Integrand size = 9, antiderivative size = 127 \[ \int \frac {1}{(\sin (x)+\sin (5 x))^3} \, dx=-\frac {7}{144} \text {arctanh}(\cos (x))-\frac {55}{9} \text {arctanh}(2 \cos (x))+\frac {139 \text {arctanh}\left (\sqrt {2} \cos (x)\right )}{16 \sqrt {2}}-\frac {1}{18 (1-2 \cos (x))^2}+\frac {35}{54 (1-2 \cos (x))}-\frac {1}{864 (1-\cos (x))}+\frac {1}{864 (1+\cos (x))}+\frac {1}{18 (1+2 \cos (x))^2}-\frac {35}{54 (1+2 \cos (x))}-\frac {19}{16} \cos (x) \sec (2 x)+\frac {1}{8} \cos (x) \sec ^2(2 x) \] Output:
-7/144*arctanh(cos(x))-55/9*arctanh(2*cos(x))+139/32*arctanh(cos(x)*2^(1/2 ))*2^(1/2)-1/18/(1-2*cos(x))^2+35/(54-108*cos(x))-1/(864-864*cos(x))+1/(86 4+864*cos(x))+1/18/(1+2*cos(x))^2-35/(54+108*cos(x))-19/16*cos(x)*sec(2*x) +1/8*cos(x)*sec(2*x)^2
Time = 0.71 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.47 \[ \int \frac {1}{(\sin (x)+\sin (5 x))^3} \, dx=\frac {-7506 \sqrt {2} \text {arctanh}\left (\frac {-1+\tan \left (\frac {x}{2}\right )}{\sqrt {2}}\right )+7506 \sqrt {2} \text {arctanh}\left (\frac {1+\tan \left (\frac {x}{2}\right )}{\sqrt {2}}\right )-\frac {96}{(1-2 \cos (x))^2}+\frac {1120}{1-2 \cos (x)}+\frac {96}{(1+2 \cos (x))^2}-\frac {1120}{1+2 \cos (x)}-\csc ^2\left (\frac {x}{2}\right )-84 \log \left (\cos \left (\frac {x}{2}\right )\right )+5280 \log (1-2 \cos (x))-5280 \log (1+2 \cos (x))+84 \log \left (\sin \left (\frac {x}{2}\right )\right )+\sec ^2\left (\frac {x}{2}\right )-\frac {918}{\cos (x)-\sin (x)}+\frac {108 \sin (x)}{(\cos (x)-\sin (x))^2}-\frac {108 \sin (x)}{(\cos (x)+\sin (x))^2}-\frac {918}{\cos (x)+\sin (x)}}{1728} \] Input:
Integrate[(Sin[x] + Sin[5*x])^(-3),x]
Output:
(-7506*Sqrt[2]*ArcTanh[(-1 + Tan[x/2])/Sqrt[2]] + 7506*Sqrt[2]*ArcTanh[(1 + Tan[x/2])/Sqrt[2]] - 96/(1 - 2*Cos[x])^2 + 1120/(1 - 2*Cos[x]) + 96/(1 + 2*Cos[x])^2 - 1120/(1 + 2*Cos[x]) - Csc[x/2]^2 - 84*Log[Cos[x/2]] + 5280* Log[1 - 2*Cos[x]] - 5280*Log[1 + 2*Cos[x]] + 84*Log[Sin[x/2]] + Sec[x/2]^2 - 918/(Cos[x] - Sin[x]) + (108*Sin[x])/(Cos[x] - Sin[x])^2 - (108*Sin[x]) /(Cos[x] + Sin[x])^2 - 918/(Cos[x] + Sin[x]))/1728
Time = 0.36 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.21, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {3042, 4824, 27, 1567, 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.
