\(\int \frac {1}{(\sin (3 x)+\sin (5 x))^3} \, dx\) [32]

Optimal result

Integrand size = 11, antiderivative size = 92 \[ \int \frac {1}{(\sin (3 x)+\sin (5 x))^3} \, dx=-\frac {19 \text {arctanh}(\cos (x))}{1024}+\frac {31 \text {arctanh}\left (\sqrt {2} \cos (x)\right )}{128 \sqrt {2}}-\frac {105 \sec (x)}{1024}-\frac {43 \sec ^3(x)}{3072}-\frac {3 \sec ^5(x)}{1280}+\frac {\sec ^5(x) \sec (2 x)}{1024}+\frac {1}{256} \sec ^5(x) \sec ^2(2 x)-\frac {\csc ^2(x) \sec ^5(x) \sec ^2(2 x)}{1024} \] Output:

-19/1024*arctanh(cos(x))+31/256*arctanh(cos(x)*2^(1/2))*2^(1/2)-105/1024*s 
ec(x)-43/3072*sec(x)^3-3/1280*sec(x)^5+1/1024*sec(x)^5*sec(2*x)+1/256*sec( 
x)^5*sec(2*x)^2-1/1024*csc(x)^2*sec(x)^5*sec(2*x)^2
 

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(277\) vs. \(2(92)=184\).

Time = 2.96 (sec) , antiderivative size = 277, normalized size of antiderivative = 3.01 \[ \int \frac {1}{(\sin (3 x)+\sin (5 x))^3} \, dx=\frac {\cos ^6(x) \left (-14880 \sqrt {2} \text {arctanh}\left (\frac {-1+\tan \left (\frac {x}{2}\right )}{\sqrt {2}}\right )+14880 \sqrt {2} \text {arctanh}\left (\frac {1+\tan \left (\frac {x}{2}\right )}{\sqrt {2}}\right )+\frac {1}{32} \csc ^2(x) \left (-7444-8604 \cos (2 x)-3264 \cos (4 x)+4302 \cos (6 x)+8020 \cos (8 x)+3150 \cos (10 x)-1710 \cos (x) \log \left (\cos \left (\frac {x}{2}\right )\right )-570 \cos (3 x) \log \left (\cos \left (\frac {x}{2}\right )\right )+285 \cos (5 x) \log \left (\cos \left (\frac {x}{2}\right )\right )+855 \cos (7 x) \log \left (\cos \left (\frac {x}{2}\right )\right )+855 \cos (9 x) \log \left (\cos \left (\frac {x}{2}\right )\right )+285 \cos (11 x) \log \left (\cos \left (\frac {x}{2}\right )\right )+1710 \cos (x) \log \left (\sin \left (\frac {x}{2}\right )\right )+570 \cos (3 x) \log \left (\sin \left (\frac {x}{2}\right )\right )-285 \cos (5 x) \log \left (\sin \left (\frac {x}{2}\right )\right )-855 \cos (7 x) \log \left (\sin \left (\frac {x}{2}\right )\right )-855 \cos (9 x) \log \left (\sin \left (\frac {x}{2}\right )\right )-285 \cos (11 x) \log \left (\sin \left (\frac {x}{2}\right )\right )\right ) \sec ^5(x) \sec ^2(2 x)\right ) (-\sin (x)+\sin (3 x))^3}{1920 (\sin (3 x)+\sin (5 x))^3} \] Input:

Integrate[(Sin[3*x] + Sin[5*x])^(-3),x]
 

Output:

(Cos[x]^6*(-14880*Sqrt[2]*ArcTanh[(-1 + Tan[x/2])/Sqrt[2]] + 14880*Sqrt[2] 
*ArcTanh[(1 + Tan[x/2])/Sqrt[2]] + (Csc[x]^2*(-7444 - 8604*Cos[2*x] - 3264 
*Cos[4*x] + 4302*Cos[6*x] + 8020*Cos[8*x] + 3150*Cos[10*x] - 1710*Cos[x]*L 
og[Cos[x/2]] - 570*Cos[3*x]*Log[Cos[x/2]] + 285*Cos[5*x]*Log[Cos[x/2]] + 8 
55*Cos[7*x]*Log[Cos[x/2]] + 855*Cos[9*x]*Log[Cos[x/2]] + 285*Cos[11*x]*Log 
[Cos[x/2]] + 1710*Cos[x]*Log[Sin[x/2]] + 570*Cos[3*x]*Log[Sin[x/2]] - 285* 
Cos[5*x]*Log[Sin[x/2]] - 855*Cos[7*x]*Log[Sin[x/2]] - 855*Cos[9*x]*Log[Sin 
[x/2]] - 285*Cos[11*x]*Log[Sin[x/2]])*Sec[x]^5*Sec[2*x]^2)/32)*(-Sin[x] + 
Sin[3*x])^3)/(1920*(Sin[3*x] + Sin[5*x])^3)
 

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.42, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.455, Rules used = {3042, 4824, 27, 374, 27, 441, 441, 27, 445, 27, 445, 27, 445, 25, 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.

