\(\int \frac {\sec ^3(c+d x)}{(a+b \sin (c+d x))^8} \, dx\) [463]

Optimal result

Integrand size = 21, antiderivative size = 474 \[ \int \frac {\sec ^3(c+d x)}{(a+b \sin (c+d x))^8} \, dx=-\frac {(a+9 b) \log (1-\sin (c+d x))}{4 (a+b)^9 d}+\frac {(a-9 b) \log (1+\sin (c+d x))}{4 (a-b)^9 d}+\frac {8 a b^3 \left (15 a^6+63 a^4 b^2+45 a^2 b^4+5 b^6\right ) \log (a+b \sin (c+d x))}{\left (a^2-b^2\right )^9 d}+\frac {1}{4 (a+b)^8 d (1-\sin (c+d x))}-\frac {1}{4 (a-b)^8 d (1+\sin (c+d x))}-\frac {b^3}{7 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^7}-\frac {2 a b^3}{3 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^6}-\frac {2 b^3 \left (5 a^2+b^2\right )}{5 \left (a^2-b^2\right )^4 d (a+b \sin (c+d x))^5}-\frac {a b^3 \left (5 a^2+3 b^2\right )}{\left (a^2-b^2\right )^5 d (a+b \sin (c+d x))^4}-\frac {b^3 \left (35 a^4+42 a^2 b^2+3 b^4\right )}{3 \left (a^2-b^2\right )^6 d (a+b \sin (c+d x))^3}-\frac {4 a b^3 \left (7 a^4+14 a^2 b^2+3 b^4\right )}{\left (a^2-b^2\right )^7 d (a+b \sin (c+d x))^2}-\frac {4 b^3 \left (21 a^6+63 a^4 b^2+27 a^2 b^4+b^6\right )}{\left (a^2-b^2\right )^8 d (a+b \sin (c+d x))} \] Output:

-1/4*(a+9*b)*ln(1-sin(d*x+c))/(a+b)^9/d+1/4*(a-9*b)*ln(1+sin(d*x+c))/(a-b) 
^9/d+8*a*b^3*(15*a^6+63*a^4*b^2+45*a^2*b^4+5*b^6)*ln(a+b*sin(d*x+c))/(a^2- 
b^2)^9/d+1/4/(a+b)^8/d/(1-sin(d*x+c))-1/4/(a-b)^8/d/(1+sin(d*x+c))-1/7*b^3 
/(a^2-b^2)^2/d/(a+b*sin(d*x+c))^7-2/3*a*b^3/(a^2-b^2)^3/d/(a+b*sin(d*x+c)) 
^6-2/5*b^3*(5*a^2+b^2)/(a^2-b^2)^4/d/(a+b*sin(d*x+c))^5-a*b^3*(5*a^2+3*b^2 
)/(a^2-b^2)^5/d/(a+b*sin(d*x+c))^4-1/3*b^3*(35*a^4+42*a^2*b^2+3*b^4)/(a^2- 
b^2)^6/d/(a+b*sin(d*x+c))^3-4*a*b^3*(7*a^4+14*a^2*b^2+3*b^4)/(a^2-b^2)^7/d 
/(a+b*sin(d*x+c))^2-4*b^3*(21*a^6+63*a^4*b^2+27*a^2*b^4+b^6)/(a^2-b^2)^8/d 
/(a+b*sin(d*x+c))
 

Mathematica [A] (verified)

