Integrand size = 11, antiderivative size = 92 \[ \int \frac {1}{(\cos (5 x)+\sin (4 x))^3} \, dx=-\frac {19 \text {arctanh}(\cos (x))}{1024}+\frac {31 \text {arctanh}\left (\sqrt {2} \cos (x)\right )}{128 \sqrt {2}}-\frac {105 \sec (x)}{1024}-\frac {43 \sec ^3(x)}{3072}-\frac {3 \sec ^5(x)}{1280}+\frac {\sec ^5(x) \sec (2 x)}{1024}+\frac {1}{256} \sec ^5(x) \sec ^2(2 x)-\frac {\csc ^2(x) \sec ^5(x) \sec ^2(2 x)}{1024} \] Output:
-19/1024*arctanh(cos(x))+31/256*arctanh(cos(x)*2^(1/2))*2^(1/2)-105/1024*s ec(x)-43/3072*sec(x)^3-3/1280*sec(x)^5+1/1024*sec(x)^5*sec(2*x)+1/256*sec( x)^5*sec(2*x)^2-1/1024*csc(x)^2*sec(x)^5*sec(2*x)^2
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 1.39 (sec) , antiderivative size = 223, normalized size of antiderivative = 2.42 \[ \int \frac {1}{(\cos (5 x)+\sin (4 x))^3} \, dx=\frac {-88938 \log \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )+42 \log \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )-480 \log (1-2 \sin (x))-1152 \left (39 \log \left (\sec ^2\left (\frac {x}{2}\right )\right )-2 \text {RootSum}\left [269+5976 \text {$\#$1}-269568 \text {$\#$1}^2+13824 \text {$\#$1}^3\&,\log \left (-\sec ^2\left (\frac {x}{2}\right ) \left (-39007-109078 \sin (x)-2595120 \sin (x) \text {$\#$1}+133632 \sin (x) \text {$\#$1}^2\right )\right ) \text {$\#$1}\&\right ]\right )+\frac {729}{\left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )^2}-\frac {1}{\left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^2}+\frac {48}{(1-2 \sin (x))^2}+\frac {128}{-1+2 \sin (x)}+\frac {864 (-4+3 \cos (2 x)-6 \sin (x))}{(1+2 \sin (3 x))^2}-\frac {288 (-35+26 \cos (2 x)-50 \sin (x))}{1+2 \sin (3 x)}}{2916} \] Input:
Integrate[(Cos[5*x] + Sin[4*x])^(-3),x]
Output:
(-88938*Log[Cos[x/2] - Sin[x/2]] + 42*Log[Cos[x/2] + Sin[x/2]] - 480*Log[1 - 2*Sin[x]] - 1152*(39*Log[Sec[x/2]^2] - 2*RootSum[269 + 5976*#1 - 269568 *#1^2 + 13824*#1^3 & , Log[-(Sec[x/2]^2*(-39007 - 109078*Sin[x] - 2595120* Sin[x]*#1 + 133632*Sin[x]*#1^2))]*#1 & ]) + 729/(Cos[x/2] - Sin[x/2])^2 - (Cos[x/2] + Sin[x/2])^(-2) + 48/(1 - 2*Sin[x])^2 + 128/(-1 + 2*Sin[x]) + ( 864*(-4 + 3*Cos[2*x] - 6*Sin[x]))/(1 + 2*Sin[3*x])^2 - (288*(-35 + 26*Cos[ 2*x] - 50*Sin[x]))/(1 + 2*Sin[3*x]))/2916
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.
