\(\int \frac {\sec (c+d x)}{(a+b \sin (c+d x))^8} \, dx\) [462]

Optimal result

Integrand size = 19, antiderivative size = 385 \[ \int \frac {\sec (c+d x)}{(a+b \sin (c+d x))^8} \, dx=-\frac {\log (1-\sin (c+d x))}{2 (a+b)^8 d}+\frac {\log (1+\sin (c+d x))}{2 (a-b)^8 d}-\frac {8 a b \left (a^2+b^2\right ) \left (a^4+6 a^2 b^2+b^4\right ) \log (a+b \sin (c+d x))}{\left (a^2-b^2\right )^8 d}+\frac {b}{7 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^7}+\frac {a b}{3 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^6}+\frac {b \left (3 a^2+b^2\right )}{5 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^5}+\frac {a b \left (a^2+b^2\right )}{\left (a^2-b^2\right )^4 d (a+b \sin (c+d x))^4}+\frac {b \left (5 a^4+10 a^2 b^2+b^4\right )}{3 \left (a^2-b^2\right )^5 d (a+b \sin (c+d x))^3}+\frac {a b \left (3 a^2+b^2\right ) \left (a^2+3 b^2\right )}{\left (a^2-b^2\right )^6 d (a+b \sin (c+d x))^2}+\frac {b \left (7 a^6+35 a^4 b^2+21 a^2 b^4+b^6\right )}{\left (a^2-b^2\right )^7 d (a+b \sin (c+d x))} \] Output:

-1/2*ln(1-sin(d*x+c))/(a+b)^8/d+1/2*ln(1+sin(d*x+c))/(a-b)^8/d-8*a*b*(a^2+ 
b^2)*(a^4+6*a^2*b^2+b^4)*ln(a+b*sin(d*x+c))/(a^2-b^2)^8/d+1/7*b/(a^2-b^2)/ 
d/(a+b*sin(d*x+c))^7+1/3*a*b/(a^2-b^2)^2/d/(a+b*sin(d*x+c))^6+1/5*b*(3*a^2 
+b^2)/(a^2-b^2)^3/d/(a+b*sin(d*x+c))^5+a*b*(a^2+b^2)/(a^2-b^2)^4/d/(a+b*si 
n(d*x+c))^4+1/3*b*(5*a^4+10*a^2*b^2+b^4)/(a^2-b^2)^5/d/(a+b*sin(d*x+c))^3+ 
a*b*(3*a^2+b^2)*(a^2+3*b^2)/(a^2-b^2)^6/d/(a+b*sin(d*x+c))^2+b*(7*a^6+35*a 
^4*b^2+21*a^2*b^4+b^6)/(a^2-b^2)^7/d/(a+b*sin(d*x+c))
 

Mathematica [A] (verified)

Time = 2.68 (sec) , antiderivative size = 365, normalized size of antiderivative = 0.95 \[ \int \frac {\sec (c+d x)}{(a+b \sin (c+d x))^8} \, dx=\frac {b \left (-\frac {\log (1-\sin (c+d x))}{2 b (a+b)^8}+\frac {\log (1+\sin (c+d x))}{2 (a-b)^8 b}-\frac {8 a \left (a^2+b^2\right ) \left (a^4+6 a^2 b^2+b^4\right ) \log (a+b \sin (c+d x))}{(a-b)^8 (a+b)^8}+\frac {1}{7 \left (a^2-b^2\right ) (a+b \sin (c+d x))^7}+\frac {a}{3 (a-b)^2 (a+b)^2 (a+b \sin (c+d x))^6}+\frac {3 a^2+b^2}{5 (a-b)^3 (a+b)^3 (a+b \sin (c+d x))^5}+\frac {a \left (a^2+b^2\right )}{(a-b)^4 (a+b)^4 (a+b \sin (c+d x))^4}+\frac {5 a^4+10 a^2 b^2+b^4}{3 (a-b)^5 (a+b)^5 (a+b \sin (c+d x))^3}+\frac {a \left (3 a^2+b^2\right ) \left (a^2+3 b^2\right )}{(a-b)^6 (a+b)^6 (a+b \sin (c+d x))^2}+\frac {7 a^6+35 a^4 b^2+21 a^2 b^4+b^6}{(a-b)^7 (a+b)^7 (a+b \sin (c+d x))}\right )}{d} \] Input:

