Integrand size = 9, antiderivative size = 60 \[ \int \frac {1}{(\cos (x)+\cos (3 x))^3} \, dx=-\frac {13}{16} \text {arctanh}(\sin (x))+\frac {19 \text {arctanh}\left (\sqrt {2} \sin (x)\right )}{16 \sqrt {2}}-\frac {7}{32} \sec (x) \tan (x)-\frac {3}{32} \sec (x) \sec (2 x) \tan (x)+\frac {1}{16} \sec (x) \sec ^2(2 x) \tan (x) \] Output:
-13/16*arctanh(sin(x))+19/32*arctanh(sin(x)*2^(1/2))*2^(1/2)-7/32*sec(x)*t an(x)-3/32*sec(x)*sec(2*x)*tan(x)+1/16*sec(x)*sec(2*x)^2*tan(x)
Leaf count is larger than twice the leaf count of optimal. \(141\) vs. \(2(60)=120\).
Time = 0.38 (sec) , antiderivative size = 141, normalized size of antiderivative = 2.35 \[ \int \frac {1}{(\cos (x)+\cos (3 x))^3} \, dx=\frac {1}{64} \left (52 \log \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )-52 \log \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )-19 \sqrt {2} \log \left (\sqrt {2}-2 \sin (x)\right )+19 \sqrt {2} \log \left (\sqrt {2}+2 \sin (x)\right )-\frac {2}{\left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )^2}+\frac {2}{\left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^2}-\frac {10}{\cos (x)-\sin (x)}+8 \sec ^2(2 x) \sin (x)+\frac {10}{\cos (x)+\sin (x)}\right ) \] Input:
Integrate[(Cos[x] + Cos[3*x])^(-3),x]
Output:
(52*Log[Cos[x/2] - Sin[x/2]] - 52*Log[Cos[x/2] + Sin[x/2]] - 19*Sqrt[2]*Lo g[Sqrt[2] - 2*Sin[x]] + 19*Sqrt[2]*Log[Sqrt[2] + 2*Sin[x]] - 2/(Cos[x/2] - Sin[x/2])^2 + 2/(Cos[x/2] + Sin[x/2])^2 - 10/(Cos[x] - Sin[x]) + 8*Sec[2* x]^2*Sin[x] + 10/(Cos[x] + Sin[x]))/64
Time = 0.27 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.75, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.111, Rules used = {3042, 4825, 27, 316, 27, 402, 402, 27, 397, 219}
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.
| \(\displaystyle \int \frac {1}{(\cos (x)+\cos (3 x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(\cos (x)+\cos (3 x))^3}dx\) |
| \(\Big \downarrow \) 4825 |
| \(\displaystyle \int \frac {1}{8 \left (1-2 \sin ^2(x)\right )^3 \left (1-\sin ^2(x)\right )^2}d\sin (x)\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{8} \int \frac {1}{\left (1-2 \sin ^2(x)\right )^3 \left (1-\sin ^2(x)\right )^2}d\sin (x)\) |
| \(\Big \downarrow \) 316 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{4} \int \frac {2 \left (1-5 \sin ^2(x)\right )}{\left (1-2 \sin ^2(x)\right )^2 \left (1-\sin ^2(x)\right )^2}d\sin (x)+\frac {\sin (x)}{2 \left (1-2 \sin ^2(x)\right )^2 \left (1-\sin ^2(x)\right )}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{2} \int \frac {1-5 \sin ^2(x)}{\left (1-2 \sin ^2(x)\right )^2 \left (1-\sin ^2(x)\right )^2}d\sin (x)+\frac {\sin (x)}{2 \left (1-2 \sin ^2(x)\right )^2 \left (1-\sin ^2(x)\right )}\right )\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {9 \sin ^2(x)+5}{\left (1-2 \sin ^2(x)\right ) \left (1-\sin ^2(x)\right )^2}d\sin (x)-\frac {3 \sin (x)}{2 \left (1-2 \sin ^2(x)\right ) \left (1-\sin ^2(x)\right )}\right )+\frac {\sin (x)}{2 \left (1-2 \sin ^2(x)\right )^2 \left (1-\sin ^2(x)\right )}\right )\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{2} \left (\frac {1}{2} \left (-\frac {1}{2} \int -\frac {4 \left (7 \sin ^2(x)+6\right )}{\left (1-2 \sin ^2(x)\right ) \left (1-\sin ^2(x)\right )}d\sin (x)-\frac {7 \sin (x)}{1-\sin ^2(x)}\right )-\frac {3 \sin (x)}{2 \left (1-2 \sin ^2(x)\right ) \left (1-\sin ^2(x)\right )}\right )+\frac {\sin (x)}{2 \left (1-2 \sin ^2(x)\right )^2 \left (1-\sin ^2(x)\right )}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{2} \left (\frac {1}{2} \left (2 \int \frac {7 \sin ^2(x)+6}{\left (1-2 \sin ^2(x)\right ) \left (1-\sin ^2(x)\right )}d\sin (x)-\frac {7 \sin (x)}{1-\sin ^2(x)}\right )-\frac {3 \sin (x)}{2 \left (1-2 \sin ^2(x)\right ) \left (1-\sin ^2(x)\right )}\right )+\frac {\sin (x)}{2 \left (1-2 \sin ^2(x)\right )^2 \left (1-\sin ^2(x)\right )}\right )\) |
| \(\Big \downarrow \) 397 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{2} \left (\frac {1}{2} \left (2 \left (19 \int \frac {1}{1-2 \sin ^2(x)}d\sin (x)-13 \int \frac {1}{1-\sin ^2(x)}d\sin (x)\right )-\frac {7 \sin (x)}{1-\sin ^2(x)}\right )-\frac {3 \sin (x)}{2 \left (1-2 \sin ^2(x)\right ) \left (1-\sin ^2(x)\right )}\right )+\frac {\sin (x)}{2 \left (1-2 \sin ^2(x)\right )^2 \left (1-\sin ^2(x)\right )}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{8} \left (\frac {1}{2} \left (\frac {1}{2} \left (2 \left (\frac {19 \text {arctanh}\left (\sqrt {2} \sin (x)\right )}{\sqrt {2}}-13 \text {arctanh}(\sin (x))\right )-\frac {7 \sin (x)}{1-\sin ^2(x)}\right )-\frac {3 \sin (x)}{2 \left (1-2 \sin ^2(x)\right ) \left (1-\sin ^2(x)\right )}\right )+\frac {\sin (x)}{2 \left (1-2 \sin ^2(x)\right )^2 \left (1-\sin ^2(x)\right )}\right )\) |
Input:
Int[(Cos[x] + Cos[3*x])^(-3),x]
Output:
(Sin[x]/(2*(1 - 2*Sin[x]^2)^2*(1 - Sin[x]^2)) + ((-3*Sin[x])/(2*(1 - 2*Sin [x]^2)*(1 - Sin[x]^2)) + (2*(-13*ArcTanh[Sin[x]] + (19*ArcTanh[Sqrt[2]*Sin [x]])/Sqrt[2]) - (7*Sin[x])/(1 - Sin[x]^2))/2)/2)/8
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[(-b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(2*a*(p + 1)*(b*c - a*d)) ), x] + Simp[1/(2*a*(p + 1)*(b*c - a*d)) Int[(a + b*x^2)^(p + 1)*(c + d*x ^2)^q*Simp[b*c + 2*(p + 1)*(b*c - a*d) + d*b*(2*(p + q + 2) + 1)*x^2, x], x ], x] /; FreeQ[{a, b, c, d, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && ! ( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomialQ[a, b, c, d, 2, p, q, x]
Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*((c_) + (d_.)*(x_)^2)), x_ Symbol] :> Simp[(b*e - a*f)/(b*c - a*d) Int[1/(a + b*x^2), x], x] - Simp[ (d*e - c*f)/(b*c - a*d) Int[1/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d, e , f}, x]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x _)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ (q + 1)/(a*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f) + e*2*(b*c - a*d) *(p + 1) + d*(b*e - a*f)*(2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b , c, d, e, f, q}, x] && LtQ[p, -1]
Int[(cos[(m_.)*((c_.) + (d_.)*(x_))]*(a_.) + cos[(n_.)*((c_.) + (d_.)*(x_)) ]*(b_.))^(p_), x_Symbol] :> Simp[1/d Subst[Int[Simplify[TrigExpand[a*Cos[ m*ArcSin[x]] + b*Cos[n*ArcSin[x]]]]^p/Sqrt[1 - x^2], x], x, Sin[c + d*x]], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[(p - 1)/2, 0] && IntegerQ[(m - 1)/2] & & IntegerQ[(n - 1)/2]
