Integrand size = 21, antiderivative size = 269 \[ \int \frac {\sec ^5(c+d x)}{(a+b \sin (c+d x))^2} \, dx=-\frac {3 \left (a^2+4 a b+5 b^2\right ) \log (1-\sin (c+d x))}{16 (a+b)^4 d}+\frac {3 \left (a^2-4 a b+5 b^2\right ) \log (1+\sin (c+d x))}{16 (a-b)^4 d}-\frac {6 a b^5 \log (a+b \sin (c+d x))}{\left (a^2-b^2\right )^4 d}-\frac {3 b \left (a^4-4 a^2 b^2-5 b^4\right )}{8 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))}-\frac {\sec ^4(c+d x) (b-a \sin (c+d x))}{4 \left (a^2-b^2\right ) d (a+b \sin (c+d x))}+\frac {\sec ^2(c+d x) \left (b \left (a^2+5 b^2\right )+3 a \left (a^2-3 b^2\right ) \sin (c+d x)\right )}{8 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))} \]
-3/16*(a^2+4*a*b+5*b^2)*ln(1-sin(d*x+c))/(a+b)^4/d+3/16*(a^2-4*a*b+5*b^2)* ln(1+sin(d*x+c))/(a-b)^4/d-6*a*b^5*ln(a+b*sin(d*x+c))/(a^2-b^2)^4/d-3/8*b* (a^4-4*a^2*b^2-5*b^4)/(a^2-b^2)^3/d/(a+b*sin(d*x+c))-1/4*sec(d*x+c)^4*(b-a *sin(d*x+c))/(a^2-b^2)/d/(a+b*sin(d*x+c))+1/8*sec(d*x+c)^2*(b*(a^2+5*b^2)+ 3*a*(a^2-3*b^2)*sin(d*x+c))/(a^2-b^2)^2/d/(a+b*sin(d*x+c))
Time = 6.07 (sec) , antiderivative size = 406, normalized size of antiderivative = 1.51 \[ \int \frac {\sec ^5(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {b^5 \left (\frac {\sec ^4(c+d x) \left (b^2-a b \sin (c+d x)\right )}{4 b^6 \left (-a^2+b^2\right ) (a+b \sin (c+d x))}-\frac {-\frac {\sec ^2(c+d x) \left (4 a^2 b^2-b^2 \left (3 a^2-5 b^2\right )-b \left (4 a b^2-a \left (3 a^2-5 b^2\right )\right ) \sin (c+d x)\right )}{2 b^4 \left (-a^2+b^2\right ) (a+b \sin (c+d x))}+\frac {-6 a \left (a^2-3 b^2\right ) \left (-\frac {\log (1-\sin (c+d x))}{2 b (a+b)}+\frac {\log (1+\sin (c+d x))}{2 (a-b) b}-\frac {\log (a+b \sin (c+d x))}{a^2-b^2}\right )+\left (6 a^2 \left (a^2-3 b^2\right )-3 \left (a^4-2 a^2 b^2+5 b^4\right )\right ) \left (-\frac {\log (1-\sin (c+d x))}{2 b (a+b)^2}+\frac {\log (1+\sin (c+d x))}{2 (a-b)^2 b}-\frac {2 a \log (a+b \sin (c+d x))}{(a-b)^2 (a+b)^2}+\frac {1}{\left (a^2-b^2\right ) (a+b \sin (c+d x))}\right )}{2 b^2 \left (-a^2+b^2\right )}}{4 b^2 \left (-a^2+b^2\right )}\right )}{d} \]
(b^5*((Sec[c + d*x]^4*(b^2 - a*b*Sin[c + d*x]))/(4*b^6*(-a^2 + b^2)*(a + b *Sin[c + d*x])) - (-1/2*(Sec[c + d*x]^2*(4*a^2*b^2 - b^2*(3*a^2 - 5*b^2) - b*(4*a*b^2 - a*(3*a^2 - 5*b^2))*Sin[c + d*x]))/(b^4*(-a^2 + b^2)*(a + b*S in[c + d*x])) + (-6*a*(a^2 - 3*b^2)*(-1/2*Log[1 - Sin[c + d*x]]/(b*(a + b) ) + Log[1 + Sin[c + d*x]]/(2*(a - b)*b) - Log[a + b*Sin[c + d*x]]/(a^2 - b ^2)) + (6*a^2*(a^2 - 3*b^2) - 3*(a^4 - 2*a^2*b^2 + 5*b^4))*(-1/2*Log[1 - S in[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))))/d
Time = 0.54 (sec) , antiderivative size = 250, normalized size of antiderivative = 0.93, 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.