Integrate[Sec[c + d*x]/(a + b*Sin[c + d*x])^8,x]
 

Output:

(b*(-1/2*Log[1 - Sin[c + d*x]]/(b*(a + b)^8) + Log[1 + Sin[c + d*x]]/(2*(a 
 - b)^8*b) - (8*a*(a^2 + b^2)*(a^4 + 6*a^2*b^2 + b^4)*Log[a + b*Sin[c + d* 
x]])/((a - b)^8*(a + b)^8) + 1/(7*(a^2 - b^2)*(a + b*Sin[c + d*x])^7) + a/ 
(3*(a - b)^2*(a + b)^2*(a + b*Sin[c + d*x])^6) + (3*a^2 + b^2)/(5*(a - b)^ 
3*(a + b)^3*(a + b*Sin[c + d*x])^5) + (a*(a^2 + b^2))/((a - b)^4*(a + b)^4 
*(a + b*Sin[c + d*x])^4) + (5*a^4 + 10*a^2*b^2 + b^4)/(3*(a - b)^5*(a + b) 
^5*(a + b*Sin[c + d*x])^3) + (a*(3*a^2 + b^2)*(a^2 + 3*b^2))/((a - b)^6*(a 
 + b)^6*(a + b*Sin[c + d*x])^2) + (7*a^6 + 35*a^4*b^2 + 21*a^2*b^4 + b^6)/ 
((a - b)^7*(a + b)^7*(a + b*Sin[c + d*x]))))/d
 

Rubi [A] (verified)

Time = 0.86 (sec) , antiderivative size = 383, normalized size of antiderivative = 0.99, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {3042, 3147, 477, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\sec (c+d x)}{(a+b \sin (c+d x))^8} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\cos (c+d x) (a+b \sin (c+d x))^8}dx\)

\(\Big \downarrow \) 3147

\(\displaystyle \frac {b \int \frac {1}{(a+b \sin (c+d x))^8 \left (b^2-b^2 \sin ^2(c+d x)\right )}d(b \sin (c+d x))}{d}\)