Time = 3.23 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.12
| method | result | size |
| default | \(-\frac {4 \left (-\frac {5 \sin \left (x \right )^{3}}{32}+\frac {3 \sin \left (x \right )}{64}\right )}{\left (2 \sin \left (x \right )^{2}-1\right )^{2}}+\frac {19 \,\operatorname {arctanh}\left (\sqrt {2}\, \sin \left (x \right )\right ) \sqrt {2}}{32}+\frac {1}{32+32 \sin \left (x \right )}-\frac {13 \ln \left (1+\sin \left (x \right )\right )}{32}+\frac {1}{32 \sin \left (x \right )-32}+\frac {13 \ln \left (\sin \left (x \right )-1\right )}{32}\) | \(67\) |
| risch | \(\frac {i \left (7 \,{\mathrm e}^{11 i x}-{\mathrm e}^{9 i x}+{\mathrm e}^{3 i x}-7 \,{\mathrm e}^{i x}\right )}{16 \left ({\mathrm e}^{6 i x}+{\mathrm e}^{4 i x}+{\mathrm e}^{2 i x}+1\right )^{2}}-\frac {13 \ln \left ({\mathrm e}^{i x}+i\right )}{16}+\frac {13 \ln \left ({\mathrm e}^{i x}-i\right )}{16}+\frac {19 \sqrt {2}\, \ln \left ({\mathrm e}^{2 i x}+i \sqrt {2}\, {\mathrm e}^{i x}-1\right )}{64}-\frac {19 \sqrt {2}\, \ln \left ({\mathrm e}^{2 i x}-i \sqrt {2}\, {\mathrm e}^{i x}-1\right )}{64}\) | \(121\) |
Input:
int(1/(cos(x)+cos(3*x))^3,x,method=_RETURNVERBOSE)
Output:
-4*(-5/32*sin(x)^3+3/64*sin(x))/(2*sin(x)^2-1)^2+19/32*arctanh(2^(1/2)*sin (x))*2^(1/2)+1/32/(1+sin(x))-13/32*ln(1+sin(x))+1/32/(sin(x)-1)+13/32*ln(s in(x)-1)
Leaf count of result is larger than twice the leaf count of optimal. 147 vs. \(2 (46) = 92\).
Time = 0.09 (sec) , antiderivative size = 147, normalized size of antiderivative = 2.45 \[ \int \frac {1}{(\cos (x)+\cos (3 x))^3} \, dx=\frac {19 \, {\left (4 \, \sqrt {2} \cos \left (x\right )^{6} - 4 \, \sqrt {2} \cos \left (x\right )^{4} + \sqrt {2} \cos \left (x\right )^{2}\right )} \log \left (-\frac {2 \, \cos \left (x\right )^{2} - 2 \, \sqrt {2} \sin \left (x\right ) - 3}{2 \, \cos \left (x\right )^{2} - 1}\right ) - 26 \, {\left (4 \, \cos \left (x\right )^{6} - 4 \, \cos \left (x\right )^{4} + \cos \left (x\right )^{2}\right )} \log \left (\sin \left (x\right ) + 1\right ) + 26 \, {\left (4 \, \cos \left (x\right )^{6} - 4 \, \cos \left (x\right )^{4} + \cos \left (x\right )^{2}\right )} \log \left (-\sin \left (x\right ) + 1\right ) - 4 \, {\left (14 \, \cos \left (x\right )^{4} - 11 \, \cos \left (x\right )^{2} + 1\right )} \sin \left (x\right )}{64 \, {\left (4 \, \cos \left (x\right )^{6} - 4 \, \cos \left (x\right )^{4} + \cos \left (x\right )^{2}\right )}} \] Input:
integrate(1/(cos(x)+cos(3*x))^3,x, algorithm="fricas")
Output:
1/64*(19*(4*sqrt(2)*cos(x)^6 - 4*sqrt(2)*cos(x)^4 + sqrt(2)*cos(x)^2)*log( -(2*cos(x)^2 - 2*sqrt(2)*sin(x) - 3)/(2*cos(x)^2 - 1)) - 26*(4*cos(x)^6 - 4*cos(x)^4 + cos(x)^2)*log(sin(x) + 1) + 26*(4*cos(x)^6 - 4*cos(x)^4 + cos (x)^2)*log(-sin(x) + 1) - 4*(14*cos(x)^4 - 11*cos(x)^2 + 1)*sin(x))/(4*cos (x)^6 - 4*cos(x)^4 + cos(x)^2)
\[ \int \frac {1}{(\cos (x)+\cos (3 x))^3} \, dx=\int \frac {1}{\left (\cos {\left (x \right )} + \cos {\left (3 x \right )}\right )^{3}}\, dx \] Input:
integrate(1/(cos(x)+cos(3*x))**3,x)
Output:
Integral((cos(x) + cos(3*x))**(-3), x)
Leaf count of result is larger than twice the leaf count of optimal. 3567 vs. \(2 (46) = 92\).
