Integrand size = 28, antiderivative size = 248 \[ \int \frac {\cos ^3(c+d x)}{(a \sin (c+d x)+b \tan (c+d x))^3} \, dx=\frac {b^6}{2 a^3 \left (a^2-b^2\right )^2 d (b+a \cos (c+d x))^2}-\frac {2 b^5 \left (3 a^2-b^2\right )}{a^3 \left (a^2-b^2\right )^3 d (b+a \cos (c+d x))}-\frac {\left (a \left (a^2+3 b^2\right )-b \left (3 a^2+b^2\right ) \cos (c+d x)\right ) \csc ^2(c+d x)}{2 \left (a^2-b^2\right )^3 d}-\frac {(2 a+5 b) \log (1-\cos (c+d x))}{4 (a+b)^4 d}-\frac {(2 a-5 b) \log (1+\cos (c+d x))}{4 (a-b)^4 d}-\frac {b^4 \left (15 a^4-4 a^2 b^2+b^4\right ) \log (b+a \cos (c+d x))}{a^3 \left (a^2-b^2\right )^4 d} \]
1/2*b^6/a^3/(a^2-b^2)^2/d/(b+a*cos(d*x+c))^2-2*b^5*(3*a^2-b^2)/a^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-1/4*(2*a+5*b)*ln(1-cos(d*x+c))/(a+b)^4/d-1/4*(2*a-5* b)*ln(1+cos(d*x+c))/(a-b)^4/d-b^4*(15*a^4-4*a^2*b^2+b^4)*ln(b+a*cos(d*x+c) )/a^3/(a^2-b^2)^4/d
Result contains complex when optimal does not.
Time = 7.14 (sec) , antiderivative size = 713, normalized size of antiderivative = 2.88 \[ \int \frac {\cos ^3(c+d x)}{(a \sin (c+d x)+b \tan (c+d x))^3} \, dx=\frac {b^6 (b+a \cos (c+d x)) \tan ^3(c+d x)}{2 a^3 (-a+b)^2 (a+b)^2 d (a \sin (c+d x)+b \tan (c+d x))^3}-\frac {2 b^5 \left (-3 a^2+b^2\right ) (b+a \cos (c+d x))^2 \tan ^3(c+d x)}{a^3 (-a+b)^3 (a+b)^3 d (a \sin (c+d x)+b \tan (c+d x))^3}-\frac {2 i \left (a^5-4 a^3 b^2-9 a b^4\right ) (c+d x) (b+a \cos (c+d x))^3 \tan ^3(c+d x)}{(a-b)^4 (a+b)^4 d (a \sin (c+d x)+b \tan (c+d x))^3}-\frac {i (-2 a-5 b) \arctan (\tan (c+d x)) (b+a \cos (c+d x))^3 \tan ^3(c+d x)}{2 (a+b)^4 d (a \sin (c+d x)+b \tan (c+d x))^3}-\frac {i (-2 a+5 b) \arctan (\tan (c+d x)) (b+a \cos (c+d x))^3 \tan ^3(c+d x)}{2 (-a+b)^4 d (a \sin (c+d x)+b \tan (c+d x))^3}-\frac {(b+a \cos (c+d x))^3 \csc ^2\left (\frac {1}{2} (c+d x)\right ) \tan ^3(c+d x)}{8 (a+b)^3 d (a \sin (c+d x)+b \tan (c+d x))^3}+\frac {(-2 a+5 b) (b+a \cos (c+d x))^3 \log \left (\cos ^2\left (\frac {1}{2} (c+d x)\right )\right ) \tan ^3(c+d x)}{4 (-a+b)^4 