\(\Big \downarrow \) 477

\(\displaystyle \frac {\int \left (-\frac {8 a \left (a^2+b^2\right ) \left (a^4+6 b^2 a^2+b^4\right ) b^2}{\left (a^2-b^2\right )^8 (a+b \sin (c+d x))}-\frac {\left (7 a^6+35 b^2 a^4+21 b^4 a^2+b^6\right ) b^2}{\left (a^2-b^2\right )^7 (a+b \sin (c+d x))^2}-\frac {2 a \left (3 a^2+b^2\right ) \left (a^2+3 b^2\right ) b^2}{\left (a^2-b^2\right )^6 (a+b \sin (c+d x))^3}-\frac {\left (5 a^4+10 b^2 a^2+b^4\right ) b^2}{\left (a^2-b^2\right )^5 (a+b \sin (c+d x))^4}-\frac {4 a \left (a^2+b^2\right ) b^2}{\left (a^2-b^2\right )^4 (a+b \sin (c+d x))^5}-\frac {\left (3 a^2+b^2\right ) b^2}{\left (a^2-b^2\right )^3 (a+b \sin (c+d x))^6}-\frac {2 a b^2}{\left (a^2-b^2\right )^2 (a+b \sin (c+d x))^7}-\frac {b^2}{\left (a^2-b^2\right ) (a+b \sin (c+d x))^8}+\frac {b}{2 (a+b)^8 (b-b \sin (c+d x))}+\frac {b}{2 (a-b)^8 (\sin (c+d x) b+b)}\right )d(b \sin (c+d x))}{b d}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\frac {a b^2 \left (3 a^2+b^2\right ) \left (a^2+3 b^2\right )}{\left (a^2-b^2\right )^6 (a+b \sin (c+d x))^2}+\frac {a b^2 \left (a^2+b^2\right )}{\left (a^2-b^2\right )^4 (a+b \sin (c+d x))^4}+\frac {b^2 \left (3 a^2+b^2\right )}{5 \left (a^2-b^2\right )^3 (a+b \sin (c+d x))^5}+\frac {a b^2}{3 \left (a^2-b^2\right )^2 (a+b \sin (c+d x))^6}+\frac {b^2}{7 \left (a^2-b^2\right ) (a+b \sin (c+d x))^7}+\frac {b^2 \left (5 a^4+10 a^2 b^2+b^4\right )}{3 \left (a^2-b^2\right )^5 (a+b \sin (c+d x))^3}-\frac {8 a b^2 \left (a^2+b^2\right ) \left (a^4+6 a^2 b^2+b^4\right ) \log (a+b \sin (c+d x))}{\left (a^2-b^2\right )^8}+\frac {b^2 \left (7 a^6+35 a^4 b^2+21 a^2 b^4+b^6\right )}{\left (a^2-b^2\right )^7 (a+b \sin (c+d x))}-\frac {b \log (b-b \sin (c+d x))}{2 (a+b)^8}+\frac {b \log (b \sin (c+d x)+b)}{2 (a-b)^8}}{b d}\)

Input:

Int[Sec[c + d*x]/(a + b*Sin[c + d*x])^8,x]
 

Output:

(-1/2*(b*Log[b - b*Sin[c + d*x]])/(a + b)^8 - (8*a*b^2*(a^2 + b^2)*(a^4 + 
6*a^2*b^2 + b^4)*Log[a + b*Sin[c + d*x]])/(a^2 - b^2)^8 + (b*Log[b + b*Sin 
[c + d*x]])/(2*(a - b)^8) + b^2/(7*(a^2 - b^2)*(a + b*Sin[c + d*x])^7) + ( 
a*b^2)/(3*(a^2 - b^2)^2*(a + b*Sin[c + d*x])^6) + (b^2*(3*a^2 + b^2))/(5*( 
a^2 - b^2)^3*(a + b*Sin[c + d*x])^5) + (a*b^2*(a^2 + b^2))/((a^2 - b^2)^4* 
(a + b*Sin[c + d*x])^4) + (b^2*(5*a^4 + 10*a^2*b^2 + b^4))/(3*(a^2 - b^2)^ 
5*(a + b*Sin[c + d*x])^3) + (a*b^2*(3*a^2 + b^2)*(a^2 + 3*b^2))/((a^2 - b^ 
2)^6*(a + b*Sin[c + d*x])^2) + (b^2*(7*a^6 + 35*a^4*b^2 + 21*a^2*b^4 + b^6 
))/((a^2 - b^2)^7*(a + b*Sin[c + d*x])))/(b*d)
 

Defintions of rubi rules used

rule 477

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ 
a^p   Int[ExpandIntegrand[(c + d*x)^n*(1 - Rt[-b/a, 2]*x)^p*(1 + Rt[-b/a, 2 
]*x)^p, x], x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[p, 0] && IntegerQ[n] & 
& NiceSqrtQ[-b/a] &&  !FractionalPowerFactorQ[Rt[-b/a, 2]]
 

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 3147

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m 
_.), x_Symbol] :> Simp[1/(b^p*f)   Subst[Int[(a + x)^m*(b^2 - x^2)^((p - 1) 
/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && IntegerQ[(p 
 - 1)/2] && NeQ[a^2 - b^2, 0]
 

