Integrand size = 21, antiderivative size = 226 \[ \int \frac {\sec ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx=-\frac {(a+4 b) \log (1-\sin (c+d x))}{4 (a+b)^4 d}+\frac {(a-4 b) \log (1+\sin (c+d x))}{4 (a-b)^4 d}+\frac {2 b^3 \left (5 a^2+b^2\right ) \log (a+b \sin (c+d x))}{\left (a^2-b^2\right )^4 d}-\frac {b \left (a^2+2 b^2\right )}{2 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))^2}-\frac {\sec ^2(c+d x) (b-a \sin (c+d x))}{2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^2}-\frac {a b \left (a^2+11 b^2\right )}{2 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))} \]
-1/4*(a+4*b)*ln(1-sin(d*x+c))/(a+b)^4/d+1/4*(a-4*b)*ln(1+sin(d*x+c))/(a-b) ^4/d+2*b^3*(5*a^2+b^2)*ln(a+b*sin(d*x+c))/(a^2-b^2)^4/d-1/2*b*(a^2+2*b^2)/ (a^2-b^2)^2/d/(a+b*sin(d*x+c))^2-1/2*sec(d*x+c)^2*(b-a*sin(d*x+c))/(a^2-b^ 2)/d/(a+b*sin(d*x+c))^2-1/2*a*b*(a^2+11*b^2)/(a^2-b^2)^3/d/(a+b*sin(d*x+c) )
Time = 2.87 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.25 \[ \int \frac {\sec ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {\frac {\sec ^2(c+d x) (b-a \sin (c+d x))}{(a+b \sin (c+d x))^2}+b \left (a^2+2 b^2\right ) \left (-\frac {\log (1-\sin (c+d x))}{b (a+b)^3}+\frac {\log (1+\sin (c+d x))}{(a-b)^3 b}-\frac {2 \left (3 a^2+b^2\right ) \log (a+b \sin (c+d x))}{(a-b)^3 (a+b)^3}+\frac {1}{\left (a^2-b^2\right ) (a+b \sin (c+d x))^2}+\frac {4 a}{(a-b)^2 (a+b)^2 (a+b \sin (c+d x))}\right )+\frac {3}{2} a \left (\frac {\log (1-\sin (c+d x))}{(a+b)^2}-\frac {\log (1+\sin (c+d x))}{(a-b)^2}+\frac {4 a b \log (a+b \sin (c+d x))}{(a-b)^2 (a+b)^2}+\frac {2 b}{\left (-a^2+b^2\right ) (a+b \sin (c+d x))}\right )}{2 \left (-a^2+b^2\right ) d} \]
((Sec[c + d*x]^2*(b - a*Sin[c + d*x]))/(a + b*Sin[c + d*x])^2 + b*(a^2 + 2 *b^2)*(-(Log[1 - Sin[c + d*x]]/(b*(a + b)^3)) + Log[1 + Sin[c + d*x]]/((a - b)^3*b) - (2*(3*a^2 + b^2)*Log[a + b*Sin[c + d*x]])/((a - b)^3*(a + b)^3 ) + 1/((a^2 - b^2)*(a + b*Sin[c + d*x])^2) + (4*a)/((a - b)^2*(a + b)^2*(a + b*Sin[c + d*x]))) + (3*a*(Log[1 - Sin[c + d*x]]/(a + b)^2 - Log[1 + Sin [c + d*x]]/(a - b)^2 + (4*a*b*Log[a + b*Sin[c + d*x]])/((a - b)^2*(a + b)^ 2) + (2*b)/((-a^2 + b^2)*(a + b*Sin[c + d*x]))))/2)/(2*(-a^2 + b^2)*d)
Time = 0.46 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.92, 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.
