Integrand size = 28, antiderivative size = 227 \[ \int \frac {\cos ^5(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx=\frac {a b^4 x}{\left (a^2+b^2\right )^3}+\frac {a b^2 x}{2 \left (a^2+b^2\right )^2}+\frac {3 a x}{8 \left (a^2+b^2\right )}+\frac {b^3 \cos ^2(c+d x)}{2 \left (a^2+b^2\right )^2 d}+\frac {b \cos ^4(c+d x)}{4 \left (a^2+b^2\right ) d}+\frac {b^5 \log (a \cos (c+d x)+b \sin (c+d x))}{\left (a^2+b^2\right )^3 d}+\frac {a b^2 \cos (c+d x) \sin (c+d x)}{2 \left (a^2+b^2\right )^2 d}+\frac {3 a \cos (c+d x) \sin (c+d x)}{8 \left (a^2+b^2\right ) d}+\frac {a \cos ^3(c+d x) \sin (c+d x)}{4 \left (a^2+b^2\right ) d} \]
a*b^4*x/(a^2+b^2)^3+1/2*a*b^2*x/(a^2+b^2)^2+3/8*a*x/(a^2+b^2)+1/2*b^3*cos( d*x+c)^2/(a^2+b^2)^2/d+1/4*b*cos(d*x+c)^4/(a^2+b^2)/d+b^5*ln(a*cos(d*x+c)+ b*sin(d*x+c))/(a^2+b^2)^3/d+1/2*a*b^2*cos(d*x+c)*sin(d*x+c)/(a^2+b^2)^2/d+ 3/8*a*cos(d*x+c)*sin(d*x+c)/(a^2+b^2)/d+1/4*a*cos(d*x+c)^3*sin(d*x+c)/(a^2 +b^2)/d
Time = 0.88 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.96 \[ \int \frac {\cos ^5(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx=\frac {12 a^5 c+40 a^3 b^2 c+60 a b^4 c+12 a^5 d x+40 a^3 b^2 d x+60 a b^4 d x+4 b \left (a^4+4 a^2 b^2+3 b^4\right ) \cos (2 (c+d x))+b \left (a^2+b^2\right )^2 \cos (4 (c+d x))+32 b^5 \log (a \cos (c+d x)+b \sin (c+d x))+8 a^5 \sin (2 (c+d x))+24 a^3 b^2 \sin (2 (c+d x))+16 a b^4 \sin (2 (c+d x))+a^5 \sin (4 (c+d x))+2 a^3 b^2 \sin (4 (c+d x))+a b^4 \sin (4 (c+d x))}{32 \left (a^2+b^2\right )^3 d} \]
(12*a^5*c + 40*a^3*b^2*c + 60*a*b^4*c + 12*a^5*d*x + 40*a^3*b^2*d*x + 60*a *b^4*d*x + 4*b*(a^4 + 4*a^2*b^2 + 3*b^4)*Cos[2*(c + d*x)] + b*(a^2 + b^2)^ 2*Cos[4*(c + d*x)] + 32*b^5*Log[a*Cos[c + d*x] + b*Sin[c + d*x]] + 8*a^5*S in[2*(c + d*x)] + 24*a^3*b^2*Sin[2*(c + d*x)] + 16*a*b^4*Sin[2*(c + d*x)] + a^5*Sin[4*(c + d*x)] + 2*a^3*b^2*Sin[4*(c + d*x)] + a*b^4*Sin[4*(c + d*x )])/(32*(a^2 + b^2)^3*d)
Time = 1.04 (sec) , antiderivative size = 221, normalized size of antiderivative = 0.97, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3042, 3579, 3042, 3115, 3042, 3115, 24, 3579, 3042, 3115, 24, 3577, 3042, 3612}
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.