Time = 6.79 (sec) , antiderivative size = 770, normalized size of antiderivative = 1.62 \[ \int \frac {\sec ^3(c+d x)}{(a+b \sin (c+d x))^8} \, dx=\frac {b^3 \left (\frac {\sec ^2(c+d x) \left (b^2-a b \sin (c+d x)\right )}{2 b^4 \left (-a^2+b^2\right ) (a+b \sin (c+d x))^7}-\frac {8 a \left (-\frac {\log (1-\sin (c+d x))}{2 b (a+b)^7}+\frac {\log (1+\sin (c+d x))}{2 (a-b)^7 b}-\frac {\left (7 a^6+35 a^4 b^2+21 a^2 b^4+b^6\right ) \log (a+b \sin (c+d x))}{(a-b)^7 (a+b)^7}+\frac {1}{6 \left (a^2-b^2\right ) (a+b \sin (c+d x))^6}+\frac {2 a}{5 (a-b)^2 (a+b)^2 (a+b \sin (c+d x))^5}+\frac {3 a^2+b^2}{4 (a-b)^3 (a+b)^3 (a+b \sin (c+d x))^4}+\frac {4 a \left (a^2+b^2\right )}{3 (a-b)^4 (a+b)^4 (a+b \sin (c+d x))^3}+\frac {5 a^4+10 a^2 b^2+b^4}{2 (a-b)^5 (a+b)^5 (a+b \sin (c+d x))^2}+\frac {2 a \left (3 a^2+b^2\right ) \left (a^2+3 b^2\right )}{(a-b)^6 (a+b)^6 (a+b \sin (c+d x))}\right )+\left (-7 a^2-9 b^2\right ) \left (-\frac {\log (1-\sin (c+d x))}{2 b (a+b)^8}+\frac {\log (1+\sin (c+d x))}{2 (a-b)^8 b}-\frac {8 a \left (a^2+b^2\right ) \left (a^4+6 a^2 b^2+b^4\right ) \log (a+b \sin (c+d x))}{(a-b)^8 (a+b)^8}+\frac {1}{7 \left (a^2-b^2\right ) (a+b \sin (c+d x))^7}+\frac {a}{3 (a-b)^2 (a+b)^2 (a+b \sin (c+d x))^6}+\frac {3 a^2+b^2}{5 (a-b)^3 (a+b)^3 (a+b \sin (c+d x))^5}+\frac {a \left (a^2+b^2\right )}{(a-b)^4 (a+b)^4 (a+b \sin (c+d x))^4}+\frac {5 a^4+10 a^2 b^2+b^4}{3 (a-b)^5 (a+b)^5 (a+b \sin (c+d x))^3}+\frac {a \left (3 a^2+b^2\right ) \left (a^2+3 b^2\right )}{(a-b)^6 (a+b)^6 (a+b \sin (c+d x))^2}+\frac {7 a^6+35 a^4 b^2+21 a^2 b^4+b^6}{(a-b)^7 (a+b)^7 (a+b \sin (c+d x))}\right )}{2 b^2 \left (-a^2+b^2\right )}\right )}{d} \] Input:

Integrate[Sec[c + d*x]^3/(a + b*Sin[c + d*x])^8,x]
 

Output:

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

Rubi [A] (verified)

Time = 1.06 (sec) , antiderivative size = 459, normalized size of antiderivative = 0.97, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, 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.

\(\displaystyle \int \frac {\sec ^3(c+d x)}{(a+b \sin (c+d x))^8} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\cos (c+d x)^3 (a+b \sin (c+d x))^8}dx\)

\(\Big \downarrow \) 3147

\(\displaystyle \frac {b^3 \int \frac {1}{(a+b \sin (c+d x))^8 \left (b^2-b^2 \sin ^2(c+d x)\right )^2}d(b \sin (c+d x))}{d}\)