d (a \sin (c+d x)+b \tan (c+d x))^3}+\frac {\left (-15 a^4 b^4+4 a^2 b^6-b^8\right ) (b+a \cos (c+d x))^3 \log (b+a \cos (c+d x)) \tan ^3(c+d x)}{a^3 \left (-a^2+b^2\right )^4 d (a \sin (c+d x)+b \tan (c+d x))^3}+\frac {(-2 a-5 b) (b+a \cos (c+d x))^3 \log \left (\sin ^2\left (\frac {1}{2} (c+d x)\right )\right ) \tan ^3(c+d x)}{4 (a+b)^4 d (a \sin (c+d x)+b \tan (c+d x))^3}+\frac {(b+a \cos (c+d x))^3 \sec ^2\left (\frac {1}{2} (c+d x)\right ) \tan ^3(c+d x)}{8 (-a+b)^3 d (a \sin (c+d x)+b \tan (c+d x))^3} \]
(b^6*(b + a*Cos[c + d*x])*Tan[c + d*x]^3)/(2*a^3*(-a + b)^2*(a + b)^2*d*(a *Sin[c + d*x] + b*Tan[c + d*x])^3) - (2*b^5*(-3*a^2 + b^2)*(b + a*Cos[c + d*x])^2*Tan[c + d*x]^3)/(a^3*(-a + b)^3*(a + b)^3*d*(a*Sin[c + d*x] + b*Ta n[c + d*x])^3) - ((2*I)*(a^5 - 4*a^3*b^2 - 9*a*b^4)*(c + d*x)*(b + a*Cos[c + d*x])^3*Tan[c + d*x]^3)/((a - b)^4*(a + b)^4*d*(a*Sin[c + d*x] + b*Tan[ c + d*x])^3) - ((I/2)*(-2*a - 5*b)*ArcTan[Tan[c + d*x]]*(b + a*Cos[c + d*x ])^3*Tan[c + d*x]^3)/((a + b)^4*d*(a*Sin[c + d*x] + b*Tan[c + d*x])^3) - ( (I/2)*(-2*a + 5*b)*ArcTan[Tan[c + d*x]]*(b + a*Cos[c + d*x])^3*Tan[c + d*x ]^3)/((-a + b)^4*d*(a*Sin[c + d*x] + b*Tan[c + d*x])^3) - ((b + a*Cos[c + d*x])^3*Csc[(c + d*x)/2]^2*Tan[c + d*x]^3)/(8*(a + b)^3*d*(a*Sin[c + d*x] + b*Tan[c + d*x])^3) + ((-2*a + 5*b)*(b + a*Cos[c + d*x])^3*Log[Cos[(c + d *x)/2]^2]*Tan[c + d*x]^3)/(4*(-a + b)^4*d*(a*Sin[c + d*x] + b*Tan[c + d*x] )^3) + ((-15*a^4*b^4 + 4*a^2*b^6 - b^8)*(b + a*Cos[c + d*x])^3*Log[b + a*C os[c + d*x]]*Tan[c + d*x]^3)/(a^3*(-a^2 + b^2)^4*d*(a*Sin[c + d*x] + b*Tan [c + d*x])^3) + ((-2*a - 5*b)*(b + a*Cos[c + d*x])^3*Log[Sin[(c + d*x)/2]^ 2]*Tan[c + d*x]^3)/(4*(a + b)^4*d*(a*Sin[c + d*x] + b*Tan[c + d*x])^3) + ( (b + a*Cos[c + d*x])^3*Sec[(c + d*x)/2]^2*Tan[c + d*x]^3)/(8*(-a + b)^3*d* (a*Sin[c + d*x] + b*Tan[c + d*x])^3)
Time = 1.19 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.09, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, 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.
