77
| method | result | size |
| derivativedivides | \(\frac {\frac {1}{16 \left (a +b \right )^{3} \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {3 a +9 b}{16 \left (a +b \right )^{4} \left (\sin \left (d x +c \right )-1\right )}+\frac {\left (-3 a^{2}-15 a b -24 b^{2}\right ) \ln \left (\sin \left (d x +c \right )-1\right )}{16 \left (a +b \right )^{5}}+\frac {b^{5}}{2 \left (a +b \right )^{3} \left (a -b \right )^{3} \left (a +b \sin \left (d x +c \right )\right )^{2}}+\frac {6 a \,b^{5}}{\left (a +b \right )^{4} \left (a -b \right )^{4} \left (a +b \sin \left (d x +c \right )\right )}-\frac {3 b^{5} \left (7 a^{2}+b^{2}\right ) \ln \left (a +b \sin \left (d x +c \right )\right )}{\left (a +b \right )^{5} \left (a -b \right )^{5}}-\frac {1}{16 \left (a -b \right )^{3} \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {3 a -9 b}{16 \left (a -b \right )^{4} \left (1+\sin \left (d x +c \right )\right )}+\frac {\left (3 a^{2}-15 a b +24 b^{2}\right ) \ln \left (1+\sin \left (d x +c \right )\right )}{16 \left (a -b \right )^{5}}}{d}\) | \(252\) |
| default | \(\frac {\frac {1}{16 \left (a +b \right )^{3} \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {3 a +9 b}{16 \left (a +b \right )^{4} \left (\sin \left (d x +c \right )-1\right )}+\frac {\left (-3 a^{2}-15 a b -24 b^{2}\right ) \ln \left (\sin \left (d x +c \right )-1\right )}{16 \left (a +b \right )^{5}}+\frac {b^{5}}{2 \left (a +b \right )^{3} \left (a -b \right )^{3} \left (a +b \sin \left (d x +c \right )\right )^{2}}+\frac {6 a \,b^{5}}{\left (a +b \right )^{4} \left (a -b \right )^{4} \left (a +b \sin \left (d x +c \right )\right )}-\frac {3 b^{5} \left (7 a^{2}+b^{2}\right ) \ln \left (a +b \sin \left (d x +c \right )\right )}{\left (a +b \right )^{5} \left (a -b \right )^{5}}-\frac {1}{16 \left (a -b \right )^{3} \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {3 a -9 b}{16 \left (a -b \right )^{4} \left (1+\sin \left (d x +c \right )\right )}+\frac {\left (3 a^{2}-15 a b +24 b^{2}\right ) \ln \left (1+\sin \left (d x +c \right )\right )}{16 \left (a -b \right )^{5}}}{d}\) | \(252\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(840\) |
| norman | \(\text {Expression too large to display}\) | \(1214\) |
| risch | \(\text {Expression too large to display}\) | \(2027\) |
1/d*(1/16/(a+b)^3/(sin(d*x+c)-1)^2-1/16*(3*a+9*b)/(a+b)^4/(sin(d*x+c)-1)+1 /16/(a+b)^5*(-3*a^2-15*a*b-24*b^2)*ln(sin(d*x+c)-1)+1/2*b^5/(a+b)^3/(a-b)^ 3/(a+b*sin(d*x+c))^2+6*a*b^5/(a+b)^4/(a-b)^4/(a+b*sin(d*x+c))-3*b^5*(7*a^2 +b^2)/(a+b)^5/(a-b)^5*ln(a+b*sin(d*x+c))-1/16/(a-b)^3/(1+sin(d*x+c))^2-1/1 6*(3*a-9*b)/(a-b)^4/(1+sin(d*x+c))+1/16/(a-b)^5*(3*a^2-15*a*b+24*b^2)*ln(1 +sin(d*x+c)))
Leaf count of result is larger than twice the leaf count of optimal. 895 vs. \(2 (317) = 634\).
