\(\int \frac {\sin ^7(c+d x)}{(a+b \sec (c+d x))^2} \, dx\) [209]

Optimal result

Integrand size = 21, antiderivative size = 267 \[ \int \frac {\sin ^7(c+d x)}{(a+b \sec (c+d x))^2} \, dx=-\frac {\left (a^2-7 b^2\right ) \left (a^2-b^2\right )^2 \cos (c+d x)}{a^8 d}-\frac {3 b \left (a^2-b^2\right )^2 \cos ^2(c+d x)}{a^7 d}+\frac {\left (3 a^4-9 a^2 b^2+5 b^4\right ) \cos ^3(c+d x)}{3 a^6 d}+\frac {b \left (3 a^2-2 b^2\right ) \cos ^4(c+d x)}{2 a^5 d}-\frac {3 \left (a^2-b^2\right ) \cos ^5(c+d x)}{5 a^4 d}-\frac {b \cos ^6(c+d x)}{3 a^3 d}+\frac {\cos ^7(c+d x)}{7 a^2 d}+\frac {b^2 \left (a^2-b^2\right )^3}{a^9 d (b+a \cos (c+d x))}+\frac {2 b \left (a^2-4 b^2\right ) \left (a^2-b^2\right )^2 \log (b+a \cos (c+d x))}{a^9 d} \] Output:

-(a^2-7*b^2)*(a^2-b^2)^2*cos(d*x+c)/a^8/d-3*b*(a^2-b^2)^2*cos(d*x+c)^2/a^7 
/d+1/3*(3*a^4-9*a^2*b^2+5*b^4)*cos(d*x+c)^3/a^6/d+1/2*b*(3*a^2-2*b^2)*cos( 
d*x+c)^4/a^5/d-3/5*(a^2-b^2)*cos(d*x+c)^5/a^4/d-1/3*b*cos(d*x+c)^6/a^3/d+1 
/7*cos(d*x+c)^7/a^2/d+b^2*(a^2-b^2)^3/a^9/d/(b+a*cos(d*x+c))+2*b*(a^2-4*b^ 
2)*(a^2-b^2)^2*ln(b+a*cos(d*x+c))/a^9/d
 

Mathematica [A] (verified)

Time = 2.61 (sec) , antiderivative size = 417, normalized size of antiderivative = 1.56 \[ \int \frac {\sin ^7(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\frac {-3675 a^8+61320 a^6 b^2-132720 a^4 b^4+87360 a^2 b^6-13440 b^8-140 \left (21 a^8-228 a^6 b^2+400 a^4 b^4-192 a^2 b^6\right ) \cos (2 (c+d x))-3780 a^7 b \cos (3 (c+d x))+8400 a^5 b^3 \cos (3 (c+d x))-4480 a^3 b^5 \cos (3 (c+d x))+588 a^8 \cos (4 (c+d x))-1848 a^6 b^2 \cos (4 (c+d x))+1120 a^4 b^4 \cos (4 (c+d x))+476 a^7 b \cos (5 (c+d x))-336 a^5 b^3 \cos (5 (c+d x))-132 a^8 \cos (6 (c+d x))+112 a^6 b^2 \cos (6 (c+d x))-40 a^7 b \cos (7 (c+d x))+15 a^8 \cos (8 (c+d x))+26880 a^6 b^2 \log (b+a \cos (c+d x))-161280 a^4 b^4 \log (b+a \cos (c+d x))+241920 a^2 b^6 \log (b+a \cos (c+d x))-107520 b^8 \log (b+a \cos (c+d x))+1680 a b \cos (c+d x) \left (-8 a^6+67 a^4 b^2-116 a^2 b^4+56 b^6+16 \left (a^2-4 b^2\right ) \left (a^2-b^2\right )^2 \log (b+a \cos (c+d x))\right )}{13440 a^9 d (b+a \cos (c+d x))} \] Input:

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

Output:

(-3675*a^8 + 61320*a^6*b^2 - 132720*a^4*b^4 + 87360*a^2*b^6 - 13440*b^8 - 
140*(21*a^8 - 228*a^6*b^2 + 400*a^4*b^4 - 192*a^2*b^6)*Cos[2*(c + d*x)] - 
3780*a^7*b*Cos[3*(c + d*x)] + 8400*a^5*b^3*Cos[3*(c + d*x)] - 4480*a^3*b^5 
*Cos[3*(c + d*x)] + 588*a^8*Cos[4*(c + d*x)] - 1848*a^6*b^2*Cos[4*(c + d*x 
)] + 1120*a^4*b^4*Cos[4*(c + d*x)] + 476*a^7*b*Cos[5*(c + d*x)] - 336*a^5* 
b^3*Cos[5*(c + d*x)] - 132*a^8*Cos[6*(c + d*x)] + 112*a^6*b^2*Cos[6*(c + d 
*x)] - 40*a^7*b*Cos[7*(c + d*x)] + 15*a^8*Cos[8*(c + d*x)] + 26880*a^6*b^2 
*Log[b + a*Cos[c + d*x]] - 161280*a^4*b^4*Log[b + a*Cos[c + d*x]] + 241920 
*a^2*b^6*Log[b + a*Cos[c + d*x]] - 107520*b^8*Log[b + a*Cos[c + d*x]] + 16 
80*a*b*Cos[c + d*x]*(-8*a^6 + 67*a^4*b^2 - 116*a^2*b^4 + 56*b^6 + 16*(a^2 
- 4*b^2)*(a^2 - b^2)^2*Log[b + a*Cos[c + d*x]]))/(13440*a^9*d*(b + a*Cos[c 
 + d*x]))
 

Rubi [A] (verified)

Time = 0.60 (sec) , antiderivative size = 240, normalized size of antiderivative = 0.90, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {3042, 4360, 3042, 25, 3316, 27, 522, 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.

