Integrand size = 11, antiderivative size = 147 \[ \int \frac {1}{(\cos (3 x)+\sin (2 x))^3} \, dx=-\frac {8}{125} \left (37-17 \sqrt {5}\right ) \log \left (1-\sqrt {5}-4 \sin (x)\right )-\frac {8}{125} \left (37+17 \sqrt {5}\right ) \log \left (1+\sqrt {5}-4 \sin (x)\right )+\frac {19}{4} \log (1-\sin (x))-\frac {7}{500} \log (1+\sin (x))+\frac {\sec ^2(x) (4+5 \sin (x))}{10 \left (1+2 \sin (x)-4 \sin ^2(x)\right )}-\frac {37+86 \sin (x)}{25 \left (1+2 \sin (x)-4 \sin ^2(x)\right )}+\frac {2 \sec (x) \tan (x)}{5 \left (1+2 \sin (x)-4 \sin ^2(x)\right )^2} \] Output:
-8/125*(37-17*5^(1/2))*ln(1-5^(1/2)-4*sin(x))-8/125*(37+17*5^(1/2))*ln(1+5 ^(1/2)-4*sin(x))+19/4*ln(1-sin(x))-7/500*ln(1+sin(x))+sec(x)^2*(4+5*sin(x) )/(10+20*sin(x)-40*sin(x)^2)-(37+86*sin(x))/(25+50*sin(x)-100*sin(x)^2)+2/ 5*sec(x)*tan(x)/(1+2*sin(x)-4*sin(x)^2)^2
Time = 0.42 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.20 \[ \int \frac {1}{(\cos (3 x)+\sin (2 x))^3} \, dx=\frac {23750 \log \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )-70 \log \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )-32 \sqrt {5} \left (-85+37 \sqrt {5}\right ) \log \left (1-\sqrt {5}-4 \sin (x)\right )-32 \sqrt {5} \left (85+37 \sqrt {5}\right ) \log \left (1+\sqrt {5}-4 \sin (x)\right )-\frac {625}{\left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )^2}+\frac {5}{\left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^2}+\frac {400 (1+6 \sin (x))}{(-1+2 \cos (2 x)+2 \sin (x))^2}-\frac {80 (31+76 \sin (x))}{-1+2 \cos (2 x)+2 \sin (x)}}{2500} \] Input:
Integrate[(Cos[3*x] + Sin[2*x])^(-3),x]
Output:
(23750*Log[Cos[x/2] - Sin[x/2]] - 70*Log[Cos[x/2] + Sin[x/2]] - 32*Sqrt[5] *(-85 + 37*Sqrt[5])*Log[1 - Sqrt[5] - 4*Sin[x]] - 32*Sqrt[5]*(85 + 37*Sqrt [5])*Log[1 + Sqrt[5] - 4*Sin[x]] - 625/(Cos[x/2] - Sin[x/2])^2 + 5/(Cos[x/ 2] + Sin[x/2])^2 + (400*(1 + 6*Sin[x]))/(-1 + 2*Cos[2*x] + 2*Sin[x])^2 - ( 80*(31 + 76*Sin[x]))/(-1 + 2*Cos[2*x] + 2*Sin[x]))/2500
Time = 0.46 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.44, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {3042, 4829, 1301, 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.