Time = 0.21 (sec) , antiderivative size = 3567, normalized size of antiderivative = 59.45 \[ \int \frac {1}{(\cos (x)+\cos (3 x))^3} \, dx=\text {Too large to display} \] Input:
integrate(1/(cos(x)+cos(3*x))^3,x, algorithm="maxima")
Output:
-1/128*(8*(7*sin(11*x) - sin(9*x) + sin(3*x) - 7*sin(x))*cos(12*x) - 56*(2 *sin(10*x) + 3*sin(8*x) + 4*sin(6*x) + 3*sin(4*x) + 2*sin(2*x))*cos(11*x) - 16*(sin(9*x) - sin(3*x) + 7*sin(x))*cos(10*x) + 8*(3*sin(8*x) + 4*sin(6* x) + 3*sin(4*x) + 2*sin(2*x))*cos(9*x) + 24*(sin(3*x) - 7*sin(x))*cos(8*x) + 32*(sin(3*x) - 7*sin(x))*cos(6*x) + 24*(sin(3*x) - 7*sin(x))*cos(4*x) - 19*(sqrt(2)*cos(12*x)^2 + 4*sqrt(2)*cos(10*x)^2 + 9*sqrt(2)*cos(8*x)^2 + 16*sqrt(2)*cos(6*x)^2 + 9*sqrt(2)*cos(4*x)^2 + 4*sqrt(2)*cos(2*x)^2 + sqrt (2)*sin(12*x)^2 + 4*sqrt(2)*sin(10*x)^2 + 9*sqrt(2)*sin(8*x)^2 + 16*sqrt(2 )*sin(6*x)^2 + 9*sqrt(2)*sin(4*x)^2 + 12*sqrt(2)*sin(4*x)*sin(2*x) + 4*sqr t(2)*sin(2*x)^2 + 2*(2*sqrt(2)*cos(10*x) + 3*sqrt(2)*cos(8*x) + 4*sqrt(2)* cos(6*x) + 3*sqrt(2)*cos(4*x) + 2*sqrt(2)*cos(2*x) + sqrt(2))*cos(12*x) + 4*(3*sqrt(2)*cos(8*x) + 4*sqrt(2)*cos(6*x) + 3*sqrt(2)*cos(4*x) + 2*sqrt(2 )*cos(2*x) + sqrt(2))*cos(10*x) + 6*(4*sqrt(2)*cos(6*x) + 3*sqrt(2)*cos(4* x) + 2*sqrt(2)*cos(2*x) + sqrt(2))*cos(8*x) + 8*(3*sqrt(2)*cos(4*x) + 2*sq rt(2)*cos(2*x) + sqrt(2))*cos(6*x) + 6*(2*sqrt(2)*cos(2*x) + sqrt(2))*cos( 4*x) + 2*(2*sqrt(2)*sin(10*x) + 3*sqrt(2)*sin(8*x) + 4*sqrt(2)*sin(6*x) + 3*sqrt(2)*sin(4*x) + 2*sqrt(2)*sin(2*x))*sin(12*x) + 4*(3*sqrt(2)*sin(8*x) + 4*sqrt(2)*sin(6*x) + 3*sqrt(2)*sin(4*x) + 2*sqrt(2)*sin(2*x))*sin(10*x) + 6*(4*sqrt(2)*sin(6*x) + 3*sqrt(2)*sin(4*x) + 2*sqrt(2)*sin(2*x))*sin(8* x) + 8*(3*sqrt(2)*sin(4*x) + 2*sqrt(2)*sin(2*x))*sin(6*x) + 4*sqrt(2)*c...