\(\displaystyle \int \frac {\sin ^7(c+d x)}{(a+b \sec (c+d x))^2} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\cos \left (c+d x-\frac {\pi }{2}\right )^7}{\left (a-b \csc \left (c+d x-\frac {\pi }{2}\right )\right )^2}dx\)

\(\Big \downarrow \) 4360

\(\displaystyle \int \frac {\sin ^7(c+d x) \cos ^2(c+d x)}{(-a \cos (c+d x)-b)^2}dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int -\frac {\sin \left (c+d x+\frac {\pi }{2}\right )^2 \cos \left (c+d x+\frac {\pi }{2}\right )^7}{\left (-a \sin \left (c+d x+\frac {\pi }{2}\right )-b\right )^2}dx\)

\(\Big \downarrow \) 25

\(\displaystyle -\int \frac {\cos \left (\frac {1}{2} (2 c+\pi )+d x\right )^7 \sin \left (\frac {1}{2} (2 c+\pi )+d x\right )^2}{\left (b+a \sin \left (\frac {1}{2} (2 c+\pi )+d x\right )\right )^2}dx\)

\(\Big \downarrow \) 3316

\(\displaystyle -\frac {\int \frac {\cos ^2(c+d x) \left (a^2-a^2 \cos ^2(c+d x)\right )^3}{(b+a \cos (c+d x))^2}d(a \cos (c+d x))}{a^7 d}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {a^2 \cos ^2(c+d x) \left (a^2-a^2 \cos ^2(c+d x)\right )^3}{(b+a \cos (c+d x))^2}d(a \cos (c+d x))}{a^9 d}\)

\(\Big \downarrow \) 522

\(\displaystyle -\frac {\int \left (-a^6 \cos ^6(c+d x)+2 a^5 b \cos ^5(c+d x)+3 a^4 \left (a^2-b^2\right ) \cos ^4(c+d x)+2 a^3 b \left (2 b^2-3 a^2\right ) \cos ^3(c+d x)-a^2 \left (3 a^4-9 b^2 a^2+5 b^4\right ) \cos ^2(c+d x)+6 a b \left (b^2-a^2\right )^2 \cos (c+d x)+\left (a^2-7 b^2\right ) \left (a^2-b^2\right )^2+\frac {2 b \left (b^2-a^2\right )^2 \left (4 b^2-a^2\right )}{b+a \cos (c+d x)}-\frac {b^2 \left (b^2-a^2\right )^3}{(b+a \cos (c+d x))^2}\right )d(a \cos (c+d x))}{a^9 d}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {-\frac {1}{7} a^7 \cos ^7(c+d x)+\frac {1}{3} a^6 b \cos ^6(c+d x)+3 a^2 b \left (a^2-b^2\right )^2 \cos ^2(c+d x)+a \left (a^2-7 b^2\right ) \left (a^2-b^2\right )^2 \cos (c+d x)-\frac {b^2 \left (a^2-b^2\right )^3}{a \cos (c+d x)+b}-2 b \left (a^2-4 b^2\right ) \left (a^2-b^2\right )^2 \log (a \cos (c+d x)+b)+\frac {3}{5} a^5 \left (a^2-b^2\right ) \cos ^5(c+d x)-\frac {1}{2} a^4 b \left (3 a^2-2 b^2\right ) \cos ^4(c+d x)-\frac {1}{3} a^3 \left (3 a^4-9 a^2 b^2+5 b^4\right ) \cos ^3(c+d x)}{a^9 d}\)

Input:

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

Output:

-((a*(a^2 - 7*b^2)*(a^2 - b^2)^2*Cos[c + d*x] + 3*a^2*b*(a^2 - b^2)^2*Cos[ 
c + d*x]^2 - (a^3*(3*a^4 - 9*a^2*b^2 + 5*b^4)*Cos[c + d*x]^3)/3 - (a^4*b*( 
3*a^2 - 2*b^2)*Cos[c + d*x]^4)/2 + (3*a^5*(a^2 - b^2)*Cos[c + d*x]^5)/5 + 
(a^6*b*Cos[c + d*x]^6)/3 - (a^7*Cos[c + d*x]^7)/7 - (b^2*(a^2 - b^2)^3)/(b 
 + a*Cos[c + d*x]) - 2*b*(a^2 - 4*b^2)*(a^2 - b^2)^2*Log[b + a*Cos[c + d*x 
]])/(a^9*d))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 522

Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_. 
), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x], 
x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]
 

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]
 

rule 4360

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + 
(a_))^(m_.), x_Symbol] :> Int[(g*Cos[e + f*x])^p*((b + a*Sin[e + f*x])^m/Si 
n[e + f*x]^m), x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]
 

Maple [A] (verified)

