Integrand size = 28, antiderivative size = 119 \[ \int \frac {\cos ^3(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx=\frac {a b^2 x}{\left (a^2+b^2\right )^2}+\frac {a x}{2 \left (a^2+b^2\right )}+\frac {b \cos ^2(c+d x)}{2 \left (a^2+b^2\right ) d}+\frac {b^3 \log (a \cos (c+d x)+b \sin (c+d x))}{\left (a^2+b^2\right )^2 d}+\frac {a \cos (c+d x) \sin (c+d x)}{2 \left (a^2+b^2\right ) d} \] Output:
a*b^2*x/(a^2+b^2)^2+a*x/(2*a^2+2*b^2)+1/2*b*cos(d*x+c)^2/(a^2+b^2)/d+b^3*l n(a*cos(d*x+c)+b*sin(d*x+c))/(a^2+b^2)^2/d+1/2*a*cos(d*x+c)*sin(d*x+c)/(a^ 2+b^2)/d
Result contains complex when optimal does not.
Time = 0.41 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.20 \[ \int \frac {\cos ^3(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx=\frac {2 a^3 c+6 a b^2 c+4 i b^3 c+2 a^3 d x+6 a b^2 d x+4 i b^3 d x-4 i b^3 \arctan (\tan (c+d x))+b \left (a^2+b^2\right ) \cos (2 (c+d x))+2 b^3 \log \left ((a \cos (c+d x)+b \sin (c+d x))^2\right )+a^3 \sin (2 (c+d x))+a b^2 \sin (2 (c+d x))}{4 \left (a^2+b^2\right )^2 d} \] Input:
Integrate[Cos[c + d*x]^3/(a*Cos[c + d*x] + b*Sin[c + d*x]),x]
Output:
(2*a^3*c + 6*a*b^2*c + (4*I)*b^3*c + 2*a^3*d*x + 6*a*b^2*d*x + (4*I)*b^3*d *x - (4*I)*b^3*ArcTan[Tan[c + d*x]] + b*(a^2 + b^2)*Cos[2*(c + d*x)] + 2*b ^3*Log[(a*Cos[c + d*x] + b*Sin[c + d*x])^2] + a^3*Sin[2*(c + d*x)] + a*b^2 *Sin[2*(c + d*x)])/(4*(a^2 + b^2)^2*d)
Time = 0.55 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.01, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3042, 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.
| \(\displaystyle \int \frac {\cos ^3(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^3}{a \cos (c+d x)+b \sin (c+d x)}dx\) |
| \(\Big \downarrow \) 3579 |
| \(\displaystyle \frac {a \int \cos ^2(c+d x)dx}{a^2+b^2}+\frac {b^2 \int \frac {\cos (c+d x)}{a \cos (c+d x)+b \sin (c+d x)}dx}{a^2+b^2}+\frac {b \cos ^2(c+d x)}{2 d \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {a \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx}{a^2+b^2}+\frac {b^2 \int \frac {\cos (c+d x)}{a \cos (c+d x)+b \sin (c+d x)}dx}{a^2+b^2}+\frac {b \cos ^2(c+d x)}{2 d \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {b^2 \int \frac {\cos (c+d x)}{a \cos (c+d x)+b \sin (c+d x)}dx}{a^2+b^2}+\frac {a \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )}{a^2+b^2}+\frac {b \cos ^2(c+d x)}{2 d \left (a^2+b^2\right )}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {b^2 \int \frac {\cos (c+d x)}{a \cos (c+d x)+b \sin (c+d x)}dx}{a^2+b^2}+\frac {b \cos ^2(c+d x)}{2 d \left (a^2+b^2\right )}+\frac {a \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 3577 |
| \(\displaystyle \frac {b^2 \left (\frac {b \int \frac {b \cos (c+d x)-a \sin (c+d x)}{a \cos (c+d x)+b \sin (c+d x)}dx}{a^2+b^2}+\frac {a x}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \cos ^2(c+d x)}{2 d \left (a^2+b^2\right )}+\frac {a \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {b^2 \left (\frac {b \int \frac {b \cos (c+d x)-a \sin (c+d x)}{a \cos (c+d x)+b \sin (c+d x)}dx}{a^2+b^2}+\frac {a x}{a^2+b^2}\right )}{a^2+b^2}+\frac {b \cos ^2(c+d x)}{2 d \left (a^2+b^2\right )}+\frac {a \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )}{a^2+b^2}\) |
| \(\Big \downarrow \) 3612 |
