\(\int (\cos ^{12}(x) \sin ^{10}(x)-\cos ^{10}(x) \sin ^{12}(x)) \, dx\) [759]

Optimal result

Integrand size = 20, antiderivative size = 12 \[ \int \left (\cos ^{12}(x) \sin ^{10}(x)-\cos ^{10}(x) \sin ^{12}(x)\right ) \, dx=\frac {1}{11} \cos ^{11}(x) \sin ^{11}(x) \]

[Out]

1/11*cos(x)^11*sin(x)^11

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(129\) vs. \(2(12)=24\).

Time = 0.41 (sec) , antiderivative size = 129, normalized size of antiderivative = 10.75, number of steps used = 25, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2648, 2715, 8} \[ \int \left (\cos ^{12}(x) \sin ^{10}(x)-\cos ^{10}(x) \sin ^{12}(x)\right ) \, dx=-\frac {1}{22} \sin ^9(x) \cos ^{13}(x)-\frac {9}{440} \sin ^7(x) \cos ^{13}(x)-\frac {7}{880} \sin ^5(x) \cos ^{13}(x)-\frac {7 \sin ^3(x) \cos ^{13}(x)}{2816}-\frac {3 \sin (x) \cos ^{13}(x)}{5632}+\frac {1}{22} \sin ^{11}(x) \cos ^{11}(x)+\frac {1}{40} \sin ^9(x) \cos ^{11}(x)+\frac {1}{80} \sin ^7(x) \cos ^{11}(x)+\frac {7 \sin ^5(x) \cos ^{11}(x)}{1280}+\frac {1}{512} \sin ^3(x) \cos ^{11}(x)+\frac {3 \sin (x) \cos ^{11}(x)}{5632} \]

[In]

Int[Cos[x]^12*Sin[x]^10 - Cos[x]^10*Sin[x]^12,x]

[Out]

(3*Cos[x]^11*Sin[x])/5632 - (3*Cos[x]^13*Sin[x])/5632 + (Cos[x]^11*Sin[x]^3)/512 - (7*Cos[x]^13*Sin[x]^3)/2816
+ (7*Cos[x]^11*Sin[x]^5)/1280 - (7*Cos[x]^13*Sin[x]^5)/880 + (Cos[x]^11*Sin[x]^7)/80 - (9*Cos[x]^13*Sin[x]^7)
/440 + (Cos[x]^11*Sin[x]^9)/40 - (Cos[x]^13*Sin[x]^9)/22 + (Cos[x]^11*Sin[x]^11)/22

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2648

Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(-a)*(b*Cos[e
+ f*x])^(n + 1)*((a*Sin[e + f*x])^(m - 1)/(b*f*(m + n))), x] + Dist[a^2*((m - 1)/(m + n)), Int[(b*Cos[e + f*x
])^n*(a*Sin[e + f*x])^(m - 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] && NeQ[m + n, 0] && IntegersQ[
2*m, 2*n]

Rule 2715

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Sin[c + d*x])^(n - 1)/(d*n))
, x] + Dist[b^2*((n - 1)/n), Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integ
erQ[2*n]