Time = 2.49 (sec) , antiderivative size = 321, normalized size of antiderivative = 1.20

method	result	size
derivativedivides	\(\frac {\frac {\frac {\cos \left (d x +c \right )^{7} a^{6}}{7}-\frac {b \cos \left (d x +c \right )^{6} a^{5}}{3}-\frac {3 a^{6} \cos \left (d x +c \right )^{5}}{5}+\frac {3 a^{4} b^{2} \cos \left (d x +c \right )^{5}}{5}+\frac {3 a^{5} b \cos \left (d x +c \right )^{4}}{2}-a^{3} b^{3} \cos \left (d x +c \right )^{4}+a^{6} \cos \left (d x +c \right )^{3}-3 a^{4} b^{2} \cos \left (d x +c \right )^{3}+\frac {5 a^{2} b^{4} \cos \left (d x +c \right )^{3}}{3}-3 a^{5} b \cos \left (d x +c \right )^{2}+6 a^{3} b^{3} \cos \left (d x +c \right )^{2}-3 a \,b^{5} \cos \left (d x +c \right )^{2}-a^{6} \cos \left (d x +c \right )+9 \cos \left (d x +c \right ) a^{4} b^{2}-15 \cos \left (d x +c \right ) a^{2} b^{4}+7 \cos \left (d x +c \right ) b^{6}}{a^{8}}+\frac {2 b \left (a^{6}-6 a^{4} b^{2}+9 a^{2} b^{4}-4 b^{6}\right ) \ln \left (b +a \cos \left (d x +c \right )\right )}{a^{9}}+\frac {b^{2} \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right )}{a^{9} \left (b +a \cos \left (d x +c \right )\right )}}{d}\)	\(321\)
default	\(\frac {\frac {\frac {\cos \left (d x +c \right )^{7} a^{6}}{7}-\frac {b \cos \left (d x +c \right )^{6} a^{5}}{3}-\frac {3 a^{6} \cos \left (d x +c \right )^{5}}{5}+\frac {3 a^{4} b^{2} \cos \left (d x +c \right )^{5}}{5}+\frac {3 a^{5} b \cos \left (d x +c \right )^{4}}{2}-a^{3} b^{3} \cos \left (d x +c \right )^{4}+a^{6} \cos \left (d x +c \right )^{3}-3 a^{4} b^{2} \cos \left (d x +c \right )^{3}+\frac {5 a^{2} b^{4} \cos \left (d x +c \right )^{3}}{3}-3 a^{5} b \cos \left (d x +c \right )^{2}+6 a^{3} b^{3} \cos \left (d x +c \right )^{2}-3 a \,b^{5} \cos \left (d x +c \right )^{2}-a^{6} \cos \left (d x +c \right )+9 \cos \left (d x +c \right ) a^{4} b^{2}-15 \cos \left (d x +c \right ) a^{2} b^{4}+7 \cos \left (d x +c \right ) b^{6}}{a^{8}}+\frac {2 b \left (a^{6}-6 a^{4} b^{2}+9 a^{2} b^{4}-4 b^{6}\right ) \ln \left (b +a \cos \left (d x +c \right )\right )}{a^{9}}+\frac {b^{2} \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right )}{a^{9} \left (b +a \cos \left (d x +c \right )\right )}}{d}\)	\(321\)
parallelrisch	\(\frac {2240 b \left (a -2 b \right ) \left (a +2 b \right ) \left (a -b \right )^{2} \left (a +b \right )^{2} \left (b +a \cos \left (d x +c \right )\right ) \ln \left (-2 a +\sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \left (a -b \right )\right )-2240 b \left (a -2 b \right ) \left (a +2 b \right ) \left (a -b \right )^{2} \left (a +b \right )^{2} \left (b +a \cos \left (d x +c \right )\right ) \ln \left (\sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )-11 a \left (\frac {245 a \left (a^{6}-\frac {76}{7} a^{4} b^{2}+\frac {400}{21} a^{2} b^{4}-\frac {64}{7} b^{6}\right ) \cos \left (2 d x +2 c \right )}{11}+\frac {35 \left (9 a^{6} b -20 a^{4} b^{3}+\frac {32}{3} a^{2} b^{5}\right ) \cos \left (3 d x +3 c \right )}{11}+7 \left (2 a^{5} b^{2}-\frac {40}{33} a^{3} b^{4}-\frac {7}{11} a^{7}\right ) \cos \left (4 d x +4 c \right )+\frac {7 \left (-\frac {17}{3} a^{6} b +4 a^{4} b^{3}\right ) \cos \left (5 d x +5 c \right )}{11}+\left (-\frac {28}{33} a^{5} b^{2}+a^{7}\right ) \cos \left (6 d x +6 c \right )+\frac {10 a^{6} b \cos \left (7 d x +7 c \right )}{33}-\frac {5 a^{7} \cos \left (8 d x +8 c \right )}{44}+\frac {4 \left (-2240 a \,b^{6}-2240 b^{7}-2128 a^{5} b^{2}+\frac {12880}{3} a^{3} b^{4}+\frac {175}{3} a^{6} b -1960 a^{4} b^{3}+4200 a^{2} b^{5}+128 a^{7}\right ) \cos \left (d x +c \right )}{11}-\frac {8960 b^{7}}{11}+\frac {1225 a^{7}}{44}-\frac {17990 a^{5} b^{2}}{33}-\frac {6720 a \,b^{6}}{11}+\frac {12600 a^{3} b^{4}}{11}+\frac {512 a^{6} b}{11}-\frac {8512 a^{4} b^{3}}{11}+\frac {51520 a^{2} b^{5}}{33}\right )}{1120 \left (b +a \cos \left (d x +c \right )\right ) d \,a^{9}}\)	\(428\)