| \(\displaystyle \int \frac {\sec ^5(c+d x)}{(a+b \sin (c+d x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\cos (c+d x)^5 (a+b \sin (c+d x))^2}dx\) |
| \(\Big \downarrow \) 3147 |
| \(\displaystyle \frac {b^5 \int \frac {1}{(a+b \sin (c+d x))^2 \left (b^2-b^2 \sin ^2(c+d x)\right )^3}d(b \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 477 |
| \(\displaystyle \frac {\int \left (-\frac {6 a b^6}{\left (a^2-b^2\right )^4 (a+b \sin (c+d x))}-\frac {b^6}{\left (a^2-b^2\right )^3 (a+b \sin (c+d x))^2}+\frac {b^3}{8 (a+b)^2 (b-b \sin (c+d x))^3}+\frac {b^3}{8 (a-b)^2 (\sin (c+d x) b+b)^3}+\frac {(3 a+7 b) b^2}{16 (a+b)^3 (b-b \sin (c+d x))^2}+\frac {(3 a-7 b) b^2}{16 (a-b)^3 (\sin (c+d x) b+b)^2}+\frac {3 \left (a^2+4 b a+5 b^2\right ) b}{16 (a+b)^4 (b-b \sin (c+d x))}+\frac {3 \left (a^2-4 b a+5 b^2\right ) b}{16 (a-b)^4 (\sin (c+d x) b+b)}\right )d(b \sin (c+d x))}{b d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {3 b \left (a^2+4 a b+5 b^2\right ) \log (b-b \sin (c+d x))}{16 (a+b)^4}+\frac {3 b \left (a^2-4 a b+5 b^2\right ) \log (b \sin (c+d x)+b)}{16 (a-b)^4}+\frac {b^6}{\left (a^2-b^2\right )^3 (a+b \sin (c+d x))}-\frac {6 a b^6 \log (a+b \sin (c+d x))}{\left (a^2-b^2\right )^4}+\frac {b^3}{16 (a+b)^2 (b-b \sin (c+d x))^2}-\frac {b^3}{16 (a-b)^2 (b \sin (c+d x)+b)^2}+\frac {b^2 (3 a+7 b)}{16 (a+b)^3 (b-b \sin (c+d x))}-\frac {b^2 (3 a-7 b)}{16 (a-b)^3 (b \sin (c+d x)+b)}}{b d}\) |
((-3*b*(a^2 + 4*a*b + 5*b^2)*Log[b - b*Sin[c + d*x]])/(16*(a + b)^4) - (6* a*b^6*Log[a + b*Sin[c + d*x]])/(a^2 - b^2)^4 + (3*b*(a^2 - 4*a*b + 5*b^2)* Log[b + b*Sin[c + d*x]])/(16*(a - b)^4) + b^3/(16*(a + b)^2*(b - b*Sin[c + d*x])^2) + (b^2*(3*a + 7*b))/(16*(a + b)^3*(b - b*Sin[c + d*x])) + b^6/(( a^2 - b^2)^3*(a + b*Sin[c + d*x])) - b^3/(16*(a - b)^2*(b + b*Sin[c + d*x] )^2) - ((3*a - 7*b)*b^2)/(16*(a - b)^3*(b + b*Sin[c + d*x])))/(b*d)
3.5.43.3.1 Defintions of rubi rules used
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]]
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]
Time = 4.30 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.79
| method | result | size |
| derivativedivides | \(\frac {\frac {1}{16 \left (a +b \right )^{2} \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {7 b +3 a}{16 \left (a +b \right )^{3} \left (\sin \left (d x +c \right )-1\right )}+\frac {\left (-3 a^{2}-12 a b -15 b^{2}\right ) \ln \left (\sin \left (d x +c \right )-1\right )}{16 \left (a +b \right )^{4}}-\frac {1}{16 \left (a -b \right )^{2} \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {-7 b +3 a}{16 \left (a -b \right )^{3} \left (1+\sin \left (d x +c \right )\right )}+\frac {\left (3 a^{2}-12 a b +15 b^{2}\right ) \ln \left (1+\sin \left (d x +c \right )\right )}{16 \left (a -b \right )^{4}}+\frac {b^{5}}{\left (a +b \right )^{3} \left (a -b \right )^{3} \left (a +b \sin \left (d x +c \right )\right )}-\frac {6 a \,b^{5} \ln \left (a +b \sin \left (d x +c \right )\right )}{\left (a +b \right )^{4} \left (a -b \right )^{4}}}{d}\) | \(213\) |
