Integrand size = 11, antiderivative size = 267 \[ \int \frac {1}{(\cos (3 x)+\cos (5 x))^4} \, dx=\frac {1}{32} \sqrt {\frac {1}{2} \left (21977-15540 \sqrt {2}\right )} \log \left (\sqrt {3-2 \sqrt {2}} \cos (x)-\sin (x)\right )+\frac {1}{32} \sqrt {\frac {1}{2} \left (21977+15540 \sqrt {2}\right )} \log \left (\sqrt {3+2 \sqrt {2}} \cos (x)-\sin (x)\right )-\frac {1}{32} \sqrt {\frac {1}{2} \left (21977-15540 \sqrt {2}\right )} \log \left (\sqrt {3-2 \sqrt {2}} \cos (x)+\sin (x)\right )-\frac {1}{32} \sqrt {\frac {1}{2} \left (21977+15540 \sqrt {2}\right )} \log \left (\sqrt {3+2 \sqrt {2}} \cos (x)+\sin (x)\right )+\frac {33 \tan (x)}{16}+\frac {\tan ^3(x)}{48}+\frac {32 \tan (x) \left (29-169 \tan ^2(x)\right )}{3 \left (1-6 \tan ^2(x)+\tan ^4(x)\right )^3}-\frac {28 \tan (x) \left (29+30 \tan ^2(x)\right )}{3 \left (1-6 \tan ^2(x)+\tan ^4(x)\right )^2}-\frac {\tan (x) \left (883+375 \tan ^2(x)\right )}{24 \left (1-6 \tan ^2(x)+\tan ^4(x)\right )} \] Output:
1/32*(105/2*2^(1/2)-74)*ln((2^(1/2)-1)*cos(x)-sin(x))+1/32*(105/2*2^(1/2)+ 74)*ln((1+2^(1/2))*cos(x)-sin(x))-1/32*(105/2*2^(1/2)-74)*ln((2^(1/2)-1)*c os(x)+sin(x))-1/32*(105/2*2^(1/2)+74)*ln((1+2^(1/2))*cos(x)+sin(x))+33/16* tan(x)+1/48*tan(x)^3+32/3*tan(x)*(29-169*tan(x)^2)/(1-6*tan(x)^2+tan(x)^4) ^3-28/3*tan(x)*(29+30*tan(x)^2)/(1-6*tan(x)^2+tan(x)^4)^2-tan(x)*(883+375* tan(x)^2)/(24-144*tan(x)^2+24*tan(x)^4)
Time = 0.86 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.80 \[ \int \frac {1}{(\cos (3 x)+\cos (5 x))^4} \, dx=\frac {1}{192} \left (888 \text {arctanh}\left (\sqrt {2}-\tan (x)\right )-888 \text {arctanh}\left (\sqrt {2}+\tan (x)\right )+315 \sqrt {2} \log \left (\sqrt {2}-2 \sin (2 x)\right )-315 \sqrt {2} \log \left (\sqrt {2}+2 \sin (2 x)\right )+\frac {12 (4-5 \sin (2 x))}{(\cos (2 x)-\sin (2 x))^2}+\frac {4 (-2+3 \sin (2 x))}{(\cos (2 x)-\sin (2 x))^3}+\frac {4 (2+3 \sin (2 x))}{(\cos (2 x)+\sin (2 x))^3}-\frac {12 (4+5 \sin (2 x))}{(\cos (2 x)+\sin (2 x))^2}+\frac {6 (-41+52 \sin (2 x))}{\cos (2 x)-\sin (2 x)}+\frac {6 (41+52 \sin (2 x))}{\cos (2 x)+\sin (2 x)}+392 \tan (x)+4 \sec ^2(x) \tan (x)\right ) \] Input:
Integrate[(Cos[3*x] + Cos[5*x])^(-4),x]
Output:
(888*ArcTanh[Sqrt[2] - Tan[x]] - 888*ArcTanh[Sqrt[2] + Tan[x]] + 315*Sqrt[ 2]*Log[Sqrt[2] - 2*Sin[2*x]] - 315*Sqrt[2]*Log[Sqrt[2] + 2*Sin[2*x]] + (12 *(4 - 5*Sin[2*x]))/(Cos[2*x] - Sin[2*x])^2 + (4*(-2 + 3*Sin[2*x]))/(Cos[2* x] - Sin[2*x])^3 + (4*(2 + 3*Sin[2*x]))/(Cos[2*x] + Sin[2*x])^3 - (12*(4 + 5*Sin[2*x]))/(Cos[2*x] + Sin[2*x])^2 + (6*(-41 + 52*Sin[2*x]))/(Cos[2*x] - Sin[2*x]) + (6*(41 + 52*Sin[2*x]))/(Cos[2*x] + Sin[2*x]) + 392*Tan[x] + 4*Sec[x]^2*Tan[x])/192
Time = 0.62 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.66, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {3042, 4823, 27, 1517, 27, 2206, 27, 2206, 27, 2205, 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.
