Integrand size = 31, antiderivative size = 372 \[ \int \frac {\sec ^5(c+d x) (A+B \sin (c+d x))}{(a+b \sin (c+d x))^2} \, dx=-\frac {\left (3 a^2 A+2 a b (6 A+B)+b^2 (15 A+8 B)\right ) \log (1-\sin (c+d x))}{16 (a+b)^4 d}+\frac {\left (3 a^2 A+b^2 (15 A-8 B)-2 a b (6 A-B)\right ) \log (1+\sin (c+d x))}{16 (a-b)^4 d}-\frac {b^4 \left (6 a A b-5 a^2 B-b^2 B\right ) \log (a+b \sin (c+d x))}{\left (a^2-b^2\right )^4 d}-\frac {b \left (3 a^4 A-12 a^2 A b^2-15 A b^4+2 a^3 b B+22 a b^3 B\right )}{8 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))}-\frac {\sec ^4(c+d x) (A b-a B-(a A-b B) \sin (c+d x))}{4 \left (a^2-b^2\right ) d (a+b \sin (c+d x))}+\frac {\sec ^2(c+d x) \left (b \left (a^2 A+5 A b^2-6 a b B\right )+\left (3 a^3 A-9 a A b^2+2 a^2 b B+4 b^3 B\right ) \sin (c+d x)\right )}{8 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))} \]
-1/16*(3*A*a^2+2*a*b*(6*A+B)+b^2*(15*A+8*B))*ln(1-sin(d*x+c))/(a+b)^4/d+1/ 16*(3*A*a^2+b^2*(15*A-8*B)-2*a*b*(6*A-B))*ln(1+sin(d*x+c))/(a-b)^4/d-b^4*( 6*A*a*b-5*B*a^2-B*b^2)*ln(a+b*sin(d*x+c))/(a^2-b^2)^4/d-1/8*b*(3*A*a^4-12* A*a^2*b^2-15*A*b^4+2*B*a^3*b+22*B*a*b^3)/(a^2-b^2)^3/d/(a+b*sin(d*x+c))-1/ 4*sec(d*x+c)^4*(A*b-B*a-(A*a-B*b)*sin(d*x+c))/(a^2-b^2)/d/(a+b*sin(d*x+c)) +1/8*sec(d*x+c)^2*(b*(A*a^2+5*A*b^2-6*B*a*b)+(3*A*a^3-9*A*a*b^2+2*B*a^2*b+ 4*B*b^3)*sin(d*x+c))/(a^2-b^2)^2/d/(a+b*sin(d*x+c))
Time = 2.72 (sec) , antiderivative size = 370, normalized size of antiderivative = 0.99 \[ \int \frac {\sec ^5(c+d x) (A+B \sin (c+d x))}{(a+b \sin (c+d x))^2} \, dx=\frac {-\frac {\left (3 a^3 A-9 a A b^2+2 a^2 b B+4 b^3 B\right ) ((a-b) \log (1-\sin (c+d x))-(a+b) \log (1+\sin (c+d x))+2 b \log (a+b \sin (c+d x)))}{(a-b) (a+b)}+\frac {2 \left (-a^2+b^2\right ) \sec ^4(c+d x) (A b-a B+(-a A+b B) \sin (c+d x))}{a+b \sin (c+d x)}+\frac {\sec ^2(c+d x) \left (b \left (a^2 A+5 A b^2-6 a b B\right )+\left (3 a^3 A-9 a A b^2+2 a^2 b B+4 b^3 B\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)}+b \left (-3 a^4 A+12 a^2 A b^2+15 A b^4-2 a^3 b B-22 a b^3 B\right ) \left (-\frac {\log (1-\sin (c+d x))}{2 b (a+b)^2}+\frac {\log (1+\sin (c+d x))}{2 (a-b)^2 b}-\frac {2 a \log (a+b \sin (c+d x))}{(a-b)^2 (a+b)^2}+\frac {1}{\left (a^2-b^2\right ) (a+b \sin (c+d x))}\right )}{8 \left (a^2-b^2\right )^2 d} \]
(-(((3*a^3*A - 9*a*A*b^2 + 2*a^2*b*B + 4*b^3*B)*((a - b)*Log[1 - Sin[c + d *x]] - (a + b)*Log[1 + Sin[c + d*x]] + 2*b*Log[a + b*Sin[c + d*x]]))/((a - b)*(a + b))) + (2*(-a^2 + b^2)*Sec[c + d*x]^4*(A*b - a*B + (-(a*A) + b*B) *Sin[c + d*x]))/(a + b*Sin[c + d*x]) + (Sec[c + d*x]^2*(b*(a^2*A + 5*A*b^2 - 6*a*b*B) + (3*a^3*A - 9*a*A*b^2 + 2*a^2*b*B + 4*b^3*B)*Sin[c + d*x]))/( a + b*Sin[c + d*x]) + b*(-3*a^4*A + 12*a^2*A*b^2 + 15*A*b^4 - 2*a^3*b*B - 22*a*b^3*B)*(-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]))))/(8*(a^2 - b^2)^2*d)
Time = 0.81 (sec) , antiderivative size = 331, normalized size of antiderivative = 0.89, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {3042, 3316, 27, 663, 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.
