Integrand size = 9, antiderivative size = 218 \[ \int \frac {1}{(\sin (x)+\sin (5 x))^6} \, dx=-\frac {33243}{128} \text {arctanh}(2 \cos (x) \sin (x))-\frac {715 \cot (x)}{139968}-\frac {11 \cot ^3(x)}{69984}-\frac {\cot ^5(x)}{233280}-\frac {109312 \log \left (\sqrt {3} \cos (x)-\sin (x)\right )}{243 \sqrt {3}}+\frac {109312 \log \left (\sqrt {3} \cos (x)+\sin (x)\right )}{243 \sqrt {3}}+\frac {32 \tan (x) \left (16627-16465 \tan ^2(x)\right )}{405 \left (3-4 \tan ^2(x)+\tan ^4(x)\right )^5}-\frac {2 \tan (x) \left (76795+61567 \tan ^2(x)\right )}{405 \left (3-4 \tan ^2(x)+\tan ^4(x)\right )^4}+\frac {\tan (x) \left (778363-606643 \tan ^2(x)\right )}{2430 \left (3-4 \tan ^2(x)+\tan ^4(x)\right )^3}-\frac {\tan (x) \left (91032631-43437157 \tan ^2(x)\right )}{174960 \left (3-4 \tan ^2(x)+\tan ^4(x)\right )^2}+\frac {\tan (x) \left (111184863-53226455 \tan ^2(x)\right )}{139968 \left (3-4 \tan ^2(x)+\tan ^4(x)\right )} \] Output:
-33243/128*arctanh(2*cos(x)*sin(x))-715/139968*cot(x)-11/69984*cot(x)^3-1/ 233280*cot(x)^5-109312/729*ln(3^(1/2)*cos(x)-sin(x))*3^(1/2)+109312/729*ln (3^(1/2)*cos(x)+sin(x))*3^(1/2)+32/405*tan(x)*(16627-16465*tan(x)^2)/(3-4* tan(x)^2+tan(x)^4)^5-2/405*tan(x)*(76795+61567*tan(x)^2)/(3-4*tan(x)^2+tan (x)^4)^4+1/2430*tan(x)*(778363-606643*tan(x)^2)/(3-4*tan(x)^2+tan(x)^4)^3- 1/174960*tan(x)*(91032631-43437157*tan(x)^2)/(3-4*tan(x)^2+tan(x)^4)^2+tan (x)*(111184863-53226455*tan(x)^2)/(419904-559872*tan(x)^2+139968*tan(x)^4)
Time = 6.05 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.33 \[ \int \frac {1}{(\sin (x)+\sin (5 x))^6} \, dx=\frac {218624 \text {arctanh}\left (\frac {\tan (x)}{\sqrt {3}}\right )}{243 \sqrt {3}}-\frac {289 \cot (x)}{58320}-\frac {13 \cot (x) \csc ^2(x)}{87480}-\frac {\cot (x) \csc ^4(x)}{233280}+\frac {33243}{128} \log (\cos (x)-\sin (x))-\frac {33243}{128} \log (\cos (x)+\sin (x))-\frac {43}{1280 (\cos (x)-\sin (x))^4}-\frac {7393}{1920 (\cos (x)-\sin (x))^2}+\frac {\sin (x)}{320 (\cos (x)-\sin (x))^5}+\frac {1099 \sin (x)}{1920 (\cos (x)-\sin (x))^3}+\frac {16613 \sin (x)}{240 (\cos (x)-\sin (x))}+\frac {\sin (x)}{320 (\cos (x)+\sin (x))^5}+\frac {43}{1280 (\cos (x)+\sin (x))^4}+\frac {1099 \sin (x)}{1920 (\cos (x)+\sin (x))^3}+\frac {7393}{1920 (\cos (x)+\sin (x))^2}+\frac {16613 \sin (x)}{240 (\cos (x)+\sin (x))}+\frac {64 \sin (2 x)}{405 (1+2 \cos (2 x))^5}+\frac {512 \sin (2 x)}{405 (1+2 \cos (2 x))^4}+\frac {7744 \sin (2 x)}{1215 (1+2 \cos (2 x))^3}+\frac {300352 \sin (2 x)}{10935 (1+2 \cos (2 x))^2}+\frac {486784 \sin (2 x)}{3645 (1+2 \cos (2 x))} \] Input:
Integrate[(Sin[x] + Sin[5*x])^(-6),x]
Output:
(218624*ArcTanh[Tan[x]/Sqrt[3]])/(243*Sqrt[3]) - (289*Cot[x])/58320 - (13* Cot[x]*Csc[x]^2)/87480 - (Cot[x]*Csc[x]^4)/233280 + (33243*Log[Cos[x] - Si n[x]])/128 - (33243*Log[Cos[x] + Sin[x]])/128 - 43/(1280*(Cos[x] - Sin[x]) ^4) - 7393/(1920*(Cos[x] - Sin[x])^2) + Sin[x]/(320*(Cos[x] - Sin[x])^5) + (1099*Sin[x])/(1920*(Cos[x] - Sin[x])^3) + (16613*Sin[x])/(240*(Cos[x] - Sin[x])) + Sin[x]/(320*(Cos[x] + Sin[x])^5) + 43/(1280*(Cos[x] + Sin[x])^4 ) + (1099*Sin[x])/(1920*(Cos[x] + Sin[x])^3) + 7393/(1920*(Cos[x] + Sin[x] )^2) + (16613*Sin[x])/(240*(Cos[x] + Sin[x])) + (64*Sin[2*x])/(405*(1 + 2* Cos[2*x])^5) + (512*Sin[2*x])/(405*(1 + 2*Cos[2*x])^4) + (7744*Sin[2*x])/( 1215*(1 + 2*Cos[2*x])^3) + (300352*Sin[2*x])/(10935*(1 + 2*Cos[2*x])^2) + (486784*Sin[2*x])/(3645*(1 + 2*Cos[2*x]))
Time = 0.93 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.89, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.667, Rules used = {3042, 4822, 27, 1673, 27, 2198, 27, 2198, 27, 2198, 27, 2198, 27, 2195, 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.
| \(\displaystyle \int \frac {1}{(\sin (x)+\sin (5 x))^6} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(\sin (x)+\sin (5 x))^6}dx\) |
| \(\Big \downarrow \) 4822 |
| \(\displaystyle \int \frac {\left (\tan ^2(x)+1\right )^{14} \cot ^6(x)}{64 \left (\tan ^4(x)-4 \tan ^2(x)+3\right )^6}d\tan (x)\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{64} \int \frac {\cot ^6(x) \left (\tan ^2(x)+1\right )^{14}}{\left (\tan ^4(x)-4 \tan ^2(x)+3\right )^6}d\tan (x)\) |
| \(\Big \downarrow \) 1673 |
| \(\displaystyle \frac {1}{64} \left (\frac {2048 \tan (x) \left (16627-16465 \tan ^2(x)\right )}{405 \left (\tan ^4(x)-4 \tan ^2(x)+3\right )^5}-\frac {1}{120} \int -\frac {8 \cot ^6(x) \left (405 \tan ^{24}(x)+7290 \tan ^{22}(x)+64800 \tan ^{20}(x)+384750 \tan ^{18}(x)+1750005 \tan ^{16}(x)+6656580 \tan ^{14}(x)+22592520 \tan ^{12}(x)+71790300 \tan ^{10}(x)-352645585 \tan ^8(x)-33983646 \tan ^6(x)+15000 \tan ^4(x)+2070 \tan ^2(x)+135\right )}{27 \left (\tan ^4(x)-4 \tan ^2(x)+3\right )^5}d\tan (x)\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{64} \left (\frac {1}{405} \int \frac {\cot ^6(x) \left (405 \tan ^{24}(x)+7290 \tan ^{22}(x)+64800 \tan ^{20}(x)+384750 \tan ^{18}(x)+1750005 \tan ^{16}(x)+6656580 \tan ^{14}(x)+22592520 \tan ^{12}(x)+71790300 \tan ^{10}(x)-352645585 \tan ^8(x)-33983646 \tan ^6(x)+15000 \tan ^4(x)+2070 \tan ^2(x)+135\right )}{\left (\tan ^4(x)-4 \tan ^2(x)+3\right )^5}d\tan (x)+\frac {2048 \tan (x) \left (16627-16465 \tan ^2(x)\right )}{405 \left (\tan ^4(x)-4 \tan ^2(x)+3\right )^5}\right )\) |
| \(\Big \downarrow \) 2198 |
| \(\displaystyle \frac {1}{64} \left (\frac {1}{405} \left (-\frac {1}{96} \int -\frac {96 \cot ^6(x) \left (405 \tan ^{20}(x)+8910 \tan ^{18}(x)+99225 \tan ^{16}(x)+754920 \tan ^{14}(x)+4472010 \tan ^{12}(x)+22279860 \tan ^{10}(x)-4151558 \tan ^8(x)-1490392 \tan ^6(x)+5985 \tan ^4(x)+750 \tan ^2(x)+45\right )}{\left (\tan ^4(x)-4 \tan ^2(x)+3\right )^4}d\tan (x)-\frac {128 \tan (x) \left (61567 \tan ^2(x)+76795\right )}{\left (\tan ^4(x)-4 \tan ^2(x)+3\right )^4}\right )+\frac {2048 \tan (x) \left (16627-16465 \tan ^2(x)\right )}{405 \left (\tan ^4(x)-4 \tan ^2(x)+3\right )^5}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{64} \left (\frac {1}{405} \left (\int \frac {\cot ^6(x) \left (405 \tan ^{20}(x)+8910 \tan ^{18}(x)+99225 \tan ^{16}(x)+754920 \tan ^{14}(x)+4472010 \tan ^{12}(x)+22279860 \tan ^{10}(x)-4151558 \tan ^8(x)-1490392 \tan ^6(x)+5985 \tan ^4(x)+750 \tan ^2(x)+45\right )}{\left (\tan ^4(x)-4 \tan ^2(x)+3\right )^4}d\tan (x)-\frac {128 \tan (x) \left (61567 \tan ^2(x)+76795\right )}{\left (\tan ^4(x)-4 \tan ^2(x)+3\right )^4}\right )+\frac {2048 \tan (x) \left (16627-16465 \tan ^2(x)\right )}{405 \left (\tan ^4(x)-4 \tan ^2(x)+3\right )^5}\right )\) |
| \(\Big \downarrow \) 2198 |
| \(\displaystyle \frac {1}{64} \left (\frac {1}{405} \left (-\frac {1}{72} \int -\frac {24 \cot ^6(x) \left (1215 \tan ^{16}(x)+31590 \tan ^{14}(x)+420390 \tan ^{12}(x)+3851550 \tan ^{10}(x)-147152124 \tan ^8(x)-26388878 \tan ^6(x)+7050 \tan ^4(x)+810 \tan ^2(x)+45\right )}{\left (\tan ^4(x)-4 \tan ^2(x)+3\right )^3}d\tan (x)+\frac {32 \tan (x) \left (778363-606643 \tan ^2(x)\right )}{3 \left (\tan ^4(x)-4 \tan ^2(x)+3\right )^3}-\frac {128 \tan (x) \left (61567 \tan ^2(x)+76795\right )}{\left (\tan ^4(x)-4 \tan ^2(x)+3\right )^4}\right )+\frac {2048 \tan (x) \left (16627-16465 \tan ^2(x)\right )}{405 \left (\tan ^4(x)-4 \tan ^2(x)+3\right )^5}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{64} \left (\frac {1}{405} \left (\frac {1}{3} \int \frac {\cot ^6(x) \left (1215 \tan ^{16}(x)+31590 \tan ^{14}(x)+420390 \tan ^{12}(x)+3851550 \tan ^{10}(x)-147152124 \tan ^8(x)-26388878 \tan ^6(x)+7050 \tan ^4(x)+810 \tan ^2(x)+45\right )}{\left (\tan ^4(x)-4 \tan ^2(x)+3\right )^3}d\tan (x)+\frac {32 \tan (x) \left (778363-606643 \tan ^2(x)\right )}{3 \left (\tan ^4(x)-4 \tan ^2(x)+3\right )^3}-\frac {128 \tan (x) \left (61567 \tan ^2(x)+76795\right )}{\left (\tan ^4(x)-4 \tan ^2(x)+3\right )^4}\right )+\frac {2048 \tan (x) \left (16627-16465 \tan ^2(x)\right )}{405 \left (\tan ^4(x)-4 \tan ^2(x)+3\right )^5}\right )\) |
| \(\Big \downarrow \) 2198 |
| \(\displaystyle \frac {1}{64} \left (\frac {1}{405} \left (\frac {1}{3} \left (-\frac {1}{48} \int -\frac {80 \cot ^6(x) \left (2187 \tan ^{12}(x)+65610 \tan ^{10}(x)+174761209 \tan ^8(x)+56999160 \tan ^6(x)+4917 \tan ^4(x)+522 \tan ^2(x)+27\right )}{3 \left (\tan ^4(x)-4 \tan ^2(x)+3\right )^2}d\tan (x)-\frac {4 \tan (x) \left (91032631-43437157 \tan ^2(x)\right )}{9 \left (\tan ^4(x)-4 \tan ^2(x)+3\right )^2}\right )+\frac {32 \tan (x) \left (778363-606643 \tan ^2(x)\right )}{3 \left (\tan ^4(x)-4 \tan ^2(x)+3\right )^3}-\frac {128 \tan (x) \left (61567 \tan ^2(x)+76795\right )}{\left (\tan ^4(x)-4 \tan ^2(x)+3\right )^4}\right )+\frac {2048 \tan (x) \left (16627-16465 \tan ^2(x)\right )}{405 \left (\tan ^4(x)-4 \tan ^2(x)+3\right )^5}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{64} \left (\frac {1}{405} \left (\frac {1}{3} \left (\frac {5}{9} \int \frac {\cot ^6(x) \left (2187 \tan ^{12}(x)+65610 \tan ^{10}(x)+174761209 \tan ^8(x)+56999160 \tan ^6(x)+4917 \tan ^4(x)+522 \tan ^2(x)+27\right )}{\left (\tan ^4(x)-4 \tan ^2(x)+3\right )^2}d\tan (x)-\frac {4 \tan (x) \left (91032631-43437157 \tan ^2(x)\right )}{9 \left (\tan ^4(x)-4 \tan ^2(x)+3\right )^2}\right )+\frac {32 \tan (x) \left (778363-606643 \tan ^2(x)\right )}{3 \left (\tan ^4(x)-4 \tan ^2(x)+3\right )^3}-\frac {128 \tan (x) \left (61567 \tan ^2(x)+76795\right )}{\left (\tan ^4(x)-4 \tan ^2(x)+3\right )^4}\right )+\frac {2048 \tan (x) \left (16627-16465 \tan ^2(x)\right )}{405 \left (\tan ^4(x)-4 \tan ^2(x)+3\right )^5}\right )\) |
| \(\Big \downarrow \) 2198 |
| \(\displaystyle \frac {1}{64} \left (\frac {1}{405} \left (\frac {1}{3} \left (\frac {5}{9} \left (\frac {\tan (x) \left (111184863-53226455 \tan ^2(x)\right )}{\tan ^4(x)-4 \tan ^2(x)+3}-\frac {1}{24} \int -\frac {24 \cot ^6(x) \left (-53224268 \tan ^8(x)-92182693 \tan ^6(x)+1884 \tan ^4(x)+186 \tan ^2(x)+9\right )}{\tan ^4(x)-4 \tan ^2(x)+3}d\tan (x)\right )-\frac {4 \tan (x) \left (91032631-43437157 \tan ^2(x)\right )}{9 \left (\tan ^4(x)-4 \tan ^2(x)+3\right )^2}\right )+\frac {32 \tan (x) \left (778363-606643 \tan ^2(x)\right )}{3 \left (\tan ^4(x)-4 \tan ^2(x)+3\right )^3}-\frac {128 \tan (x) \left (61567 \tan ^2(x)+76795\right )}{\left (\tan ^4(x)-4 \tan ^2(x)+3\right )^4}\right )+\frac {2048 \tan (x) \left (16627-16465 \tan ^2(x)\right )}{405 \left (\tan ^4(x)-4 \tan ^2(x)+3\right )^5}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{64} \left (\frac {1}{405} \left (\frac {1}{3} \left (\frac {5}{9} \left (\int \frac {\cot ^6(x) \left (-53224268 \tan ^8(x)-92182693 \tan ^6(x)+1884 \tan ^4(x)+186 \tan ^2(x)+9\right )}{\tan ^4(x)-4 \tan ^2(x)+3}d\tan (x)+\frac {\tan (x) \left (111184863-53226455 \tan ^2(x)\right )}{\tan ^4(x)-4 \tan ^2(x)+3}\right )-\frac {4 \tan (x) \left (91032631-43437157 \tan ^2(x)\right )}{9 \left (\tan ^4(x)-4 \tan ^2(x)+3\right )^2}\right )+\frac {32 \tan (x) \left (778363-606643 \tan ^2(x)\right )}{3 \left (\tan ^4(x)-4 \tan ^2(x)+3\right )^3}-\frac {128 \tan (x) \left (61567 \tan ^2(x)+76795\right )}{\left (\tan ^4(x)-4 \tan ^2(x)+3\right )^4}\right )+\frac {2048 \tan (x) \left (16627-16465 \tan ^2(x)\right )}{405 \left (\tan ^4(x)-4 \tan ^2(x)+3\right )^5}\right )\) |
| \(\Big \downarrow \) 2195 |
| \(\displaystyle \frac {1}{64} \left (\frac {1}{405} \left (\frac {1}{3} \left (\frac {5}{9} \left (\int \left (3 \cot ^6(x)+66 \cot ^4(x)+715 \cot ^2(x)-\frac {125927424}{\tan ^2(x)-3}+\frac {72702441}{\tan ^2(x)-1}\right )d\tan (x)+\frac {\tan (x) \left (111184863-53226455 \tan ^2(x)\right )}{\tan ^4(x)-4 \tan ^2(x)+3}\right )-\frac {4 \tan (x) \left (91032631-43437157 \tan ^2(x)\right )}{9 \left (\tan ^4(x)-4 \tan ^2(x)+3\right )^2}\right )+\frac {32 \tan (x) \left (778363-606643 \tan ^2(x)\right )}{3 \left (\tan ^4(x)-4 \tan ^2(x)+3\right )^3}-\frac {128 \tan (x) \left (61567 \tan ^2(x)+76795\right )}{\left (\tan ^4(x)-4 \tan ^2(x)+3\right )^4}\right )+\frac {2048 \tan (x) \left (16627-16465 \tan ^2(x)\right )}{405 \left (\tan ^4(x)-4 \tan ^2(x)+3\right )^5}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{64} \left (\frac {1}{405} \left (\frac {1}{3} \left (\frac {5}{9} \left (-72702441 \text {arctanh}(\tan (x))+41975808 \sqrt {3} \text {arctanh}\left (\frac {\tan (x)}{\sqrt {3}}\right )+\frac {\tan (x) \left (111184863-53226455 \tan ^2(x)\right )}{\tan ^4(x)-4 \tan ^2(x)+3}-\frac {3}{5} \cot ^5(x)-22 \cot ^3(x)-715 \cot (x)\right )-\frac {4 \tan (x) \left (91032631-43437157 \tan ^2(x)\right )}{9 \left (\tan ^4(x)-4 \tan ^2(x)+3\right )^2}\right )+\frac {32 \tan (x) \left (778363-606643 \tan ^2(x)\right )}{3 \left (\tan ^4(x)-4 \tan ^2(x)+3\right )^3}-\frac {128 \tan (x) \left (61567 \tan ^2(x)+76795\right )}{\left (\tan ^4(x)-4 \tan ^2(x)+3\right )^4}\right )+\frac {2048 \tan (x) \left (16627-16465 \tan ^2(x)\right )}{405 \left (\tan ^4(x)-4 \tan ^2(x)+3\right )^5}\right )\) |
Input:
Int[(Sin[x] + Sin[5*x])^(-6),x]
Output:
((2048*Tan[x]*(16627 - 16465*Tan[x]^2))/(405*(3 - 4*Tan[x]^2 + Tan[x]^4)^5 ) + ((-128*Tan[x]*(76795 + 61567*Tan[x]^2))/(3 - 4*Tan[x]^2 + Tan[x]^4)^4 + (32*Tan[x]*(778363 - 606643*Tan[x]^2))/(3*(3 - 4*Tan[x]^2 + Tan[x]^4)^3) + ((-4*Tan[x]*(91032631 - 43437157*Tan[x]^2))/(9*(3 - 4*Tan[x]^2 + Tan[x] ^4)^2) + (5*(-72702441*ArcTanh[Tan[x]] + 41975808*Sqrt[3]*ArcTanh[Tan[x]/S qrt[3]] - 715*Cot[x] - 22*Cot[x]^3 - (3*Cot[x]^5)/5 + (Tan[x]*(111184863 - 53226455*Tan[x]^2))/(3 - 4*Tan[x]^2 + Tan[x]^4)))/9)/3)/405)/64
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^ 4)^(p_), x_Symbol] :> With[{f = Coeff[PolynomialRemainder[x^m*(d + e*x^2)^q , a + b*x^2 + c*x^4, x], x, 0], g = Coeff[PolynomialRemainder[x^m*(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[x^m*(a + b*x^2 + c*x^4)^(p + 1)*Simp[ExpandToSum[(2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuotient[x^m*(d + e*x^2)^q, a + b*x^2 + c*x^4, x])/x^m + (b^2*f*(2*p + 3) - 2*a*c*f*(4*p + 5 ) - a*b*g)/x^m + c*(4*p + 7)*(b*f - 2*a*g)*x^(2 - m), x], x], x], x]] /; Fr eeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && IGtQ[q, 1] && ILtQ[m/2, 0]
Int[(Pq_)*((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_ Symbol] :> Int[ExpandIntegrand[(d*x)^m*Pq*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && PolyQ[Pq, x^2] && IGtQ[p, -2]
Int[(Pq_)*(x_)^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{Qx = PolynomialQuotient[x^m*Pq, a + b*x^2 + c*x^4, x], d = Coeff[Pol ynomialRemainder[x^m*Pq, a + b*x^2 + c*x^4, x], x, 0], e = Coeff[Polynomial Remainder[x^m*Pq, 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[x^m*(a + b*x^2 + c*x^4)^(p + 1)*ExpandToSum[(2*a*(p + 1)*(b^2 - 4*a*c)*Qx)/x^m + (b^2*d*(2* p + 3) - 2*a*c*d*(4*p + 5) - a*b*e)/x^m + c*(4*p + 7)*(b*d - 2*a*e)*x^(2 - m), x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x^2] && GtQ[Expon[Pq, x ^2], 1] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && ILtQ[m/2, 0]