| \(\displaystyle \int \frac {\cos ^3(c+d x)}{(a \sin (c+d x)+b \tan (c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^3}{(a \sin (c+d x)+b \tan (c+d x))^3}dx\) |
| \(\Big \downarrow \) 4897 |
| \(\displaystyle \int \frac {\cos ^3(c+d x) \cot ^3(c+d x)}{(a \cos (c+d x)+b)^3}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin \left (c+d x-\frac {\pi }{2}\right )^6}{\cos \left (c+d x-\frac {\pi }{2}\right )^3 \left (b-a \sin \left (c+d x-\frac {\pi }{2}\right )\right )^3}dx\) |
| \(\Big \downarrow \) 3316 |
| \(\displaystyle -\frac {a^3 \int \frac {\cos ^6(c+d x)}{(b+a \cos (c+d x))^3 \left (a^2-a^2 \cos ^2(c+d x)\right )^2}d(a \cos (c+d x))}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {a^6 \cos ^6(c+d x)}{(b+a \cos (c+d x))^3 \left (a^2-a^2 \cos ^2(c+d x)\right )^2}d(a \cos (c+d x))}{a^3 d}\) |
| \(\Big \downarrow \) 601 |
| \(\displaystyle -\frac {\frac {a^4 \left (a^2 \left (a^2+3 b^2\right )-a b \left (3 a^2+b^2\right ) \cos (c+d x)\right )}{2 \left (a^2-b^2\right )^3 \left (a^2-a^2 \cos ^2(c+d x)\right )}-\frac {\int -\frac {-\frac {b \left (3 a^2+b^2\right ) \cos ^3(c+d x) a^9}{\left (a^2-b^2\right )^3}+\frac {b^3 \left (7 a^2-3 b^2\right ) \cos (c+d x) a^7}{\left (a^2-b^2\right )^3}-2 \cos ^4(c+d x) a^6+\frac {b^2 \left (3 a^4-9 b^2 a^2+2 b^4\right ) \cos ^2(c+d x) a^6}{\left (a^2-b^2\right )^3}+\frac {b^4 \left (3 a^2+b^2\right ) a^6}{\left (a^2-b^2\right )^3}}{(b+a \cos (c+d x))^3 \left (a^2-a^2 \cos ^2(c+d x)\right )}d(a \cos (c+d x))}{2 a^2}}{a^3 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {\int \frac {-\frac {b \left (3 a^2+b^2\right ) \cos ^3(c+d x) a^9}{\left (a^2-b^2\right )^3}+\frac {b^3 \left (7 a^2-3 b^2\right ) \cos (c+d x) a^7}{\left (a^2-b^2\right )^3}-2 \cos ^4(c+d x) a^6+\frac {b^2 \left (3 a^4-9 b^2 a^2+2 b^4\right ) \cos ^2(c+d x) a^6}{\left (a^2-b^2\right )^3}+\frac {b^4 \left (3 a^2+b^2\right ) a^6}{\left (a^2-b^2\right )^3}}{(b+a \cos (c+d x))^3 \left (a^2-a^2 \cos ^2(c+d x)\right )}d(a \cos (c+d x))}{2 a^2}+\frac {a^4 \left (a^2 \left (a^2+3 b^2\right )-a b \left (3 a^2+b^2\right ) \cos (c+d x)\right )}{2 \left (a^2-b^2\right )^3 \left (a^2-a^2 \cos ^2(c+d x)\right )}}{a^3 d}\) |
| \(\Big \downarrow \) 2160 |
| \(\displaystyle -\frac {\frac {\int \left (\frac {2 a^2 b^6}{(a-b)^2 (a+b)^2 (b+a \cos (c+d x))^3}-\frac {4 a^2 \left (3 a^2-b^2\right ) b^5}{(a-b)^3 (a+b)^3 (b+a \cos (c+d x))^2}+\frac {2 a^2 \left (15 a^4-4 b^2 a^2+b^4\right ) b^4}{(a-b)^4 (a+b)^4 (b+a \cos (c+d x))}-\frac {a^5 (2 a+5 b)}{2 (a+b)^4 (a-a \cos (c+d x))}+\frac {a^5 (2 a-5 b)}{2 (a-b)^4 (\cos (c+d x) a+a)}\right )d(a \cos (c+d x))}{2 a^2}+\frac {a^4 \left (a^2 \left (a^2+3 b^2\right )-a b \left (3 a^2+b^2\right ) \cos (c+d x)\right )}{2 \left (a^2-b^2\right )^3 \left (a^2-a^2 \cos ^2(c+d x)\right )}}{a^3 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\frac {a^4 \left (a^2 \left (a^2+3 b^2\right )-a b \left (3 a^2+b^2\right ) \cos (c+d x)\right )}{2 \left (a^2-b^2\right )^3 \left (a^2-a^2 \cos ^2(c+d x)\right )}+\frac {\frac {a^5 (2 a+5 b) \log (a-a \cos (c+d x))}{2 (a+b)^4}+\frac {a^5 (2 a-5 b) \log (a \cos (c+d x)+a)}{2 (a-b)^4}-\frac {a^2 b^6}{\left (a^2-b^2\right )^2 (a \cos (c+d x)+b)^2}+\frac {4 a^2 b^5 \left (3 a^2-b^2\right )}{\left (a^2-b^2\right )^3 (a \cos (c+d x)+b)}+\frac {2 a^2 b^4 \left (15 a^4-4 a^2 b^2+b^4\right ) \log (a \cos (c+d x)+b)}{\left (a^2-b^2\right )^4}}{2 a^2}}{a^3 d}\) |
