\(\int \frac {\sec ^7(c+d x) (A+B \sin (c+d x))}{(a+b \sin (c+d x))^2} \, dx\) [1555]

Optimal result

Integrand size = 31, antiderivative size = 469 \[ \int \frac {\sec ^7(c+d x) (A+B \sin (c+d x))}{(a+b \sin (c+d x))^2} \, dx=-\frac {\left (5 a^3 A+a^2 b (25 A+2 B)+a b^2 (47 A+10 B)+b^3 (35 A+16 B)\right ) \log (1-\sin (c+d x))}{32 (a+b)^5 d}+\frac {\left (5 a^3 A-b^3 (35 A-16 B)+a b^2 (47 A-10 B)-a^2 (25 A b-2 b B)\right ) \log (1+\sin (c+d x))}{32 (a-b)^5 d}+\frac {b^6 \left (8 a A b-7 a^2 B-b^2 B\right ) \log (a+b \sin (c+d x))}{\left (a^2-b^2\right )^5 d}+\frac {A+B}{48 (a+b)^2 d (1-\sin (c+d x))^3}+\frac {2 a A+4 A b+a B+3 b B}{32 (a+b)^3 d (1-\sin (c+d x))^2}+\frac {6 a b (3 A+B)+a^2 (5 A+B)+b^2 (19 A+11 B)}{32 (a+b)^4 d (1-\sin (c+d x))}-\frac {A-B}{48 (a-b)^2 d (1+\sin (c+d x))^3}-\frac {2 a A-4 A b-a B+3 b B}{32 (a-b)^3 d (1+\sin (c+d x))^2}-\frac {b^2 (19 A-11 B)-6 a b (3 A-B)+a^2 (5 A-B)}{32 (a-b)^4 d (1+\sin (c+d x))}-\frac {b^6 (A b-a B)}{\left (a^2-b^2\right )^4 d (a+b \sin (c+d x))} \] Output:

-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))*ln(1-si 
n(d*x+c))/(a+b)^5/d+1/32*(5*a^3*A-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+1/48*(A+B)/(a+b)^2/d/(1-sin(d*x+c))^3+1/32*(2* 
A*a+4*A*b+B*a+3*B*b)/(a+b)^3/d/(1-sin(d*x+c))^2+1/32*(6*a*b*(3*A+B)+a^2*(5 
*A+B)+b^2*(19*A+11*B))/(a+b)^4/d/(1-sin(d*x+c))-1/48*(A-B)/(a-b)^2/d/(1+si 
n(d*x+c))^3-1/32*(2*A*a-4*A*b-B*a+3*B*b)/(a-b)^3/d/(1+sin(d*x+c))^2-1/32*( 
b^2*(19*A-11*B)-6*a*b*(3*A-B)+a^2*(5*A-B))/(a-b)^4/d/(1+sin(d*x+c))-b^6*(A 
*b-B*a)/(a^2-b^2)^4/d/(a+b*sin(d*x+c))
 

Mathematica [A] (verified)

