Integrand size = 9, antiderivative size = 154 \[ \int \frac {1}{(\sin (x)+\sin (5 x))^4} \, dx=-\frac {103}{8} \text {arctanh}(2 \cos (x) \sin (x))-\frac {43 \cot (x)}{3888}-\frac {\cot ^3(x)}{3888}-\frac {1808 \log \left (\sqrt {3} \cos (x)-\sin (x)\right )}{81 \sqrt {3}}+\frac {1808 \log \left (\sqrt {3} \cos (x)+\sin (x)\right )}{81 \sqrt {3}}+\frac {4 \tan (x) \left (593-539 \tan ^2(x)\right )}{81 \left (3-4 \tan ^2(x)+\tan ^4(x)\right )^3}-\frac {2 \tan (x) \left (3502-1237 \tan ^2(x)\right )}{243 \left (3-4 \tan ^2(x)+\tan ^4(x)\right )^2}+\frac {\tan (x) \left (38097-18413 \tan ^2(x)\right )}{972 \left (3-4 \tan ^2(x)+\tan ^4(x)\right )} \] Output:
-103/8*arctanh(2*cos(x)*sin(x))-43/3888*cot(x)-1/3888*cot(x)^3-1808/243*ln (3^(1/2)*cos(x)-sin(x))*3^(1/2)+1808/243*ln(3^(1/2)*cos(x)+sin(x))*3^(1/2) +4/81*tan(x)*(593-539*tan(x)^2)/(3-4*tan(x)^2+tan(x)^4)^3-2/243*tan(x)*(35 02-1237*tan(x)^2)/(3-4*tan(x)^2+tan(x)^4)^2+tan(x)*(38097-18413*tan(x)^2)/ (2916-3888*tan(x)^2+972*tan(x)^4)
Time = 0.64 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.07 \[ \int \frac {1}{(\sin (x)+\sin (5 x))^4} \, dx=\frac {115712 \sqrt {3} \text {arctanh}\left (\frac {\tan (x)}{\sqrt {3}}\right )-2 \cot (x) \left (42+\csc ^2(x)\right )+100116 \log (\cos (x)-\sin (x))-100116 \log (\cos (x)+\sin (x))-\frac {1377}{(\cos (x)-\sin (x))^2}+\frac {162 \sin (x)}{(\cos (x)-\sin (x))^3}+\frac {26892 \sin (x)}{\cos (x)-\sin (x)}+\frac {162 \sin (x)}{(\cos (x)+\sin (x))^3}+\frac {1377}{(\cos (x)+\sin (x))^2}+\frac {26892 \sin (x)}{\cos (x)+\sin (x)}+\frac {1536 \sin (2 x)}{(1+2 \cos (2 x))^3}+\frac {9472 \sin (2 x)}{(1+2 \cos (2 x))^2}+\frac {51456 \sin (2 x)}{1+2 \cos (2 x)}}{7776} \] Input:
Integrate[(Sin[x] + Sin[5*x])^(-4),x]
Output:
(115712*Sqrt[3]*ArcTanh[Tan[x]/Sqrt[3]] - 2*Cot[x]*(42 + Csc[x]^2) + 10011 6*Log[Cos[x] - Sin[x]] - 100116*Log[Cos[x] + Sin[x]] - 1377/(Cos[x] - Sin[ x])^2 + (162*Sin[x])/(Cos[x] - Sin[x])^3 + (26892*Sin[x])/(Cos[x] - Sin[x] ) + (162*Sin[x])/(Cos[x] + Sin[x])^3 + 1377/(Cos[x] + Sin[x])^2 + (26892*S in[x])/(Cos[x] + Sin[x]) + (1536*Sin[2*x])/(1 + 2*Cos[2*x])^3 + (9472*Sin[ 2*x])/(1 + 2*Cos[2*x])^2 + (51456*Sin[2*x])/(1 + 2*Cos[2*x]))/7776
Time = 0.57 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.83, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.222, Rules used = {3042, 4822, 27, 1673, 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.