| \(\displaystyle \int \frac {\cos ^5(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^5}{a \cos (c+d x)+b \sin (c+d x)}dx\) |
| \(\Big \downarrow \) 3579 |
| \(\displaystyle \frac {a \int \cos ^4(c+d x)dx}{a^2+b^2}+\frac {b^2 \int \frac {\cos ^3(c+d x)}{a \cos (c+d x)+b \sin (c+d x)}dx}{a^2+b^2}+\frac {b \cos ^4(c+d x)}{4 d \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a \int \sin \left (c+d x+\frac {\pi }{2}\right )^4dx}{a^2+b^2}+\frac {b^2 \int \frac {\cos (c+d x)^3}{a \cos (c+d x)+b \sin (c+d x)}dx}{a^2+b^2}+\frac {b \cos ^4(c+d x)}{4 d \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {a \left (\frac {3}{4} \int \cos ^2(c+d x)dx+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )}{a^2+b^2}+\frac {b^2 \int \frac {\cos (c+d x)^3}{a \cos (c+d x)+b \sin (c+d x)}dx}{a^2+b^2}+\frac {b \cos ^4(c+d x)}{4 d \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a \left (\frac {3}{4} \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )}{a^2+b^2}+\frac {b^2 \int \frac {\cos (c+d x)^3}{a \cos (c+d x)+b \sin (c+d x)}dx}{a^2+b^2}+\frac {b \cos ^4(c+d x)}{4 d \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {a \left (\frac {3}{4} \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )}{a^2+b^2}+\frac {b^2 \int \frac {\cos (c+d x)^3}{a \cos (c+d x)+b \sin (c+d x)}dx}{a^2+b^2}+\frac {b \cos ^4(c+d x)}{4 d \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {b^2 \int \frac {\cos (c+d x)^3}{a \cos (c+d x)+b \sin (c+d x)}dx}{a^2+b^2}+\frac {b \cos ^4(c+d x)}{4 d \left (a^2+b^2\right )}+\frac {a \left (\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 3579 |
| \(\displaystyle \frac {b^2 \left (\frac {a \int \cos ^2(c+d x)dx}{a^2+b^2}+\frac {b^2 \int \frac {\cos (c+d x)}{a \cos (c+d x)+b \sin (c+d x)}dx}{a^2+b^2}+\frac {b \cos ^2(c+d x)}{2 d \left (a^2+b^2\right )}\right )}{a^2+b^2}+\frac {b \cos ^4(c+d x)}{4 d \left (a^2+b^2\right )}+\frac {a \left (\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {b^2 \left (\frac {a \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx}{a^2+b^2}+\frac {b^2 \int \frac {\cos (c+d x)}{a \cos (c+d x)+b \sin (c+d x)}dx}{a^2+b^2}+\frac {b \cos ^2(c+d x)}{2 d \left (a^2+b^2\right )}\right )}{a^2+b^2}+\frac {b \cos ^4(c+d x)}{4 d \left (a^2+b^2\right )}+\frac {a \left (\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {b^2 \left (\frac {b^2 \int \frac {\cos (c+d x)}{a \cos (c+d x)+b \sin (c+d x)}dx}{a^2+b^2}+\frac {a \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )}{a^2+b^2}+\frac {b \cos ^2(c+d x)}{2 d \left (a^2+b^2\right )}\right )}{a^2+b^2}+\frac {b \cos ^4(c+d x)}{4 d \left (a^2+b^2\right )}+\frac {a \left (\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {b^2 \left (\frac {b^2 \int \frac {\cos (c+d x)}{a \cos (c+d x)+b \sin (c+d x)}dx}{a^2+b^2}+\frac {b \cos ^2(c+d x)}{2 d \left (a^2+b^2\right )}+\frac {a \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \cos ^4(c+d x)}{4 d \left (a^2+b^2\right )}+\frac {a \left (\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 3577 |