| \(\displaystyle \int \frac {1}{(\sin (4 x)+\cos (5 x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(\sin (4 x)+\cos (5 x))^3}dx\) |
| \(\Big \downarrow \) 4829 |
| \(\displaystyle \int \frac {1}{\left (1-\sin ^2(x)\right )^2 \left (16 \sin ^4(x)-8 \sin ^3(x)-12 \sin ^2(x)+4 \sin (x)+1\right )^3}d\sin (x)\) |
| \(\Big \downarrow \) 2462 |
| \(\displaystyle \int \left (\frac {32 \left (16 \sin ^2(x)+15 \sin (x)+2\right )}{9 \left (8 \sin ^3(x)-6 \sin (x)-1\right )^3}+\frac {64 \left (52 \sin ^2(x)+51 \sin (x)+11\right )}{27 \left (8 \sin ^3(x)-6 \sin (x)-1\right )}+\frac {16 \left (140 \sin ^2(x)+138 \sin (x)+31\right )}{27 \left (8 \sin ^3(x)-6 \sin (x)-1\right )^2}-\frac {61}{4 (\sin (x)-1)}+\frac {7}{972 (\sin (x)+1)}-\frac {80}{243 (2 \sin (x)-1)}+\frac {1}{4 (\sin (x)-1)^2}+\frac {1}{2916 (\sin (x)+1)^2}-\frac {64}{729 (2 \sin (x)-1)^2}-\frac {16}{243 (2 \sin (x)-1)^3}\right )d\sin (x)\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{(1-2 \sin (x))^3 \left (1-\sin ^2(x)\right )^2 \left (-8 \sin ^3(x)+6 \sin (x)+1\right )^3}d\sin (x)\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {32 \left (16 \sin ^2(x)+15 \sin (x)+2\right )}{9 \left (8 \sin ^3(x)-6 \sin (x)-1\right )^3}+\frac {64 \left (52 \sin ^2(x)+51 \sin (x)+11\right )}{27 \left (8 \sin ^3(x)-6 \sin (x)-1\right )}+\frac {16 \left (140 \sin ^2(x)+138 \sin (x)+31\right )}{27 \left (8 \sin ^3(x)-6 \sin (x)-1\right )^2}-\frac {61}{4 (\sin (x)-1)}+\frac {7}{972 (\sin (x)+1)}-\frac {80}{243 (2 \sin (x)-1)}+\frac {1}{4 (\sin (x)-1)^2}+\frac {1}{2916 (\sin (x)+1)^2}-\frac {64}{729 (2 \sin (x)-1)^2}-\frac {16}{243 (2 \sin (x)-1)^3}\right )d\sin (x)\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{(1-2 \sin (x))^3 \left (1-\sin ^2(x)\right )^2 \left (-8 \sin ^3(x)+6 \sin (x)+1\right )^3}d\sin (x)\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {32 \left (16 \sin ^2(x)+15 \sin (x)+2\right )}{9 \left (8 \sin ^3(x)-6 \sin (x)-1\right )^3}+\frac {64 \left (52 \sin ^2(x)+51 \sin (x)+11\right )}{27 \left (8 \sin ^3(x)-6 \sin (x)-1\right )}+\frac {16 \left (140 \sin ^2(x)+138 \sin (x)+31\right )}{27 \left (8 \sin ^3(x)-6 \sin (x)-1\right )^2}-\frac {61}{4 (\sin (x)-1)}+\frac {7}{972 (\sin (x)+1)}-\frac {80}{243 (2 \sin (x)-1)}+\frac {1}{4 (\sin (x)-1)^2}+\frac {1}{2916 (\sin (x)+1)^2}-\frac {64}{729 (2 \sin (x)-1)^2}-\frac {16}{243 (2 \sin (x)-1)^3}\right )d\sin (x)\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{(1-2 \sin (x))^3 \left (1-\sin ^2(x)\right )^2 \left (-8 \sin ^3(x)+6 \sin (x)+1\right )^3}d\sin (x)\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {32 \left (16 \sin ^2(x)+15 \sin (x)+2\right )}{9 \left (8 \sin ^3(x)-6 \sin (x)-1\right )^3}+\frac {64 \left (52 \sin ^2(x)+51 \sin (x)+11\right )}{27 \left (8 \sin ^3(x)-6 \sin (x)-1\right )}+\frac {16 \left (140 \sin ^2(x)+138 \sin (x)+31\right )}{27 \left (8 \sin ^3(x)-6 \sin (x)-1\right )^2}-\frac {61}{4 (\sin (x)-1)}+\frac {7}{972 (\sin (x)+1)}-\frac {80}{243 (2 \sin (x)-1)}+\frac {1}{4 (\sin (x)-1)^2}+\frac {1}{2916 (\sin (x)+1)^2}-\frac {64}{729 (2 \sin (x)-1)^2}-\frac {16}{243 (2 \sin (x)-1)^3}\right )d\sin (x)\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{(1-2 \sin (x))^3 \left (1-\sin ^2(x)\right )^2 \left (-8 \sin ^3(x)+6 \sin (x)+1\right )^3}d\sin (x)\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {32 \left (16 \sin ^2(x)+15 \sin (x)+2\right )}{9 \left (8 \sin ^3(x)-6 \sin (x)-1\right )^3}+\frac {64 \left (52 \sin ^2(x)+51 \sin (x)+11\right )}{27 \left (8 \sin ^3(x)-6 \sin (x)-1\right )}+\frac {16 \left (140 \sin ^2(x)+138 \sin (x)+31\right )}{27 \left (8 \sin ^3(x)-6 \sin (x)-1\right )^2}-\frac {61}{4 (\sin (x)-1)}+\frac {7}{972 (\sin (x)+1)}-\frac {80}{243 (2 \sin (x)-1)}+\frac {1}{4 (\sin (x)-1)^2}+\frac {1}{2916 (\sin (x)+1)^2}-\frac {64}{729 (2 \sin (x)-1)^2}-\frac {16}{243 (2 \sin (x)-1)^3}\right )d\sin (x)\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{(1-2 \sin (x))^3 \left (1-\sin ^2(x)\right )^2 \left (-8 \sin ^3(x)+6 \sin (x)+1\right )^3}d\sin (x)\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {32 \left (16 \sin ^2(x)+15 \sin (x)+2\right )}{9 \left (8 \sin ^3(x)-6 \sin (x)-1\right )^3}+\frac {64 \left (52 \sin ^2(x)+51 \sin (x)+11\right )}{27 \left (8 \sin ^3(x)-6 \sin (x)-1\right )}+\frac {16 \left (140 \sin ^2(x)+138 \sin (x)+31\right )}{27 \left (8 \sin ^3(x)-6 \sin (x)-1\right )^2}-\frac {61}{4 (\sin (x)-1)}+\frac {7}{972 (\sin (x)+1)}-\frac {80}{243 (2 \sin (x)-1)}+\frac {1}{4 (\sin (x)-1)^2}+\frac {1}{2916 (\sin (x)+1)^2}-\frac {64}{729 (2 \sin (x)-1)^2}-\frac {16}{243 (2 \sin (x)-1)^3}\right )d\sin (x)\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{(1-2 \sin (x))^3 \left (1-\sin ^2(x)\right )^2 \left (-8 \sin ^3(x)+6 \sin (x)+1\right )^3}d\sin (x)\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {32 \left (16 \sin ^2(x)+15 \sin (x)+2\right )}{9 \left (8 \sin ^3(x)-6 \sin (x)-1\right )^3}+\frac {64 \left (52 \sin ^2(x)+51 \sin (x)+11\right )}{27 \left (8 \sin ^3(x)-6 \sin (x)-1\right )}+\frac {16 \left (140 \sin ^2(x)+138 \sin (x)+31\right )}{27 \left (8 \sin ^3(x)-6 \sin (x)-1\right )^2}-\frac {61}{4 (\sin (x)-1)}+\frac {7}{972 (\sin (x)+1)}-\frac {80}{243 (2 \sin (x)-1)}+\frac {1}{4 (\sin (x)-1)^2}+\frac {1}{2916 (\sin (x)+1)^2}-\frac {64}{729 (2 \sin (x)-1)^2}-\frac {16}{243 (2 \sin (x)-1)^3}\right )d\sin (x)\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{(1-2 \sin (x))^3 \left (1-\sin ^2(x)\right )^2 \left (-8 \sin ^3(x)+6 \sin (x)+1\right )^3}d\sin (x)\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {32 \left (16 \sin ^2(x)+15 \sin (x)+2\right )}{9 \left (8 \sin ^3(x)-6 \sin (x)-1\right )^3}+\frac {64 \left (52 \sin ^2(x)+51 \sin (x)+11\right )}{27 \left (8 \sin ^3(x)-6 \sin (x)-1\right )}+\frac {16 \left (140 \sin ^2(x)+138 \sin (x)+31\right )}{27 \left (8 \sin ^3(x)-6 \sin (x)-1\right )^2}-\frac {61}{4 (\sin (x)-1)}+\frac {7}{972 (\sin (x)+1)}-\frac {80}{243 (2 \sin (x)-1)}+\frac {1}{4 (\sin (x)-1)^2}+\frac {1}{2916 (\sin (x)+1)^2}-\frac {64}{729 (2 \sin (x)-1)^2}-\frac {16}{243 (2 \sin (x)-1)^3}\right )d\sin (x)\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{(1-2 \sin (x))^3 \left (1-\sin ^2(x)\right )^2 \left (-8 \sin ^3(x)+6 \sin (x)+1\right )^3}d\sin (x)\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {32 \left (16 \sin ^2(x)+15 \sin (x)+2\right )}{9 \left (8 \sin ^3(x)-6 \sin (x)-1\right )^3}+\frac {64 \left (52 \sin ^2(x)+51 \sin (x)+11\right )}{27 \left (8 \sin ^3(x)-6 \sin (x)-1\right )}+\frac {16 \left (140 \sin ^2(x)+138 \sin (x)+31\right )}{27 \left (8 \sin ^3(x)-6 \sin (x)-1\right )^2}-\frac {61}{4 (\sin (x)-1)}+\frac {7}{972 (\sin (x)+1)}-\frac {80}{243 (2 \sin (x)-1)}+\frac {1}{4 (\sin (x)-1)^2}+\frac {1}{2916 (\sin (x)+1)^2}-\frac {64}{729 (2 \sin (x)-1)^2}-\frac {16}{243 (2 \sin (x)-1)^3}\right )d\sin (x)\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{(1-2 \sin (x))^3 \left (1-\sin ^2(x)\right )^2 \left (-8 \sin ^3(x)+6 \sin (x)+1\right )^3}d\sin (x)\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {32 \left (16 \sin ^2(x)+15 \sin (x)+2\right )}{9 \left (8 \sin ^3(x)-6 \sin (x)-1\right )^3}+\frac {64 \left (52 \sin ^2(x)+51 \sin (x)+11\right )}{27 \left (8 \sin ^3(x)-6 \sin (x)-1\right )}+\frac {16 \left (140 \sin ^2(x)+138 \sin (x)+31\right )}{27 \left (8 \sin ^3(x)-6 \sin (x)-1\right )^2}-\frac {61}{4 (\sin (x)-1)}+\frac {7}{972 (\sin (x)+1)}-\frac {80}{243 (2 \sin (x)-1)}+\frac {1}{4 (\sin (x)-1)^2}+\frac {1}{2916 (\sin (x)+1)^2}-\frac {64}{729 (2 \sin (x)-1)^2}-\frac {16}{243 (2 \sin (x)-1)^3}\right )d\sin (x)\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{(1-2 \sin (x))^3 \left (1-\sin ^2(x)\right )^2 \left (-8 \sin ^3(x)+6 \sin (x)+1\right )^3}d\sin (x)\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {32 \left (16 \sin ^2(x)+15 \sin (x)+2\right )}{9 \left (8 \sin ^3(x)-6 \sin (x)-1\right )^3}+\frac {64 \left (52 \sin ^2(x)+51 \sin (x)+11\right )}{27 \left (8 \sin ^3(x)-6 \sin (x)-1\right )}+\frac {16 \left (140 \sin ^2(x)+138 \sin (x)+31\right )}{27 \left (8 \sin ^3(x)-6 \sin (x)-1\right )^2}-\frac {61}{4 (\sin (x)-1)}+\frac {7}{972 (\sin (x)+1)}-\frac {80}{243 (2 \sin (x)-1)}+\frac {1}{4 (\sin (x)-1)^2}+\frac {1}{2916 (\sin (x)+1)^2}-\frac {64}{729 (2 \sin (x)-1)^2}-\frac {16}{243 (2 \sin (x)-1)^3}\right )d\sin (x)\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{(1-2 \sin (x))^3 \left (1-\sin ^2(x)\right )^2 \left (-8 \sin ^3(x)+6 \sin (x)+1\right )^3}d\sin (x)\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {32 \left (16 \sin ^2(x)+15 \sin (x)+2\right )}{9 \left (8 \sin ^3(x)-6 \sin (x)-1\right )^3}+\frac {64 \left (52 \sin ^2(x)+51 \sin (x)+11\right )}{27 \left (8 \sin ^3(x)-6 \sin (x)-1\right )}+\frac {16 \left (140 \sin ^2(x)+138 \sin (x)+31\right )}{27 \left (8 \sin ^3(x)-6 \sin (x)-1\right )^2}-\frac {61}{4 (\sin (x)-1)}+\frac {7}{972 (\sin (x)+1)}-\frac {80}{243 (2 \sin (x)-1)}+\frac {1}{4 (\sin (x)-1)^2}+\frac {1}{2916 (\sin (x)+1)^2}-\frac {64}{729 (2 \sin (x)-1)^2}-\frac {16}{243 (2 \sin (x)-1)^3}\right )d\sin (x)\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{(1-2 \sin (x))^3 \left (1-\sin ^2(x)\right )^2 \left (-8 \sin ^3(x)+6 \sin (x)+1\right )^3}d\sin (x)\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {32 \left (16 \sin ^2(x)+15 \sin (x)+2\right )}{9 \left (8 \sin ^3(x)-6 \sin (x)-1\right )^3}+\frac {64 \left (52 \sin ^2(x)+51 \sin (x)+11\right )}{27 \left (8 \sin ^3(x)-6 \sin (x)-1\right )}+\frac {16 \left (140 \sin ^2(x)+138 \sin (x)+31\right )}{27 \left (8 \sin ^3(x)-6 \sin (x)-1\right )^2}-\frac {61}{4 (\sin (x)-1)}+\frac {7}{972 (\sin (x)+1)}-\frac {80}{243 (2 \sin (x)-1)}+\frac {1}{4 (\sin (x)-1)^2}+\frac {1}{2916 (\sin (x)+1)^2}-\frac {64}{729 (2 \sin (x)-1)^2}-\frac {16}{243 (2 \sin (x)-1)^3}\right )d\sin (x)\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{(1-2 \sin (x))^3 \left (1-\sin ^2(x)\right )^2 \left (-8 \sin ^3(x)+6 \sin (x)+1\right )^3}d\sin (x)\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {32 \left (16 \sin ^2(x)+15 \sin (x)+2\right )}{9 \left (8 \sin ^3(x)-6 \sin (x)-1\right )^3}+\frac {64 \left (52 \sin ^2(x)+51 \sin (x)+11\right )}{27 \left (8 \sin ^3(x)-6 \sin (x)-1\right )}+\frac {16 \left (140 \sin ^2(x)+138 \sin (x)+31\right )}{27 \left (8 \sin ^3(x)-6 \sin (x)-1\right )^2}-\frac {61}{4 (\sin (x)-1)}+\frac {7}{972 (\sin (x)+1)}-\frac {80}{243 (2 \sin (x)-1)}+\frac {1}{4 (\sin (x)-1)^2}+\frac {1}{2916 (\sin (x)+1)^2}-\frac {64}{729 (2 \sin (x)-1)^2}-\frac {16}{243 (2 \sin (x)-1)^3}\right )d\sin (x)\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {1}{(1-2 \sin (x))^3 \left (1-\sin ^2(x)\right )^2 \left (-8 \sin ^3(x)+6 \sin (x)+1\right )^3}d\sin (x)\) |
Input:
Int[(Cos[5*x] + Sin[4*x])^(-3),x]
Output:
$Aborted