| \(\displaystyle \int \frac {1}{(\sin (x)+\sin (5 x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(\sin (x)+\sin (5 x))^3}dx\) |
| \(\Big \downarrow \) 4824 |
| \(\displaystyle -\int \frac {1}{8 \left (1-\cos ^2(x)\right )^2 \left (8 \cos ^4(x)-6 \cos ^2(x)+1\right )^3}d\cos (x)\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{8} \int \frac {1}{\left (1-\cos ^2(x)\right )^2 \left (8 \cos ^4(x)-6 \cos ^2(x)+1\right )^3}d\cos (x)\) |
| \(\Big \downarrow \) 1567 |
| \(\displaystyle -\frac {1}{8} \int \left (\frac {60}{2 \cos ^2(x)-1}-\frac {880}{9 \left (4 \cos ^2(x)-1\right )}+\frac {1}{108 (\cos (x)-1)^2}+\frac {1}{108 (\cos (x)+1)^2}-\frac {280}{27 (2 \cos (x)-1)^2}-\frac {280}{27 (2 \cos (x)+1)^2}-\frac {16}{\left (2 \cos ^2(x)-1\right )^2}-\frac {16}{9 (2 \cos (x)-1)^3}+\frac {16}{9 (2 \cos (x)+1)^3}+\frac {4}{\left (2 \cos ^2(x)-1\right )^3}-\frac {7}{18 \left (\cos ^2(x)-1\right )}\right )d\cos (x)\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{8} \left (-\frac {7}{18} \text {arctanh}(\cos (x))-\frac {440}{9} \text {arctanh}(2 \cos (x))+34 \sqrt {2} \text {arctanh}\left (\sqrt {2} \cos (x)\right )+\frac {3 \text {arctanh}\left (\sqrt {2} \cos (x)\right )}{2 \sqrt {2}}+\frac {19 \cos (x)}{2 \left (1-2 \cos ^2(x)\right )}+\frac {\cos (x)}{\left (1-2 \cos ^2(x)\right )^2}+\frac {140}{27 (1-2 \cos (x))}-\frac {1}{108 (1-\cos (x))}+\frac {1}{108 (\cos (x)+1)}-\frac {140}{27 (2 \cos (x)+1)}-\frac {4}{9 (1-2 \cos (x))^2}+\frac {4}{9 (2 \cos (x)+1)^2}\right )\) |
Input:
Int[(Sin[x] + Sin[5*x])^(-3),x]
Output:
((-7*ArcTanh[Cos[x]])/18 - (440*ArcTanh[2*Cos[x]])/9 + (3*ArcTanh[Sqrt[2]* Cos[x]])/(2*Sqrt[2]) + 34*Sqrt[2]*ArcTanh[Sqrt[2]*Cos[x]] - 4/(9*(1 - 2*Co s[x])^2) + 140/(27*(1 - 2*Cos[x])) - 1/(108*(1 - Cos[x])) + 1/(108*(1 + Co s[x])) + 4/(9*(1 + 2*Cos[x])^2) - 140/(27*(1 + 2*Cos[x])) + Cos[x]/(1 - 2* Cos[x]^2)^2 + (19*Cos[x])/(2*(1 - 2*Cos[x]^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[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x _Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b^2 - 4*a*c, 0] && ((IntegerQ[p] && IntegerQ[q]) || IGtQ[p, 0] || IGtQ[q, 0])
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]
Time = 5.68 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.98
| method | result | size |
| default | \(\frac {1}{18 \left (2 \cos \left (x \right )+1\right )^{2}}-\frac {35}{54 \left (2 \cos \left (x \right )+1\right )}-\frac {55 \ln \left (2 \cos \left (x \right )+1\right )}{18}-\frac {1}{18 \left (2 \cos \left (x \right )-1\right )^{2}}-\frac {35}{54 \left (2 \cos \left (x \right )-1\right )}+\frac {55 \ln \left (2 \cos \left (x \right )-1\right )}{18}+\frac {1}{864+864 \cos \left (x \right )}-\frac {7 \ln \left (1+\cos \left (x \right )\right )}{288}-\frac {8 \left (\frac {19 \cos \left (x \right )^{3}}{64}-\frac {21 \cos \left (x \right )}{128}\right )}{\left (2 \cos \left (x \right )^{2}-1\right )^{2}}+\frac {139 \,\operatorname {arctanh}\left (\sqrt {2}\, \cos \left (x \right )\right ) \sqrt {2}}{32}+\frac {1}{864 \cos \left (x \right )-864}+\frac {7 \ln \left (\cos \left (x \right )-1\right )}{288}\) | \(125\) |