Rubi steps \begin{align*} \text {integral}& = \int \cos ^{12}(x) \sin ^{10}(x) \, dx-\int \cos ^{10}(x) \sin ^{12}(x) \, dx \\ & = -\frac {1}{22} \cos ^{13}(x) \sin ^9(x)+\frac {1}{22} \cos ^{11}(x) \sin ^{11}(x)+\frac {9}{22} \int \cos ^{12}(x) \sin ^8(x) \, dx-\frac {1}{2} \int \cos ^{10}(x) \sin ^{10}(x) \, dx \\ & = -\frac {9}{440} \cos ^{13}(x) \sin ^7(x)+\frac {1}{40} \cos ^{11}(x) \sin ^9(x)-\frac {1}{22} \cos ^{13}(x) \sin ^9(x)+\frac {1}{22} \cos ^{11}(x) \sin ^{11}(x)+\frac {63}{440} \int \cos ^{12}(x) \sin ^6(x) \, dx-\frac {9}{40} \int \cos ^{10}(x) \sin ^8(x) \, dx \\ & = -\frac {7}{880} \cos ^{13}(x) \sin ^5(x)+\frac {1}{80} \cos ^{11}(x) \sin ^7(x)-\frac {9}{440} \cos ^{13}(x) \sin ^7(x)+\frac {1}{40} \cos ^{11}(x) \sin ^9(x)-\frac {1}{22} \cos ^{13}(x) \sin ^9(x)+\frac {1}{22} \cos ^{11}(x) \sin ^{11}(x)+\frac {7}{176} \int \cos ^{12}(x) \sin ^4(x) \, dx-\frac {7}{80} \int \cos ^{10}(x) \sin ^6(x) \, dx \\ & = -\frac {7 \cos ^{13}(x) \sin ^3(x)}{2816}+\frac {7 \cos ^{11}(x) \sin ^5(x)}{1280}-\frac {7}{880} \cos ^{13}(x) \sin ^5(x)+\frac {1}{80} \cos ^{11}(x) \sin ^7(x)-\frac {9}{440} \cos ^{13}(x) \sin ^7(x)+\frac {1}{40} \cos ^{11}(x) \sin ^9(x)-\frac {1}{22} \cos ^{13}(x) \sin ^9(x)+\frac {1}{22} \cos ^{11}(x) \sin ^{11}(x)+\frac {21 \int \cos ^{12}(x) \sin ^2(x) \, dx}{2816}-\frac {7}{256} \int \cos ^{10}(x) \sin ^4(x) \, dx \\ & = -\frac {3 \cos ^{13}(x) \sin (x)}{5632}+\frac {1}{512} \cos ^{11}(x) \sin ^3(x)-\frac {7 \cos ^{13}(x) \sin ^3(x)}{2816}+\frac {7 \cos ^{11}(x) \sin ^5(x)}{1280}-\frac {7}{880} \cos ^{13}(x) \sin ^5(x)+\frac {1}{80} \cos ^{11}(x) \sin ^7(x)-\frac {9}{440} \cos ^{13}(x) \sin ^7(x)+\frac {1}{40} \cos ^{11}(x) \sin ^9(x)-\frac {1}{22} \cos ^{13}(x) \sin ^9(x)+\frac {1}{22} \cos ^{11}(x) \sin ^{11}(x)+\frac {3 \int \cos ^{12}(x) \, dx}{5632}-\frac {3}{512} \int \cos ^{10}(x) \sin ^2(x) \, dx \\ & = \frac {3 \cos ^{11}(x) \sin (x)}{5632}-\frac {3 \cos ^{13}(x) \sin (x)}{5632}+\frac {1}{512} \cos ^{11}(x) \sin ^3(x)-\frac {7 \cos ^{13}(x) \sin ^3(x)}{2816}+\frac {7 \cos ^{11}(x) \sin ^5(x)}{1280}-\frac {7}{880} \cos ^{13}(x) \sin ^5(x)+\frac {1}{80} \cos ^{11}(x) \sin ^7(x)-\frac {9}{440} \cos ^{13}(x) \sin ^7(x)+\frac {1}{40} \cos ^{11}(x) \sin ^9(x)-\frac {1}{22} \cos ^{13}(x) \sin ^9(x)+\frac {1}{22} \cos ^{11}(x) \sin ^{11}(x) \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(49\) vs. \(2(12)=24\).