| \(\displaystyle \frac {b^2 \left (\frac {b^2 \left (\frac {b \int \frac {b \cos (c+d x)-a \sin (c+d x)}{a \cos (c+d x)+b \sin (c+d x)}dx}{a^2+b^2}+\frac {a x}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \cos ^2(c+d x)}{2 d \left (a^2+b^2\right )}+\frac {a \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \cos ^4(c+d x)}{4 d \left (a^2+b^2\right )}+\frac {a \left (\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {b^2 \left (\frac {b^2 \left (\frac {b \int \frac {b \cos (c+d x)-a \sin (c+d x)}{a \cos (c+d x)+b \sin (c+d x)}dx}{a^2+b^2}+\frac {a x}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \cos ^2(c+d x)}{2 d \left (a^2+b^2\right )}+\frac {a \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \cos ^4(c+d x)}{4 d \left (a^2+b^2\right )}+\frac {a \left (\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 3612 |
| \(\displaystyle \frac {b \cos ^4(c+d x)}{4 d \left (a^2+b^2\right )}+\frac {a \left (\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{a^2+b^2}+\frac {b^2 \left (\frac {b \cos ^2(c+d x)}{2 d \left (a^2+b^2\right )}+\frac {a \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )}{a^2+b^2}+\frac {b^2 \left (\frac {b \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )}+\frac {a x}{a^2+b^2}\right )}{a^2+b^2}\right )}{a^2+b^2}\) |
(b*Cos[c + d*x]^4)/(4*(a^2 + b^2)*d) + (a*((Cos[c + d*x]^3*Sin[c + d*x])/( 4*d) + (3*(x/2 + (Cos[c + d*x]*Sin[c + d*x])/(2*d)))/4))/(a^2 + b^2) + (b^ 2*((b*Cos[c + d*x]^2)/(2*(a^2 + b^2)*d) + (b^2*((a*x)/(a^2 + b^2) + (b*Log [a*Cos[c + d*x] + b*Sin[c + d*x]])/((a^2 + b^2)*d)))/(a^2 + b^2) + (a*(x/2 + (Cos[c + d*x]*Sin[c + d*x])/(2*d)))/(a^2 + b^2)))/(a^2 + b^2)
3.2.10.3.1 Defintions of rubi rules used
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
Int[cos[(c_.) + (d_.)*(x_)]/(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_. ) + (d_.)*(x_)]), x_Symbol] :> Simp[a*(x/(a^2 + b^2)), x] + Simp[b/(a^2 + b ^2) Int[(b*Cos[c + d*x] - a*Sin[c + d*x])/(a*Cos[c + d*x] + b*Sin[c + d*x ]), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0]
Int[cos[(c_.) + (d_.)*(x_)]^(m_)/(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin [(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[b*(Cos[c + d*x]^(m - 1)/(d*(a^2 + b^2)*(m - 1))), x] + (Simp[a/(a^2 + b^2) Int[Cos[c + d*x]^(m - 1), x], x] + Simp[b^2/(a^2 + b^2) Int[Cos[c + d*x]^(m - 2)/(a*Cos[c + d*x] + b*Sin[ c + d*x]), x], x]) /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && GtQ[m, 1]
Int[((A_.) + cos[(d_.) + (e_.)*(x_)]*(B_.) + (C_.)*sin[(d_.) + (e_.)*(x_)]) /((a_.) + cos[(d_.) + (e_.)*(x_)]*(b_.) + (c_.)*sin[(d_.) + (e_.)*(x_)]), x _Symbol] :> Simp[(b*B + c*C)*(x/(b^2 + c^2)), x] + Simp[(c*B - b*C)*(Log[a + b*Cos[d + e*x] + c*Sin[d + e*x]]/(e*(b^2 + c^2))), x] /; FreeQ[{a, b, c, d, e, A, B, C}, x] && NeQ[b^2 + c^2, 0] && EqQ[A*(b^2 + c^2) - a*(b*B + c*C ), 0]
Time = 0.85 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.87
| method | result | size |