| \(\displaystyle \int \frac {\sec ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\cos (c+d x)^3 (a+b \sin (c+d x))^3}dx\) |
| \(\Big \downarrow \) 3147 |
| \(\displaystyle \frac {b^3 \int \frac {1}{(a+b \sin (c+d x))^3 \left (b^2-b^2 \sin ^2(c+d x)\right )^2}d(b \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 477 |
| \(\displaystyle \frac {\int \left (\frac {2 \left (5 a^2+b^2\right ) b^4}{\left (a^2-b^2\right )^4 (a+b \sin (c+d x))}+\frac {4 a b^4}{\left (a^2-b^2\right )^3 (a+b \sin (c+d x))^2}+\frac {b^4}{\left (a^2-b^2\right )^2 (a+b \sin (c+d x))^3}+\frac {b^2}{4 (a+b)^3 (b-b \sin (c+d x))^2}+\frac {b^2}{4 (a-b)^3 (\sin (c+d x) b+b)^2}+\frac {(a+4 b) b}{4 (a+b)^4 (b-b \sin (c+d x))}+\frac {(a-4 b) b}{4 (a-b)^4 (\sin (c+d x) b+b)}\right )d(b \sin (c+d x))}{b d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {4 a b^4}{\left (a^2-b^2\right )^3 (a+b \sin (c+d x))}-\frac {b^4}{2 \left (a^2-b^2\right )^2 (a+b \sin (c+d x))^2}+\frac {2 b^4 \left (5 a^2+b^2\right ) \log (a+b \sin (c+d x))}{\left (a^2-b^2\right )^4}+\frac {b^2}{4 (a+b)^3 (b-b \sin (c+d x))}-\frac {b^2}{4 (a-b)^3 (b \sin (c+d x)+b)}-\frac {b (a+4 b) \log (b-b \sin (c+d x))}{4 (a+b)^4}+\frac {b (a-4 b) \log (b \sin (c+d x)+b)}{4 (a-b)^4}}{b d}\) |
(-1/4*(b*(a + 4*b)*Log[b - b*Sin[c + d*x]])/(a + b)^4 + (2*b^4*(5*a^2 + b^ 2)*Log[a + b*Sin[c + d*x]])/(a^2 - b^2)^4 + ((a - 4*b)*b*Log[b + b*Sin[c + d*x]])/(4*(a - b)^4) + b^2/(4*(a + b)^3*(b - b*Sin[c + d*x])) - b^4/(2*(a ^2 - b^2)^2*(a + b*Sin[c + d*x])^2) - (4*a*b^4)/((a^2 - b^2)^3*(a + b*Sin[ c + d*x])) - b^2/(4*(a - b)^3*(b + b*Sin[c + d*x])))/(b*d)
3.5.54.3.1 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 = 3.31 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.81
| method | result | size |
| derivativedivides | \(\frac {-\frac {1}{4 \left (a -b \right )^{3} \left (1+\sin \left (d x +c \right )\right )}+\frac {\left (a -4 b \right ) \ln \left (1+\sin \left (d x +c \right )\right )}{4 \left (a -b \right )^{4}}-\frac {1}{4 \left (a +b \right )^{3} \left (\sin \left (d x +c \right )-1\right )}+\frac {\left (-a -4 b \right ) \ln \left (\sin \left (d x +c \right )-1\right )}{4 \left (a +b \right )^{4}}-\frac {b^{3}}{2 \left (a +b \right )^{2} \left (a -b \right )^{2} \left (a +b \sin \left (d x +c \right )\right )^{2}}-\frac {4 a \,b^{3}}{\left (a +b \right )^{3} \left (a -b \right )^{3} \left (a +b \sin \left (d x +c \right )\right )}+\frac {2 b^{3} \left (5 a^{2}+b^{2}\right ) \ln \left (a +b \sin \left (d x +c \right )\right )}{\left (a +b \right )^{4} \left (a -b \right )^{4}}}{d}\) | \(184\) |