risch	\(-\frac {2 b^{2} \left (-a^{6}+3 a^{4} b^{2}-3 a^{2} b^{4}+b^{6}\right ) {\mathrm e}^{i \left (d x +c \right )}}{a^{9} d \left (a \,{\mathrm e}^{2 i \left (d x +c \right )}+2 b \,{\mathrm e}^{i \left (d x +c \right )}+a \right )}-\frac {35 \,{\mathrm e}^{i \left (d x +c \right )}}{128 d \,a^{2}}-\frac {35 \,{\mathrm e}^{-i \left (d x +c \right )}}{128 d \,a^{2}}-\frac {9 \cos \left (3 d x +3 c \right ) b^{2}}{16 d \,a^{4}}+\frac {5 \cos \left (3 d x +3 c \right ) b^{4}}{12 d \,a^{6}}+\frac {57 \,{\mathrm e}^{-i \left (d x +c \right )} b^{2}}{16 a^{4} d}-\frac {55 \,{\mathrm e}^{-i \left (d x +c \right )} b^{4}}{8 a^{6} d}+\frac {7 \,{\mathrm e}^{-i \left (d x +c \right )} b^{6}}{2 a^{8} d}-\frac {29 b \,{\mathrm e}^{-2 i \left (d x +c \right )}}{64 a^{3} d}+\frac {5 b^{3} {\mathrm e}^{-2 i \left (d x +c \right )}}{4 a^{5} d}-\frac {3 b^{5} {\mathrm e}^{-2 i \left (d x +c \right )}}{4 a^{7} d}+\frac {2 b \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 b \,{\mathrm e}^{i \left (d x +c \right )}}{a}+1\right )}{a^{3} d}-\frac {12 b^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 b \,{\mathrm e}^{i \left (d x +c \right )}}{a}+1\right )}{a^{5} d}+\frac {18 b^{5} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 b \,{\mathrm e}^{i \left (d x +c \right )}}{a}+1\right )}{a^{7} d}-\frac {8 b^{7} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 b \,{\mathrm e}^{i \left (d x +c \right )}}{a}+1\right )}{a^{9} d}-\frac {2 i x b}{a^{3}}+\frac {12 i x \,b^{3}}{a^{5}}-\frac {18 i x \,b^{5}}{a^{7}}+\frac {8 i x \,b^{7}}{a^{9}}-\frac {29 b \,{\mathrm e}^{2 i \left (d x +c \right )}}{64 a^{3} d}+\frac {5 b^{3} {\mathrm e}^{2 i \left (d x +c \right )}}{4 a^{5} d}-\frac {3 b^{5} {\mathrm e}^{2 i \left (d x +c \right )}}{4 a^{7} d}+\frac {57 \,{\mathrm e}^{i \left (d x +c \right )} b^{2}}{16 a^{4} d}-\frac {55 \,{\mathrm e}^{i \left (d x +c \right )} b^{4}}{8 a^{6} d}+\frac {7 \,{\mathrm e}^{i \left (d x +c \right )} b^{6}}{2 a^{8} d}-\frac {b \cos \left (6 d x +6 c \right )}{96 a^{3} d}+\frac {7 \cos \left (3 d x +3 c \right )}{64 a^{2} d}-\frac {7 \cos \left (5 d x +5 c \right )}{320 a^{2} d}-\frac {36 i b^{5} c}{a^{7} d}+\frac {16 i b^{7} c}{a^{9} d}+\frac {\cos \left (7 d x +7 c \right )}{448 a^{2} d}+\frac {3 \cos \left (5 d x +5 c \right ) b^{2}}{80 d \,a^{4}}+\frac {b \cos \left (4 d x +4 c \right )}{8 d \,a^{3}}-\frac {b^{3} \cos \left (4 d x +4 c \right )}{8 d \,a^{5}}-\frac {4 i b c}{a^{3} d}+\frac {24 i b^{3} c}{a^{5} d}\)	\(749\)
norman	\(\frac {\frac {\left (96 a^{7}+96 a^{6} b +380 a^{5} b^{2}-1596 a^{4} b^{3}-1188 a^{3} b^{4}+3220 a^{2} b^{5}+720 a \,b^{6}-1680 b^{7}\right ) \left (a +b \right )}{210 a^{8} d b}-\frac {\left (32 a^{8}-160 a^{7} b -32 a^{6} b^{2}+172 a^{5} b^{3}-644 a^{4} b^{4}-196 a^{3} b^{5}+1180 a^{2} b^{6}+80 a \,b^{7}-560 b^{8}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{5 a^{8} d b}+\frac {\left (32 a^{8}+160 a^{7} b -32 a^{6} b^{2}-172 a^{5} b^{3}-644 a^{4} b^{4}+196 a^{3} b^{5}+1180 a^{2} b^{6}-80 a \,b^{7}-560 b^{8}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{5 a^{8} d b}-\frac {\left (96 a^{8}-288 a^{7} b +212 a^{6} b^{2}+896 a^{5} b^{3}-2160 a^{4} b^{4}-1144 a^{3} b^{5}+3620 a^{2} b^{6}+480 a \,b^{7}-1680 b^{8}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{15 a^{8} d b}-\frac {\left (96 a^{8}-192 a^{7} b +476 a^{6} b^{2}+1216 a^{5} b^{3}-2784 a^{4} b^{4}-2032 a^{3} b^{5}+3940 a^{2} b^{6}+960 a \,b^{7}-1680 b^{8}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{16}}{210 a^{8} d b}+\frac {\left (96 a^{8}+288 a^{7} b +212 a^{6} b^{2}-896 a^{5} b^{3}-2160 a^{4} b^{4}+1144 a^{3} b^{5}+3620 a^{2} b^{6}-480 a \,b^{7}-1680 b^{8}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{15 a^{8} d b}-\frac {\left (288 a^{8}-672 a^{7} b +1104 a^{6} b^{2}+3268 a^{5} b^{3}-7428 a^{4} b^{4}-4908 a^{3} b^{5}+11260 a^{2} b^{6}+2160 a \,b^{7}-5040 b^{8}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}{105 a^{8} d b}+\frac {\left (288 a^{8}+672 a^{7} b +1104 a^{6} b^{2}-3268 a^{5} b^{3}-7428 a^{4} b^{4}+4908 a^{3} b^{5}+11260 a^{2} b^{6}-2160 a \,b^{7}-5040 b^{8}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{105 a^{8} d b}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{7} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )}-\frac {2 b \left (a^{6}-6 a^{4} b^{2}+9 a^{2} b^{4}-4 b^{6}\right ) \ln \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}{a^{9} d}+\frac {2 b \left (a^{6}-6 a^{4} b^{2}+9 a^{2} b^{4}-4 b^{6}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b -a -b \right )}{a^{9} d}\)	\(836\)