| derivativedivides | \(\frac {\frac {\frac {\left (\frac {3}{8} a^{5}+\frac {5}{4} a^{3} b^{2}+\frac {7}{8} a \,b^{4}\right ) \tan \left (d x +c \right )^{3}+\left (\frac {1}{2} a^{2} b^{3}+\frac {1}{2} b^{5}\right ) \tan \left (d x +c \right )^{2}+\left (\frac {7}{4} a^{3} b^{2}+\frac {9}{8} a \,b^{4}+\frac {5}{8} a^{5}\right ) \tan \left (d x +c \right )+\frac {a^{4} b}{4}+a^{2} b^{3}+\frac {3 b^{5}}{4}}{\left (1+\tan \left (d x +c \right )^{2}\right )^{2}}-\frac {b^{5} \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+\frac {\left (3 a^{5}+10 a^{3} b^{2}+15 a \,b^{4}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{8}}{\left (a^{2}+b^{2}\right )^{3}}+\frac {b^{5} \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3}}}{d}\) | \(197\) |
| default | \(\frac {\frac {\frac {\left (\frac {3}{8} a^{5}+\frac {5}{4} a^{3} b^{2}+\frac {7}{8} a \,b^{4}\right ) \tan \left (d x +c \right )^{3}+\left (\frac {1}{2} a^{2} b^{3}+\frac {1}{2} b^{5}\right ) \tan \left (d x +c \right )^{2}+\left (\frac {7}{4} a^{3} b^{2}+\frac {9}{8} a \,b^{4}+\frac {5}{8} a^{5}\right ) \tan \left (d x +c \right )+\frac {a^{4} b}{4}+a^{2} b^{3}+\frac {3 b^{5}}{4}}{\left (1+\tan \left (d x +c \right )^{2}\right )^{2}}-\frac {b^{5} \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+\frac {\left (3 a^{5}+10 a^{3} b^{2}+15 a \,b^{4}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{8}}{\left (a^{2}+b^{2}\right )^{3}}+\frac {b^{5} \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{3}}}{d}\) | \(197\) |
| parallelrisch | \(\frac {32 b^{5} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )-32 b^{5} \ln \left (\sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )+4 \left (a^{4} b +4 a^{2} b^{3}+3 b^{5}\right ) \cos \left (2 d x +2 c \right )+b \left (a^{2}+b^{2}\right )^{2} \cos \left (4 d x +4 c \right )+8 \left (a^{5}+3 a^{3} b^{2}+2 a \,b^{4}\right ) \sin \left (2 d x +2 c \right )+a \left (a^{2}+b^{2}\right )^{2} \sin \left (4 d x +4 c \right )+12 a^{5} d x +40 a^{3} b^{2} d x +60 a \,b^{4} d x -5 a^{4} b -18 a^{2} b^{3}-13 b^{5}}{32 d \left (a^{2}+b^{2}\right )^{3}}\) | \(211\) |
| risch | \(\frac {9 x a b}{8 i a^{3}-24 i a \,b^{2}+24 a^{2} b -8 b^{3}}+\frac {3 i x \,a^{2}}{8 i a^{3}-24 i a \,b^{2}+24 a^{2} b -8 b^{3}}-\frac {8 i x \,b^{2}}{8 i a^{3}-24 i a \,b^{2}+24 a^{2} b -8 b^{3}}-\frac {3 \,{\mathrm e}^{2 i \left (d x +c \right )} b}{16 \left (-2 i b a +a^{2}-b^{2}\right ) d}-\frac {i {\mathrm e}^{2 i \left (d x +c \right )} a}{8 \left (-2 i b a +a^{2}-b^{2}\right ) d}-\frac {3 \,{\mathrm e}^{-2 i \left (d x +c \right )} b}{16 \left (i b +a \right )^{2} d}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )} a}{8 \left (i b +a \right )^{2} d}-\frac {2 i b^{5} x}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}-\frac {2 i b^{5} c}{d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {b^{5} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {b \cos \left (4 d x +4 c \right )}{32 d \left (-a^{2}-b^{2}\right )}-\frac {a \sin \left (4 d x +4 c \right )}{32 d \left (-a^{2}-b^{2}\right )}\) | \(396\) |
