Integrand size = 31, antiderivative size = 550 \[ \int \frac {\sec ^7(c+d x) (A+B \sin (c+d x))}{(a+b \sin (c+d x))^2} \, dx=-\frac {\left (5 a^3 A+a^2 b (25 A+2 B)+a b^2 (47 A+10 B)+b^3 (35 A+16 B)\right ) \log (1-\sin (c+d x))}{32 (a+b)^5 d}+\frac {\left (5 a^3 A-b^3 (35 A-16 B)+a b^2 (47 A-10 B)-a^2 (25 A b-2 b B)\right ) \log (1+\sin (c+d x))}{32 (a-b)^5 d}+\frac {b^6 \left (8 a A b-7 a^2 B-b^2 B\right ) \log (a+b \sin (c+d x))}{\left (a^2-b^2\right )^5 d}-\frac {b \left (5 a^6 A-23 a^4 A b^2+47 a^2 A b^4+35 A b^6+2 a^5 b B-12 a^3 b^3 B-54 a b^5 B\right )}{16 \left (a^2-b^2\right )^4 d (a+b \sin (c+d x))}-\frac {\sec ^6(c+d x) (A b-a B-(a A-b B) \sin (c+d x))}{6 \left (a^2-b^2\right ) d (a+b \sin (c+d x))}+\frac {\sec ^4(c+d x) \left (b \left (a^2 A+7 A b^2-8 a b B\right )+\left (5 a^3 A-13 a A b^2+2 a^2 b B+6 b^3 B\right ) \sin (c+d x)\right )}{24 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))}+\frac {\sec ^2(c+d x) \left (b \left (5 a^4 A-18 a^2 A b^2-35 A b^4+2 a^3 b B+46 a b^3 B\right )+3 \left (5 a^5 A-18 a^3 A b^2+29 a A b^4+2 a^4 b B-10 a^2 b^3 B-8 b^5 B\right ) \sin (c+d x)\right )}{48 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))} \]
-1/32*(5*A*a^3+a^2*b*(25*A+2*B)+a*b^2*(47*A+10*B)+b^3*(35*A+16*B))*ln(1-si n(d*x+c))/(a+b)^5/d+1/32*(5*A*a^3-b^3*(35*A-16*B)+a*b^2*(47*A-10*B)-a^2*(2 5*A*b-2*B*b))*ln(1+sin(d*x+c))/(a-b)^5/d+b^6*(8*A*a*b-7*B*a^2-B*b^2)*ln(a+ b*sin(d*x+c))/(a^2-b^2)^5/d-1/16*b*(5*A*a^6-23*A*a^4*b^2+47*A*a^2*b^4+35*A *b^6+2*B*a^5*b-12*B*a^3*b^3-54*B*a*b^5)/(a^2-b^2)^4/d/(a+b*sin(d*x+c))-1/6 *sec(d*x+c)^6*(A*b-B*a-(A*a-B*b)*sin(d*x+c))/(a^2-b^2)/d/(a+b*sin(d*x+c))+ 1/24*sec(d*x+c)^4*(b*(A*a^2+7*A*b^2-8*B*a*b)+(5*A*a^3-13*A*a*b^2+2*B*a^2*b +6*B*b^3)*sin(d*x+c))/(a^2-b^2)^2/d/(a+b*sin(d*x+c))+1/48*sec(d*x+c)^2*(b* (5*A*a^4-18*A*a^2*b^2-35*A*b^4+2*B*a^3*b+46*B*a*b^3)+3*(5*A*a^5-18*A*a^3*b ^2+29*A*a*b^4+2*B*a^4*b-10*B*a^2*b^3-8*B*b^5)*sin(d*x+c))/(a^2-b^2)^3/d/(a +b*sin(d*x+c))
Time = 5.74 (sec) , antiderivative size = 542, normalized size of antiderivative = 0.99 \[ \int \frac {\sec ^7(c+d x) (A+B \sin (c+d x))}{(a+b \sin (c+d x))^2} \, dx=\frac {\frac {b \sec ^6(c+d x) (A b-a B+(-a A+b B) \sin (c+d x))}{a+b \sin (c+d x)}+\frac {b \left (\frac {2 \left (-a^2+b^2\right ) \sec ^4(c+d x) \left (b \left (a^2 A+7 A b^2-8 a b B\right )+\left (5 a^3 A-13 a A b^2+2 a^2 b B+6 b^3 B\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)}+\frac {\sec ^2(c+d x) \left (b \left (-5 a^4 A+18 a^2 A b^2+35 A b^4-2 a^3 b B-46 a b^3 B\right )-3 \left (5 a^5 A-18 a^3 A b^2+29 a A b^4+2 a^4 b B-10 a^2 b^3 B-8 b^5 B\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)}+3 \left (\frac {\left (5 a^5 A-18 a^3 A b^2+29 a A b^4+2 a^4 b B-10 a^2 b^3 B-8 b^5 B\right ) ((a-b) \log (1-\sin (c+d x))-(a+b) \log (1+\sin (c+d