Time = 0.11 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.38 \[ \int \frac {1}{(\cos (x)+\cos (3 x))^3} \, dx=-\frac {19}{64} \, \sqrt {2} \log \left (\frac {{\left | -2 \, \sqrt {2} + 4 \, \sin \left (x\right ) \right |}}{{\left | 2 \, \sqrt {2} + 4 \, \sin \left (x\right ) \right |}}\right ) + \frac {\sin \left (x\right )}{16 \, {\left (\sin \left (x\right )^{2} - 1\right )}} + \frac {10 \, \sin \left (x\right )^{3} - 3 \, \sin \left (x\right )}{16 \, {\left (2 \, \sin \left (x\right )^{2} - 1\right )}^{2}} - \frac {13}{32} \, \log \left (\sin \left (x\right ) + 1\right ) + \frac {13}{32} \, \log \left (-\sin \left (x\right ) + 1\right ) \] Input:
integrate(1/(cos(x)+cos(3*x))^3,x, algorithm="giac")
Output:
-19/64*sqrt(2)*log(abs(-2*sqrt(2) + 4*sin(x))/abs(2*sqrt(2) + 4*sin(x))) + 1/16*sin(x)/(sin(x)^2 - 1) + 1/16*(10*sin(x)^3 - 3*sin(x))/(2*sin(x)^2 - 1)^2 - 13/32*log(sin(x) + 1) + 13/32*log(-sin(x) + 1)
Time = 20.87 (sec) , antiderivative size = 164, normalized size of antiderivative = 2.73 \[ \int \frac {1}{(\cos (x)+\cos (3 x))^3} \, dx=\frac {2\,\sin \left (3\,x\right )-14\,\sin \left (5\,x\right )-104\,\mathrm {atanh}\left (\frac {\sin \left (\frac {x}{2}\right )}{\cos \left (\frac {x}{2}\right )}\right )-156\,\cos \left (2\,x\right )\,\mathrm {atanh}\left (\frac {\sin \left (\frac {x}{2}\right )}{\cos \left (\frac {x}{2}\right )}\right )-104\,\cos \left (4\,x\right )\,\mathrm {atanh}\left (\frac {\sin \left (\frac {x}{2}\right )}{\cos \left (\frac {x}{2}\right )}\right )-52\,\cos \left (6\,x\right )\,\mathrm {atanh}\left (\frac {\sin \left (\frac {x}{2}\right )}{\cos \left (\frac {x}{2}\right )}\right )+38\,\sqrt {2}\,\mathrm {atanh}\left (\sqrt {2}\,\sin \left (x\right )\right )+57\,\sqrt {2}\,\mathrm {atanh}\left (\sqrt {2}\,\sin \left (x\right )\right )\,\cos \left (2\,x\right )+38\,\sqrt {2}\,\mathrm {atanh}\left (\sqrt {2}\,\sin \left (x\right )\right )\,\cos \left (4\,x\right )+19\,\sqrt {2}\,\mathrm {atanh}\left (\sqrt {2}\,\sin \left (x\right )\right )\,\cos \left (6\,x\right )}{96\,\cos \left (2\,x\right )+64\,\cos \left (4\,x\right )+32\,\cos \left (6\,x\right )+64} \] Input:
int(1/(cos(3*x) + cos(x))^3,x)
Output:
(2*sin(3*x) - 14*sin(5*x) - 104*atanh(sin(x/2)/cos(x/2)) - 156*cos(2*x)*at anh(sin(x/2)/cos(x/2)) - 104*cos(4*x)*atanh(sin(x/2)/cos(x/2)) - 52*cos(6* x)*atanh(sin(x/2)/cos(x/2)) + 38*2^(1/2)*atanh(2^(1/2)*sin(x)) + 57*2^(1/2 )*atanh(2^(1/2)*sin(x))*cos(2*x) + 38*2^(1/2)*atanh(2^(1/2)*sin(x))*cos(4* x) + 19*2^(1/2)*atanh(2^(1/2)*sin(x))*cos(6*x))/(96*cos(2*x) + 64*cos(4*x) + 32*cos(6*x) + 64)
\[ \int \frac {1}{(\cos (x)+\cos (3 x))^3} \, dx=\int \frac {1}{\cos \left (3 x \right )^{3}+3 \cos \left (3 x \right )^{2} \cos \left (x \right )+3 \cos \left (3 x \right ) \cos \left (x \right )^{2}+\cos \left (x \right )^{3}}d x \] Input:
int(1/(cos(x)+cos(3*x))^3,x)
Output:
int(1/(cos(3*x)**3 + 3*cos(3*x)**2*cos(x) + 3*cos(3*x)*cos(x)**2 + cos(x)* *3),x)