| norman | \(\frac {\frac {\left (-2 a^{2} b -4 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {\left (-2 a^{2} b -4 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {a \left (3 a^{4}+10 a^{2} b^{2}+15 b^{4}\right ) x}{8 a^{6}+24 a^{4} b^{2}+24 a^{2} b^{4}+8 b^{6}}+\frac {2 \left (-a^{2} b -4 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {2 \left (-a^{2} b -4 b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {a \left (a^{2}+5 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{2 d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {a \left (a^{2}+5 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{2 d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {\left (5 a^{2}+9 b^{2}\right ) a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {\left (5 a^{2}+9 b^{2}\right ) a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{4 d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {5 \left (3 a^{4}+10 a^{2} b^{2}+15 b^{4}\right ) a x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{8 \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {5 \left (3 a^{4}+10 a^{2} b^{2}+15 b^{4}\right ) a x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{4 \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {5 \left (3 a^{4}+10 a^{2} b^{2}+15 b^{4}\right ) a x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{4 \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {5 \left (3 a^{4}+10 a^{2} b^{2}+15 b^{4}\right ) a x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{8 \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {\left (3 a^{4}+10 a^{2} b^{2}+15 b^{4}\right ) a x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{8 a^{6}+24 a^{4} b^{2}+24 a^{2} b^{4}+8 b^{6}}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{5}}+\frac {b^{5} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )}{d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {b^{5} \ln \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}{d \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}\) | \(823\) |
1/d*(1/(a^2+b^2)^3*(((3/8*a^5+5/4*a^3*b^2+7/8*a*b^4)*tan(d*x+c)^3+(1/2*a^2 *b^3+1/2*b^5)*tan(d*x+c)^2+(7/4*a^3*b^2+9/8*a*b^4+5/8*a^5)*tan(d*x+c)+1/4* a^4*b+a^2*b^3+3/4*b^5)/(1+tan(d*x+c)^2)^2-1/2*b^5*ln(1+tan(d*x+c)^2)+1/8*( 3*a^5+10*a^3*b^2+15*a*b^4)*arctan(tan(d*x+c)))+b^5/(a^2+b^2)^3*ln(a+b*tan( d*x+c)))
Time = 0.30 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.92 \[ \int \frac {\cos ^5(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx=\frac {4 \, b^{5} \log \left (2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + b^{2}\right ) + 2 \, {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{4} + {\left (3 \, a^{5} + 10 \, a^{3} b^{2} + 15 \, a b^{4}\right )} d x + 4 \, {\left (a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{2} + {\left (2 \, {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} \cos \left (d x + c\right )^{3} + {\left (3 \, a^{5} + 10 \, a^{3} b^{2} + 7 \, a b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{8 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} d} \]
1/8*(4*b^5*log(2*a*b*cos(d*x + c)*sin(d*x + c) + (a^2 - b^2)*cos(d*x + c)^ 2 + b^2) + 2*(a^4*b + 2*a^2*b^3 + b^5)*cos(d*x + c)^4 + (3*a^5 + 10*a^3*b^ 2 + 15*a*b^4)*d*x + 4*(a^2*b^3 + b^5)*cos(d*x + c)^2 + (2*(a^5 + 2*a^3*b^2 + a*b^4)*cos(d*x + c)^3 + (3*a^5 + 10*a^3*b^2 + 7*a*b^4)*cos(d*x + c))*si n(d*x + c))/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*d)
Timed out. \[ \int \frac {\cos ^5(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 564 vs. \(2 (213) = 426\).