| \(\displaystyle \int \frac {1}{(\sin (2 x)+\cos (3 x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(\sin (2 x)+\cos (3 x))^3}dx\) |
| \(\Big \downarrow \) 4829 |
| \(\displaystyle \int \frac {1}{\left (-4 \sin ^2(x)+2 \sin (x)+1\right )^3 \left (1-\sin ^2(x)\right )^2}d\sin (x)\) |
| \(\Big \downarrow \) 1301 |
| \(\displaystyle \int \left (\frac {16 (2 \sin (x)+1)}{5 \left (-4 \sin ^2(x)+2 \sin (x)+1\right )^3}+\frac {16 (148 \sin (x)+67)}{125 \left (-4 \sin ^2(x)+2 \sin (x)+1\right )}-\frac {16 (20 \sin (x)+9)}{25 \left (-4 \sin ^2(x)+2 \sin (x)+1\right )^2}-\frac {19}{4 (1-\sin (x))}-\frac {7}{500 (\sin (x)+1)}-\frac {1}{4 (1-\sin (x))^2}-\frac {1}{500 (\sin (x)+1)^2}\right )d\sin (x)\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {36 (1-4 \sin (x))}{125 \left (-4 \sin ^2(x)+2 \sin (x)+1\right )}-\frac {8 (56 \sin (x)+11)}{125 \left (-4 \sin ^2(x)+2 \sin (x)+1\right )}+\frac {4 (6 \sin (x)+1)}{25 \left (-4 \sin ^2(x)+2 \sin (x)+1\right )^2}-\frac {1}{4 (1-\sin (x))}+\frac {1}{500 (\sin (x)+1)}-\frac {8}{625} \left (185-104 \sqrt {5}\right ) \log \left (-4 \sin (x)-\sqrt {5}+1\right )-\frac {152 \log \left (-4 \sin (x)-\sqrt {5}+1\right )}{125 \sqrt {5}}-\frac {8}{625} \left (185+104 \sqrt {5}\right ) \log \left (-4 \sin (x)+\sqrt {5}+1\right )+\frac {152 \log \left (-4 \sin (x)+\sqrt {5}+1\right )}{125 \sqrt {5}}+\frac {19}{4} \log (1-\sin (x))-\frac {7}{500} \log (\sin (x)+1)\) |
Input:
Int[(Cos[3*x] + Sin[2*x])^(-3),x]
Output:
(-152*Log[1 - Sqrt[5] - 4*Sin[x]])/(125*Sqrt[5]) - (8*(185 - 104*Sqrt[5])* Log[1 - Sqrt[5] - 4*Sin[x]])/625 + (152*Log[1 + Sqrt[5] - 4*Sin[x]])/(125* Sqrt[5]) - (8*(185 + 104*Sqrt[5])*Log[1 + Sqrt[5] - 4*Sin[x]])/625 + (19*L og[1 - Sin[x]])/4 - (7*Log[1 + Sin[x]])/500 - 1/(4*(1 - Sin[x])) + 1/(500* (1 + Sin[x])) + (4*(1 + 6*Sin[x]))/(25*(1 + 2*Sin[x] - 4*Sin[x]^2)^2) - (3 6*(1 - 4*Sin[x]))/(125*(1 + 2*Sin[x] - 4*Sin[x]^2)) - (8*(11 + 56*Sin[x])) /(125*(1 + 2*Sin[x] - 4*Sin[x]^2))
Int[((a_.) + (c_.)*(x_)^2)^(p_)*((d_.) + (e_.)*(x_) + (f_.)*(x_)^2)^(q_), x _Symbol] :> With[{r = Rt[(-a)*c, 2]}, Simp[1/c^p Int[ExpandIntegrand[(-r + c*x)^p*(r + c*x)^p*(d + e*x + f*x^2)^q, x], x], x] /; EqQ[p, -1] || !Fra ctionalPowerFactorQ[r]] /; FreeQ[{a, c, d, e, f}, x] && ILtQ[p, 0] && Integ erQ[q] && NiceSqrtQ[(-a)*c]
Int[(cos[(n_.)*((c_.) + (d_.)*(x_))]*(b_.) + (a_.)*sin[(m_.)*((c_.) + (d_.) *(x_))])^(p_), x_Symbol] :> Simp[1/d Subst[Int[Simplify[TrigExpand[a*Sin[ m*ArcSin[x]] + b*Cos[n*ArcSin[x]]]]^p/Sqrt[1 - x^2], x], x, Sin[c + d*x]], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[(p - 1)/2, 0] && IntegerQ[m/2] && Inte gerQ[(n - 1)/2]
Time = 14.73 (sec) , antiderivative size = 2, normalized size of antiderivative = 0.01
| method | result | size |
| parallelrisch | \(0\) | \(2\) |
| default | \(-\frac {256 \left (-\frac {19 \sin \left (x \right )^{3}}{4}+\frac {7 \sin \left (x \right )^{2}}{16}+\frac {27 \sin \left (x \right )}{16}+\frac {13}{32}\right )}{125 \left (4 \sin \left (x \right )^{2}-2 \sin \left (x \right )-1\right )^{2}}-\frac {296 \ln \left (4 \sin \left (x \right )^{2}-2 \sin \left (x \right )-1\right )}{125}+\frac {272 \sqrt {5}\, \operatorname {arctanh}\left (\frac {\left (-2+8 \sin \left (x \right )\right ) \sqrt {5}}{10}\right )}{125}+\frac {1}{500+500 \sin \left (x \right )}-\frac {7 \ln \left (1+\sin \left (x \right )\right )}{500}+\frac {1}{4 \sin \left (x \right )-4}+\frac {19 \ln \left (\sin \left (x \right )-1\right )}{4}\) | \(98\) |