Time = 6.22 (sec) , antiderivative size = 766, normalized size of antiderivative = 1.63 \[ \int \frac {\sec ^7(c+d x) (A+B \sin (c+d x))}{(a+b \sin (c+d x))^2} \, dx=\frac {b^7 \left (-\frac {\sec ^6(c+d x) \left (-A b^2+a b B-b (-a A+b B) \sin (c+d x)\right )}{6 b^8 \left (-a^2+b^2\right ) (a+b \sin (c+d x))}+\frac {-\frac {\sec ^4(c+d x) \left (-6 a b^2 (a A-b B)-b^2 \left (-5 a^2 A+7 A b^2-2 a b B\right )-b \left (-6 b^2 (a A-b B)-a \left (-5 a^2 A+7 A b^2-2 a b B\right )\right ) \sin (c+d x)\right )}{4 b^6 \left (-a^2+b^2\right ) (a+b \sin (c+d x))}+\frac {-\frac {\sec ^2(c+d x) \left (4 a b^2 \left (5 a^3 A-13 a A b^2+2 a^2 b B+6 b^3 B\right )-b^2 \left (15 a^4 A-34 a^2 A b^2+35 A b^4+6 a^3 b B-22 a b^3 B\right )-b \left (4 b^2 \left (5 a^3 A-13 a A b^2+2 a^2 b B+6 b^3 B\right )-a \left (15 a^4 A-34 a^2 A b^2+35 A b^4+6 a^3 b B-22 a b^3 B\right )\right ) \sin (c+d x)\right )}{2 b^4 \left (-a^2+b^2\right ) (a+b \sin (c+d x))}+\frac {-6 \left (5 a^5 A-18 a^3 A b^2+29 a A b^4+2 a^4 b B-10 a^2 b^3 B-8 b^5 B\right ) \left (-\frac {\log (1-\sin (c+d x))}{2 b (a+b)}+\frac {\log (1+\sin (c+d x))}{2 (a-b) b}-\frac {\log (a+b \sin (c+d x))}{a^2-b^2}\right )+\left (6 a \left (5 a^5 A-18 a^3 A b^2+29 a A b^4+2 a^4 b B-10 a^2 b^3 B-8 b^5 B\right )-3 \left (5 a^6 A-13 a^4 A b^2+11 a^2 A b^4-35 A b^6+2 a^5 b B-8 a^3 b^3 B+38 a b^5 B\right )\right ) \left (-\frac {\log (1-\sin (c+d x))}{2 b (a+b)^2}+\frac {\log (1+\sin (c+d x))}{2 (a-b)^2 b}-\frac {2 a \log (a+b \sin (c+d x))}{(a-b)^2 (a+b)^2}+\frac {1}{\left (a^2-b^2\right ) (a+b \sin (c+d x))}\right )}{2 b^2 \left (-a^2+b^2\right )}}{4 b^2 \left (-a^2+b^2\right )}}{6 b^2 \left (-a^2+b^2\right )}\right )}{d} \] Input:

Integrate[(Sec[c + d*x]^7*(A + B*Sin[c + d*x]))/(a + b*Sin[c + d*x])^2,x]
 

Output:

(b^7*(-1/6*(Sec[c + d*x]^6*(-(A*b^2) + a*b*B - b*(-(a*A) + b*B)*Sin[c + d* 
x]))/(b^8*(-a^2 + b^2)*(a + b*Sin[c + d*x])) + (-1/4*(Sec[c + d*x]^4*(-6*a 
*b^2*(a*A - b*B) - b^2*(-5*a^2*A + 7*A*b^2 - 2*a*b*B) - b*(-6*b^2*(a*A - b 
*B) - a*(-5*a^2*A + 7*A*b^2 - 2*a*b*B))*Sin[c + d*x]))/(b^6*(-a^2 + b^2)*( 
a + b*Sin[c + d*x])) + (-1/2*(Sec[c + d*x]^2*(4*a*b^2*(5*a^3*A - 13*a*A*b^ 
2 + 2*a^2*b*B + 6*b^3*B) - b^2*(15*a^4*A - 34*a^2*A*b^2 + 35*A*b^4 + 6*a^3 
*b*B - 22*a*b^3*B) - b*(4*b^2*(5*a^3*A - 13*a*A*b^2 + 2*a^2*b*B + 6*b^3*B) 
 - a*(15*a^4*A - 34*a^2*A*b^2 + 35*A*b^4 + 6*a^3*b*B - 22*a*b^3*B))*Sin[c 
+ d*x]))/(b^4*(-a^2 + b^2)*(a + b*Sin[c + d*x])) + (-6*(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)*(-1/2*Log[1 - Sin[ 
c + d*x]]/(b*(a + b)) + Log[1 + Sin[c + d*x]]/(2*(a - b)*b) - Log[a + b*Si 
n[c + d*x]]/(a^2 - b^2)) + (6*a*(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) - 3*(5*a^6*A - 13*a^4*A*b^2 + 11*a^2*A*b^ 
4 - 35*A*b^6 + 2*a^5*b*B - 8*a^3*b^3*B + 38*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]))))/(2*b^2*(-a^2 + b^2)))/(4*b^2*(-a^2 + b^2)))/(6*b^2*(-a^2 + b^2))) 
)/d
 

Rubi [A] (verified)

Time = 1.23 (sec) , antiderivative size = 476, normalized size of antiderivative = 1.01, 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

