Integrand size = 28, antiderivative size = 212 \[ \int \frac {\sec ^2(c+d x)}{(a \sin (c+d x)+b \tan (c+d x))^3} \, dx=-\frac {3 a^2 b}{2 \left (a^2-b^2\right )^2 d (b+a \cos (c+d x))^2}+\frac {3 a^2 \left (a^2+3 b^2\right )}{2 \left (a^2-b^2\right )^3 d (b+a \cos (c+d x))}+\frac {(b-a \cos (c+d x)) \csc ^2(c+d x)}{2 \left (a^2-b^2\right ) d (b+a \cos (c+d x))^2}+\frac {3 a \log (1-\cos (c+d x))}{4 (a+b)^4 d}-\frac {3 a \log (1+\cos (c+d x))}{4 (a-b)^4 d}+\frac {6 a^2 b \left (a^2+b^2\right ) \log (b+a \cos (c+d x))}{\left (a^2-b^2\right )^4 d} \]
-3/2*a^2*b/(a^2-b^2)^2/d/(b+a*cos(d*x+c))^2+3/2*a^2*(a^2+3*b^2)/(a^2-b^2)^ 3/d/(b+a*cos(d*x+c))+1/2*(b-a*cos(d*x+c))*csc(d*x+c)^2/(a^2-b^2)/d/(b+a*co s(d*x+c))^2+3/4*a*ln(1-cos(d*x+c))/(a+b)^4/d-3/4*a*ln(1+cos(d*x+c))/(a-b)^ 4/d+6*a^2*b*(a^2+b^2)*ln(b+a*cos(d*x+c))/(a^2-b^2)^4/d
Time = 6.64 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.02 \[ \int \frac {\sec ^2(c+d x)}{(a \sin (c+d x)+b \tan (c+d x))^3} \, dx=-\frac {a^2 b}{2 (-a+b)^2 (a+b)^2 d (b+a \cos (c+d x))^2}-\frac {a^2 \left (a^2+3 b^2\right )}{(-a+b)^3 (a+b)^3 d (b+a \cos (c+d x))}-\frac {\csc ^2\left (\frac {1}{2} (c+d x)\right )}{8 (a+b)^3 d}-\frac {3 a \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{2 (-a+b)^4 d}+\frac {6 \left (a^4 b+a^2 b^3\right ) \log (b+a \cos (c+d x))}{\left (-a^2+b^2\right )^4 d}+\frac {3 a \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{2 (a+b)^4 d}-\frac {\sec ^2\left (\frac {1}{2} (c+d x)\right )}{8 (-a+b)^3 d} \]
-1/2*(a^2*b)/((-a + b)^2*(a + b)^2*d*(b + a*Cos[c + d*x])^2) - (a^2*(a^2 + 3*b^2))/((-a + b)^3*(a + b)^3*d*(b + a*Cos[c + d*x])) - Csc[(c + d*x)/2]^ 2/(8*(a + b)^3*d) - (3*a*Log[Cos[(c + d*x)/2]])/(2*(-a + b)^4*d) + (6*(a^4 *b + a^2*b^3)*Log[b + a*Cos[c + d*x]])/((-a^2 + b^2)^4*d) + (3*a*Log[Sin[( c + d*x)/2]])/(2*(a + b)^4*d) - Sec[(c + d*x)/2]^2/(8*(-a + b)^3*d)
Time = 0.64 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.08, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.393, Rules used = {3042, 4897, 3042, 25, 3316, 25, 27, 593, 27, 657, 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.