| default | \(\frac {\frac {1}{16 \left (a +b \right )^{2} \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {7 b +3 a}{16 \left (a +b \right )^{3} \left (\sin \left (d x +c \right )-1\right )}+\frac {\left (-3 a^{2}-12 a b -15 b^{2}\right ) \ln \left (\sin \left (d x +c \right )-1\right )}{16 \left (a +b \right )^{4}}-\frac {1}{16 \left (a -b \right )^{2} \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {-7 b +3 a}{16 \left (a -b \right )^{3} \left (1+\sin \left (d x +c \right )\right )}+\frac {\left (3 a^{2}-12 a b +15 b^{2}\right ) \ln \left (1+\sin \left (d x +c \right )\right )}{16 \left (a -b \right )^{4}}+\frac {b^{5}}{\left (a +b \right )^{3} \left (a -b \right )^{3} \left (a +b \sin \left (d x +c \right )\right )}-\frac {6 a \,b^{5} \ln \left (a +b \sin \left (d x +c \right )\right )}{\left (a +b \right )^{4} \left (a -b \right )^{4}}}{d}\) | \(213\) |
| parallelrisch | \(\frac {-48 a^{2} \left (\frac {b \sin \left (5 d x +5 c \right )}{8}+\frac {3 b \sin \left (3 d x +3 c \right )}{8}+\frac {b \sin \left (d x +c \right )}{4}+\frac {\cos \left (4 d x +4 c \right ) a}{4}+\cos \left (2 d x +2 c \right ) a +\frac {3 a}{4}\right ) b^{5} \ln \left (2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )-3 a \left (a -b \right )^{4} \left (a^{2}+4 a b +5 b^{2}\right ) \left (\frac {b \sin \left (5 d x +5 c \right )}{8}+\frac {3 b \sin \left (3 d x +3 c \right )}{8}+\frac {b \sin \left (d x +c \right )}{4}+\frac {\cos \left (4 d x +4 c \right ) a}{4}+\cos \left (2 d x +2 c \right ) a +\frac {3 a}{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+3 \left (a +b \right ) \left (\left (a^{2}-4 a b +5 b^{2}\right ) \left (a +b \right )^{3} a \left (\frac {b \sin \left (5 d x +5 c \right )}{8}+\frac {3 b \sin \left (3 d x +3 c \right )}{8}+\frac {b \sin \left (d x +c \right )}{4}+\frac {\cos \left (4 d x +4 c \right ) a}{4}+\cos \left (2 d x +2 c \right ) a +\frac {3 a}{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\frac {11 \left (\frac {4 a b \left (a -b \right )^{2} \left (a +b \right )^{2} \cos \left (2 d x +2 c \right )}{11}+\frac {\left (\frac {1}{2} a^{5} b -4 a^{3} b^{3}+\frac {7}{2} a \,b^{5}\right ) \cos \left (4 d x +4 c \right )}{11}+\frac {3 \left (a^{6}-3 a^{4} b^{2}-2 a^{2} b^{4}-2 b^{6}\right ) \sin \left (3 d x +3 c \right )}{11}+\frac {b^{2} \left (a^{4}-5 a^{2} b^{2}-2 b^{4}\right ) \sin \left (5 d x +5 c \right )}{11}+\left (a^{6}-\frac {26}{11} a^{4} b^{2}+\frac {7}{11} a^{2} b^{4}-\frac {4}{11} b^{6}\right ) \sin \left (d x +c \right )-\frac {9 a^{5} b}{22}+\frac {12 a^{3} b^{3}}{11}-\frac {15 a \,b^{5}}{22}\right ) \left (a -b \right )}{6}\right )}{\left (a +b \right )^{4} a \left (a -b \right )^{4} d \left (3 b \sin \left (3 d x +3 c \right )+b \sin \left (5 d x +5 c \right )+2 \cos \left (4 d x +4 c \right ) a +8 \cos \left (2 d x +2 c \right ) a +2 b \sin \left (d x +c \right )+6 a \right )}\) | \(555\) |
| norman | \(\frac {-\frac {3 \left (a^{2} b -3 b^{3}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}-\frac {3 \left (a^{2} b -3 b^{3}\right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}+\frac {\left (-5 a^{2} b -b^{3}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}+\frac {\left (-5 a^{2} b -b^{3}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}+\frac {\left (5 a^{6}-12 a^{4} b^{2}-9 a^{2} b^{4}-8 b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d a \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right )}+\frac {\left (5 a^{6}-12 a^{4} b^{2}-9 a^{2} b^{4}-8 b^{6}\right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d a \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right )}-\frac {2 \left (-a^{6}+4 a^{4} b^{2}-11 a^{2} b^{4}-4 b^{6}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right )}-\frac {2 \left (-a^{6}+4 a^{4} b^{2}-11 a^{2} b^{4}-4 b^{6}\right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right )}+\frac {3 \left (a^{6}+4 a^{4} b^{2}-21 a^{2} b^{4}-8 b^{6}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a d \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right )}}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4} \left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )}+\frac {3 \left (a^{2}-4 a b +5 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8 d \left (a^{4}-4 a^{3} b +6 a^{2} b^{2}-4 a \,b^{3}+b^{4}\right )}-\frac {3 \left (a^{2}+4 a b +5 b^{2}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 \left (a^{4}+4 a^{3} b +6 a^{2} b^{2}+4 a \,b^{3}+b^{4}\right ) d}-\frac {6 a \,b^{5} \ln \left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )}{d \left (a^{8}-4 a^{6} b^{2}+6 a^{4} b^{4}-4 a^{2} b^{6}+b^{8}\right )}\) | \(775\) |
| risch | \(\text {Expression too large to display}\) | \(1409\) |
1/d*(1/16/(a+b)^2/(sin(d*x+c)-1)^2-1/16*(7*b+3*a)/(a+b)^3/(sin(d*x+c)-1)+1 /16/(a+b)^4*(-3*a^2-12*a*b-15*b^2)*ln(sin(d*x+c)-1)-1/16/(a-b)^2/(1+sin(d* x+c))^2-1/16*(-7*b+3*a)/(a-b)^3/(1+sin(d*x+c))+1/16/(a-b)^4*(3*a^2-12*a*b+ 15*b^2)*ln(1+sin(d*x+c))+b^5/(a+b)^3/(a-b)^3/(a+b*sin(d*x+c))-6*a*b^5/(a+b )^4/(a-b)^4*ln(a+b*sin(d*x+c)))
Leaf count of result is larger than twice the leaf count of optimal. 527 vs. \(2 (260) = 520\).
