Integrand size = 9, antiderivative size = 42 \[ \int \frac {1}{(\sin (x)+\sin (3 x))^3} \, dx=-\frac {7}{128} \text {arctanh}(\cos (x))+\frac {7 \sec (x)}{128}+\frac {7 \sec ^3(x)}{384}+\frac {7 \sec ^5(x)}{640}-\frac {1}{128} \csc ^2(x) \sec ^5(x) \] Output:
-7/128*arctanh(cos(x))+7/128*sec(x)+7/384*sec(x)^3+7/640*sec(x)^5-1/128*cs c(x)^2*sec(x)^5
Leaf count is larger than twice the leaf count of optimal. \(132\) vs. \(2(42)=84\).
Time = 0.14 (sec) , antiderivative size = 132, normalized size of antiderivative = 3.14 \[ \int \frac {1}{(\sin (x)+\sin (3 x))^3} \, dx=-\frac {\csc ^2(x) \left (-412+462 \cos (2 x)+700 \cos (4 x)+210 \cos (6 x)+525 \cos (x) \log \left (\cos \left (\frac {x}{2}\right )\right )-105 \cos (3 x) \log \left (\cos \left (\frac {x}{2}\right )\right )-315 \cos (5 x) \log \left (\cos \left (\frac {x}{2}\right )\right )-105 \cos (7 x) \log \left (\cos \left (\frac {x}{2}\right )\right )-525 \cos (x) \log \left (\sin \left (\frac {x}{2}\right )\right )+105 \cos (3 x) \log \left (\sin \left (\frac {x}{2}\right )\right )+315 \cos (5 x) \log \left (\sin \left (\frac {x}{2}\right )\right )+105 \cos (7 x) \log \left (\sin \left (\frac {x}{2}\right )\right )\right ) \sec ^5(x)}{122880} \] Input:
Integrate[(Sin[x] + Sin[3*x])^(-3),x]
Output:
-1/122880*(Csc[x]^2*(-412 + 462*Cos[2*x] + 700*Cos[4*x] + 210*Cos[6*x] + 5 25*Cos[x]*Log[Cos[x/2]] - 105*Cos[3*x]*Log[Cos[x/2]] - 315*Cos[5*x]*Log[Co s[x/2]] - 105*Cos[7*x]*Log[Cos[x/2]] - 525*Cos[x]*Log[Sin[x/2]] + 105*Cos[ 3*x]*Log[Sin[x/2]] + 315*Cos[5*x]*Log[Sin[x/2]] + 105*Cos[7*x]*Log[Sin[x/2 ]])*Sec[x]^5)
Time = 0.22 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.21, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.889, Rules used = {3042, 4824, 27, 253, 264, 264, 264, 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.
\(\displaystyle \int \frac {1}{(\sin (x)+\sin (3 x))^3} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{(\sin (x)+\sin (3 x))^3}dx\) |
\(\Big \downarrow \) 4824 |
\(\displaystyle -\int \frac {\sec ^6(x)}{64 \left (1-\cos ^2(x)\right )^2}d\cos (x)\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {1}{64} \int \frac {\sec ^6(x)}{\left (1-\cos ^2(x)\right )^2}d\cos (x)\) |
\(\Big \downarrow \) 253 |
\(\displaystyle \frac {1}{64} \left (-\frac {7}{2} \int \frac {\sec ^6(x)}{1-\cos ^2(x)}d\cos (x)-\frac {\sec ^5(x)}{2 \left (1-\cos ^2(x)\right )}\right )\) |
\(\Big \downarrow \) 264 |
\(\displaystyle \frac {1}{64} \left (-\frac {7}{2} \left (\int \frac {\sec ^4(x)}{1-\cos ^2(x)}d\cos (x)-\frac {\sec ^5(x)}{5}\right )-\frac {\sec ^5(x)}{2 \left (1-\cos ^2(x)\right )}\right )\) |
\(\Big \downarrow \) 264 |
\(\displaystyle \frac {1}{64} \left (-\frac {7}{2} \left (\int \frac {\sec ^2(x)}{1-\cos ^2(x)}d\cos (x)-\frac {1}{5} \sec ^5(x)-\frac {\sec ^3(x)}{3}\right )-\frac {\sec ^5(x)}{2 \left (1-\cos ^2(x)\right )}\right )\) |
\(\Big \downarrow \) 264 |
\(\displaystyle \frac {1}{64} \left (-\frac {7}{2} \left (\int \frac {1}{1-\cos ^2(x)}d\cos (x)-\frac {1}{5} \sec ^5(x)-\frac {\sec ^3(x)}{3}-\sec (x)\right )-\frac {\sec ^5(x)}{2 \left (1-\cos ^2(x)\right )}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{64} \left (-\frac {7}{2} \left (\text {arctanh}(\cos (x))-\frac {1}{5} \sec ^5(x)-\frac {\sec ^3(x)}{3}-\sec (x)\right )-\frac {\sec ^5(x)}{2 \left (1-\cos ^2(x)\right )}\right )\) |
Input:
Int[(Sin[x] + Sin[3*x])^(-3),x]
Output:
(-1/2*Sec[x]^5/(1 - Cos[x]^2) - (7*(ArcTanh[Cos[x]] - Sec[x] - Sec[x]^3/3 - Sec[x]^5/5))/2)/64