Input:

int(sin(d*x+c)^7/(a+b*sec(d*x+c))^2,x,method=_RETURNVERBOSE)
 

Output:

1/d*(1/a^8*(1/7*cos(d*x+c)^7*a^6-1/3*b*cos(d*x+c)^6*a^5-3/5*a^6*cos(d*x+c) 
^5+3/5*a^4*b^2*cos(d*x+c)^5+3/2*a^5*b*cos(d*x+c)^4-a^3*b^3*cos(d*x+c)^4+a^ 
6*cos(d*x+c)^3-3*a^4*b^2*cos(d*x+c)^3+5/3*a^2*b^4*cos(d*x+c)^3-3*a^5*b*cos 
(d*x+c)^2+6*a^3*b^3*cos(d*x+c)^2-3*a*b^5*cos(d*x+c)^2-a^6*cos(d*x+c)+9*cos 
(d*x+c)*a^4*b^2-15*cos(d*x+c)*a^2*b^4+7*cos(d*x+c)*b^6)+2/a^9*b*(a^6-6*a^4 
*b^2+9*a^2*b^4-4*b^6)*ln(b+a*cos(d*x+c))+b^2*(a^6-3*a^4*b^2+3*a^2*b^4-b^6) 
/a^9/(b+a*cos(d*x+c)))
 

Fricas [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 344, normalized size of antiderivative = 1.29 \[ \int \frac {\sin ^7(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\frac {120 \, a^{8} \cos \left (d x + c\right )^{8} - 160 \, a^{7} b \cos \left (d x + c\right )^{7} + 1715 \, a^{6} b^{2} - 4725 \, a^{4} b^{4} + 3780 \, a^{2} b^{6} - 840 \, b^{8} - 56 \, {\left (9 \, a^{8} - 4 \, a^{6} b^{2}\right )} \cos \left (d x + c\right )^{6} + 84 \, {\left (9 \, a^{7} b - 4 \, a^{5} b^{3}\right )} \cos \left (d x + c\right )^{5} + 140 \, {\left (6 \, a^{8} - 9 \, a^{6} b^{2} + 4 \, a^{4} b^{4}\right )} \cos \left (d x + c\right )^{4} - 280 \, {\left (6 \, a^{7} b - 9 \, a^{5} b^{3} + 4 \, a^{3} b^{5}\right )} \cos \left (d x + c\right )^{3} - 840 \, {\left (a^{8} - 6 \, a^{6} b^{2} + 9 \, a^{4} b^{4} - 4 \, a^{2} b^{6}\right )} \cos \left (d x + c\right )^{2} + 35 \, {\left (a^{7} b + 153 \, a^{5} b^{3} - 324 \, a^{3} b^{5} + 168 \, a b^{7}\right )} \cos \left (d x + c\right ) + 1680 \, {\left (a^{6} b^{2} - 6 \, a^{4} b^{4} + 9 \, a^{2} b^{6} - 4 \, b^{8} + {\left (a^{7} b - 6 \, a^{5} b^{3} + 9 \, a^{3} b^{5} - 4 \, a b^{7}\right )} \cos \left (d x + c\right )\right )} \log \left (a \cos \left (d x + c\right ) + b\right )}{840 \, {\left (a^{10} d \cos \left (d x + c\right ) + a^{9} b d\right )}} \] Input:

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

Output:

1/840*(120*a^8*cos(d*x + c)^8 - 160*a^7*b*cos(d*x + c)^7 + 1715*a^6*b^2 - 
4725*a^4*b^4 + 3780*a^2*b^6 - 840*b^8 - 56*(9*a^8 - 4*a^6*b^2)*cos(d*x + c 
)^6 + 84*(9*a^7*b - 4*a^5*b^3)*cos(d*x + c)^5 + 140*(6*a^8 - 9*a^6*b^2 + 4 
*a^4*b^4)*cos(d*x + c)^4 - 280*(6*a^7*b - 9*a^5*b^3 + 4*a^3*b^5)*cos(d*x + 
 c)^3 - 840*(a^8 - 6*a^6*b^2 + 9*a^4*b^4 - 4*a^2*b^6)*cos(d*x + c)^2 + 35* 
(a^7*b + 153*a^5*b^3 - 324*a^3*b^5 + 168*a*b^7)*cos(d*x + c) + 1680*(a^6*b 
^2 - 6*a^4*b^4 + 9*a^2*b^6 - 4*b^8 + (a^7*b - 6*a^5*b^3 + 9*a^3*b^5 - 4*a* 
b^7)*cos(d*x + c))*log(a*cos(d*x + c) + b))/(a^10*d*cos(d*x + c) + a^9*b*d 
)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.01 \[ \int \frac {\sin ^7(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\frac {\frac {210 \, {\left (a^{6} b^{2} - 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} - b^{8}\right )}}{a^{10} \cos \left (d x + c\right ) + a^{9} b} + \frac {30 \, a^{6} \cos \left (d x + c\right )^{7} - 70 \, a^{5} b \cos \left (d x + c\right )^{6} - 126 \, {\left (a^{6} - a^{4} b^{2}\right )} \cos \left (d x + c\right )^{5} + 105 \, {\left (3 \, a^{5} b - 2 \, a^{3} b^{3}\right )} \cos \left (d x + c\right )^{4} + 70 \, {\left (3 \, a^{6} - 9 \, a^{4} b^{2} + 5 \, a^{2} b^{4}\right )} \cos \left (d x + c\right )^{3} - 630 \, {\left (a^{5} b - 2 \, a^{3} b^{3} + a b^{5}\right )} \cos \left (d x + c\right )^{2} - 210 \, {\left (a^{6} - 9 \, a^{4} b^{2} + 15 \, a^{2} b^{4} - 7 \, b^{6}\right )} \cos \left (d x + c\right )}{a^{8}} + \frac {420 \, {\left (a^{6} b - 6 \, a^{4} b^{3} + 9 \, a^{2} b^{5} - 4 \, b^{7}\right )} \log \left (a \cos \left (d x + c\right ) + b\right )}{a^{9}}}{210 \, d} \] Input:

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

Output:

1/210*(210*(a^6*b^2 - 3*a^4*b^4 + 3*a^2*b^6 - b^8)/(a^10*cos(d*x + c) + a^ 
9*b) + (30*a^6*cos(d*x + c)^7 - 70*a^5*b*cos(d*x + c)^6 - 126*(a^6 - a^4*b 
^2)*cos(d*x + c)^5 + 105*(3*a^5*b - 2*a^3*b^3)*cos(d*x + c)^4 + 70*(3*a^6 
- 9*a^4*b^2 + 5*a^2*b^4)*cos(d*x + c)^3 - 630*(a^5*b - 2*a^3*b^3 + a*b^5)* 
cos(d*x + c)^2 - 210*(a^6 - 9*a^4*b^2 + 15*a^2*b^4 - 7*b^6)*cos(d*x + c))/ 
a^8 + 420*(a^6*b - 6*a^4*b^3 + 9*a^2*b^5 - 4*b^7)*log(a*cos(d*x + c) + b)/ 
a^9)/d
 

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 383, normalized size of antiderivative = 1.43 \[ \int \frac {\sin ^7(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\frac {2 \, {\left (a^{6} b - 6 \, a^{4} b^{3} + 9 \, a^{2} b^{5} - 4 \, b^{7}\right )} \log \left ({\left | a \cos \left (d x + c\right ) + b \right |}\right )}{a^{9} d} + \frac {a^{6} b^{2} - 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} - b^{8}}{{\left (a \cos \left (d x + c\right ) + b\right )} a^{9} d} + \frac {30 \, a^{12} d^{6} \cos \left (d x + c\right )^{7} - 70 \, a^{11} b d^{6} \cos \left (d x + c\right )^{6} - 126 \, a^{12} d^{6} \cos \left (d x + c\right )^{5} + 126 \, a^{10} b^{2} d^{6} \cos \left (d x + c\right )^{5} + 315 \, a^{11} b d^{6} \cos \left (d x + c\right )^{4} - 210 \, a^{9} b^{3} d^{6} \cos \left (d x + c\right )^{4} + 210 \, a^{12} d^{6} \cos \left (d x + c\right )^{3} - 630 \, a^{10} b^{2} d^{6} \cos \left (d x + c\right )^{3} + 350 \, a^{8} b^{4} d^{6} \cos \left (d x + c\right )^{3} - 630 \, a^{11} b d^{6} \cos \left (d x + c\right )^{2} + 1260 \, a^{9} b^{3} d^{6} \cos \left (d x + c\right )^{2} - 630 \, a^{7} b^{5} d^{6} \cos \left (d x + c\right )^{2} - 210 \, a^{12} d^{6} \cos \left (d x + c\right ) + 1890 \, a^{10} b^{2} d^{6} \cos \left (d x + c\right ) - 3150 \, a^{8} b^{4} d^{6} \cos \left (d x + c\right ) + 1470 \, a^{6} b^{6} d^{6} \cos \left (d x + c\right )}{210 \, a^{14} d^{7}} \] Input:

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

Output:

2*(a^6*b - 6*a^4*b^3 + 9*a^2*b^5 - 4*b^7)*log(abs(a*cos(d*x + c) + b))/(a^ 
9*d) + (a^6*b^2 - 3*a^4*b^4 + 3*a^2*b^6 - b^8)/((a*cos(d*x + c) + b)*a^9*d 
) + 1/210*(30*a^12*d^6*cos(d*x + c)^7 - 70*a^11*b*d^6*cos(d*x + c)^6 - 126 
*a^12*d^6*cos(d*x + c)^5 + 126*a^10*b^2*d^6*cos(d*x + c)^5 + 315*a^11*b*d^ 
6*cos(d*x + c)^4 - 210*a^9*b^3*d^6*cos(d*x + c)^4 + 210*a^12*d^6*cos(d*x + 
 c)^3 - 630*a^10*b^2*d^6*cos(d*x + c)^3 + 350*a^8*b^4*d^6*cos(d*x + c)^3 - 
 630*a^11*b*d^6*cos(d*x + c)^2 + 1260*a^9*b^3*d^6*cos(d*x + c)^2 - 630*a^7 
*b^5*d^6*cos(d*x + c)^2 - 210*a^12*d^6*cos(d*x + c) + 1890*a^10*b^2*d^6*co 
s(d*x + c) - 3150*a^8*b^4*d^6*cos(d*x + c) + 1470*a^6*b^6*d^6*cos(d*x + c) 
)/(a^14*d^7)
 

Mupad [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 588, normalized size of antiderivative = 2.20 \[ \int \frac {\sin ^7(c+d x)}{(a+b \sec (c+d x))^2} \, dx=\frac {{\cos \left (c+d\,x\right )}^4\,\left (\frac {b^3}{2\,a^5}+\frac {b\,\left (\frac {3}{a^2}-\frac {3\,b^2}{a^4}\right )}{2\,a}\right )}{d}-\frac {{\cos \left (c+d\,x\right )}^2\,\left (\frac {b^2\,\left (\frac {2\,b^3}{a^5}+\frac {2\,b\,\left (\frac {3}{a^2}-\frac {3\,b^2}{a^4}\right )}{a}\right )}{2\,a^2}+\frac {b\,\left (\frac {3}{a^2}+\frac {b^2\,\left (\frac {3}{a^2}-\frac {3\,b^2}{a^4}\right )}{a^2}-\frac {2\,b\,\left (\frac {2\,b^3}{a^5}+\frac {2\,b\,\left (\frac {3}{a^2}-\frac {3\,b^2}{a^4}\right )}{a}\right )}{a}\right )}{a}\right )}{d}-\frac {{\cos \left (c+d\,x\right )}^5\,\left (\frac {3}{5\,a^2}-\frac {3\,b^2}{5\,a^4}\right )}{d}+\frac {{\cos \left (c+d\,x\right )}^7}{7\,a^2\,d}-\frac {\cos \left (c+d\,x\right )\,\left (\frac {1}{a^2}+\frac {b^2\,\left (\frac {3}{a^2}+\frac {b^2\,\left (\frac {3}{a^2}-\frac {3\,b^2}{a^4}\right )}{a^2}-\frac {2\,b\,\left (\frac {2\,b^3}{a^5}+\frac {2\,b\,\left (\frac {3}{a^2}-\frac {3\,b^2}{a^4}\right )}{a}\right )}{a}\right )}{a^2}-\frac {2\,b\,\left (\frac {b^2\,\left (\frac {2\,b^3}{a^5}+\frac {2\,b\,\left (\frac {3}{a^2}-\frac {3\,b^2}{a^4}\right )}{a}\right )}{a^2}+\frac {2\,b\,\left (\frac {3}{a^2}+\frac {b^2\,\left (\frac {3}{a^2}-\frac {3\,b^2}{a^4}\right )}{a^2}-\frac {2\,b\,\left (\frac {2\,b^3}{a^5}+\frac {2\,b\,\left (\frac {3}{a^2}-\frac {3\,b^2}{a^4}\right )}{a}\right )}{a}\right )}{a}\right )}{a}\right )}{d}+\frac {{\cos \left (c+d\,x\right )}^3\,\left (\frac {1}{a^2}+\frac {b^2\,\left (\frac {3}{a^2}-\frac {3\,b^2}{a^4}\right )}{3\,a^2}-\frac {2\,b\,\left (\frac {2\,b^3}{a^5}+\frac {2\,b\,\left (\frac {3}{a^2}-\frac {3\,b^2}{a^4}\right )}{a}\right )}{3\,a}\right )}{d}-\frac {b\,{\cos \left (c+d\,x\right )}^6}{3\,a^3\,d}-\frac {-a^6\,b^2+3\,a^4\,b^4-3\,a^2\,b^6+b^8}{a\,d\,\left (\cos \left (c+d\,x\right )\,a^9+b\,a^8\right )}+\frac {\ln \left (b+a\,\cos \left (c+d\,x\right )\right )\,\left (2\,a^6\,b-12\,a^4\,b^3+18\,a^2\,b^5-8\,b^7\right )}{a^9\,d} \] Input:

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

Output:

(cos(c + d*x)^4*(b^3/(2*a^5) + (b*(3/a^2 - (3*b^2)/a^4))/(2*a)))/d - (cos( 
c + d*x)^2*((b^2*((2*b^3)/a^5 + (2*b*(3/a^2 - (3*b^2)/a^4))/a))/(2*a^2) + 
(b*(3/a^2 + (b^2*(3/a^2 - (3*b^2)/a^4))/a^2 - (2*b*((2*b^3)/a^5 + (2*b*(3/ 
a^2 - (3*b^2)/a^4))/a))/a))/a))/d - (cos(c + d*x)^5*(3/(5*a^2) - (3*b^2)/( 
5*a^4)))/d + cos(c + d*x)^7/(7*a^2*d) - (cos(c + d*x)*(1/a^2 + (b^2*(3/a^2 
 + (b^2*(3/a^2 - (3*b^2)/a^4))/a^2 - (2*b*((2*b^3)/a^5 + (2*b*(3/a^2 - (3* 
b^2)/a^4))/a))/a))/a^2 - (2*b*((b^2*((2*b^3)/a^5 + (2*b*(3/a^2 - (3*b^2)/a 
^4))/a))/a^2 + (2*b*(3/a^2 + (b^2*(3/a^2 - (3*b^2)/a^4))/a^2 - (2*b*((2*b^ 
3)/a^5 + (2*b*(3/a^2 - (3*b^2)/a^4))/a))/a))/a))/a))/d + (cos(c + d*x)^3*( 
1/a^2 + (b^2*(3/a^2 - (3*b^2)/a^4))/(3*a^2) - (2*b*((2*b^3)/a^5 + (2*b*(3/ 
a^2 - (3*b^2)/a^4))/a))/(3*a)))/d - (b*cos(c + d*x)^6)/(3*a^3*d) - (b^8 - 
3*a^2*b^6 + 3*a^4*b^4 - a^6*b^2)/(a*d*(a^9*cos(c + d*x) + a^8*b)) + (log(b 
 + a*cos(c + d*x))*(2*a^6*b - 8*b^7 + 18*a^2*b^5 - 12*a^4*b^3))/(a^9*d)
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 1011, normalized size of antiderivative = 3.79 \[ \int \frac {\sin ^7(c+d x)}{(a+b \sec (c+d x))^2} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 420*cos(c + d*x)*log(tan((c + d*x)/2)**2 + 1)*a**7*b + 2520*cos(c + d* 
x)*log(tan((c + d*x)/2)**2 + 1)*a**5*b**3 - 3780*cos(c + d*x)*log(tan((c + 
 d*x)/2)**2 + 1)*a**3*b**5 + 1680*cos(c + d*x)*log(tan((c + d*x)/2)**2 + 1 
)*a*b**7 + 420*cos(c + d*x)*log(tan((c + d*x)/2)**2*a - tan((c + d*x)/2)** 
2*b - a - b)*a**7*b - 2520*cos(c + d*x)*log(tan((c + d*x)/2)**2*a - tan((c 
 + d*x)/2)**2*b - a - b)*a**5*b**3 + 3780*cos(c + d*x)*log(tan((c + d*x)/2 
)**2*a - tan((c + d*x)/2)**2*b - a - b)*a**3*b**5 - 1680*cos(c + d*x)*log( 
tan((c + d*x)/2)**2*a - tan((c + d*x)/2)**2*b - a - b)*a*b**7 + 40*cos(c + 
 d*x)*sin(c + d*x)**6*a**7*b + 69*cos(c + d*x)*sin(c + d*x)**4*a**7*b - 84 
*cos(c + d*x)*sin(c + d*x)**4*a**5*b**3 + 162*cos(c + d*x)*sin(c + d*x)**2 
*a**7*b - 462*cos(c + d*x)*sin(c + d*x)**2*a**5*b**3 + 280*cos(c + d*x)*si 
n(c + d*x)**2*a**3*b**5 + 96*cos(c + d*x)*a**8 - 96*cos(c + d*x)*a**7*b - 
1596*cos(c + d*x)*a**6*b**2 + 1596*cos(c + d*x)*a**5*b**3 + 3220*cos(c + d 
*x)*a**4*b**4 - 3220*cos(c + d*x)*a**3*b**5 - 1680*cos(c + d*x)*a**2*b**6 
+ 1680*cos(c + d*x)*a*b**7 - 420*log(tan((c + d*x)/2)**2 + 1)*a**6*b**2 + 
2520*log(tan((c + d*x)/2)**2 + 1)*a**4*b**4 - 3780*log(tan((c + d*x)/2)**2 
 + 1)*a**2*b**6 + 1680*log(tan((c + d*x)/2)**2 + 1)*b**8 + 420*log(tan((c 
+ d*x)/2)**2*a - tan((c + d*x)/2)**2*b - a - b)*a**6*b**2 - 2520*log(tan(( 
c + d*x)/2)**2*a - tan((c + d*x)/2)**2*b - a - b)*a**4*b**4 + 3780*log(tan 
((c + d*x)/2)**2*a - tan((c + d*x)/2)**2*b - a - b)*a**2*b**6 - 1680*lo...