| \(\displaystyle \int \frac {\sec ^2(c+d x)}{(a \sin (c+d x)+b \tan (c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sec (c+d x)^2}{(a \sin (c+d x)+b \tan (c+d x))^3}dx\) |
| \(\Big \downarrow \) 4897 |
| \(\displaystyle \int \frac {\cot (c+d x) \csc ^2(c+d x)}{(a \cos (c+d x)+b)^3}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {\sin \left (c+d x-\frac {\pi }{2}\right )}{\cos \left (c+d x-\frac {\pi }{2}\right )^3 \left (b-a \sin \left (c+d x-\frac {\pi }{2}\right )\right )^3}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {\sin \left (\frac {1}{2} (2 c-\pi )+d x\right )}{\cos \left (\frac {1}{2} (2 c-\pi )+d x\right )^3 \left (b-a \sin \left (\frac {1}{2} (2 c-\pi )+d x\right )\right )^3}dx\) |
| \(\Big \downarrow \) 3316 |
| \(\displaystyle \frac {a^3 \int -\frac {\cos (c+d x)}{(b+a \cos (c+d x))^3 \left (a^2-a^2 \cos ^2(c+d x)\right )^2}d(a \cos (c+d x))}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {a^3 \int \frac {\cos (c+d x)}{(b+a \cos (c+d x))^3 \left (a^2-a^2 \cos ^2(c+d x)\right )^2}d(a \cos (c+d x))}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a^2 \int \frac {a \cos (c+d x)}{(b+a \cos (c+d x))^3 \left (a^2-a^2 \cos ^2(c+d x)\right )^2}d(a \cos (c+d x))}{d}\) |
| \(\Big \downarrow \) 593 |
| \(\displaystyle -\frac {a^2 \left (\frac {\int -\frac {3 (b-a \cos (c+d x))}{(b+a \cos (c+d x))^3 \left (a^2-a^2 \cos ^2(c+d x)\right )}d(a \cos (c+d x))}{2 \left (a^2-b^2\right )}-\frac {b-a \cos (c+d x)}{2 \left (a^2-b^2\right ) \left (a^2-a^2 \cos ^2(c+d x)\right ) (a \cos (c+d x)+b)^2}\right )}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {a^2 \left (-\frac {3 \int \frac {b-a \cos (c+d x)}{(b+a \cos (c+d x))^3 \left (a^2-a^2 \cos ^2(c+d x)\right )}d(a \cos (c+d x))}{2 \left (a^2-b^2\right )}-\frac {b-a \cos (c+d x)}{2 \left (a^2-b^2\right ) \left (a^2-a^2 \cos ^2(c+d x)\right ) (a \cos (c+d x)+b)^2}\right )}{d}\) |
| \(\Big \downarrow \) 657 |
| \(\displaystyle -\frac {a^2 \left (-\frac {3 \int \left (\frac {-a-b}{2 a (a-b)^3 (\cos (c+d x) a+a)}+\frac {b-a}{2 a (a+b)^3 (a-a \cos (c+d x))}+\frac {4 b \left (a^2+b^2\right )}{(a-b)^3 (a+b)^3 (b+a \cos (c+d x))}+\frac {-a^2-3 b^2}{(a-b)^2 (a+b)^2 (b+a \cos (c+d x))^2}+\frac {2 b}{(a-b) (a+b) (b+a \cos (c+d x))^3}\right )d(a \cos (c+d x))}{2 \left (a^2-b^2\right )}-\frac {b-a \cos (c+d x)}{2 \left (a^2-b^2\right ) \left (a^2-a^2 \cos ^2(c+d x)\right ) (a \cos (c+d x)+b)^2}\right )}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a^2 \left (-\frac {b-a \cos (c+d x)}{2 \left (a^2-b^2\right ) \left (a^2-a^2 \cos ^2(c+d x)\right ) (a \cos (c+d x)+b)^2}-\frac {3 \left (-\frac {b}{\left (a^2-b^2\right ) (a \cos (c+d x)+b)^2}+\frac {a^2+3 b^2}{\left (a^2-b^2\right )^2 (a \cos (c+d x)+b)}+\frac {4 b \left (a^2+b^2\right ) \log (a \cos (c+d x)+b)}{\left (a^2-b^2\right )^3}+\frac {(a-b) \log (a-a \cos (c+d x))}{2 a (a+b)^3}-\frac {(a+b) \log (a \cos (c+d x)+a)}{2 a (a-b)^3}\right )}{2 \left (a^2-b^2\right )}\right )}{d}\) |