| risch | \(\frac {i \left (-6 i {\mathrm e}^{10 i x}+43 \,{\mathrm e}^{11 i x}+22 i {\mathrm e}^{8 i x}+12 \,{\mathrm e}^{9 i x}+76 i {\mathrm e}^{6 i x}-16 \,{\mathrm e}^{7 i x}+22 i {\mathrm e}^{4 i x}+16 \,{\mathrm e}^{5 i x}-6 i {\mathrm e}^{2 i x}-12 \,{\mathrm e}^{3 i x}-43 \,{\mathrm e}^{i x}\right )}{25 \left ({\mathrm e}^{i x}+i\right )^{2} \left (-2 i {\mathrm e}^{4 i x}+{\mathrm e}^{5 i x}+2 i {\mathrm e}^{2 i x}-2 \,{\mathrm e}^{3 i x}-i+2 \,{\mathrm e}^{i x}\right )^{2}}-\frac {7 \ln \left ({\mathrm e}^{i x}+i\right )}{250}+\frac {19 \ln \left ({\mathrm e}^{i x}-i\right )}{2}-\frac {296 \ln \left ({\mathrm e}^{2 i x}+\left (-\frac {i}{2}+\frac {i \sqrt {5}}{2}\right ) {\mathrm e}^{i x}-1\right )}{125}+\frac {136 \sqrt {5}\, \ln \left ({\mathrm e}^{2 i x}+\left (-\frac {i}{2}+\frac {i \sqrt {5}}{2}\right ) {\mathrm e}^{i x}-1\right )}{125}-\frac {296 \ln \left ({\mathrm e}^{2 i x}+\left (-\frac {i}{2}-\frac {i \sqrt {5}}{2}\right ) {\mathrm e}^{i x}-1\right )}{125}-\frac {136 \sqrt {5}\, \ln \left ({\mathrm e}^{2 i x}+\left (-\frac {i}{2}-\frac {i \sqrt {5}}{2}\right ) {\mathrm e}^{i x}-1\right )}{125}\) | \(266\) |
Input:
int(1/(cos(3*x)+sin(2*x))^3,x,method=_RETURNVERBOSE)
Output:
0
Leaf count of result is larger than twice the leaf count of optimal. 300 vs. \(2 (125) = 250\).
Time = 0.10 (sec) , antiderivative size = 300, normalized size of antiderivative = 2.04 \[ \int \frac {1}{(\cos (3 x)+\sin (2 x))^3} \, dx=\frac {480 \, \cos \left (x\right )^{4} - 920 \, \cos \left (x\right )^{2} - 1184 \, {\left (16 \, \cos \left (x\right )^{6} - 28 \, \cos \left (x\right )^{4} + 13 \, \cos \left (x\right )^{2} + 4 \, {\left (4 \, \cos \left (x\right )^{4} - 3 \, \cos \left (x\right )^{2}\right )} \sin \left (x\right )\right )} \log \left (4 \, \cos \left (x\right )^{2} + 2 \, \sin \left (x\right ) - 3\right ) + 544 \, {\left (16 \, \sqrt {5} \cos \left (x\right )^{6} - 28 \, \sqrt {5} \cos \left (x\right )^{4} + 13 \, \sqrt {5} \cos \left (x\right )^{2} + 4 \, {\left (4 \, \sqrt {5} \cos \left (x\right )^{4} - 3 \, \sqrt {5} \cos \left (x\right )^{2}\right )} \sin \left (x\right )\right )} \log \left (\frac {8 \, \cos \left (x\right )^{2} - 4 \, {\left (\sqrt {5} - 1\right )} \sin \left (x\right ) + \sqrt {5} - 11}{4 \, \cos \left (x\right )^{2} + 2 \, \sin \left (x\right ) - 3}\right ) - 7 \, {\left (16 \, \cos \left (x\right )^{6} - 28 \, \cos \left (x\right )^{4} + 13 \, \cos \left (x\right )^{2} + 4 \, {\left (4 \, \cos \left (x\right )^{4} - 3 \, \cos \left (x\right )^{2}\right )} \sin \left (x\right )\right )} \log \left (\sin \left (x\right ) + 1\right ) + 2375 \, {\left (16 \, \cos \left (x\right )^{6} - 28 \, \cos \left (x\right )^{4} + 13 \, \cos \left (x\right )^{2} + 4 \, {\left (4 \, \cos \left (x\right )^{4} - 3 \, \cos \left (x\right )^{2}\right )} \sin \left (x\right )\right )} \log \left (-\sin \left (x\right ) + 1\right ) - 10 \, {\left (688 \, \cos \left (x\right )^{4} - 468 \, \cos \left (x\right )^{2} + 15\right )} \sin \left (x\right ) - 100}{500 \, {\left (16 \, \cos \left (x\right )^{6} - 28 \, \cos \left (x\right )^{4} + 13 \, \cos \left (x\right )^{2} + 4 \, {\left (4 \, \cos \left (x\right )^{4} - 3 \, \cos \left (x\right )^{2}\right )} \sin \left (x\right )\right )}} \] Input:
integrate(1/(cos(3*x)+sin(2*x))^3,x, algorithm="fricas")
Output:
1/500*(480*cos(x)^4 - 920*cos(x)^2 - 1184*(16*cos(x)^6 - 28*cos(x)^4 + 13* cos(x)^2 + 4*(4*cos(x)^4 - 3*cos(x)^2)*sin(x))*log(4*cos(x)^2 + 2*sin(x) - 3) + 544*(16*sqrt(5)*cos(x)^6 - 28*sqrt(5)*cos(x)^4 + 13*sqrt(5)*cos(x)^2 + 4*(4*sqrt(5)*cos(x)^4 - 3*sqrt(5)*cos(x)^2)*sin(x))*log((8*cos(x)^2 - 4 *(sqrt(5) - 1)*sin(x) + sqrt(5) - 11)/(4*cos(x)^2 + 2*sin(x) - 3)) - 7*(16 *cos(x)^6 - 28*cos(x)^4 + 13*cos(x)^2 + 4*(4*cos(x)^4 - 3*cos(x)^2)*sin(x) )*log(sin(x) + 1) + 2375*(16*cos(x)^6 - 28*cos(x)^4 + 13*cos(x)^2 + 4*(4*c os(x)^4 - 3*cos(x)^2)*sin(x))*log(-sin(x) + 1) - 10*(688*cos(x)^4 - 468*co s(x)^2 + 15)*sin(x) - 100)/(16*cos(x)^6 - 28*cos(x)^4 + 13*cos(x)^2 + 4*(4 *cos(x)^4 - 3*cos(x)^2)*sin(x))
\[ \int \frac {1}{(\cos (3 x)+\sin (2 x))^3} \, dx=\int \frac {1}{\left (\sin {\left (2 x \right )} + \cos {\left (3 x \right )}\right )^{3}}\, dx \] Input:
integrate(1/(cos(3*x)+sin(2*x))**3,x)
Output:
Integral((sin(2*x) + cos(3*x))**(-3), x)
Exception generated. \[ \int \frac {1}{(\cos (3 x)+\sin (2 x))^3} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(1/(cos(3*x)+sin(2*x))^3,x, algorithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is un defined.
Time = 0.12 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.87 \[ \int \frac {1}{(\cos (3 x)+\sin (2 x))^3} \, dx=-\frac {136}{125} \, \sqrt {5} \log \left (\frac {{\left | -2 \, \sqrt {5} + 8 \, \sin \left (x\right ) - 2 \right |}}{{\left | 2 \, \sqrt {5} + 8 \, \sin \left (x\right ) - 2 \right |}}\right ) - \frac {592 \, \sin \left (x\right )^{2} - 63 \, \sin \left (x\right ) - 654}{250 \, {\left (\sin \left (x\right )^{2} - 1\right )}} + \frac {4 \, {\left (1776 \, \sin \left (x\right )^{4} - 1472 \, \sin \left (x\right )^{3} - 472 \, \sin \left (x\right )^{2} + 336 \, \sin \left (x\right ) + 85\right )}}{125 \, {\left (4 \, \sin \left (x\right )^{2} - 2 \, \sin \left (x\right ) - 1\right )}^{2}} - \frac {7}{500} \, \log \left (\sin \left (x\right ) + 1\right ) + \frac {19}{4} \, \log \left (-\sin \left (x\right ) + 1\right ) - \frac {296}{125} \, \log \left ({\left | 4 \, \sin \left (x\right )^{2} - 2 \, \sin \left (x\right ) - 1 \right |}\right ) \] Input:
integrate(1/(cos(3*x)+sin(2*x))^3,x, algorithm="giac")
Output:
-136/125*sqrt(5)*log(abs(-2*sqrt(5) + 8*sin(x) - 2)/abs(2*sqrt(5) + 8*sin( x) - 2)) - 1/250*(592*sin(x)^2 - 63*sin(x) - 654)/(sin(x)^2 - 1) + 4/125*( 1776*sin(x)^4 - 1472*sin(x)^3 - 472*sin(x)^2 + 336*sin(x) + 85)/(4*sin(x)^ 2 - 2*sin(x) - 1)^2 - 7/500*log(sin(x) + 1) + 19/4*log(-sin(x) + 1) - 296/ 125*log(abs(4*sin(x)^2 - 2*sin(x) - 1))