Maple [A] (verified)

Time = 13.66 (sec) , antiderivative size = 356, normalized size of antiderivative = 0.92

method	result	size
derivativedivides	\(\frac {\frac {\ln \left (1+\sin \left (d x +c \right )\right )}{2 \left (a -b \right )^{8}}+\frac {b}{7 \left (a -b \right ) \left (a +b \right ) \left (a +b \sin \left (d x +c \right )\right )^{7}}+\frac {a b}{3 \left (a +b \right )^{2} \left (a -b \right )^{2} \left (a +b \sin \left (d x +c \right )\right )^{6}}+\frac {b \left (3 a^{2}+b^{2}\right )}{5 \left (a +b \right )^{3} \left (a -b \right )^{3} \left (a +b \sin \left (d x +c \right )\right )^{5}}+\frac {b \left (5 a^{4}+10 b^{2} a^{2}+b^{4}\right )}{3 \left (a +b \right )^{5} \left (a -b \right )^{5} \left (a +b \sin \left (d x +c \right )\right )^{3}}+\frac {b \left (7 a^{6}+35 a^{4} b^{2}+21 a^{2} b^{4}+b^{6}\right )}{\left (a -b \right )^{7} \left (a +b \right )^{7} \left (a +b \sin \left (d x +c \right )\right )}+\frac {b a \left (a^{2}+b^{2}\right )}{\left (a +b \right )^{4} \left (a -b \right )^{4} \left (a +b \sin \left (d x +c \right )\right )^{4}}+\frac {b a \left (3 a^{4}+10 b^{2} a^{2}+3 b^{4}\right )}{\left (a +b \right )^{6} \left (a -b \right )^{6} \left (a +b \sin \left (d x +c \right )\right )^{2}}-\frac {8 b a \left (a^{6}+7 a^{4} b^{2}+7 a^{2} b^{4}+b^{6}\right ) \ln \left (a +b \sin \left (d x +c \right )\right )}{\left (a +b \right )^{8} \left (a -b \right )^{8}}-\frac {\ln \left (\sin \left (d x +c \right )-1\right )}{2 \left (a +b \right )^{8}}}{d}\)	\(356\)
default	\(\frac {\frac {\ln \left (1+\sin \left (d x +c \right )\right )}{2 \left (a -b \right )^{8}}+\frac {b}{7 \left (a -b \right ) \left (a +b \right ) \left (a +b \sin \left (d x +c \right )\right )^{7}}+\frac {a b}{3 \left (a +b \right )^{2} \left (a -b \right )^{2} \left (a +b \sin \left (d x +c \right )\right )^{6}}+\frac {b \left (3 a^{2}+b^{2}\right )}{5 \left (a +b \right )^{3} \left (a -b \right )^{3} \left (a +b \sin \left (d x +c \right )\right )^{5}}+\frac {b \left (5 a^{4}+10 b^{2} a^{2}+b^{4}\right )}{3 \left (a +b \right )^{5} \left (a -b \right )^{5} \left (a +b \sin \left (d x +c \right )\right )^{3}}+\frac {b \left (7 a^{6}+35 a^{4} b^{2}+21 a^{2} b^{4}+b^{6}\right )}{\left (a -b \right )^{7} \left (a +b \right )^{7} \left (a +b \sin \left (d x +c \right )\right )}+\frac {b a \left (a^{2}+b^{2}\right )}{\left (a +b \right )^{4} \left (a -b \right )^{4} \left (a +b \sin \left (d x +c \right )\right )^{4}}+\frac {b a \left (3 a^{4}+10 b^{2} a^{2}+3 b^{4}\right )}{\left (a +b \right )^{6} \left (a -b \right )^{6} \left (a +b \sin \left (d x +c \right )\right )^{2}}-\frac {8 b a \left (a^{6}+7 a^{4} b^{2}+7 a^{2} b^{4}+b^{6}\right ) \ln \left (a +b \sin \left (d x +c \right )\right )}{\left (a +b \right )^{8} \left (a -b \right )^{8}}-\frac {\ln \left (\sin \left (d x +c \right )-1\right )}{2 \left (a +b \right )^{8}}}{d}\)	\(356\)
parallelrisch	\(\text {Expression too large to display}\)	\(1529\)
norman	\(\text {Expression too large to display}\)	\(2136\)
risch	\(\text {Expression too large to display}\)	\(2620\)

