Integrand size = 26, antiderivative size = 211 \[ \int \frac {\cos (c+d x)}{(a \sin (c+d x)+b \tan (c+d x))^3} \, dx=\frac {b^4}{2 a \left (a^2-b^2\right )^2 d (b+a \cos (c+d x))^2}-\frac {4 a b^3}{\left (a^2-b^2\right )^3 d (b+a \cos (c+d x))}-\frac {\left (a \left (a^2+3 b^2\right )-b \left (3 a^2+b^2\right ) \cos (c+d x)\right ) \csc ^2(c+d x)}{2 \left (a^2-b^2\right )^3 d}-\frac {3 b \log (1-\cos (c+d x))}{4 (a+b)^4 d}+\frac {3 b \log (1+\cos (c+d x))}{4 (a-b)^4 d}-\frac {6 a b^2 \left (a^2+b^2\right ) \log (b+a \cos (c+d x))}{\left (a^2-b^2\right )^4 d} \]
1/2*b^4/a/(a^2-b^2)^2/d/(b+a*cos(d*x+c))^2-4*a*b^3/(a^2-b^2)^3/d/(b+a*cos( d*x+c))-1/2*(a*(a^2+3*b^2)-b*(3*a^2+b^2)*cos(d*x+c))*csc(d*x+c)^2/(a^2-b^2 )^3/d-3/4*b*ln(1-cos(d*x+c))/(a+b)^4/d+3/4*b*ln(1+cos(d*x+c))/(a-b)^4/d-6* a*b^2*(a^2+b^2)*ln(b+a*cos(d*x+c))/(a^2-b^2)^4/d
Time = 6.29 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.87 \[ \int \frac {\cos (c+d x)}{(a \sin (c+d x)+b \tan (c+d x))^3} \, dx=\frac {\frac {4 b^4}{a (a-b)^2 (a+b)^2 (b+a \cos (c+d x))^2}+\frac {32 a b^3}{(-a+b)^3 (a+b)^3 (b+a \cos (c+d x))}-\frac {\csc ^2\left (\frac {1}{2} (c+d x)\right )}{(a+b)^3}+\frac {12 b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{(a-b)^4}-\frac {48 a b^2 \left (a^2+b^2\right ) \log (b+a \cos (c+d x))}{\left (a^2-b^2\right )^4}-\frac {12 b \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{(a+b)^4}+\frac {\sec ^2\left (\frac {1}{2} (c+d x)\right )}{(-a+b)^3}}{8 d} \]
((4*b^4)/(a*(a - b)^2*(a + b)^2*(b + a*Cos[c + d*x])^2) + (32*a*b^3)/((-a + b)^3*(a + b)^3*(b + a*Cos[c + d*x])) - Csc[(c + d*x)/2]^2/(a + b)^3 + (1 2*b*Log[Cos[(c + d*x)/2]])/(a - b)^4 - (48*a*b^2*(a^2 + b^2)*Log[b + a*Cos [c + d*x]])/(a^2 - b^2)^4 - (12*b*Log[Sin[(c + d*x)/2]])/(a + b)^4 + Sec[( c + d*x)/2]^2/(-a + b)^3)/(8*d)
Time = 0.98 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.12, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {3042, 4897, 3042, 3316, 27, 601, 25, 2160, 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.