Time = 0.49 (sec) , antiderivative size = 527, normalized size of antiderivative = 1.96 \[ \int \frac {\sec ^5(c+d x)}{(a+b \sin (c+d x))^2} \, dx=-\frac {4 \, a^{6} b - 12 \, a^{4} b^{3} + 12 \, a^{2} b^{5} - 4 \, b^{7} + 6 \, {\left (a^{6} b - 5 \, a^{4} b^{3} - a^{2} b^{5} + 5 \, b^{7}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (a^{6} b + 3 \, a^{4} b^{3} - 9 \, a^{2} b^{5} + 5 \, b^{7}\right )} \cos \left (d x + c\right )^{2} + 96 \, {\left (a b^{6} \cos \left (d x + c\right )^{4} \sin \left (d x + c\right ) + a^{2} b^{5} \cos \left (d x + c\right )^{4}\right )} \log \left (b \sin \left (d x + c\right ) + a\right ) - 3 \, {\left ({\left (a^{6} b - 5 \, a^{4} b^{3} + 15 \, a^{2} b^{5} + 16 \, a b^{6} + 5 \, b^{7}\right )} \cos \left (d x + c\right )^{4} \sin \left (d x + c\right ) + {\left (a^{7} - 5 \, a^{5} b^{2} + 15 \, a^{3} b^{4} + 16 \, a^{2} b^{5} + 5 \, a b^{6}\right )} \cos \left (d x + c\right )^{4}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, {\left ({\left (a^{6} b - 5 \, a^{4} b^{3} + 15 \, a^{2} b^{5} - 16 \, a b^{6} + 5 \, b^{7}\right )} \cos \left (d x + c\right )^{4} \sin \left (d x + c\right ) + {\left (a^{7} - 5 \, a^{5} b^{2} + 15 \, a^{3} b^{4} - 16 \, a^{2} b^{5} + 5 \, a b^{6}\right )} \cos \left (d x + c\right )^{4}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (2 \, a^{7} - 6 \, a^{5} b^{2} + 6 \, a^{3} b^{4} - 2 \, a b^{6} + 3 \, {\left (a^{7} - 5 \, a^{5} b^{2} + 7 \, a^{3} b^{4} - 3 \, a b^{6}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{16 \, {\left ({\left (a^{8} b - 4 \, a^{6} b^{3} + 6 \, a^{4} b^{5} - 4 \, a^{2} b^{7} + b^{9}\right )} d \cos \left (d x + c\right )^{4} \sin \left (d x + c\right ) + {\left (a^{9} - 4 \, a^{7} b^{2} + 6 \, a^{5} b^{4} - 4 \, a^{3} b^{6} + a b^{8}\right )} d \cos \left (d x + c\right )^{4}\right )}} \]
-1/16*(4*a^6*b - 12*a^4*b^3 + 12*a^2*b^5 - 4*b^7 + 6*(a^6*b - 5*a^4*b^3 - a^2*b^5 + 5*b^7)*cos(d*x + c)^4 - 2*(a^6*b + 3*a^4*b^3 - 9*a^2*b^5 + 5*b^7 )*cos(d*x + c)^2 + 96*(a*b^6*cos(d*x + c)^4*sin(d*x + c) + a^2*b^5*cos(d*x + c)^4)*log(b*sin(d*x + c) + a) - 3*((a^6*b - 5*a^4*b^3 + 15*a^2*b^5 + 16 *a*b^6 + 5*b^7)*cos(d*x + c)^4*sin(d*x + c) + (a^7 - 5*a^5*b^2 + 15*a^3*b^ 4 + 16*a^2*b^5 + 5*a*b^6)*cos(d*x + c)^4)*log(sin(d*x + c) + 1) + 3*((a^6* b - 5*a^4*b^3 + 15*a^2*b^5 - 16*a*b^6 + 5*b^7)*cos(d*x + c)^4*sin(d*x + c) + (a^7 - 5*a^5*b^2 + 15*a^3*b^4 - 16*a^2*b^5 + 5*a*b^6)*cos(d*x + c)^4)*l og(-sin(d*x + c) + 1) - 2*(2*a^7 - 6*a^5*b^2 + 6*a^3*b^4 - 2*a*b^6 + 3*(a^ 7 - 5*a^5*b^2 + 7*a^3*b^4 - 3*a*b^6)*cos(d*x + c)^2)*sin(d*x + c))/((a^8*b - 4*a^6*b^3 + 6*a^4*b^5 - 4*a^2*b^7 + b^9)*d*cos(d*x + c)^4*sin(d*x + c) + (a^9 - 4*a^7*b^2 + 6*a^5*b^4 - 4*a^3*b^6 + a*b^8)*d*cos(d*x + c)^4)