-((a^2*(-1/2*(b - a*Cos[c + d*x])/((a^2 - b^2)*(b + a*Cos[c + d*x])^2*(a^2 - a^2*Cos[c + d*x]^2)) - (3*(-(b/((a^2 - b^2)*(b + a*Cos[c + d*x])^2)) + (a^2 + 3*b^2)/((a^2 - b^2)^2*(b + a*Cos[c + d*x])) + ((a - b)*Log[a - a*Co s[c + d*x]])/(2*a*(a + b)^3) - ((a + b)*Log[a + a*Cos[c + d*x]])/(2*a*(a - b)^3) + (4*b*(a^2 + b^2)*Log[b + a*Cos[c + d*x]])/(a^2 - b^2)^3))/(2*(a^2 - b^2))))/d)
3.3.69.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(x_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c + d*x)^(n + 1)*(c - d*x)*((a + b*x^2)^(p + 1)/(2*(p + 1)*(b*c^2 + a*d^2))), x] - Simp[d/(2*(p + 1)*(b*c^2 + a*d^2)) Int[(c + d*x)^n*(a + b* x^2)^(p + 1)*(c*n - d*(n + 2*p + 4)*x), x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[p, -1] && NeQ[b*c^2 + a*d^2, 0]
Int[(((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.))/((a_) + (c_.)*( x_)^2), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*((f + g*x)^n/(a + c*x^ 2)), x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && IntegersQ[n]
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]
Time = 31.82 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.85
| method | result | size |
| derivativedivides | \(\frac {-\frac {b \,a^{2}}{2 \left (a +b \right )^{2} \left (a -b \right )^{2} \left (b +\cos \left (d x +c \right ) a \right )^{2}}+\frac {a^{2} \left (a^{2}+3 b^{2}\right )}{\left (a +b \right )^{3} \left (a -b \right )^{3} \left (b +\cos \left (d x +c \right ) a \right )}+\frac {6 a^{2} b \left (a^{2}+b^{2}\right ) \ln \left (b +\cos \left (d x +c \right ) a \right )}{\left (a +b \right )^{4} \left (a -b \right )^{4}}+\frac {1}{4 \left (a -b \right )^{3} \left (\cos \left (d x +c \right )+1\right )}-\frac {3 a \ln \left (\cos \left (d x +c \right )+1\right )}{4 \left (a -b \right )^{4}}+\frac {1}{4 \left (a +b \right )^{3} \left (\cos \left (d x +c \right )-1\right )}+\frac {3 a \ln \left (\cos \left (d x +c \right )-1\right )}{4 \left (a +b \right )^{4}}}{d}\) | \(181\) |
| default | \(\frac {-\frac {b \,a^{2}}{2 \left (a +b \right )^{2} \left (a -b \right )^{2} \left (b +\cos \left (d x +c \right ) a \right )^{2}}+\frac {a^{2} \left (a^{2}+3 b^{2}\right )}{\left (a +b \right )^{3} \left (a -b \right )^{3} \left (b +\cos \left (d x +c \right ) a \right )}+\frac {6 a^{2} b \left (a^{2}+b^{2}\right ) \ln \left (b +\cos \left (d x +c \right ) a \right )}{\left (a +b \right )^{4} \left (a -b \right )^{4}}+\frac {1}{4 \left (a -b \right )^{3} \left (\cos \left (d x +c \right )+1\right )}-\frac {3 a \ln \left (\cos \left (d x +c \right )+1\right )}{4 \left (a -b \right )^{4}}+\frac {1}{4 \left (a +b \right )^{3} \left (\cos \left (d x +c \right )-1\right )}+\frac {3 a \ln \left (\cos \left (d x +c \right )-1\right )}{4 \left (a +b \right )^{4}}}{d}\) | \(181\) |