| \(\displaystyle \int \frac {\cos (c+d x)}{(a \sin (c+d x)+b \tan (c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)}{(a \sin (c+d x)+b \tan (c+d x))^3}dx\) |
| \(\Big \downarrow \) 4897 |
| \(\displaystyle \int \frac {\cos (c+d x) \cot ^3(c+d x)}{(a \cos (c+d x)+b)^3}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin \left (c+d x-\frac {\pi }{2}\right )^4}{\cos \left (c+d x-\frac {\pi }{2}\right )^3 \left (b-a \sin \left (c+d x-\frac {\pi }{2}\right )\right )^3}dx\) |
| \(\Big \downarrow \) 3316 |
| \(\displaystyle -\frac {a^3 \int \frac {\cos ^4(c+d x)}{(b+a \cos (c+d x))^3 \left (a^2-a^2 \cos ^2(c+d x)\right )^2}d(a \cos (c+d x))}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {a^4 \cos ^4(c+d x)}{(b+a \cos (c+d x))^3 \left (a^2-a^2 \cos ^2(c+d x)\right )^2}d(a \cos (c+d x))}{a d}\) |
| \(\Big \downarrow \) 601 |
| \(\displaystyle -\frac {\frac {a^2 \left (a^2 \left (a^2+3 b^2\right )-a b \left (3 a^2+b^2\right ) \cos (c+d x)\right )}{2 \left (a^2-b^2\right )^3 \left (a^2-a^2 \cos ^2(c+d x)\right )}-\frac {\int -\frac {-\frac {b \left (3 a^2+b^2\right ) \cos ^3(c+d x) a^7}{\left (a^2-b^2\right )^3}+\frac {b^3 \left (7 a^2-3 b^2\right ) \cos (c+d x) a^5}{\left (a^2-b^2\right )^3}+\frac {b^2 \left (3 a^4-9 b^2 a^2+2 b^4\right ) \cos ^2(c+d x) a^4}{\left (a^2-b^2\right )^3}+\frac {b^4 \left (3 a^2+b^2\right ) a^4}{\left (a^2-b^2\right )^3}}{(b+a \cos (c+d x))^3 \left (a^2-a^2 \cos ^2(c+d x)\right )}d(a \cos (c+d x))}{2 a^2}}{a d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {\int \frac {-\frac {b \left (3 a^2+b^2\right ) \cos ^3(c+d x) a^7}{\left (a^2-b^2\right )^3}+\frac {b^3 \left (7 a^2-3 b^2\right ) \cos (c+d x) a^5}{\left (a^2-b^2\right )^3}+\frac {b^2 \left (3 a^4-9 b^2 a^2+2 b^4\right ) \cos ^2(c+d x) a^4}{\left (a^2-b^2\right )^3}+\frac {b^4 \left (3 a^2+b^2\right ) a^4}{\left (a^2-b^2\right )^3}}{(b+a \cos (c+d x))^3 \left (a^2-a^2 \cos ^2(c+d x)\right )}d(a \cos (c+d x))}{2 a^2}+\frac {a^2 \left (a^2 \left (a^2+3 b^2\right )-a b \left (3 a^2+b^2\right ) \cos (c+d x)\right )}{2 \left (a^2-b^2\right )^3 \left (a^2-a^2 \cos ^2(c+d x)\right )}}{a d}\) |
| \(\Big \downarrow \) 2160 |
| \(\displaystyle -\frac {\frac {\int \left (\frac {12 b^2 \left (a^2+b^2\right ) a^4}{\left (a^2-b^2\right )^4 (b+a \cos (c+d x))}-\frac {8 b^3 a^4}{\left (a^2-b^2\right )^3 (b+a \cos (c+d x))^2}-\frac {3 b a^3}{2 (a+b)^4 (a-a \cos (c+d x))}-\frac {3 b a^3}{2 (a-b)^4 (\cos (c+d x) a+a)}+\frac {2 b^4 a^2}{\left (a^2-b^2\right )^2 (b+a \cos (c+d x))^3}\right )d(a \cos (c+d x))}{2 a^2}+\frac {a^2 \left (a^2 \left (a^2+3 b^2\right )-a b \left (3 a^2+b^2\right ) \cos (c+d x)\right )}{2 \left (a^2-b^2\right )^3 \left (a^2-a^2 \cos ^2(c+d x)\right )}}{a d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\frac {a^2 \left (a^2 \left (a^2+3 b^2\right )-a b \left (3 a^2+b^2\right ) \cos (c+d x)\right )}{2 \left (a^2-b^2\right )^3 \left (a^2-a^2 \cos ^2(c+d x)\right )}+\frac {\frac {3 a^3 b \log (a-a \cos (c+d x))}{2 (a+b)^4}-\frac {3 a^3 b \log (a \cos (c+d x)+a)}{2 (a-b)^4}-\frac {a^2 b^4}{\left (a^2-b^2\right )^2 (a \cos (c+d x)+b)^2}+\frac {12 a^4 b^2 \left (a^2+b^2\right ) \log (a \cos (c+d x)+b)}{\left (a^2-b^2\right )^4}+\frac {8 a^4 b^3}{\left (a^2-b^2\right )^3 (a \cos (c+d x)+b)}}{2 a^2}}{a d}\) |
