\(\int \frac {1}{(\cos (3 x)+\sin (2 x))^4} \, dx\) [5]

Optimal result

Integrand size = 11, antiderivative size = 1 \[ \int \frac {1}{(\cos (3 x)+\sin (2 x))^4} \, dx=0 \] Output:

0
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 3 vs. order 1 in optimal.

Time = 4.37 (sec) , antiderivative size = 266, normalized size of antiderivative = 266.00 \[ \int \frac {1}{(\cos (3 x)+\sin (2 x))^4} \, dx=\frac {192 \sqrt {\frac {2}{5+\sqrt {5}}} \left (-825+367 \sqrt {5}\right ) \text {arctanh}\left (\frac {4+\left (-1+\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )}{\sqrt {2 \left (5+\sqrt {5}\right )}}\right )-96 \sqrt {2 \left (5+\sqrt {5}\right )} \left (367+165 \sqrt {5}\right ) \text {arctanh}\left (\frac {4-\left (1+\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )}{\sqrt {10-2 \sqrt {5}}}\right )+\frac {2 (1360+17802 \cos (x)+34305 \cos (2 x)-53406 \cos (3 x)+32985 \cos (4 x)-11535 \cos (6 x)+17802 \cos (7 x)-12075 \cos (8 x)-5934 \cos (9 x)+34395 \sin (x)-53406 \sin (2 x)+35315 \sin (3 x)+17802 \sin (4 x)+1110 \sin (5 x)+5934 \sin (6 x)+11565 \sin (7 x)-17802 \sin (8 x)+10625 \sin (9 x))}{\left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )^3 \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^3 (-1+2 \cos (2 x)+2 \sin (x))^3}}{7500} \] Input:

Integrate[(Cos[3*x] + Sin[2*x])^(-4),x]
 

Output:

(192*Sqrt[2/(5 + Sqrt[5])]*(-825 + 367*Sqrt[5])*ArcTanh[(4 + (-1 + Sqrt[5] 
)*Tan[x/2])/Sqrt[2*(5 + Sqrt[5])]] - 96*Sqrt[2*(5 + Sqrt[5])]*(367 + 165*S 
qrt[5])*ArcTanh[(4 - (1 + Sqrt[5])*Tan[x/2])/Sqrt[10 - 2*Sqrt[5]]] + (2*(1 
360 + 17802*Cos[x] + 34305*Cos[2*x] - 53406*Cos[3*x] + 32985*Cos[4*x] - 11 
535*Cos[6*x] + 17802*Cos[7*x] - 12075*Cos[8*x] - 5934*Cos[9*x] + 34395*Sin 
[x] - 53406*Sin[2*x] + 35315*Sin[3*x] + 17802*Sin[4*x] + 1110*Sin[5*x] + 5 
934*Sin[6*x] + 11565*Sin[7*x] - 17802*Sin[8*x] + 10625*Sin[9*x]))/((Cos[x/ 
2] - Sin[x/2])^3*(Cos[x/2] + Sin[x/2])^3*(-1 + 2*Cos[2*x] + 2*Sin[x])^3))/ 
7500
 

Rubi [C] (verified)

Result contains higher order function than in optimal. Order 3 vs. order 1 in optimal.

Time = 4.58 (sec) , antiderivative size = 2393, normalized size of antiderivative = 2393.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {3042, 4830, 2462, 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.

\(\displaystyle \int \frac {1}{(\sin (2 x)+\cos (3 x))^4} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{(\sin (2 x)+\cos (3 x))^4}dx\)

\(\Big \downarrow \) 4830

\(\displaystyle 2 \int \frac {\left (\tan ^2\left (\frac {x}{2}\right )+1\right )^{11}}{\left (-\tan ^6\left (\frac {x}{2}\right )-4 \tan ^5\left (\frac {x}{2}\right )+15 \tan ^4\left (\frac {x}{2}\right )-15 \tan ^2\left (\frac {x}{2}\right )+4 \tan \left (\frac {x}{2}\right )+1\right )^4}d\tan \left (\frac {x}{2}\right )\)