| \(\displaystyle \frac {b \cos ^2(c+d x)}{2 d \left (a^2+b^2\right )}+\frac {a \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )}{a^2+b^2}+\frac {b^2 \left (\frac {b \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )}+\frac {a x}{a^2+b^2}\right )}{a^2+b^2}\) |
Input:
Int[Cos[c + d*x]^3/(a*Cos[c + d*x] + b*Sin[c + d*x]),x]
Output:
(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)
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 = 0.38 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.01
| method | result | size |
| derivativedivides | \(\frac {\frac {b^{3} \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}+\frac {\frac {\left (\frac {1}{2} a^{3}+\frac {1}{2} a \,b^{2}\right ) \tan \left (d x +c \right )+\frac {a^{2} b}{2}+\frac {b^{3}}{2}}{1+\tan \left (d x +c \right )^{2}}-\frac {b^{3} \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+\frac {\left (a^{3}+3 a \,b^{2}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{2}}{\left (a^{2}+b^{2}\right )^{2}}}{d}\) | \(120\) |
| default | \(\frac {\frac {b^{3} \ln \left (a +b \tan \left (d x +c \right )\right )}{\left (a^{2}+b^{2}\right )^{2}}+\frac {\frac {\left (\frac {1}{2} a^{3}+\frac {1}{2} a \,b^{2}\right ) \tan \left (d x +c \right )+\frac {a^{2} b}{2}+\frac {b^{3}}{2}}{1+\tan \left (d x +c \right )^{2}}-\frac {b^{3} \ln \left (1+\tan \left (d x +c \right )^{2}\right )}{2}+\frac {\left (a^{3}+3 a \,b^{2}\right ) \arctan \left (\tan \left (d x +c \right )\right )}{2}}{\left (a^{2}+b^{2}\right )^{2}}}{d}\) | \(120\) |
| parallelrisch | \(\frac {4 b^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )-4 b^{3} \ln \left (\sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )+\left (a^{2} b +b^{3}\right ) \cos \left (2 d x +2 c \right )+\left (a^{3}+a \,b^{2}\right ) \sin \left (2 d x +2 c \right )+2 a^{3} d x +6 a \,b^{2} d x -a^{2} b -b^{3}}{4 \left (a^{2}+b^{2}\right )^{2} d}\) | \(132\) |
| risch | \(\frac {2 i x b}{4 i a b -2 a^{2}+2 b^{2}}-\frac {x a}{4 i a b -2 a^{2}+2 b^{2}}-\frac {i {\mathrm e}^{2 i \left (d x +c \right )}}{8 \left (-i b +a \right ) d}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )}}{8 \left (i b +a \right ) d}-\frac {2 i b^{3} x}{a^{4}+2 a^{2} b^{2}+b^{4}}-\frac {2 i b^{3} c}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {b^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {i b +a}{i b -a}\right )}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}\) | \(197\) |
| norman | \(\frac {\frac {a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d \left (a^{2}+b^{2}\right )}-\frac {2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{d \left (a^{2}+b^{2}\right )}-\frac {2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{d \left (a^{2}+b^{2}\right )}-\frac {a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{d \left (a^{2}+b^{2}\right )}+\frac {a \left (a^{2}+3 b^{2}\right ) x}{2 a^{4}+4 a^{2} b^{2}+2 b^{4}}+\frac {3 \left (a^{2}+3 b^{2}\right ) a x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{2 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {3 \left (a^{2}+3 b^{2}\right ) a x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{2 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {\left (a^{2}+3 b^{2}\right ) a x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{2 a^{4}+4 a^{2} b^{2}+2 b^{4}}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3}}+\frac {b^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {b^{3} \ln \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}\) | \(366\) |