Input:

int(sec(d*x+c)/(a+b*sin(d*x+c))^8,x,method=_RETURNVERBOSE)
 

Output:

1/d*(1/2/(a-b)^8*ln(1+sin(d*x+c))+1/7*b/(a-b)/(a+b)/(a+b*sin(d*x+c))^7+1/3 
*a*b/(a+b)^2/(a-b)^2/(a+b*sin(d*x+c))^6+1/5*b*(3*a^2+b^2)/(a+b)^3/(a-b)^3/ 
(a+b*sin(d*x+c))^5+1/3*b*(5*a^4+10*a^2*b^2+b^4)/(a+b)^5/(a-b)^5/(a+b*sin(d 
*x+c))^3+b*(7*a^6+35*a^4*b^2+21*a^2*b^4+b^6)/(a-b)^7/(a+b)^7/(a+b*sin(d*x+ 
c))+b*a*(a^2+b^2)/(a+b)^4/(a-b)^4/(a+b*sin(d*x+c))^4+b*a*(3*a^4+10*a^2*b^2 
+3*b^4)/(a+b)^6/(a-b)^6/(a+b*sin(d*x+c))^2-8*b*a*(a^6+7*a^4*b^2+7*a^2*b^4+ 
b^6)/(a+b)^8/(a-b)^8*ln(a+b*sin(d*x+c))-1/2/(a+b)^8*ln(sin(d*x+c)-1))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3165 vs. \(2 (373) = 746\).

Time = 1.46 (sec) , antiderivative size = 3165, normalized size of antiderivative = 8.22 \[ \int \frac {\sec (c+d x)}{(a+b \sin (c+d x))^8} \, dx=\text {Too large to display} \] Input:

integrate(sec(d*x+c)/(a+b*sin(d*x+c))^8,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\sec (c+d x)}{(a+b \sin (c+d x))^8} \, dx=\text {Timed out} \] Input:

integrate(sec(d*x+c)/(a+b*sin(d*x+c))**8,x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1160 vs. \(2 (373) = 746\).

Time = 0.08 (sec) , antiderivative size = 1160, normalized size of antiderivative = 3.01 \[ \int \frac {\sec (c+d x)}{(a+b \sin (c+d x))^8} \, dx=\text {Too large to display} \] Input:

integrate(sec(d*x+c)/(a+b*sin(d*x+c))^8,x, algorithm="maxima")
 

Output:

-1/210*(1680*(a^7*b + 7*a^5*b^3 + 7*a^3*b^5 + a*b^7)*log(b*sin(d*x + c) + 
a)/(a^16 - 8*a^14*b^2 + 28*a^12*b^4 - 56*a^10*b^6 + 70*a^8*b^8 - 56*a^6*b^ 
10 + 28*a^4*b^12 - 8*a^2*b^14 + b^16) - 2*(1443*a^12*b + 3704*a^10*b^3 + 1 
849*a^8*b^5 - 496*a^6*b^7 + 309*a^4*b^9 - 104*a^2*b^11 + 15*b^13 + 105*(7* 
a^6*b^7 + 35*a^4*b^9 + 21*a^2*b^11 + b^13)*sin(d*x + c)^6 + 105*(45*a^7*b^ 
6 + 217*a^5*b^8 + 119*a^3*b^10 + 3*a*b^12)*sin(d*x + c)^5 + 35*(365*a^8*b^ 
5 + 1680*a^6*b^7 + 826*a^4*b^9 + 8*a^2*b^11 + b^13)*sin(d*x + c)^4 + 35*(5 
33*a^9*b^4 + 2304*a^7*b^6 + 994*a^5*b^8 + 8*a^3*b^10 + a*b^12)*sin(d*x + c 
)^3 + 21*(743*a^10*b^3 + 2934*a^8*b^5 + 1099*a^6*b^7 + 29*a^4*b^9 - 6*a^2* 
b^11 + b^13)*sin(d*x + c)^2 + 7*(1023*a^11*b^2 + 3494*a^9*b^4 + 1219*a^7*b 
^6 + 29*a^5*b^8 - 6*a^3*b^10 + a*b^12)*sin(d*x + c))/(a^21 - 7*a^19*b^2 + 
21*a^17*b^4 - 35*a^15*b^6 + 35*a^13*b^8 - 21*a^11*b^10 + 7*a^9*b^12 - a^7* 
b^14 + (a^14*b^7 - 7*a^12*b^9 + 21*a^10*b^11 - 35*a^8*b^13 + 35*a^6*b^15 - 
 21*a^4*b^17 + 7*a^2*b^19 - b^21)*sin(d*x + c)^7 + 7*(a^15*b^6 - 7*a^13*b^ 
8 + 21*a^11*b^10 - 35*a^9*b^12 + 35*a^7*b^14 - 21*a^5*b^16 + 7*a^3*b^18 - 
a*b^20)*sin(d*x + c)^6 + 21*(a^16*b^5 - 7*a^14*b^7 + 21*a^12*b^9 - 35*a^10 
*b^11 + 35*a^8*b^13 - 21*a^6*b^15 + 7*a^4*b^17 - a^2*b^19)*sin(d*x + c)^5 
+ 35*(a^17*b^4 - 7*a^15*b^6 + 21*a^13*b^8 - 35*a^11*b^10 + 35*a^9*b^12 - 2 
1*a^7*b^14 + 7*a^5*b^16 - a^3*b^18)*sin(d*x + c)^4 + 35*(a^18*b^3 - 7*a^16 
*b^5 + 21*a^14*b^7 - 35*a^12*b^9 + 35*a^10*b^11 - 21*a^8*b^13 + 7*a^6*b...
 

Giac [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 719, normalized size of antiderivative = 1.87 \[ \int \frac {\sec (c+d x)}{(a+b \sin (c+d x))^8} \, dx =\text {Too large to display} \] Input:

integrate(sec(d*x+c)/(a+b*sin(d*x+c))^8,x, algorithm="giac")
 

Output:

-8*(a^7*b^2 + 7*a^5*b^4 + 7*a^3*b^6 + a*b^8)*log(abs(b*sin(d*x + c) + a))/ 
(a^16*b*d - 8*a^14*b^3*d + 28*a^12*b^5*d - 56*a^10*b^7*d + 70*a^8*b^9*d - 
56*a^6*b^11*d + 28*a^4*b^13*d - 8*a^2*b^15*d + b^17*d) - 1/2*log(abs(-sin( 
d*x + c) + 1))/(a^8*d + 8*a^7*b*d + 28*a^6*b^2*d + 56*a^5*b^3*d + 70*a^4*b 
^4*d + 56*a^3*b^5*d + 28*a^2*b^6*d + 8*a*b^7*d + b^8*d) + 1/2*log(abs(-sin 
(d*x + c) - 1))/(a^8*d - 8*a^7*b*d + 28*a^6*b^2*d - 56*a^5*b^3*d + 70*a^4* 
b^4*d - 56*a^3*b^5*d + 28*a^2*b^6*d - 8*a*b^7*d + b^8*d) + 1/105*(1443*a^1 
4*b + 2261*a^12*b^3 - 1855*a^10*b^5 - 2345*a^8*b^7 + 805*a^6*b^9 - 413*a^4 
*b^11 + 119*a^2*b^13 - 15*b^15 + 105*(7*a^8*b^7 + 28*a^6*b^9 - 14*a^4*b^11 
 - 20*a^2*b^13 - b^15)*sin(d*x + c)^6 + 105*(45*a^9*b^6 + 172*a^7*b^8 - 98 
*a^5*b^10 - 116*a^3*b^12 - 3*a*b^14)*sin(d*x + c)^5 + 35*(365*a^10*b^5 + 1 
315*a^8*b^7 - 854*a^6*b^9 - 818*a^4*b^11 - 7*a^2*b^13 - b^15)*sin(d*x + c) 
^4 + 35*(533*a^11*b^4 + 1771*a^9*b^6 - 1310*a^7*b^8 - 986*a^5*b^10 - 7*a^3 
*b^12 - a*b^14)*sin(d*x + c)^3 + 21*(743*a^12*b^3 + 2191*a^10*b^5 - 1835*a 
^8*b^7 - 1070*a^6*b^9 - 35*a^4*b^11 + 7*a^2*b^13 - b^15)*sin(d*x + c)^2 + 
7*(1023*a^13*b^2 + 2471*a^11*b^4 - 2275*a^9*b^6 - 1190*a^7*b^8 - 35*a^5*b^ 
10 + 7*a^3*b^12 - a*b^14)*sin(d*x + c))/((b*sin(d*x + c) + a)^7*(a + b)^8* 
(a - b)^8*d)
 

Mupad [B] (verification not implemented)

Time = 18.55 (sec) , antiderivative size = 937, normalized size of antiderivative = 2.43 \[ \int \frac {\sec (c+d x)}{(a+b \sin (c+d x))^8} \, dx =\text {Too large to display} \] Input:

int(1/(cos(c + d*x)*(a + b*sin(c + d*x))^8),x)
 

Output:

(log(a + b*sin(c + d*x))*(1/(2*(a + b)^8) - 1/(2*(a - b)^8)))/d + ((1443*a 
^12*b + 15*b^13 - 104*a^2*b^11 + 309*a^4*b^9 - 496*a^6*b^7 + 1849*a^8*b^5 
+ 3704*a^10*b^3)/(105*(a^14 - b^14 + 7*a^2*b^12 - 21*a^4*b^10 + 35*a^6*b^8 
 - 35*a^8*b^6 + 21*a^10*b^4 - 7*a^12*b^2)) + (sin(c + d*x)*(a*b^12 - 6*a^3 
*b^10 + 29*a^5*b^8 + 1219*a^7*b^6 + 3494*a^9*b^4 + 1023*a^11*b^2))/(15*(a^ 
14 - b^14 + 7*a^2*b^12 - 21*a^4*b^10 + 35*a^6*b^8 - 35*a^8*b^6 + 21*a^10*b 
^4 - 7*a^12*b^2)) + (sin(c + d*x)^3*(a*b^12 + 8*a^3*b^10 + 994*a^5*b^8 + 2 
304*a^7*b^6 + 533*a^9*b^4))/(3*(a^14 - b^14 + 7*a^2*b^12 - 21*a^4*b^10 + 3 
5*a^6*b^8 - 35*a^8*b^6 + 21*a^10*b^4 - 7*a^12*b^2)) + (sin(c + d*x)^5*(3*a 
*b^12 + 119*a^3*b^10 + 217*a^5*b^8 + 45*a^7*b^6))/(a^14 - b^14 + 7*a^2*b^1 
2 - 21*a^4*b^10 + 35*a^6*b^8 - 35*a^8*b^6 + 21*a^10*b^4 - 7*a^12*b^2) + (s 
in(c + d*x)^2*(b^13 - 6*a^2*b^11 + 29*a^4*b^9 + 1099*a^6*b^7 + 2934*a^8*b^ 
5 + 743*a^10*b^3))/(5*(a^14 - b^14 + 7*a^2*b^12 - 21*a^4*b^10 + 35*a^6*b^8 
 - 35*a^8*b^6 + 21*a^10*b^4 - 7*a^12*b^2)) + (sin(c + d*x)^4*(b^13 + 8*a^2 
*b^11 + 826*a^4*b^9 + 1680*a^6*b^7 + 365*a^8*b^5))/(3*(a^14 - b^14 + 7*a^2 
*b^12 - 21*a^4*b^10 + 35*a^6*b^8 - 35*a^8*b^6 + 21*a^10*b^4 - 7*a^12*b^2)) 
 + (sin(c + d*x)^6*(b^13 + 21*a^2*b^11 + 35*a^4*b^9 + 7*a^6*b^7))/(a^14 - 
b^14 + 7*a^2*b^12 - 21*a^4*b^10 + 35*a^6*b^8 - 35*a^8*b^6 + 21*a^10*b^4 - 
7*a^12*b^2))/(d*(a^7 + b^7*sin(c + d*x)^7 + 7*a*b^6*sin(c + d*x)^6 + 21*a^ 
5*b^2*sin(c + d*x)^2 + 35*a^4*b^3*sin(c + d*x)^3 + 35*a^3*b^4*sin(c + d...
 