-(((a^4*(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^6)/((a^2 - b^2)^2*(b + a*Cos[ c + d*x])^2)) + (4*a^2*b^5*(3*a^2 - b^2))/((a^2 - b^2)^3*(b + a*Cos[c + d* x])) + (a^5*(2*a + 5*b)*Log[a - a*Cos[c + d*x]])/(2*(a + b)^4) + (a^5*(2*a - 5*b)*Log[a + a*Cos[c + d*x]])/(2*(a - b)^4) + (2*a^2*b^4*(15*a^4 - 4*a^ 2*b^2 + b^4)*Log[b + a*Cos[c + d*x]])/(a^2 - b^2)^4)/(2*a^2))/(a^3*d))
3.3.64.3.1 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 = 5.24 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.86
| method | result | size |
| derivativedivides | \(\frac {\frac {1}{4 \left (a +b \right )^{3} \left (\cos \left (d x +c \right )-1\right )}+\frac {\left (-2 a -5 b \right ) \ln \left (\cos \left (d x +c \right )-1\right )}{4 \left (a +b \right )^{4}}+\frac {b^{6}}{2 a^{3} \left (a +b \right )^{2} \left (a -b \right )^{2} \left (b +\cos \left (d x +c \right ) a \right )^{2}}-\frac {2 b^{5} \left (3 a^{2}-b^{2}\right )}{a^{3} \left (a +b \right )^{3} \left (a -b \right )^{3} \left (b +\cos \left (d x +c \right ) a \right )}-\frac {b^{4} \left (15 a^{4}-4 a^{2} b^{2}+b^{4}\right ) \ln \left (b +\cos \left (d x +c \right ) a \right )}{\left (a +b \right )^{4} \left (a -b \right )^{4} a^{3}}-\frac {1}{4 \left (a -b \right )^{3} \left (\cos \left (d x +c \right )+1\right )}+\frac {\left (-2 a +5 b \right ) \ln \left (\cos \left (d x +c \right )+1\right )}{4 \left (a -b \right )^{4}}}{d}\) | \(213\) |
| default | \(\frac {\frac {1}{4 \left (a +b \right )^{3} \left (\cos \left (d x +c \right )-1\right )}+\frac {\left (-2 a -5 b \right ) \ln \left (\cos \left (d x +c \right )-1\right )}{4 \left (a +b \right )^{4}}+\frac {b^{6}}{2 a^{3} \left (a +b \right )^{2} \left (a -b \right )^{2} \left (b +\cos \left (d x +c \right ) a \right )^{2}}-\frac {2 b^{5} \left (3 a^{2}-b^{2}\right )}{a^{3} \left (a +b \right )^{3} \left (a -b \right )^{3} \left (b +\cos \left (d x +c \right ) a \right )}-\frac {b^{4} \left (15 a^{4}-4 a^{2} b^{2}+b^{4}\right ) \ln \left (b +\cos \left (d x +c \right ) a \right )}{\left (a +b \right )^{4} \left (a -b \right )^{4} a^{3}}-\frac {1}{4 \left (a -b \right )^{3} \left (\cos \left (d x +c \right )+1\right )}+\frac {\left (-2 a +5 b \right ) \ln \left (\cos \left (d x +c \right )+1\right )}{4 \left (a -b \right )^{4}}}{d}\) | \(213\) |
| risch | \(\text {Expression too large to display}\) | \(1524\) |
1/d*(1/4/(a+b)^3/(cos(d*x+c)-1)+1/4/(a+b)^4*(-2*a-5*b)*ln(cos(d*x+c)-1)+1/ 2*b^6/a^3/(a+b)^2/(a-b)^2/(b+cos(d*x+c)*a)^2-2/a^3*b^5*(3*a^2-b^2)/(a+b)^3 /(a-b)^3/(b+cos(d*x+c)*a)-b^4*(15*a^4-4*a^2*b^2+b^4)/(a+b)^4/(a-b)^4/a^3*l n(b+cos(d*x+c)*a)-1/4/(a-b)^3/(cos(d*x+c)+1)+1/4/(a-b)^4*(-2*a+5*b)*ln(cos (d*x+c)+1))
Leaf count of result is larger than twice the leaf count of optimal. 1180 vs. \(2 (240) = 480\).