| risch | \(-\frac {3 i a x}{2 \left (a^{4}+4 a^{3} b +6 a^{2} b^{2}+4 a \,b^{3}+b^{4}\right )}-\frac {3 i a c}{2 d \left (a^{4}+4 a^{3} b +6 a^{2} b^{2}+4 a \,b^{3}+b^{4}\right )}+\frac {3 i a x}{2 \left (a^{4}-4 a^{3} b +6 a^{2} b^{2}-4 a \,b^{3}+b^{4}\right )}+\frac {3 i a c}{2 d \left (a^{4}-4 a^{3} b +6 a^{2} b^{2}-4 a \,b^{3}+b^{4}\right )}-\frac {12 i a^{4} b x}{a^{8}-4 a^{6} b^{2}+6 a^{4} b^{4}-4 a^{2} b^{6}+b^{8}}-\frac {12 i a^{4} b c}{d \left (a^{8}-4 a^{6} b^{2}+6 a^{4} b^{4}-4 a^{2} b^{6}+b^{8}\right )}-\frac {12 i a^{2} b^{3} x}{a^{8}-4 a^{6} b^{2}+6 a^{4} b^{4}-4 a^{2} b^{6}+b^{8}}-\frac {12 i a^{2} b^{3} c}{d \left (a^{8}-4 a^{6} b^{2}+6 a^{4} b^{4}-4 a^{2} b^{6}+b^{8}\right )}+\frac {3 a^{5} {\mathrm e}^{7 i \left (d x +c \right )}+9 a^{3} b^{2} {\mathrm e}^{7 i \left (d x +c \right )}+24 a^{2} b^{3} {\mathrm e}^{6 i \left (d x +c \right )}+a^{5} {\mathrm e}^{5 i \left (d x +c \right )}-17 a^{3} b^{2} {\mathrm e}^{5 i \left (d x +c \right )}+4 a \,b^{4} {\mathrm e}^{5 i \left (d x +c \right )}-8 a^{4} b \,{\mathrm e}^{4 i \left (d x +c \right )}-32 a^{2} b^{3} {\mathrm e}^{4 i \left (d x +c \right )}-8 b^{5} {\mathrm e}^{4 i \left (d x +c \right )}+a^{5} {\mathrm e}^{3 i \left (d x +c \right )}-17 a^{3} b^{2} {\mathrm e}^{3 i \left (d x +c \right )}+4 a \,b^{4} {\mathrm e}^{3 i \left (d x +c \right )}+24 a^{2} b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+3 a^{5} {\mathrm e}^{i \left (d x +c \right )}+9 a^{3} b^{2} {\mathrm e}^{i \left (d x +c \right )}}{\left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \left ({\mathrm e}^{2 i \left (d x +c \right )} a +2 b \,{\mathrm e}^{i \left (d x +c \right )}+a \right )^{2} \left (a^{2}-b^{2}\right ) \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2} d}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right ) a}{2 d \left (a^{4}+4 a^{3} b +6 a^{2} b^{2}+4 a \,b^{3}+b^{4}\right )}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) a}{2 d \left (a^{4}-4 a^{3} b +6 a^{2} b^{2}-4 a \,b^{3}+b^{4}\right )}+\frac {6 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^{4}}{d \left (a^{8}-4 a^{6} b^{2}+6 a^{4} b^{4}-4 a^{2} b^{6}+b^{8}\right )}+\frac {6 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^{2}}{d \left (a^{8}-4 a^{6} b^{2}+6 a^{4} b^{4}-4 a^{2} b^{6}+b^{8}\right )}\) | \(857\) |
1/d*(-1/2*b*a^2/(a+b)^2/(a-b)^2/(b+cos(d*x+c)*a)^2+a^2*(a^2+3*b^2)/(a+b)^3 /(a-b)^3/(b+cos(d*x+c)*a)+6*a^2*b*(a^2+b^2)/(a+b)^4/(a-b)^4*ln(b+cos(d*x+c )*a)+1/4/(a-b)^3/(cos(d*x+c)+1)-3/4/(a-b)^4*a*ln(cos(d*x+c)+1)+1/4/(a+b)^3 /(cos(d*x+c)-1)+3/4/(a+b)^4*a*ln(cos(d*x+c)-1))
Leaf count of result is larger than twice the leaf count of optimal. 939 vs. \(2 (203) = 406\).