Time = 0.64 (sec) , antiderivative size = 895, normalized size of antiderivative = 2.73 \[ \int \frac {\sec ^5(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {4 \, a^{8} b - 16 \, a^{6} b^{3} + 24 \, a^{4} b^{5} - 16 \, a^{2} b^{7} + 4 \, b^{9} + 12 \, {\left (a^{8} b - 7 \, a^{6} b^{3} - 7 \, a^{4} b^{5} + 15 \, a^{2} b^{7} - 2 \, b^{9}\right )} \cos \left (d x + c\right )^{4} - 4 \, {\left (a^{8} b - 6 \, a^{4} b^{5} + 8 \, a^{2} b^{7} - 3 \, b^{9}\right )} \cos \left (d x + c\right )^{2} - 48 \, {\left ({\left (7 \, a^{2} b^{7} + b^{9}\right )} \cos \left (d x + c\right )^{6} - 2 \, {\left (7 \, a^{3} b^{6} + a b^{8}\right )} \cos \left (d x + c\right )^{4} \sin \left (d x + c\right ) - {\left (7 \, a^{4} b^{5} + 8 \, a^{2} b^{7} + b^{9}\right )} \cos \left (d x + c\right )^{4}\right )} \log \left (b \sin \left (d x + c\right ) + a\right ) + 3 \, {\left ({\left (a^{7} b^{2} - 7 \, a^{5} b^{4} + 35 \, a^{3} b^{6} + 56 \, a^{2} b^{7} + 35 \, a b^{8} + 8 \, b^{9}\right )} \cos \left (d x + c\right )^{6} - 2 \, {\left (a^{8} b - 7 \, a^{6} b^{3} + 35 \, a^{4} b^{5} + 56 \, a^{3} b^{6} + 35 \, a^{2} b^{7} + 8 \, a b^{8}\right )} \cos \left (d x + c\right )^{4} \sin \left (d x + c\right ) - {\left (a^{9} - 6 \, a^{7} b^{2} + 28 \, a^{5} b^{4} + 56 \, a^{4} b^{5} + 70 \, a^{3} b^{6} + 64 \, a^{2} b^{7} + 35 \, a b^{8} + 8 \, b^{9}\right )} \cos \left (d x + c\right )^{4}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left ({\left (a^{7} b^{2} - 7 \, a^{5} b^{4} + 35 \, a^{3} b^{6} - 56 \, a^{2} b^{7} + 35 \, a b^{8} - 8 \, b^{9}\right )} \cos \left (d x + c\right )^{6} - 2 \, {\left (a^{8} b - 7 \, a^{6} b^{3} + 35 \, a^{4} b^{5} - 56 \, a^{3} b^{6} + 35 \, a^{2} b^{7} - 8 \, a b^{8}\right )} \cos \left (d x + c\right )^{4} \sin \left (d x + c\right ) - {\left (a^{9} - 6 \, a^{7} b^{2} + 28 \, a^{5} b^{4} - 56 \, a^{4} b^{5} + 70 \, a^{3} b^{6} - 64 \, a^{2} b^{7} + 35 \, a b^{8} - 8 \, b^{9}\right )} \cos \left (d x + c\right )^{4}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (2 \, a^{9} - 8 \, a^{7} b^{2} + 12 \, a^{5} b^{4} - 8 \, a^{3} b^{6} + 2 \, a b^{8} - 3 \, {\left (a^{7} b^{2} - 7 \, a^{5} b^{4} - 21 \, a^{3} b^{6} + 27 \, a b^{8}\right )} \cos \left (d x + c\right )^{4} + {\left (3 \, a^{9} - 20 \, a^{7} b^{2} + 42 \, a^{5} b^{4} - 36 \, a^{3} b^{6} + 11 \, a b^{8}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{16 \, {\left ({\left (a^{10} b^{2} - 5 \, a^{8} b^{4} + 10 \, a^{6} b^{6} - 10 \, a^{4} b^{8} + 5 \, a^{2} b^{10} - b^{12}\right )} d \cos \left (d x + c\right )^{6} - 2 \, {\left (a^{11} b - 5 \, a^{9} b^{3} + 10 \, a^{7} b^{5} - 10 \, a^{5} b^{7} + 5 \, a^{3} b^{9} - a b^{11}\right )} d \cos \left (d x + c\right )^{4} \sin \left (d x + c\right ) - {\left (a^{12} - 4 \, a^{10} b^{2} + 5 \, a^{8} b^{4} - 5 \, a^{4} b^{8} + 4 \, a^{2} b^{10} - b^{12}\right )} d \cos \left (d x + c\right )^{4}\right )}} \]
1/16*(4*a^8*b - 16*a^6*b^3 + 24*a^4*b^5 - 16*a^2*b^7 + 4*b^9 + 12*(a^8*b - 7*a^6*b^3 - 7*a^4*b^5 + 15*a^2*b^7 - 2*b^9)*cos(d*x + c)^4 - 4*(a^8*b - 6 *a^4*b^5 + 8*a^2*b^7 - 3*b^9)*cos(d*x + c)^2 - 48*((7*a^2*b^7 + b^9)*cos(d *x + c)^6 - 2*(7*a^3*b^6 + a*b^8)*cos(d*x + c)^4*sin(d*x + c) - (7*a^4*b^5 + 8*a^2*b^7 + b^9)*cos(d*x + c)^4)*log(b*sin(d*x + c) + a) + 3*((a^7*b^2 - 7*a^5*b^4 + 35*a^3*b^6 + 56*a^2*b^7 + 35*a*b^8 + 8*b^9)*cos(d*x + c)^6 - 2*(a^8*b - 7*a^6*b^3 + 35*a^4*b^5 + 56*a^3*b^6 + 35*a^2*b^7 + 8*a*b^8)*co s(d*x + c)^4*sin(d*x + c) - (a^9 - 6*a^7*b^2 + 28*a^5*b^4 + 56*a^4*b^5 + 7 0*a^3*b^6 + 64*a^2*b^7 + 35*a*b^8 + 8*b^9)*cos(d*x + c)^4)*log(sin(d*x + c ) + 1) - 3*((a^7*b^2 - 7*a^5*b^4 + 35*a^3*b^6 - 56*a^2*b^7 + 35*a*b^8 - 8* b^9)*cos(d*x + c)^6 - 2*(a^8*b - 7*a^6*b^3 + 35*a^4*b^5 - 56*a^3*b^6 + 35* a^2*b^7 - 8*a*b^8)*cos(d*x + c)^4*sin(d*x + c) - (a^9 - 6*a^7*b^2 + 28*a^5 *b^4 - 56*a^4*b^5 + 70*a^3*b^6 - 64*a^2*b^7 + 35*a*b^8 - 8*b^9)*cos(d*x + c)^4)*log(-sin(d*x + c) + 1) - 2*(2*a^9 - 8*a^7*b^2 + 12*a^5*b^4 - 8*a^3*b ^6 + 2*a*b^8 - 3*(a^7*b^2 - 7*a^5*b^4 - 21*a^3*b^6 + 27*a*b^8)*cos(d*x + c )^4 + (3*a^9 - 20*a^7*b^2 + 42*a^5*b^4 - 36*a^3*b^6 + 11*a*b^8)*cos(d*x + c)^2)*sin(d*x + c))/((a^10*b^2 - 5*a^8*b^4 + 10*a^6*b^6 - 10*a^4*b^8 + 5*a ^2*b^10 - b^12)*d*cos(d*x + c)^6 - 2*(a^11*b - 5*a^9*b^3 + 10*a^7*b^5 - 10 *a^5*b^7 + 5*a^3*b^9 - a*b^11)*d*cos(d*x + c)^4*sin(d*x + c) - (a^12 - 4*a ^10*b^2 + 5*a^8*b^4 - 5*a^4*b^8 + 4*a^2*b^10 - b^12)*d*cos(d*x + c)^4)
\[ \int \frac {\sec ^5(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\int \frac {\sec ^{5}{\left (c + d x \right )}}{\left (a + b \sin {\left (c + d x \right )}\right )^{3}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 725 vs. \(2 (317) = 634\).