\(\Big \downarrow \) 2462

\(\displaystyle 2 \int \left (-\frac {16 \left (215 \tan ^2\left (\frac {x}{2}\right )+2012 \tan \left (\frac {x}{2}\right )-6585\right )}{625 \left (\tan ^4\left (\frac {x}{2}\right )+4 \tan ^3\left (\frac {x}{2}\right )-14 \tan ^2\left (\frac {x}{2}\right )+4 \tan \left (\frac {x}{2}\right )+1\right )}+\frac {13}{2 \left (\tan \left (\frac {x}{2}\right )-1\right )^2}+\frac {1}{250 \left (\tan \left (\frac {x}{2}\right )+1\right )^2}-\frac {256 \left (712 \tan ^3\left (\frac {x}{2}\right )-4075 \tan ^2\left (\frac {x}{2}\right )+22100 \tan \left (\frac {x}{2}\right )-160955\right )}{125 \left (\tan ^4\left (\frac {x}{2}\right )+4 \tan ^3\left (\frac {x}{2}\right )-14 \tan ^2\left (\frac {x}{2}\right )+4 \tan \left (\frac {x}{2}\right )+1\right )^2}+\frac {1}{2 \left (\tan \left (\frac {x}{2}\right )-1\right )^3}-\frac {1}{1250 \left (\tan \left (\frac {x}{2}\right )+1\right )^3}-\frac {4096 \left (14560 \tan ^3\left (\frac {x}{2}\right )-45475 \tan ^2\left (\frac {x}{2}\right )+110180 \tan \left (\frac {x}{2}\right )-644787\right )}{25 \left (\tan ^4\left (\frac {x}{2}\right )+4 \tan ^3\left (\frac {x}{2}\right )-14 \tan ^2\left (\frac {x}{2}\right )+4 \tan \left (\frac {x}{2}\right )+1\right )^3}+\frac {1}{2 \left (\tan \left (\frac {x}{2}\right )-1\right )^4}+\frac {1}{1250 \left (\tan \left (\frac {x}{2}\right )+1\right )^4}-\frac {65536 \left (51040 \tan ^3\left (\frac {x}{2}\right )-118107 \tan ^2\left (\frac {x}{2}\right )+31060 \tan \left (\frac {x}{2}\right )+8085\right )}{5 \left (\tan ^4\left (\frac {x}{2}\right )+4 \tan ^3\left (\frac {x}{2}\right )-14 \tan ^2\left (\frac {x}{2}\right )+4 \tan \left (\frac {x}{2}\right )+1\right )^4}\right )d\tan \left (\frac {x}{2}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle 2 \left (\frac {32}{625} \sqrt {5496395760940885+755342688027722 \sqrt {5}} \text {arctanh}\left (\frac {\tan \left (\frac {x}{2}\right )-\sqrt {5}+1}{\sqrt {5-2 \sqrt {5}}}\right )-\frac {16 \sqrt {15903211069805+2287334301082 \sqrt {5}} \text {arctanh}\left (\frac {\tan \left (\frac {x}{2}\right )-\sqrt {5}+1}{\sqrt {5-2 \sqrt {5}}}\right )}{3125}+\frac {64 \sqrt {127439782405+54770507458 \sqrt {5}} \text {arctanh}\left (\frac {\tan \left (\frac {x}{2}\right )-\sqrt {5}+1}{\sqrt {5-2 \sqrt {5}}}\right )}{3125}-\frac {5328 \sqrt {2 \left (81545744785+36416503171 \sqrt {5}\right )} \text {arctanh}\left (\frac {\tan \left (\frac {x}{2}\right )-\sqrt {5}+1}{\sqrt {5-2 \sqrt {5}}}\right )}{3125}+\frac {96 \left (4566082+2562235 \sqrt {5}\right ) \text {arctanh}\left (\frac {\tan \left (\frac {x}{2}\right )-\sqrt {5}+1}{\sqrt {5-2 \sqrt {5}}}\right )}{625 \sqrt {5 \left (5-2 \sqrt {5}\right )}}-\frac {8 \sqrt {2 \left (37230665+1543931 \sqrt {5}\right )} \text {arctanh}\left (\frac {\tan \left (\frac {x}{2}\right )-\sqrt {5}+1}{\sqrt {5-2 \sqrt {5}}}\right )}{3125}-\frac {128 \left (6462366+833905 \sqrt {5}\right ) \text {arctanh}\left (\frac {\tan \left (\frac {x}{2}\right )-\sqrt {5}+1}{\sqrt {5-2 \sqrt {5}}}\right )}{625 \sqrt {5-2 \sqrt {5}}}+\frac {128}{625} \sqrt {845+358 \sqrt {5}} \text {arctanh}\left (\frac {\tan \left (\frac {x}{2}\right )-\sqrt {5}+1}{\sqrt {5-2 \sqrt {5}}}\right )+\frac {16 \left (285412903-250146680 \sqrt {5}\right ) \text {arctanh}\left (\frac {\tan \left (\frac {x}{2}\right )-\sqrt {5}+1}{\sqrt {5-2 \sqrt {5}}}\right )}{625 \sqrt {5 \left (5-2 \sqrt {5}\right )}}+\frac {32 \left (95442211-34854353 \sqrt {5}\right ) \text {arctanh}\left (\frac {\tan \left (\frac {x}{2}\right )-\sqrt {5}+1}{\sqrt {5-2 \sqrt {5}}}\right )}{625 \left (5-2 \sqrt {5}\right )^{3/2}}-\frac {16 \left (285412903+250146680 \sqrt {5}\right ) \text {arctanh}\left (\frac {\tan \left (\frac {x}{2}\right )+\sqrt {5}+1}{\sqrt {5+2 \sqrt {5}}}\right )}{625 \sqrt {5 \left (5+2 \sqrt {5}\right )}}+\frac {32 \left (95442211+34854353 \sqrt {5}\right ) \text {arctanh}\left (\frac {\tan \left (\frac {x}{2}\right )+\sqrt {5}+1}{\sqrt {5+2 \sqrt {5}}}\right )}{625 \left (5+2 \sqrt {5}\right )^{3/2}}+\frac {32 \left (28693031+25149236 \sqrt {5}\right ) \text {arctanh}\left (\frac {\tan \left (\frac {x}{2}\right )+\sqrt {5}+1}{\sqrt {5+2 \sqrt {5}}}\right )}{125 \sqrt {5 \left (5+2 \sqrt {5}\right )}}+\frac {96 \left (23978935+10149962 \sqrt {5}\right ) \text {arctanh}\left (\frac {\tan \left (\frac {x}{2}\right )+\sqrt {5}+1}{\sqrt {5+2 \sqrt {5}}}\right )}{625 \left (5+2 \sqrt {5}\right )^{5/2}}-\frac {5328 \left (432095+182901 \sqrt {5}\right ) \text {arctanh}\left (\frac {\tan \left (\frac {x}{2}\right )+\sqrt {5}+1}{\sqrt {5+2 \sqrt {5}}}\right )}{625 \left (5+2 \sqrt {5}\right )^{5/2}}+\frac {64 \left (224674+83329 \sqrt {5}\right ) \text {arctanh}\left (\frac {\tan \left (\frac {x}{2}\right )+\sqrt {5}+1}{\sqrt {5+2 \sqrt {5}}}\right )}{625 \left (5+2 \sqrt {5}\right )^{3/2}}-\frac {128 \left (6462366-833905 \sqrt {5}\right ) \text {arctanh}\left (\frac {\tan \left (\frac {x}{2}\right )+\sqrt {5}+1}{\sqrt {5+2 \sqrt {5}}}\right )}{625 \sqrt {5+2 \sqrt {5}}}+\frac {128}{625} \sqrt {845-358 \sqrt {5}} \text {arctanh}\left (\frac {\tan \left (\frac {x}{2}\right )+\sqrt {5}+1}{\sqrt {5+2 \sqrt {5}}}\right )-\frac {8 \sqrt {2 \left (37230665-1543931 \sqrt {5}\right )} \text {arctanh}\left (\frac {\tan \left (\frac {x}{2}\right )+\sqrt {5}+1}{\sqrt {5+2 \sqrt {5}}}\right )}{3125}-\frac {16 \sqrt {15903211069805-2287334301082 \sqrt {5}} \text {arctanh}\left (\frac {\tan \left (\frac {x}{2}\right )+\sqrt {5}+1}{\sqrt {5+2 \sqrt {5}}}\right )}{3125}+\frac {13}{2 \left (1-\tan \left (\frac {x}{2}\right )\right )}-\frac {1}{250 \left (\tan \left (\frac {x}{2}\right )+1\right )}-\frac {96 \left (12811175+4566082 \sqrt {5}\right ) \left (\tan \left (\frac {x}{2}\right )-\sqrt {5}+1\right )}{3125 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1-\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )}+\frac {5328 \left (230835+82271 \sqrt {5}\right ) \left (\tan \left (\frac {x}{2}\right )-\sqrt {5}+1\right )}{3125 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1-\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )}-\frac {128 \left (25+2 \sqrt {5}\right ) \left (\tan \left (\frac {x}{2}\right )-\sqrt {5}+1\right )}{625 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1-\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )}-\frac {16 \left (5+2 \sqrt {5}\right ) \left (2 \left (95442211-34854353 \sqrt {5}\right ) \tan \left (\frac {x}{2}\right )-489891349 \sqrt {5}[phone]\right )}{3125 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1-\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )}+\frac {32 \left (5+2 \sqrt {5}\right ) \left (4 \left (23972780-8755207 \sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+7 \left (74165855-35162189 \sqrt {5}\right )\right )}{3125 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1-\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )}-\frac {16 \left (5+2 \sqrt {5}\right ) \left (4 \left (224674-83329 \sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+17 \left (296063-140005 \sqrt {5}\right )\right )}{3125 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1-\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )}-\frac {128 \left (25-2 \sqrt {5}\right ) \left (\tan \left (\frac {x}{2}\right )+\sqrt {5}+1\right )}{625 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1+\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )}+\frac {5328 \left (230835-82271 \sqrt {5}\right ) \left (\tan \left (\frac {x}{2}\right )+\sqrt {5}+1\right )}{3125 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1+\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )}-\frac {96 \left (12811175-4566082 \sqrt {5}\right ) \left (\tan \left (\frac {x}{2}\right )+\sqrt {5}+1\right )}{3125 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1+\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )}-\frac {16 \left (5-2 \sqrt {5}\right ) \left (4 \left (224674+83329 \sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+17 \left (296063+140005 \sqrt {5}\right )\right )}{3125 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1+\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )}+\frac {32 \left (5-2 \sqrt {5}\right ) \left (4 \left (23972780+8755207 \sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+7 \left (74165855+35162189 \sqrt {5}\right )\right )}{3125 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1+\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )}-\frac {16 \left (5-2 \sqrt {5}\right ) \left (2 \left (95442211+34854353 \sqrt {5}\right ) \tan \left (\frac {x}{2}\right )[phone] \sqrt {5}[phone]\right )}{3125 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1+\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )}-\frac {1}{4 \left (1-\tan \left (\frac {x}{2}\right )\right )^2}+\frac {1}{2500 \left (\tan \left (\frac {x}{2}\right )+1\right )^2}+\frac {256 \left (21-8 \sqrt {5}\right ) \left (\tan \left (\frac {x}{2}\right )-\sqrt {5}+1\right )}{375 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1-\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )^2}-\frac {64 \left (2+\sqrt {5}\right ) \left (\left (10149962-4795787 \sqrt {5}\right ) \tan \left (\frac {x}{2}\right )-29218021 \sqrt {5}[phone]\right )}{625 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1-\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )^2}+\frac {32 \left (2+\sqrt {5}\right ) \left (111 \left (182901-86419 \sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+4 \left (33224419-14609274 \sqrt {5}\right )\right )}{625 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1-\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )^2}+\frac {256 \left (349+156 \sqrt {5}\right ) \left (\tan \left (\frac {x}{2}\right )+\sqrt {5}+1\right )}{75 \left (5+2 \sqrt {5}\right )^2 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1+\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )^2}+\frac {32 \left (2-\sqrt {5}\right ) \left (111 \left (182901+86419 \sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+4 \left (33224419+14609274 \sqrt {5}\right )\right )}{625 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1+\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )^2}-\frac {64 \left (2-\sqrt {5}\right ) \left (\left (10149962+4795787 \sqrt {5}\right ) \tan \left (\frac {x}{2}\right )[phone] \sqrt {5}[phone]\right )}{625 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1+\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )^2}+\frac {1}{6 \left (1-\tan \left (\frac {x}{2}\right )\right )^3}-\frac {1}{3750 \left (\tan \left (\frac {x}{2}\right )+1\right )^3}-\frac {256 \left (5+2 \sqrt {5}\right ) \left (4 \left (349-156 \sqrt {5}\right ) \tan \left (\frac {x}{2}\right )-99 \sqrt {5}+221\right )}{1875 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1-\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )^3}-\frac {256 \left (5-2 \sqrt {5}\right ) \left (4 \left (349+156 \sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+99 \sqrt {5}+221\right )}{1875 \left (\tan ^2\left (\frac {x}{2}\right )+2 \left (1+\sqrt {5}\right ) \tan \left (\frac {x}{2}\right )+1\right )^3}\right )\)