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]
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]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 15.67 (sec) , antiderivative size = 2, normalized size of antiderivative = 0.02
| method | result | size |
| parallelrisch | \(0\) | \(2\) |
| default | \(-\frac {1}{2916 \left (1+\sin \left (x \right )\right )}+\frac {7 \ln \left (1+\sin \left (x \right )\right )}{972}+\frac {-\frac {3328 \sin \left (x \right )^{5}}{81}-\frac {3200 \sin \left (x \right )^{4}}{81}+\frac {640 \sin \left (x \right )^{3}}{27}+\frac {2672 \sin \left (x \right )^{2}}{81}+\frac {688 \sin \left (x \right )}{81}+\frac {16}{27}}{\left (8 \sin \left (x \right )^{3}-6 \sin \left (x \right )-1\right )^{2}}+\frac {16 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (8 \textit {\_Z}^{3}-6 \textit {\_Z} -1\right )}{\sum }\frac {\left (312 \textit {\_R}^{2}+293 \textit {\_R} +41\right ) \ln \left (\sin \left (x \right )-\textit {\_R} \right )}{4 \textit {\_R}^{2}-1}\right )}{243}-\frac {1}{4 \left (\sin \left (x \right )-1\right )}-\frac {61 \ln \left (\sin \left (x \right )-1\right )}{4}+\frac {4}{243 \left (2 \sin \left (x \right )-1\right )^{2}}+\frac {32}{729 \left (2 \sin \left (x \right )-1\right )}-\frac {40 \ln \left (2 \sin \left (x \right )-1\right )}{243}\) | \(150\) |
| risch | \(-\frac {i \left (-2 i {\mathrm e}^{18 i x}+47 \,{\mathrm e}^{19 i x}-2 i {\mathrm e}^{2 i x}+4 \,{\mathrm e}^{17 i x}+88 i {\mathrm e}^{10 i x}-8 \,{\mathrm e}^{15 i x}-14 i {\mathrm e}^{14 i x}+12 \,{\mathrm e}^{13 i x}+10 i {\mathrm e}^{12 i x}-8 \,{\mathrm e}^{11 i x}+10 i {\mathrm e}^{8 i x}+8 \,{\mathrm e}^{9 i x}+6 i {\mathrm e}^{4 i x}-12 \,{\mathrm e}^{7 i x}-14 i {\mathrm e}^{6 i x}+8 \,{\mathrm e}^{5 i x}+6 i {\mathrm e}^{16 i x}-4 \,{\mathrm e}^{3 i x}-47 \,{\mathrm e}^{i x}\right )}{27 \left ({\mathrm e}^{3 i x}-i\right )^{2} \left (-i {\mathrm e}^{6 i x}+{\mathrm e}^{7 i x}+i {\mathrm e}^{4 i x}+{\mathrm e}^{3 i x}+i-{\mathrm e}^{i x}\right )^{2}}+\frac {7 \ln \left ({\mathrm e}^{i x}+i\right )}{486}-\frac {61 \ln \left ({\mathrm e}^{i x}-i\right )}{2}+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (14348907 \textit {\_Z}^{3}-221079456 \textit {\_Z}^{2}+3872448 \textit {\_Z} +137728\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i x}+\left (\frac {3601989}{2496448} i \textit {\_R}^{2}-\frac {6990381}{312056} i \textit {\_R} +\frac {44290}{39007} i\right ) {\mathrm e}^{i x}-1\right )\right )-\frac {40 \ln \left (-i {\mathrm e}^{i x}+{\mathrm e}^{2 i x}-1\right )}{243}\) | \(284\) |
Input:
int(1/(cos(5*x)+sin(4*x))^3,x,method=_RETURNVERBOSE)
Output:
0
Result contains complex when optimal does not.