Time = 0.34 (sec) , antiderivative size = 564, normalized size of antiderivative = 2.48 \[ \int \frac {\cos ^5(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx=\frac {\frac {4 \, b^{5} \log \left (-a - \frac {2 \, b \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {4 \, b^{5} \log \left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {{\left (3 \, a^{5} + 10 \, a^{3} b^{2} + 15 \, a b^{4}\right )} \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {\frac {16 \, b^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {{\left (5 \, a^{3} + 9 \, a b^{2}\right )} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {8 \, {\left (a^{2} b + 2 \, b^{3}\right )} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {{\left (3 \, a^{3} - a b^{2}\right )} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {{\left (3 \, a^{3} - a b^{2}\right )} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {8 \, {\left (a^{2} b + 2 \, b^{3}\right )} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {{\left (5 \, a^{3} + 9 \, a b^{2}\right )} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4} + 2 \, a^{2} b^{2} + b^{4} + \frac {4 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {6 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {4 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}}}{4 \, d} \]
1/4*(4*b^5*log(-a - 2*b*sin(d*x + c)/(cos(d*x + c) + 1) + a*sin(d*x + c)^2 /(cos(d*x + c) + 1)^2)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) - 4*b^5*log(sin (d*x + c)^2/(cos(d*x + c) + 1)^2 + 1)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) + (3*a^5 + 10*a^3*b^2 + 15*a*b^4)*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/ (a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) - (16*b^3*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - (5*a^3 + 9*a*b^2)*sin(d*x + c)/(cos(d*x + c) + 1) + 8*(a^2*b + 2 *b^3)*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + (3*a^3 - a*b^2)*sin(d*x + c)^3 /(cos(d*x + c) + 1)^3 - (3*a^3 - a*b^2)*sin(d*x + c)^5/(cos(d*x + c) + 1)^ 5 + 8*(a^2*b + 2*b^3)*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + (5*a^3 + 9*a*b ^2)*sin(d*x + c)^7/(cos(d*x + c) + 1)^7)/(a^4 + 2*a^2*b^2 + b^4 + 4*(a^4 + 2*a^2*b^2 + b^4)*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 6*(a^4 + 2*a^2*b^2 + b^4)*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 4*(a^4 + 2*a^2*b^2 + b^4)*si n(d*x + c)^6/(cos(d*x + c) + 1)^6 + (a^4 + 2*a^2*b^2 + b^4)*sin(d*x + c)^8 /(cos(d*x + c) + 1)^8))/d