| risch | \(-\frac {119 \,{\mathrm e}^{19 i x}+55 \,{\mathrm e}^{17 i x}+95 \,{\mathrm e}^{15 i x}-111 \,{\mathrm e}^{13 i x}-166 \,{\mathrm e}^{11 i x}-166 \,{\mathrm e}^{9 i x}-111 \,{\mathrm e}^{7 i x}+95 \,{\mathrm e}^{5 i x}+55 \,{\mathrm e}^{3 i x}+119 \,{\mathrm e}^{i x}}{48 \left ({\mathrm e}^{10 i x}+{\mathrm e}^{6 i x}-{\mathrm e}^{4 i x}-1\right )^{2}}+\frac {7 \ln \left ({\mathrm e}^{i x}-1\right )}{144}-\frac {7 \ln \left ({\mathrm e}^{i x}+1\right )}{144}+\frac {139 \sqrt {2}\, \ln \left ({\mathrm e}^{2 i x}+\sqrt {2}\, {\mathrm e}^{i x}+1\right )}{64}-\frac {139 \sqrt {2}\, \ln \left ({\mathrm e}^{2 i x}-\sqrt {2}\, {\mathrm e}^{i x}+1\right )}{64}+\frac {55 \ln \left ({\mathrm e}^{2 i x}-{\mathrm e}^{i x}+1\right )}{18}-\frac {55 \ln \left ({\mathrm e}^{2 i x}+{\mathrm e}^{i x}+1\right )}{18}\) | \(193\) |
Input:
int(1/(sin(x)+sin(5*x))^3,x,method=_RETURNVERBOSE)
Output:
1/18/(2*cos(x)+1)^2-35/54/(2*cos(x)+1)-55/18*ln(2*cos(x)+1)-1/18/(2*cos(x) -1)^2-35/54/(2*cos(x)-1)+55/18*ln(2*cos(x)-1)+1/864/(1+cos(x))-7/288*ln(1+ cos(x))-8*(19/64*cos(x)^3-21/128*cos(x))/(2*cos(x)^2-1)^2+139/32*arctanh(2 ^(1/2)*cos(x))*2^(1/2)+1/864/(cos(x)-1)+7/288*ln(cos(x)-1)
Leaf count of result is larger than twice the leaf count of optimal. 310 vs. \(2 (99) = 198\).
Time = 0.11 (sec) , antiderivative size = 310, normalized size of antiderivative = 2.44 \[ \int \frac {1}{(\sin (x)+\sin (5 x))^3} \, dx=-\frac {45696 \, \cos \left (x\right )^{9} - 97536 \, \cos \left (x\right )^{7} + 70152 \, \cos \left (x\right )^{5} - 20316 \, \cos \left (x\right )^{3} - 1251 \, {\left (64 \, \sqrt {2} \cos \left (x\right )^{10} - 160 \, \sqrt {2} \cos \left (x\right )^{8} + 148 \, \sqrt {2} \cos \left (x\right )^{6} - 64 \, \sqrt {2} \cos \left (x\right )^{4} + 13 \, \sqrt {2} \cos \left (x\right )^{2} - \sqrt {2}\right )} \log \left (-\frac {2 \, \cos \left (x\right )^{2} + 2 \, \sqrt {2} \cos \left (x\right ) + 1}{2 \, \cos \left (x\right )^{2} - 1}\right ) + 14 \, {\left (64 \, \cos \left (x\right )^{10} - 160 \, \cos \left (x\right )^{8} + 148 \, \cos \left (x\right )^{6} - 64 \, \cos \left (x\right )^{4} + 13 \, \cos \left (x\right )^{2} - 1\right )} \log \left (\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) - 14 \, {\left (64 \, \cos \left (x\right )^{10} - 160 \, \cos \left (x\right )^{8} + 148 \, \cos \left (x\right )^{6} - 64 \, \cos \left (x\right )^{4} + 13 \, \cos \left (x\right )^{2} - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) - 1760 \, {\left (64 \, \cos \left (x\right )^{10} - 160 \, \cos \left (x\right )^{8} + 148 \, \cos \left (x\right )^{6} - 64 \, \cos \left (x\right )^{4} + 13 \, \cos \left (x\right )^{2} - 1\right )} \log \left (-2 \, \cos \left (x\right ) + 1\right ) + 1760 \, {\left (64 \, \cos \left (x\right )^{10} - 160 \, \cos \left (x\right )^{8} + 148 \, \cos \left (x\right )^{6} - 64 \, \cos \left (x\right )^{4} + 13 \, \cos \left (x\right )^{2} - 1\right )} \log \left (-2 \, \cos \left (x\right ) - 1\right ) + 1992 \, \cos \left (x\right )}{576 \, {\left (64 \, \cos \left (x\right )^{10} - 160 \, \cos \left (x\right )^{8} + 148 \, \cos \left (x\right )^{6} - 64 \, \cos \left (x\right )^{4} + 13 \, \cos \left (x\right )^{2} - 1\right )}} \] Input:
integrate(1/(sin(x)+sin(5*x))^3,x, algorithm="fricas")
Output:
-1/576*(45696*cos(x)^9 - 97536*cos(x)^7 + 70152*cos(x)^5 - 20316*cos(x)^3 - 1251*(64*sqrt(2)*cos(x)^10 - 160*sqrt(2)*cos(x)^8 + 148*sqrt(2)*cos(x)^6 - 64*sqrt(2)*cos(x)^4 + 13*sqrt(2)*cos(x)^2 - sqrt(2))*log(-(2*cos(x)^2 + 2*sqrt(2)*cos(x) + 1)/(2*cos(x)^2 - 1)) + 14*(64*cos(x)^10 - 160*cos(x)^8 + 148*cos(x)^6 - 64*cos(x)^4 + 13*cos(x)^2 - 1)*log(1/2*cos(x) + 1/2) - 1 4*(64*cos(x)^10 - 160*cos(x)^8 + 148*cos(x)^6 - 64*cos(x)^4 + 13*cos(x)^2 - 1)*log(-1/2*cos(x) + 1/2) - 1760*(64*cos(x)^10 - 160*cos(x)^8 + 148*cos( x)^6 - 64*cos(x)^4 + 13*cos(x)^2 - 1)*log(-2*cos(x) + 1) + 1760*(64*cos(x) ^10 - 160*cos(x)^8 + 148*cos(x)^6 - 64*cos(x)^4 + 13*cos(x)^2 - 1)*log(-2* cos(x) - 1) + 1992*cos(x))/(64*cos(x)^10 - 160*cos(x)^8 + 148*cos(x)^6 - 6 4*cos(x)^4 + 13*cos(x)^2 - 1)
\[ \int \frac {1}{(\sin (x)+\sin (5 x))^3} \, dx=\int \frac {1}{\left (\sin {\left (x \right )} + \sin {\left (5 x \right )}\right )^{3}}\, dx \] Input:
integrate(1/(sin(x)+sin(5*x))**3,x)
Output:
Integral((sin(x) + sin(5*x))**(-3), x)
Exception generated. \[ \int \frac {1}{(\sin (x)+\sin (5 x))^3} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(1/(sin(x)+sin(5*x))^3,x, algorithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is un defined.
Leaf count of result is larger than twice the leaf count of optimal. 297 vs. \(2 (99) = 198\).
Time = 0.13 (sec) , antiderivative size = 297, normalized size of antiderivative = 2.34 \[ \int \frac {1}{(\sin (x)+\sin (5 x))^3} \, dx =\text {Too large to display} \] Input:
integrate(1/(sin(x)+sin(5*x))^3,x, algorithm="giac")
Output:
139/64*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)) - 1/1728*(42*(cos(x) - 1)/(co s(x) + 1) - 1)*(cos(x) + 1)/(cos(x) - 1) - 1/1728*(cos(x) - 1)/(cos(x) + 1 ) - 1/648*(379605*(cos(x) - 1)/(cos(x) + 1) + 2276125*(cos(x) - 1)^2/(cos( x) + 1)^2 + 6174297*(cos(x) - 1)^3/(cos(x) + 1)^3 + 7608731*(cos(x) - 1)^4 /(cos(x) + 1)^4 + 3721407*(cos(x) - 1)^5/(cos(x) + 1)^5 + 745855*(cos(x) - 1)^6/(cos(x) + 1)^6 + 50643*(cos(x) - 1)^7/(cos(x) + 1)^7 + 23049)/(28*(c os(x) - 1)/(cos(x) + 1) + 66*(cos(x) - 1)^2/(cos(x) + 1)^2 + 28*(cos(x) - 1)^3/(cos(x) + 1)^3 + 3*(cos(x) - 1)^4/(cos(x) + 1)^4 + 3)^2 + 7/288*log(- (cos(x) - 1)/(cos(x) + 1)) - 55/18*log(abs(-(cos(x) - 1)/(cos(x) + 1) - 3) ) + 55/18*log(abs(-3*(cos(x) - 1)/(cos(x) + 1) - 1))