Time = 0.85 (sec) , antiderivative size = 1534, normalized size of antiderivative = 16.67 \[ \int \frac {1}{(\cos (5 x)+\sin (4 x))^3} \, dx=\text {Too large to display} \] Input:
integrate(1/(cos(5*x)+sin(4*x))^3,x, algorithm="fricas")
Output:
-1/236196*(1119744*cos(x)^8 - 3079296*cos(x)^6 + 3149280*cos(x)^4 + 2*(256 *cos(x)^10 - 704*cos(x)^8 + 688*cos(x)^6 - 280*cos(x)^4 + 41*cos(x)^2 + 8* (32*cos(x)^8 - 56*cos(x)^6 + 30*cos(x)^4 - 5*cos(x)^2)*sin(x))*(59049*(998 5792/14348907*I*sqrt(3) + 1933752064/14348907)^(1/3)*(-I*sqrt(3) + 1) + 15 52192*(I*sqrt(3) + 1)/(9985792/14348907*I*sqrt(3) + 1933752064/14348907)^( 1/3) - 606528)*log(61/236196*(59049*(9985792/14348907*I*sqrt(3) + 19337520 64/14348907)^(1/3)*(-I*sqrt(3) + 1) + 1552192*(I*sqrt(3) + 1)/(9985792/143 48907*I*sqrt(3) + 1933752064/14348907)^(1/3) - 606528)^2 + 27961524*(99857 92/14348907*I*sqrt(3) + 1933752064/14348907)^(1/3)*(-I*sqrt(3) + 1) + 5953 5876352/81*(I*sqrt(3) + 1)/(9985792/14348907*I*sqrt(3) + 1933752064/143489 07)^(1/3) + 4992896*sin(x) - 284375168) - 1329696*cos(x)^2 - (465813504*co s(x)^10 - 1280987136*cos(x)^8 + 1251873792*cos(x)^6 - 509483520*cos(x)^4 + (256*cos(x)^10 - 704*cos(x)^8 + 688*cos(x)^6 - 280*cos(x)^4 + 41*cos(x)^2 + 8*(32*cos(x)^8 - 56*cos(x)^6 + 30*cos(x)^4 - 5*cos(x)^2)*sin(x))*(59049 *(9985792/14348907*I*sqrt(3) + 1933752064/14348907)^(1/3)*(-I*sqrt(3) + 1) + 1552192*(I*sqrt(3) + 1)/(9985792/14348907*I*sqrt(3) + 1933752064/143489 07)^(1/3) - 606528) + 74602944*cos(x)^2 + 14556672*(32*cos(x)^8 - 56*cos(x )^6 + 30*cos(x)^4 - 5*cos(x)^2)*sin(x) + 1458*(256*cos(x)^10 - 704*cos(x)^ 8 + 688*cos(x)^6 - 280*cos(x)^4 + 41*cos(x)^2 + 8*(32*cos(x)^8 - 56*cos(x) ^6 + 30*cos(x)^4 - 5*cos(x)^2)*sin(x))*sqrt(-1/708588*(59049*(9985792/1...
\[ \int \frac {1}{(\cos (5 x)+\sin (4 x))^3} \, dx=\int \frac {1}{\left (\sin {\left (4 x \right )} + \cos {\left (5 x \right )}\right )^{3}}\, dx \] Input:
integrate(1/(cos(5*x)+sin(4*x))**3,x)
Output:
Integral((sin(4*x) + cos(5*x))**(-3), x)
Exception generated. \[ \int \frac {1}{(\cos (5 x)+\sin (4 x))^3} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(1/(cos(5*x)+sin(4*x))^3,x, algorithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is un defined.