Time = 0.43 (sec) , antiderivative size = 939, normalized size of antiderivative = 4.43 \[ \int \frac {\sec ^2(c+d x)}{(a \sin (c+d x)+b \tan (c+d x))^3} \, dx=-\frac {2 \, a^{6} b + 18 \, a^{4} b^{3} - 18 \, a^{2} b^{5} - 2 \, b^{7} - 6 \, {\left (a^{7} + 2 \, a^{5} b^{2} - 3 \, a^{3} b^{4}\right )} \cos \left (d x + c\right )^{3} - 24 \, {\left (a^{4} b^{3} - a^{2} b^{5}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (2 \, a^{7} + 9 \, a^{5} b^{2} - 12 \, a^{3} b^{4} + a b^{6}\right )} \cos \left (d x + c\right ) + 24 \, {\left (a^{4} b^{3} + a^{2} b^{5} - {\left (a^{6} b + a^{4} b^{3}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (a^{5} b^{2} + a^{3} b^{4}\right )} \cos \left (d x + c\right )^{3} + {\left (a^{6} b - a^{2} b^{5}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{5} b^{2} + a^{3} b^{4}\right )} \cos \left (d x + c\right )\right )} \log \left (a \cos \left (d x + c\right ) + b\right ) - 3 \, {\left (a^{5} b^{2} + 4 \, a^{4} b^{3} + 6 \, a^{3} b^{4} + 4 \, a^{2} b^{5} + a b^{6} - {\left (a^{7} + 4 \, a^{6} b + 6 \, a^{5} b^{2} + 4 \, a^{4} b^{3} + a^{3} b^{4}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (a^{6} b + 4 \, a^{5} b^{2} + 6 \, a^{4} b^{3} + 4 \, a^{3} b^{4} + a^{2} b^{5}\right )} \cos \left (d x + c\right )^{3} + {\left (a^{7} + 4 \, a^{6} b + 5 \, a^{5} b^{2} - 5 \, a^{3} b^{4} - 4 \, a^{2} b^{5} - a b^{6}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{6} b + 4 \, a^{5} b^{2} + 6 \, a^{4} b^{3} + 4 \, a^{3} b^{4} + a^{2} b^{5}\right )} \cos \left (d x + c\right )\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 3 \, {\left (a^{5} b^{2} - 4 \, a^{4} b^{3} + 6 \, a^{3} b^{4} - 4 \, a^{2} b^{5} + a b^{6} - {\left (a^{7} - 4 \, a^{6} b + 6 \, a^{5} b^{2} - 4 \, a^{4} b^{3} + a^{3} b^{4}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (a^{6} b - 4 \, a^{5} b^{2} + 6 \, a^{4} b^{3} - 4 \, a^{3} b^{4} + a^{2} b^{5}\right )} \cos \left (d x + c\right )^{3} + {\left (a^{7} - 4 \, a^{6} b + 5 \, a^{5} b^{2} - 5 \, a^{3} b^{4} + 4 \, a^{2} b^{5} - a b^{6}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{6} b - 4 \, a^{5} b^{2} + 6 \, a^{4} b^{3} - 4 \, a^{3} b^{4} + a^{2} b^{5}\right )} \cos \left (d x + c\right )\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{4 \, {\left ({\left (a^{10} - 4 \, a^{8} b^{2} + 6 \, a^{6} b^{4} - 4 \, a^{4} b^{6} + a^{2} b^{8}\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 )^{3} - {\left (a^{10} - 5 \, a^{8} b^{2} + 10 \, a^{6} b^{4} - 10 \, a^{4} b^{6} + 5 \, a^{2} b^{8} - b^{10}\right )} d \cos \left (d x + c\right )^{2} - 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 ) - {\left (a^{8} b^{2} - 4 \, a^{6} b^{4} + 6 \, a^{4} b^{6} - 4 \, a^{2} b^{8} + b^{10}\right )} d\right )}} \]