Time = 0.60 (sec) , antiderivative size = 1180, normalized size of antiderivative = 4.76 \[ \int \frac {\cos ^3(c+d x)}{(a \sin (c+d x)+b \tan (c+d x))^3} \, dx=\text {Too large to display} \]
1/4*(2*a^8*b^2 + 4*a^6*b^4 + 16*a^4*b^6 - 28*a^2*b^8 + 6*b^10 - 2*(3*a^9*b - 2*a^7*b^3 + 11*a^5*b^5 - 16*a^3*b^7 + 4*a*b^9)*cos(d*x + c)^3 + 2*(a^10 - 4*a^8*b^2 + a^6*b^4 - 9*a^4*b^6 + 14*a^2*b^8 - 3*b^10)*cos(d*x + c)^2 + 2*(2*a^9*b + a^7*b^3 + 8*a^5*b^5 - 15*a^3*b^7 + 4*a*b^9)*cos(d*x + c) + 4 *(15*a^4*b^6 - 4*a^2*b^8 + b^10 - (15*a^6*b^4 - 4*a^4*b^6 + a^2*b^8)*cos(d *x + c)^4 - 2*(15*a^5*b^5 - 4*a^3*b^7 + a*b^9)*cos(d*x + c)^3 + (15*a^6*b^ 4 - 19*a^4*b^6 + 5*a^2*b^8 - b^10)*cos(d*x + c)^2 + 2*(15*a^5*b^5 - 4*a^3* b^7 + a*b^9)*cos(d*x + c))*log(a*cos(d*x + c) + b) + (2*a^8*b^2 + 3*a^7*b^ 3 - 8*a^6*b^4 - 22*a^5*b^5 - 18*a^4*b^6 - 5*a^3*b^7 - (2*a^10 + 3*a^9*b - 8*a^8*b^2 - 22*a^7*b^3 - 18*a^6*b^4 - 5*a^5*b^5)*cos(d*x + c)^4 - 2*(2*a^9 *b + 3*a^8*b^2 - 8*a^7*b^3 - 22*a^6*b^4 - 18*a^5*b^5 - 5*a^4*b^6)*cos(d*x + c)^3 + (2*a^10 + 3*a^9*b - 10*a^8*b^2 - 25*a^7*b^3 - 10*a^6*b^4 + 17*a^5 *b^5 + 18*a^4*b^6 + 5*a^3*b^7)*cos(d*x + c)^2 + 2*(2*a^9*b + 3*a^8*b^2 - 8 *a^7*b^3 - 22*a^6*b^4 - 18*a^5*b^5 - 5*a^4*b^6)*cos(d*x + c))*log(1/2*cos( d*x + c) + 1/2) + (2*a^8*b^2 - 3*a^7*b^3 - 8*a^6*b^4 + 22*a^5*b^5 - 18*a^4 *b^6 + 5*a^3*b^7 - (2*a^10 - 3*a^9*b - 8*a^8*b^2 + 22*a^7*b^3 - 18*a^6*b^4 + 5*a^5*b^5)*cos(d*x + c)^4 - 2*(2*a^9*b - 3*a^8*b^2 - 8*a^7*b^3 + 22*a^6 *b^4 - 18*a^5*b^5 + 5*a^4*b^6)*cos(d*x + c)^3 + (2*a^10 - 3*a^9*b - 10*a^8 *b^2 + 25*a^7*b^3 - 10*a^6*b^4 - 17*a^5*b^5 + 18*a^4*b^6 - 5*a^3*b^7)*cos( d*x + c)^2 + 2*(2*a^9*b - 3*a^8*b^2 - 8*a^7*b^3 + 22*a^6*b^4 - 18*a^5*b...
Timed out. \[ \int \frac {\cos ^3(c+d x)}{(a \sin (c+d x)+b \tan (c+d x))^3} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 684 vs. \(2 (240) = 480\).