-(((a^2*(a^2*(a^2 + 3*b^2) - a*b*(3*a^2 + b^2)*Cos[c + d*x]))/(2*(a^2 - b^ 2)^3*(a^2 - a^2*Cos[c + d*x]^2)) + (-((a^2*b^4)/((a^2 - b^2)^2*(b + a*Cos[ c + d*x])^2)) + (8*a^4*b^3)/((a^2 - b^2)^3*(b + a*Cos[c + d*x])) + (3*a^3* b*Log[a - a*Cos[c + d*x]])/(2*(a + b)^4) - (3*a^3*b*Log[a + a*Cos[c + d*x] ])/(2*(a - b)^4) + (12*a^4*b^2*(a^2 + b^2)*Log[b + a*Cos[c + d*x]])/(a^2 - b^2)^4)/(2*a^2))/(a*d))
3.3.66.3.1 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_)^(m_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> With[{Qx = PolynomialQuotient[x^m*(c + d*x)^n, a + b*x^2, x], e = Coe ff[PolynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 0], f = Coeff[Pol ynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 1]}, Simp[(a*f - b*e*x) *((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1)) Int[(c + d*x)^n*(a + b*x^2)^(p + 1)*ExpandToSum[(2*a*(p + 1)*Qx)/(c + d*x)^n + (e* (2*p + 3))/(c + d*x)^n, x], x], x]] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 1] && LtQ[p, -1] && ILtQ[n, 0] && NeQ[b*c^2 + a*d^2, 0]
Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*Pq*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]
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 = 5.15 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.83
| method | result | size |
| derivativedivides | \(\frac {\frac {b^{4}}{2 \left (a +b \right )^{2} \left (a -b \right )^{2} a \left (b +\cos \left (d x +c \right ) a \right )^{2}}-\frac {4 a \,b^{3}}{\left (a +b \right )^{3} \left (a -b \right )^{3} \left (b +\cos \left (d x +c \right ) a \right )}-\frac {6 a \,b^{2} \left (a^{2}+b^{2}\right ) \ln \left (b +\cos \left (d x +c \right ) a \right )}{\left (a +b \right )^{4} \left (a -b \right )^{4}}-\frac {1}{4 \left (a -b \right )^{3} \left (\cos \left (d x +c \right )+1\right )}+\frac {3 b \ln \left (\cos \left (d x +c \right )+1\right )}{4 \left (a -b \right )^{4}}+\frac {1}{4 \left (a +b \right )^{3} \left (\cos \left (d x +c \right )-1\right )}-\frac {3 b \ln \left (\cos \left (d x +c \right )-1\right )}{4 \left (a +b \right )^{4}}}{d}\) | \(176\) |
| default | \(\frac {\frac {b^{4}}{2 \left (a +b \right )^{2} \left (a -b \right )^{2} a \left (b +\cos \left (d x +c \right ) a \right )^{2}}-\frac {4 a \,b^{3}}{\left (a +b \right )^{3} \left (a -b \right )^{3} \left (b +\cos \left (d x +c \right ) a \right )}-\frac {6 a \,b^{2} \left (a^{2}+b^{2}\right ) \ln \left (b +\cos \left (d x +c \right ) a \right )}{\left (a +b \right )^{4} \left (a -b \right )^{4}}-\frac {1}{4 \left (a -b \right )^{3} \left (\cos \left (d x +c \right )+1\right )}+\frac {3 b \ln \left (\cos \left (d x +c \right )+1\right )}{4 \left (a -b \right )^{4}}+\frac {1}{4 \left (a +b \right )^{3} \left (\cos \left (d x +c \right )-1\right )}-\frac {3 b \ln \left (\cos \left (d x +c \right )-1\right )}{4 \left (a +b \right )^{4}}}{d}\) | \(176\) |