| \(\displaystyle \int \frac {1}{(\cos (3 x)+\cos (5 x))^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(\cos (3 x)+\cos (5 x))^4}dx\) |
| \(\Big \downarrow \) 4823 |
| \(\displaystyle \int \frac {\left (\tan ^2(x)+1\right )^9}{16 \left (\tan ^4(x)-6 \tan ^2(x)+1\right )^4}d\tan (x)\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{16} \int \frac {\left (\tan ^2(x)+1\right )^9}{\left (\tan ^4(x)-6 \tan ^2(x)+1\right )^4}d\tan (x)\) |
| \(\Big \downarrow \) 1517 |
| \(\displaystyle \frac {1}{16} \left (\frac {512 \tan (x) \left (29-169 \tan ^2(x)\right )}{3 \left (\tan ^4(x)-6 \tan ^2(x)+1\right )^3}-\frac {1}{192} \int \frac {64 \left (-3 \tan ^{14}(x)-45 \tan ^{12}(x)-375 \tan ^{10}(x)-2457 \tan ^8(x)-14745 \tan ^6(x)-86391 \tan ^4(x)+274899 \tan ^2(x)+14845\right )}{\left (\tan ^4(x)-6 \tan ^2(x)+1\right )^3}d\tan (x)\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{16} \left (\frac {512 \tan (x) \left (29-169 \tan ^2(x)\right )}{3 \left (\tan ^4(x)-6 \tan ^2(x)+1\right )^3}-\frac {1}{3} \int \frac {-3 \tan ^{14}(x)-45 \tan ^{12}(x)-375 \tan ^{10}(x)-2457 \tan ^8(x)-14745 \tan ^6(x)-86391 \tan ^4(x)+274899 \tan ^2(x)+14845}{\left (\tan ^4(x)-6 \tan ^2(x)+1\right )^3}d\tan (x)\right )\) |
| \(\Big \downarrow \) 2206 |
| \(\displaystyle \frac {1}{16} \left (\frac {1}{3} \left (\frac {1}{128} \int -\frac {128 \left (-3 \tan ^{10}(x)-63 \tan ^8(x)-750 \tan ^6(x)-6894 \tan ^4(x)+11841 \tan ^2(x)+1853\right )}{\left (\tan ^4(x)-6 \tan ^2(x)+1\right )^2}d\tan (x)-\frac {448 \tan (x) \left (30 \tan ^2(x)+29\right )}{\left (\tan ^4(x)-6 \tan ^2(x)+1\right )^2}\right )+\frac {512 \tan (x) \left (29-169 \tan ^2(x)\right )}{3 \left (\tan ^4(x)-6 \tan ^2(x)+1\right )^3}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{16} \left (\frac {1}{3} \left (-\int \frac {-3 \tan ^{10}(x)-63 \tan ^8(x)-750 \tan ^6(x)-6894 \tan ^4(x)+11841 \tan ^2(x)+1853}{\left (\tan ^4(x)-6 \tan ^2(x)+1\right )^2}d\tan (x)-\frac {448 \tan (x) \left (30 \tan ^2(x)+29\right )}{\left (\tan ^4(x)-6 \tan ^2(x)+1\right )^2}\right )+\frac {512 \tan (x) \left (29-169 \tan ^2(x)\right )}{3 \left (\tan ^4(x)-6 \tan ^2(x)+1\right )^3}\right )\) |
| \(\Big \downarrow \) 2206 |