| \(\displaystyle \int \frac {\sec ^5(c+d x) (A+B \sin (c+d x))}{(a+b \sin (c+d x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+B \sin (c+d x)}{\cos (c+d x)^5 (a+b \sin (c+d x))^2}dx\) |
| \(\Big \downarrow \) 3316 |
| \(\displaystyle \frac {b^5 \int \frac {A b+B \sin (c+d x) b}{b (a+b \sin (c+d x))^2 \left (b^2-b^2 \sin ^2(c+d x)\right )^3}d(b \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b^4 \int \frac {A b+B \sin (c+d x) b}{(a+b \sin (c+d x))^2 \left (b^2-b^2 \sin ^2(c+d x)\right )^3}d(b \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 663 |
| \(\displaystyle -\frac {b^4 \int \left (-\frac {3 A a^2-2 b (6 A-B) a+b^2 (15 A-8 B)}{16 (a-b)^4 b^4 (\sin (c+d x) b+b)}-\frac {3 A a^2+2 b (6 A+B) a+b^2 (15 A+8 B)}{16 b^4 (a+b)^4 (b-b \sin (c+d x))}+\frac {-5 B a^2+6 A b a-b^2 B}{\left (a^2-b^2\right )^4 (a+b \sin (c+d x))}-\frac {3 a A+7 b A+a B+5 b B}{16 b^3 (a+b)^3 (b-b \sin (c+d x))^2}+\frac {A b-a B}{\left (a^2-b^2\right )^3 (a+b \sin (c+d x))^2}-\frac {3 a A-7 b A-a B+5 b B}{16 (a-b)^3 b^3 (\sin (c+d x) b+b)^2}-\frac {A+B}{8 b^2 (a+b)^2 (b-b \sin (c+d x))^3}-\frac {A-B}{8 (a-b)^2 b^2 (\sin (c+d x) b+b)^3}\right )d(b \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {b^4 \left (-\frac {A b-a B}{\left (a^2-b^2\right )^3 (a+b \sin (c+d x))}+\frac {\left (-5 a^2 B+6 a A b-b^2 B\right ) \log (a+b \sin (c+d x))}{\left (a^2-b^2\right )^4}+\frac {\left (3 a^2 A+2 a b (6 A+B)+b^2 (15 A+8 B)\right ) \log (b-b \sin (c+d x))}{16 b^4 (a+b)^4}-\frac {\left (3 a^2 A-2 a b (6 A-B)+b^2 (15 A-8 B)\right ) \log (b \sin (c+d x)+b)}{16 b^4 (a-b)^4}-\frac {3 a A+a B+7 A b+5 b B}{16 b^3 (a+b)^3 (b-b \sin (c+d x))}+\frac {3 a A-a B-7 A b+5 b B}{16 b^3 (a-b)^3 (b \sin (c+d x)+b)}+\frac {A-B}{16 b^2 (a-b)^2 (b \sin (c+d x)+b)^2}-\frac {A+B}{16 b^2 (a+b)^2 (b-b \sin (c+d x))^2}\right )}{d}\) |
-((b^4*(((3*a^2*A + 2*a*b*(6*A + B) + b^2*(15*A + 8*B))*Log[b - b*Sin[c + d*x]])/(16*b^4*(a + b)^4) + ((6*a*A*b - 5*a^2*B - b^2*B)*Log[a + b*Sin[c + d*x]])/(a^2 - b^2)^4 - ((3*a^2*A + b^2*(15*A - 8*B) - 2*a*b*(6*A - B))*Lo g[b + b*Sin[c + d*x]])/(16*(a - b)^4*b^4) - (A + B)/(16*b^2*(a + b)^2*(b - b*Sin[c + d*x])^2) - (3*a*A + 7*A*b + a*B + 5*b*B)/(16*b^3*(a + b)^3*(b - b*Sin[c + d*x])) - (A*b - a*B)/((a^2 - b^2)^3*(a + b*Sin[c + d*x])) + (A - B)/(16*(a - b)^2*b^2*(b + b*Sin[c + d*x])^2) + (3*a*A - 7*A*b - a*B + 5* b*B)/(16*(a - b)^3*b^3*(b + b*Sin[c + d*x]))))/d)
3.16.58.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (c_.)*(x_ )^2)^(p_.), x_Symbol] :> With[{q = Rt[(-a)*c, 2]}, Simp[1/c^p Int[ExpandI ntegrand[(d + e*x)^m*(f + g*x)^n*(-q + c*x)^p*(q + c*x)^p, x], x], x] /; ! FractionalPowerFactorQ[q]] /; FreeQ[{a, c, d, e, f, g}, x] && ILtQ[p, -1] & & IntegersQ[m, n] && NiceSqrtQ[(-a)*c]
Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_ .)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[1/(b^p* f) Subst[Int[(a + x)^m*(c + (d/b)*x)^n*(b^2 - x^2)^((p - 1)/2), x], x, b* Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IntegerQ[(p - 1) /2] && NeQ[a^2 - b^2, 0]
Time = 4.37 (sec) , antiderivative size = 297, normalized size of antiderivative = 0.80
| method | result | size |