\(\displaystyle \int \frac {1}{(\sin (3 x)+\sin (5 x))^3} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{(\sin (3 x)+\sin (5 x))^3}dx\)

\(\Big \downarrow \) 4824

\(\displaystyle -\int -\frac {\sec ^6(x)}{512 \left (1-2 \cos ^2(x)\right )^3 \left (1-\cos ^2(x)\right )^2}d\cos (x)\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{512} \int \frac {\sec ^6(x)}{\left (1-2 \cos ^2(x)\right )^3 \left (1-\cos ^2(x)\right )^2}d\cos (x)\)

\(\Big \downarrow \) 374

\(\displaystyle \frac {1}{512} \left (\frac {1}{4} \int \frac {2 \left (7-11 \cos ^2(x)\right ) \sec ^6(x)}{\left (1-2 \cos ^2(x)\right )^2 \left (1-\cos ^2(x)\right )^2}d\cos (x)+\frac {\sec ^5(x)}{2 \left (1-2 \cos ^2(x)\right )^2 \left (1-\cos ^2(x)\right )}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{512} \left (\frac {1}{2} \int \frac {\left (7-11 \cos ^2(x)\right ) \sec ^6(x)}{\left (1-2 \cos ^2(x)\right )^2 \left (1-\cos ^2(x)\right )^2}d\cos (x)+\frac {\sec ^5(x)}{2 \left (1-2 \cos ^2(x)\right )^2 \left (1-\cos ^2(x)\right )}\right )\)

\(\Big \downarrow \) 441

\(\displaystyle \frac {1}{512} \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {\left (29-27 \cos ^2(x)\right ) \sec ^6(x)}{\left (1-2 \cos ^2(x)\right ) \left (1-\cos ^2(x)\right )^2}d\cos (x)+\frac {3 \sec ^5(x)}{2 \left (1-2 \cos ^2(x)\right ) \left (1-\cos ^2(x)\right )}\right )+\frac {\sec ^5(x)}{2 \left (1-2 \cos ^2(x)\right )^2 \left (1-\cos ^2(x)\right )}\right )\)

\(\Big \downarrow \) 441

\(\displaystyle \frac {1}{512} \left (\frac {1}{2} \left (\frac {1}{2} \left (-\frac {1}{2} \int -\frac {4 \left (7 \cos ^2(x)+12\right ) \sec ^6(x)}{\left (1-2 \cos ^2(x)\right ) \left (1-\cos ^2(x)\right )}d\cos (x)-\frac {\sec ^5(x)}{1-\cos ^2(x)}\right )+\frac {3 \sec ^5(x)}{2 \left (1-2 \cos ^2(x)\right ) \left (1-\cos ^2(x)\right )}\right )+\frac {\sec ^5(x)}{2 \left (1-2 \cos ^2(x)\right )^2 \left (1-\cos ^2(x)\right )}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{512} \left (\frac {1}{2} \left (\frac {1}{2} \left (2 \int \frac {\left (7 \cos ^2(x)+12\right ) \sec ^6(x)}{\left (1-2 \cos ^2(x)\right ) \left (1-\cos ^2(x)\right )}d\cos (x)-\frac {\sec ^5(x)}{1-\cos ^2(x)}\right )+\frac {3 \sec ^5(x)}{2 \left (1-2 \cos ^2(x)\right ) \left (1-\cos ^2(x)\right )}\right )+\frac {\sec ^5(x)}{2 \left (1-2 \cos ^2(x)\right )^2 \left (1-\cos ^2(x)\right )}\right )\)