| \(\displaystyle \frac {1}{16} \left (\frac {1}{3} \left (\frac {1}{64} \int -\frac {192 \left (-\tan ^6(x)-27 \tan ^4(x)-161 \tan ^2(x)+29\right )}{\tan ^4(x)-6 \tan ^2(x)+1}d\tan (x)-\frac {448 \tan (x) \left (30 \tan ^2(x)+29\right )}{\left (\tan ^4(x)-6 \tan ^2(x)+1\right )^2}-\frac {2 \tan (x) \left (375 \tan ^2(x)+883\right )}{\tan ^4(x)-6 \tan ^2(x)+1}\right )+\frac {512 \tan (x) \left (29-169 \tan ^2(x)\right )}{3 \left (\tan ^4(x)-6 \tan ^2(x)+1\right )^3}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{16} \left (\frac {1}{3} \left (-3 \int \frac {-\tan ^6(x)-27 \tan ^4(x)-161 \tan ^2(x)+29}{\tan ^4(x)-6 \tan ^2(x)+1}d\tan (x)-\frac {448 \tan (x) \left (30 \tan ^2(x)+29\right )}{\left (\tan ^4(x)-6 \tan ^2(x)+1\right )^2}-\frac {2 \tan (x) \left (375 \tan ^2(x)+883\right )}{\tan ^4(x)-6 \tan ^2(x)+1}\right )+\frac {512 \tan (x) \left (29-169 \tan ^2(x)\right )}{3 \left (\tan ^4(x)-6 \tan ^2(x)+1\right )^3}\right )\) |
| \(\Big \downarrow \) 2205 |
| \(\displaystyle \frac {1}{16} \left (\frac {1}{3} \left (-3 \int \left (-\tan ^2(x)+\frac {2 \left (31-179 \tan ^2(x)\right )}{\tan ^4(x)-6 \tan ^2(x)+1}-33\right )d\tan (x)-\frac {448 \tan (x) \left (30 \tan ^2(x)+29\right )}{\left (\tan ^4(x)-6 \tan ^2(x)+1\right )^2}-\frac {2 \tan (x) \left (375 \tan ^2(x)+883\right )}{\tan ^4(x)-6 \tan ^2(x)+1}\right )+\frac {512 \tan (x) \left (29-169 \tan ^2(x)\right )}{3 \left (\tan ^4(x)-6 \tan ^2(x)+1\right )^3}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{16} \left (\frac {1}{3} \left (-3 \left (\sqrt {\frac {1}{2} \left (21977-15540 \sqrt {2}\right )} \text {arctanh}\left (\frac {\tan (x)}{\sqrt {3-2 \sqrt {2}}}\right )+\sqrt {\frac {1}{2} \left (21977+15540 \sqrt {2}\right )} \text {arctanh}\left (\frac {\tan (x)}{\sqrt {3+2 \sqrt {2}}}\right )-\frac {1}{3} \tan ^3(x)-33 \tan (x)\right )-\frac {448 \tan (x) \left (30 \tan ^2(x)+29\right )}{\left (\tan ^4(x)-6 \tan ^2(x)+1\right )^2}-\frac {2 \tan (x) \left (375 \tan ^2(x)+883\right )}{\tan ^4(x)-6 \tan ^2(x)+1}\right )+\frac {512 \tan (x) \left (29-169 \tan ^2(x)\right )}{3 \left (\tan ^4(x)-6 \tan ^2(x)+1\right )^3}\right )\) |
Input:
Int[(Cos[3*x] + Cos[5*x])^(-4),x]
Output:
((512*Tan[x]*(29 - 169*Tan[x]^2))/(3*(1 - 6*Tan[x]^2 + Tan[x]^4)^3) + (-3* (Sqrt[(21977 - 15540*Sqrt[2])/2]*ArcTanh[Tan[x]/Sqrt[3 - 2*Sqrt[2]]] + Sqr t[(21977 + 15540*Sqrt[2])/2]*ArcTanh[Tan[x]/Sqrt[3 + 2*Sqrt[2]]] - 33*Tan[ x] - Tan[x]^3/3) - (448*Tan[x]*(29 + 30*Tan[x]^2))/(1 - 6*Tan[x]^2 + Tan[x ]^4)^2 - (2*Tan[x]*(883 + 375*Tan[x]^2))/(1 - 6*Tan[x]^2 + Tan[x]^4))/3)/1 6