| derivativedivides | \(\frac {-\frac {b^{4} \left (6 A a b -5 B \,a^{2}-B \,b^{2}\right ) \ln \left (a +b \sin \left (d x +c \right )\right )}{\left (a +b \right )^{4} \left (a -b \right )^{4}}+\frac {\left (A b -B a \right ) b^{4}}{\left (a +b \right )^{3} \left (a -b \right )^{3} \left (a +b \sin \left (d x +c \right )\right )}-\frac {A -B}{16 \left (a -b \right )^{2} \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {3 a A -7 A b -B a +5 B b}{16 \left (a -b \right )^{3} \left (1+\sin \left (d x +c \right )\right )}+\frac {\left (3 A \,a^{2}-12 A a b +15 A \,b^{2}+2 B a b -8 B \,b^{2}\right ) \ln \left (1+\sin \left (d x +c \right )\right )}{16 \left (a -b \right )^{4}}-\frac {-A -B}{16 \left (a +b \right )^{2} \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {3 a A +7 A b +B a +5 B b}{16 \left (a +b \right )^{3} \left (\sin \left (d x +c \right )-1\right )}+\frac {\left (-3 A \,a^{2}-12 A a b -15 A \,b^{2}-2 B a b -8 B \,b^{2}\right ) \ln \left (\sin \left (d x +c \right )-1\right )}{16 \left (a +b \right )^{4}}}{d}\) | \(297\) |
| default | \(\frac {-\frac {b^{4} \left (6 A a b -5 B \,a^{2}-B \,b^{2}\right ) \ln \left (a +b \sin \left (d x +c \right )\right )}{\left (a +b \right )^{4} \left (a -b \right )^{4}}+\frac {\left (A b -B a \right ) b^{4}}{\left (a +b \right )^{3} \left (a -b \right )^{3} \left (a +b \sin \left (d x +c \right )\right )}-\frac {A -B}{16 \left (a -b \right )^{2} \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {3 a A -7 A b -B a +5 B b}{16 \left (a -b \right )^{3} \left (1+\sin \left (d x +c \right )\right )}+\frac {\left (3 A \,a^{2}-12 A a b +15 A \,b^{2}+2 B a b -8 B \,b^{2}\right ) \ln \left (1+\sin \left (d x +c \right )\right )}{16 \left (a -b \right )^{4}}-\frac {-A -B}{16 \left (a +b \right )^{2} \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {3 a A +7 A b +B a +5 B b}{16 \left (a +b \right )^{3} \left (\sin \left (d x +c \right )-1\right )}+\frac {\left (-3 A \,a^{2}-12 A a b -15 A \,b^{2}-2 B a b -8 B \,b^{2}\right ) \ln \left (\sin \left (d x +c \right )-1\right )}{16 \left (a +b \right )^{4}}}{d}\) | \(297\) |
| parallelrisch | \(\frac {-6 b^{4} \left (A a b -\frac {5}{6} B \,a^{2}-\frac {1}{6} B \,b^{2}\right ) a \left (\frac {b \sin \left (5 d x +5 c \right )}{2}+\frac {3 b \sin \left (3 d x +3 c \right )}{2}+b \sin \left (d x +c \right )+\cos \left (4 d x +4 c \right ) a +4 a \cos \left (2 d x +2 c \right )+3 a \right ) \ln \left (2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )-\frac {3 \left (A \,a^{2}+4 \left (A +\frac {B}{6}\right ) b a +5 b^{2} \left (A +\frac {8 B}{15}\right )\right ) a \left (\frac {b \sin \left (5 d x +5 c \right )}{2}+\frac {3 b \sin \left (3 d x +3 c \right )}{2}+b \sin \left (d x +c \right )+\cos \left (4 d x +4 c \right ) a +4 a \cos \left (2 d x +2 c \right )+3 a \right ) \left (a -b \right )^{4} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8}+\frac {3 \left (\left (A \,a^{2}-4 b \left (A -\frac {B}{6}\right ) a +5 b^{2} \left (A -\frac {8 B}{15}\right )\right ) a \left (\frac {b \sin \left (5 d x +5 c \right )}{2}+\frac {3 b \sin \left (3 d x +3 c \right )}{2}+b \sin \left (d x +c \right )+\cos \left (4 d x +4 c \right ) a +4 a \cos \left (2 d x +2 c \right )+3 a \right ) \left (a +b \right )^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\frac {22 \left (\frac {4 a \left (a -b \right )^{2} \left (a +b \right )^{2} \left (A b -B a \right ) \cos \left (2 d x +2 c \right )}{11}+\frac {a \left (a -b \right ) \left (a +b \right ) \left (A \,a^{2} b -7 A \,b^{3}-2 B \,a^{3}+8 B a \,b^{2}\right ) \cos \left (4 d x +4 c \right )}{22}+\left (-\frac {9}{11} A \,a^{4} b^{2}-\frac {6}{11} A \,a^{2} b^{4}-\frac {6}{11} A \,b^{6}+\frac {13}{22} B a \,b^{5}+\frac {3}{11} A \,a^{6}+\frac {1}{22} B \,a^{5} b +B \,a^{3} b^{3}\right ) \sin \left (3 d x +3 c \right )+\frac {\left (A \,a^{4} b -5 A \,a^{2} b^{3}-2 A \,b^{5}-\frac {1}{2} B \,a^{5}+3 B \,a^{3} b^{2}+\frac {7}{2} B a \,b^{4}\right ) b \sin \left (5 d x +5 c \right )}{11}+\left (-\frac {5}{11} B a \,b^{5}-\frac {26}{11} A \,a^{4} b^{2}+\frac {7}{11} A \,a^{2} b^{4}+A \,a^{6}-\frac {7}{11} B \,a^{5} b +\frac {24}{11} B \,a^{3} b^{3}-\frac {4}{11} A \,b^{6}\right ) \sin \left (d x +c \right )-\frac {9 \left (A \,a^{2} b -\frac {5}{3} A \,b^{3}-\frac {10}{9} B \,a^{3}+\frac {16}{9} B a \,b^{2}\right ) a \left (a +b \right ) \left (a -b \right )}{22}\right ) \left (a -b \right )}{3}\right ) \left (a +b \right )}{8}}{d a \left (\frac {b \sin \left (5 d x +5 c \right )}{2}+\frac {3 b \sin \left (3 d x +3 c \right )}{2}+b \sin \left (d x +c \right )+\cos \left (4 d x +4 c \right ) a +4 a \cos \left (2 d x +2 c \right )+3 a \right ) \left (a +b \right )^{4} \left (a -b \right )^{4}}\) | \(708\) |
| norman | \(\text {Expression too large to display}\) | \(1149\) |
| risch | \(\text {Expression too large to display}\) | \(2657\) |
1/d*(-b^4*(6*A*a*b-5*B*a^2-B*b^2)/(a+b)^4/(a-b)^4*ln(a+b*sin(d*x+c))+(A*b- B*a)*b^4/(a+b)^3/(a-b)^3/(a+b*sin(d*x+c))-1/16*(A-B)/(a-b)^2/(1+sin(d*x+c) )^2-1/16*(3*A*a-7*A*b-B*a+5*B*b)/(a-b)^3/(1+sin(d*x+c))+1/16*(3*A*a^2-12*A *a*b+15*A*b^2+2*B*a*b-8*B*b^2)/(a-b)^4*ln(1+sin(d*x+c))-1/16*(-A-B)/(a+b)^ 2/(sin(d*x+c)-1)^2-1/16*(3*A*a+7*A*b+B*a+5*B*b)/(a+b)^3/(sin(d*x+c)-1)+1/1 6/(a+b)^4*(-3*A*a^2-12*A*a*b-15*A*b^2-2*B*a*b-8*B*b^2)*ln(sin(d*x+c)-1))
Leaf count of result is larger than twice the leaf count of optimal. 881 vs. \(2 (359) = 718\).
Time = 1.82 (sec) , antiderivative size = 881, normalized size of antiderivative = 2.37 \[ \int \frac {\sec ^5(c+d x) (A+B \sin (c+d x))}{(a+b \sin (c+d x))^2} \, dx=\frac {4 \, B a^{7} - 4 \, A a^{6} b - 12 \, B a^{5} b^{2} + 12 \, A a^{4} b^{3} + 12 \, B a^{3} b^{4} - 12 \, A a^{2} b^{5} - 4 \, B a b^{6} + 4 \, A b^{7} - 2 \, {\left (3 \, A a^{6} b + 2 \, B a^{5} b^{2} - 15 \, A a^{4} b^{3} + 20 \, B a^{3} b^{4} - 3 \, A a^{2} b^{5} - 22 \, B a b^{6} + 15 \, A b^{7}\right )} \cos \left (d x + c\right )^{4} + 2 \, {\left (A a^{6} b - 6 \, B a^{5} b^{2} + 3 \, A a^{4} b^{3} + 12 \, B a^{3} b^{4} - 9 \, A a^{2} b^{5} - 6 \, B a b^{6} + 5 \, A b^{7}\right )} \cos \left (d x + c\right )^{2} + 16 \, {\left ({\left (5 \, B a^{2} b^{5} - 6 \, A a b^{6} + B b^{7}\right )} \cos \left (d x + c\right )^{4} \sin \left (d x + c\right ) + {\left (5 \, B a^{3} b^{4} - 6 \, A a^{2} b^{5} + B a b^{6}\right )} \cos \left (d x + c\right )^{4}\right )} \log \left (b \sin \left (d x + c\right ) + a\right ) + {\left ({\left (3 \, A a^{6} b + 2 \, B a^{5} b^{2} - 15 \, A a^{4} b^{3} - 20 \, B a^{3} b^{4} + 5 \, {\left (9 \, A - 8 \, B\right )} a^{2} b^{5} + 6 \, {\left (8 \, A - 5 \, B\right )} a b^{6} + {\left (15 \, A - 8 \, B\right )} b^{7}\right )} \cos \left (d x + c\right )^{4} \sin \left (d x + c\right ) + {\left (3 \, A a^{7} + 2 \, B