\(\displaystyle \frac {b^6 \left (-\frac {A b-a B}{\left (a^2-b^2\right )^4 (a+b \sin (c+d x))}+\frac {\left (-7 a^2 B+8 a A b-b^2 B\right ) \log (a+b \sin (c+d x))}{\left (a^2-b^2\right )^5}-\frac {a^2 (5 A-B)-6 a b (3 A-B)+b^2 (19 A-11 B)}{32 b^5 (a-b)^4 (b \sin (c+d x)+b)}+\frac {a^2 (5 A+B)+6 a b (3 A+B)+b^2 (19 A+11 B)}{32 b^5 (a+b)^4 (b-b \sin (c+d x))}-\frac {\left (5 a^3 A+a^2 b (25 A+2 B)+a b^2 (47 A+10 B)+b^3 (35 A+16 B)\right ) \log (b-b \sin (c+d x))}{32 b^6 (a+b)^5}+\frac {\left (5 a^3 A-a^2 (25 A b-2 b B)+a b^2 (47 A-10 B)-b^3 (35 A-16 B)\right ) \log (b \sin (c+d x)+b)}{32 b^6 (a-b)^5}+\frac {2 a A+a B+4 A b+3 b B}{32 b^4 (a+b)^3 (b-b \sin (c+d x))^2}-\frac {2 a A-a B-4 A b+3 b B}{32 b^4 (a-b)^3 (b \sin (c+d x)+b)^2}+\frac {A+B}{48 b^3 (a+b)^2 (b-b \sin (c+d x))^3}-\frac {A-B}{48 b^3 (a-b)^2 (b \sin (c+d x)+b)^3}\right )}{d}\)

Input:

Int[(Sec[c + d*x]^7*(A + B*Sin[c + d*x]))/(a + b*Sin[c + d*x])^2,x]
 

Output:

(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
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 663

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]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3316

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]
 

Maple [A] (verified)

Time = 13.27 (sec) , antiderivative size = 435, normalized size of antiderivative = 0.93

method	result	size
derivativedivides	\(\frac {-\frac {A -B}{48 \left (a -b \right )^{2} \left (1+\sin \left (d x +c \right )\right )^{3}}-\frac {2 a A -4 A b -B a +3 B b}{32 \left (a -b \right )^{3} \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {5 A \,a^{2}-18 A a b +19 b^{2} A -a^{2} B +6 a b B -11 B \,b^{2}}{32 \left (a -b \right )^{4} \left (1+\sin \left (d x +c \right )\right )}+\frac {\left (5 a^{3} A -25 A \,a^{2} b +47 A a \,b^{2}-35 A \,b^{3}+2 B \,a^{2} b -10 B a \,b^{2}+16 B \,b^{3}\right ) \ln \left (1+\sin \left (d x +c \right )\right )}{32 \left (a -b \right )^{5}}+\frac {b^{6} \left (8 A a b -7 a^{2} B -B \,b^{2}\right ) \ln \left (a +b \sin \left (d x +c \right )\right )}{\left (a +b \right )^{5} \left (a -b \right )^{5}}-\frac {\left (A b -B a \right ) b^{6}}{\left (a +b \right )^{4} \left (a -b \right )^{4} \left (a +b \sin \left (d x +c \right )\right )}-\frac {A +B}{48 \left (a +b \right )^{2} \left (\sin \left (d x +c \right )-1\right )^{3}}-\frac {-2 a A -4 A b -B a -3 B b}{32 \left (a +b \right )^{3} \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {5 A \,a^{2}+18 A a b +19 b^{2} A +a^{2} B +6 a b B +11 B \,b^{2}}{32 \left (a +b \right )^{4} \left (\sin \left (d x +c \right )-1\right )}+\frac {\left (-5 a^{3} A -25 A \,a^{2} b -47 A a \,b^{2}-35 A \,b^{3}-2 B \,a^{2} b -10 B a \,b^{2}-16 B \,b^{3}\right ) \ln \left (\sin \left (d x +c \right )-1\right )}{32 \left (a +b \right )^{5}}}{d}\)	\(435\)
default	\(\frac {-\frac {A -B}{48 \left (a -b \right )^{2} \left (1+\sin \left (d x +c \right )\right )^{3}}-\frac {2 a A -4 A b -B a +3 B b}{32 \left (a -b \right )^{3} \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {5 A \,a^{2}-18 A a b +19 b^{2} A -a^{2} B +6 a b B -11 B \,b^{2}}{32 \left (a -b \right )^{4} \left (1+\sin \left (d x +c \right )\right )}+\frac {\left (5 a^{3} A -25 A \,a^{2} b +47 A a \,b^{2}-35 A \,b^{3}+2 B \,a^{2} b -10 B a \,b^{2}+16 B \,b^{3}\right ) \ln \left (1+\sin \left (d x +c \right )\right )}{32 \left (a -b \right )^{5}}+\frac {b^{6} \left (8 A a b -7 a^{2} B -B \,b^{2}\right ) \ln \left (a +b \sin \left (d x +c \right )\right )}{\left (a +b \right )^{5} \left (a -b \right )^{5}}-\frac {\left (A b -B a \right ) b^{6}}{\left (a +b \right )^{4} \left (a -b \right )^{4} \left (a +b \sin \left (d x +c \right )\right )}-\frac {A +B}{48 \left (a +b \right )^{2} \left (\sin \left (d x +c \right )-1\right )^{3}}-\frac {-2 a A -4 A b -B a -3 B b}{32 \left (a +b \right )^{3} \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {5 A \,a^{2}+18 A a b +19 b^{2} A +a^{2} B +6 a b B +11 B \,b^{2}}{32 \left (a +b \right )^{4} \left (\sin \left (d x +c \right )-1\right )}+\frac {\left (-5 a^{3} A -25 A \,a^{2} b -47 A a \,b^{2}-35 A \,b^{3}-2 B \,a^{2} b -10 B a \,b^{2}-16 B \,b^{3}\right ) \ln \left (\sin \left (d x +c \right )-1\right )}{32 \left (a +b \right )^{5}}}{d}\)	\(435\)
parallelrisch	\(\text {Expression too large to display}\)	\(1118\)
risch	\(\text {Expression too large to display}\)	\(4554\)