Int[((a_.)*sin[(m_.)*((c_.) + (d_.)*(x_))] + (b_.)*sin[(n_.)*((c_.) + (d_.) *(x_))])^(p_), x_Symbol] :> Simp[1/d Subst[Int[Simplify[TrigExpand[a*Sin[ m*ArcTan[x]] + b*Sin[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 = 421.25 (sec) , antiderivative size = 2, normalized size of antiderivative = 0.01
| method | result | size |
| parallelrisch | \(0\) | \(2\) |
| default | \(-\frac {1}{80 \left (\tan \left (x \right )+1\right )^{5}}+\frac {11}{64 \left (\tan \left (x \right )+1\right )^{4}}-\frac {281}{192 \left (\tan \left (x \right )+1\right )^{3}}+\frac {659}{64 \left (\tan \left (x \right )+1\right )^{2}}-\frac {10085}{128 \left (\tan \left (x \right )+1\right )}-\frac {33243 \ln \left (\tan \left (x \right )+1\right )}{128}-\frac {1}{80 \left (\tan \left (x \right )-1\right )^{5}}-\frac {11}{64 \left (\tan \left (x \right )-1\right )^{4}}-\frac {281}{192 \left (\tan \left (x \right )-1\right )^{3}}-\frac {659}{64 \left (\tan \left (x \right )-1\right )^{2}}-\frac {10085}{128 \left (\tan \left (x \right )-1\right )}+\frac {33243 \ln \left (\tan \left (x \right )-1\right )}{128}-\frac {1}{233280 \tan \left (x \right )^{5}}-\frac {11}{69984 \tan \left (x \right )^{3}}-\frac {715}{139968 \tan \left (x \right )}-\frac {128 \left (3805 \tan \left (x \right )^{9}-48104 \tan \left (x \right )^{7}+\frac {1146978 \tan \left (x \right )^{5}}{5}-489888 \tan \left (x \right )^{3}+397089 \tan \left (x \right )\right )}{2187 \left (\tan \left (x \right )^{2}-3\right )^{5}}+\frac {218624 \sqrt {3}\, \operatorname {arctanh}\left (\frac {\tan \left (x \right ) \sqrt {3}}{3}\right )}{729}\) | \(166\) |
| risch | \(\frac {i \left (-21150720+5410405 \,{\mathrm e}^{2 i x}-88263680 \,{\mathrm e}^{4 i x}+702189680 \,{\mathrm e}^{14 i x}-359783050 \,{\mathrm e}^{12 i x}-905873280 \,{\mathrm e}^{16 i x}-1305280104 \,{\mathrm e}^{20 i x}+443179392 \,{\mathrm e}^{10 i x}-159854585 \,{\mathrm e}^{8 i x}+119342210 \,{\mathrm e}^{6 i x}+742867375 \,{\mathrm e}^{18 i x}+1206807525 \,{\mathrm e}^{26 i x}-612290560 \,{\mathrm e}^{28 i x}-1014018075 \,{\mathrm e}^{24 i x}+974013440 \,{\mathrm e}^{22 i x}+830942616 \,{\mathrm e}^{30 i x}+257416320 \,{\mathrm e}^{34 i x}+63231350 \,{\mathrm e}^{38 i x}+5410405 \,{\mathrm e}^{48 i x}-526175825 \,{\mathrm e}^{32 i x}+17489920 \,{\mathrm e}^{46 i x}-355346320 \,{\mathrm e}^{36 i x}+51652615 \,{\mathrm e}^{42 i x}+13588610 \,{\mathrm e}^{44 i x}-85588608 \,{\mathrm e}^{40 i x}\right )}{77760 \left ({\mathrm e}^{10 i x}+{\mathrm e}^{6 i x}-{\mathrm e}^{4 i x}-1\right )^{5}}+\frac {33243 \ln \left ({\mathrm e}^{2 i x}-i\right )}{128}-\frac {33243 \ln \left ({\mathrm e}^{2 i x}+i\right )}{128}+\frac {109312 \sqrt {3}\, \ln \left ({\mathrm e}^{2 i x}+\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{729}-\frac {109312 \sqrt {3}\, \ln \left ({\mathrm e}^{2 i x}+\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{729}\) | \(256\) |
Input:
int(1/(sin(x)+sin(5*x))^6,x,method=_RETURNVERBOSE)
Output:
0
Leaf count of result is larger than twice the leaf count of optimal. 502 vs. \(2 (188) = 376\).