a^{6} b - 15 \, A a^{5} b^{2} - 20 \, B a^{4} b^{3} + 5 \, {\left (9 \, A - 8 \, B\right )} a^{3} b^{4} + 6 \, {\left (8 \, A - 5 \, B\right )} a^{2} b^{5} + {\left (15 \, A - 8 \, B\right )} a b^{6}\right )} \cos \left (d x + c\right )^{4}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left ({\left (3 \, A a^{6} b + 2 \, B a^{5} b^{2} - 15 \, A a^{4} b^{3} - 20 \, B a^{3} b^{4} + 5 \, {\left (9 \, A + 8 \, B\right )} a^{2} b^{5} - 6 \, {\left (8 \, A + 5 \, B\right )} a b^{6} + {\left (15 \, A + 8 \, B\right )} b^{7}\right )} \cos \left (d x + c\right )^{4} \sin \left (d x + c\right ) + {\left (3 \, A a^{7} + 2 \, B a^{6} b - 15 \, A a^{5} b^{2} - 20 \, B a^{4} b^{3} + 5 \, {\left (9 \, A + 8 \, B\right )} a^{3} b^{4} - 6 \, {\left (8 \, A + 5 \, B\right )} a^{2} b^{5} + {\left (15 \, A + 8 \, B\right )} a b^{6}\right )} \cos \left (d x + c\right )^{4}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (2 \, A a^{7} - 2 \, B a^{6} b - 6 \, A a^{5} b^{2} + 6 \, B a^{4} b^{3} + 6 \, A a^{3} b^{4} - 6 \, B a^{2} b^{5} - 2 \, A a b^{6} + 2 \, B b^{7} + {\left (3 \, A a^{7} + 2 \, B a^{6} b - 15 \, A a^{5} b^{2} + 21 \, A a^{3} b^{4} - 6 \, B a^{2} b^{5} - 9 \, A a b^{6} + 4 \, B b^{7}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{16 \, {\left ({\left (a^{8} b - 4 \, a^{6} b^{3} + 6 \, a^{4} b^{5} - 4 \, a^{2} b^{7} + b^{9}\right )} d \cos \left (d x + c\right )^{4} \sin \left (d x + c\right ) + {\left (a^{9} - 4 \, a^{7} b^{2} + 6 \, a^{5} b^{4} - 4 \, a^{3} b^{6} + a b^{8}\right )} d \cos \left (d x + c\right )^{4}\right )}} \]
1/16*(4*B*a^7 - 4*A*a^6*b - 12*B*a^5*b^2 + 12*A*a^4*b^3 + 12*B*a^3*b^4 - 1 2*A*a^2*b^5 - 4*B*a*b^6 + 4*A*b^7 - 2*(3*A*a^6*b + 2*B*a^5*b^2 - 15*A*a^4* b^3 + 20*B*a^3*b^4 - 3*A*a^2*b^5 - 22*B*a*b^6 + 15*A*b^7)*cos(d*x + c)^4 + 2*(A*a^6*b - 6*B*a^5*b^2 + 3*A*a^4*b^3 + 12*B*a^3*b^4 - 9*A*a^2*b^5 - 6*B *a*b^6 + 5*A*b^7)*cos(d*x + c)^2 + 16*((5*B*a^2*b^5 - 6*A*a*b^6 + B*b^7)*c os(d*x + c)^4*sin(d*x + c) + (5*B*a^3*b^4 - 6*A*a^2*b^5 + B*a*b^6)*cos(d*x + c)^4)*log(b*sin(d*x + c) + a) + ((3*A*a^6*b + 2*B*a^5*b^2 - 15*A*a^4*b^ 3 - 20*B*a^3*b^4 + 5*(9*A - 8*B)*a^2*b^5 + 6*(8*A - 5*B)*a*b^6 + (15*A - 8 *B)*b^7)*cos(d*x + c)^4*sin(d*x + c) + (3*A*a^7 + 2*B*a^6*b - 15*A*a^5*b^2 - 20*B*a^4*b^3 + 5*(9*A - 8*B)*a^3*b^4 + 6*(8*A - 5*B)*a^2*b^5 + (15*A - 8*B)*a*b^6)*cos(d*x + c)^4)*log(sin(d*x + c) + 1) - ((3*A*a^6*b + 2*B*a^5* b^2 - 15*A*a^4*b^3 - 20*B*a^3*b^4 + 5*(9*A + 8*B)*a^2*b^5 - 6*(8*A + 5*B)* a*b^6 + (15*A + 8*B)*b^7)*cos(d*x + c)^4*sin(d*x + c) + (3*A*a^7 + 2*B*a^6 *b - 15*A*a^5*b^2 - 20*B*a^4*b^3 + 5*(9*A + 8*B)*a^3*b^4 - 6*(8*A + 5*B)*a ^2*b^5 + (15*A + 8*B)*a*b^6)*cos(d*x + c)^4)*log(-sin(d*x + c) + 1) + 2*(2 *A*a^7 - 2*B*a^6*b - 6*A*a^5*b^2 + 6*B*a^4*b^3 + 6*A*a^3*b^4 - 6*B*a^2*b^5 - 2*A*a*b^6 + 2*B*b^7 + (3*A*a^7 + 2*B*a^6*b - 15*A*a^5*b^2 + 21*A*a^3*b^ 4 - 6*B*a^2*b^5 - 9*A*a*b^6 + 4*B*b^7)*cos(d*x + c)^2)*sin(d*x + c))/((a^8 *b - 4*a^6*b^3 + 6*a^4*b^5 - 4*a^2*b^7 + b^9)*d*cos(d*x + c)^4*sin(d*x + c ) + (a^9 - 4*a^7*b^2 + 6*a^5*b^4 - 4*a^3*b^6 + a*b^8)*d*cos(d*x + c)^4)
\[ \int \frac {\sec ^5(c+d x) (A+B \sin (c+d x))}{(a+b \sin (c+d x))^2} \, dx=\int \frac {\left (A + B \sin {\left (c + d x \right )}\right ) \sec ^{5}{\left (c + d x \right )}}{\left (a + b \sin {\left (c + d x \right )}\right )^{2}}\, dx \]