x))+2 b \log (a+b \sin (c+d x)))}{(a-b) (a+b)}+b \left (5 a^6 A-23 a^4 A b^2+47 a^2 A b^4+35 A b^6+2 a^5 b B-12 a^3 b^3 B-54 a b^5 B\right ) \left (-\frac {\log (1-\sin (c+d x))}{2 b (a+b)^2}+\frac {\log (1+\sin (c+d x))}{2 (a-b)^2 b}-\frac {2 a \log (a+b \sin (c+d x))}{(a-b)^2 (a+b)^2}+\frac {1}{\left (a^2-b^2\right ) (a+b \sin (c+d x))}\right )\right )\right )}{8 \left (a^2-b^2\right )^2}}{6 b \left (-a^2+b^2\right ) d} \]
((b*Sec[c + d*x]^6*(A*b - a*B + (-(a*A) + b*B)*Sin[c + d*x]))/(a + b*Sin[c + d*x]) + (b*((2*(-a^2 + b^2)*Sec[c + d*x]^4*(b*(a^2*A + 7*A*b^2 - 8*a*b* B) + (5*a^3*A - 13*a*A*b^2 + 2*a^2*b*B + 6*b^3*B)*Sin[c + d*x]))/(a + b*Si n[c + d*x]) + (Sec[c + d*x]^2*(b*(-5*a^4*A + 18*a^2*A*b^2 + 35*A*b^4 - 2*a ^3*b*B - 46*a*b^3*B) - 3*(5*a^5*A - 18*a^3*A*b^2 + 29*a*A*b^4 + 2*a^4*b*B - 10*a^2*b^3*B - 8*b^5*B)*Sin[c + d*x]))/(a + b*Sin[c + d*x]) + 3*(((5*a^5 *A - 18*a^3*A*b^2 + 29*a*A*b^4 + 2*a^4*b*B - 10*a^2*b^3*B - 8*b^5*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)) + b*(5*a^6*A - 23*a^4*A*b^2 + 47*a^2*A*b ^4 + 35*A*b^6 + 2*a^5*b*B - 12*a^3*b^3*B - 54*a*b^5*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))/(6*b*(-a^2 + b^2)*d)
Time = 1.20 (sec) , antiderivative size = 476, normalized size of antiderivative = 0.87, 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.
| \(\displaystyle \int \frac {\sec ^7(c+d x) (A+B \sin (c+d x))}{(a+b \sin (c+d x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+B \sin (c+d x)}{\cos (c+d x)^7 (a+b \sin (c+d x))^2}dx\) |
| \(\Big \downarrow \) 3316 |
| \(\displaystyle \frac {b^7 \int \frac {A b+B \sin (c+d x) b}{b (a+b \sin (c+d x))^2 \left (b^2-b^2 \sin ^2(c+d x)\right )^4}d(b \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b^6 \int \frac {A b+B \sin (c+d x) b}{(a+b \sin (c+d x))^2 \left (b^2-b^2 \sin ^2(c+d x)\right )^4}d(b \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 663 |
| \(\displaystyle \frac {b^6 \int \left (\frac {(5 A-B) a^2-6 b (3 A-B) a+b^2 (19 A-11 B)}{32 (a-b)^4 b^5 (\sin (c+d x) b+b)^2}+\frac {5 A a^3+b (25 A+2 B) a^2+b^2 (47 A+10 B) a+b^3 (35 A+16 B)}{32 b^6 (a+b)^5 (b-b \sin (c+d x))}+\frac {-7 B a^2+8 A b a-b^2 B}{\left (a^2-b^2\right )^5 (a+b \sin (c+d x))}+\frac {5 A a^3-(25 A b-2 b B) a^2+b^2 (47 A-10 B) a-b^3 (35 A-16 B)}{32 (a-b)^5 b^6 (\sin (c+d x) b+b)}+\frac {(5 A+B) a^2+6 b (3 A+B) a+b^2 (19 A+11 B)}{32 b^5 (a+b)^4 (b-b \sin (c+d x))^2}+\frac {A b-a B}{\left (a^2-b^2\right )^4 (a+b \sin (c+d x))^2}+\frac {2 a A+4 b A+a B+3 b B}{16 b^4 (a+b)^3 (b-b \sin (c+d x))^3}+\frac {2 a A-4 b A-a B+3 b B}{16 (a-b)^3 b^4 (\sin (c+d x) b+b)^3}+\frac {A+B}{16 b^3 (a+b)^2 (b-b \sin (c+d x))^4}+\frac {A-B}{16 (a-b)^2 b^3 (\sin (c+d x) b+b)^4}\right )d(b \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {b^6 \left (-\frac {A b-a B}{\left (a^2-b^2\right )^4 (a+b \sin (c+d x))}+\frac {\left (-7 a^2 B+8 a A b-b^2 B\right ) \log (a+b \sin (c+d x))}{\left (a^2-b^2\right )^5}-\frac {a^2 (5 A-B)-6 a b (3 A-B)+b^2 (19 A-11 B)}{32 b^5 (a-b)^4 (b \sin (c+d x)+b)}+\frac {a^2 (5 A+B)+6 a b (3 A+B)+b^2 (19 A+11 B)}{32 b^5 (a+b)^4 (b-b \sin (c+d x))}-\frac {\left (5 a^3 A+a^2 b (25 A+2 B)+a b^2 (47 A+10 B)+b^3 (35 A+16 B)\right ) \log (b-b \sin (c+d x))}{32 b^6 (a+b)^5}+\frac {\left (5 a^3 A-a^2 (25 A b-2 b B)+a b^2 (47 A-10 B)-b^3 (35 A-16 B)\right ) \log (b \sin (c+d x)+b)}{32 b^6 (a-b)^5}+\frac {2 a A+a B+4 A b+3 b B}{32 b^4 (a+b)^3 (b-b \sin (c+d x))^2}-\frac {2 a A-a B-4 A b+3 b B}{32 b^4 (a-b)^3 (b \sin (c+d x)+b)^2}+\frac {A+B}{48 b^3 (a+b)^2 (b-b \sin (c+d x))^3}-\frac {A-B}{48 b^3 (a-b)^2 (b \sin (c+d x)+b)^3}\right )}{d}\) |
(b^6*(-1/32*((5*a^3*A + a^2*b*(25*A + 2*B) + a*b^2*(47*A + 10*B) + b^3*(35 *A + 16*B))*Log[b - b*Sin[c + d*x]])/(b^6*(a + b)^5) + ((8*a*A*b - 7*a^2*B - b^2*B)*Log[a + b*Sin[c + d*x]])/(a^2 - b^2)^5 + ((5*a^3*A - b^3*(35*A - 16*B) + a*b^2*(47*A - 10*B) - a^2*(25*A*b - 2*b*B))*Log[b + b*Sin[c + d*x ]])/(32*(a - b)^5*b^6) + (A + B)/(48*b^3*(a + b)^2*(b - b*Sin[c + d*x])^3) + (2*a*A + 4*A*b + a*B + 3*b*B)/(32*b^4*(a + b)^3*(b - b*Sin[c + d*x])^2) + (6*a*b*(3*A + B) + a^2*(5*A + B) + b^2*(19*A + 11*B))/(32*b^5*(a + b)^4 *(b - b*Sin[c + d*x])) - (A*b - a*B)/((a^2 - b^2)^4*(a + b*Sin[c + d*x])) - (A - B)/(48*(a - b)^2*b^3*(b + b*Sin[c + d*x])^3) - (2*a*A - 4*A*b - a*B + 3*b*B)/(32*(a - b)^3*b^4*(b + b*Sin[c + d*x])^2) - (b^2*(19*A - 11*B) - 6*a*b*(3*A - B) + a^2*(5*A - B))/(32*(a - b)^4*b^5*(b + b*Sin[c + d*x]))) )/d
3.16.59.3.1 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 = 6.84 (sec) , antiderivative size = 435, normalized size of antiderivative = 0.79
| method | result | size |
| derivativedivides | \(\frac {-\frac {A +B}{48 \left (a +b \right )^{2} \left (\sin \left (d x +c \right )-1\right )^{3}}-\frac {-2 a A -4 A b -B a -3 B b}{32 \left (a +b \right )^{3} \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {5 A \,a^{2}+18 A a b +19 A \,b^{2}+B \,a^{2}+6 B a b +11 B \,b^{2}}{32 \left (a +b \right )^{4} \left (\sin \left (d x +c \right )-1\right )}+\frac {\left (-5 A \,a^{3}-25 A \,a^{2} b -47 A a \,b^{2}-35 A \,b^{3}-2 B \,a^{2} b -10 B a \,b^{2}-16 B \,b^{3}\right ) \ln \left (\sin \left (d x +c \right )-1\right )}{32 \left (a +b \right )^{5}}+\frac {b^{6} \left (8 A a b -7 B \,a^{2}-B \,b^{2}\right ) \ln \left (a +b \sin \left (d x +c \right )\right )}{\left (a +b \right )^{5} \left (a -b \right )^{5}}-\frac {\left (A b -B a \right ) b^{6}}{\left (a +b \right )^{4} \left (a -b \right )^{4} \left (a +b \sin \left (d x +c \right )\right )}-\frac {A -B}{48 \left (a -b \right )^{2} \left (1+\sin \left (d x +c \right )\right )^{3}}-\frac {2 a A -4 A b -B