\(\Big \downarrow \) 477

\(\displaystyle \frac {\int \left (\frac {8 a \left (15 a^6+63 b^2 a^4+45 b^4 a^2+5 b^6\right ) b^4}{\left (a^2-b^2\right )^9 (a+b \sin (c+d x))}+\frac {4 \left (21 a^6+63 b^2 a^4+27 b^4 a^2+b^6\right ) b^4}{\left (a^2-b^2\right )^8 (a+b \sin (c+d x))^2}+\frac {8 a \left (7 a^4+14 b^2 a^2+3 b^4\right ) b^4}{\left (a^2-b^2\right )^7 (a+b \sin (c+d x))^3}+\frac {\left (35 a^4+42 b^2 a^2+3 b^4\right ) b^4}{\left (a^2-b^2\right )^6 (a+b \sin (c+d x))^4}+\frac {4 a \left (5 a^2+3 b^2\right ) b^4}{\left (a^2-b^2\right )^5 (a+b \sin (c+d x))^5}+\frac {2 \left (5 a^2+b^2\right ) b^4}{\left (a^2-b^2\right )^4 (a+b \sin (c+d x))^6}+\frac {4 a b^4}{\left (a^2-b^2\right )^3 (a+b \sin (c+d x))^7}+\frac {b^4}{\left (a^2-b^2\right )^2 (a+b \sin (c+d x))^8}+\frac {b^2}{4 (a+b)^8 (b-b \sin (c+d x))^2}+\frac {b^2}{4 (a-b)^8 (\sin (c+d x) b+b)^2}+\frac {(a+9 b) b}{4 (a+b)^9 (b-b \sin (c+d x))}+\frac {(a-9 b) b}{4 (a-b)^9 (\sin (c+d x) b+b)}\right )d(b \sin (c+d x))}{b d}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {-\frac {a b^4 \left (5 a^2+3 b^2\right )}{\left (a^2-b^2\right )^5 (a+b \sin (c+d x))^4}-\frac {2 b^4 \left (5 a^2+b^2\right )}{5 \left (a^2-b^2\right )^4 (a+b \sin (c+d x))^5}-\frac {2 a b^4}{3 \left (a^2-b^2\right )^3 (a+b \sin (c+d x))^6}-\frac {b^4}{7 \left (a^2-b^2\right )^2 (a+b \sin (c+d x))^7}-\frac {4 a b^4 \left (7 a^4+14 a^2 b^2+3 b^4\right )}{\left (a^2-b^2\right )^7 (a+b \sin (c+d x))^2}-\frac {b^4 \left (35 a^4+42 a^2 b^2+3 b^4\right )}{3 \left (a^2-b^2\right )^6 (a+b \sin (c+d x))^3}-\frac {4 b^4 \left (21 a^6+63 a^4 b^2+27 a^2 b^4+b^6\right )}{\left (a^2-b^2\right )^8 (a+b \sin (c+d x))}+\frac {8 a b^4 \left (15 a^6+63 a^4 b^2+45 a^2 b^4+5 b^6\right ) \log (a+b \sin (c+d x))}{\left (a^2-b^2\right )^9}+\frac {b^2}{4 (a+b)^8 (b-b \sin (c+d x))}-\frac {b^2}{4 (a-b)^8 (b \sin (c+d x)+b)}-\frac {b (a+9 b) \log (b-b \sin (c+d x))}{4 (a+b)^9}+\frac {b (a-9 b) \log (b \sin (c+d x)+b)}{4 (a-b)^9}}{b d}\)

Input:

Int[Sec[c + d*x]^3/(a + b*Sin[c + d*x])^8,x]
 

Output:

(-1/4*(b*(a + 9*b)*Log[b - b*Sin[c + d*x]])/(a + b)^9 + (8*a*b^4*(15*a^6 + 
 63*a^4*b^2 + 45*a^2*b^4 + 5*b^6)*Log[a + b*Sin[c + d*x]])/(a^2 - b^2)^9 + 
 ((a - 9*b)*b*Log[b + b*Sin[c + d*x]])/(4*(a - b)^9) + b^2/(4*(a + b)^8*(b 
 - b*Sin[c + d*x])) - b^4/(7*(a^2 - b^2)^2*(a + b*Sin[c + d*x])^7) - (2*a* 
b^4)/(3*(a^2 - b^2)^3*(a + b*Sin[c + d*x])^6) - (2*b^4*(5*a^2 + b^2))/(5*( 
a^2 - b^2)^4*(a + b*Sin[c + d*x])^5) - (a*b^4*(5*a^2 + 3*b^2))/((a^2 - b^2 
)^5*(a + b*Sin[c + d*x])^4) - (b^4*(35*a^4 + 42*a^2*b^2 + 3*b^4))/(3*(a^2 
- b^2)^6*(a + b*Sin[c + d*x])^3) - (4*a*b^4*(7*a^4 + 14*a^2*b^2 + 3*b^4))/ 
((a^2 - b^2)^7*(a + b*Sin[c + d*x])^2) - (4*b^4*(21*a^6 + 63*a^4*b^2 + 27* 
a^2*b^4 + b^6))/((a^2 - b^2)^8*(a + b*Sin[c + d*x])) - b^2/(4*(a - b)^8*(b 
 + 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 = 28.03 (sec) , antiderivative size = 433, normalized size of antiderivative = 0.91