\(\Big \downarrow \) 445

\(\displaystyle \frac {1}{512} \left (\frac {1}{2} \left (\frac {1}{2} \left (2 \left (-\frac {1}{5} \int -\frac {5 \left (43-24 \cos ^2(x)\right ) \sec ^4(x)}{\left (1-2 \cos ^2(x)\right ) \left (1-\cos ^2(x)\right )}d\cos (x)-\frac {12}{5} \sec ^5(x)\right )-\frac {\sec ^5(x)}{1-\cos ^2(x)}\right )+\frac {3 \sec ^5(x)}{2 \left (1-2 \cos ^2(x)\right ) \left (1-\cos ^2(x)\right )}\right )+\frac {\sec ^5(x)}{2 \left (1-2 \cos ^2(x)\right )^2 \left (1-\cos ^2(x)\right )}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{512} \left (\frac {1}{2} \left (\frac {1}{2} \left (2 \left (\int \frac {\left (43-24 \cos ^2(x)\right ) \sec ^4(x)}{\left (1-2 \cos ^2(x)\right ) \left (1-\cos ^2(x)\right )}d\cos (x)-\frac {12 \sec ^5(x)}{5}\right )-\frac {\sec ^5(x)}{1-\cos ^2(x)}\right )+\frac {3 \sec ^5(x)}{2 \left (1-2 \cos ^2(x)\right ) \left (1-\cos ^2(x)\right )}\right )+\frac {\sec ^5(x)}{2 \left (1-2 \cos ^2(x)\right )^2 \left (1-\cos ^2(x)\right )}\right )\)

\(\Big \downarrow \) 445

\(\displaystyle \frac {1}{512} \left (\frac {1}{2} \left (\frac {1}{2} \left (2 \left (-\frac {1}{3} \int -\frac {3 \left (105-86 \cos ^2(x)\right ) \sec ^2(x)}{\left (1-2 \cos ^2(x)\right ) \left (1-\cos ^2(x)\right )}d\cos (x)-\frac {12}{5} \sec ^5(x)-\frac {43 \sec ^3(x)}{3}\right )-\frac {\sec ^5(x)}{1-\cos ^2(x)}\right )+\frac {3 \sec ^5(x)}{2 \left (1-2 \cos ^2(x)\right ) \left (1-\cos ^2(x)\right )}\right )+\frac {\sec ^5(x)}{2 \left (1-2 \cos ^2(x)\right )^2 \left (1-\cos ^2(x)\right )}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{512} \left (\frac {1}{2} \left (\frac {1}{2} \left (2 \left (\int \frac {\left (105-86 \cos ^2(x)\right ) \sec ^2(x)}{\left (1-2 \cos ^2(x)\right ) \left (1-\cos ^2(x)\right )}d\cos (x)-\frac {12}{5} \sec ^5(x)-\frac {43 \sec ^3(x)}{3}\right )-\frac {\sec ^5(x)}{1-\cos ^2(x)}\right )+\frac {3 \sec ^5(x)}{2 \left (1-2 \cos ^2(x)\right ) \left (1-\cos ^2(x)\right )}\right )+\frac {\sec ^5(x)}{2 \left (1-2 \cos ^2(x)\right )^2 \left (1-\cos ^2(x)\right )}\right )\)

\(\Big \downarrow \) 445

\(\displaystyle \frac {1}{512} \left (\frac {1}{2} \left (\frac {1}{2} \left (2 \left (-\int -\frac {229-210 \cos ^2(x)}{\left (1-2 \cos ^2(x)\right ) \left (1-\cos ^2(x)\right )}d\cos (x)-\frac {12}{5} \sec ^5(x)-\frac {43 \sec ^3(x)}{3}-105 \sec (x)\right )-\frac {\sec ^5(x)}{1-\cos ^2(x)}\right )+\frac {3 \sec ^5(x)}{2 \left (1-2 \cos ^2(x)\right ) \left (1-\cos ^2(x)\right )}\right )+\frac {\sec ^5(x)}{2 \left (1-2 \cos ^2(x)\right )^2 \left (1-\cos ^2(x)\right )}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{512} \left (\frac {1}{2} \left (\frac {1}{2} \left (2 \left (\int \frac {229-210 \cos ^2(x)}{\left (1-2 \cos ^2(x)\right ) \left (1-\cos ^2(x)\right )}d\cos (x)-\frac {12}{5} \sec ^5(x)-\frac {43 \sec ^3(x)}{3}-105 \sec (x)\right )-\frac {\sec ^5(x)}{1-\cos ^2(x)}\right )+\frac {3 \sec ^5(x)}{2 \left (1-2 \cos ^2(x)\right ) \left (1-\cos ^2(x)\right )}\right )+\frac {\sec ^5(x)}{2 \left (1-2 \cos ^2(x)\right )^2 \left (1-\cos ^2(x)\right )}\right )\)