-1/4*(2*a^6*b + 18*a^4*b^3 - 18*a^2*b^5 - 2*b^7 - 6*(a^7 + 2*a^5*b^2 - 3*a ^3*b^4)*cos(d*x + c)^3 - 24*(a^4*b^3 - a^2*b^5)*cos(d*x + c)^2 + 2*(2*a^7 + 9*a^5*b^2 - 12*a^3*b^4 + a*b^6)*cos(d*x + c) + 24*(a^4*b^3 + a^2*b^5 - ( a^6*b + a^4*b^3)*cos(d*x + c)^4 - 2*(a^5*b^2 + a^3*b^4)*cos(d*x + c)^3 + ( a^6*b - a^2*b^5)*cos(d*x + c)^2 + 2*(a^5*b^2 + a^3*b^4)*cos(d*x + c))*log( a*cos(d*x + c) + b) - 3*(a^5*b^2 + 4*a^4*b^3 + 6*a^3*b^4 + 4*a^2*b^5 + a*b ^6 - (a^7 + 4*a^6*b + 6*a^5*b^2 + 4*a^4*b^3 + a^3*b^4)*cos(d*x + c)^4 - 2* (a^6*b + 4*a^5*b^2 + 6*a^4*b^3 + 4*a^3*b^4 + a^2*b^5)*cos(d*x + c)^3 + (a^ 7 + 4*a^6*b + 5*a^5*b^2 - 5*a^3*b^4 - 4*a^2*b^5 - a*b^6)*cos(d*x + c)^2 + 2*(a^6*b + 4*a^5*b^2 + 6*a^4*b^3 + 4*a^3*b^4 + a^2*b^5)*cos(d*x + c))*log( 1/2*cos(d*x + c) + 1/2) + 3*(a^5*b^2 - 4*a^4*b^3 + 6*a^3*b^4 - 4*a^2*b^5 + a*b^6 - (a^7 - 4*a^6*b + 6*a^5*b^2 - 4*a^4*b^3 + a^3*b^4)*cos(d*x + c)^4 - 2*(a^6*b - 4*a^5*b^2 + 6*a^4*b^3 - 4*a^3*b^4 + a^2*b^5)*cos(d*x + c)^3 + (a^7 - 4*a^6*b + 5*a^5*b^2 - 5*a^3*b^4 + 4*a^2*b^5 - a*b^6)*cos(d*x + c)^ 2 + 2*(a^6*b - 4*a^5*b^2 + 6*a^4*b^3 - 4*a^3*b^4 + a^2*b^5)*cos(d*x + c))* log(-1/2*cos(d*x + c) + 1/2))/((a^10 - 4*a^8*b^2 + 6*a^6*b^4 - 4*a^4*b^6 + a^2*b^8)*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)^3 - (a^10 - 5*a^8*b^2 + 10*a^6*b^4 - 10*a^4*b^6 + 5*a^2*b^8 - b^10)*d*cos(d*x + c)^2 - 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) - (a^8*b^2 - 4*a^6*b^4 + 6*a^4*b^6 - 4*...
\[ \int \frac {\sec ^2(c+d x)}{(a \sin (c+d x)+b \tan (c+d x))^3} \, dx=\int \frac {\sec ^{2}{\left (c + d x \right )}}{\left (a \sin {\left (c + d x \right )} + b \tan {\left (c + d x \right )}\right )^{3}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 596 vs. \(2 (203) = 406\).
Time = 0.25 (sec) , antiderivative size = 596, normalized size of antiderivative = 2.81 \[ \int \frac {\sec ^2(c+d x)}{(a \sin (c+d x)+b \tan (c+d x))^3} \, dx=\frac {\frac {48 \, {\left (a^{4} b + a^{2} b^{3}\right )} \log \left (a + b - \frac {{\left (a - b\right )} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )}{a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}} + \frac {12 \, a \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}} - \frac {a^{6} - 2 \, a^{5} b - a^{4} b^{2} + 4 \, a^{3} b^{3} - a^{2} b^{4} - 2 \, a b^{5} + b^{6} - \frac {2 \, {\left (9 \, a^{6} + 4 \, a^{5} b + 37 \, a^{4} b^{2} + 32 \, a^{3} b^{3} - 5 \, a^{2} b^{4} + 4 \, a b^{5} - b^{6}\right )} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {{\left (17 \, a^{6} - 6 \, a^{5} b + 63 \, a^{4} b^{2} - 84 \, a^{3} b^{3} + 15 \, a^{2} b^{4} - 6 \, a b^{5} + b^{6}\right )} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}}{\frac {{\left (a^{9} + a^{8} b - 4 \, a^{7} b^{2} - 4 \, a^{6} b^{3} + 6 \, a^{5} b^{4} + 6 \, a^{4} b^{5} - 4 \, a^{3} b^{6} - 4 \, a^{2} b^{7} + a b^{8} + b^{9}\right )} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {2 \, {\left (a^{9} - a^{8} b - 4 \, a^{7} b^{2} + 4 \, a^{6} b^{3} + 6 \, a^{5} b^{4} - 6 \, a^{4} b^{5} - 4 \, a^{3} b^{6} + 4 \, a^{2} b^{7} + a b^{8} - b^{9}\right )} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {{\left (a^{9} - 3 \, a^{8} b + 8 \, a^{6} b^{3} - 6 \, a^{5} b^{4} - 6 \, a^{4} b^{5} + 8 \, a^{3} b^{6} - 3 \, a b^{8} + b^{9}\right )} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}} + \frac {\sin \left (d x + c\right )^{2}}{{\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{2}}}{8 \, d} \]