Input:
int(cos(d*x+c)^3/(a*cos(d*x+c)+b*sin(d*x+c)),x,method=_RETURNVERBOSE)
Output:
1/d*(b^3/(a^2+b^2)^2*ln(a+b*tan(d*x+c))+1/(a^2+b^2)^2*(((1/2*a^3+1/2*a*b^2 )*tan(d*x+c)+1/2*a^2*b+1/2*b^3)/(1+tan(d*x+c)^2)-1/2*b^3*ln(1+tan(d*x+c)^2 )+1/2*(a^3+3*a*b^2)*arctan(tan(d*x+c))))
Time = 0.08 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.00 \[ \int \frac {\cos ^3(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx=\frac {b^{3} \log \left (2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + b^{2}\right ) + {\left (a^{3} + 3 \, a b^{2}\right )} d x + {\left (a^{2} b + b^{3}\right )} \cos \left (d x + c\right )^{2} + {\left (a^{3} + a b^{2}\right )} \cos \left (d x + c\right ) \sin \left (d x + c\right )}{2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d} \] Input:
integrate(cos(d*x+c)^3/(a*cos(d*x+c)+b*sin(d*x+c)),x, algorithm="fricas")
Output:
1/2*(b^3*log(2*a*b*cos(d*x + c)*sin(d*x + c) + (a^2 - b^2)*cos(d*x + c)^2 + b^2) + (a^3 + 3*a*b^2)*d*x + (a^2*b + b^3)*cos(d*x + c)^2 + (a^3 + a*b^2 )*cos(d*x + c)*sin(d*x + c))/((a^4 + 2*a^2*b^2 + b^4)*d)
Timed out. \[ \int \frac {\cos ^3(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**3/(a*cos(d*x+c)+b*sin(d*x+c)),x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 284 vs. \(2 (113) = 226\).
Time = 0.12 (sec) , antiderivative size = 284, normalized size of antiderivative = 2.39 \[ \int \frac {\cos ^3(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx=\frac {\frac {b^{3} \log \left (-a - \frac {2 \, b \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {b^{3} \log \left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {{\left (a^{3} + 3 \, a b^{2}\right )} \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {\frac {a \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {2 \, b \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {a \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a^{2} + b^{2} + \frac {2 \, {\left (a^{2} + b^{2}\right )} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {{\left (a^{2} + b^{2}\right )} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}}}{d} \] Input:
integrate(cos(d*x+c)^3/(a*cos(d*x+c)+b*sin(d*x+c)),x, algorithm="maxima")
Output:
(b^3*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^4 + 2*a^2*b^2 + b^4) - b^3*log(sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 1)/(a^4 + 2*a^2*b^2 + b^4) + (a^3 + 3*a*b^2)*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/(a^4 + 2*a^2*b^2 + b^4) + (a*sin(d*x + c)/(cos(d *x + c) + 1) - 2*b*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 - a*sin(d*x + c)^3/ (cos(d*x + c) + 1)^3)/(a^2 + b^2 + 2*(a^2 + b^2)*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + (a^2 + b^2)*sin(d*x + c)^4/(cos(d*x + c) + 1)^4))/d
Time = 0.13 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.53 \[ \int \frac {\cos ^3(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx=\frac {\frac {2 \, b^{4} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{4} b + 2 \, a^{2} b^{3} + b^{5}} - \frac {b^{3} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {{\left (a^{3} + 3 \, a b^{2}\right )} {\left (d x + c\right )}}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {b^{3} \tan \left (d x + c\right )^{2} + a^{3} \tan \left (d x + c\right ) + a b^{2} \tan \left (d x + c\right ) + a^{2} b + 2 \, b^{3}}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} {\left (\tan \left (d x + c\right )^{2} + 1\right )}}}{2 \, d} \] Input:
integrate(cos(d*x+c)^3/(a*cos(d*x+c)+b*sin(d*x+c)),x, algorithm="giac")
Output:
1/2*(2*b^4*log(abs(b*tan(d*x + c) + a))/(a^4*b + 2*a^2*b^3 + b^5) - b^3*lo g(tan(d*x + c)^2 + 1)/(a^4 + 2*a^2*b^2 + b^4) + (a^3 + 3*a*b^2)*(d*x + c)/ (a^4 + 2*a^2*b^2 + b^4) + (b^3*tan(d*x + c)^2 + a^3*tan(d*x + c) + a*b^2*t an(d*x + c) + a^2*b + 2*b^3)/((a^4 + 2*a^2*b^2 + b^4)*(tan(d*x + c)^2 + 1) ))/d
Time = 21.76 (sec) , antiderivative size = 3572, normalized size of antiderivative = 30.02 \[ \int \frac {\cos ^3(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx=\text {Too large to display} \] Input:
int(cos(c + d*x)^3/(a*cos(c + d*x) + b*sin(c + d*x)),x)
Output:
(b^3*log(a + 2*b*tan(c/2 + (d*x)/2) - a*tan(c/2 + (d*x)/2)^2))/(d*(a^4 + b ^4 + 2*a^2*b^2)) - (4*b^3*log(1/(cos(c + d*x) + 1)))/(d*(4*a^4 + 4*b^4 + 8 *a^2*b^2)) - ((a*tan(c/2 + (d*x)/2)^3)/(a^2 + b^2) + (2*b*tan(c/2 + (d*x)/ 2)^2)/(a^2 + b^2) - (a*tan(c/2 + (d*x)/2))/(a^2 + b^2))/(d*(2*tan(c/2 + (d *x)/2)^2 + tan(c/2 + (d*x)/2)^4 + 1)) - (a*atan((tan(c/2 + (d*x)/2)*((((4* b^3*((a*((8*(4*a*b^9 + 4*a^9*b + 28*a^3*b^7 + 48*a^5*b^5 + 28*a^7*b^3))/(a ^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2) - (32*b^3*(12*a*b^10 + 48*a^3*b^8 + 72*a ^5*b^6 + 48*a^7*b^4 + 12*a^9*b^2))/((4*a^4 + 4*b^4 + 8*a^2*b^2)*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)))*(a^2 + 3*b^2))/(2*(a^4 + b^4 + 2*a^2*b^2)) - ( 16*a*b^3*(a^2 + 3*b^2)*(12*a*b^10 + 48*a^3*b^8 + 72*a^5*b^6 + 48*a^7*b^4 + 12*a^9*b^2))/((4*a^4 + 4*b^4 + 8*a^2*b^2)*(a^4 + b^4 + 2*a^2*b^2)*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2))))/(4*a^4 + 4*b^4 + 8*a^2*b^2) - (a*((8*(a^9 - 12*a*b^8 - 6*a^3*b^6 + 13*a^5*b^4 + 8*a^7*b^2))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2) - (4*b^3*((8*(4*a*b^9 + 4*a^9*b + 28*a^3*b^7 + 48*a^5*b^5 + 28 *a^7*b^3))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2) - (32*b^3*(12*a*b^10 + 48*a ^3*b^8 + 72*a^5*b^6 + 48*a^7*b^4 + 12*a^9*b^2))/((4*a^4 + 4*b^4 + 8*a^2*b^ 2)*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2))))/(4*a^4 + 4*b^4 + 8*a^2*b^2))*(a^ 2 + 3*b^2))/(2*(a^4 + b^4 + 2*a^2*b^2)) + (a^3*(a^2 + 3*b^2)^3*(12*a*b^10 + 48*a^3*b^8 + 72*a^5*b^6 + 48*a^7*b^4 + 12*a^9*b^2))/((a^4 + b^4 + 2*a^2* b^2)^3*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)))*(a^8 + 16*b^8 - 73*a^2*b^6...
Time = 0.18 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.46 \[ \int \frac {\cos ^3(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx=\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right ) a^{3}+\cos \left (d x +c \right ) \sin \left (d x +c \right ) a \,b^{2}-2 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right ) b^{3}+2 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b -a \right ) b^{3}-\sin \left (d x +c \right )^{2} a^{2} b -\sin \left (d x +c \right )^{2} b^{3}+a^{3} c +a^{3} d x +2 a^{2} b +3 a \,b^{2} c +3 a \,b^{2} d x +2 b^{3}}{2 d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )} \] Input:
int(cos(d*x+c)^3/(a*cos(d*x+c)+b*sin(d*x+c)),x)
Output:
(cos(c + d*x)*sin(c + d*x)*a**3 + cos(c + d*x)*sin(c + d*x)*a*b**2 - 2*log (tan((c + d*x)/2)**2 + 1)*b**3 + 2*log(tan((c + d*x)/2)**2*a - 2*tan((c + d*x)/2)*b - a)*b**3 - sin(c + d*x)**2*a**2*b - sin(c + d*x)**2*b**3 + a**3 *c + a**3*d*x + 2*a**2*b + 3*a*b**2*c + 3*a*b**2*d*x + 2*b**3)/(2*d*(a**4 + 2*a**2*b**2 + b**4))