method	result	size
derivativedivides	\(\frac {-\frac {1}{4 \left (a -b \right )^{8} \left (1+\sin \left (d x +c \right )\right )}+\frac {\left (a -9 b \right ) \ln \left (1+\sin \left (d x +c \right )\right )}{4 \left (a -b \right )^{9}}-\frac {b^{3}}{7 \left (a +b \right )^{2} \left (a -b \right )^{2} \left (a +b \sin \left (d x +c \right )\right )^{7}}-\frac {b^{3} \left (35 a^{4}+42 b^{2} a^{2}+3 b^{4}\right )}{3 \left (a +b \right )^{6} \left (a -b \right )^{6} \left (a +b \sin \left (d x +c \right )\right )^{3}}-\frac {2 a \,b^{3}}{3 \left (a +b \right )^{3} \left (a -b \right )^{3} \left (a +b \sin \left (d x +c \right )\right )^{6}}-\frac {2 b^{3} \left (5 a^{2}+b^{2}\right )}{5 \left (a +b \right )^{4} \left (a -b \right )^{4} \left (a +b \sin \left (d x +c \right )\right )^{5}}-\frac {4 b^{3} \left (21 a^{6}+63 a^{4} b^{2}+27 a^{2} b^{4}+b^{6}\right )}{\left (a +b \right )^{8} \left (a -b \right )^{8} \left (a +b \sin \left (d x +c \right )\right )}-\frac {b^{3} a \left (5 a^{2}+3 b^{2}\right )}{\left (a +b \right )^{5} \left (a -b \right )^{5} \left (a +b \sin \left (d x +c \right )\right )^{4}}-\frac {4 b^{3} a \left (7 a^{4}+14 b^{2} a^{2}+3 b^{4}\right )}{\left (a +b \right )^{7} \left (a -b \right )^{7} \left (a +b \sin \left (d x +c \right )\right )^{2}}+\frac {8 b^{3} a \left (15 a^{6}+63 a^{4} b^{2}+45 a^{2} b^{4}+5 b^{6}\right ) \ln \left (a +b \sin \left (d x +c \right )\right )}{\left (a +b \right )^{9} \left (a -b \right )^{9}}-\frac {1}{4 \left (a +b \right )^{8} \left (\sin \left (d x +c \right )-1\right )}+\frac {\left (-a -9 b \right ) \ln \left (\sin \left (d x +c \right )-1\right )}{4 \left (a +b \right )^{9}}}{d}\)	\(433\)
default	\(\frac {-\frac {1}{4 \left (a -b \right )^{8} \left (1+\sin \left (d x +c \right )\right )}+\frac {\left (a -9 b \right ) \ln \left (1+\sin \left (d x +c \right )\right )}{4 \left (a -b \right )^{9}}-\frac {b^{3}}{7 \left (a +b \right )^{2} \left (a -b \right )^{2} \left (a +b \sin \left (d x +c \right )\right )^{7}}-\frac {b^{3} \left (35 a^{4}+42 b^{2} a^{2}+3 b^{4}\right )}{3 \left (a +b \right )^{6} \left (a -b \right )^{6} \left (a +b \sin \left (d x +c \right )\right )^{3}}-\frac {2 a \,b^{3}}{3 \left (a +b \right )^{3} \left (a -b \right )^{3} \left (a +b \sin \left (d x +c \right )\right )^{6}}-\frac {2 b^{3} \left (5 a^{2}+b^{2}\right )}{5 \left (a +b \right )^{4} \left (a -b \right )^{4} \left (a +b \sin \left (d x +c \right )\right )^{5}}-\frac {4 b^{3} \left (21 a^{6}+63 a^{4} b^{2}+27 a^{2} b^{4}+b^{6}\right )}{\left (a +b \right )^{8} \left (a -b \right )^{8} \left (a +b \sin \left (d x +c \right )\right )}-\frac {b^{3} a \left (5 a^{2}+3 b^{2}\right )}{\left (a +b \right )^{5} \left (a -b \right )^{5} \left (a +b \sin \left (d x +c \right )\right )^{4}}-\frac {4 b^{3} a \left (7 a^{4}+14 b^{2} a^{2}+3 b^{4}\right )}{\left (a +b \right )^{7} \left (a -b \right )^{7} \left (a +b \sin \left (d x +c \right )\right )^{2}}+\frac {8 b^{3} a \left (15 a^{6}+63 a^{4} b^{2}+45 a^{2} b^{4}+5 b^{6}\right ) \ln \left (a +b \sin \left (d x +c \right )\right )}{\left (a +b \right )^{9} \left (a -b \right )^{9}}-\frac {1}{4 \left (a +b \right )^{8} \left (\sin \left (d x +c \right )-1\right )}+\frac {\left (-a -9 b \right ) \ln \left (\sin \left (d x +c \right )-1\right )}{4 \left (a +b \right )^{9}}}{d}\)	\(433\)
parallelrisch	\(\text {Expression too large to display}\)	\(2067\)
risch	\(\text {Expression too large to display}\)	\(4084\)