\[ \int \frac {\sec ^5(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\int \frac {\sec ^{5}{\left (c + d x \right )}}{\left (a + b \sin {\left (c + d x \right )}\right )^{2}}\, dx \]
Time = 0.21 (sec) , antiderivative size = 505, normalized size of antiderivative = 1.88 \[ \int \frac {\sec ^5(c+d x)}{(a+b \sin (c+d x))^2} \, dx=-\frac {\frac {96 \, a b^{5} \log \left (b \sin \left (d x + c\right ) + a\right )}{a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}} - \frac {3 \, {\left (a^{2} - 4 \, a b + 5 \, b^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}} + \frac {3 \, {\left (a^{2} + 4 \, a b + 5 \, b^{2}\right )} \log \left (\sin \left (d x + c\right ) - 1\right )}{a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}} + \frac {2 \, {\left (4 \, a^{4} b - 20 \, a^{2} b^{3} - 8 \, b^{5} + 3 \, {\left (a^{4} b - 4 \, a^{2} b^{3} - 5 \, b^{5}\right )} \sin \left (d x + c\right )^{4} + 3 \, {\left (a^{5} - 4 \, a^{3} b^{2} + 3 \, a b^{4}\right )} \sin \left (d x + c\right )^{3} - {\left (5 \, a^{4} b - 28 \, a^{2} b^{3} - 25 \, b^{5}\right )} \sin \left (d x + c\right )^{2} - {\left (5 \, a^{5} - 16 \, a^{3} b^{2} + 11 \, a b^{4}\right )} \sin \left (d x + c\right )\right )}}{a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6} + {\left (a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7}\right )} \sin \left (d x + c\right )^{5} + {\left (a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}\right )} \sin \left (d x + c\right )^{4} - 2 \, {\left (a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7}\right )} \sin \left (d x + c\right )^{3} - 2 \, {\left (a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}\right )} \sin \left (d x + c\right )^{2} + {\left (a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7}\right )} \sin \left (d x + c\right )}}{16 \, d} \]
-1/16*(96*a*b^5*log(b*sin(d*x + c) + a)/(a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a ^2*b^6 + b^8) - 3*(a^2 - 4*a*b + 5*b^2)*log(sin(d*x + c) + 1)/(a^4 - 4*a^3 *b + 6*a^2*b^2 - 4*a*b^3 + b^4) + 3*(a^2 + 4*a*b + 5*b^2)*log(sin(d*x + c) - 1)/(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4) + 2*(4*a^4*b - 20*a^2*b^ 3 - 8*b^5 + 3*(a^4*b - 4*a^2*b^3 - 5*b^5)*sin(d*x + c)^4 + 3*(a^5 - 4*a^3* b^2 + 3*a*b^4)*sin(d*x + c)^3 - (5*a^4*b - 28*a^2*b^3 - 25*b^5)*sin(d*x + c)^2 - (5*a^5 - 16*a^3*b^2 + 11*a*b^4)*sin(d*x + c))/(a^7 - 3*a^5*b^2 + 3* a^3*b^4 - a*b^6 + (a^6*b - 3*a^4*b^3 + 3*a^2*b^5 - b^7)*sin(d*x + c)^5 + ( a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*sin(d*x + c)^4 - 2*(a^6*b - 3*a^4*b^3 + 3*a^2*b^5 - b^7)*sin(d*x + c)^3 - 2*(a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^ 6)*sin(d*x + c)^2 + (a^6*b - 3*a^4*b^3 + 3*a^2*b^5 - b^7)*sin(d*x + c)))/d
Time = 0.36 (sec) , antiderivative size = 460, normalized size of antiderivative = 1.71 \[ \int \frac {\sec ^5(c+d x)}{(a+b \sin (c+d x))^2} \, dx=-\frac {\frac {96 \, a b^{6} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{a^{8} b - 4 \, a^{6} b^{3} + 6 \, a^{4} b^{5} - 4 \, a^{2} b^{7} + b^{9}} - \frac {3 \, {\left (a^{2} - 4 \, a b + 5 \, b^{2}\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}} + \frac {3 \, {\left (a^{2} + 4 \, a b + 5 \, b^{2}\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}} - \frac {16 \, {\left (6 \, a b^{6} \sin \left (d x + c\right ) + 7 \, a^{2} b^{5} - b^{7}\right )}}{{\left (a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}\right )} {\left (b \sin \left (d x + c\right ) + a\right )}} + \frac {2 \, {\left (36 \, a b^{5} \sin \left (d x + c\right )^{4} + 3 \, a^{6} \sin \left (d x + c\right )^{3} - 15 \, a^{4} b^{2} \sin \left (d x + c\right )^{3} + 5 \, a^{2} b^{4} \sin \left (d x + c\right )^{3} + 7 \, b^{6} \sin \left (d x + c\right )^{3} + 16 \, a^{3} b^{3} \sin \left (d x + c\right )^{2} - 88 \, a b^{5} \sin \left (d x + c\right )^{2} - 5 \, a^{6} \sin \left (d x + c\right ) + 17 \, a^{4} b^{2} \sin \left (d x + c\right ) - 3 \, a^{2} b^{4} \sin \left (d x + c\right ) - 9 \, b^{6} \sin \left (d x + c\right ) + 4 \, a^{5} b - 24 \, a^{3} b^{3} + 56 \, a b^{5}\right )}}{{\left (a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}\right )} {\left (\sin \left (d x + c\right )^{2} - 1\right )}^{2}}}{16 \, d} \]
-1/16*(96*a*b^6*log(abs(b*sin(d*x + c) + a))/(a^8*b - 4*a^6*b^3 + 6*a^4*b^ 5 - 4*a^2*b^7 + b^9) - 3*(a^2 - 4*a*b + 5*b^2)*log(abs(sin(d*x + c) + 1))/ (a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4) + 3*(a^2 + 4*a*b + 5*b^2)*log( abs(sin(d*x + c) - 1))/(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4) - 16*(6 *a*b^6*sin(d*x + c) + 7*a^2*b^5 - b^7)/((a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a ^2*b^6 + b^8)*(b*sin(d*x + c) + a)) + 2*(36*a*b^5*sin(d*x + c)^4 + 3*a^6*s in(d*x + c)^3 - 15*a^4*b^2*sin(d*x + c)^3 + 5*a^2*b^4*sin(d*x + c)^3 + 7*b ^6*sin(d*x + c)^3 + 16*a^3*b^3*sin(d*x + c)^2 - 88*a*b^5*sin(d*x + c)^2 - 5*a^6*sin(d*x + c) + 17*a^4*b^2*sin(d*x + c) - 3*a^2*b^4*sin(d*x + c) - 9* b^6*sin(d*x + c) + 4*a^5*b - 24*a^3*b^3 + 56*a*b^5)/((a^8 - 4*a^6*b^2 + 6* a^4*b^4 - 4*a^2*b^6 + b^8)*(sin(d*x + c)^2 - 1)^2))/d
