Integrand size = 11, antiderivative size = 225 \[ \int \frac {1}{(\cos (3 x)+\sin (2 x))^5} \, dx=-\frac {16}{625} \left (1383-619 \sqrt {5}\right ) \log \left (1-\sqrt {5}-4 \sin (x)\right )-\frac {16}{625} \left (1383+619 \sqrt {5}\right ) \log \left (1+\sqrt {5}-4 \sin (x)\right )+\frac {1133}{16} \log (1-\sin (x))-\frac {29 \log (1+\sin (x))}{10000}+\frac {\sec ^4(x) (4+5 \sin (x))}{20 \left (1+2 \sin (x)-4 \sin ^2(x)\right )^3}+\frac {\sec ^2(x) (118+127 \sin (x))}{40 \left (1+2 \sin (x)-4 \sin ^2(x)\right )^3}-\frac {2269+5014 \sin (x)}{300 \left (1+2 \sin (x)-4 \sin ^2(x)\right )^3}+\frac {3367+7754 \sin (x)}{300 \left (1+2 \sin (x)-4 \sin ^2(x)\right )^2}-\frac {10951+26994 \sin (x)}{500 \left (1+2 \sin (x)-4 \sin ^2(x)\right )}+\frac {\sec ^3(x) \tan (x)}{5 \left (1+2 \sin (x)-4 \sin ^2(x)\right )^4} \] Output:
-16/625*(1383-619*5^(1/2))*ln(1-5^(1/2)-4*sin(x))-16/625*(1383+619*5^(1/2) )*ln(1+5^(1/2)-4*sin(x))+1133/16*ln(1-sin(x))-29/10000*ln(1+sin(x))+1/20*s ec(x)^4*(4+5*sin(x))/(1+2*sin(x)-4*sin(x)^2)^3+1/40*sec(x)^2*(118+127*sin( x))/(1+2*sin(x)-4*sin(x)^2)^3-1/300*(2269+5014*sin(x))/(1+2*sin(x)-4*sin(x )^2)^3+1/300*(3367+7754*sin(x))/(1+2*sin(x)-4*sin(x)^2)^2-(10951+26994*sin (x))/(500+1000*sin(x)-2000*sin(x)^2)+1/5*sec(x)^3*tan(x)/(1+2*sin(x)-4*sin (x)^2)^4
Time = 6.07 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.23 \[ \int \frac {1}{(\cos (3 x)+\sin (2 x))^5} \, dx=\frac {1133}{8} \log \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )-\frac {29 \log \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )}{5000}-\frac {16 \left (-3095+1383 \sqrt {5}\right ) \log \left (1-\sqrt {5}-4 \sin (x)\right )}{625 \sqrt {5}}-\frac {16 \left (3095+1383 \sqrt {5}\right ) \log \left (1+\sqrt {5}-4 \sin (x)\right )}{625 \sqrt {5}}-\frac {1}{16 \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )^4}-\frac {63}{16 \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )^2}+\frac {1}{50000 \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^4}+\frac {23}{50000 \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^2}+\frac {8 (1+4 \sin (x))}{25 (-1+2 \cos (2 x)+2 \sin (x))^4}-\frac {16 (35+92 \sin (x))}{375 (-1+2 \cos (2 x)+2 \sin (x))^3}+\frac {112 (73+194 \sin (x))}{1875 (-1+2 \cos (2 x)+2 \sin (x))^2}-\frac {16 (2789+7468 \sin (x))}{3125 (-1+2 \cos (2 x)+2 \sin (x))} \] Input:
Integrate[(Cos[3*x] + Sin[2*x])^(-5),x]
Output:
(1133*Log[Cos[x/2] - Sin[x/2]])/8 - (29*Log[Cos[x/2] + Sin[x/2]])/5000 - ( 16*(-3095 + 1383*Sqrt[5])*Log[1 - Sqrt[5] - 4*Sin[x]])/(625*Sqrt[5]) - (16 *(3095 + 1383*Sqrt[5])*Log[1 + Sqrt[5] - 4*Sin[x]])/(625*Sqrt[5]) - 1/(16* (Cos[x/2] - Sin[x/2])^4) - 63/(16*(Cos[x/2] - Sin[x/2])^2) + 1/(50000*(Cos [x/2] + Sin[x/2])^4) + 23/(50000*(Cos[x/2] + Sin[x/2])^2) + (8*(1 + 4*Sin[ x]))/(25*(-1 + 2*Cos[2*x] + 2*Sin[x])^4) - (16*(35 + 92*Sin[x]))/(375*(-1 + 2*Cos[2*x] + 2*Sin[x])^3) + (112*(73 + 194*Sin[x]))/(1875*(-1 + 2*Cos[2* x] + 2*Sin[x])^2) - (16*(2789 + 7468*Sin[x]))/(3125*(-1 + 2*Cos[2*x] + 2*S in[x]))
Time = 0.63 (sec) , antiderivative size = 329, normalized size of antiderivative = 1.46, 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.