1/8*(48*(a^4*b + a^2*b^3)*log(a + b - (a - b)*sin(d*x + c)^2/(cos(d*x + c) + 1)^2)/(a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8) + 12*a*log(sin(d* x + c)/(cos(d*x + c) + 1))/(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4) - ( a^6 - 2*a^5*b - a^4*b^2 + 4*a^3*b^3 - a^2*b^4 - 2*a*b^5 + b^6 - 2*(9*a^6 + 4*a^5*b + 37*a^4*b^2 + 32*a^3*b^3 - 5*a^2*b^4 + 4*a*b^5 - b^6)*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + (17*a^6 - 6*a^5*b + 63*a^4*b^2 - 84*a^3*b^3 + 15*a^2*b^4 - 6*a*b^5 + b^6)*sin(d*x + c)^4/(cos(d*x + c) + 1)^4)/((a^9 + a ^8*b - 4*a^7*b^2 - 4*a^6*b^3 + 6*a^5*b^4 + 6*a^4*b^5 - 4*a^3*b^6 - 4*a^2*b ^7 + a*b^8 + b^9)*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 2*(a^9 - a^8*b - 4 *a^7*b^2 + 4*a^6*b^3 + 6*a^5*b^4 - 6*a^4*b^5 - 4*a^3*b^6 + 4*a^2*b^7 + a*b ^8 - b^9)*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + (a^9 - 3*a^8*b + 8*a^6*b^3 - 6*a^5*b^4 - 6*a^4*b^5 + 8*a^3*b^6 - 3*a*b^8 + b^9)*sin(d*x + c)^6/(cos( d*x + c) + 1)^6) + sin(d*x + c)^2/((a^3 - 3*a^2*b + 3*a*b^2 - b^3)*(cos(d* x + c) + 1)^2))/d
Leaf count of result is larger than twice the leaf count of optimal. 689 vs. \(2 (203) = 406\).
Time = 0.73 (sec) , antiderivative size = 689, normalized size of antiderivative = 3.25 \[ \int \frac {\sec ^2(c+d x)}{(a \sin (c+d x)+b \tan (c+d x))^3} \, dx=\frac {\frac {6 \, a \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \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 {48 \, {\left (a^{4} b + a^{2} b^{3}\right )} \log \left ({\left | -a - b - \frac {a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} \right |}\right )}{a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}} + \frac {{\left (a + b - \frac {6 \, a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}}{{\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} {\left (\cos \left (d x + c\right ) - 1\right )}} - \frac {\cos \left (d x + c\right ) - 1}{{\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}} + \frac {8 \, {\left (2 \, a^{7} - 5 \, a^{6} b - 8 \, a^{5} b^{2} - 2 \, a^{4} b^{3} - 10 \, a^{3} b^{4} - 9 \, a^{2} b^{5} + \frac {2 \, a^{7} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {16 \, a^{6} b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {6 \, a^{5} b^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {2 \, a^{4} b^{3} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {8 \, a^{3} b^{4} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {18 \, a^{2} b^{5} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {9 \, a^{6} b {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {18 \, a^{5} b^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {18 \, a^{4} b^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {18 \, a^{3} b^{4} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {9 \, a^{2} b^{5} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )}}{{\left (a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}\right )} {\left (a + b + \frac {a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}\right )}^{2}}}{8 \, d} \]