Time = 21.09 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.83 \[ \int \frac {1}{(\cos (3 x)+\sin (2 x))^3} \, dx=\frac {19\,\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )-1\right )}{2}-\frac {7\,\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )+1\right )}{250}-\frac {\frac {19\,{\mathrm {tan}\left (\frac {x}{2}\right )}^{11}}{25}+\frac {548\,{\mathrm {tan}\left (\frac {x}{2}\right )}^{10}}{25}+\frac {783\,{\mathrm {tan}\left (\frac {x}{2}\right )}^9}{25}-\frac {6832\,{\mathrm {tan}\left (\frac {x}{2}\right )}^8}{25}-\frac {562\,{\mathrm {tan}\left (\frac {x}{2}\right )}^7}{25}+\frac {12888\,{\mathrm {tan}\left (\frac {x}{2}\right )}^6}{25}-\frac {562\,{\mathrm {tan}\left (\frac {x}{2}\right )}^5}{25}-\frac {6832\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4}{25}+\frac {783\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3}{25}+\frac {548\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{25}+\frac {19\,\mathrm {tan}\left (\frac {x}{2}\right )}{25}}{{\mathrm {tan}\left (\frac {x}{2}\right )}^{12}+8\,{\mathrm {tan}\left (\frac {x}{2}\right )}^{11}-14\,{\mathrm {tan}\left (\frac {x}{2}\right )}^{10}-120\,{\mathrm {tan}\left (\frac {x}{2}\right )}^9+255\,{\mathrm {tan}\left (\frac {x}{2}\right )}^8+112\,{\mathrm {tan}\left (\frac {x}{2}\right )}^7-484\,{\mathrm {tan}\left (\frac {x}{2}\right )}^6+112\,{\mathrm {tan}\left (\frac {x}{2}\right )}^5+255\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4-120\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3-14\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2+8\,\mathrm {tan}\left (\frac {x}{2}\right )+1}-\ln \left (2\,\mathrm {tan}\left (\frac {x}{2}\right )-2\,\sqrt {5}\,\mathrm {tan}\left (\frac {x}{2}\right )+{\mathrm {tan}\left (\frac {x}{2}\right )}^2+1\right )\,\left (\frac {136\,\sqrt {5}}{125}+\frac {296}{125}\right )+\ln \left (2\,\mathrm {tan}\left (\frac {x}{2}\right )+2\,\sqrt {5}\,\mathrm {tan}\left (\frac {x}{2}\right )+{\mathrm {tan}\left (\frac {x}{2}\right )}^2+1\right )\,\left (\frac {136\,\sqrt {5}}{125}-\frac {296}{125}\right ) \] Input:
int(1/(cos(3*x) + sin(2*x))^3,x)
Output:
(19*log(tan(x/2) - 1))/2 - (7*log(tan(x/2) + 1))/250 - ((19*tan(x/2))/25 + (548*tan(x/2)^2)/25 + (783*tan(x/2)^3)/25 - (6832*tan(x/2)^4)/25 - (562*t an(x/2)^5)/25 + (12888*tan(x/2)^6)/25 - (562*tan(x/2)^7)/25 - (6832*tan(x/ 2)^8)/25 + (783*tan(x/2)^9)/25 + (548*tan(x/2)^10)/25 + (19*tan(x/2)^11)/2 5)/(8*tan(x/2) - 14*tan(x/2)^2 - 120*tan(x/2)^3 + 255*tan(x/2)^4 + 112*tan (x/2)^5 - 484*tan(x/2)^6 + 112*tan(x/2)^7 + 255*tan(x/2)^8 - 120*tan(x/2)^ 9 - 14*tan(x/2)^10 + 8*tan(x/2)^11 + tan(x/2)^12 + 1) - log(2*tan(x/2) - 2 *5^(1/2)*tan(x/2) + tan(x/2)^2 + 1)*((136*5^(1/2))/125 + 296/125) + log(2* tan(x/2) + 2*5^(1/2)*tan(x/2) + tan(x/2)^2 + 1)*((136*5^(1/2))/125 - 296/1 25)
\[ \int \frac {1}{(\cos (3 x)+\sin (2 x))^3} \, dx=\int \frac {1}{\cos \left (3 x \right )^{3}+3 \cos \left (3 x \right )^{2} \sin \left (2 x \right )+3 \cos \left (3 x \right ) \sin \left (2 x \right )^{2}+\sin \left (2 x \right )^{3}}d x \] Input:
int(1/(cos(3*x)+sin(2*x))^3,x)
Output:
int(1/(cos(3*x)**3 + 3*cos(3*x)**2*sin(2*x) + 3*cos(3*x)*sin(2*x)**2 + sin (2*x)**3),x)