| risch | \(-\frac {3 i b x}{2 \left (a^{4}-4 a^{3} b +6 a^{2} b^{2}-4 a \,b^{3}+b^{4}\right )}-\frac {3 i b 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 b x}{2 \left (a^{4}+4 a^{3} b +6 a^{2} b^{2}+4 a \,b^{3}+b^{4}\right )}+\frac {3 i b 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 b^{2} a^{3} x}{a^{8}-4 a^{6} b^{2}+6 a^{4} b^{4}-4 a^{2} b^{6}+b^{8}}+\frac {12 i b^{2} a^{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 {12 i b^{4} a x}{a^{8}-4 a^{6} b^{2}+6 a^{4} b^{4}-4 a^{2} b^{6}+b^{8}}+\frac {12 i b^{4} a 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} b \,{\mathrm e}^{7 i \left (d x +c \right )}-9 a^{3} b^{3} {\mathrm e}^{7 i \left (d x +c \right )}+2 a^{6} {\mathrm e}^{6 i \left (d x +c \right )}-6 a^{4} b^{2} {\mathrm e}^{6 i \left (d x +c \right )}-18 a^{2} b^{4} {\mathrm e}^{6 i \left (d x +c \right )}-2 b^{6} {\mathrm e}^{6 i \left (d x +c \right )}-a^{5} b \,{\mathrm e}^{5 i \left (d x +c \right )}+17 a^{3} b^{3} {\mathrm e}^{5 i \left (d x +c \right )}-4 a \,b^{5} {\mathrm e}^{5 i \left (d x +c \right )}+4 a^{6} {\mathrm e}^{4 i \left (d x +c \right )}-4 a^{4} b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+44 a^{2} b^{4} {\mathrm e}^{4 i \left (d x +c \right )}+4 b^{6} {\mathrm e}^{4 i \left (d x +c \right )}-a^{5} b \,{\mathrm e}^{3 i \left (d x +c \right )}+17 a^{3} b^{3} {\mathrm e}^{3 i \left (d x +c \right )}-4 a \,b^{5} {\mathrm e}^{3 i \left (d x +c \right )}+2 a^{6} {\mathrm e}^{2 i \left (d x +c \right )}-6 a^{4} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-18 a^{2} b^{4} {\mathrm e}^{2 i \left (d x +c \right )}-2 b^{6} {\mathrm e}^{2 i \left (d x +c \right )}-3 a^{5} b \,{\mathrm e}^{i \left (d x +c \right )}-9 a^{3} b^{3} {\mathrm e}^{i \left (d x +c \right )}}{a \left (a^{2}-b^{2}\right )^{3} \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2} \left ({\mathrm e}^{2 i \left (d x +c \right )} a +2 b \,{\mathrm e}^{i \left (d x +c \right )}+a \right )^{2} d}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right ) b}{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 ) b}{2 d \left (a^{4}+4 a^{3} b +6 a^{2} b^{2}+4 a \,b^{3}+b^{4}\right )}-\frac {6 b^{2} a^{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 )}{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^{4} a \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 )}{d \left (a^{8}-4 a^{6} b^{2}+6 a^{4} b^{4}-4 a^{2} b^{6}+b^{8}\right )}\) | \(955\) |
1/d*(1/2*b^4/(a+b)^2/(a-b)^2/a/(b+cos(d*x+c)*a)^2-4*a*b^3/(a+b)^3/(a-b)^3/ (b+cos(d*x+c)*a)-6*a*b^2*(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*b/(a-b)^4*ln(cos(d*x+c)+1)+1/4/(a+b)^3/(cos(d*x +c)-1)-3/4*b/(a+b)^4*ln(cos(d*x+c)-1))
Leaf count of result is larger than twice the leaf count of optimal. 994 vs. \(2 (203) = 406\).