Time = 0.24 (sec) , antiderivative size = 659, normalized size of antiderivative = 1.77 \[ \int \frac {\sec ^5(c+d x) (A+B \sin (c+d x))}{(a+b \sin (c+d x))^2} \, dx=\frac {\frac {16 \, {\left (5 \, B a^{2} b^{4} - 6 \, A a b^{5} + B b^{6}\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}} + \frac {{\left (3 \, A a^{2} - 2 \, {\left (6 \, A - B\right )} a b + {\left (15 \, A - 8 \, B\right )} b^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}} - \frac {{\left (3 \, A a^{2} + 2 \, {\left (6 \, A + B\right )} a b + {\left (15 \, A + 8 \, B\right )} b^{2}\right )} \log \left (\sin \left (d x + c\right ) - 1\right )}{a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}} + \frac {2 \, {\left (2 \, B a^{5} - 4 \, A a^{4} b - 12 \, B a^{3} b^{2} + 20 \, A a^{2} b^{3} - 14 \, B a b^{4} + 8 \, A b^{5} - {\left (3 \, A a^{4} b + 2 \, B a^{3} b^{2} - 12 \, A a^{2} b^{3} + 22 \, B a b^{4} - 15 \, A b^{5}\right )} \sin \left (d x + c\right )^{4} - {\left (3 \, A a^{5} + 2 \, B a^{4} b - 12 \, A a^{3} b^{2} + 2 \, B a^{2} b^{3} + 9 \, A a b^{4} - 4 \, B b^{5}\right )} \sin \left (d x + c\right )^{3} + {\left (5 \, A a^{4} b + 10 \, B a^{3} b^{2} - 28 \, A a^{2} b^{3} + 38 \, B a b^{4} - 25 \, A b^{5}\right )} \sin \left (d x + c\right )^{2} + {\left (5 \, A a^{5} - 16 \, A a^{3} b^{2} + 6 \, B a^{2} b^{3} + 11 \, A a b^{4} - 6 \, B b^{5}\right )} \sin \left (d x + c\right )\right )}}{a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6} + {\left (a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7}\right )} \sin \left (d x + c\right )^{5} + {\left (a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}\right )} \sin \left (d x + c\right )^{4} - 2 \, {\left (a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7}\right )} \sin \left (d x + c\right )^{3} - 2 \, {\left (a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}\right )} \sin \left (d x + c\right )^{2} + {\left (a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7}\right )} \sin \left (d x + c\right )}}{16 \, d} \]
1/16*(16*(5*B*a^2*b^4 - 6*A*a*b^5 + B*b^6)*log(b*sin(d*x + c) + a)/(a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8) + (3*A*a^2 - 2*(6*A - B)*a*b + (1 5*A - 8*B)*b^2)*log(sin(d*x + c) + 1)/(a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4) - (3*A*a^2 + 2*(6*A + B)*a*b + (15*A + 8*B)*b^2)*log(sin(d*x + c) - 1)/(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4) + 2*(2*B*a^5 - 4*A*a^4*b - 12*B*a^3*b^2 + 20*A*a^2*b^3 - 14*B*a*b^4 + 8*A*b^5 - (3*A*a^4*b + 2*B*a^ 3*b^2 - 12*A*a^2*b^3 + 22*B*a*b^4 - 15*A*b^5)*sin(d*x + c)^4 - (3*A*a^5 + 2*B*a^4*b - 12*A*a^3*b^2 + 2*B*a^2*b^3 + 9*A*a*b^4 - 4*B*b^5)*sin(d*x + c) ^3 + (5*A*a^4*b + 10*B*a^3*b^2 - 28*A*a^2*b^3 + 38*B*a*b^4 - 25*A*b^5)*sin (d*x + c)^2 + (5*A*a^5 - 16*A*a^3*b^2 + 6*B*a^2*b^3 + 11*A*a*b^4 - 6*B*b^5 )*sin(d*x + c))/(a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6 + (a^6*b - 3*a^4*b^3 + 3*a^2*b^5 - b^7)*sin(d*x + c)^5 + (a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)* sin(d*x + c)^4 - 2*(a^6*b - 3*a^4*b^3 + 3*a^2*b^5 - b^7)*sin(d*x + c)^3 - 2*(a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*sin(d*x + c)^2 + (a^6*b - 3*a^4*b^ 3 + 3*a^2*b^5 - b^7)*sin(d*x + c)))/d
Leaf count of result is larger than twice the leaf count of optimal. 761 vs. \(2 (359) = 718\).