1/8*(6*a*log(abs(-cos(d*x + c) + 1)/abs(cos(d*x + c) + 1))/(a^4 + 4*a^3*b + 6*a^2*b^2 + 4*a*b^3 + b^4) + 48*(a^4*b + a^2*b^3)*log(abs(-a - b - a*(co s(d*x + c) - 1)/(cos(d*x + c) + 1) + b*(cos(d*x + c) - 1)/(cos(d*x + c) + 1)))/(a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8) + (a + b - 6*a*(cos(d *x + c) - 1)/(cos(d*x + c) + 1))*(cos(d*x + c) + 1)/((a^4 + 4*a^3*b + 6*a^ 2*b^2 + 4*a*b^3 + b^4)*(cos(d*x + c) - 1)) - (cos(d*x + c) - 1)/((a^3 - 3* a^2*b + 3*a*b^2 - b^3)*(cos(d*x + c) + 1)) + 8*(2*a^7 - 5*a^6*b - 8*a^5*b^ 2 - 2*a^4*b^3 - 10*a^3*b^4 - 9*a^2*b^5 + 2*a^7*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 16*a^6*b*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 6*a^5*b^2*(c os(d*x + c) - 1)/(cos(d*x + c) + 1) - 2*a^4*b^3*(cos(d*x + c) - 1)/(cos(d* x + c) + 1) - 8*a^3*b^4*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 18*a^2*b^5 *(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 9*a^6*b*(cos(d*x + c) - 1)^2/(cos (d*x + c) + 1)^2 + 18*a^5*b^2*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - 18*a^4*b^3*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 + 18*a^3*b^4*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - 9*a^2*b^5*(cos(d*x + c) - 1)^2/(cos(d* x + c) + 1)^2)/((a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8)*(a + b + a *(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - b*(cos(d*x + c) - 1)/(cos(d*x + c ) + 1))^2))/d
Time = 23.15 (sec) , antiderivative size = 492, normalized size of antiderivative = 2.32 \[ \int \frac {\sec ^2(c+d x)}{(a \sin (c+d x)+b \tan (c+d x))^3} \, dx=\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,d\,{\left (a-b\right )}^3}-\frac {\frac {a^3-3\,a^2\,b+3\,a\,b^2-b^3}{2\,\left (a+b\right )}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (9\,a^5-5\,a^4\,b+42\,a^3\,b^2-10\,a^2\,b^3+5\,a\,b^4-b^5\right )}{\left (a-b\right )\,\left (a^2+2\,a\,b+b^2\right )}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (17\,a^5+11\,a^4\,b+74\,a^3\,b^2-10\,a^2\,b^3+5\,a\,b^4-b^5\right )}{2\,\left (a+b\right )\,\left (a^2+2\,a\,b+b^2\right )}}{d\,\left (\left (4\,a^5-20\,a^4\,b+40\,a^3\,b^2-40\,a^2\,b^3+20\,a\,b^4-4\,b^5\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+\left (-8\,a^5+24\,a^4\,b-16\,a^3\,b^2-16\,a^2\,b^3+24\,a\,b^4-8\,b^5\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\left (4\,a^5-4\,a^4\,b-8\,a^3\,b^2+8\,a^2\,b^3+4\,a\,b^4-4\,b^5\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}+\frac {\ln \left (a+b-a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )\,\left (6\,a^4\,b+6\,a^2\,b^3\right )}{d\,\left (a^8-4\,a^6\,b^2+6\,a^4\,b^4-4\,a^2\,b^6+b^8\right )}+\frac {3\,a\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d\,\left (2\,a^4+8\,a^3\,b+12\,a^2\,b^2+8\,a\,b^3+2\,b^4\right )} \]
tan(c/2 + (d*x)/2)^2/(8*d*(a - b)^3) - ((3*a*b^2 - 3*a^2*b + a^3 - b^3)/(2 *(a + b)) - (tan(c/2 + (d*x)/2)^2*(5*a*b^4 - 5*a^4*b + 9*a^5 - b^5 - 10*a^ 2*b^3 + 42*a^3*b^2))/((a - b)*(2*a*b + a^2 + b^2)) + (tan(c/2 + (d*x)/2)^4 *(5*a*b^4 + 11*a^4*b + 17*a^5 - b^5 - 10*a^2*b^3 + 74*a^3*b^2))/(2*(a + b) *(2*a*b + a^2 + b^2)))/(d*(tan(c/2 + (d*x)/2)^2*(4*a*b^4 - 4*a^4*b + 4*a^5 - 4*b^5 + 8*a^2*b^3 - 8*a^3*b^2) - tan(c/2 + (d*x)/2)^4*(8*a^5 - 24*a^4*b - 24*a*b^4 + 8*b^5 + 16*a^2*b^3 + 16*a^3*b^2) + tan(c/2 + (d*x)/2)^6*(20* a*b^4 - 20*a^4*b + 4*a^5 - 4*b^5 - 40*a^2*b^3 + 40*a^3*b^2))) + (log(a + b - a*tan(c/2 + (d*x)/2)^2 + b*tan(c/2 + (d*x)/2)^2)*(6*a^4*b + 6*a^2*b^3)) /(d*(a^8 + b^8 - 4*a^2*b^6 + 6*a^4*b^4 - 4*a^6*b^2)) + (3*a*log(tan(c/2 + (d*x)/2)))/(d*(8*a*b^3 + 8*a^3*b + 2*a^4 + 2*b^4 + 12*a^2*b^2))