| default | \(\frac {-\frac {1}{4 \left (a -b \right )^{3} \left (1+\sin \left (d x +c \right )\right )}+\frac {\left (a -4 b \right ) \ln \left (1+\sin \left (d x +c \right )\right )}{4 \left (a -b \right )^{4}}-\frac {1}{4 \left (a +b \right )^{3} \left (\sin \left (d x +c \right )-1\right )}+\frac {\left (-a -4 b \right ) \ln \left (\sin \left (d x +c \right )-1\right )}{4 \left (a +b \right )^{4}}-\frac {b^{3}}{2 \left (a +b \right )^{2} \left (a -b \right )^{2} \left (a +b \sin \left (d x +c \right )\right )^{2}}-\frac {4 a \,b^{3}}{\left (a +b \right )^{3} \left (a -b \right )^{3} \left (a +b \sin \left (d x +c \right )\right )}+\frac {2 b^{3} \left (5 a^{2}+b^{2}\right ) \ln \left (a +b \sin \left (d x +c \right )\right )}{\left (a +b \right )^{4} \left (a -b \right )^{4}}}{d}\) | \(184\) |
| parallelrisch | \(\frac {80 \left (-\frac {b^{2} \cos \left (4 d x +4 c \right )}{4}+\frac {b^{2}}{4}+a b \sin \left (3 d x +3 c \right )+\sin \left (d x +c \right ) a b +\cos \left (2 d x +2 c \right ) a^{2}+a^{2}\right ) \left (a^{2}+\frac {b^{2}}{5}\right ) a^{2} b^{3} \ln \left (2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )-4 \left (a +4 b \right ) \left (-\frac {b^{2} \cos \left (4 d x +4 c \right )}{4}+\frac {b^{2}}{4}+a b \sin \left (3 d x +3 c \right )+\sin \left (d x +c \right ) a b +\cos \left (2 d x +2 c \right ) a^{2}+a^{2}\right ) a^{2} \left (a -b \right )^{4} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+4 \left (a +b \right ) \left (\left (-\frac {b^{2} \cos \left (4 d x +4 c \right )}{4}+\frac {b^{2}}{4}+a b \sin \left (3 d x +3 c \right )+\sin \left (d x +c \right ) a b +\cos \left (2 d x +2 c \right ) a^{2}+a^{2}\right ) \left (a +b \right )^{3} \left (a -4 b \right ) a^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+2 \left (\frac {a^{2} b \left (a -b \right )^{2} \left (a +b \right )^{2} \cos \left (2 d x +2 c \right )}{2}+\left (-\frac {3}{8} a^{4} b^{3}-\frac {5}{4} a^{2} b^{5}+\frac {1}{8} b^{7}\right ) \cos \left (4 d x +4 c \right )+\left (\frac {5}{4} b^{2} a^{5}+\frac {9}{4} b^{4} a^{3}-\frac {1}{2} b^{6} a \right ) \sin \left (3 d x +3 c \right )+a \left (a^{6}-\frac {3}{4} a^{4} b^{2}+\frac {13}{4} a^{2} b^{4}-\frac {1}{2} b^{6}\right ) \sin \left (d x +c \right )-\frac {b \,a^{6}}{2}+\frac {11 a^{4} b^{3}}{8}+\frac {3 a^{2} b^{5}}{4}-\frac {b^{7}}{8}\right ) \left (a -b \right )\right )}{8 \left (-\frac {b^{2} \cos \left (4 d x +4 c \right )}{4}+\frac {b^{2}}{4}+a b \sin \left (3 d x +3 c \right )+\sin \left (d x +c \right ) a b +\cos \left (2 d x +2 c \right ) a^{2}+a^{2}\right ) \left (a +b \right )^{4} a^{2} \left (a -b \right )^{4} d}\) | \(517\) |