Time = 0.20 (sec) , antiderivative size = 725, normalized size of antiderivative = 2.21 \[ \int \frac {\sec ^5(c+d x)}{(a+b \sin (c+d x))^3} \, dx=-\frac {\frac {48 \, {\left (7 \, a^{2} b^{5} + b^{7}\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{a^{10} - 5 \, a^{8} b^{2} + 10 \, a^{6} b^{4} - 10 \, a^{4} b^{6} + 5 \, a^{2} b^{8} - b^{10}} - \frac {3 \, {\left (a^{2} - 5 \, a b + 8 \, b^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{5} - 5 \, a^{4} b + 10 \, a^{3} b^{2} - 10 \, a^{2} b^{3} + 5 \, a b^{4} - b^{5}} + \frac {3 \, {\left (a^{2} + 5 \, a b + 8 \, b^{2}\right )} \log \left (\sin \left (d x + c\right ) - 1\right )}{a^{5} + 5 \, a^{4} b + 10 \, a^{3} b^{2} + 10 \, a^{2} b^{3} + 5 \, a b^{4} + b^{5}} + \frac {2 \, {\left (6 \, a^{6} b - 44 \, a^{4} b^{3} - 62 \, a^{2} b^{5} + 4 \, b^{7} + 3 \, {\left (a^{5} b^{2} - 6 \, a^{3} b^{4} - 27 \, a b^{6}\right )} \sin \left (d x + c\right )^{5} + 6 \, {\left (a^{6} b - 6 \, a^{4} b^{3} - 13 \, a^{2} b^{5} + 2 \, b^{7}\right )} \sin \left (d x + c\right )^{4} + {\left (3 \, a^{7} - 23 \, a^{5} b^{2} + 61 \, a^{3} b^{4} + 151 \, a b^{6}\right )} \sin \left (d x + c\right )^{3} - 2 \, {\left (5 \, a^{6} b - 37 \, a^{4} b^{3} - 73 \, a^{2} b^{5} + 9 \, b^{7}\right )} \sin \left (d x + c\right )^{2} - {\left (5 \, a^{7} - 26 \, a^{5} b^{2} + 49 \, a^{3} b^{4} + 68 \, a b^{6}\right )} \sin \left (d x + c\right )\right )}}{a^{10} - 4 \, a^{8} b^{2} + 6 \, a^{6} b^{4} - 4 \, a^{4} b^{6} + a^{2} b^{8} + {\left (a^{8} b^{2} - 4 \, a^{6} b^{4} + 6 \, a^{4} b^{6} - 4 \, a^{2} b^{8} + b^{10}\right )} \sin \left (d x + c\right )^{6} + 2 \, {\left (a^{9} b - 4 \, a^{7} b^{3} + 6 \, a^{5} b^{5} - 4 \, a^{3} b^{7} + a b^{9}\right )} \sin \left (d x + c\right )^{5} + {\left (a^{10} - 6 \, a^{8} b^{2} + 14 \, a^{6} b^{4} - 16 \, a^{4} b^{6} + 9 \, a^{2} b^{8} - 2 \, b^{10}\right )} \sin \left (d x + c\right )^{4} - 4 \, {\left (a^{9} b - 4 \, a^{7} b^{3} + 6 \, a^{5} b^{5} - 4 \, a^{3} b^{7} + a b^{9}\right )} \sin \left (d x + c\right )^{3} - {\left (2 \, a^{10} - 9 \, a^{8} b^{2} + 16 \, a^{6} b^{4} - 14 \, a^{4} b^{6} + 6 \, a^{2} b^{8} - b^{10}\right )} \sin \left (d x + c\right )^{2} + 2 \, {\left (a^{9} b - 4 \, a^{7} b^{3} + 6 \, a^{5} b^{5} - 4 \, a^{3} b^{7} + a b^{9}\right )} \sin \left (d x + c\right )}}{16 \, d} \]