| \(\displaystyle \int \frac {1}{(\sin (x)+\sin (5 x))^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(\sin (x)+\sin (5 x))^4}dx\) |
| \(\Big \downarrow \) 4822 |
| \(\displaystyle \int \frac {\left (\tan ^2(x)+1\right )^9 \cot ^4(x)}{16 \left (\tan ^4(x)-4 \tan ^2(x)+3\right )^4}d\tan (x)\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{16} \int \frac {\cot ^4(x) \left (\tan ^2(x)+1\right )^9}{\left (\tan ^4(x)-4 \tan ^2(x)+3\right )^4}d\tan (x)\) |
| \(\Big \downarrow \) 1673 |
| \(\displaystyle \frac {1}{16} \left (\frac {64 \tan (x) \left (593-539 \tan ^2(x)\right )}{81 \left (\tan ^4(x)-4 \tan ^2(x)+3\right )^3}-\frac {1}{72} \int -\frac {8 \cot ^4(x) \left (81 \tan ^{14}(x)+1053 \tan ^{12}(x)+6885 \tan ^{10}(x)+31185 \tan ^8(x)-196173 \tan ^6(x)-36617 \tan ^4(x)+279 \tan ^2(x)+27\right )}{9 \left (\tan ^4(x)-4 \tan ^2(x)+3\right )^3}d\tan (x)\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{16} \left (\frac {1}{81} \int \frac {\cot ^4(x) \left (81 \tan ^{14}(x)+1053 \tan ^{12}(x)+6885 \tan ^{10}(x)+31185 \tan ^8(x)-196173 \tan ^6(x)-36617 \tan ^4(x)+279 \tan ^2(x)+27\right )}{\left (\tan ^4(x)-4 \tan ^2(x)+3\right )^3}d\tan (x)+\frac {64 \tan (x) \left (593-539 \tan ^2(x)\right )}{81 \left (\tan ^4(x)-4 \tan ^2(x)+3\right )^3}\right )\) |
| \(\Big \downarrow \) 2198 |
| \(\displaystyle \frac {1}{16} \left (\frac {1}{81} \left (-\frac {1}{48} \int -\frac {16 \cot ^4(x) \left (243 \tan ^{10}(x)+4131 \tan ^8(x)+234370 \tan ^6(x)+75858 \tan ^4(x)+315 \tan ^2(x)+27\right )}{\left (\tan ^4(x)-4 \tan ^2(x)+3\right )^2}d\tan (x)-\frac {32 \tan (x) \left (3502-1237 \tan ^2(x)\right )}{3 \left (\tan ^4(x)-4 \tan ^2(x)+3\right )^2}\right )+\frac {64 \tan (x) \left (593-539 \tan ^2(x)\right )}{81 \left (\tan ^4(x)-4 \tan ^2(x)+3\right )^3}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{16} \left (\frac {1}{81} \left (\frac {1}{3} \int \frac {\cot ^4(x) \left (243 \tan ^{10}(x)+4131 \tan ^8(x)+234370 \tan ^6(x)+75858 \tan ^4(x)+315 \tan ^2(x)+27\right )}{\left (\tan ^4(x)-4 \tan ^2(x)+3\right )^2}d\tan (x)-\frac {32 \tan (x) \left (3502-1237 \tan ^2(x)\right )}{3 \left (\tan ^4(x)-4 \tan ^2(x)+3\right )^2}\right )+\frac {64 \tan (x) \left (593-539 \tan ^2(x)\right )}{81 \left (\tan ^4(x)-4 \tan ^2(x)+3\right )^3}\right )\) |
| \(\Big \downarrow \) 2198 |