Time = 0.06 (sec) , antiderivative size = 49, normalized size of antiderivative = 4.08 \[ \int \left (\cos ^{12}(x) \sin ^{10}(x)-\cos ^{10}(x) \sin ^{12}(x)\right ) \, dx=\frac {21 \sin (2 x)}{1048576}-\frac {15 \sin (6 x)}{1048576}+\frac {15 \sin (10 x)}{2097152}-\frac {5 \sin (14 x)}{2097152}+\frac {\sin (18 x)}{2097152}-\frac {\sin (22 x)}{23068672} \]

[In]

Integrate[Cos[x]^12*Sin[x]^10 - Cos[x]^10*Sin[x]^12,x]

[Out]

(21*Sin[2*x])/1048576 - (15*Sin[6*x])/1048576 + (15*Sin[10*x])/2097152 - (5*Sin[14*x])/2097152 + Sin[18*x]/209
7152 - Sin[22*x]/23068672

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(37\) vs. \(2(10)=20\).

Time = 2.08 (sec) , antiderivative size = 38, normalized size of antiderivative = 3.17

method	result	size
risch	\(-\frac {\sin \left (22 x \right )}{23068672}+\frac {\sin \left (18 x \right )}{2097152}-\frac {5 \sin \left (14 x \right )}{2097152}+\frac {15 \sin \left (10 x \right )}{2097152}-\frac {15 \sin \left (6 x \right )}{1048576}+\frac {21 \sin \left (2 x \right )}{1048576}\)	\(38\)
parallelrisch	\(-\frac {5 \left (\frac {\sin \left (11 x \right )}{55}-\frac {\sin \left (9 x \right )}{5}+\sin \left (7 x \right )-3 \sin \left (5 x \right )+6 \sin \left (3 x \right )-\frac {42 \sin \left (x \right )}{5}\right ) \left (\cos \left (11 x \right )+11 \cos \left (9 x \right )+55 \cos \left (7 x \right )+165 \cos \left (5 x \right )+330 \cos \left (3 x \right )+462 \cos \left (x \right )\right )}{1048576}\)	\(69\)
default	\(-\frac {\cos \left (x \right )^{13} \sin \left (x \right )^{9}}{22}-\frac {9 \sin \left (x \right )^{7} \cos \left (x \right )^{13}}{440}-\frac {7 \sin \left (x \right )^{5} \cos \left (x \right )^{13}}{880}-\frac {7 \sin \left (x \right )^{3} \cos \left (x \right )^{13}}{2816}-\frac {3 \sin \left (x \right ) \cos \left (x \right )^{13}}{5632}+\frac {\left (\cos \left (x \right )^{11}+\frac {11 \cos \left (x \right )^{9}}{10}+\frac {99 \cos \left (x \right )^{7}}{80}+\frac {231 \cos \left (x \right )^{5}}{160}+\frac {231 \cos \left (x \right )^{3}}{128}+\frac {693 \cos \left (x \right )}{256}\right ) \sin \left (x \right )}{22528}+\frac {\cos \left (x \right )^{11} \sin \left (x \right )^{11}}{22}+\frac {\sin \left (x \right )^{9} \cos \left (x \right )^{11}}{40}+\frac {\sin \left (x \right )^{7} \cos \left (x \right )^{11}}{80}+\frac {7 \sin \left (x \right )^{5} \cos \left (x \right )^{11}}{1280}+\frac {\sin \left (x \right )^{3} \cos \left (x \right )^{11}}{512}+\frac {\sin \left (x \right ) \cos \left (x \right )^{11}}{2048}-\frac {\left (\cos \left (x \right )^{9}+\frac {9 \cos \left (x \right )^{7}}{8}+\frac {21 \cos \left (x \right )^{5}}{16}+\frac {105 \cos \left (x \right )^{3}}{64}+\frac {315 \cos \left (x \right )}{128}\right ) \sin \left (x \right )}{20480}\)	\(176\)
parts	\(-\frac {\cos \left (x \right )^{13} \sin \left (x \right )^{9}}{22}-\frac {9 \sin \left (x \right )^{7} \cos \left (x \right )^{13}}{440}-\frac {7 \sin \left (x \right )^{5} \cos \left (x \right )^{13}}{880}-\frac {7 \sin \left (x \right )^{3} \cos \left (x \right )^{13}}{2816}-\frac {3 \sin \left (x \right ) \cos \left (x \right )^{13}}{5632}+\frac {\left (\cos \left (x \right )^{11}+\frac {11 \cos \left (x \right )^{9}}{10}+\frac {99 \cos \left (x \right )^{7}}{80}+\frac {231 \cos \left (x \right )^{5}}{160}+\frac {231 \cos \left (x \right )^{3}}{128}+\frac {693 \cos \left (x \right )}{256}\right ) \sin \left (x \right )}{22528}+\frac {\cos \left (x \right )^{11} \sin \left (x \right )^{11}}{22}+\frac {\sin \left (x \right )^{9} \cos \left (x \right )^{11}}{40}+\frac {\sin \left (x \right )^{7} \cos \left (x \right )^{11}}{80}+\frac {7 \sin \left (x \right )^{5} \cos \left (x \right )^{11}}{1280}+\frac {\sin \left (x \right )^{3} \cos \left (x \right )^{11}}{512}+\frac {\sin \left (x \right ) \cos \left (x \right )^{11}}{2048}-\frac {\left (\cos \left (x \right )^{9}+\frac {9 \cos \left (x \right )^{7}}{8}+\frac {21 \cos \left (x \right )^{5}}{16}+\frac {105 \cos \left (x \right )^{3}}{64}+\frac {315 \cos \left (x \right )}{128}\right ) \sin \left (x \right )}{20480}\)	\(176\)