Time = 22.38 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.74 \[ \int \frac {1}{(\sin (x)+\sin (5 x))^3} \, dx=\frac {7\,\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )}{144}-\frac {55\,\mathrm {atanh}\left (\frac {327840921493337120}{36687588870411\,\left (\frac {2192467318223163091000\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{282429536481}-\frac {243607059170136554200}{94143178827}\right )}-\frac {345171004}{345170855}\right )}{9}-\frac {-\frac {45013\,{\mathrm {tan}\left (\frac {x}{2}\right )}^{16}}{5184}+\frac {93224\,{\mathrm {tan}\left (\frac {x}{2}\right )}^{14}}{729}-\frac {275627\,{\mathrm {tan}\left (\frac {x}{2}\right )}^{12}}{432}+\frac {3803641\,{\mathrm {tan}\left (\frac {x}{2}\right )}^{10}}{2916}-\frac {8229425\,{\mathrm {tan}\left (\frac {x}{2}\right )}^8}{7776}+\frac {568669\,{\mathrm {tan}\left (\frac {x}{2}\right )}^6}{1458}-\frac {252775\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4}{3888}+\frac {1277\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{324}+\frac {1}{1728}}{{\mathrm {tan}\left (\frac {x}{2}\right )}^{18}-\frac {56\,{\mathrm {tan}\left (\frac {x}{2}\right )}^{16}}{3}+\frac {1180\,{\mathrm {tan}\left (\frac {x}{2}\right )}^{14}}{9}-\frac {1288\,{\mathrm {tan}\left (\frac {x}{2}\right )}^{12}}{3}+\frac {5942\,{\mathrm {tan}\left (\frac {x}{2}\right )}^{10}}{9}-\frac {1288\,{\mathrm {tan}\left (\frac {x}{2}\right )}^8}{3}+\frac {1180\,{\mathrm {tan}\left (\frac {x}{2}\right )}^6}{9}-\frac {56\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4}{3}+{\mathrm {tan}\left (\frac {x}{2}\right )}^2}+\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{1728}+\frac {139\,\sqrt {2}\,\mathrm {atanh}\left (\frac {4678477767243387713\,\sqrt {2}}{918330048\,\left (\frac {43383387569666028847\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{1033121304}-\frac {44660475234141568277}{6198727824}\right )}-\frac {736240501019846950969\,\sqrt {2}\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{24794911296\,\left (\frac {43383387569666028847\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{1033121304}-\frac {44660475234141568277}{6198727824}\right )}\right )}{32} \] Input:
int(1/(sin(5*x) + sin(x))^3,x)
Output:
(7*log(tan(x/2)))/144 - (55*atanh(327840921493337120/(36687588870411*((219 2467318223163091000*tan(x/2)^2)/282429536481 - 243607059170136554200/94143 178827)) - 345171004/345170855))/9 - ((1277*tan(x/2)^2)/324 - (252775*tan( x/2)^4)/3888 + (568669*tan(x/2)^6)/1458 - (8229425*tan(x/2)^8)/7776 + (380 3641*tan(x/2)^10)/2916 - (275627*tan(x/2)^12)/432 + (93224*tan(x/2)^14)/72 9 - (45013*tan(x/2)^16)/5184 + 1/1728)/(tan(x/2)^2 - (56*tan(x/2)^4)/3 + ( 1180*tan(x/2)^6)/9 - (1288*tan(x/2)^8)/3 + (5942*tan(x/2)^10)/9 - (1288*ta n(x/2)^12)/3 + (1180*tan(x/2)^14)/9 - (56*tan(x/2)^16)/3 + tan(x/2)^18) + tan(x/2)^2/1728 + (139*2^(1/2)*atanh((4678477767243387713*2^(1/2))/(918330 048*((43383387569666028847*tan(x/2)^2)/1033121304 - 44660475234141568277/6 198727824)) - (736240501019846950969*2^(1/2)*tan(x/2)^2)/(24794911296*((43 383387569666028847*tan(x/2)^2)/1033121304 - 44660475234141568277/619872782 4))))/32
\[ \int \frac {1}{(\sin (x)+\sin (5 x))^3} \, dx=\int \frac {1}{\sin \left (5 x \right )^{3}+3 \sin \left (5 x \right )^{2} \sin \left (x \right )+3 \sin \left (5 x \right ) \sin \left (x \right )^{2}+\sin \left (x \right )^{3}}d x \] Input:
int(1/(sin(x)+sin(5*x))^3,x)
Output:
int(1/(sin(5*x)**3 + 3*sin(5*x)**2*sin(x) + 3*sin(5*x)*sin(x)**2 + sin(x)* *3),x)