Input:

Int[(Cos[3*x] + Sin[2*x])^(-4),x]
 

Output:

2*((32*(95442211 - 34854353*Sqrt[5])*ArcTanh[(1 - Sqrt[5] + Tan[x/2])/Sqrt 
[5 - 2*Sqrt[5]]])/(625*(5 - 2*Sqrt[5])^(3/2)) + (16*(285412903 - 250146680 
*Sqrt[5])*ArcTanh[(1 - Sqrt[5] + Tan[x/2])/Sqrt[5 - 2*Sqrt[5]]])/(625*Sqrt 
[5*(5 - 2*Sqrt[5])]) + (128*Sqrt[845 + 358*Sqrt[5]]*ArcTanh[(1 - Sqrt[5] + 
 Tan[x/2])/Sqrt[5 - 2*Sqrt[5]]])/625 - (128*(6462366 + 833905*Sqrt[5])*Arc 
Tanh[(1 - Sqrt[5] + Tan[x/2])/Sqrt[5 - 2*Sqrt[5]]])/(625*Sqrt[5 - 2*Sqrt[5 
]]) - (8*Sqrt[2*(37230665 + 1543931*Sqrt[5])]*ArcTanh[(1 - Sqrt[5] + Tan[x 
/2])/Sqrt[5 - 2*Sqrt[5]]])/3125 + (96*(4566082 + 2562235*Sqrt[5])*ArcTanh[ 
(1 - Sqrt[5] + Tan[x/2])/Sqrt[5 - 2*Sqrt[5]]])/(625*Sqrt[5*(5 - 2*Sqrt[5]) 
]) - (5328*Sqrt[2*(81545744785 + 36416503171*Sqrt[5])]*ArcTanh[(1 - Sqrt[5 
] + Tan[x/2])/Sqrt[5 - 2*Sqrt[5]]])/3125 + (64*Sqrt[127439782405 + 5477050 
7458*Sqrt[5]]*ArcTanh[(1 - Sqrt[5] + Tan[x/2])/Sqrt[5 - 2*Sqrt[5]]])/3125 
- (16*Sqrt[15903211069805 + 2287334301082*Sqrt[5]]*ArcTanh[(1 - Sqrt[5] + 
Tan[x/2])/Sqrt[5 - 2*Sqrt[5]]])/3125 + (32*Sqrt[5496395760940885 + 7553426 
88027722*Sqrt[5]]*ArcTanh[(1 - Sqrt[5] + Tan[x/2])/Sqrt[5 - 2*Sqrt[5]]])/6 
25 - (16*Sqrt[15903211069805 - 2287334301082*Sqrt[5]]*ArcTanh[(1 + Sqrt[5] 
 + Tan[x/2])/Sqrt[5 + 2*Sqrt[5]]])/3125 - (8*Sqrt[2*(37230665 - 1543931*Sq 
rt[5])]*ArcTanh[(1 + Sqrt[5] + Tan[x/2])/Sqrt[5 + 2*Sqrt[5]]])/3125 + (128 
*Sqrt[845 - 358*Sqrt[5]]*ArcTanh[(1 + Sqrt[5] + Tan[x/2])/Sqrt[5 + 2*Sqrt[ 
5]]])/625 - (128*(6462366 - 833905*Sqrt[5])*ArcTanh[(1 + Sqrt[5] + Tan[...
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2462

Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr 
and[u*Qx^p, x], x] /;  !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && GtQ 
[Expon[Px, x], 2] &&  !BinomialQ[Px, x] &&  !TrinomialQ[Px, x] && ILtQ[p, 0 
] && RationalFunctionQ[u, x]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4830

Int[(cos[(n_.)*((c_.) + (d_.)*(x_))]*(b_.) + (a_.)*sin[(m_.)*((c_.) + (d_.) 
*(x_))])^(p_), x_Symbol] :> Simp[2/d   Subst[Int[Simplify[TrigExpand[a*Sin[ 
2*m*ArcTan[x]] + b*Cos[2*n*ArcTan[x]]]]^p/(1 + x^2), x], x, Tan[(1/2)*(c + 
d*x)]], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[p, 0] && IntegerQ[m] && Intege 
rQ[n]
 

Maple [A] (verified)

Time = 115.72 (sec) , antiderivative size = 2, normalized size of antiderivative = 2.00