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x _Symbol] :> With[{f = Coeff[PolynomialRemainder[(d + e*x^2)^q, a + b*x^2 + c*x^4, x], x, 0], g = Coeff[PolynomialRemainder[(d + e*x^2)^q, a + b*x^2 + c*x^4, x], x, 2]}, Simp[x*(a + b*x^2 + c*x^4)^(p + 1)*((a*b*g - f*(b^2 - 2* a*c) - c*(b*f - 2*a*g)*x^2)/(2*a*(p + 1)*(b^2 - 4*a*c))), x] + Simp[1/(2*a* (p + 1)*(b^2 - 4*a*c)) Int[(a + b*x^2 + c*x^4)^(p + 1)*ExpandToSum[2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuotient[(d + e*x^2)^q, a + b*x^2 + c*x^4, x] + b^2*f*(2*p + 3) - 2*a*c*f*(4*p + 5) - a*b*g + c*(4*p + 7)*(b*f - 2*a*g)* x^2, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ [c*d^2 - b*d*e + a*e^2, 0] && IGtQ[q, 1] && LtQ[p, -1]
Int[(Px_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Int[ExpandInte grand[Px/(a + b*x^2 + c*x^4), x], x] /; FreeQ[{a, b, c}, x] && PolyQ[Px, x^ 2] && Expon[Px, x^2] > 1
Int[(Px_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{d = Coeff[PolynomialRemainder[Px, a + b*x^2 + c*x^4, x], x, 0], e = Coeff[Poly nomialRemainder[Px, a + b*x^2 + c*x^4, x], x, 2]}, Simp[x*(a + b*x^2 + c*x^ 4)^(p + 1)*((a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2)/(2*a*(p + 1)*(b ^2 - 4*a*c))), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c)) Int[(a + b*x^2 + c *x^4)^(p + 1)*ExpandToSum[2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuotient[Px, a + b*x^2 + c*x^4, x] + b^2*d*(2*p + 3) - 2*a*c*d*(4*p + 5) - a*b*e + c*(4* p + 7)*(b*d - 2*a*e)*x^2, x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Px, x ^2] && Expon[Px, x^2] > 1 && NeQ[b^2 - 4*a*c, 0] && 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*ArcTan[x]] + b*Cos[n*ArcTan[x]]]]^p/(1 + x^2), x], x, Tan[c + d*x]], x] / ; FreeQ[{a, b, c, d}, x] && ILtQ[p/2, 0] && IntegerQ[(m - 1)/2] && IntegerQ [(n - 1)/2]
Time = 17.89 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.60
| method | result | size |
| default | \(\frac {-\frac {125 \tan \left (x \right )^{5}}{2}+\frac {387 \tan \left (x \right )^{4}}{2}+\frac {13 \tan \left (x \right )^{3}}{3}-237 \tan \left (x \right )^{2}-\frac {269 \tan \left (x \right )}{2}-\frac {127}{6}}{8 \left (\tan \left (x \right )^{2}-2 \tan \left (x \right )-1\right )^{3}}+\frac {37 \ln \left (\tan \left (x \right )^{2}-2 \tan \left (x \right )-1\right )}{16}-\frac {105 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (2 \tan \left (x \right )-2\right ) \sqrt {2}}{4}\right )}{32}+\frac {\tan \left (x \right )^{3}}{48}+\frac {33 \tan \left (x \right )}{16}-\frac {\frac {125 \tan \left (x \right )^{5}}{2}+\frac {387 \tan \left (x \right )^{4}}{2}-\frac {13 \tan \left (x \right )^{3}}{3}-237 \tan \left (x \right )^{2}+\frac {269 \tan \left (x \right )}{2}-\frac {127}{6}}{8 \left (\tan \left (x \right )^{2}+2 \tan \left (x \right )-1\right )^{3}}-\frac {37 \ln \left (\tan \left (x \right )^{2}+2 \tan \left (x \right )-1\right )}{16}-\frac {105 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (2 \tan \left (x \right )+2\right ) \sqrt {2}}{4}\right )}{32}\) | \(160\) |
| risch | \(\frac {i \left (315 \,{\mathrm e}^{28 i x}+723 \,{\mathrm e}^{26 i x}+384 \,{\mathrm e}^{24 i x}-36 \,{\mathrm e}^{22 i x}+975 \,{\mathrm e}^{20 i x}+2159 \,{\mathrm e}^{18 i x}+1140 \,{\mathrm e}^{16 i x}-84 \,{\mathrm e}^{14 i x}+1009 \,{\mathrm e}^{12 i x}+2193 \,{\mathrm e}^{10 i x}+1092 \,{\mathrm e}^{8 i x}-32 \,{\mathrm e}^{6 i x}+333 \,{\mathrm e}^{4 i x}+741 \,{\mathrm e}^{2 i x}+352\right )}{48 \left ({\mathrm e}^{10 i x}+{\mathrm e}^{8 i x}+{\mathrm e}^{2 i x}+1\right )^{3}}-\frac {37 \ln \left ({\mathrm e}^{2 i x}-\frac {i \sqrt {2}}{2}-\frac {\sqrt {2}}{2}\right )}{16}+\frac {105 \ln \left ({\mathrm e}^{2 i x}-\frac {i \sqrt {2}}{2}-\frac {\sqrt {2}}{2}\right ) \sqrt {2}}{64}-\frac {37 \ln \left ({\mathrm e}^{2 i x}+\frac {i \sqrt {2}}{2}+\frac {\sqrt {2}}{2}\right )}{16}-\frac {105 \ln \left ({\mathrm e}^{2 i x}+\frac {i \sqrt {2}}{2}+\frac {\sqrt {2}}{2}\right ) \sqrt {2}}{64}+\frac {37 \ln \left ({\mathrm e}^{2 i x}-\frac {i \sqrt {2}}{2}+\frac {\sqrt {2}}{2}\right )}{16}+\frac {105 \ln \left ({\mathrm e}^{2 i x}-\frac {i \sqrt {2}}{2}+\frac {\sqrt {2}}{2}\right ) \sqrt {2}}{64}+\frac {37 \ln \left ({\mathrm e}^{2 i x}+\frac {i \sqrt {2}}{2}-\frac {\sqrt {2}}{2}\right )}{16}-\frac {105 \ln \left ({\mathrm e}^{2 i x}+\frac {i \sqrt {2}}{2}-\frac {\sqrt {2}}{2}\right ) \sqrt {2}}{64}\) | \(296\) |
Input:
int(1/(cos(3*x)+cos(5*x))^4,x,method=_RETURNVERBOSE)
Output:
1/8*(-125/2*tan(x)^5+387/2*tan(x)^4+13/3*tan(x)^3-237*tan(x)^2-269/2*tan(x )-127/6)/(tan(x)^2-2*tan(x)-1)^3+37/16*ln(tan(x)^2-2*tan(x)-1)-105/32*2^(1 /2)*arctanh(1/4*(2*tan(x)-2)*2^(1/2))+1/48*tan(x)^3+33/16*tan(x)-1/8*(125/ 2*tan(x)^5+387/2*tan(x)^4-13/3*tan(x)^3-237*tan(x)^2+269/2*tan(x)-127/6)/( tan(x)^2+2*tan(x)-1)^3-37/16*ln(tan(x)^2+2*tan(x)-1)-105/32*2^(1/2)*arctan h(1/4*(2*tan(x)+2)*2^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 460 vs. \(2 (177) = 354\).