-1/16*(48*(7*a^2*b^5 + b^7)*log(b*sin(d*x + c) + a)/(a^10 - 5*a^8*b^2 + 10 *a^6*b^4 - 10*a^4*b^6 + 5*a^2*b^8 - b^10) - 3*(a^2 - 5*a*b + 8*b^2)*log(si n(d*x + c) + 1)/(a^5 - 5*a^4*b + 10*a^3*b^2 - 10*a^2*b^3 + 5*a*b^4 - b^5) + 3*(a^2 + 5*a*b + 8*b^2)*log(sin(d*x + c) - 1)/(a^5 + 5*a^4*b + 10*a^3*b^ 2 + 10*a^2*b^3 + 5*a*b^4 + b^5) + 2*(6*a^6*b - 44*a^4*b^3 - 62*a^2*b^5 + 4 *b^7 + 3*(a^5*b^2 - 6*a^3*b^4 - 27*a*b^6)*sin(d*x + c)^5 + 6*(a^6*b - 6*a^ 4*b^3 - 13*a^2*b^5 + 2*b^7)*sin(d*x + c)^4 + (3*a^7 - 23*a^5*b^2 + 61*a^3* b^4 + 151*a*b^6)*sin(d*x + c)^3 - 2*(5*a^6*b - 37*a^4*b^3 - 73*a^2*b^5 + 9 *b^7)*sin(d*x + c)^2 - (5*a^7 - 26*a^5*b^2 + 49*a^3*b^4 + 68*a*b^6)*sin(d* x + c))/(a^10 - 4*a^8*b^2 + 6*a^6*b^4 - 4*a^4*b^6 + a^2*b^8 + (a^8*b^2 - 4 *a^6*b^4 + 6*a^4*b^6 - 4*a^2*b^8 + b^10)*sin(d*x + c)^6 + 2*(a^9*b - 4*a^7 *b^3 + 6*a^5*b^5 - 4*a^3*b^7 + a*b^9)*sin(d*x + c)^5 + (a^10 - 6*a^8*b^2 + 14*a^6*b^4 - 16*a^4*b^6 + 9*a^2*b^8 - 2*b^10)*sin(d*x + c)^4 - 4*(a^9*b - 4*a^7*b^3 + 6*a^5*b^5 - 4*a^3*b^7 + a*b^9)*sin(d*x + c)^3 - (2*a^10 - 9*a ^8*b^2 + 16*a^6*b^4 - 14*a^4*b^6 + 6*a^2*b^8 - b^10)*sin(d*x + c)^2 + 2*(a ^9*b - 4*a^7*b^3 + 6*a^5*b^5 - 4*a^3*b^7 + a*b^9)*sin(d*x + c)))/d
Time = 0.36 (sec) , antiderivative size = 575, normalized size of antiderivative = 1.75 \[ \int \frac {\sec ^5(c+d x)}{(a+b \sin (c+d x))^3} \, dx=-\frac {\frac {48 \, {\left (7 \, a^{2} b^{6} + b^{8}\right )} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{a^{10} b - 5 \, a^{8} b^{3} + 10 \, a^{6} b^{5} - 10 \, a^{4} b^{7} + 5 \, a^{2} b^{9} - b^{11}} - \frac {3 \, {\left (a^{2} - 5 \, a b + 8 \, b^{2}\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{5} - 5 \, a^{4} b + 10 \, a^{3} b^{2} - 10 \, a^{2} b^{3} + 5 \, a b^{4} - b^{5}} + \frac {3 \, {\left (a^{2} + 5 \, a b + 8 \, b^{2}\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a^{5} + 5 \, a^{4} b + 10 \, a^{3} b^{2} + 10 \, a^{2} b^{3} + 5 \, a b^{4} + b^{5}} + \frac {2 \, {\left (3 \, a^{5} b^{2} \sin \left (d x + c\right )^{5} - 18 \, a^{3} b^{4} \sin \left (d x + c\right )^{5} - 81 \, a b^{6} \sin \left (d x + c\right )^{5} + 6 \, a^{6} b \sin \left (d x + c\right )^{4} - 36 \, a^{4} b^{3} \sin \left (d x + c\right )^{4} - 78 \, a^{2} b^{5} \sin \left (d x + c\right )^{4} + 12 \, b^{7} \sin \left (d x + c\right )^{4} + 3 \, a^{7} \sin \left (d x + c\right )^{3} - 23 \, a^{5} b^{2} \sin \left (d x + c\right )^{3} + 61 \, a^{3} b^{4} \sin \left (d x + c\right )^{3} + 151 \, a b^{6} \sin \left (d x + c\right )^{3} - 10 \, a^{6} b \sin \left (d x + c\right )^{2} + 74 \, a^{4} b^{3} \sin \left (d x + c\right )^{2} + 146 \, a^{2} b^{5} \sin \left (d x + c\right )^{2} - 18 \, b^{7} \sin \left (d x + c\right )^{2} - 5 \, a^{7} \sin \left (d x + c\right ) + 26 \, a^{5} b^{2} \sin \left (d x + c\right ) - 49 \, a^{3} b^{4} \sin \left (d x + c\right ) - 68 \, a b^{6} \sin \left (d x + c\right ) + 6 \, a^{6} b - 44 \, a^{4} b^{3} - 62 \, a^{2} b^{5} + 4 \, b^{7}\right )}}{{\left (a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}\right )} {\left (b \sin \left (d x + c\right )^{3} + a \sin \left (d x + c\right )^{2} - b \sin \left (d x + c\right ) - a\right )}^{2}}}{16 \, d} \]
-1/16*(48*(7*a^2*b^6 + b^8)*log(abs(b*sin(d*x + c) + a))/(a^10*b - 5*a^8*b ^3 + 10*a^6*b^5 - 10*a^4*b^7 + 5*a^2*b^9 - b^11) - 3*(a^2 - 5*a*b + 8*b^2) *log(abs(sin(d*x + c) + 1))/(a^5 - 5*a^4*b + 10*a^3*b^2 - 10*a^2*b^3 + 5*a *b^4 - b^5) + 3*(a^2 + 5*a*b + 8*b^2)*log(abs(sin(d*x + c) - 1))/(a^5 + 5* a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4 + b^5) + 2*(3*a^5*b^2*sin(d*x + c)^5 - 18*a^3*b^4*sin(d*x + c)^5 - 81*a*b^6*sin(d*x + c)^5 + 6*a^6*b*sin(d *x + c)^4 - 36*a^4*b^3*sin(d*x + c)^4 - 78*a^2*b^5*sin(d*x + c)^4 + 12*b^7 *sin(d*x + c)^4 + 3*a^7*sin(d*x + c)^3 - 23*a^5*b^2*sin(d*x + c)^3 + 61*a^ 3*b^4*sin(d*x + c)^3 + 151*a*b^6*sin(d*x + c)^3 - 10*a^6*b*sin(d*x + c)^2 + 74*a^4*b^3*sin(d*x + c)^2 + 146*a^2*b^5*sin(d*x + c)^2 - 18*b^7*sin(d*x + c)^2 - 5*a^7*sin(d*x + c) + 26*a^5*b^2*sin(d*x + c) - 49*a^3*b^4*sin(d*x + c) - 68*a*b^6*sin(d*x + c) + 6*a^6*b - 44*a^4*b^3 - 62*a^2*b^5 + 4*b^7) /((a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8)*(b*sin(d*x + c)^3 + a*si n(d*x + c)^2 - b*sin(d*x + c) - a)^2))/d
Time = 6.16 (sec) , antiderivative size = 688, normalized size of antiderivative = 2.10 \[ \int \frac {\sec ^5(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {\ln \left (\sin \left (c+d\,x\right )+1\right )\,\left (\frac {3\,b^2}{4\,{\left (a-b\right )}^5}-\frac {9\,b}{16\,{\left (a-b\right )}^4}+\frac {3}{16\,{\left (a-b\right )}^3}\right )}{d}-\frac {\ln \left (\sin \left (c+d\,x\right )-1\right )\,\left (\frac {9\,b}{16\,{\left (a+b\right )}^4}+\frac {3}{16\,{\left (a+b\right )}^3}+\frac {3\,b^2}{4\,{\left (a+b\right )}^5}\right )}{d}-\frac {\frac {3\,a^6\,b-22\,a^4\,b^3-31\,a^2\,b^5+2\,b^7}{4\,\left (a^8-4\,a^6\,b^2+6\,a^4\,b^4-4\,a^2\,b^6+b^8\right )}-\frac {\sin \left (c+d\,x\right )\,\left (5\,a^7-26\,a^5\,b^2+49\,a^3\,b^4+68\,a\,b^6\right )}{8\,\left (a^8-4\,a^6\,b^2+6\,a^4\,b^4-4\,a^2\,b^6+b^8\right )}-\frac {3\,{\sin \left (c+d\,x\right )}^5\,\left (-a^5\,b^2+6\,a^3\,b^4+27\,a\,b^6\right )}{8\,\left (a^8-4\,a^6\,b^2+6\,a^4\,b^4-4\,a^2\,b^6+b^8\right )}+\frac {{\sin \left (c+d\,x\right )}^3\,\left (3\,a^7-23\,a^5\,b^2+61\,a^3\,b^4+151\,a\,b^6\right )}{8\,\left (a^8-4\,a^6\,b^2+6\,a^4\,b^4-4\,a^2\,b^6+b^8\right )}+\frac {3\,{\sin \left (c+d\,x\right )}^4\,\left (a^6\,b-6\,a^4\,b^3-13\,a^2\,b^5+2\,b^7\right )}{4\,\left (a^8-4\,a^6\,b^2+6\,a^4\,b^4-4\,a^2\,b^6+b^8\right )}-\frac {{\sin \left (c+d\,x\right )}^2\,\left (5\,a^6\,b-37\,a^4\,b^3-73\,a^2\,b^5+9\,b^7\right )}{4\,\left (a^8-4\,a^6\,b^2+6\,a^4\,b^4-4\,a^2\,b^6+b^8\right )}}{d\,\left ({\sin \left (c+d\,x\right )}^4\,\left (a^2-2\,b^2\right )+a^2-{\sin \left (c+d\,x\right )}^2\,\left (2\,a^2-b^2\right )+b^2\,{\sin \left (c+d\,x\right )}^6+2\,a\,b\,\sin \left (c+d\,x\right )-4\,a\,b\,{\sin \left (c+d\,x\right )}^3+2\,a\,b\,{\sin \left (c+d\,x\right )}^5\right )}-\frac {\ln \left (a+b\,\sin \left (c+d\,x\right )\right )\,\left (21\,a^2\,b^5+3\,b^7\right )}{d\,\left (a^{10}-5\,a^8\,b^2+10\,a^6\,b^4-10\,a^4\,b^6+5\,a^2\,b^8-b^{10}\right )} \]
(log(sin(c + d*x) + 1)*((3*b^2)/(4*(a - b)^5) - (9*b)/(16*(a - b)^4) + 3/( 16*(a - b)^3)))/d - (log(sin(c + d*x) - 1)*((9*b)/(16*(a + b)^4) + 3/(16*( a + b)^3) + (3*b^2)/(4*(a + b)^5)))/d - ((3*a^6*b + 2*b^7 - 31*a^2*b^5 - 2 2*a^4*b^3)/(4*(a^8 + b^8 - 4*a^2*b^6 + 6*a^4*b^4 - 4*a^6*b^2)) - (sin(c + d*x)*(68*a*b^6 + 5*a^7 + 49*a^3*b^4 - 26*a^5*b^2))/(8*(a^8 + b^8 - 4*a^2*b ^6 + 6*a^4*b^4 - 4*a^6*b^2)) - (3*sin(c + d*x)^5*(27*a*b^6 + 6*a^3*b^4 - a ^5*b^2))/(8*(a^8 + b^8 - 4*a^2*b^6 + 6*a^4*b^4 - 4*a^6*b^2)) + (sin(c + d* x)^3*(151*a*b^6 + 3*a^7 + 61*a^3*b^4 - 23*a^5*b^2))/(8*(a^8 + b^8 - 4*a^2* b^6 + 6*a^4*b^4 - 4*a^6*b^2)) + (3*sin(c + d*x)^4*(a^6*b + 2*b^7 - 13*a^2* b^5 - 6*a^4*b^3))/(4*(a^8 + b^8 - 4*a^2*b^6 + 6*a^4*b^4 - 4*a^6*b^2)) - (s in(c + d*x)^2*(5*a^6*b + 9*b^7 - 73*a^2*b^5 - 37*a^4*b^3))/(4*(a^8 + b^8 - 4*a^2*b^6 + 6*a^4*b^4 - 4*a^6*b^2)))/(d*(sin(c + d*x)^4*(a^2 - 2*b^2) + a ^2 - sin(c + d*x)^2*(2*a^2 - b^2) + b^2*sin(c + d*x)^6 + 2*a*b*sin(c + d*x ) - 4*a*b*sin(c + d*x)^3 + 2*a*b*sin(c + d*x)^5)) - (log(a + b*sin(c + d*x ))*(3*b^7 + 21*a^2*b^5))/(d*(a^10 - b^10 + 5*a^2*b^8 - 10*a^4*b^6 + 10*a^6 *b^4 - 5*a^8*b^2))