\(\Big \downarrow \) 397

\(\displaystyle \frac {1}{512} \left (\frac {1}{2} \left (\frac {1}{2} \left (2 \left (248 \int \frac {1}{1-2 \cos ^2(x)}d\cos (x)-19 \int \frac {1}{1-\cos ^2(x)}d\cos (x)-\frac {12}{5} \sec ^5(x)-\frac {43 \sec ^3(x)}{3}-105 \sec (x)\right )-\frac {\sec ^5(x)}{1-\cos ^2(x)}\right )+\frac {3 \sec ^5(x)}{2 \left (1-2 \cos ^2(x)\right ) \left (1-\cos ^2(x)\right )}\right )+\frac {\sec ^5(x)}{2 \left (1-2 \cos ^2(x)\right )^2 \left (1-\cos ^2(x)\right )}\right )\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{512} \left (\frac {1}{2} \left (\frac {1}{2} \left (2 \left (-19 \text {arctanh}(\cos (x))+124 \sqrt {2} \text {arctanh}\left (\sqrt {2} \cos (x)\right )-\frac {12}{5} \sec ^5(x)-\frac {43 \sec ^3(x)}{3}-105 \sec (x)\right )-\frac {\sec ^5(x)}{1-\cos ^2(x)}\right )+\frac {3 \sec ^5(x)}{2 \left (1-2 \cos ^2(x)\right ) \left (1-\cos ^2(x)\right )}\right )+\frac {\sec ^5(x)}{2 \left (1-2 \cos ^2(x)\right )^2 \left (1-\cos ^2(x)\right )}\right )\)

Input:

Int[(Sin[3*x] + Sin[5*x])^(-3),x]
 

Output:

(Sec[x]^5/(2*(1 - 2*Cos[x]^2)^2*(1 - Cos[x]^2)) + ((3*Sec[x]^5)/(2*(1 - 2* 
Cos[x]^2)*(1 - Cos[x]^2)) + (-(Sec[x]^5/(1 - Cos[x]^2)) + 2*(-19*ArcTanh[C 
os[x]] + 124*Sqrt[2]*ArcTanh[Sqrt[2]*Cos[x]] - 105*Sec[x] - (43*Sec[x]^3)/ 
3 - (12*Sec[x]^5)/5))/2)/2)/512
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 219

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])
 

rule 374

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ 
), x_Symbol] :> Simp[(-b)*(e*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q 
 + 1)/(a*e*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) 
 Int[(e*x)^m*(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[b*c*(m + 1) + 2*(b*c - 
a*d)*(p + 1) + d*b*(m + 2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b, 
c, d, e, m, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && IntBinomialQ[a, b, 
 c, d, e, m, 2, p, q, x]
 

rule 397

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]
 

rule 441

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ 
)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[(-(b*e - a*f))*(g*x)^(m + 1)*(a 
+ b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(a*g*2*(b*c - a*d)*(p + 1))), x] + Si 
mp[1/(a*2*(b*c - a*d)*(p + 1))   Int[(g*x)^m*(a + b*x^2)^(p + 1)*(c + d*x^2 
)^q*Simp[c*(b*e - a*f)*(m + 1) + e*2*(b*c - a*d)*(p + 1) + d*(b*e - a*f)*(m 
 + 2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m, q}, 
 x] && LtQ[p, -1]
 

rule 445

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_ 
.)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^2)^(p 
+ 1)*((c + d*x^2)^(q + 1)/(a*c*g*(m + 1))), x] + Simp[1/(a*c*g^2*(m + 1)) 
 Int[(g*x)^(m + 2)*(a + b*x^2)^p*(c + d*x^2)^q*Simp[a*f*c*(m + 1) - e*(b*c 
+ a*d)*(m + 2 + 1) - e*2*(b*c*p + a*d*q) - b*e*d*(m + 2*(p + q + 2) + 1)*x^ 
2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && LtQ[m, -1]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4824

Int[((a_.)*sin[(m_.)*((c_.) + (d_.)*(x_))] + (b_.)*sin[(n_.)*((c_.) + (d_.) 
*(x_))])^(p_), x_Symbol] :> Simp[-d^(-1)   Subst[Int[Simplify[TrigExpand[a* 
Sin[m*ArcCos[x]] + b*Sin[n*ArcCos[x]]]]^p/Sqrt[1 - x^2], x], x, Cos[c + d*x 
]], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[(p - 1)/2, 0] && IntegerQ[(m - 1)/ 
2] && IntegerQ[(n - 1)/2]
 