a +3 B b}{32 \left (a -b \right )^{3} \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {5 A \,a^{2}-18 A a b +19 A \,b^{2}-B \,a^{2}+6 B a b -11 B \,b^{2}}{32 \left (a -b \right )^{4} \left (1+\sin \left (d x +c \right )\right )}+\frac {\left (5 A \,a^{3}-25 A \,a^{2} b +47 A a \,b^{2}-35 A \,b^{3}+2 B \,a^{2} b -10 B a \,b^{2}+16 B \,b^{3}\right ) \ln \left (1+\sin \left (d x +c \right )\right )}{32 \left (a -b \right )^{5}}}{d}\) | \(435\) |
| default | \(\frac {-\frac {A +B}{48 \left (a +b \right )^{2} \left (\sin \left (d x +c \right )-1\right )^{3}}-\frac {-2 a A -4 A b -B a -3 B b}{32 \left (a +b \right )^{3} \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {5 A \,a^{2}+18 A a b +19 A \,b^{2}+B \,a^{2}+6 B a b +11 B \,b^{2}}{32 \left (a +b \right )^{4} \left (\sin \left (d x +c \right )-1\right )}+\frac {\left (-5 A \,a^{3}-25 A \,a^{2} b -47 A a \,b^{2}-35 A \,b^{3}-2 B \,a^{2} b -10 B a \,b^{2}-16 B \,b^{3}\right ) \ln \left (\sin \left (d x +c \right )-1\right )}{32 \left (a +b \right )^{5}}+\frac {b^{6} \left (8 A a b -7 B \,a^{2}-B \,b^{2}\right ) \ln \left (a +b \sin \left (d x +c \right )\right )}{\left (a +b \right )^{5} \left (a -b \right )^{5}}-\frac {\left (A b -B a \right ) b^{6}}{\left (a +b \right )^{4} \left (a -b \right )^{4} \left (a +b \sin \left (d x +c \right )\right )}-\frac {A -B}{48 \left (a -b \right )^{2} \left (1+\sin \left (d x +c \right )\right )^{3}}-\frac {2 a A -4 A b -B a +3 B b}{32 \left (a -b \right )^{3} \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {5 A \,a^{2}-18 A a b +19 A \,b^{2}-B \,a^{2}+6 B a b -11 B \,b^{2}}{32 \left (a -b \right )^{4} \left (1+\sin \left (d x +c \right )\right )}+\frac {\left (5 A \,a^{3}-25 A \,a^{2} b +47 A a \,b^{2}-35 A \,b^{3}+2 B \,a^{2} b -10 B a \,b^{2}+16 B \,b^{3}\right ) \ln \left (1+\sin \left (d x +c \right )\right )}{32 \left (a -b \right )^{5}}}{d}\) | \(435\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(1118\) |
| risch | \(\text {Expression too large to display}\) | \(4554\) |
1/d*(-1/48*(A+B)/(a+b)^2/(sin(d*x+c)-1)^3-1/32*(-2*A*a-4*A*b-B*a-3*B*b)/(a +b)^3/(sin(d*x+c)-1)^2-1/32*(5*A*a^2+18*A*a*b+19*A*b^2+B*a^2+6*B*a*b+11*B* b^2)/(a+b)^4/(sin(d*x+c)-1)+1/32/(a+b)^5*(-5*A*a^3-25*A*a^2*b-47*A*a*b^2-3 5*A*b^3-2*B*a^2*b-10*B*a*b^2-16*B*b^3)*ln(sin(d*x+c)-1)+b^6*(8*A*a*b-7*B*a ^2-B*b^2)/(a+b)^5/(a-b)^5*ln(a+b*sin(d*x+c))-(A*b-B*a)*b^6/(a+b)^4/(a-b)^4 /(a+b*sin(d*x+c))-1/48*(A-B)/(a-b)^2/(1+sin(d*x+c))^3-1/32*(2*A*a-4*A*b-B* a+3*B*b)/(a-b)^3/(1+sin(d*x+c))^2-1/32*(5*A*a^2-18*A*a*b+19*A*b^2-B*a^2+6* B*a*b-11*B*b^2)/(a-b)^4/(1+sin(d*x+c))+1/32*(5*A*a^3-25*A*a^2*b+47*A*a*b^2 -35*A*b^3+2*B*a^2*b-10*B*a*b^2+16*B*b^3)/(a-b)^5*ln(1+sin(d*x+c)))
Leaf count of result is larger than twice the leaf count of optimal. 1244 vs. \(2 (537) = 1074\).