| norman | \(\frac {\frac {\left (a^{6}+3 a^{4} b^{2}+10 a^{2} b^{4}-2 b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d a \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right )}+\frac {\left (a^{6}+3 a^{4} b^{2}+10 a^{2} b^{4}-2 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 {\left (-3 a^{6}+11 a^{4} b^{2}+6 a^{2} b^{4}-2 b^{6}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right )}-\frac {\left (-3 a^{6}+11 a^{4} b^{2}+6 a^{2} b^{4}-2 b^{6}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right )}-\frac {2 \left (2 b \,a^{6}+2 a^{4} b^{3}+22 a^{2} b^{5}-2 b^{7}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{2} \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right )}+\frac {2 b \left (-a^{6}+5 a^{4} b^{2}+9 a^{2} b^{4}-b^{6}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{2} \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right )}+\frac {2 b \left (-a^{6}+5 a^{4} b^{2}+9 a^{2} b^{4}-b^{6}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{2} \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 )^{2} {\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 )}^{2}}+\frac {\left (a -4 b \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 d \left (a^{4}-4 a^{3} b +6 a^{2} b^{2}-4 a \,b^{3}+b^{4}\right )}-\frac {\left (a +4 b \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 \left (a^{4}+4 a^{3} b +6 a^{2} b^{2}+4 a \,b^{3}+b^{4}\right ) d}+\frac {2 b^{3} \left (5 a^{2}+b^{2}\right ) \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 )}\) | \(734\) |
| risch | \(\text {Expression too large to display}\) | \(1141\) |
1/d*(-1/4/(a-b)^3/(1+sin(d*x+c))+1/4*(a-4*b)/(a-b)^4*ln(1+sin(d*x+c))-1/4/ (a+b)^3/(sin(d*x+c)-1)+1/4/(a+b)^4*(-a-4*b)*ln(sin(d*x+c)-1)-1/2*b^3/(a+b) ^2/(a-b)^2/(a+b*sin(d*x+c))^2-4*a*b^3/(a+b)^3/(a-b)^3/(a+b*sin(d*x+c))+2*b ^3*(5*a^2+b^2)/(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. 707 vs. \(2 (217) = 434\).
Time = 0.45 (sec) , antiderivative size = 707, normalized size of antiderivative = 3.13 \[ \int \frac {\sec ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {2 \, a^{6} b - 6 \, a^{4} b^{3} + 6 \, a^{2} b^{5} - 2 \, b^{7} + 4 \, {\left (a^{6} b + 5 \, a^{4} b^{3} - 7 \, a^{2} b^{5} + b^{7}\right )} \cos \left (d x + c\right )^{2} + 8 \, {\left ({\left (5 \, a^{2} b^{5} + b^{7}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (5 \, a^{3} b^{4} + a b^{6}\right )} \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) - {\left (5 \, a^{4} b^{3} + 6 \, a^{2} b^{5} + b^{7}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (b \sin \left (d x + c\right ) + a\right ) + {\left ({\left (a^{5} b^{2} - 10 \, a^{3} b^{4} - 20 \, a^{2} b^{5} - 15 \, a b^{6} - 4 \, b^{7}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (a^{6} b - 10 \, a^{4} b^{3} - 20 \, a^{3} b^{4} - 15 \, a^{2} b^{5} - 4 \, a b^{6}\right )} \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) - {\left (a^{7} - 9 \, a^{5} b^{2} - 20 \, a^{4} b^{3} - 25 \, a^{3} b^{4} - 24 \, a^{2} b^{5} - 15 \, a b^{6} - 4 \, b^{7}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left ({\left (a^{5} b^{2} - 10 \, a^{3} b^{4} + 20 \, a^{2} b^{5} - 15 \, a b^{6} + 4 \, b^{7}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (a^{6} b - 10 \, a^{4} b^{3} + 20 \, a^{3} b^{4} - 15 \, a^{2} b^{5} + 4 \, a b^{6}\right )} \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) - {\left (a^{7} - 9 \, a^{5} b^{2} + 20 \, a^{4} b^{3} - 25 \, a^{3} b^{4} + 24 \, a^{2} b^{5} - 15 \, a b^{6} + 4 \, b^{7}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6} - {\left (a^{5} b^{2} + 10 \, a^{3} b^{4} - 11 \, a b^{6}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{4 \, {\left ({\left (a^{8} b^{2} - 4 \, a^{6} b^{4} + 6 \, a^{4} b^{6} - 4 \, a^{2} b^{8} + b^{10}\right )} d \cos \left (d x + c\right )^{4} - 2 \, {\left (a^{9} b - 4 \, a^{7} b^{3} + 6 \, a^{5} b^{5} - 4 \, a^{3} b^{7} + a b^{9}\right )} d \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) - {\left (a^{10} - 3 \, a^{8} b^{2} + 2 \, a^{6} b^{4} + 2 \, a^{4} b^{6} - 3 \, a^{2} b^{8} + b^{10}\right )} d \cos \left (d x + c\right )^{2}\right )}} \]
1/4*(2*a^6*b - 6*a^4*b^3 + 6*a^2*b^5 - 2*b^7 + 4*(a^6*b + 5*a^4*b^3 - 7*a^ 2*b^5 + b^7)*cos(d*x + c)^2 + 8*((5*a^2*b^5 + b^7)*cos(d*x + c)^4 - 2*(5*a ^3*b^4 + a*b^6)*cos(d*x + c)^2*sin(d*x + c) - (5*a^4*b^3 + 6*a^2*b^5 + b^7 )*cos(d*x + c)^2)*log(b*sin(d*x + c) + a) + ((a^5*b^2 - 10*a^3*b^4 - 20*a^ 2*b^5 - 15*a*b^6 - 4*b^7)*cos(d*x + c)^4 - 2*(a^6*b - 10*a^4*b^3 - 20*a^3* b^4 - 15*a^2*b^5 - 4*a*b^6)*cos(d*x + c)^2*sin(d*x + c) - (a^7 - 9*a^5*b^2 - 20*a^4*b^3 - 25*a^3*b^4 - 24*a^2*b^5 - 15*a*b^6 - 4*b^7)*cos(d*x + c)^2 )*log(sin(d*x + c) + 1) - ((a^5*b^2 - 10*a^3*b^4 + 20*a^2*b^5 - 15*a*b^6 + 4*b^7)*cos(d*x + c)^4 - 2*(a^6*b - 10*a^4*b^3 + 20*a^3*b^4 - 15*a^2*b^5 + 4*a*b^6)*cos(d*x + c)^2*sin(d*x + c) - (a^7 - 9*a^5*b^2 + 20*a^4*b^3 - 25 *a^3*b^4 + 24*a^2*b^5 - 15*a*b^6 + 4*b^7)*cos(d*x + c)^2)*log(-sin(d*x + c ) + 1) - 2*(a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6 - (a^5*b^2 + 10*a^3*b^4 - 11*a*b^6)*cos(d*x + c)^2)*sin(d*x + c))/((a^8*b^2 - 4*a^6*b^4 + 6*a^4*b^6 - 4*a^2*b^8 + b^10)*d*cos(d*x + c)^4 - 2*(a^9*b - 4*a^7*b^3 + 6*a^5*b^5 - 4*a^3*b^7 + a*b^9)*d*cos(d*x + c)^2*sin(d*x + c) - (a^10 - 3*a^8*b^2 + 2*a ^6*b^4 + 2*a^4*b^6 - 3*a^2*b^8 + b^10)*d*cos(d*x + c)^2)
\[ \int \frac {\sec ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\int \frac {\sec ^{3}{\left (c + d x \right )}}{\left (a + b \sin {\left (c + d x \right )}\right )^{3}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 438 vs. \(2 (217) = 434\).