Maple [A] (verified)

Time = 6.58 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.92

method	result	size
default	\(\frac {1}{2048+2048 \cos \left (x \right )}-\frac {19 \ln \left (1+\cos \left (x \right )\right )}{2048}+\frac {1}{2048 \cos \left (x \right )-2048}+\frac {19 \ln \left (\cos \left (x \right )-1\right )}{2048}-\frac {\frac {7 \cos \left (x \right )^{3}}{16}-\frac {9 \cos \left (x \right )}{32}}{4 \left (2 \cos \left (x \right )^{2}-1\right )^{2}}+\frac {31 \,\operatorname {arctanh}\left (\sqrt {2}\, \cos \left (x \right )\right ) \sqrt {2}}{256}-\frac {1}{2560 \cos \left (x \right )^{5}}-\frac {1}{192 \cos \left (x \right )^{3}}-\frac {39}{512 \cos \left (x \right )}\)	\(85\)
risch	\(-\frac {1575 \,{\mathrm e}^{21 i x}+4010 \,{\mathrm e}^{19 i x}+2151 \,{\mathrm e}^{17 i x}-1632 \,{\mathrm e}^{15 i x}-4302 \,{\mathrm e}^{13 i x}-7444 \,{\mathrm e}^{11 i x}-4302 \,{\mathrm e}^{9 i x}-1632 \,{\mathrm e}^{7 i x}+2151 \,{\mathrm e}^{5 i x}+4010 \,{\mathrm e}^{3 i x}+1575 \,{\mathrm e}^{i x}}{7680 \left ({\mathrm e}^{2 i x}+1\right )^{5} \left ({\mathrm e}^{6 i x}-{\mathrm e}^{4 i x}+{\mathrm e}^{2 i x}-1\right )^{2}}+\frac {19 \ln \left ({\mathrm e}^{i x}-1\right )}{1024}-\frac {19 \ln \left ({\mathrm e}^{i x}+1\right )}{1024}+\frac {31 \sqrt {2}\, \ln \left ({\mathrm e}^{2 i x}+\sqrt {2}\, {\mathrm e}^{i x}+1\right )}{512}-\frac {31 \sqrt {2}\, \ln \left ({\mathrm e}^{2 i x}-\sqrt {2}\, {\mathrm e}^{i x}+1\right )}{512}\)	\(177\)

Input:

int(1/(sin(3*x)+sin(5*x))^3,x,method=_RETURNVERBOSE)
 

Output:

1/2048/(1+cos(x))-19/2048*ln(1+cos(x))+1/2048/(cos(x)-1)+19/2048*ln(cos(x) 
-1)-1/4*(7/16*cos(x)^3-9/32*cos(x))/(2*cos(x)^2-1)^2+31/256*arctanh(2^(1/2 
)*cos(x))*2^(1/2)-1/2560/cos(x)^5-1/192/cos(x)^3-39/512/cos(x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 196 vs. \(2 (72) = 144\).

Time = 0.10 (sec) , antiderivative size = 196, normalized size of antiderivative = 2.13 \[ \int \frac {1}{(\sin (3 x)+\sin (5 x))^3} \, dx=-\frac {12600 \, \cos \left (x\right )^{10} - 23480 \, \cos \left (x\right )^{8} + 12598 \, \cos \left (x\right )^{6} - 1636 \, \cos \left (x\right )^{4} - 100 \, \cos \left (x\right )^{2} - 1860 \, {\left (4 \, \sqrt {2} \cos \left (x\right )^{11} - 8 \, \sqrt {2} \cos \left (x\right )^{9} + 5 \, \sqrt {2} \cos \left (x\right )^{7} - \sqrt {2} \cos \left (x\right )^{5}\right )} \log \left (-\frac {2 \, \cos \left (x\right )^{2} + 2 \, \sqrt {2} \cos \left (x\right ) + 1}{2 \, \cos \left (x\right )^{2} - 1}\right ) + 285 \, {\left (4 \, \cos \left (x\right )^{11} - 8 \, \cos \left (x\right )^{9} + 5 \, \cos \left (x\right )^{7} - \cos \left (x\right )^{5}\right )} \log \left (\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) - 285 \, {\left (4 \, \cos \left (x\right )^{11} - 8 \, \cos \left (x\right )^{9} + 5 \, \cos \left (x\right )^{7} - \cos \left (x\right )^{5}\right )} \log \left (-\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) - 12}{30720 \, {\left (4 \, \cos \left (x\right )^{11} - 8 \, \cos \left (x\right )^{9} + 5 \, \cos \left (x\right )^{7} - \cos \left (x\right )^{5}\right )}} \] Input:

integrate(1/(sin(3*x)+sin(5*x))^3,x, algorithm="fricas")
 