Time = 4.60 (sec) , antiderivative size = 1244, normalized size of antiderivative = 2.26 \[ \int \frac {\sec ^7(c+d x) (A+B \sin (c+d x))}{(a+b \sin (c+d x))^2} \, dx=\text {Too large to display} \]
1/96*(16*B*a^9 - 16*A*a^8*b - 64*B*a^7*b^2 + 64*A*a^6*b^3 + 96*B*a^5*b^4 - 96*A*a^4*b^5 - 64*B*a^3*b^6 + 64*A*a^2*b^7 + 16*B*a*b^8 - 16*A*b^9 - 6*(5 *A*a^8*b + 2*B*a^7*b^2 - 28*A*a^6*b^3 - 14*B*a^5*b^4 + 70*A*a^4*b^5 - 42*B *a^3*b^6 - 12*A*a^2*b^7 + 54*B*a*b^8 - 35*A*b^9)*cos(d*x + c)^6 + 2*(5*A*a ^8*b + 2*B*a^7*b^2 - 28*A*a^6*b^3 + 42*B*a^5*b^4 + 6*A*a^4*b^5 - 90*B*a^3* b^6 + 52*A*a^2*b^7 + 46*B*a*b^8 - 35*A*b^9)*cos(d*x + c)^4 + 4*(A*a^8*b - 8*B*a^7*b^2 + 4*A*a^6*b^3 + 24*B*a^5*b^4 - 18*A*a^4*b^5 - 24*B*a^3*b^6 + 2 0*A*a^2*b^7 + 8*B*a*b^8 - 7*A*b^9)*cos(d*x + c)^2 - 96*((7*B*a^2*b^7 - 8*A *a*b^8 + B*b^9)*cos(d*x + c)^6*sin(d*x + c) + (7*B*a^3*b^6 - 8*A*a^2*b^7 + B*a*b^8)*cos(d*x + c)^6)*log(b*sin(d*x + c) + a) + 3*((5*A*a^8*b + 2*B*a^ 7*b^2 - 28*A*a^6*b^3 - 14*B*a^5*b^4 + 70*A*a^4*b^5 + 70*B*a^3*b^6 - 28*(5* A - 4*B)*a^2*b^7 - 2*(64*A - 35*B)*a*b^8 - (35*A - 16*B)*b^9)*cos(d*x + c) ^6*sin(d*x + c) + (5*A*a^9 + 2*B*a^8*b - 28*A*a^7*b^2 - 14*B*a^6*b^3 + 70* A*a^5*b^4 + 70*B*a^4*b^5 - 28*(5*A - 4*B)*a^3*b^6 - 2*(64*A - 35*B)*a^2*b^ 7 - (35*A - 16*B)*a*b^8)*cos(d*x + c)^6)*log(sin(d*x + c) + 1) - 3*((5*A*a ^8*b + 2*B*a^7*b^2 - 28*A*a^6*b^3 - 14*B*a^5*b^4 + 70*A*a^4*b^5 + 70*B*a^3 *b^6 - 28*(5*A + 4*B)*a^2*b^7 + 2*(64*A + 35*B)*a*b^8 - (35*A + 16*B)*b^9) *cos(d*x + c)^6*sin(d*x + c) + (5*A*a^9 + 2*B*a^8*b - 28*A*a^7*b^2 - 14*B* a^6*b^3 + 70*A*a^5*b^4 + 70*B*a^4*b^5 - 28*(5*A + 4*B)*a^3*b^6 + 2*(64*A + 35*B)*a^2*b^7 - (35*A + 16*B)*a*b^8)*cos(d*x + c)^6)*log(-sin(d*x + c)...