Time = 0.13 (sec) , antiderivative size = 460, normalized size of antiderivative = 1.72 \[ \int \frac {1}{(\cos (3 x)+\cos (5 x))^4} \, dx =\text {Too large to display} \] Input:
integrate(1/(cos(3*x)+cos(5*x))^4,x, algorithm="fricas")
Output:
1/384*(444*(512*cos(x)^15 - 1536*cos(x)^13 + 1728*cos(x)^11 - 896*cos(x)^9 + 216*cos(x)^7 - 24*cos(x)^5 + cos(x)^3)*log(4*(2*cos(x)^3 - cos(x))*sin( x) + 1) - 444*(512*cos(x)^15 - 1536*cos(x)^13 + 1728*cos(x)^11 - 896*cos(x )^9 + 216*cos(x)^7 - 24*cos(x)^5 + cos(x)^3)*log(-4*(2*cos(x)^3 - cos(x))* sin(x) + 1) + 315*(512*sqrt(2)*cos(x)^15 - 1536*sqrt(2)*cos(x)^13 + 1728*s qrt(2)*cos(x)^11 - 896*sqrt(2)*cos(x)^9 + 216*sqrt(2)*cos(x)^7 - 24*sqrt(2 )*cos(x)^5 + sqrt(2)*cos(x)^3)*log((4*sqrt(2)*cos(x)^2 - 4*(2*cos(x)^3 + ( sqrt(2) - 1)*cos(x))*sin(x) - 2*sqrt(2) + 3)/(4*(2*cos(x)^3 - cos(x))*sin( x) + 1)) + 315*(512*sqrt(2)*cos(x)^15 - 1536*sqrt(2)*cos(x)^13 + 1728*sqrt (2)*cos(x)^11 - 896*sqrt(2)*cos(x)^9 + 216*sqrt(2)*cos(x)^7 - 24*sqrt(2)*c os(x)^5 + sqrt(2)*cos(x)^3)*log(-(4*sqrt(2)*cos(x)^2 - 4*(2*cos(x)^3 - (sq rt(2) + 1)*cos(x))*sin(x) - 2*sqrt(2) - 3)/(4*(2*cos(x)^3 - cos(x))*sin(x) - 1)) + 8*(90112*cos(x)^14 - 265600*cos(x)^12 + 290496*cos(x)^10 - 143264 *cos(x)^8 + 31256*cos(x)^6 - 2886*cos(x)^4 + 74*cos(x)^2 + 1)*sin(x))/(512 *cos(x)^15 - 1536*cos(x)^13 + 1728*cos(x)^11 - 896*cos(x)^9 + 216*cos(x)^7 - 24*cos(x)^5 + cos(x)^3)
\[ \int \frac {1}{(\cos (3 x)+\cos (5 x))^4} \, dx=\int \frac {1}{\left (\cos {\left (3 x \right )} + \cos {\left (5 x \right )}\right )^{4}}\, dx \] Input:
integrate(1/(cos(3*x)+cos(5*x))**4,x)
Output:
Integral((cos(3*x) + cos(5*x))**(-4), x)
Leaf count of result is larger than twice the leaf count of optimal. 17029 vs. \(2 (177) = 354\).