Input:

int(sec(d*x+c)^3/(a+b*sin(d*x+c))^8,x,method=_RETURNVERBOSE)
 

Output:

1/d*(-1/4/(a-b)^8/(1+sin(d*x+c))+1/4*(a-9*b)/(a-b)^9*ln(1+sin(d*x+c))-1/7* 
b^3/(a+b)^2/(a-b)^2/(a+b*sin(d*x+c))^7-1/3*b^3*(35*a^4+42*a^2*b^2+3*b^4)/( 
a+b)^6/(a-b)^6/(a+b*sin(d*x+c))^3-2/3*a*b^3/(a+b)^3/(a-b)^3/(a+b*sin(d*x+c 
))^6-2/5*b^3*(5*a^2+b^2)/(a+b)^4/(a-b)^4/(a+b*sin(d*x+c))^5-4*b^3*(21*a^6+ 
63*a^4*b^2+27*a^2*b^4+b^6)/(a+b)^8/(a-b)^8/(a+b*sin(d*x+c))-b^3*a*(5*a^2+3 
*b^2)/(a+b)^5/(a-b)^5/(a+b*sin(d*x+c))^4-4*b^3*a*(7*a^4+14*a^2*b^2+3*b^4)/ 
(a+b)^7/(a-b)^7/(a+b*sin(d*x+c))^2+8*b^3*a*(15*a^6+63*a^4*b^2+45*a^2*b^4+5 
*b^6)/(a+b)^9/(a-b)^9*ln(a+b*sin(d*x+c))-1/4/(a+b)^8/(sin(d*x+c)-1)+1/4/(a 
+b)^9*(-a-9*b)*ln(sin(d*x+c)-1))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3678 vs. \(2 (456) = 912\).

Time = 2.63 (sec) , antiderivative size = 3678, normalized size of antiderivative = 7.76 \[ \int \frac {\sec ^3(c+d x)}{(a+b \sin (c+d x))^8} \, dx=\text {Too large to display} \] Input:

integrate(sec(d*x+c)^3/(a+b*sin(d*x+c))^8,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\sec ^3(c+d x)}{(a+b \sin (c+d x))^8} \, dx=\text {Timed out} \] Input:

integrate(sec(d*x+c)**3/(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. 1670 vs. \(2 (456) = 912\).