Timed out. \[ \int \frac {\sec ^7(c+d x) (A+B \sin (c+d x))}{(a+b \sin (c+d x))^2} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 1083 vs. \(2 (537) = 1074\).
Time = 0.27 (sec) , antiderivative size = 1083, normalized size of antiderivative = 1.97 \[ \int \frac {\sec ^7(c+d x) (A+B \sin (c+d x))}{(a+b \sin (c+d x))^2} \, dx=\text {Too large to display} \]
-1/96*(96*(7*B*a^2*b^6 - 8*A*a*b^7 + B*b^8)*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*(5*A*a^3 - ( 25*A - 2*B)*a^2*b + (47*A - 10*B)*a*b^2 - (35*A - 16*B)*b^3)*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) + 3*(5* A*a^3 + (25*A + 2*B)*a^2*b + (47*A + 10*B)*a*b^2 + (35*A + 16*B)*b^3)*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*(8*B*a^7 - 16*A*a^6*b - 44*B*a^5*b^2 + 80*A*a^4*b^3 + 136*B*a^3*b^4 - 208*A*a^2*b^5 + 92*B*a*b^6 - 48*A*b^7 + 3*(5*A*a^6*b + 2*B*a^5*b^2 - 23* A*a^4*b^3 - 12*B*a^3*b^4 + 47*A*a^2*b^5 - 54*B*a*b^6 + 35*A*b^7)*sin(d*x + c)^6 + 3*(5*A*a^7 + 2*B*a^6*b - 23*A*a^5*b^2 - 12*B*a^4*b^3 + 47*A*a^3*b^ 4 + 2*B*a^2*b^5 - 29*A*a*b^6 + 8*B*b^7)*sin(d*x + c)^5 - 8*(5*A*a^6*b + 2* B*a^5*b^2 - 23*A*a^4*b^3 - 19*B*a^3*b^4 + 55*A*a^2*b^5 - 55*B*a*b^6 + 35*A *b^7)*sin(d*x + c)^4 - 4*(10*A*a^7 + 4*B*a^6*b - 46*A*a^5*b^2 - 17*B*a^4*b ^3 + 86*A*a^3*b^4 - 2*B*a^2*b^5 - 50*A*a*b^6 + 15*B*b^7)*sin(d*x + c)^3 + 3*(11*A*a^6*b + 10*B*a^5*b^2 - 57*A*a^4*b^3 - 76*B*a^3*b^4 + 161*A*a^2*b^5 - 126*B*a*b^6 + 77*A*b^7)*sin(d*x + c)^2 + (33*A*a^7 + 2*B*a^6*b - 139*A* a^5*b^2 - 8*B*a^4*b^3 + 227*A*a^3*b^4 - 38*B*a^2*b^5 - 121*A*a*b^6 + 44*B* b^7)*sin(d*x + c))/(a^9 - 4*a^7*b^2 + 6*a^5*b^4 - 4*a^3*b^6 + a*b^8 - (a^8 *b - 4*a^6*b^3 + 6*a^4*b^5 - 4*a^2*b^7 + b^9)*sin(d*x + c)^7 - (a^9 - 4*a^ 7*b^2 + 6*a^5*b^4 - 4*a^3*b^6 + a*b^8)*sin(d*x + c)^6 + 3*(a^8*b - 4*a^...
Leaf count of result is larger than twice the leaf count of optimal. 1185 vs. \(2 (537) = 1074\).