Time = 0.30 (sec) , antiderivative size = 502, normalized size of antiderivative = 2.30 \[ \int \frac {1}{(\sin (x)+\sin (5 x))^6} \, dx =\text {Too large to display} \] Input:
integrate(1/(sin(x)+sin(5*x))^6,x, algorithm="fricas")
Output:
-1/933120*(4158400757760*cos(x)^25 - 26521869189120*cos(x)^23 + 7540040159 2320*cos(x)^21 - 126325942517760*cos(x)^19 + 138927429196800*cos(x)^17 - 1 05686653930368*cos(x)^15 + 57050747014560*cos(x)^13 - 22030071327120*cos(x )^11 + 6043251410040*cos(x)^9 - 1149349729260*cos(x)^7 + 143978865060*cos( x)^5 - 10676989740*cos(x)^3 + 121170735*(32768*cos(x)^24 - 188416*cos(x)^2 2 + 483328*cos(x)^20 - 732160*cos(x)^18 + 730240*cos(x)^16 - 505696*cos(x) ^14 + 249552*cos(x)^12 - 88496*cos(x)^10 + 22400*cos(x)^8 - 3950*cos(x)^6 + 461*cos(x)^4 - 32*cos(x)^2 + 1)*log(2*cos(x)*sin(x) + 1)*sin(x) - 121170 735*(32768*cos(x)^24 - 188416*cos(x)^22 + 483328*cos(x)^20 - 732160*cos(x) ^18 + 730240*cos(x)^16 - 505696*cos(x)^14 + 249552*cos(x)^12 - 88496*cos(x )^10 + 22400*cos(x)^8 - 3950*cos(x)^6 + 461*cos(x)^4 - 32*cos(x)^2 + 1)*lo g(-2*cos(x)*sin(x) + 1)*sin(x) - 69959680*(32768*sqrt(3)*cos(x)^24 - 18841 6*sqrt(3)*cos(x)^22 + 483328*sqrt(3)*cos(x)^20 - 732160*sqrt(3)*cos(x)^18 + 730240*sqrt(3)*cos(x)^16 - 505696*sqrt(3)*cos(x)^14 + 249552*sqrt(3)*cos (x)^12 - 88496*sqrt(3)*cos(x)^10 + 22400*sqrt(3)*cos(x)^8 - 3950*sqrt(3)*c os(x)^6 + 461*sqrt(3)*cos(x)^4 - 32*sqrt(3)*cos(x)^2 + sqrt(3))*log(-(8*co s(x)^4 - 16*cos(x)^2 - 4*(2*sqrt(3)*cos(x)^3 + sqrt(3)*cos(x))*sin(x) - 1) /(16*cos(x)^4 - 8*cos(x)^2 + 1))*sin(x) + 354847800*cos(x))/((32768*cos(x) ^24 - 188416*cos(x)^22 + 483328*cos(x)^20 - 732160*cos(x)^18 + 730240*cos( x)^16 - 505696*cos(x)^14 + 249552*cos(x)^12 - 88496*cos(x)^10 + 22400*c...
Timed out. \[ \int \frac {1}{(\sin (x)+\sin (5 x))^6} \, dx=\text {Timed out} \] Input:
integrate(1/(sin(x)+sin(5*x))**6,x)
Output:
Timed out
Exception generated. \[ \int \frac {1}{(\sin (x)+\sin (5 x))^6} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(1/(sin(x)+sin(5*x))^6,x, algorithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is un defined.