Time = 0.42 (sec) , antiderivative size = 994, normalized size of antiderivative = 4.71 \[ \int \frac {\cos (c+d x)}{(a \sin (c+d x)+b \tan (c+d x))^3} \, dx=\frac {2 \, a^{6} b^{2} + 18 \, a^{4} b^{4} - 18 \, a^{2} b^{6} - 2 \, b^{8} - 6 \, {\left (a^{7} b + 2 \, a^{5} b^{3} - 3 \, a^{3} b^{5}\right )} \cos \left (d x + c\right )^{3} + 2 \, {\left (a^{8} - 4 \, a^{6} b^{2} - 6 \, a^{4} b^{4} + 8 \, a^{2} b^{6} + b^{8}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (2 \, a^{7} b + 9 \, a^{5} b^{3} - 12 \, a^{3} b^{5} + a b^{7}\right )} \cos \left (d x + c\right ) + 24 \, {\left (a^{4} b^{4} + a^{2} b^{6} - {\left (a^{6} b^{2} + a^{4} b^{4}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (a^{5} b^{3} + a^{3} b^{5}\right )} \cos \left (d x + c\right )^{3} + {\left (a^{6} b^{2} - a^{2} b^{6}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{5} b^{3} + a^{3} b^{5}\right )} \cos \left (d x + c\right )\right )} \log \left (a \cos \left (d x + c\right ) + b\right ) - 3 \, {\left (a^{5} b^{3} + 4 \, a^{4} b^{4} + 6 \, a^{3} b^{5} + 4 \, a^{2} b^{6} + a b^{7} - {\left (a^{7} b + 4 \, a^{6} b^{2} + 6 \, a^{5} b^{3} + 4 \, a^{4} b^{4} + a^{3} b^{5}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (a^{6} b^{2} + 4 \, a^{5} b^{3} + 6 \, a^{4} b^{4} + 4 \, a^{3} b^{5} + a^{2} b^{6}\right )} \cos \left (d x + c\right )^{3} + {\left (a^{7} b + 4 \, a^{6} b^{2} + 5 \, a^{5} b^{3} - 5 \, a^{3} b^{5} - 4 \, a^{2} b^{6} - a b^{7}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{6} b^{2} + 4 \, a^{5} b^{3} + 6 \, a^{4} b^{4} + 4 \, a^{3} b^{5} + a^{2} b^{6}\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^{3} - 4 \, a^{4} b^{4} + 6 \, a^{3} b^{5} - 4 \, a^{2} b^{6} + a b^{7} - {\left (a^{7} b - 4 \, a^{6} b^{2} + 6 \, a^{5} b^{3} - 4 \, a^{4} b^{4} + a^{3} b^{5}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (a^{6} b^{2} - 4 \, a^{5} b^{3} + 6 \, a^{4} b^{4} - 4 \, a^{3} b^{5} + a^{2} b^{6}\right )} \cos \left (d x + c\right )^{3} + {\left (a^{7} b - 4 \, a^{6} b^{2} + 5 \, a^{5} b^{3} - 5 \, a^{3} b^{5} + 4 \, a^{2} b^{6} - a b^{7}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{6} b^{2} - 4 \, a^{5} b^{3} + 6 \, a^{4} b^{4} - 4 \, a^{3} b^{5} + a^{2} b^{6}\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^{11} - 4 \, a^{9} b^{2} + 6 \, a^{7} b^{4} - 4 \, a^{5} b^{6} + a^{3} b^{8}\right )} d \cos \left (d x + c\right )^{4} + 2 \, {\left (a^{10} b - 4 \, a^{8} b^{3} + 6 \, a^{6} b^{5} - 4 \, a^{4} b^{7} + a^{2} b^{9}\right )} d \cos \left (d x + c\right )^{3} - {\left (a^{11} - 5 \, a^{9} b^{2} + 10 \, a^{7} b^{4} - 10 \, a^{5} b^{6} + 5 \, a^{3} b^{8} - a b^{10}\right )} d \cos \left (d x + c\right )^{2} - 2 \, {\left (a^{10} b - 4 \, a^{8} b^{3} + 6 \, a^{6} b^{5} - 4 \, a^{4} b^{7} + a^{2} b^{9}\right )} d \cos \left (d x + c\right ) - {\left (a^{9} b^{2} - 4 \, a^{7} b^{4} + 6 \, a^{5} b^{6} - 4 \, a^{3} b^{8} + a b^{10}\right )} d\right )}} \]
1/4*(2*a^6*b^2 + 18*a^4*b^4 - 18*a^2*b^6 - 2*b^8 - 6*(a^7*b + 2*a^5*b^3 - 3*a^3*b^5)*cos(d*x + c)^3 + 2*(a^8 - 4*a^6*b^2 - 6*a^4*b^4 + 8*a^2*b^6 + b ^8)*cos(d*x + c)^2 + 2*(2*a^7*b + 9*a^5*b^3 - 12*a^3*b^5 + a*b^7)*cos(d*x + c) + 24*(a^4*b^4 + a^2*b^6 - (a^6*b^2 + a^4*b^4)*cos(d*x + c)^4 - 2*(a^5 *b^3 + a^3*b^5)*cos(d*x + c)^3 + (a^6*b^2 - a^2*b^6)*cos(d*x + c)^2 + 2*(a ^5*b^3 + a^3*b^5)*cos(d*x + c))*log(a*cos(d*x + c) + b) - 3*(a^5*b^3 + 4*a ^4*b^4 + 6*a^3*b^5 + 4*a^2*b^6 + a*b^7 - (a^7*b + 4*a^6*b^2 + 6*a^5*b^3 + 4*a^4*b^4 + a^3*b^5)*cos(d*x + c)^4 - 2*(a^6*b^2 + 4*a^5*b^3 + 6*a^4*b^4 + 4*a^3*b^5 + a^2*b^6)*cos(d*x + c)^3 + (a^7*b + 4*a^6*b^2 + 5*a^5*b^3 - 5* a^3*b^5 - 4*a^2*b^6 - a*b^7)*cos(d*x + c)^2 + 2*(a^6*b^2 + 4*a^5*b^3 + 6*a ^4*b^4 + 4*a^3*b^5 + a^2*b^6)*cos(d*x + c))*log(1/2*cos(d*x + c) + 1/2) + 3*(a^5*b^3 - 4*a^4*b^4 + 6*a^3*b^5 - 4*a^2*b^6 + a*b^7 - (a^7*b - 4*a^6*b^ 2 + 6*a^5*b^3 - 4*a^4*b^4 + a^3*b^5)*cos(d*x + c)^4 - 2*(a^6*b^2 - 4*a^5*b ^3 + 6*a^4*b^4 - 4*a^3*b^5 + a^2*b^6)*cos(d*x + c)^3 + (a^7*b - 4*a^6*b^2 + 5*a^5*b^3 - 5*a^3*b^5 + 4*a^2*b^6 - a*b^7)*cos(d*x + c)^2 + 2*(a^6*b^2 - 4*a^5*b^3 + 6*a^4*b^4 - 4*a^3*b^5 + a^2*b^6)*cos(d*x + c))*log(-1/2*cos(d *x + c) + 1/2))/((a^11 - 4*a^9*b^2 + 6*a^7*b^4 - 4*a^5*b^6 + a^3*b^8)*d*co s(d*x + c)^4 + 2*(a^10*b - 4*a^8*b^3 + 6*a^6*b^5 - 4*a^4*b^7 + a^2*b^9)*d* cos(d*x + c)^3 - (a^11 - 5*a^9*b^2 + 10*a^7*b^4 - 10*a^5*b^6 + 5*a^3*b^8 - a*b^10)*d*cos(d*x + c)^2 - 2*(a^10*b - 4*a^8*b^3 + 6*a^6*b^5 - 4*a^4*b...