| \(\displaystyle \int \frac {1}{(\sin (2 x)+\cos (3 x))^5} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(\sin (2 x)+\cos (3 x))^5}dx\) |
| \(\Big \downarrow \) 4829 |
| \(\displaystyle \int \frac {1}{\left (-4 \sin ^2(x)+2 \sin (x)+1\right )^5 \left (1-\sin ^2(x)\right )^3}d\sin (x)\) |
| \(\Big \downarrow \) 1301 |
| \(\displaystyle -\int \left (-\frac {64 (8 \sin (x)+3)}{25 \left (-4 \sin ^2(x)+2 \sin (x)+1\right )^5}+\frac {1133}{16 (1-\sin (x))}+\frac {29}{10000 (\sin (x)+1)}-\frac {192 (4610 \sin (x)+2049)}{3125 \left (-4 \sin ^2(x)+2 \sin (x)+1\right )}+\frac {63}{16 (1-\sin (x))^2}+\frac {23}{50000 (\sin (x)+1)^2}+\frac {64 (9312 \sin (x)+4037)}{3125 \left (-4 \sin ^2(x)+2 \sin (x)+1\right )^2}+\frac {1}{8 (1-\sin (x))^3}+\frac {1}{25000 (\sin (x)+1)^3}-\frac {384 (189 \sin (x)+79)}{625 \left (-4 \sin ^2(x)+2 \sin (x)+1\right )^3}+\frac {128 (58 \sin (x)+23)}{125 \left (-4 \sin ^2(x)+2 \sin (x)+1\right )^4}\right )d\sin (x)\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {10864 (1-4 \sin (x))}{3125 \left (-4 \sin ^2(x)+2 \sin (x)+1\right )}-\frac {32 (5092 \sin (x)+1055)}{3125 \left (-4 \sin ^2(x)+2 \sin (x)+1\right )}+\frac {368 (1-4 \sin (x))}{375 \left (-4 \sin ^2(x)+2 \sin (x)+1\right )^2}+\frac {96 (101 \sin (x)+22)}{625 \left (-4 \sin ^2(x)+2 \sin (x)+1\right )^2}-\frac {112 (1-4 \sin (x))}{375 \left (-4 \sin ^2(x)+2 \sin (x)+1\right )^3}-\frac {64 (30 \sin (x)+7)}{375 \left (-4 \sin ^2(x)+2 \sin (x)+1\right )^3}+\frac {8 (4 \sin (x)+1)}{25 \left (-4 \sin ^2(x)+2 \sin (x)+1\right )^4}-\frac {63}{16 (1-\sin (x))}+\frac {23}{50000 (\sin (x)+1)}-\frac {1}{16 (1-\sin (x))^2}+\frac {1}{50000 (\sin (x)+1)^2}-\frac {48 \left (11525-6403 \sqrt {5}\right ) \log \left (-4 \sin (x)-\sqrt {5}+1\right )}{15625}-\frac {59744 \log \left (-4 \sin (x)-\sqrt {5}+1\right )}{3125 \sqrt {5}}-\frac {48 \left (11525+6403 \sqrt {5}\right ) \log \left (-4 \sin (x)+\sqrt {5}+1\right )}{15625}+\frac {59744 \log \left (-4 \sin (x)+\sqrt {5}+1\right )}{3125 \sqrt {5}}+\frac {1133}{16} \log (1-\sin (x))-\frac {29 \log (\sin (x)+1)}{10000}\) |
Input:
Int[(Cos[3*x] + Sin[2*x])^(-5),x]
Output:
(-59744*Log[1 - Sqrt[5] - 4*Sin[x]])/(3125*Sqrt[5]) - (48*(11525 - 6403*Sq rt[5])*Log[1 - Sqrt[5] - 4*Sin[x]])/15625 + (59744*Log[1 + Sqrt[5] - 4*Sin [x]])/(3125*Sqrt[5]) - (48*(11525 + 6403*Sqrt[5])*Log[1 + Sqrt[5] - 4*Sin[ x]])/15625 + (1133*Log[1 - Sin[x]])/16 - (29*Log[1 + Sin[x]])/10000 - 1/(1 6*(1 - Sin[x])^2) - 63/(16*(1 - Sin[x])) + 1/(50000*(1 + Sin[x])^2) + 23/( 50000*(1 + Sin[x])) + (8*(1 + 4*Sin[x]))/(25*(1 + 2*Sin[x] - 4*Sin[x]^2)^4 ) - (112*(1 - 4*Sin[x]))/(375*(1 + 2*Sin[x] - 4*Sin[x]^2)^3) - (64*(7 + 30 *Sin[x]))/(375*(1 + 2*Sin[x] - 4*Sin[x]^2)^3) + (368*(1 - 4*Sin[x]))/(375* (1 + 2*Sin[x] - 4*Sin[x]^2)^2) + (96*(22 + 101*Sin[x]))/(625*(1 + 2*Sin[x] - 4*Sin[x]^2)^2) - (10864*(1 - 4*Sin[x]))/(3125*(1 + 2*Sin[x] - 4*Sin[x]^ 2)) - (32*(1055 + 5092*Sin[x]))/(3125*(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 = 0.60 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.61
\[\frac {1}{50000 \left (1+\sin \left (x \right )\right )^{2}}+\frac {23}{50000 \left (1+\sin \left (x \right )\right )}-\frac {29 \ln \left (1+\sin \left (x \right )\right )}{10000}-\frac {16384 \left (-\frac {1867 \sin \left (x \right )^{7}}{4}+\frac {8413 \sin \left (x \right )^{6}}{16}+\frac {10853 \sin \left (x \right )^{5}}{48}-\frac {51775 \sin \left (x \right )^{4}}{192}-\frac {2155 \sin \left (x \right )^{3}}{24}+\frac {4941 \sin \left (x \right )^{2}}{128}+\frac {1227 \sin \left (x \right )}{64}+\frac {4333}{2048}\right )}{3125 \left (4 \sin \left (x \right )^{2}-2 \sin \left (x \right )-1\right )^{4}}-\frac {22128 \ln \left (4 \sin \left (x \right )^{2}-2 \sin \left (x \right )-1\right )}{625}+\frac {19808 \sqrt {5}\, \operatorname {arctanh}\left (\frac {\left (-2+8 \sin \left (x \right )\right ) \sqrt {5}}{10}\right )}{625}-\frac {1}{16 \left (\sin \left (x \right )-1\right )^{2}}+\frac {63}{16 \left (\sin \left (x \right )-1\right )}+\frac {1133 \ln \left (\sin \left (x \right )-1\right )}{16}\]
Input:
int(1/(cos(3*x)+sin(2*x))^5,x)
Output:
1/50000/(1+sin(x))^2+23/50000/(1+sin(x))-29/10000*ln(1+sin(x))-16384/3125* (-1867/4*sin(x)^7+8413/16*sin(x)^6+10853/48*sin(x)^5-51775/192*sin(x)^4-21 55/24*sin(x)^3+4941/128*sin(x)^2+1227/64*sin(x)+4333/2048)/(4*sin(x)^2-2*s in(x)-1)^4-22128/625*ln(4*sin(x)^2-2*sin(x)-1)+19808/625*5^(1/2)*arctanh(1 /10*(-2+8*sin(x))*5^(1/2))-1/16/(sin(x)-1)^2+63/16/(sin(x)-1)+1133/16*ln(s in(x)-1)
Leaf count of result is larger than twice the leaf count of optimal. 468 vs. \(2 (197) = 394\).