Time = 0.45 (sec) , antiderivative size = 1185, normalized size of antiderivative = 2.15 \[ \int \frac {\sec ^7(c+d x) (A+B \sin (c+d x))}{(a+b \sin (c+d x))^2} \, dx=\text {Too large to display} \]
-1/96*(96*(7*B*a^2*b^7 - 8*A*a*b^8 + B*b^9)*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*(5*A* a^3 + 25*A*a^2*b + 2*B*a^2*b + 47*A*a*b^2 + 10*B*a*b^2 + 35*A*b^3 + 16*B*b ^3)*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*(5*A*a^3 - 25*A*a^2*b + 2*B*a^2*b + 47*A*a*b^2 - 10*B* a*b^2 - 35*A*b^3 + 16*B*b^3)*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) - 96*(7*B*a^2*b^7*sin(d*x + c) - 8*A*a*b^8*sin(d*x + c) + B*b^9*sin(d*x + c) + 8*B*a^3*b^6 - 9*A*a^2*b^7 + A*b^9)/((a^10 - 5*a^8*b^2 + 10*a^6*b^4 - 10*a^4*b^6 + 5*a^2*b^8 - b^10)*(b *sin(d*x + c) + a)) + 2*(308*B*a^2*b^6*sin(d*x + c)^6 - 352*A*a*b^7*sin(d* x + c)^6 + 44*B*b^8*sin(d*x + c)^6 + 15*A*a^8*sin(d*x + c)^5 + 6*B*a^7*b*s in(d*x + c)^5 - 84*A*a^6*b^2*sin(d*x + c)^5 - 42*B*a^5*b^3*sin(d*x + c)^5 + 210*A*a^4*b^4*sin(d*x + c)^5 - 78*B*a^3*b^5*sin(d*x + c)^5 - 84*A*a^2*b^ 6*sin(d*x + c)^5 + 114*B*a*b^7*sin(d*x + c)^5 - 57*A*b^8*sin(d*x + c)^5 + 120*B*a^4*b^4*sin(d*x + c)^4 - 144*A*a^3*b^5*sin(d*x + c)^4 - 1020*B*a^2*b ^6*sin(d*x + c)^4 + 1200*A*a*b^7*sin(d*x + c)^4 - 156*B*b^8*sin(d*x + c)^4 - 40*A*a^8*sin(d*x + c)^3 - 16*B*a^7*b*sin(d*x + c)^3 + 224*A*a^6*b^2*sin (d*x + c)^3 + 48*B*a^5*b^3*sin(d*x + c)^3 - 480*A*a^4*b^4*sin(d*x + c)^3 + 240*B*a^3*b^5*sin(d*x + c)^3 + 160*A*a^2*b^6*sin(d*x + c)^3 - 272*B*a*b^7 *sin(d*x + c)^3 + 136*A*b^8*sin(d*x + c)^3 + 36*B*a^6*b^2*sin(d*x + c)^...
Time = 13.94 (sec) , antiderivative size = 1024, normalized size of antiderivative = 1.86 \[ \int \frac {\sec ^7(c+d x) (A+B \sin (c+d x))}{(a+b \sin (c+d x))^2} \, dx=\frac {\frac {\sin \left (c+d\,x\right )\,\left (33\,A\,a^5+2\,B\,a^4\,b-106\,A\,a^3\,b^2-6\,B\,a^2\,b^3+121\,A\,a\,b^4-44\,B\,b^5\right )}{48\,\left (a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6\right )}+\frac {{\sin \left (c+d\,x\right )}^5\,\left (5\,A\,a^5+2\,B\,a^4\,b-18\,A\,a^3\,b^2-10\,B\,a^2\,b^3+29\,A\,a\,b^4-8\,B\,b^5\right )}{16\,\left (a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6\right )}-\frac {{\sin \left (c+d\,x\right )}^3\,\left (10\,A\,a^5+4\,B\,a^4\,b-36\,A\,a^3\,b^2-13\,B\,a^2\,b^3+50\,A\,a\,b^4-15\,B\,b^5\right )}{12\,\left (a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6\right )}-\frac {-2\,B\,a^7+4\,A\,a^6\,b+11\,B\,a^5\,b^2-20\,A\,a^4\,b^3-34\,B\,a^3\,b^4+52\,A\,a^2\,b^5-23\,B\,a\,b^6+12\,A\,b^7}{12\,\left (a^2-b^2\right )\,\left (a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6\right )}+\frac {{\sin \left (c+d\,x\right )}^6\,\left (5\,A\,a^6\,b+2\,B\,a^5\,b^2-23\,A\,a^4\,b^3-12\,B\,a^3\,b^4+47\,A\,a^2\,b^5-54\,B\,a\,b^6+35\,A\,b^7\right )}{16\,\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 )}^4\,\left (5\,A\,a^6\,b+2\,B\,a^5\,b^2-23\,A\,a^4\,b^3-19\,B\,a^3\,b^4+55\,A\,a^2\,b^5-55\,B\,a\,b^6+35\,A\,b^7\right )}{6\,\left (a^2-b^2\right )\,\left (a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6\right )}+\frac {{\sin \left (c+d\,x\right )}^2\,\left (11\,A\,a^6\,b+10\,B\,a^5\,b^2-57\,A\,a^4\,b^3-76\,B\,a^3\,b^4+161\,A\,a^2\,b^5-126\,B\,a\,b^6+77\,A\,b^7\right )}{16\,\left (a^2-b^2\right )\,\left (a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6\right )}}{d\,\left (-b\,{\sin \left (c+d\,x\right )}^7-a\,{\sin \left (c+d\,x\right )}^6+3\,b\,{\sin \left (c+d\,x\right )}^5+3\,a\,{\sin \left (c+d\,x\right )}^4-3\,b\,{\sin \left (c+d\,x\right )}^3-3\,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 (7\,B\,a^2\,b^6-8\,A\,a\,b^7+B\,b^8\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 )}-\frac {\ln \left (\sin \left (c+d\,x\right )-1\right )\,\left (5\,A\,a^3+\left (25\,A+2\,B\right )\,a^2\,b+\left (47\,A+10\,B\right )\,a\,b^2+\left (35\,A+16\,B\right )\,b^3\right )}{d\,\left (32\,a^5+160\,a^4\,b+320\,a^3\,b^2+320\,a^2\,b^3+160\,a\,b^4+32\,b^5\right )}+\frac {\ln \left (\sin \left (c+d\,x\right )+1\right )\,\left (5\,A\,a^3+\left (2\,B-25\,A\right )\,a^2\,b+\left (47\,A-10\,B\right )\,a\,b^2+\left (16\,B-35\,A\right )\,b^3\right )}{d\,\left (32\,a^5-160\,a^4\,b+320\,a^3\,b^2-320\,a^2\,b^3+160\,a\,b^4-32\,b^5\right )} \]
((sin(c + d*x)*(33*A*a^5 - 44*B*b^5 - 106*A*a^3*b^2 - 6*B*a^2*b^3 + 121*A* a*b^4 + 2*B*a^4*b))/(48*(a^6 - b^6 + 3*a^2*b^4 - 3*a^4*b^2)) + (sin(c + d* x)^5*(5*A*a^5 - 8*B*b^5 - 18*A*a^3*b^2 - 10*B*a^2*b^3 + 29*A*a*b^4 + 2*B*a ^4*b))/(16*(a^6 - b^6 + 3*a^2*b^4 - 3*a^4*b^2)) - (sin(c + d*x)^3*(10*A*a^ 5 - 15*B*b^5 - 36*A*a^3*b^2 - 13*B*a^2*b^3 + 50*A*a*b^4 + 4*B*a^4*b))/(12* (a^6 - b^6 + 3*a^2*b^4 - 3*a^4*b^2)) - (12*A*b^7 - 2*B*a^7 + 52*A*a^2*b^5 - 20*A*a^4*b^3 - 34*B*a^3*b^4 + 11*B*a^5*b^2 + 4*A*a^6*b - 23*B*a*b^6)/(12 *(a^2 - b^2)*(a^6 - b^6 + 3*a^2*b^4 - 3*a^4*b^2)) + (sin(c + d*x)^6*(35*A* b^7 + 47*A*a^2*b^5 - 23*A*a^4*b^3 - 12*B*a^3*b^4 + 2*B*a^5*b^2 + 5*A*a^6*b - 54*B*a*b^6))/(16*(a^8 + b^8 - 4*a^2*b^6 + 6*a^4*b^4 - 4*a^6*b^2)) - (si n(c + d*x)^4*(35*A*b^7 + 55*A*a^2*b^5 - 23*A*a^4*b^3 - 19*B*a^3*b^4 + 2*B* a^5*b^2 + 5*A*a^6*b - 55*B*a*b^6))/(6*(a^2 - b^2)*(a^6 - b^6 + 3*a^2*b^4 - 3*a^4*b^2)) + (sin(c + d*x)^2*(77*A*b^7 + 161*A*a^2*b^5 - 57*A*a^4*b^3 - 76*B*a^3*b^4 + 10*B*a^5*b^2 + 11*A*a^6*b - 126*B*a*b^6))/(16*(a^2 - b^2)*( a^6 - b^6 + 3*a^2*b^4 - 3*a^4*b^2)))/(d*(a + b*sin(c + d*x) - 3*a*sin(c + d*x)^2 + 3*a*sin(c + d*x)^4 - a*sin(c + d*x)^6 - 3*b*sin(c + d*x)^3 + 3*b* sin(c + d*x)^5 - b*sin(c + d*x)^7)) - (log(a + b*sin(c + d*x))*(B*b^8 + 7* B*a^2*b^6 - 8*A*a*b^7))/(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)) - (log(sin(c + d*x) - 1)*(5*A*a^3 + b^3*(35*A + 16*B) + a^2*b*(25*A + 2*B) + a*b^2*(47*A + 10*B)))/(d*(160*a*b^4 + 160*a^4*b + ...