Time = 0.34 (sec) , antiderivative size = 684, normalized size of antiderivative = 2.76 \[ \int \frac {\cos ^3(c+d x)}{(a \sin (c+d x)+b \tan (c+d x))^3} \, dx=-\frac {\frac {8 \, {\left (15 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}\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^{11} - 4 \, a^{9} b^{2} + 6 \, a^{7} b^{4} - 4 \, a^{5} b^{6} + a^{3} b^{8}} + \frac {4 \, {\left (2 \, a + 5 \, b\right )} \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^{8} - 2 \, a^{7} b - a^{6} b^{2} + 4 \, a^{5} b^{3} - a^{4} b^{4} - 2 \, a^{3} b^{5} + a^{2} b^{6} - \frac {2 \, {\left (a^{8} - 4 \, a^{7} b + 5 \, a^{6} b^{2} - 5 \, a^{4} b^{4} - 44 \, a^{3} b^{5} - 49 \, a^{2} b^{6} + 8 \, a b^{7} + 8 \, b^{8}\right )} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {{\left (a^{8} - 6 \, a^{7} b + 15 \, a^{6} b^{2} - 20 \, a^{5} b^{3} + 15 \, a^{4} b^{4} - 102 \, a^{3} b^{5} + 81 \, a^{2} b^{6} + 32 \, a b^{7} - 16 \, b^{8}\right )} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}}{\frac {{\left (a^{11} + a^{10} b - 4 \, a^{9} b^{2} - 4 \, a^{8} b^{3} + 6 \, a^{7} b^{4} + 6 \, a^{6} b^{5} - 4 \, a^{5} b^{6} - 4 \, a^{4} b^{7} + a^{3} b^{8} + a^{2} b^{9}\right )} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {2 \, {\left (a^{11} - a^{10} b - 4 \, a^{9} b^{2} + 4 \, a^{8} b^{3} + 6 \, a^{7} b^{4} - 6 \, a^{6} b^{5} - 4 \, a^{5} b^{6} + 4 \, a^{4} b^{7} + a^{3} b^{8} - a^{2} b^{9}\right )} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {{\left (a^{11} - 3 \, a^{10} b + 8 \, a^{8} b^{3} - 6 \, a^{7} b^{4} - 6 \, a^{6} b^{5} + 8 \, a^{5} b^{6} - 3 \, a^{3} b^{8} + a^{2} 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}} - \frac {8 \, \log \left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}{a^{3}}}{8 \, d} \]
-1/8*(8*(15*a^4*b^4 - 4*a^2*b^6 + b^8)*log(a + b - (a - b)*sin(d*x + c)^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) + 4*(2*a + 5*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^8 - 2*a^7*b - a^6*b^2 + 4*a^5*b^3 - a^4*b^4 - 2*a^3*b^5 + a^2*b^6 - 2*(a^8 - 4*a^7*b + 5*a^6*b^2 - 5*a^4*b^4 - 44*a^3*b ^5 - 49*a^2*b^6 + 8*a*b^7 + 8*b^8)*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + ( a^8 - 6*a^7*b + 15*a^6*b^2 - 20*a^5*b^3 + 15*a^4*b^4 - 102*a^3*b^5 + 81*a^ 2*b^6 + 32*a*b^7 - 16*b^8)*sin(d*x + c)^4/(cos(d*x + c) + 1)^4)/((a^11 + a ^10*b - 4*a^9*b^2 - 4*a^8*b^3 + 6*a^7*b^4 + 6*a^6*b^5 - 4*a^5*b^6 - 4*a^4* b^7 + a^3*b^8 + a^2*b^9)*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - 2*(a^11 - a ^10*b - 4*a^9*b^2 + 4*a^8*b^3 + 6*a^7*b^4 - 6*a^6*b^5 - 4*a^5*b^6 + 4*a^4* b^7 + a^3*b^8 - a^2*b^9)*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + (a^11 - 3*a ^10*b + 8*a^8*b^3 - 6*a^7*b^4 - 6*a^6*b^5 + 8*a^5*b^6 - 3*a^3*b^8 + a^2*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) - 8*log(sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 1)/a^3)/d
Leaf count of result is larger than twice the leaf count of optimal. 848 vs. \(2 (240) = 480\).