Time = 0.20 (sec) , antiderivative size = 438, normalized size of antiderivative = 1.94 \[ \int \frac {\sec ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {\frac {8 \, {\left (5 \, a^{2} b^{3} + b^{5}\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}} + \frac {{\left (a - 4 \, b\right )} \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}} - \frac {{\left (a + 4 \, b\right )} \log \left (\sin \left (d x + c\right ) - 1\right )}{a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}} - \frac {2 \, {\left (3 \, a^{4} b + 10 \, a^{2} b^{3} - b^{5} - {\left (a^{3} b^{2} + 11 \, a b^{4}\right )} \sin \left (d x + c\right )^{3} - 2 \, {\left (a^{4} b + 6 \, a^{2} b^{3} - b^{5}\right )} \sin \left (d x + c\right )^{2} - {\left (a^{5} - 3 \, a^{3} b^{2} - 10 \, a b^{4}\right )} \sin \left (d x + c\right )\right )}}{a^{8} - 3 \, a^{6} b^{2} + 3 \, a^{4} b^{4} - a^{2} b^{6} - {\left (a^{6} b^{2} - 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} - b^{8}\right )} \sin \left (d x + c\right )^{4} - 2 \, {\left (a^{7} b - 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} - a b^{7}\right )} \sin \left (d x + c\right )^{3} - {\left (a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}\right )} \sin \left (d x + c\right )^{2} + 2 \, {\left (a^{7} b - 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} - a b^{7}\right )} \sin \left (d x + c\right )}}{4 \, d} \]
1/4*(8*(5*a^2*b^3 + 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) + (a - 4*b)*log(sin(d*x + c) + 1)/(a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4) - (a + 4*b)*log(sin(d*x + c) - 1)/(a^4 + 4*a^3* b + 6*a^2*b^2 + 4*a*b^3 + b^4) - 2*(3*a^4*b + 10*a^2*b^3 - b^5 - (a^3*b^2 + 11*a*b^4)*sin(d*x + c)^3 - 2*(a^4*b + 6*a^2*b^3 - b^5)*sin(d*x + c)^2 - (a^5 - 3*a^3*b^2 - 10*a*b^4)*sin(d*x + c))/(a^8 - 3*a^6*b^2 + 3*a^4*b^4 - a^2*b^6 - (a^6*b^2 - 3*a^4*b^4 + 3*a^2*b^6 - b^8)*sin(d*x + c)^4 - 2*(a^7* b - 3*a^5*b^3 + 3*a^3*b^5 - a*b^7)*sin(d*x + c)^3 - (a^8 - 4*a^6*b^2 + 6*a ^4*b^4 - 4*a^2*b^6 + b^8)*sin(d*x + c)^2 + 2*(a^7*b - 3*a^5*b^3 + 3*a^3*b^ 5 - a*b^7)*sin(d*x + c)))/d
Time = 0.37 (sec) , antiderivative size = 413, normalized size of antiderivative = 1.83 \[ \int \frac {\sec ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {\frac {8 \, {\left (5 \, a^{2} b^{4} + b^{6}\right )} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{a^{8} b - 4 \, a^{6} b^{3} + 6 \, a^{4} b^{5} - 4 \, a^{2} b^{7} + b^{9}} + \frac {{\left (a - 4 \, b\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}} - \frac {{\left (a + 4 \, b\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}} + \frac {2 \, {\left (10 \, a^{2} b^{3} \sin \left (d x + c\right )^{2} + 2 \, b^{5} \sin \left (d x + c\right )^{2} - a^{5} \sin \left (d x + c\right ) - 2 \, a^{3} b^{2} \sin \left (d x + c\right ) + 3 \, a b^{4} \sin \left (d x + c\right ) + 3 \, a^{4} b - 12 \, a^{2} b^{3} - 3 \, b^{5}\right )}}{{\left (a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}\right )} {\left (\sin \left (d x + c\right )^{2} - 1\right )}} - \frac {2 \, {\left (30 \, a^{2} b^{5} \sin \left (d x + c\right )^{2} + 6 \, b^{7} \sin \left (d x + c\right )^{2} + 68 \, a^{3} b^{4} \sin \left (d x + c\right ) + 4 \, a b^{6} \sin \left (d x + c\right ) + 39 \, a^{4} b^{3} - 4 \, a^{2} b^{5} + b^{7}\right )}}{{\left (a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}\right )} {\left (b \sin \left (d x + c\right ) + a\right )}^{2}}}{4 \, d} \]