Time = 1.19 (sec) , antiderivative size = 17029, normalized size of antiderivative = 63.78 \[ \int \frac {1}{(\cos (3 x)+\cos (5 x))^4} \, dx=\text {Too large to display} \] Input:
integrate(1/(cos(3*x)+cos(5*x))^4,x, algorithm="maxima")
Output:
-1/384*(8*(315*sin(28*x) + 723*sin(26*x) + 384*sin(24*x) - 36*sin(22*x) + 975*sin(20*x) + 2159*sin(18*x) + 1140*sin(16*x) - 84*sin(14*x) + 1009*sin( 12*x) + 2193*sin(10*x) + 1092*sin(8*x) - 32*sin(6*x) + 333*sin(4*x) + 741* sin(2*x))*cos(30*x) + 24*(408*sin(26*x) + 279*sin(24*x) - 351*sin(22*x) + 30*sin(20*x) + 1214*sin(18*x) + 825*sin(16*x) - 399*sin(14*x) + 64*sin(12* x) + 1248*sin(10*x) + 777*sin(8*x) - 137*sin(6*x) + 18*sin(4*x) + 426*sin( 2*x))*cos(28*x) + 24*(143*sin(24*x) - 759*sin(22*x) - 1194*sin(20*x) - 10* sin(18*x) + 417*sin(16*x) - 807*sin(14*x) - 1160*sin(12*x) + 24*sin(10*x) + 369*sin(8*x) - 273*sin(6*x) - 390*sin(4*x) + 18*sin(2*x))*cos(26*x) - 8* (1188*sin(22*x) + 2481*sin(20*x) + 1297*sin(18*x) + 12*sin(16*x) + 1236*si n(14*x) + 2447*sin(12*x) + 1263*sin(10*x) + 60*sin(8*x) + 416*sin(6*x) + 8 19*sin(4*x) + 411*sin(2*x))*cos(24*x) + 24*(1083*sin(20*x) + 2267*sin(18*x ) + 1176*sin(16*x) - 48*sin(14*x) + 1117*sin(12*x) + 2301*sin(10*x) + 1128 *sin(8*x) - 20*sin(6*x) + 369*sin(4*x) + 777*sin(2*x))*cos(22*x) + 24*(355 2*sin(18*x) + 2445*sin(16*x) - 1227*sin(14*x) + 102*sin(12*x) + 3654*sin(1 0*x) + 2301*sin(8*x) - 421*sin(6*x) + 24*sin(4*x) + 1248*sin(2*x))*cos(20* x) + 8*(3783*sin(16*x) - 7233*sin(14*x) - 10350*sin(12*x) + 306*sin(10*x) + 3351*sin(8*x) - 2447*sin(6*x) - 3480*sin(4*x) + 192*sin(2*x))*cos(18*x) - 24*(1224*sin(14*x) + 2411*sin(12*x) + 1227*sin(10*x) + 48*sin(8*x) + 412 *sin(6*x) + 807*sin(4*x) + 399*sin(2*x))*cos(16*x) + 24*(1261*sin(12*x)...
Time = 0.15 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.58 \[ \int \frac {1}{(\cos (3 x)+\cos (5 x))^4} \, dx=\frac {1}{48} \, \tan \left (x\right )^{3} + \frac {105}{64} \, \sqrt {2} \log \left (\frac {{\left | -2 \, \sqrt {2} + 2 \, \tan \left (x\right ) + 2 \right |}}{{\left | 2 \, \sqrt {2} + 2 \, \tan \left (x\right ) + 2 \right |}}\right ) + \frac {105}{64} \, \sqrt {2} \log \left (\frac {{\left | -2 \, \sqrt {2} + 2 \, \tan \left (x\right ) - 2 \right |}}{{\left | 2 \, \sqrt {2} + 2 \, \tan \left (x\right ) - 2 \right |}}\right ) - \frac {375 \, \tan \left (x\right )^{11} - 3617 \, \tan \left (x\right )^{9} + 10374 \, \tan \left (x\right )^{7} - 4770 \, \tan \left (x\right )^{5} + 787 \, \tan \left (x\right )^{3} - 45 \, \tan \left (x\right )}{24 \, {\left (\tan \left (x\right )^{4} - 6 \, \tan \left (x\right )^{2} + 1\right )}^{3}} - \frac {37}{16} \, \log \left ({\left | \tan \left (x\right )^{2} + 2 \, \tan \left (x\right ) - 1 \right |}\right ) + \frac {37}{16} \, \log \left ({\left | \tan \left (x\right )^{2} - 2 \, \tan \left (x\right ) - 1 \right |}\right ) + \frac {33}{16} \, \tan \left (x\right ) \] Input:
integrate(1/(cos(3*x)+cos(5*x))^4,x, algorithm="giac")
Output:
1/48*tan(x)^3 + 105/64*sqrt(2)*log(abs(-2*sqrt(2) + 2*tan(x) + 2)/abs(2*sq rt(2) + 2*tan(x) + 2)) + 105/64*sqrt(2)*log(abs(-2*sqrt(2) + 2*tan(x) - 2) /abs(2*sqrt(2) + 2*tan(x) - 2)) - 1/24*(375*tan(x)^11 - 3617*tan(x)^9 + 10 374*tan(x)^7 - 4770*tan(x)^5 + 787*tan(x)^3 - 45*tan(x))/(tan(x)^4 - 6*tan (x)^2 + 1)^3 - 37/16*log(abs(tan(x)^2 + 2*tan(x) - 1)) + 37/16*log(abs(tan (x)^2 - 2*tan(x) - 1)) + 33/16*tan(x)
Time = 22.22 (sec) , antiderivative size = 753, normalized size of antiderivative = 2.82 \[ \int \frac {1}{(\cos (3 x)+\cos (5 x))^4} \, dx=\text {Too large to display} \] Input:
int(1/(cos(3*x) + cos(5*x))^4,x)
Output:
atan((tan(x/2)*163633626628096i + 2^(1/2)*tan(x/2)*163633626628096i - 2^(1 /2)*tan(x/2)^3*163633626628096i - tan(x/2)^3*163633626628096i)/(8181681331 4048*tan(x/2)^4 - 163633626628096*tan(x/2)^2 + 81816813314048))*((2^(1/2)* 105i)/32 - 37i/8) - ((63*tan(x/2))/8 - (8033*tan(x/2)^3)/12 + (168805*tan( x/2)^5)/8 - (596143*tan(x/2)^7)/2 + (45849605*tan(x/2)^9)/24 - (26362213*t an(x/2)^11)/4 + (107582213*tan(x/2)^13)/8 - (50937899*tan(x/2)^15)/3 + (10 7582213*tan(x/2)^17)/8 - (26362213*tan(x/2)^19)/4 + (45849605*tan(x/2)^21) /24 - (596143*tan(x/2)^23)/2 + (168805*tan(x/2)^25)/8 - (8033*tan(x/2)^27) /12 + (63*tan(x/2)^29)/8)/(87*tan(x/2)^2 - 2817*tan(x/2)^4 + 41735*tan(x/2 )^6 - 293205*tan(x/2)^8 + 1145475*tan(x/2)^10 - 2731493*tan(x/2)^12 + 4173 795*tan(x/2)^14 - 4173795*tan(x/2)^16 + 2731493*tan(x/2)^18 - 1145475*tan( x/2)^20 + 293205*tan(x/2)^22 - 41735*tan(x/2)^24 + 2817*tan(x/2)^26 - 87*t an(x/2)^28 + tan(x/2)^30 - 1) - atan(-(((105*2^(1/2))/64 + 37/16)*(8932066 246656*tan(x/2) - ((105*2^(1/2))/64 + 37/16)*(((105*2^(1/2))/64 + 37/16)*( 152251111003127808*tan(x/2) - ((105*2^(1/2))/64 + 37/16)*(((105*2^(1/2))/6 4 + 37/16)*(6971659634343936*tan(x/2) + ((105*2^(1/2))/64 + 37/16)*(533373 0906341376*tan(x/2)^2 - 484884627849216)) - 152256050819497984*tan(x/2)^2 + 11255239629864960)) - 825607983117369344*tan(x/2)^2 + 32899822855913472) )*1i + ((105*2^(1/2))/64 + 37/16)*(8932066246656*tan(x/2) - ((105*2^(1/2)) /64 + 37/16)*(((105*2^(1/2))/64 + 37/16)*(152251111003127808*tan(x/2) -...
\[ \int \frac {1}{(\cos (3 x)+\cos (5 x))^4} \, dx=\int \frac {1}{\cos \left (5 x \right )^{4}+4 \cos \left (5 x \right )^{3} \cos \left (3 x \right )+6 \cos \left (5 x \right )^{2} \cos \left (3 x \right )^{2}+4 \cos \left (5 x \right ) \cos \left (3 x \right )^{3}+\cos \left (3 x \right )^{4}}d x \] Input:
int(1/(cos(3*x)+cos(5*x))^4,x)
Output:
int(1/(cos(5*x)**4 + 4*cos(5*x)**3*cos(3*x) + 6*cos(5*x)**2*cos(3*x)**2 + 4*cos(5*x)*cos(3*x)**3 + cos(3*x)**4),x)