Time = 0.95 (sec) , antiderivative size = 848, normalized size of antiderivative = 3.42 \[ \int \frac {\cos ^3(c+d x)}{(a \sin (c+d x)+b \tan (c+d x))^3} \, dx=\text {Too large to display} \]
-1/8*(2*(2*a + 5*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) + 8*(15*a^4*b^4 - 4*a^2*b^6 + b^8) *log(abs(-a - b - a*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + b*(cos(d*x + c ) - 1)/(cos(d*x + c) + 1)))/(a^11 - 4*a^9*b^2 + 6*a^7*b^4 - 4*a^5*b^6 + a^ 3*b^8) - (a + b + 4*a*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 10*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)) - 4*(45*a^6*b^4 + 66*a^5*b^5 - 1 5*a^4*b^6 - 44*a^3*b^7 - a^2*b^8 + 10*a*b^9 + 3*b^10 + 90*a^6*b^4*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 24*a^5*b^5*(cos(d*x + c) - 1)/(cos(d*x + c ) + 1) - 118*a^4*b^6*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 28*a^3*b^7*(c os(d*x + c) - 1)/(cos(d*x + c) + 1) + 34*a^2*b^8*(cos(d*x + c) - 1)/(cos(d *x + c) + 1) - 4*a*b^9*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 6*b^10*(cos (d*x + c) - 1)/(cos(d*x + c) + 1) + 45*a^6*b^4*(cos(d*x + c) - 1)^2/(cos(d *x + c) + 1)^2 - 90*a^5*b^5*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 + 33 *a^4*b^6*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 + 24*a^3*b^7*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - 9*a^2*b^8*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - 6*a*b^9*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 + 3*b^10*( cos(d*x + c) - 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)*(a + b + a*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) ...
Time = 25.34 (sec) , antiderivative size = 527, normalized size of antiderivative = 2.12 \[ \int \frac {\cos ^3(c+d x)}{(a \sin (c+d x)+b \tan (c+d x))^3} \, dx=\frac {\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (-a^7+5\,a^6\,b-10\,a^5\,b^2+10\,a^4\,b^3-5\,a^3\,b^4+97\,a^2\,b^5+16\,a\,b^6-16\,b^7\right )}{2\,a^2\,\left (a+b\right )\,\left (a^2+2\,a\,b+b^2\right )}-\frac {a^3-3\,a^2\,b+3\,a\,b^2-b^3}{2\,\left (a+b\right )}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (a^7-5\,a^6\,b+10\,a^5\,b^2-10\,a^4\,b^3+5\,a^3\,b^4-49\,a^2\,b^5+8\,b^7\right )}{a^2\,{\left (a+b\right )}^2\,\left (a-b\right )}}{d\,\left (\left (4\,a^5-20\,a^4\,b+40\,a^3\,b^2-40\,a^2\,b^3+20\,a\,b^4-4\,b^5\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+\left (-8\,a^5+24\,a^4\,b-16\,a^3\,b^2-16\,a^2\,b^3+24\,a\,b^4-8\,b^5\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\left (4\,a^5-4\,a^4\,b-8\,a^3\,b^2+8\,a^2\,b^3+4\,a\,b^4-4\,b^5\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,d\,{\left (a-b\right )}^3}+\frac {\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{a^3\,d}-\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (2\,a+5\,b\right )}{d\,\left (2\,a^4+8\,a^3\,b+12\,a^2\,b^2+8\,a\,b^3+2\,b^4\right )}-\frac {b^4\,\ln \left (a+b-a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )\,\left (15\,a^4-4\,a^2\,b^2+b^4\right )}{a^3\,d\,{\left (a^2-b^2\right )}^4} \]
((tan(c/2 + (d*x)/2)^4*(16*a*b^6 + 5*a^6*b - a^7 - 16*b^7 + 97*a^2*b^5 - 5 *a^3*b^4 + 10*a^4*b^3 - 10*a^5*b^2))/(2*a^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)^2*(a^7 - 5*a^6*b + 8*b^7 - 49*a^2*b^5 + 5*a^3*b^4 - 10*a^4*b^3 + 10*a^5*b^2))/(a^ 2*(a + b)^2*(a - b)))/(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(tan(c/2 + (d*x)/2)^2 + 1)/(a^3*d) - (log(t an(c/2 + (d*x)/2))*(2*a + 5*b))/(d*(8*a*b^3 + 8*a^3*b + 2*a^4 + 2*b^4 + 12 *a^2*b^2)) - (b^4*log(a + b - a*tan(c/2 + (d*x)/2)^2 + b*tan(c/2 + (d*x)/2 )^2)*(15*a^4 + b^4 - 4*a^2*b^2))/(a^3*d*(a^2 - b^2)^4)