Time = 5.39 (sec) , antiderivative size = 449, normalized size of antiderivative = 1.67 \[ \int \frac {\sec ^5(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {\ln \left (\sin \left (c+d\,x\right )+1\right )\,\left (\frac {3\,b^2}{8\,{\left (a-b\right )}^4}-\frac {3\,b}{8\,{\left (a-b\right )}^3}+\frac {3}{16\,{\left (a-b\right )}^2}\right )}{d}-\frac {\ln \left (\sin \left (c+d\,x\right )-1\right )\,\left (\frac {3\,b}{8\,{\left (a+b\right )}^3}+\frac {3}{16\,{\left (a+b\right )}^2}+\frac {3\,b^2}{8\,{\left (a+b\right )}^4}\right )}{d}+\frac {\frac {-a^4\,b+5\,a^2\,b^3+2\,b^5}{2\,\left (a^2-b^2\right )\,\left (a^4-2\,a^2\,b^2+b^4\right )}+\frac {3\,{\sin \left (c+d\,x\right )}^3\,\left (3\,a\,b^2-a^3\right )}{8\,\left (a^4-2\,a^2\,b^2+b^4\right )}+\frac {3\,{\sin \left (c+d\,x\right )}^4\,\left (-a^4\,b+4\,a^2\,b^3+5\,b^5\right )}{8\,\left (a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6\right )}-\frac {\sin \left (c+d\,x\right )\,\left (11\,a\,b^2-5\,a^3\right )}{8\,\left (a^4-2\,a^2\,b^2+b^4\right )}-\frac {{\sin \left (c+d\,x\right )}^2\,\left (-5\,a^4\,b+28\,a^2\,b^3+25\,b^5\right )}{8\,\left (a^2-b^2\right )\,\left (a^4-2\,a^2\,b^2+b^4\right )}}{d\,\left (b\,{\sin \left (c+d\,x\right )}^5+a\,{\sin \left (c+d\,x\right )}^4-2\,b\,{\sin \left (c+d\,x\right )}^3-2\,a\,{\sin \left (c+d\,x\right )}^2+b\,\sin \left (c+d\,x\right )+a\right )}-\frac {6\,a\,b^5\,\ln \left (a+b\,\sin \left (c+d\,x\right )\right )}{d\,\left (a^8-4\,a^6\,b^2+6\,a^4\,b^4-4\,a^2\,b^6+b^8\right )} \]
(log(sin(c + d*x) + 1)*((3*b^2)/(8*(a - b)^4) - (3*b)/(8*(a - b)^3) + 3/(1 6*(a - b)^2)))/d - (log(sin(c + d*x) - 1)*((3*b)/(8*(a + b)^3) + 3/(16*(a + b)^2) + (3*b^2)/(8*(a + b)^4)))/d + ((2*b^5 - a^4*b + 5*a^2*b^3)/(2*(a^2 - b^2)*(a^4 + b^4 - 2*a^2*b^2)) + (3*sin(c + d*x)^3*(3*a*b^2 - a^3))/(8*( a^4 + b^4 - 2*a^2*b^2)) + (3*sin(c + d*x)^4*(5*b^5 - a^4*b + 4*a^2*b^3))/( 8*(a^6 - b^6 + 3*a^2*b^4 - 3*a^4*b^2)) - (sin(c + d*x)*(11*a*b^2 - 5*a^3)) /(8*(a^4 + b^4 - 2*a^2*b^2)) - (sin(c + d*x)^2*(25*b^5 - 5*a^4*b + 28*a^2* b^3))/(8*(a^2 - b^2)*(a^4 + b^4 - 2*a^2*b^2)))/(d*(a + b*sin(c + d*x) - 2* a*sin(c + d*x)^2 + a*sin(c + d*x)^4 - 2*b*sin(c + d*x)^3 + b*sin(c + d*x)^ 5)) - (6*a*b^5*log(a + b*sin(c + d*x)))/(d*(a^8 + b^8 - 4*a^2*b^6 + 6*a^4* b^4 - 4*a^6*b^2))