| \(\displaystyle \frac {1}{16} \left (\frac {1}{81} \left (\frac {1}{3} \left (\frac {4 \tan (x) \left (38097-18413 \tan ^2(x)\right )}{\tan ^4(x)-4 \tan ^2(x)+3}-\frac {1}{24} \int -\frac {24 \cot ^4(x) \left (-73409 \tan ^6(x)-126949 \tan ^4(x)+117 \tan ^2(x)+9\right )}{\tan ^4(x)-4 \tan ^2(x)+3}d\tan (x)\right )-\frac {32 \tan (x) \left (3502-1237 \tan ^2(x)\right )}{3 \left (\tan ^4(x)-4 \tan ^2(x)+3\right )^2}\right )+\frac {64 \tan (x) \left (593-539 \tan ^2(x)\right )}{81 \left (\tan ^4(x)-4 \tan ^2(x)+3\right )^3}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{16} \left (\frac {1}{81} \left (\frac {1}{3} \left (\int \frac {\cot ^4(x) \left (-73409 \tan ^6(x)-126949 \tan ^4(x)+117 \tan ^2(x)+9\right )}{\tan ^4(x)-4 \tan ^2(x)+3}d\tan (x)+\frac {4 \tan (x) \left (38097-18413 \tan ^2(x)\right )}{\tan ^4(x)-4 \tan ^2(x)+3}\right )-\frac {32 \tan (x) \left (3502-1237 \tan ^2(x)\right )}{3 \left (\tan ^4(x)-4 \tan ^2(x)+3\right )^2}\right )+\frac {64 \tan (x) \left (593-539 \tan ^2(x)\right )}{81 \left (\tan ^4(x)-4 \tan ^2(x)+3\right )^3}\right )\) |
| \(\Big \downarrow \) 2195 |
| \(\displaystyle \frac {1}{16} \left (\frac {1}{81} \left (\frac {1}{3} \left (\int \left (3 \cot ^4(x)+43 \cot ^2(x)-\frac {173568}{\tan ^2(x)-3}+\frac {100116}{\tan ^2(x)-1}\right )d\tan (x)+\frac {4 \tan (x) \left (38097-18413 \tan ^2(x)\right )}{\tan ^4(x)-4 \tan ^2(x)+3}\right )-\frac {32 \tan (x) \left (3502-1237 \tan ^2(x)\right )}{3 \left (\tan ^4(x)-4 \tan ^2(x)+3\right )^2}\right )+\frac {64 \tan (x) \left (593-539 \tan ^2(x)\right )}{81 \left (\tan ^4(x)-4 \tan ^2(x)+3\right )^3}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{16} \left (\frac {1}{81} \left (\frac {1}{3} \left (-100116 \text {arctanh}(\tan (x))+57856 \sqrt {3} \text {arctanh}\left (\frac {\tan (x)}{\sqrt {3}}\right )+\frac {4 \tan (x) \left (38097-18413 \tan ^2(x)\right )}{\tan ^4(x)-4 \tan ^2(x)+3}-\cot ^3(x)-43 \cot (x)\right )-\frac {32 \tan (x) \left (3502-1237 \tan ^2(x)\right )}{3 \left (\tan ^4(x)-4 \tan ^2(x)+3\right )^2}\right )+\frac {64 \tan (x) \left (593-539 \tan ^2(x)\right )}{81 \left (\tan ^4(x)-4 \tan ^2(x)+3\right )^3}\right )\) |
Input:
Int[(Sin[x] + Sin[5*x])^(-4),x]
Output:
((64*Tan[x]*(593 - 539*Tan[x]^2))/(81*(3 - 4*Tan[x]^2 + Tan[x]^4)^3) + ((- 32*Tan[x]*(3502 - 1237*Tan[x]^2))/(3*(3 - 4*Tan[x]^2 + Tan[x]^4)^2) + (-10 0116*ArcTanh[Tan[x]] + 57856*Sqrt[3]*ArcTanh[Tan[x]/Sqrt[3]] - 43*Cot[x] - Cot[x]^3 + (4*Tan[x]*(38097 - 18413*Tan[x]^2))/(3 - 4*Tan[x]^2 + Tan[x]^4 ))/3)/81)/16
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 = 25.68 (sec) , antiderivative size = 2, normalized size of antiderivative = 0.01
| method | result | size |
| parallelrisch | \(0\) | \(2\) |