Input:

int(sec(d*x+c)^7*(A+B*sin(d*x+c))/(a+b*sin(d*x+c))^2,x,method=_RETURNVERBO 
SE)
 

Output:

1/d*(-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))+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/(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^2-35*A*b^3-2*B*a^2*b-10*B*a*b^2-16*B*b^3)*ln(sin(d*x+c)-1))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1244 vs. \(2 (446) = 892\).

Time = 3.91 (sec) , antiderivative size = 1244, normalized size of antiderivative = 2.65 \[ \int \frac {\sec ^7(c+d x) (A+B \sin (c+d x))}{(a+b \sin (c+d x))^2} \, dx=\text {Too large to display} \] Input:

integrate(sec(d*x+c)^7*(A+B*sin(d*x+c))/(a+b*sin(d*x+c))^2,x, algorithm="f 
ricas")
 

Output:

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)...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\sec ^7(c+d x) (A+B \sin (c+d x))}{(a+b \sin (c+d x))^2} \, dx=\text {Timed out} \] Input:

integrate(sec(d*x+c)**7*(A+B*sin(d*x+c))/(a+b*sin(d*x+c))**2,x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1083 vs. \(2 (446) = 892\).

Time = 0.08 (sec) , antiderivative size = 1083, normalized size of antiderivative = 2.31 \[ \int \frac {\sec ^7(c+d x) (A+B \sin (c+d x))}{(a+b \sin (c+d x))^2} \, dx=\text {Too large to display} \] Input:

integrate(sec(d*x+c)^7*(A+B*sin(d*x+c))/(a+b*sin(d*x+c))^2,x, algorithm="m 
axima")
 

Output:

-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^...
                                                                                    
                                                                                    
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 956 vs. \(2 (446) = 892\).

Time = 0.45 (sec) , antiderivative size = 956, normalized size of antiderivative = 2.04 \[ \int \frac {\sec ^7(c+d x) (A+B \sin (c+d x))}{(a+b \sin (c+d x))^2} \, dx =\text {Too large to display} \] Input:

integrate(sec(d*x+c)^7*(A+B*sin(d*x+c))/(a+b*sin(d*x+c))^2,x, algorithm="g 
iac")
 

Output:

-(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*d 
- 5*a^8*b^3*d + 10*a^6*b^5*d - 10*a^4*b^7*d + 5*a^2*b^9*d - b^11*d) - 1/32 
*(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*d + 5*a^4*b*d + 10*a^3*b^2*d + 
10*a^2*b^3*d + 5*a*b^4*d + b^5*d) + 1/32*(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*d - 5*a^4*b*d + 10*a^3*b^2*d - 10*a^2*b^3*d + 5*a*b^4*d - b^5*d) 
- 1/48*(8*B*a^9 - 16*A*a^8*b - 52*B*a^7*b^2 + 96*A*a^6*b^3 + 180*B*a^5*b^4 
 - 288*A*a^4*b^5 - 44*B*a^3*b^6 + 160*A*a^2*b^7 - 92*B*a*b^8 + 48*A*b^9 + 
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 - 
42*B*a^3*b^6 - 12*A*a^2*b^7 + 54*B*a*b^8 - 35*A*b^9)*sin(d*x + c)^6 + 3*(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 + 14*B*a^4 
*b^5 - 76*A*a^3*b^6 + 6*B*a^2*b^7 + 29*A*a*b^8 - 8*B*b^9)*sin(d*x + c)^5 - 
 8*(5*A*a^8*b + 2*B*a^7*b^2 - 28*A*a^6*b^3 - 21*B*a^5*b^4 + 78*A*a^4*b^5 - 
 36*B*a^3*b^6 - 20*A*a^2*b^7 + 55*B*a*b^8 - 35*A*b^9)*sin(d*x + c)^4 - 4*( 
10*A*a^9 + 4*B*a^8*b - 56*A*a^7*b^2 - 21*B*a^6*b^3 + 132*A*a^5*b^4 + 15*B* 
a^4*b^5 - 136*A*a^3*b^6 + 17*B*a^2*b^7 + 50*A*a*b^8 - 15*B*b^9)*sin(d*x + 
c)^3 + 3*(11*A*a^8*b + 10*B*a^7*b^2 - 68*A*a^6*b^3 - 86*B*a^5*b^4 + 218*A* 
a^4*b^5 - 50*B*a^3*b^6 - 84*A*a^2*b^7 + 126*B*a*b^8 - 77*A*b^9)*sin(d*x + 
c)^2 + (33*A*a^9 + 2*B*a^8*b - 172*A*a^7*b^2 - 10*B*a^6*b^3 + 366*A*a^5...
 

Mupad [B] (verification not implemented)

Time = 39.16 (sec) , antiderivative size = 1024, normalized size of antiderivative = 2.18 \[ \int \frac {\sec ^7(c+d x) (A+B \sin (c+d x))}{(a+b \sin (c+d x))^2} \, dx =\text {Too large to display} \] Input:

int((A + B*sin(c + d*x))/(cos(c + d*x)^7*(a + b*sin(c + d*x))^2),x)
 

Output:

((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 + ...
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 1768, normalized size of antiderivative = 3.77 \[ \int \frac {\sec ^7(c+d x) (A+B \sin (c+d x))}{(a+b \sin (c+d x))^2} \, dx =\text {Too large to display} \] Input:

int(sec(d*x+c)^7*(A+B*sin(d*x+c))/(a+b*sin(d*x+c))^2,x)
 

Output:

( - 15*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**6*a**7 + 63*log(tan((c + d* 
x)/2) - 1)*sin(c + d*x)**6*a**5*b**2 - 105*log(tan((c + d*x)/2) - 1)*sin(c 
 + d*x)**6*a**3*b**4 + 105*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**6*a*b** 
6 - 48*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**6*b**7 + 45*log(tan((c + d* 
x)/2) - 1)*sin(c + d*x)**4*a**7 - 189*log(tan((c + d*x)/2) - 1)*sin(c + d* 
x)**4*a**5*b**2 + 315*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**4*a**3*b**4 
- 315*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**4*a*b**6 + 144*log(tan((c + 
d*x)/2) - 1)*sin(c + d*x)**4*b**7 - 45*log(tan((c + d*x)/2) - 1)*sin(c + d 
*x)**2*a**7 + 189*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**2*a**5*b**2 - 31 
5*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**2*a**3*b**4 + 315*log(tan((c + d 
*x)/2) - 1)*sin(c + d*x)**2*a*b**6 - 144*log(tan((c + d*x)/2) - 1)*sin(c + 
 d*x)**2*b**7 + 15*log(tan((c + d*x)/2) - 1)*a**7 - 63*log(tan((c + d*x)/2 
) - 1)*a**5*b**2 + 105*log(tan((c + d*x)/2) - 1)*a**3*b**4 - 105*log(tan(( 
c + d*x)/2) - 1)*a*b**6 + 48*log(tan((c + d*x)/2) - 1)*b**7 + 15*log(tan(( 
c + d*x)/2) + 1)*sin(c + d*x)**6*a**7 - 63*log(tan((c + d*x)/2) + 1)*sin(c 
 + d*x)**6*a**5*b**2 + 105*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**6*a**3* 
b**4 - 105*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**6*a*b**6 - 48*log(tan(( 
c + d*x)/2) + 1)*sin(c + d*x)**6*b**7 - 45*log(tan((c + d*x)/2) + 1)*sin(c 
 + d*x)**4*a**7 + 189*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**4*a**5*b**2 
- 315*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**4*a**3*b**4 + 315*log(tan...