Output:

-1/30720*(12600*cos(x)^10 - 23480*cos(x)^8 + 12598*cos(x)^6 - 1636*cos(x)^ 
4 - 100*cos(x)^2 - 1860*(4*sqrt(2)*cos(x)^11 - 8*sqrt(2)*cos(x)^9 + 5*sqrt 
(2)*cos(x)^7 - sqrt(2)*cos(x)^5)*log(-(2*cos(x)^2 + 2*sqrt(2)*cos(x) + 1)/ 
(2*cos(x)^2 - 1)) + 285*(4*cos(x)^11 - 8*cos(x)^9 + 5*cos(x)^7 - cos(x)^5) 
*log(1/2*cos(x) + 1/2) - 285*(4*cos(x)^11 - 8*cos(x)^9 + 5*cos(x)^7 - cos( 
x)^5)*log(-1/2*cos(x) + 1/2) - 12)/(4*cos(x)^11 - 8*cos(x)^9 + 5*cos(x)^7 
- cos(x)^5)
 

Sympy [F]

\[ \int \frac {1}{(\sin (3 x)+\sin (5 x))^3} \, dx=\int \frac {1}{\left (\sin {\left (3 x \right )} + \sin {\left (5 x \right )}\right )^{3}}\, dx \] Input:

integrate(1/(sin(3*x)+sin(5*x))**3,x)
 

Output:

Integral((sin(3*x) + sin(5*x))**(-3), x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 7473 vs. \(2 (72) = 144\).

Time = 0.53 (sec) , antiderivative size = 7473, normalized size of antiderivative = 81.23 \[ \int \frac {1}{(\sin (3 x)+\sin (5 x))^3} \, dx=\text {Too large to display} \] Input:

integrate(1/(sin(3*x)+sin(5*x))^3,x, algorithm="maxima")
 

Output:

-1/30720*(4*(1575*cos(21*x) + 4010*cos(19*x) + 2151*cos(17*x) - 1632*cos(1 
5*x) - 4302*cos(13*x) - 7444*cos(11*x) - 4302*cos(9*x) - 1632*cos(7*x) + 2 
151*cos(5*x) + 4010*cos(3*x) + 1575*cos(x))*cos(22*x) + 6300*(3*cos(20*x) 
+ 3*cos(18*x) + cos(16*x) - 2*cos(14*x) - 6*cos(12*x) - 6*cos(10*x) - 2*co 
s(8*x) + cos(6*x) + 3*cos(4*x) + 3*cos(2*x) + 1)*cos(21*x) + 12*(4010*cos( 
19*x) + 2151*cos(17*x) - 1632*cos(15*x) - 4302*cos(13*x) - 7444*cos(11*x) 
- 4302*cos(9*x) - 1632*cos(7*x) + 2151*cos(5*x) + 4010*cos(3*x) + 1575*cos 
(x))*cos(20*x) + 16040*(3*cos(18*x) + cos(16*x) - 2*cos(14*x) - 6*cos(12*x 
) - 6*cos(10*x) - 2*cos(8*x) + cos(6*x) + 3*cos(4*x) + 3*cos(2*x) + 1)*cos 
(19*x) + 12*(2151*cos(17*x) - 1632*cos(15*x) - 4302*cos(13*x) - 7444*cos(1 
1*x) - 4302*cos(9*x) - 1632*cos(7*x) + 2151*cos(5*x) + 4010*cos(3*x) + 157 
5*cos(x))*cos(18*x) + 8604*(cos(16*x) - 2*cos(14*x) - 6*cos(12*x) - 6*cos( 
10*x) - 2*cos(8*x) + cos(6*x) + 3*cos(4*x) + 3*cos(2*x) + 1)*cos(17*x) - 4 
*(1632*cos(15*x) + 4302*cos(13*x) + 7444*cos(11*x) + 4302*cos(9*x) + 1632* 
cos(7*x) - 2151*cos(5*x) - 4010*cos(3*x) - 1575*cos(x))*cos(16*x) + 6528*( 
2*cos(14*x) + 6*cos(12*x) + 6*cos(10*x) + 2*cos(8*x) - cos(6*x) - 3*cos(4* 
x) - 3*cos(2*x) - 1)*cos(15*x) + 8*(4302*cos(13*x) + 7444*cos(11*x) + 4302 
*cos(9*x) + 1632*cos(7*x) - 2151*cos(5*x) - 4010*cos(3*x) - 1575*cos(x))*c 
os(14*x) + 17208*(6*cos(12*x) + 6*cos(10*x) + 2*cos(8*x) - cos(6*x) - 3*co 
s(4*x) - 3*cos(2*x) - 1)*cos(13*x) + 24*(7444*cos(11*x) + 4302*cos(9*x)...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 249 vs. \(2 (72) = 144\).