| default | \(-\frac {64 \left (\frac {85 \tan \left (x \right )^{5}}{2}-280 \tan \left (x \right )^{3}+\frac {963 \tan \left (x \right )}{2}\right )}{243 \left (\tan \left (x \right )^{2}-3\right )^{3}}+\frac {3616 \sqrt {3}\, \operatorname {arctanh}\left (\frac {\tan \left (x \right ) \sqrt {3}}{3}\right )}{243}-\frac {1}{3888 \tan \left (x \right )^{3}}-\frac {43}{3888 \tan \left (x \right )}-\frac {1}{24 \left (\tan \left (x \right )-1\right )^{3}}-\frac {7}{16 \left (\tan \left (x \right )-1\right )^{2}}-\frac {31}{8 \left (\tan \left (x \right )-1\right )}+\frac {103 \ln \left (\tan \left (x \right )-1\right )}{8}-\frac {1}{24 \left (\tan \left (x \right )+1\right )^{3}}+\frac {7}{16 \left (\tan \left (x \right )+1\right )^{2}}-\frac {31}{8 \left (\tan \left (x \right )+1\right )}-\frac {103 \ln \left (\tan \left (x \right )+1\right )}{8}\) | \(116\) |
| risch | \(\frac {i \left (1111 \,{\mathrm e}^{28 i x}+3616 \,{\mathrm e}^{26 i x}+552 \,{\mathrm e}^{24 i x}+5707 \,{\mathrm e}^{22 i x}-12743 \,{\mathrm e}^{20 i x}+3768 \,{\mathrm e}^{18 i x}-23899 \,{\mathrm e}^{16 i x}+15629 \,{\mathrm e}^{14 i x}-13800 \,{\mathrm e}^{12 i x}+26785 \,{\mathrm e}^{10 i x}-7469 \,{\mathrm e}^{8 i x}+13728 \,{\mathrm e}^{6 i x}-9560 \,{\mathrm e}^{4 i x}+1111 \,{\mathrm e}^{2 i x}-4392\right )}{324 \left ({\mathrm e}^{10 i x}+{\mathrm e}^{6 i x}-{\mathrm e}^{4 i x}-1\right )^{3}}+\frac {1808 \sqrt {3}\, \ln \left ({\mathrm e}^{2 i x}+\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{243}-\frac {1808 \sqrt {3}\, \ln \left ({\mathrm e}^{2 i x}+\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{243}+\frac {103 \ln \left ({\mathrm e}^{2 i x}-i\right )}{8}-\frac {103 \ln \left ({\mathrm e}^{2 i x}+i\right )}{8}\) | \(186\) |
Input:
int(1/(sin(x)+sin(5*x))^4,x,method=_RETURNVERBOSE)
Output:
0
Leaf count of result is larger than twice the leaf count of optimal. 339 vs. \(2 (130) = 260\).
Time = 0.13 (sec) , antiderivative size = 339, normalized size of antiderivative = 2.20 \[ \int \frac {1}{(\sin (x)+\sin (5 x))^4} \, dx=-\frac {13492224 \, \cos \left (x\right )^{15} - 52302336 \, \cos \left (x\right )^{13} + 82581120 \, \cos \left (x\right )^{11} - 68730912 \, \cos \left (x\right )^{9} + 32578200 \, \cos \left (x\right )^{7} - 8802612 \, \cos \left (x\right )^{5} + 1257984 \, \cos \left (x\right )^{3} + 25029 \, {\left (512 \, \cos \left (x\right )^{14} - 1664 \, \cos \left (x\right )^{12} + 2208 \, \cos \left (x\right )^{10} - 1560 \, \cos \left (x\right )^{8} + 636 \, \cos \left (x\right )^{6} - 150 \, \cos \left (x\right )^{4} + 19 \, \cos \left (x\right )^{2} - 1\right )} \log \left (2 \, \cos \left (x\right ) \sin \left (x\right ) + 1\right ) \sin \left (x\right ) - 25029 \, {\left (512 \, \cos \left (x\right )^{14} - 1664 \, \cos \left (x\right )^{12} + 2208 \, \cos \left (x\right )^{10} - 1560 \, \cos \left (x\right )^{8} + 636 \, \cos \left (x\right )^{6} - 150 \, \cos \left (x\right )^{4} + 19 \, \cos \left (x\right )^{2} - 1\right )} \log \left (-2 \, \cos \left (x\right ) \sin \left (x\right ) + 1\right ) \sin \left (x\right ) - 14464 \, {\left (512 \, \sqrt {3} \cos \left (x\right )^{14} - 1664 \, \sqrt {3} \cos \left (x\right )^{12} + 2208 \, \sqrt {3} \cos \left (x\right )^{10} - 1560 \, \sqrt {3} \cos \left (x\right )^{8} + 636 \, \sqrt {3} \cos \left (x\right )^{6} - 150 \, \sqrt {3} \cos \left (x\right )^{4} + 19 \, \sqrt {3} \cos \left (x\right )^{2} - \sqrt {3}\right )} \log \left (-\frac {8 \, \cos \left (x\right )^{4} - 16 \, \cos \left (x\right )^{2} - 4 \, {\left (2 \, \sqrt {3} \cos \left (x\right )^{3} + \sqrt {3} \cos \left (x\right )\right )} \sin \left (x\right ) - 1}{16 \, \cos \left (x\right )^{4} - 8 \, \cos \left (x\right )^{2} + 1}\right ) \sin \left (x\right ) - 73695 \, \cos \left (x\right )}{3888 \, {\left (512 \, \cos \left (x\right )^{14} - 1664 \, \cos \left (x\right )^{12} + 2208 \, \cos \left (x\right )^{10} - 1560 \, \cos \left (x\right )^{8} + 636 \, \cos \left (x\right )^{6} - 150 \, \cos \left (x\right )^{4} + 19 \, \cos \left (x\right )^{2} - 1\right )} \sin \left (x\right )} \] Input:
integrate(1/(sin(x)+sin(5*x))^4,x, algorithm="fricas")
Output:
-1/3888*(13492224*cos(x)^15 - 52302336*cos(x)^13 + 82581120*cos(x)^11 - 68 730912*cos(x)^9 + 32578200*cos(x)^7 - 8802612*cos(x)^5 + 1257984*cos(x)^3 + 25029*(512*cos(x)^14 - 1664*cos(x)^12 + 2208*cos(x)^10 - 1560*cos(x)^8 + 636*cos(x)^6 - 150*cos(x)^4 + 19*cos(x)^2 - 1)*log(2*cos(x)*sin(x) + 1)*s in(x) - 25029*(512*cos(x)^14 - 1664*cos(x)^12 + 2208*cos(x)^10 - 1560*cos( x)^8 + 636*cos(x)^6 - 150*cos(x)^4 + 19*cos(x)^2 - 1)*log(-2*cos(x)*sin(x) + 1)*sin(x) - 14464*(512*sqrt(3)*cos(x)^14 - 1664*sqrt(3)*cos(x)^12 + 220 8*sqrt(3)*cos(x)^10 - 1560*sqrt(3)*cos(x)^8 + 636*sqrt(3)*cos(x)^6 - 150*s qrt(3)*cos(x)^4 + 19*sqrt(3)*cos(x)^2 - sqrt(3))*log(-(8*cos(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) - 73695*cos(x))/((512*cos(x)^14 - 1664*cos(x)^12 + 2208*cos(x)^10 - 1560*cos(x)^8 + 636*cos(x)^6 - 150*cos(x)^4 + 19*cos(x) ^2 - 1)*sin(x))
\[ \int \frac {1}{(\sin (x)+\sin (5 x))^4} \, dx=\int \frac {1}{\left (\sin {\left (x \right )} + \sin {\left (5 x \right )}\right )^{4}}\, dx \] Input:
integrate(1/(sin(x)+sin(5*x))**4,x)
Output:
Integral((sin(x) + sin(5*x))**(-4), x)
Exception generated. \[ \int \frac {1}{(\sin (x)+\sin (5 x))^4} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(1/(sin(x)+sin(5*x))^4,x, algorithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is un defined.