Time = 0.31 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.42 \[ \int \frac {\cos ^5(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx=\frac {\frac {8 \, b^{6} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}} - \frac {4 \, b^{5} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {{\left (3 \, a^{5} + 10 \, a^{3} b^{2} + 15 \, a b^{4}\right )} {\left (d x + c\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {6 \, b^{5} \tan \left (d x + c\right )^{4} + 3 \, a^{5} \tan \left (d x + c\right )^{3} + 10 \, a^{3} b^{2} \tan \left (d x + c\right )^{3} + 7 \, a b^{4} \tan \left (d x + c\right )^{3} + 4 \, a^{2} b^{3} \tan \left (d x + c\right )^{2} + 16 \, b^{5} \tan \left (d x + c\right )^{2} + 5 \, a^{5} \tan \left (d x + c\right ) + 14 \, a^{3} b^{2} \tan \left (d x + c\right ) + 9 \, a b^{4} \tan \left (d x + c\right ) + 2 \, a^{4} b + 8 \, a^{2} b^{3} + 12 \, b^{5}}{{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} {\left (\tan \left (d x + c\right )^{2} + 1\right )}^{2}}}{8 \, d} \]
1/8*(8*b^6*log(abs(b*tan(d*x + c) + a))/(a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b ^7) - 4*b^5*log(tan(d*x + c)^2 + 1)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) + (3*a^5 + 10*a^3*b^2 + 15*a*b^4)*(d*x + c)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b ^6) + (6*b^5*tan(d*x + c)^4 + 3*a^5*tan(d*x + c)^3 + 10*a^3*b^2*tan(d*x + c)^3 + 7*a*b^4*tan(d*x + c)^3 + 4*a^2*b^3*tan(d*x + c)^2 + 16*b^5*tan(d*x + c)^2 + 5*a^5*tan(d*x + c) + 14*a^3*b^2*tan(d*x + c) + 9*a*b^4*tan(d*x + c) + 2*a^4*b + 8*a^2*b^3 + 12*b^5)/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*(t an(d*x + c)^2 + 1)^2))/d
Time = 35.84 (sec) , antiderivative size = 6099, normalized size of antiderivative = 26.87 \[ \int \frac {\cos ^5(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx=\text {Too large to display} \]
(b^5*log(a + 2*b*tan(c/2 + (d*x)/2) - a*tan(c/2 + (d*x)/2)^2))/(d*(a^6 + b ^6 + 3*a^2*b^4 + 3*a^4*b^2)) - (64*b^5*log(1/(cos(c + d*x) + 1)))/(d*(64*a ^6 + 64*b^6 + 192*a^2*b^4 + 192*a^4*b^2)) - ((4*b^3*tan(c/2 + (d*x)/2)^4)/ (a^4 + b^4 + 2*a^2*b^2) - (tan(c/2 + (d*x)/2)*(9*a*b^2 + 5*a^3))/(4*(a^4 + b^4 + 2*a^2*b^2)) - (tan(c/2 + (d*x)/2)^3*(a*b^2 - 3*a^3))/(4*(a^4 + b^4 + 2*a^2*b^2)) + (tan(c/2 + (d*x)/2)^5*(a*b^2 - 3*a^3))/(4*(a^4 + b^4 + 2*a ^2*b^2)) + (tan(c/2 + (d*x)/2)^7*(9*a*b^2 + 5*a^3))/(4*(a^4 + b^4 + 2*a^2* b^2)) + (2*b*tan(c/2 + (d*x)/2)^2*(a^2 + 2*b^2))/(a^4 + b^4 + 2*a^2*b^2) + (2*b*tan(c/2 + (d*x)/2)^6*(a^2 + 2*b^2))/(a^4 + b^4 + 2*a^2*b^2))/(d*(4*t an(c/2 + (d*x)/2)^2 + 6*tan(c/2 + (d*x)/2)^4 + 4*tan(c/2 + (d*x)/2)^6 + ta n(c/2 + (d*x)/2)^8 + 1)) - (a*atan((tan(c/2 + (d*x)/2)*((((64*b^5*((a*((64 *a*b^15 + 48*a^15*b + 624*a^3*b^13 + 2016*a^5*b^11 + 3152*a^7*b^9 + 2688*a ^9*b^7 + 1296*a^11*b^5 + 352*a^13*b^3)/(2*(a^12 + b^12 + 6*a^2*b^10 + 15*a ^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2)) - (32*b^5*(192*a*b^16 + 13 44*a^3*b^14 + 4032*a^5*b^12 + 6720*a^7*b^10 + 6720*a^9*b^8 + 4032*a^11*b^6 + 1344*a^13*b^4 + 192*a^15*b^2))/((64*a^6 + 64*b^6 + 192*a^2*b^4 + 192*a^ 4*b^2)*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2)))*(3*a^4 + 15*b^4 + 10*a^2*b^2))/(8*(a^6 + b^6 + 3*a^2*b^4 + 3 *a^4*b^2)) - (4*a*b^5*(3*a^4 + 15*b^4 + 10*a^2*b^2)*(192*a*b^16 + 1344*a^3 *b^14 + 4032*a^5*b^12 + 6720*a^7*b^10 + 6720*a^9*b^8 + 4032*a^11*b^6 + ...