Time = 0.13 (sec) , antiderivative size = 249, normalized size of antiderivative = 2.71 \[ \int \frac {1}{(\sin (3 x)+\sin (5 x))^3} \, dx=\frac {31}{512} \, \sqrt {2} \log \left (\frac {{\left | -4 \, \sqrt {2} - \frac {2 \, {\left (\cos \left (x\right ) - 1\right )}}{\cos \left (x\right ) + 1} - 6 \right |}}{{\left | 4 \, \sqrt {2} - \frac {2 \, {\left (\cos \left (x\right ) - 1\right )}}{\cos \left (x\right ) + 1} - 6 \right |}}\right ) - \frac {{\left (\frac {38 \, {\left (\cos \left (x\right ) - 1\right )}}{\cos \left (x\right ) + 1} - 1\right )} {\left (\cos \left (x\right ) + 1\right )}}{4096 \, {\left (\cos \left (x\right ) - 1\right )}} - \frac {\cos \left (x\right ) - 1}{4096 \, {\left (\cos \left (x\right ) + 1\right )}} - \frac {\frac {53 \, {\left (\cos \left (x\right ) - 1\right )}}{\cos \left (x\right ) + 1} + \frac {95 \, {\left (\cos \left (x\right ) - 1\right )}^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {7 \, {\left (\cos \left (x\right ) - 1\right )}^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + 5}{64 \, {\left (\frac {6 \, {\left (\cos \left (x\right ) - 1\right )}}{\cos \left (x\right ) + 1} + \frac {{\left (\cos \left (x\right ) - 1\right )}^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 1\right )}^{2}} - \frac {\frac {605 \, {\left (\cos \left (x\right ) - 1\right )}}{\cos \left (x\right ) + 1} + \frac {925 \, {\left (\cos \left (x\right ) - 1\right )}^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {645 \, {\left (\cos \left (x\right ) - 1\right )}^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {180 \, {\left (\cos \left (x\right ) - 1\right )}^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + 157}{960 \, {\left (\frac {\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1} + 1\right )}^{5}} + \frac {19}{2048} \, \log \left (-\frac {\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1}\right ) \] Input:

integrate(1/(sin(3*x)+sin(5*x))^3,x, algorithm="giac")
 

Output:

31/512*sqrt(2)*log(abs(-4*sqrt(2) - 2*(cos(x) - 1)/(cos(x) + 1) - 6)/abs(4 
*sqrt(2) - 2*(cos(x) - 1)/(cos(x) + 1) - 6)) - 1/4096*(38*(cos(x) - 1)/(co 
s(x) + 1) - 1)*(cos(x) + 1)/(cos(x) - 1) - 1/4096*(cos(x) - 1)/(cos(x) + 1 
) - 1/64*(53*(cos(x) - 1)/(cos(x) + 1) + 95*(cos(x) - 1)^2/(cos(x) + 1)^2 
+ 7*(cos(x) - 1)^3/(cos(x) + 1)^3 + 5)/(6*(cos(x) - 1)/(cos(x) + 1) + (cos 
(x) - 1)^2/(cos(x) + 1)^2 + 1)^2 - 1/960*(605*(cos(x) - 1)/(cos(x) + 1) + 
925*(cos(x) - 1)^2/(cos(x) + 1)^2 + 645*(cos(x) - 1)^3/(cos(x) + 1)^3 + 18 
0*(cos(x) - 1)^4/(cos(x) + 1)^4 + 157)/((cos(x) - 1)/(cos(x) + 1) + 1)^5 + 
 19/2048*log(-(cos(x) - 1)/(cos(x) + 1))
 