Time = 0.13 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.72 \[ \int \frac {1}{(\sin (x)+\sin (5 x))^4} \, dx=-\frac {1808}{243} \, \sqrt {3} \log \left (\frac {{\left | -2 \, \sqrt {3} + 2 \, \tan \left (x\right ) \right |}}{{\left | 2 \, \sqrt {3} + 2 \, \tan \left (x\right ) \right |}}\right ) - \frac {24565 \, \tan \left (x\right )^{14} - 247373 \, \tan \left (x\right )^{12} + 934101 \, \tan \left (x\right )^{10} - 1618525 \, \tan \left (x\right )^{8} + 1287959 \, \tan \left (x\right )^{6} - 384543 \, \tan \left (x\right )^{4} + 351 \, \tan \left (x\right )^{2} + 9}{1296 \, {\left (\tan \left (x\right )^{5} - 4 \, \tan \left (x\right )^{3} + 3 \, \tan \left (x\right )\right )}^{3}} - \frac {103}{8} \, \log \left ({\left | \tan \left (x\right ) + 1 \right |}\right ) + \frac {103}{8} \, \log \left ({\left | \tan \left (x\right ) - 1 \right |}\right ) \] Input:
integrate(1/(sin(x)+sin(5*x))^4,x, algorithm="giac")
Output:
-1808/243*sqrt(3)*log(abs(-2*sqrt(3) + 2*tan(x))/abs(2*sqrt(3) + 2*tan(x)) ) - 1/1296*(24565*tan(x)^14 - 247373*tan(x)^12 + 934101*tan(x)^10 - 161852 5*tan(x)^8 + 1287959*tan(x)^6 - 384543*tan(x)^4 + 351*tan(x)^2 + 9)/(tan(x )^5 - 4*tan(x)^3 + 3*tan(x))^3 - 103/8*log(abs(tan(x) + 1)) + 103/8*log(ab s(tan(x) - 1))
Time = 22.26 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.78 \[ \int \frac {1}{(\sin (x)+\sin (5 x))^4} \, dx =\text {Too large to display} \] Input:
int(1/(sin(5*x) + sin(x))^4,x)
Output:
(169*tan(x/2))/31104 + (103*atanh((211880270124391989248*tan(x/2))/(254186 5828329*((105940135062195994624*tan(x/2)^2)/2541865828329 - 10594013506219 5994624/2541865828329))))/4 - ((47*tan(x/2)^2)/10368 - (38077*tan(x/2)^4)/ 1728 + (75085891*tan(x/2)^6)/139968 - (1494496261*tan(x/2)^8)/279936 + (23 512181909*tan(x/2)^10)/839808 - (17581132309*tan(x/2)^12)/209952 + (101785 38305*tan(x/2)^14)/69984 - (40705839889*tan(x/2)^16)/279936 + (23427167539 *tan(x/2)^18)/279936 - (3914473429*tan(x/2)^20)/139968 + (2237865505*tan(x /2)^22)/419904 - (449205433*tan(x/2)^24)/839808 + (2043451*tan(x/2)^26)/93 312 + 1/31104)/(tan(x/2)^3 - 28*tan(x/2)^5 + (982*tan(x/2)^7)/3 - (55972*t an(x/2)^9)/27 + 7727*tan(x/2)^11 - (155512*tan(x/2)^13)/9 + (68404*tan(x/2 )^15)/3 - (155512*tan(x/2)^17)/9 + 7727*tan(x/2)^19 - (55972*tan(x/2)^21)/ 27 + (982*tan(x/2)^23)/3 - 28*tan(x/2)^25 + tan(x/2)^27) + tan(x/2)^3/3110 4 - (3616*3^(1/2)*atanh((2089117654278142695571456*3^(1/2)*tan(x/2))/(5559 060566555523*((1044558827139071347785728*tan(x/2)^2)/1853020188851841 - 10 44558827139071347785728/1853020188851841))))/243
\[ \int \frac {1}{(\sin (x)+\sin (5 x))^4} \, dx=\int \frac {1}{\sin \left (5 x \right )^{4}+4 \sin \left (5 x \right )^{3} \sin \left (x \right )+6 \sin \left (5 x \right )^{2} \sin \left (x \right )^{2}+4 \sin \left (5 x \right ) \sin \left (x \right )^{3}+\sin \left (x \right )^{4}}d x \] Input:
int(1/(sin(x)+sin(5*x))^4,x)
Output:
int(1/(sin(5*x)**4 + 4*sin(5*x)**3*sin(x) + 6*sin(5*x)**2*sin(x)**2 + 4*si n(5*x)*sin(x)**3 + sin(x)**4),x)