Time = 0.41 (sec) , antiderivative size = 761, normalized size of antiderivative = 2.05 \[ \int \frac {\sec ^5(c+d x) (A+B \sin (c+d x))}{(a+b \sin (c+d x))^2} \, dx=\frac {\frac {16 \, {\left (5 \, B a^{2} b^{5} - 6 \, A a b^{6} + B b^{7}\right )} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{a^{8} b - 4 \, a^{6} b^{3} + 6 \, a^{4} b^{5} - 4 \, a^{2} b^{7} + b^{9}} - \frac {{\left (3 \, A a^{2} + 12 \, A a b + 2 \, B a b + 15 \, A b^{2} + 8 \, B b^{2}\right )} \log \left ({\left | -\sin \left (d x + c\right ) + 1 \right |}\right )}{a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}} + \frac {{\left (3 \, A a^{2} - 12 \, A a b + 2 \, B a b + 15 \, A b^{2} - 8 \, B b^{2}\right )} \log \left ({\left | -\sin \left (d x + c\right ) - 1 \right |}\right )}{a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}} - \frac {16 \, {\left (5 \, B a^{2} b^{5} \sin \left (d x + c\right ) - 6 \, A a b^{6} \sin \left (d x + c\right ) + B b^{7} \sin \left (d x + c\right ) + 6 \, B a^{3} b^{4} - 7 \, A a^{2} b^{5} + A 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 ) + a\right )}} + \frac {2 \, {\left (30 \, B a^{2} b^{4} \sin \left (d x + c\right )^{4} - 36 \, A a b^{5} \sin \left (d x + c\right )^{4} + 6 \, B b^{6} \sin \left (d x + c\right )^{4} - 3 \, A a^{6} \sin \left (d x + c\right )^{3} - 2 \, B a^{5} b \sin \left (d x + c\right )^{3} + 15 \, A a^{4} b^{2} \sin \left (d x + c\right )^{3} - 12 \, B a^{3} b^{3} \sin \left (d x + c\right )^{3} - 5 \, A a^{2} b^{4} \sin \left (d x + c\right )^{3} + 14 \, B a b^{5} \sin \left (d x + c\right )^{3} - 7 \, A b^{6} \sin \left (d x + c\right )^{3} + 12 \, B a^{4} b^{2} \sin \left (d x + c\right )^{2} - 16 \, A a^{3} b^{3} \sin \left (d x + c\right )^{2} - 68 \, B a^{2} b^{4} \sin \left (d x + c\right )^{2} + 88 \, A a b^{5} \sin \left (d x + c\right )^{2} - 16 \, B b^{6} \sin \left (d x + c\right )^{2} + 5 \, A a^{6} \sin \left (d x + c\right ) - 2 \, B a^{5} b \sin \left (d x + c\right ) - 17 \, A a^{4} b^{2} \sin \left (d x + c\right ) + 20 \, B a^{3} b^{3} \sin \left (d x + c\right ) + 3 \, A a^{2} b^{4} \sin \left (d x + c\right ) - 18 \, B a b^{5} \sin \left (d x + c\right ) + 9 \, A b^{6} \sin \left (d x + c\right ) + 2 \, B a^{6} - 4 \, A a^{5} b - 14 \, B a^{4} b^{2} + 24 \, A a^{3} b^{3} + 36 \, B a^{2} b^{4} - 56 \, A a b^{5} + 12 \, B b^{6}\right )}}{{\left (a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}\right )} {\left (\sin \left (d x + c\right )^{2} - 1\right )}^{2}}}{16 \, d} \]
1/16*(16*(5*B*a^2*b^5 - 6*A*a*b^6 + B*b^7)*log(abs(b*sin(d*x + c) + a))/(a ^8*b - 4*a^6*b^3 + 6*a^4*b^5 - 4*a^2*b^7 + b^9) - (3*A*a^2 + 12*A*a*b + 2* B*a*b + 15*A*b^2 + 8*B*b^2)*log(abs(-sin(d*x + c) + 1))/(a^4 + 4*a^3*b + 6 *a^2*b^2 + 4*a*b^3 + b^4) + (3*A*a^2 - 12*A*a*b + 2*B*a*b + 15*A*b^2 - 8*B *b^2)*log(abs(-sin(d*x + c) - 1))/(a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b ^4) - 16*(5*B*a^2*b^5*sin(d*x + c) - 6*A*a*b^6*sin(d*x + c) + B*b^7*sin(d* x + c) + 6*B*a^3*b^4 - 7*A*a^2*b^5 + A*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) + a)) + 2*(30*B*a^2*b^4*sin(d*x + c)^4 - 36*A*a*b^5*sin(d*x + c)^4 + 6*B*b^6*sin(d*x + c)^4 - 3*A*a^6*sin(d*x + c )^3 - 2*B*a^5*b*sin(d*x + c)^3 + 15*A*a^4*b^2*sin(d*x + c)^3 - 12*B*a^3*b^ 3*sin(d*x + c)^3 - 5*A*a^2*b^4*sin(d*x + c)^3 + 14*B*a*b^5*sin(d*x + c)^3 - 