\[ \int \frac {\cos (c+d x)}{(a \sin (c+d x)+b \tan (c+d x))^3} \, dx=\int \frac {\cos {\left (c + d x \right )}}{\left (a \sin {\left (c + d x \right )} + b \tan {\left (c + d x \right )}\right )^{3}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 593 vs. \(2 (203) = 406\).
Time = 0.25 (sec) , antiderivative size = 593, normalized size of antiderivative = 2.81 \[ \int \frac {\cos (c+d x)}{(a \sin (c+d x)+b \tan (c+d x))^3} \, dx=-\frac {\frac {48 \, {\left (a^{3} b^{2} + a b^{4}\right )} \log \left (a + b - \frac {{\left (a - b\right )} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )}{a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}} + \frac {12 \, b \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}} + \frac {a^{6} - 2 \, a^{5} b - a^{4} b^{2} + 4 \, a^{3} b^{3} - a^{2} b^{4} - 2 \, a b^{5} + b^{6} - \frac {2 \, {\left (a^{6} - 4 \, a^{5} b + 5 \, a^{4} b^{2} - 32 \, a^{3} b^{3} - 37 \, a^{2} b^{4} - 4 \, a b^{5} - 9 \, b^{6}\right )} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {{\left (a^{6} - 6 \, a^{5} b + 15 \, a^{4} b^{2} - 84 \, a^{3} b^{3} + 63 \, a^{2} b^{4} - 6 \, a b^{5} + 17 \, b^{6}\right )} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}}{\frac {{\left (a^{9} + a^{8} b - 4 \, a^{7} b^{2} - 4 \, a^{6} b^{3} + 6 \, a^{5} b^{4} + 6 \, a^{4} b^{5} - 4 \, a^{3} b^{6} - 4 \, a^{2} b^{7} + a b^{8} + b^{9}\right )} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {2 \, {\left (a^{9} - a^{8} b - 4 \, a^{7} b^{2} + 4 \, a^{6} b^{3} + 6 \, a^{5} b^{4} - 6 \, a^{4} b^{5} - 4 \, a^{3} b^{6} + 4 \, a^{2} b^{7} + a b^{8} - b^{9}\right )} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {{\left (a^{9} - 3 \, a^{8} b + 8 \, a^{6} b^{3} - 6 \, a^{5} b^{4} - 6 \, a^{4} b^{5} + 8 \, a^{3} b^{6} - 3 \, a b^{8} + b^{9}\right )} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}} + \frac {\sin \left (d x + c\right )^{2}}{{\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{2}}}{8 \, d} \]
-1/8*(48*(a^3*b^2 + a*b^4)*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*b*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*(a^6 - 4*a^5*b + 5*a^4*b^2 - 32*a^3*b^3 - 37*a^2*b^4 - 4*a*b^5 - 9*b^6)*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + (a^6 - 6*a^5*b + 15*a^4*b^2 - 84*a^3*b^3 + 63 *a^2*b^4 - 6*a*b^5 + 17*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. 690 vs. \(2 (203) = 406\).