1/4*(8*(5*a^2*b^4 + 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) + (a - 4*b)*log(abs(sin(d*x + c) + 1))/(a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4) - (a + 4*b)*log(abs(sin(d*x + c) - 1))/(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4) + 2*(10*a^2*b^3*sin(d*x + c)^2 + 2*b^5*sin(d*x + c)^2 - a^5*sin(d*x + c) - 2*a^3*b^2*sin(d*x + c) + 3*a*b^4*sin(d*x + c) + 3*a^4*b - 12*a^2*b^3 - 3*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*(30*a^2*b^5*sin(d*x + c)^2 + 6*b^7*sin(d*x + c)^2 + 68*a^3*b^4*sin(d*x + c) + 4*a*b^6*sin(d*x + c) + 39*a^4*b^3 - 4*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))/d
Time = 5.03 (sec) , antiderivative size = 388, normalized size of antiderivative = 1.72 \[ \int \frac {\sec ^3(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {\ln \left (a+b\,\sin \left (c+d\,x\right )\right )\,\left (\frac {3\,b}{4\,{\left (a+b\right )}^4}+\frac {1}{4\,{\left (a+b\right )}^3}+\frac {3\,b}{4\,{\left (a-b\right )}^4}-\frac {1}{4\,{\left (a-b\right )}^3}\right )}{d}-\frac {\ln \left (\sin \left (c+d\,x\right )-1\right )\,\left (\frac {3\,b}{4\,{\left (a+b\right )}^4}+\frac {1}{4\,{\left (a+b\right )}^3}\right )}{d}+\frac {\frac {3\,a^4\,b+10\,a^2\,b^3-b^5}{2\,\left (a^2-b^2\right )\,\left (a^4-2\,a^2\,b^2+b^4\right )}-\frac {{\sin \left (c+d\,x\right )}^3\,\left (a^3\,b^2+11\,a\,b^4\right )}{2\,\left (a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6\right )}-\frac {{\sin \left (c+d\,x\right )}^2\,\left (a^4\,b+6\,a^2\,b^3-b^5\right )}{\left (a^2-b^2\right )\,\left (a^4-2\,a^2\,b^2+b^4\right )}+\frac {a\,\sin \left (c+d\,x\right )\,\left (-a^4+3\,a^2\,b^2+10\,b^4\right )}{2\,\left (a^2-b^2\right )\,\left (a^4-2\,a^2\,b^2+b^4\right )}}{d\,\left ({\sin \left (c+d\,x\right )}^2\,\left (a^2-b^2\right )-a^2+b^2\,{\sin \left (c+d\,x\right )}^4-2\,a\,b\,\sin \left (c+d\,x\right )+2\,a\,b\,{\sin \left (c+d\,x\right )}^3\right )}+\frac {\ln \left (\sin \left (c+d\,x\right )+1\right )\,\left (a-4\,b\right )}{4\,d\,{\left (a-b\right )}^4} \]
(log(a + b*sin(c + d*x))*((3*b)/(4*(a + b)^4) + 1/(4*(a + b)^3) + (3*b)/(4 *(a - b)^4) - 1/(4*(a - b)^3)))/d - (log(sin(c + d*x) - 1)*((3*b)/(4*(a + b)^4) + 1/(4*(a + b)^3)))/d + ((3*a^4*b - b^5 + 10*a^2*b^3)/(2*(a^2 - b^2) *(a^4 + b^4 - 2*a^2*b^2)) - (sin(c + d*x)^3*(11*a*b^4 + a^3*b^2))/(2*(a^6 - b^6 + 3*a^2*b^4 - 3*a^4*b^2)) - (sin(c + d*x)^2*(a^4*b - b^5 + 6*a^2*b^3 ))/((a^2 - b^2)*(a^4 + b^4 - 2*a^2*b^2)) + (a*sin(c + d*x)*(10*b^4 - a^4 + 3*a^2*b^2))/(2*(a^2 - b^2)*(a^4 + b^4 - 2*a^2*b^2)))/(d*(sin(c + d*x)^2*( a^2 - b^2) - a^2 + b^2*sin(c + d*x)^4 - 2*a*b*sin(c + d*x) + 2*a*b*sin(c + d*x)^3)) + (log(sin(c + d*x) + 1)*(a - 4*b))/(4*d*(a - b)^4)