Time = 0.16 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.65 \[ \int \frac {1}{(\sin (x)+\sin (5 x))^6} \, dx=-\frac {109312}{729} \, \sqrt {3} \log \left (\frac {{\left | -2 \, \sqrt {3} + 2 \, \tan \left (x\right ) \right |}}{{\left | 2 \, \sqrt {3} + 2 \, \tan \left (x\right ) \right |}}\right ) - \frac {29570650 \, \tan \left (x\right )^{24} - 534901345 \, \tan \left (x\right )^{22} + 4162661060 \, \tan \left (x\right )^{20} - 18227416530 \, \tan \left (x\right )^{18} + 49341594025 \, \tan \left (x\right )^{16} - 85462804130 \, \tan \left (x\right )^{14} + 94715143160 \, \tan \left (x\right )^{12} - 64892703684 \, \tan \left (x\right )^{10} + 25017153300 \, \tan \left (x\right )^{8} - 4149372645 \, \tan \left (x\right )^{6} + 78300 \, \tan \left (x\right )^{4} + 2430 \, \tan \left (x\right )^{2} + 81}{77760 \, {\left (\tan \left (x\right )^{5} - 4 \, \tan \left (x\right )^{3} + 3 \, \tan \left (x\right )\right )}^{5}} - \frac {33243}{128} \, \log \left ({\left | \tan \left (x\right ) + 1 \right |}\right ) + \frac {33243}{128} \, \log \left ({\left | \tan \left (x\right ) - 1 \right |}\right ) \] Input:
integrate(1/(sin(x)+sin(5*x))^6,x, algorithm="giac")
Output:
-109312/729*sqrt(3)*log(abs(-2*sqrt(3) + 2*tan(x))/abs(2*sqrt(3) + 2*tan(x ))) - 1/77760*(29570650*tan(x)^24 - 534901345*tan(x)^22 + 4162661060*tan(x )^20 - 18227416530*tan(x)^18 + 49341594025*tan(x)^16 - 85462804130*tan(x)^ 14 + 94715143160*tan(x)^12 - 64892703684*tan(x)^10 + 25017153300*tan(x)^8 - 4149372645*tan(x)^6 + 78300*tan(x)^4 + 2430*tan(x)^2 + 81)/(tan(x)^5 - 4 *tan(x)^3 + 3*tan(x))^5 - 33243/128*log(abs(tan(x) + 1)) + 33243/128*log(a bs(tan(x) - 1))
Time = 23.33 (sec) , antiderivative size = 418, normalized size of antiderivative = 1.92 \[ \int \frac {1}{(\sin (x)+\sin (5 x))^6} \, dx=\text {Too large to display} \] Input:
int(1/(sin(5*x) + sin(x))^6,x)
Output:
(33243*atanh((11425111033420770921670428889*tan(x/2))/(2928229434235008*(( 11425111033420770921670428889*tan(x/2)^2)/5856458868470016 - 1142511103342 0770921670428889/5856458868470016))))/64 + (5591*tan(x/2))/2239488 - (2186 24*3^(1/2)*atanh((71157089211203516607394358493184*3^(1/2)*tan(x/2))/(4052 555153018976267*((35578544605601758303697179246592*tan(x/2)^2)/13508517176 72992089 - 35578544605601758303697179246592/1350851717672992089))))/729 - ((19*tan(x/2)^2)/1492992 + (1951*tan(x/2)^4)/1119744 - (2950915285*tan(x/2 )^6)/6718464 + (254448996545*tan(x/2)^8)/13436928 - (14715707859823*tan(x/ 2)^10)/40310784 + (27922427907149*tan(x/2)^12)/6718464 - (1410378354771185 *tan(x/2)^14)/45349632 + (87714731161410877*tan(x/2)^16)/544195584 - (3234 45875817330293*tan(x/2)^18)/544195584 + (537090433481225933*tan(x/2)^20)/3 40122240 - (825564740867246567*tan(x/2)^22)/272097792 + (22913747991752548 93*tan(x/2)^24)/544195584 - (2291368397703201113*tan(x/2)^26)/544195584 + (825557780178649715*tan(x/2)^28)/272097792 - (1074165500610568903*tan(x/2) ^30)/680244480 + (646878327275255233*tan(x/2)^32)/1088391168 - (1754244943 28205713*tan(x/2)^34)/1088391168 + (8461949214736403*tan(x/2)^36)/27209779 2 - (2261600671578389*tan(x/2)^38)/544195584 + (44143935599789*tan(x/2)^40 )/120932352 - (254420598121*tan(x/2)^42)/13436928 + (8850790063*tan(x/2)^4 4)/20155392 + 1/7464960)/(tan(x/2)^5 - (140*tan(x/2)^7)/3 + (8830*tan(x/2) ^9)/9 - (331660*tan(x/2)^11)/27 + (8263805*tan(x/2)^13)/81 - (144041968...
\[ \int \frac {1}{(\sin (x)+\sin (5 x))^6} \, dx=\int \frac {1}{\sin \left (5 x \right )^{6}+6 \sin \left (5 x \right )^{5} \sin \left (x \right )+15 \sin \left (5 x \right )^{4} \sin \left (x \right )^{2}+20 \sin \left (5 x \right )^{3} \sin \left (x \right )^{3}+15 \sin \left (5 x \right )^{2} \sin \left (x \right )^{4}+6 \sin \left (5 x \right ) \sin \left (x \right )^{5}+\sin \left (x \right )^{6}}d x \] Input:
int(1/(sin(x)+sin(5*x))^6,x)
Output:
int(1/(sin(5*x)**6 + 6*sin(5*x)**5*sin(x) + 15*sin(5*x)**4*sin(x)**2 + 20* sin(5*x)**3*sin(x)**3 + 15*sin(5*x)**2*sin(x)**4 + 6*sin(5*x)*sin(x)**5 + sin(x)**6),x)