method	result	size
parallelrisch	\(0\)	\(2\)
risch	\(\frac {\frac {148 \,{\mathrm e}^{13 i x}}{125}-\frac {8864 \,{\mathrm e}^{15 i x}}{375}-\frac {1172 \,{\mathrm e}^{i x}}{25}+\frac {68 i}{3}+\frac {1088 \,{\mathrm e}^{9 i x}}{375}+\frac {528 \,{\mathrm e}^{17 i x}}{25}-\frac {364 \,{\mathrm e}^{3 i x}}{375}+\frac {17324 \,{\mathrm e}^{11 i x}}{125}-\frac {8176 \,{\mathrm e}^{7 i x}}{125}-\frac {2708 i {\mathrm e}^{2 i x}}{125}-\frac {8836 i {\mathrm e}^{10 i x}}{125}-\frac {148 i {\mathrm e}^{14 i x}}{125}+\frac {24124 i {\mathrm e}^{12 i x}}{375}+\frac {336 i {\mathrm e}^{8 i x}}{125}+\frac {148 i {\mathrm e}^{4 i x}}{125}+\frac {52376 i {\mathrm e}^{6 i x}}{375}-\frac {5792 i {\mathrm e}^{16 i x}}{125}+\frac {8648 \,{\mathrm e}^{5 i x}}{125}}{\left (-2 i {\mathrm e}^{4 i x}+{\mathrm e}^{5 i x}+2 i {\mathrm e}^{2 i x}-2 \,{\mathrm e}^{3 i x}-i+2 \,{\mathrm e}^{i x}\right )^{3} \left ({\mathrm e}^{i x}+i\right )^{3}}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (30517578125 \textit {\_Z}^{4}-9798100000000 \textit {\_Z}^{2}[phone]\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{i x}+\frac {32470703125 \textit {\_R}^{3}}{322165428224}-\frac {390625 i \textit {\_R}^{2}}{112174592}-\frac {81529051875 \textit {\_R}}{2516917408}+\frac {135407 i}{438182}\right )\right )\)	\(225\)
default	\(-\frac {1}{1875 \left (\tan \left (\frac {x}{2}\right )+1\right )^{3}}+\frac {1}{1250 \left (\tan \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {1}{125 \left (\tan \left (\frac {x}{2}\right )+1\right )}-\frac {32 \left (290 \tan \left (\frac {x}{2}\right )^{11}+3544 \tan \left (\frac {x}{2}\right )^{10}+6054 \tan \left (\frac {x}{2}\right )^{9}-57184 \tan \left (\frac {x}{2}\right )^{8}-68156 \tan \left (\frac {x}{2}\right )^{7}+\frac {1336928 \tan \left (\frac {x}{2}\right )^{6}}{3}-419908 \tan \left (\frac {x}{2}\right )^{5}+53904 \tan \left (\frac {x}{2}\right )^{4}+\frac {152398 \tan \left (\frac {x}{2}\right )^{3}}{3}-4856 \tan \left (\frac {x}{2}\right )^{2}-2914 \tan \left (\frac {x}{2}\right )-\frac {656}{3}\right )}{625 \left (\tan \left (\frac {x}{2}\right )^{4}+4 \tan \left (\frac {x}{2}\right )^{3}-14 \tan \left (\frac {x}{2}\right )^{2}+4 \tan \left (\frac {x}{2}\right )+1\right )^{3}}-\frac {32 \left (-\frac {101}{2}-\frac {229 \sqrt {5}}{10}\right ) \operatorname {arctanh}\left (\frac {2 \tan \left (\frac {x}{2}\right )-2 \sqrt {5}+2}{2 \sqrt {5-2 \sqrt {5}}}\right )}{125 \sqrt {5-2 \sqrt {5}}}+\frac {32 \left (\frac {101}{2}-\frac {229 \sqrt {5}}{10}\right ) \operatorname {arctanh}\left (\frac {2 \tan \left (\frac {x}{2}\right )+2+2 \sqrt {5}}{2 \sqrt {5+2 \sqrt {5}}}\right )}{125 \sqrt {5+2 \sqrt {5}}}-\frac {1}{3 \left (\tan \left (\frac {x}{2}\right )-1\right )^{3}}-\frac {1}{2 \left (\tan \left (\frac {x}{2}\right )-1\right )^{2}}-\frac {13}{\tan \left (\frac {x}{2}\right )-1}\)	\(270\)

Input:

int(1/(cos(3*x)+sin(2*x))^4,x,method=_RETURNVERBOSE)
 

Output:

0
 

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 0.13 (sec) , antiderivative size = 484, normalized size of antiderivative = 484.00 \[ \int \frac {1}{(\cos (3 x)+\sin (2 x))^4} \, dx =\text {Too large to display} \] Input:

integrate(1/(cos(3*x)+sin(2*x))^4,x, algorithm="fricas")
 

Output:

-1/3750*(1545600*cos(x)^8 - 2722080*cos(x)^6 + 1114440*cos(x)^4 - 24*sqrt( 
2)*(64*cos(x)^9 - 192*cos(x)^7 + 192*cos(x)^5 - 63*cos(x)^3 + 2*(48*cos(x) 
^7 - 76*cos(x)^5 + 31*cos(x)^3)*sin(x))*sqrt(219091*sqrt(5) + 489905)*log( 
-8*sqrt(2)*sqrt(219091*sqrt(5) + 489905)*(133*sqrt(5) - 298)*cos(x) + 2872 
*(sqrt(5) + 1)*sin(x) - 11488) + 24*sqrt(2)*(64*cos(x)^9 - 192*cos(x)^7 + 
192*cos(x)^5 - 63*cos(x)^3 + 2*(48*cos(x)^7 - 76*cos(x)^5 + 31*cos(x)^3)*s 
in(x))*sqrt(219091*sqrt(5) + 489905)*log(-8*sqrt(2)*sqrt(219091*sqrt(5) + 
489905)*(133*sqrt(5) - 298)*cos(x) - 2872*(sqrt(5) + 1)*sin(x) + 11488) - 
3*(64*cos(x)^9 - 192*cos(x)^7 + 192*cos(x)^5 - 63*cos(x)^3 + 2*(48*cos(x)^ 
7 - 76*cos(x)^5 + 31*cos(x)^3)*sin(x))*sqrt(-28043648*sqrt(5) + 62707840)* 
log(-(133*sqrt(5) + 298)*sqrt(-28043648*sqrt(5) + 62707840)*cos(x) + 2872* 
(sqrt(5) - 1)*sin(x) + 11488) + 3*(64*cos(x)^9 - 192*cos(x)^7 + 192*cos(x) 
^5 - 63*cos(x)^3 + 2*(48*cos(x)^7 - 76*cos(x)^5 + 31*cos(x)^3)*sin(x))*sqr 
t(-28043648*sqrt(5) + 62707840)*log(-(133*sqrt(5) + 298)*sqrt(-28043648*sq 
rt(5) + 62707840)*cos(x) - 2872*(sqrt(5) - 1)*sin(x) - 11488) + 16500*cos( 
x)^2 - 10*(272000*cos(x)^8 - 401984*cos(x)^6 + 164256*cos(x)^4 - 1950*cos( 
x)^2 - 75)*sin(x) + 500)/(64*cos(x)^9 - 192*cos(x)^7 + 192*cos(x)^5 - 63*c 
os(x)^3 + 2*(48*cos(x)^7 - 76*cos(x)^5 + 31*cos(x)^3)*sin(x))
 