Mupad [B] (verification not implemented)

Time = 22.55 (sec) , antiderivative size = 220, normalized size of antiderivative = 2.39 \[ \int \frac {1}{(\sin (3 x)+\sin (5 x))^3} \, dx=\frac {19\,\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )}{1024}+\frac {31\,\sqrt {2}\,\mathrm {atanh}\left (\frac {616001\,\sqrt {2}}{2097152\,\left (\frac {10149121\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{4194304}-\frac {1740371}{4194304}\right )}-\frac {1794187\,\sqrt {2}\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{1048576\,\left (\frac {10149121\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{4194304}-\frac {1740371}{4194304}\right )}\right )}{256}+\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{4096}-\frac {\frac {1215\,{\mathrm {tan}\left (\frac {x}{2}\right )}^{18}}{4096}-\frac {20271\,{\mathrm {tan}\left (\frac {x}{2}\right )}^{16}}{4096}+\frac {78239\,{\mathrm {tan}\left (\frac {x}{2}\right )}^{14}}{3072}-\frac {184471\,{\mathrm {tan}\left (\frac {x}{2}\right )}^{12}}{3072}+\frac {783513\,{\mathrm {tan}\left (\frac {x}{2}\right )}^{10}}{10240}-\frac {1677323\,{\mathrm {tan}\left (\frac {x}{2}\right )}^8}{30720}+\frac {107857\,{\mathrm {tan}\left (\frac {x}{2}\right )}^6}{5120}-\frac {58139\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4}{15360}+\frac {14593\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{61440}+\frac {1}{4096}}{-{\mathrm {tan}\left (\frac {x}{2}\right )}^{20}+17\,{\mathrm {tan}\left (\frac {x}{2}\right )}^{18}-108\,{\mathrm {tan}\left (\frac {x}{2}\right )}^{16}+332\,{\mathrm {tan}\left (\frac {x}{2}\right )}^{14}-566\,{\mathrm {tan}\left (\frac {x}{2}\right )}^{12}+566\,{\mathrm {tan}\left (\frac {x}{2}\right )}^{10}-332\,{\mathrm {tan}\left (\frac {x}{2}\right )}^8+108\,{\mathrm {tan}\left (\frac {x}{2}\right )}^6-17\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4+{\mathrm {tan}\left (\frac {x}{2}\right )}^2} \] Input:

int(1/(sin(3*x) + sin(5*x))^3,x)
 

Output:

(19*log(tan(x/2)))/1024 + (31*2^(1/2)*atanh((616001*2^(1/2))/(2097152*((10 
149121*tan(x/2)^2)/4194304 - 1740371/4194304)) - (1794187*2^(1/2)*tan(x/2) 
^2)/(1048576*((10149121*tan(x/2)^2)/4194304 - 1740371/4194304))))/256 + ta 
n(x/2)^2/4096 - ((14593*tan(x/2)^2)/61440 - (58139*tan(x/2)^4)/15360 + (10 
7857*tan(x/2)^6)/5120 - (1677323*tan(x/2)^8)/30720 + (783513*tan(x/2)^10)/ 
10240 - (184471*tan(x/2)^12)/3072 + (78239*tan(x/2)^14)/3072 - (20271*tan( 
x/2)^16)/4096 + (1215*tan(x/2)^18)/4096 + 1/4096)/(tan(x/2)^2 - 17*tan(x/2 
)^4 + 108*tan(x/2)^6 - 332*tan(x/2)^8 + 566*tan(x/2)^10 - 566*tan(x/2)^12 
+ 332*tan(x/2)^14 - 108*tan(x/2)^16 + 17*tan(x/2)^18 - tan(x/2)^20)
 

Reduce [F]

\[ \int \frac {1}{(\sin (3 x)+\sin (5 x))^3} \, dx=\int \frac {1}{\sin \left (5 x \right )^{3}+3 \sin \left (5 x \right )^{2} \sin \left (3 x \right )+3 \sin \left (5 x \right ) \sin \left (3 x \right )^{2}+\sin \left (3 x \right )^{3}}d x \] Input:

int(1/(sin(3*x)+sin(5*x))^3,x)
 

Output:

int(1/(sin(5*x)**3 + 3*sin(5*x)**2*sin(3*x) + 3*sin(5*x)*sin(3*x)**2 + sin 
(3*x)**3),x)