Time = 0.14 (sec) , antiderivative size = 468, normalized size of antiderivative = 2.08 \[ \int \frac {1}{(\cos (3 x)+\sin (2 x))^5} \, dx =\text {Too large to display} \] Input:
integrate(1/(cos(3*x)+sin(2*x))^5,x, algorithm="fricas")
Output:
1/30000*(113433600*cos(x)^10 - 282534400*cos(x)^8 + 232826240*cos(x)^6 - 6 4109200*cos(x)^4 - 66000*cos(x)^2 - 1062144*(256*cos(x)^12 - 1152*cos(x)^1 0 + 1840*cos(x)^8 - 1256*cos(x)^6 + 313*cos(x)^4 + 8*(64*cos(x)^10 - 160*c os(x)^8 + 136*cos(x)^6 - 39*cos(x)^4)*sin(x))*log(4*cos(x)^2 + 2*sin(x) - 3) + 475392*(256*sqrt(5)*cos(x)^12 - 1152*sqrt(5)*cos(x)^10 + 1840*sqrt(5) *cos(x)^8 - 1256*sqrt(5)*cos(x)^6 + 313*sqrt(5)*cos(x)^4 + 8*(64*sqrt(5)*c os(x)^10 - 160*sqrt(5)*cos(x)^8 + 136*sqrt(5)*cos(x)^6 - 39*sqrt(5)*cos(x) ^4)*sin(x))*log((8*cos(x)^2 - 4*(sqrt(5) - 1)*sin(x) + sqrt(5) - 11)/(4*co s(x)^2 + 2*sin(x) - 3)) - 87*(256*cos(x)^12 - 1152*cos(x)^10 + 1840*cos(x) ^8 - 1256*cos(x)^6 + 313*cos(x)^4 + 8*(64*cos(x)^10 - 160*cos(x)^8 + 136*c os(x)^6 - 39*cos(x)^4)*sin(x))*log(sin(x) + 1) + 2124375*(256*cos(x)^12 - 1152*cos(x)^10 + 1840*cos(x)^8 - 1256*cos(x)^6 + 313*cos(x)^4 + 8*(64*cos( x)^10 - 160*cos(x)^8 + 136*cos(x)^6 - 39*cos(x)^4)*sin(x))*log(-sin(x) + 1 ) - 10*(10365696*cos(x)^10 - 26029952*cos(x)^8 + 22942736*cos(x)^6 - 68770 80*cos(x)^4 + 7875*cos(x)^2 + 450)*sin(x) - 3000)/(256*cos(x)^12 - 1152*co s(x)^10 + 1840*cos(x)^8 - 1256*cos(x)^6 + 313*cos(x)^4 + 8*(64*cos(x)^10 - 160*cos(x)^8 + 136*cos(x)^6 - 39*cos(x)^4)*sin(x))
\[ \int \frac {1}{(\cos (3 x)+\sin (2 x))^5} \, dx=\int \frac {1}{\left (\sin {\left (2 x \right )} + \cos {\left (3 x \right )}\right )^{5}}\, dx \] Input:
integrate(1/(cos(3*x)+sin(2*x))**5,x)
Output:
Integral((sin(2*x) + cos(3*x))**(-5), x)
Exception generated. \[ \int \frac {1}{(\cos (3 x)+\sin (2 x))^5} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(1/(cos(3*x)+sin(2*x))^5,x, algorithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is un defined.
Time = 0.13 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.73 \[ \int \frac {1}{(\cos (3 x)+\sin (2 x))^5} \, dx=-\frac {9904}{625} \, \sqrt {5} \log \left (\frac {{\left | -2 \, \sqrt {5} + 8 \, \sin \left (x\right ) - 2 \right |}}{{\left | 2 \, \sqrt {5} + 8 \, \sin \left (x\right ) - 2 \right |}}\right ) - \frac {1327680 \, \sin \left (x\right )^{4} - 98449 \, \sin \left (x\right )^{3} - 2752224 \, \sin \left (x\right )^{2} + 101575 \, \sin \left (x\right ) + 1427668}{25000 \, {\left (\sin \left (x\right )^{2} - 1\right )}^{2}} + \frac {4 \, {\left (44256000 \, \sin \left (x\right )^{8} - 82776576 \, \sin \left (x\right )^{7} + 15666816 \, \sin \left (x\right )^{6} + 41477632 \, \sin \left (x\right )^{5} - 10516400 \, \sin \left (x\right )^{4} - 9960640 \, \sin \left (x\right )^{3} + 908664 \, \sin \left (x\right )^{2} + 1147416 \, \sin \left (x\right ) + 146877\right )}}{9375 \, {\left (4 \, \sin \left (x\right )^{2} - 2 \, \sin \left (x\right ) - 1\right )}^{4}} - \frac {29}{10000} \, \log \left (\sin \left (x\right ) + 1\right ) + \frac {1133}{16} \, \log \left (-\sin \left (x\right ) + 1\right ) - \frac {22128}{625} \, \log \left ({\left | 4 \, \sin \left (x\right )^{2} - 2 \, \sin \left (x\right ) - 1 \right |}\right ) \] Input:
integrate(1/(cos(3*x)+sin(2*x))^5,x, algorithm="giac")
Output:
-9904/625*sqrt(5)*log(abs(-2*sqrt(5) + 8*sin(x) - 2)/abs(2*sqrt(5) + 8*sin (x) - 2)) - 1/25000*(1327680*sin(x)^4 - 98449*sin(x)^3 - 2752224*sin(x)^2 + 101575*sin(x) + 1427668)/(sin(x)^2 - 1)^2 + 4/9375*(44256000*sin(x)^8 - 82776576*sin(x)^7 + 15666816*sin(x)^6 + 41477632*sin(x)^5 - 10516400*sin(x )^4 - 9960640*sin(x)^3 + 908664*sin(x)^2 + 1147416*sin(x) + 146877)/(4*sin (x)^2 - 2*sin(x) - 1)^4 - 29/10000*log(sin(x) + 1) + 1133/16*log(-sin(x) + 1) - 22128/625*log(abs(4*sin(x)^2 - 2*sin(x) - 1))
Time = 20.81 (sec) , antiderivative size = 461, normalized size of antiderivative = 2.05 \[ \int \frac {1}{(\cos (3 x)+\sin (2 x))^5} \, dx=\text {Too large to display} \] Input:
int(1/(cos(3*x) + sin(2*x))^5,x)
Output:
(1133*log(tan(x/2) - 1))/8 - (29*log(tan(x/2) + 1))/5000 - log(2*tan(x/2) - 2*5^(1/2)*tan(x/2) + tan(x/2)^2 + 1)*((9904*5^(1/2))/625 + 22128/625) + log(2*tan(x/2) + 2*5^(1/2)*tan(x/2) + tan(x/2)^2 + 1)*((9904*5^(1/2))/625 - 22128/625) - ((15839*tan(x/2))/500 + (75136*tan(x/2)^2)/125 + (3848627*t an(x/2)^3)/1500 - (3843304*tan(x/2)^4)/375 - (33523387*tan(x/2)^5)/500 + ( 14590144*tan(x/2)^6)/125 + (303233451*tan(x/2)^7)/500 - (444161888*tan(x/2 )^8)/375 - (1081143559*tan(x/2)^9)/750 + (479882176*tan(x/2)^10)/125 + (22 4953597*tan(x/2)^11)/250 - (690373584*tan(x/2)^12)/125 + (224953597*tan(x/ 2)^13)/250 + (479882176*tan(x/2)^14)/125 - (1081143559*tan(x/2)^15)/750 - (444161888*tan(x/2)^16)/375 + (303233451*tan(x/2)^17)/500 + (14590144*tan( x/2)^18)/125 - (33523387*tan(x/2)^19)/500 - (3843304*tan(x/2)^20)/375 + (3 848627*tan(x/2)^21)/1500 + (75136*tan(x/2)^22)/125 + (15839*tan(x/2)^23)/5 00)/(16*tan(x/2) + 36*tan(x/2)^2 - 464*tan(x/2)^3 - 1214*tan(x/2)^4 + 7664 *tan(x/2)^5 + 8084*tan(x/2)^6 - 71856*tan(x/2)^7 + 53999*tan(x/2)^8 + 1739 84*tan(x/2)^9 - 270264*tan(x/2)^10 - 109344*tan(x/2)^11 + 418716*tan(x/2)^ 12 - 109344*tan(x/2)^13 - 270264*tan(x/2)^14 + 173984*tan(x/2)^15 + 53999* tan(x/2)^16 - 71856*tan(x/2)^17 + 8084*tan(x/2)^18 + 7664*tan(x/2)^19 - 12 14*tan(x/2)^20 - 464*tan(x/2)^21 + 36*tan(x/2)^22 + 16*tan(x/2)^23 + tan(x /2)^24 + 1)
\[ \int \frac {1}{(\cos (3 x)+\sin (2 x))^5} \, dx=\int \frac {1}{\cos \left (3 x \right )^{5}+5 \cos \left (3 x \right )^{4} \sin \left (2 x \right )+10 \cos \left (3 x \right )^{3} \sin \left (2 x \right )^{2}+10 \cos \left (3 x \right )^{2} \sin \left (2 x \right )^{3}+5 \cos \left (3 x \right ) \sin \left (2 x \right )^{4}+\sin \left (2 x \right )^{5}}d x \] Input:
int(1/(cos(3*x)+sin(2*x))^5,x)
Output:
int(1/(cos(3*x)**5 + 5*cos(3*x)**4*sin(2*x) + 10*cos(3*x)**3*sin(2*x)**2 + 10*cos(3*x)**2*sin(2*x)**3 + 5*cos(3*x)*sin(2*x)**4 + sin(2*x)**5),x)