Time = 0.10 (sec) , antiderivative size = 1670, normalized size of antiderivative = 3.52 \[ \int \frac {\sec ^3(c+d x)}{(a+b \sin (c+d x))^8} \, dx=\text {Too large to display} \] Input:

integrate(sec(d*x+c)^3/(a+b*sin(d*x+c))^8,x, algorithm="maxima")
 

Output:

1/420*(3360*(15*a^7*b^3 + 63*a^5*b^5 + 45*a^3*b^7 + 5*a*b^9)*log(b*sin(d*x 
 + c) + a)/(a^18 - 9*a^16*b^2 + 36*a^14*b^4 - 84*a^12*b^6 + 126*a^10*b^8 - 
 126*a^8*b^10 + 84*a^6*b^12 - 36*a^4*b^14 + 9*a^2*b^16 - b^18) + 105*(a - 
9*b)*log(sin(d*x + c) + 1)/(a^9 - 9*a^8*b + 36*a^7*b^2 - 84*a^6*b^3 + 126* 
a^5*b^4 - 126*a^4*b^5 + 84*a^3*b^6 - 36*a^2*b^7 + 9*a*b^8 - b^9) - 105*(a 
+ 9*b)*log(sin(d*x + c) - 1)/(a^9 + 9*a^8*b + 36*a^7*b^2 + 84*a^6*b^3 + 12 
6*a^5*b^4 + 126*a^4*b^5 + 84*a^3*b^6 + 36*a^2*b^7 + 9*a*b^8 + b^9) - 2*(84 
0*a^14*b + 33490*a^12*b^3 + 57724*a^10*b^5 + 16354*a^8*b^7 - 1496*a^6*b^9 
+ 814*a^4*b^11 - 236*a^2*b^13 + 30*b^15 - 105*(a^8*b^7 + 196*a^6*b^9 + 574 
*a^4*b^11 + 244*a^2*b^13 + 9*b^15)*sin(d*x + c)^8 - 105*(7*a^9*b^6 + 1252* 
a^7*b^8 + 3514*a^5*b^10 + 1348*a^3*b^12 + 23*a*b^14)*sin(d*x + c)^7 - 35*( 
63*a^10*b^5 + 10066*a^8*b^7 + 26194*a^6*b^9 + 7384*a^4*b^11 - 681*a^2*b^13 
 - 18*b^15)*sin(d*x + c)^6 - 35*(105*a^11*b^4 + 14506*a^9*b^6 + 32254*a^7* 
b^8 + 160*a^5*b^10 - 3951*a^3*b^12 - 66*a*b^14)*sin(d*x + c)^5 - 7*(525*a^ 
12*b^3 + 59310*a^10*b^5 + 83812*a^8*b^7 - 98528*a^6*b^9 - 44663*a^4*b^11 - 
 438*a^2*b^13 - 18*b^15)*sin(d*x + c)^4 - 7*(315*a^13*b^2 + 25930*a^11*b^4 
 - 20896*a^9*b^6 - 166336*a^7*b^8 - 53641*a^5*b^10 - 386*a^3*b^12 - 26*a*b 
^14)*sin(d*x + c)^3 - (735*a^14*b + 30550*a^12*b^3 - 361856*a^10*b^5 - 919 
070*a^8*b^7 - 252845*a^6*b^9 - 3050*a^4*b^11 + 310*a^2*b^13 - 54*b^15)*sin 
(d*x + c)^2 - 7*(15*a^15 - 420*a^13*b^2 - 26140*a^11*b^4 - 52264*a^9*b^...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 991 vs. \(2 (456) = 912\).

Time = 0.16 (sec) , antiderivative size = 991, normalized size of antiderivative = 2.09 \[ \int \frac {\sec ^3(c+d x)}{(a+b \sin (c+d x))^8} \, dx=\text {Too large to display} \] Input:

integrate(sec(d*x+c)^3/(a+b*sin(d*x+c))^8,x, algorithm="giac")
 

Output:

8*(15*a^7*b^4 + 63*a^5*b^6 + 45*a^3*b^8 + 5*a*b^10)*log(abs(b*sin(d*x + c) 
 + a))/(a^18*b*d - 9*a^16*b^3*d + 36*a^14*b^5*d - 84*a^12*b^7*d + 126*a^10 
*b^9*d - 126*a^8*b^11*d + 84*a^6*b^13*d - 36*a^4*b^15*d + 9*a^2*b^17*d - b 
^19*d) - 1/4*(a + 9*b)*log(abs(-sin(d*x + c) + 1))/(a^9*d + 9*a^8*b*d + 36 
*a^7*b^2*d + 84*a^6*b^3*d + 126*a^5*b^4*d + 126*a^4*b^5*d + 84*a^3*b^6*d + 
 36*a^2*b^7*d + 9*a*b^8*d + b^9*d) + 1/4*(a - 9*b)*log(abs(-sin(d*x + c) - 
 1))/(a^9*d - 9*a^8*b*d + 36*a^7*b^2*d - 84*a^6*b^3*d + 126*a^5*b^4*d - 12 
6*a^4*b^5*d + 84*a^3*b^6*d - 36*a^2*b^7*d + 9*a*b^8*d - b^9*d) + 1/210*(84 
0*a^16*b + 32650*a^14*b^3 + 24234*a^12*b^5 - 41370*a^10*b^7 - 17850*a^8*b^ 
9 + 2310*a^6*b^11 - 1050*a^4*b^13 + 266*a^2*b^15 - 30*b^17 - 105*(a^10*b^7 
 + 195*a^8*b^9 + 378*a^6*b^11 - 330*a^4*b^13 - 235*a^2*b^15 - 9*b^17)*sin( 
d*x + c)^8 - 105*(7*a^11*b^6 + 1245*a^9*b^8 + 2262*a^7*b^10 - 2166*a^5*b^1 
2 - 1325*a^3*b^14 - 23*a*b^16)*sin(d*x + c)^7 - 35*(63*a^12*b^5 + 10003*a^ 
10*b^7 + 16128*a^8*b^9 - 18810*a^6*b^11 - 8065*a^4*b^13 + 663*a^2*b^15 + 1 
8*b^17)*sin(d*x + c)^6 - 35*(105*a^13*b^4 + 14401*a^11*b^6 + 17748*a^9*b^8 
 - 32094*a^7*b^10 - 4111*a^5*b^12 + 3885*a^3*b^14 + 66*a*b^16)*sin(d*x + c 
)^5 - 7*(525*a^14*b^3 + 58785*a^12*b^5 + 24502*a^10*b^7 - 182340*a^8*b^9 + 
 53865*a^6*b^11 + 44225*a^4*b^13 + 420*a^2*b^15 + 18*b^17)*sin(d*x + c)^4 
- 7*(315*a^15*b^2 + 25615*a^13*b^4 - 46826*a^11*b^6 - 145440*a^9*b^8 + 112 
695*a^7*b^10 + 53255*a^5*b^12 + 360*a^3*b^14 + 26*a*b^16)*sin(d*x + c)^...
 

Mupad [B] (verification not implemented)

Time = 20.94 (sec) , antiderivative size = 1443, normalized size of antiderivative = 3.04 \[ \int \frac {\sec ^3(c+d x)}{(a+b \sin (c+d x))^8} \, dx=\text {Too large to display} \] Input:

int(1/(cos(c + d*x)^3*(a + b*sin(c + d*x))^8),x)
 

Output:

((sin(c + d*x)^7*(23*a*b^14 + 1348*a^3*b^12 + 3514*a^5*b^10 + 1252*a^7*b^8 
 + 7*a^9*b^6))/(2*(a^16 + b^16 - 8*a^2*b^14 + 28*a^4*b^12 - 56*a^6*b^10 + 
70*a^8*b^8 - 56*a^10*b^6 + 28*a^12*b^4 - 8*a^14*b^2)) - (420*a^14*b + 15*b 
^15 - 118*a^2*b^13 + 407*a^4*b^11 - 748*a^6*b^9 + 8177*a^8*b^7 + 28862*a^1 
0*b^5 + 16745*a^12*b^3)/(105*(a^2 - b^2)*(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*(7384*a^4*b^11 - 681*a^2*b^13 - 18*b^15 + 26194*a^6*b^9 + 10066*a^8* 
b^7 + 63*a^10*b^5))/(6*(a^16 + b^16 - 8*a^2*b^14 + 28*a^4*b^12 - 56*a^6*b^ 
10 + 70*a^8*b^8 - 56*a^10*b^6 + 28*a^12*b^4 - 8*a^14*b^2)) + (sin(c + d*x) 
^8*(9*b^15 + 244*a^2*b^13 + 574*a^4*b^11 + 196*a^6*b^9 + a^8*b^7))/(2*(a^1 
6 + b^16 - 8*a^2*b^14 + 28*a^4*b^12 - 56*a^6*b^10 + 70*a^8*b^8 - 56*a^10*b 
^6 + 28*a^12*b^4 - 8*a^14*b^2)) + (sin(c + d*x)^5*(160*a^5*b^10 - 3951*a^3 
*b^12 - 66*a*b^14 + 32254*a^7*b^8 + 14506*a^9*b^6 + 105*a^11*b^4))/(6*(a^1 
6 + b^16 - 8*a^2*b^14 + 28*a^4*b^12 - 56*a^6*b^10 + 70*a^8*b^8 - 56*a^10*b 
^6 + 28*a^12*b^4 - 8*a^14*b^2)) + (sin(c + d*x)^4*(18*b^13 + 456*a^2*b^11 
+ 45119*a^4*b^9 + 143647*a^6*b^7 + 59835*a^8*b^5 + 525*a^10*b^3))/(30*(a^1 
4 - 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)^2*(54*b^15 - 735*a^14*b - 310*a^2*b^13 + 
3050*a^4*b^11 + 252845*a^6*b^9 + 919070*a^8*b^7 + 361856*a^10*b^5 - 30550* 
a^12*b^3))/(210*(a^2 - b^2)*(a^14 - b^14 + 7*a^2*b^12 - 21*a^4*b^10 + 3...
 

Reduce [B] (verification not implemented)

Time = 16.15 (sec) , antiderivative size = 10889, normalized size of antiderivative = 22.97 \[ \int \frac {\sec ^3(c+d x)}{(a+b \sin (c+d x))^8} \, dx =\text {Too large to display} \] Input:

int(sec(d*x+c)^3/(a+b*sin(d*x+c))^8,x)
 

Output:

( - 105*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**9*a**15*b**8 + 4725*log(ta 
n((c + d*x)/2) - 1)*sin(c + d*x)**9*a**13*b**10 - 25200*log(tan((c + d*x)/ 
2) - 1)*sin(c + d*x)**9*a**12*b**11 + 66150*log(tan((c + d*x)/2) - 1)*sin( 
c + d*x)**9*a**11*b**12 - 105840*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**9 
*a**10*b**13 + 110250*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**9*a**9*b**14 
 - 75600*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**9*a**8*b**15 + 33075*log( 
tan((c + d*x)/2) - 1)*sin(c + d*x)**9*a**7*b**16 - 8400*log(tan((c + d*x)/ 
2) - 1)*sin(c + d*x)**9*a**6*b**17 + 945*log(tan((c + d*x)/2) - 1)*sin(c + 
 d*x)**9*a**5*b**18 - 735*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**8*a**16* 
b**7 + 33075*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**8*a**14*b**9 - 176400 
*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**8*a**13*b**10 + 463050*log(tan((c 
 + d*x)/2) - 1)*sin(c + d*x)**8*a**12*b**11 - 740880*log(tan((c + d*x)/2) 
- 1)*sin(c + d*x)**8*a**11*b**12 + 771750*log(tan((c + d*x)/2) - 1)*sin(c 
+ d*x)**8*a**10*b**13 - 529200*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**8*a 
**9*b**14 + 231525*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**8*a**8*b**15 - 
58800*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**8*a**7*b**16 + 6615*log(tan( 
(c + d*x)/2) - 1)*sin(c + d*x)**8*a**6*b**17 - 2205*log(tan((c + d*x)/2) - 
 1)*sin(c + d*x)**7*a**17*b**6 + 99330*log(tan((c + d*x)/2) - 1)*sin(c + d 
*x)**7*a**15*b**8 - 529200*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**7*a**14 
*b**9 + 1384425*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**7*a**13*b**10 -...