Exception generated. \[ \int \frac {1}{(\cos (5 x)+\sin (4 x))^3} \, dx=\text {Exception raised: NotImplementedError} \] Input:
integrate(1/(cos(5*x)+sin(4*x))^3,x, algorithm="giac")
Output:
Exception raised: NotImplementedError >> unable to parse Giac output: 2*(( 11112*sin(sageVARx)^2-365*sin(sageVARx)-11476)*1/2916/(sin(sageVARx)^2-1)+ (-1066752*sin(sageVARx)^8+951040*sin(sageVARx)^7+1336768*sin(sageVARx)^6-1 185216*sin(sageVARx
Time = 23.56 (sec) , antiderivative size = 3808, normalized size of antiderivative = 41.39 \[ \int \frac {1}{(\cos (5 x)+\sin (4 x))^3} \, dx=\text {Too large to display} \] Input:
int(1/(cos(5*x) + sin(4*x))^3,x)
Output:
log((tan(x/2) + 1)^(7/486)) - (61*log(tan(x/2) - 1))/2 + log(1/(tan(x/2)^2 - 4*tan(x/2) + 1)^(40/243)) + symsum(log((140737488355328*(56773075803816 0310272*root(z^3 - (416*z^2)/27 + (5312*z)/19683 + 137728/14348907, z, k) - 7876471068796846080*cos(x) + 1404841151662063616*sin(x) - 90263756714586 2332416*root(z^3 - (416*z^2)/27 + (5312*z)/19683 + 137728/14348907, z, k)* cos(x) - 182569497549826129920*root(z^3 - (416*z^2)/27 + (5312*z)/19683 + 137728/14348907, z, k)*sin(x) - 90413312977791420051456*root(z^3 - (416*z^ 2)/27 + (5312*z)/19683 + 137728/14348907, z, k)^2 - 3076154487462175685162 496*root(z^3 - (416*z^2)/27 + (5312*z)/19683 + 137728/14348907, z, k)^3 + 113345302030483110355175652*root(z^3 - (416*z^2)/27 + (5312*z)/19683 + 137 728/14348907, z, k)^4 + 749811600252164300513936184*root(z^3 - (416*z^2)/2 7 + (5312*z)/19683 + 137728/14348907, z, k)^5 - 19040565612451199769099056 7*root(z^3 - (416*z^2)/27 + (5312*z)/19683 + 137728/14348907, z, k)^6 - 12 7547085639049919712*root(z^3 - (416*z^2)/27 + (5312*z)/19683 + 137728/1434 8907, z, k)^7 + 598994668650558152104380*root(z^3 - (416*z^2)/27 + (5312*z )/19683 + 137728/14348907, z, k)^8 + 79183039611028784108544*root(z^3 - (4 16*z^2)/27 + (5312*z)/19683 + 137728/14348907, z, k)^2*cos(x) + 4029718528 254535928065152*root(z^3 - (416*z^2)/27 + (5312*z)/19683 + 137728/14348907 , z, k)^3*cos(x) - 122884354394691867550879632*root(z^3 - (416*z^2)/27 + ( 5312*z)/19683 + 137728/14348907, z, k)^4*cos(x) - 841003008403170140972...
\[ \int \frac {1}{(\cos (5 x)+\sin (4 x))^3} \, dx=\int \frac {1}{\cos \left (5 x \right )^{3}+3 \cos \left (5 x \right )^{2} \sin \left (4 x \right )+3 \cos \left (5 x \right ) \sin \left (4 x \right )^{2}+\sin \left (4 x \right )^{3}}d x \] Input:
int(1/(cos(5*x)+sin(4*x))^3,x)
Output:
int(1/(cos(5*x)**3 + 3*cos(5*x)**2*sin(4*x) + 3*cos(5*x)*sin(4*x)**2 + sin (4*x)**3),x)