Sympy [F]

\[ \int \frac {1}{(\cos (3 x)+\sin (2 x))^4} \, dx=\int \frac {1}{\left (\sin {\left (2 x \right )} + \cos {\left (3 x \right )}\right )^{4}}\, dx \] Input:

integrate(1/(cos(3*x)+sin(2*x))**4,x)
 

Output:

Integral((sin(2*x) + cos(3*x))**(-4), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{(\cos (3 x)+\sin (2 x))^4} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate(1/(cos(3*x)+sin(2*x))^4,x, algorithm="maxima")
 

Output:

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is un 
defined.
 

Giac [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 0.12 (sec) , antiderivative size = 215, normalized size of antiderivative = 215.00 \[ \int \frac {1}{(\cos (3 x)+\sin (2 x))^4} \, dx=-\frac {2 \, {\left (5223 \, \tan \left (\frac {1}{2} \, x\right )^{17} + 65820 \, \tan \left (\frac {1}{2} \, x\right )^{16} + 90128 \, \tan \left (\frac {1}{2} \, x\right )^{15} - 1330212 \, \tan \left (\frac {1}{2} \, x\right )^{14} - 1522956 \, \tan \left (\frac {1}{2} \, x\right )^{13} + 12715668 \, \tan \left (\frac {1}{2} \, x\right )^{12} - 4999824 \, \tan \left (\frac {1}{2} \, x\right )^{11} - 29883540 \, \tan \left (\frac {1}{2} \, x\right )^{10} + 24059626 \, \tan \left (\frac {1}{2} \, x\right )^{9} + 25434588 \, \tan \left (\frac {1}{2} \, x\right )^{8} - 28672848 \, \tan \left (\frac {1}{2} \, x\right )^{7} - 5768748 \, \tan \left (\frac {1}{2} \, x\right )^{6} + 12205236 \, \tan \left (\frac {1}{2} \, x\right )^{5} - 1330212 \, \tan \left (\frac {1}{2} \, x\right )^{4} - 1243056 \, \tan \left (\frac {1}{2} \, x\right )^{3} + 79332 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 59271 \, \tan \left (\frac {1}{2} \, x\right ) + 4504\right )}}{375 \, {\left (\tan \left (\frac {1}{2} \, x\right )^{6} + 4 \, \tan \left (\frac {1}{2} \, x\right )^{5} - 15 \, \tan \left (\frac {1}{2} \, x\right )^{4} + 15 \, \tan \left (\frac {1}{2} \, x\right )^{2} - 4 \, \tan \left (\frac {1}{2} \, x\right ) - 1\right )}^{3}} - 0.0293605416398400 \, \log \left (\tan \left (\frac {1}{2} \, x\right ) + 6.31375151468000\right ) + 0.0293605416398400 \, \log \left (\tan \left (\frac {1}{2} \, x\right ) + 0.158384440325000\right ) + 17.9182387180800 \, \log \left (\tan \left (\frac {1}{2} \, x\right ) - 0.509525449494000\right ) - 17.9182387180800 \, \log \left (\tan \left (\frac {1}{2} \, x\right ) - 1.96261050551000\right ) \] Input:

integrate(1/(cos(3*x)+sin(2*x))^4,x, algorithm="giac")
 

Output:

-2/375*(5223*tan(1/2*x)^17 + 65820*tan(1/2*x)^16 + 90128*tan(1/2*x)^15 - 1 
330212*tan(1/2*x)^14 - 1522956*tan(1/2*x)^13 + 12715668*tan(1/2*x)^12 - 49 
99824*tan(1/2*x)^11 - 29883540*tan(1/2*x)^10 + 24059626*tan(1/2*x)^9 + 254 
34588*tan(1/2*x)^8 - 28672848*tan(1/2*x)^7 - 5768748*tan(1/2*x)^6 + 122052 
36*tan(1/2*x)^5 - 1330212*tan(1/2*x)^4 - 1243056*tan(1/2*x)^3 + 79332*tan( 
1/2*x)^2 + 59271*tan(1/2*x) + 4504)/(tan(1/2*x)^6 + 4*tan(1/2*x)^5 - 15*ta 
n(1/2*x)^4 + 15*tan(1/2*x)^2 - 4*tan(1/2*x) - 1)^3 - 0.0293605416398400*lo 
g(tan(1/2*x) + 6.31375151468000) + 0.0293605416398400*log(tan(1/2*x) + 0.1 
58384440325000) + 17.9182387180800*log(tan(1/2*x) - 0.509525449494000) - 1 
7.9182387180800*log(tan(1/2*x) - 1.96261050551000)
 

Mupad [B] (verification not implemented)

Time = 22.48 (sec) , antiderivative size = 594, normalized size of antiderivative = 594.00 \[ \int \frac {1}{(\cos (3 x)+\sin (2 x))^4} \, dx=\text {Too large to display} \] Input:

int(1/(cos(3*x) + sin(2*x))^4,x)
 

Output:

((39514*tan(x/2))/125 + (52888*tan(x/2)^2)/125 - (828704*tan(x/2)^3)/125 - 
 (886808*tan(x/2)^4)/125 + (8136824*tan(x/2)^5)/125 - (3845832*tan(x/2)^6) 
/125 - (19115232*tan(x/2)^7)/125 + (16956392*tan(x/2)^8)/125 + (48119252*t 
an(x/2)^9)/375 - (3984472*tan(x/2)^10)/25 - (3333216*tan(x/2)^11)/125 + (8 
477112*tan(x/2)^12)/125 - (1015304*tan(x/2)^13)/125 - (886808*tan(x/2)^14) 
/125 + (180256*tan(x/2)^15)/375 + (8776*tan(x/2)^16)/25 + (3482*tan(x/2)^1 
7)/125 + 9008/375)/(12*tan(x/2) + 3*tan(x/2)^2 - 296*tan(x/2)^3 + 3048*tan 
(x/2)^5 - 4104*tan(x/2)^6 - 5256*tan(x/2)^7 + 12282*tan(x/2)^8 - 12282*tan 
(x/2)^10 + 5256*tan(x/2)^11 + 4104*tan(x/2)^12 - 3048*tan(x/2)^13 + 296*ta 
n(x/2)^15 - 3*tan(x/2)^16 - 12*tan(x/2)^17 - tan(x/2)^18 + 1) - (16*atanh( 
(344600346624*(979810 - 438182*5^(1/2))^(1/2))/(9765625*((122218256269312* 
tan(x/2))/1953125 - (273842408783872*5^(1/2)*tan(x/2))/9765625 + (11061671 
1266304*5^(1/2))/9765625 - 49507583131648/1953125)) - (229733564416*tan(x/ 
2)*(979810 - 438182*5^(1/2))^(1/2))/(9765625*((122218256269312*tan(x/2))/1 
953125 - (273842408783872*5^(1/2)*tan(x/2))/9765625 + (110616711266304*5^( 
1/2))/9765625 - 49507583131648/1953125)) - (114866782208*5^(1/2)*(979810 - 
 438182*5^(1/2))^(1/2))/(9765625*((122218256269312*tan(x/2))/1953125 - (27 
3842408783872*5^(1/2)*tan(x/2))/9765625 + (110616711266304*5^(1/2))/976562 
5 - 49507583131648/1953125)) + (344600346624*5^(1/2)*tan(x/2)*(979810 - 43 
8182*5^(1/2))^(1/2))/(9765625*((122218256269312*tan(x/2))/1953125 - (27...
 

Reduce [F]

\[ \int \frac {1}{(\cos (3 x)+\sin (2 x))^4} \, dx=\int \frac {1}{\cos \left (3 x \right )^{4}+4 \cos \left (3 x \right )^{3} \sin \left (2 x \right )+6 \cos \left (3 x \right )^{2} \sin \left (2 x \right )^{2}+4 \cos \left (3 x \right ) \sin \left (2 x \right )^{3}+\sin \left (2 x \right )^{4}}d x \] Input:

int(1/(cos(3*x)+sin(2*x))^4,x)
 

Output:

int(1/(cos(3*x)**4 + 4*cos(3*x)**3*sin(2*x) + 6*cos(3*x)**2*sin(2*x)**2 + 
4*cos(3*x)*sin(2*x)**3 + sin(2*x)**4),x)