7*A*b^6*sin(d*x + c)^3 + 12*B*a^4*b^2*sin(d*x + c)^2 - 16*A*a^3*b^3*sin( d*x + c)^2 - 68*B*a^2*b^4*sin(d*x + c)^2 + 88*A*a*b^5*sin(d*x + c)^2 - 16* B*b^6*sin(d*x + c)^2 + 5*A*a^6*sin(d*x + c) - 2*B*a^5*b*sin(d*x + c) - 17* A*a^4*b^2*sin(d*x + c) + 20*B*a^3*b^3*sin(d*x + c) + 3*A*a^2*b^4*sin(d*x + c) - 18*B*a*b^5*sin(d*x + c) + 9*A*b^6*sin(d*x + c) + 2*B*a^6 - 4*A*a^5*b - 14*B*a^4*b^2 + 24*A*a^3*b^3 + 36*B*a^2*b^4 - 56*A*a*b^5 + 12*B*b^6)/((a ^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8)*(sin(d*x + c)^2 - 1)^2))/d
Time = 12.91 (sec) , antiderivative size = 615, normalized size of antiderivative = 1.65 \[ \int \frac {\sec ^5(c+d x) (A+B \sin (c+d x))}{(a+b \sin (c+d x))^2} \, dx=\frac {\frac {B\,a^5-2\,A\,a^4\,b-6\,B\,a^3\,b^2+10\,A\,a^2\,b^3-7\,B\,a\,b^4+4\,A\,b^5}{4\,\left (a^2-b^2\right )\,\left (a^4-2\,a^2\,b^2+b^4\right )}-\frac {{\sin \left (c+d\,x\right )}^4\,\left (3\,A\,a^4\,b+2\,B\,a^3\,b^2-12\,A\,a^2\,b^3+22\,B\,a\,b^4-15\,A\,b^5\right )}{8\,\left (a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6\right )}+\frac {\sin \left (c+d\,x\right )\,\left (5\,A\,a^3-11\,A\,a\,b^2+6\,B\,b^3\right )}{8\,\left (a^4-2\,a^2\,b^2+b^4\right )}-\frac {{\sin \left (c+d\,x\right )}^3\,\left (3\,A\,a^3+2\,B\,a^2\,b-9\,A\,a\,b^2+4\,B\,b^3\right )}{8\,\left (a^4-2\,a^2\,b^2+b^4\right )}+\frac {{\sin \left (c+d\,x\right )}^2\,\left (5\,A\,a^4\,b+10\,B\,a^3\,b^2-28\,A\,a^2\,b^3+38\,B\,a\,b^4-25\,A\,b^5\right )}{8\,\left (a^2-b^2\right )\,\left (a^4-2\,a^2\,b^2+b^4\right )}}{d\,\left (b\,{\sin \left (c+d\,x\right )}^5+a\,{\sin \left (c+d\,x\right )}^4-2\,b\,{\sin \left (c+d\,x\right )}^3-2\,a\,{\sin \left (c+d\,x\right )}^2+b\,\sin \left (c+d\,x\right )+a\right )}+\frac {\ln \left (a+b\,\sin \left (c+d\,x\right )\right )\,\left (5\,B\,a^2\,b^4-6\,A\,a\,b^5+B\,b^6\right )}{d\,\left (a^8-4\,a^6\,b^2+6\,a^4\,b^4-4\,a^2\,b^6+b^8\right )}-\frac {\ln \left (\sin \left (c+d\,x\right )-1\right )\,\left (3\,A\,a^2+\left (12\,A+2\,B\right )\,a\,b+\left (15\,A+8\,B\right )\,b^2\right )}{d\,\left (16\,a^4+64\,a^3\,b+96\,a^2\,b^2+64\,a\,b^3+16\,b^4\right )}+\frac {\ln \left (\sin \left (c+d\,x\right )+1\right )\,\left (3\,A\,a^2+\left (2\,B-12\,A\right )\,a\,b+\left (15\,A-8\,B\right )\,b^2\right )}{d\,\left (16\,a^4-64\,a^3\,b+96\,a^2\,b^2-64\,a\,b^3+16\,b^4\right )} \]
((4*A*b^5 + B*a^5 + 10*A*a^2*b^3 - 6*B*a^3*b^2 - 2*A*a^4*b - 7*B*a*b^4)/(4 *(a^2 - b^2)*(a^4 + b^4 - 2*a^2*b^2)) - (sin(c + d*x)^4*(2*B*a^3*b^2 - 12* A*a^2*b^3 - 15*A*b^5 + 3*A*a^4*b + 22*B*a*b^4))/(8*(a^6 - b^6 + 3*a^2*b^4 - 3*a^4*b^2)) + (sin(c + d*x)*(5*A*a^3 + 6*B*b^3 - 11*A*a*b^2))/(8*(a^4 + b^4 - 2*a^2*b^2)) - (sin(c + d*x)^3*(3*A*a^3 + 4*B*b^3 - 9*A*a*b^2 + 2*B*a ^2*b))/(8*(a^4 + b^4 - 2*a^2*b^2)) + (sin(c + d*x)^2*(10*B*a^3*b^2 - 28*A* a^2*b^3 - 25*A*b^5 + 5*A*a^4*b + 38*B*a*b^4))/(8*(a^2 - b^2)*(a^4 + b^4 - 2*a^2*b^2)))/(d*(a + b*sin(c + d*x) - 2*a*sin(c + d*x)^2 + a*sin(c + d*x)^ 4 - 2*b*sin(c + d*x)^3 + b*sin(c + d*x)^5)) + (log(a + b*sin(c + d*x))*(B* b^6 + 5*B*a^2*b^4 - 6*A*a*b^5))/(d*(a^8 + b^8 - 4*a^2*b^6 + 6*a^4*b^4 - 4* a^6*b^2)) - (log(sin(c + d*x) - 1)*(3*A*a^2 + b^2*(15*A + 8*B) + a*b*(12*A + 2*B)))/(d*(64*a*b^3 + 64*a^3*b + 16*a^4 + 16*b^4 + 96*a^2*b^2)) + (log( sin(c + d*x) + 1)*(3*A*a^2 + b^2*(15*A - 8*B) - a*b*(12*A - 2*B)))/(d*(16* a^4 - 64*a^3*b - 64*a*b^3 + 16*b^4 + 96*a^2*b^2))