Reduce [B] (verification not implemented)

Time = 1.29 (sec) , antiderivative size = 6943, normalized size of antiderivative = 18.03 \[ \int \frac {\sec (c+d x)}{(a+b \sin (c+d x))^8} \, dx =\text {Too large to display} \] Input:

int(sec(d*x+c)/(a+b*sin(d*x+c))^8,x)
 

Output:

( - 105*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**7*a**13*b**7 + 840*log(tan 
((c + d*x)/2) - 1)*sin(c + d*x)**7*a**12*b**8 - 2940*log(tan((c + d*x)/2) 
- 1)*sin(c + d*x)**7*a**11*b**9 + 5880*log(tan((c + d*x)/2) - 1)*sin(c + d 
*x)**7*a**10*b**10 - 7350*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**7*a**9*b 
**11 + 5880*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**7*a**8*b**12 - 2940*lo 
g(tan((c + d*x)/2) - 1)*sin(c + d*x)**7*a**7*b**13 + 840*log(tan((c + d*x) 
/2) - 1)*sin(c + d*x)**7*a**6*b**14 - 105*log(tan((c + d*x)/2) - 1)*sin(c 
+ d*x)**7*a**5*b**15 - 735*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**6*a**14 
*b**6 + 5880*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**6*a**13*b**7 - 20580* 
log(tan((c + d*x)/2) - 1)*sin(c + d*x)**6*a**12*b**8 + 41160*log(tan((c + 
d*x)/2) - 1)*sin(c + d*x)**6*a**11*b**9 - 51450*log(tan((c + d*x)/2) - 1)* 
sin(c + d*x)**6*a**10*b**10 + 41160*log(tan((c + d*x)/2) - 1)*sin(c + d*x) 
**6*a**9*b**11 - 20580*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**6*a**8*b**1 
2 + 5880*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**6*a**7*b**13 - 735*log(ta 
n((c + d*x)/2) - 1)*sin(c + d*x)**6*a**6*b**14 - 2205*log(tan((c + d*x)/2) 
 - 1)*sin(c + d*x)**5*a**15*b**5 + 17640*log(tan((c + d*x)/2) - 1)*sin(c + 
 d*x)**5*a**14*b**6 - 61740*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**5*a**1 
3*b**7 + 123480*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**5*a**12*b**8 - 154 
350*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**5*a**11*b**9 + 123480*log(tan( 
(c + d*x)/2) - 1)*sin(c + d*x)**5*a**10*b**10 - 61740*log(tan((c + d*x)...