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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])
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(c*x )^(m + 1))*((a + b*x^2)^(p + 1)/(2*a*c*(p + 1))), x] + Simp[(m + 2*p + 3)/( 2*a*(p + 1)) Int[(c*x)^m*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, m }, x] && LtQ[p, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c ^2*(m + 1))) Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p }, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
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 = 5.32 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.14
method | result | size |
default | \(\frac {1}{320 \sin \left (x \right )^{2} \cos \left (x \right )^{5}}+\frac {7}{960 \sin \left (x \right )^{2} \cos \left (x \right )^{3}}-\frac {7}{384 \sin \left (x \right )^{2} \cos \left (x \right )}+\frac {7}{128 \cos \left (x \right )}+\frac {7 \ln \left (-\cot \left (x \right )+\csc \left (x \right )\right )}{128}\) | \(48\) |
risch | \(\frac {105 \,{\mathrm e}^{13 i x}+350 \,{\mathrm e}^{11 i x}+231 \,{\mathrm e}^{9 i x}-412 \,{\mathrm e}^{7 i x}+231 \,{\mathrm e}^{5 i x}+350 \,{\mathrm e}^{3 i x}+105 \,{\mathrm e}^{i x}}{960 \left ({\mathrm e}^{2 i x}+1\right )^{5} \left ({\mathrm e}^{2 i x}-1\right )^{2}}-\frac {7 \ln \left ({\mathrm e}^{i x}+1\right )}{128}+\frac {7 \ln \left ({\mathrm e}^{i x}-1\right )}{128}\) | \(92\) |
Input:
int(1/(sin(x)+sin(3*x))^3,x,method=_RETURNVERBOSE)
Output:
1/320/sin(x)^2/cos(x)^5+7/960/sin(x)^2/cos(x)^3-7/384/sin(x)^2/cos(x)+7/12 8/cos(x)+7/128*ln(-cot(x)+csc(x))
Leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (32) = 64\).
Time = 0.09 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.79 \[ \int \frac {1}{(\sin (x)+\sin (3 x))^3} \, dx=\frac {210 \, \cos \left (x\right )^{6} - 140 \, \cos \left (x\right )^{4} - 28 \, \cos \left (x\right )^{2} - 105 \, {\left (\cos \left (x\right )^{7} - \cos \left (x\right )^{5}\right )} \log \left (\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) + 105 \, {\left (\cos \left (x\right )^{7} - \cos \left (x\right )^{5}\right )} \log \left (-\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) - 12}{3840 \, {\left (\cos \left (x\right )^{7} - \cos \left (x\right )^{5}\right )}} \] Input:
integrate(1/(sin(x)+sin(3*x))^3,x, algorithm="fricas")
Output:
1/3840*(210*cos(x)^6 - 140*cos(x)^4 - 28*cos(x)^2 - 105*(cos(x)^7 - cos(x) ^5)*log(1/2*cos(x) + 1/2) + 105*(cos(x)^7 - cos(x)^5)*log(-1/2*cos(x) + 1/ 2) - 12)/(cos(x)^7 - cos(x)^5)
Timed out. \[ \int \frac {1}{(\sin (x)+\sin (3 x))^3} \, dx=\text {Timed out} \] Input:
integrate(1/(sin(x)+sin(3*x))**3,x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 2043 vs. \(2 (32) = 64\).
Time = 0.12 (sec) , antiderivative size = 2043, normalized size of antiderivative = 48.64 \[ \int \frac {1}{(\sin (x)+\sin (3 x))^3} \, dx=\text {Too large to display} \] Input:
integrate(1/(sin(x)+sin(3*x))^3,x, algorithm="maxima")
Output:
1/3840*(4*(105*cos(13*x) + 350*cos(11*x) + 231*cos(9*x) - 412*cos(7*x) + 2 31*cos(5*x) + 350*cos(3*x) + 105*cos(x))*cos(14*x) + 420*(3*cos(12*x) + co s(10*x) - 5*cos(8*x) - 5*cos(6*x) + cos(4*x) + 3*cos(2*x) + 1)*cos(13*x) + 12*(350*cos(11*x) + 231*cos(9*x) - 412*cos(7*x) + 231*cos(5*x) + 350*cos( 3*x) + 105*cos(x))*cos(12*x) + 1400*(cos(10*x) - 5*cos(8*x) - 5*cos(6*x) + cos(4*x) + 3*cos(2*x) + 1)*cos(11*x) + 4*(231*cos(9*x) - 412*cos(7*x) + 2 31*cos(5*x) + 350*cos(3*x) + 105*cos(x))*cos(10*x) - 924*(5*cos(8*x) + 5*c os(6*x) - cos(4*x) - 3*cos(2*x) - 1)*cos(9*x) + 20*(412*cos(7*x) - 231*cos (5*x) - 350*cos(3*x) - 105*cos(x))*cos(8*x) + 1648*(5*cos(6*x) - cos(4*x) - 3*cos(2*x) - 1)*cos(7*x) - 140*(33*cos(5*x) + 50*cos(3*x) + 15*cos(x))*c os(6*x) + 924*(cos(4*x) + 3*cos(2*x) + 1)*cos(5*x) + 140*(10*cos(3*x) + 3* cos(x))*cos(4*x) + 1400*(3*cos(2*x) + 1)*cos(3*x) + 1260*cos(2*x)*cos(x) - 105*(2*(3*cos(12*x) + cos(10*x) - 5*cos(8*x) - 5*cos(6*x) + cos(4*x) + 3* cos(2*x) + 1)*cos(14*x) + cos(14*x)^2 + 6*(cos(10*x) - 5*cos(8*x) - 5*cos( 6*x) + cos(4*x) + 3*cos(2*x) + 1)*cos(12*x) + 9*cos(12*x)^2 - 2*(5*cos(8*x ) + 5*cos(6*x) - cos(4*x) - 3*cos(2*x) - 1)*cos(10*x) + cos(10*x)^2 + 10*( 5*cos(6*x) - cos(4*x) - 3*cos(2*x) - 1)*cos(8*x) + 25*cos(8*x)^2 - 10*(cos (4*x) + 3*cos(2*x) + 1)*cos(6*x) + 25*cos(6*x)^2 + 2*(3*cos(2*x) + 1)*cos( 4*x) + cos(4*x)^2 + 9*cos(2*x)^2 + 2*(3*sin(12*x) + sin(10*x) - 5*sin(8*x) - 5*sin(6*x) + sin(4*x) + 3*sin(2*x))*sin(14*x) + sin(14*x)^2 + 6*(sin...
Leaf count of result is larger than twice the leaf count of optimal. 127 vs. \(2 (32) = 64\).
Time = 0.12 (sec) , antiderivative size = 127, normalized size of antiderivative = 3.02 \[ \int \frac {1}{(\sin (x)+\sin (3 x))^3} \, dx=-\frac {{\left (\frac {14 \, {\left (\cos \left (x\right ) - 1\right )}}{\cos \left (x\right ) + 1} - 1\right )} {\left (\cos \left (x\right ) + 1\right )}}{512 \, {\left (\cos \left (x\right ) - 1\right )}} - \frac {\cos \left (x\right ) - 1}{512 \, {\left (\cos \left (x\right ) + 1\right )}} + \frac {\frac {100 \, {\left (\cos \left (x\right ) - 1\right )}}{\cos \left (x\right ) + 1} + \frac {170 \, {\left (\cos \left (x\right ) - 1\right )}^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {120 \, {\left (\cos \left (x\right ) - 1\right )}^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {45 \, {\left (\cos \left (x\right ) - 1\right )}^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}} + 29}{240 \, {\left (\frac {\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1} + 1\right )}^{5}} + \frac {7}{256} \, \log \left (-\frac {\cos \left (x\right ) - 1}{\cos \left (x\right ) + 1}\right ) \] Input:
integrate(1/(sin(x)+sin(3*x))^3,x, algorithm="giac")
Output:
-1/512*(14*(cos(x) - 1)/(cos(x) + 1) - 1)*(cos(x) + 1)/(cos(x) - 1) - 1/51 2*(cos(x) - 1)/(cos(x) + 1) + 1/240*(100*(cos(x) - 1)/(cos(x) + 1) + 170*( cos(x) - 1)^2/(cos(x) + 1)^2 + 120*(cos(x) - 1)^3/(cos(x) + 1)^3 + 45*(cos (x) - 1)^4/(cos(x) + 1)^4 + 29)/((cos(x) - 1)/(cos(x) + 1) + 1)^5 + 7/256* log(-(cos(x) - 1)/(cos(x) + 1))
Time = 0.13 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.95 \[ \int \frac {1}{(\sin (x)+\sin (3 x))^3} \, dx=\frac {-\frac {7\,{\cos \left (x\right )}^6}{128}+\frac {7\,{\cos \left (x\right )}^4}{192}+\frac {7\,{\cos \left (x\right )}^2}{960}+\frac {1}{320}}{{\cos \left (x\right )}^5-{\cos \left (x\right )}^7}-\frac {7\,\mathrm {atanh}\left (\cos \left (x\right )\right )}{128} \] Input:
int(1/(sin(3*x) + sin(x))^3,x)
Output:
((7*cos(x)^2)/960 + (7*cos(x)^4)/192 - (7*cos(x)^6)/128 + 1/320)/(cos(x)^5 - cos(x)^7) - (7*atanh(cos(x)))/128
\[ \int \frac {1}{(\sin (x)+\sin (3 x))^3} \, dx=\int \frac {1}{\sin \left (3 x \right )^{3}+3 \sin \left (3 x \right )^{2} \sin \left (x \right )+3 \sin \left (3 x \right ) \sin \left (x \right )^{2}+\sin \left (x \right )^{3}}d x \] Input:
int(1/(sin(x)+sin(3*x))^3,x)
Output:
int(1/(sin(3*x)**3 + 3*sin(3*x)**2*sin(x) + 3*sin(3*x)*sin(x)**2 + sin(x)* *3),x)