Time = 0.87 (sec) , antiderivative size = 690, normalized size of antiderivative = 3.27 \[ \int \frac {\cos (c+d x)}{(a \sin (c+d x)+b \tan (c+d x))^3} \, dx=-\frac {\frac {6 \, b \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^{3} b^{2} + a b^{4}\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 \, b {\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 (9 \, a^{5} b^{2} + 10 \, a^{4} b^{3} + 2 \, a^{3} b^{4} + 8 \, a^{2} b^{5} + 5 \, a b^{6} - 2 \, b^{7} + \frac {18 \, a^{5} b^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {8 \, a^{4} b^{3} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {2 \, a^{3} b^{4} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {6 \, a^{2} b^{5} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {16 \, a b^{6} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {2 \, b^{7} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {9 \, 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 {18 \, a^{2} b^{5} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {9 \, a b^{6} {\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*b*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^3*b^2 + a*b^4)*log(abs(-a - b - a*(c os(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*b*(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*(9*a^5*b^2 + 10*a^4*b^3 + 2*a^3*b^4 + 8*a^2*b^5 + 5*a*b^6 - 2*b^7 + 18*a^5*b^2*(cos(d*x + c) - 1)/(c os(d*x + c) + 1) - 8*a^4*b^3*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 2*a^3 *b^4*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 6*a^2*b^5*(cos(d*x + c) - 1)/ (cos(d*x + c) + 1) - 16*a*b^6*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 2*b^ 7*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 9*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 - 18*a^2*b^5*(cos( d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 + 9*a*b^6*(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 = 22.77 (sec) , antiderivative size = 490, normalized size of antiderivative = 2.32 \[ \int \frac {\cos (c+d x)}{(a \sin (c+d x)+b \tan (c+d x))^3} \, dx=\frac {\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (a^5-5\,a^4\,b+10\,a^3\,b^2-42\,a^2\,b^3+5\,a\,b^4-9\,b^5\right )}{\left (a-b\right )\,\left (a^2+2\,a\,b+b^2\right )}-\frac {a^3-3\,a^2\,b+3\,a\,b^2-b^3}{2\,\left (a+b\right )}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (-a^5+5\,a^4\,b-10\,a^3\,b^2+74\,a^2\,b^3+11\,a\,b^4+17\,b^5\right )}{2\,\left (a+b\right )\,\left (a^2+2\,a\,b+b^2\right )}}{d\,\left (\left (4\,a^5-20\,a^4\,b+40\,a^3\,b^2-40\,a^2\,b^3+20\,a\,b^4-4\,b^5\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+\left (-8\,a^5+24\,a^4\,b-16\,a^3\,b^2-16\,a^2\,b^3+24\,a\,b^4-8\,b^5\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\left (4\,a^5-4\,a^4\,b-8\,a^3\,b^2+8\,a^2\,b^3+4\,a\,b^4-4\,b^5\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,d\,{\left (a-b\right )}^3}-\frac {\ln \left (a+b-a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )\,\left (6\,a^3\,b^2+6\,a\,b^4\right )}{d\,\left (a^8-4\,a^6\,b^2+6\,a^4\,b^4-4\,a^2\,b^6+b^8\right )}-\frac {3\,b\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d\,\left (2\,a^4+8\,a^3\,b+12\,a^2\,b^2+8\,a\,b^3+2\,b^4\right )} \]
((tan(c/2 + (d*x)/2)^2*(5*a*b^4 - 5*a^4*b + a^5 - 9*b^5 - 42*a^2*b^3 + 10* a^3*b^2))/((a - b)*(2*a*b + a^2 + b^2)) - (3*a*b^2 - 3*a^2*b + a^3 - b^3)/ (2*(a + b)) + (tan(c/2 + (d*x)/2)^4*(11*a*b^4 + 5*a^4*b - a^5 + 17*b^5 + 7 4*a^2*b^3 - 10*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))) - tan(c/2 + (d*x)/2)^2/(8*d*(a - b)^3) - (log(a + b - a*tan(c/2 + (d*x)/2)^2 + b*tan(c/2 + (d*x)/2)^2)*(6*a*b^4 + 6*a^3*b^2)) /(d*(a^8 + b^8 - 4*a^2*b^6 + 6*a^4*b^4 - 4*a^6*b^2)) - (3*b*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))