[In]

int(cos(x)^12*sin(x)^10-cos(x)^10*sin(x)^12,x,method=_RETURNVERBOSE)

[Out]

-1/23068672*sin(22*x)+1/2097152*sin(18*x)-5/2097152*sin(14*x)+15/2097152*sin(10*x)-15/1048576*sin(6*x)+21/1048
576*sin(2*x)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 39 vs. \(2 (10) = 20\).

Time = 0.29 (sec) , antiderivative size = 39, normalized size of antiderivative = 3.25 \[ \int \left (\cos ^{12}(x) \sin ^{10}(x)-\cos ^{10}(x) \sin ^{12}(x)\right ) \, dx=-\frac {1}{11} \, {\left (\cos \left (x\right )^{21} - 5 \, \cos \left (x\right )^{19} + 10 \, \cos \left (x\right )^{17} - 10 \, \cos \left (x\right )^{15} + 5 \, \cos \left (x\right )^{13} - \cos \left (x\right )^{11}\right )} \sin \left (x\right ) \]

[In]

integrate(cos(x)^12*sin(x)^10-cos(x)^10*sin(x)^12,x, algorithm="fricas")

[Out]

-1/11*(cos(x)^21 - 5*cos(x)^19 + 10*cos(x)^17 - 10*cos(x)^15 + 5*cos(x)^13 - cos(x)^11)*sin(x)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 236 vs. \(2 (10) = 20\).

Time = 0.03 (sec) , antiderivative size = 236, normalized size of antiderivative = 19.67 \[ \int \left (\cos ^{12}(x) \sin ^{10}(x)-\cos ^{10}(x) \sin ^{12}(x)\right ) \, dx=- \frac {\sin ^{21}{\left (x \right )} \cos {\left (x \right )}}{22} + \frac {89 \sin ^{19}{\left (x \right )} \cos {\left (x \right )}}{440} - \frac {301 \sin ^{17}{\left (x \right )} \cos {\left (x \right )}}{880} + \frac {3683 \sin ^{15}{\left (x \right )} \cos {\left (x \right )}}{14080} - \frac {433 \sin ^{13}{\left (x \right )} \cos {\left (x \right )}}{5632} + \frac {\sin ^{11}{\left (x \right )} \cos {\left (x \right )}}{22528} + \frac {\sin ^{9}{\left (x \right )} \cos {\left (x \right )}}{20480} + \frac {9 \sin ^{7}{\left (x \right )} \cos {\left (x \right )}}{163840} + \frac {21 \sin ^{5}{\left (x \right )} \cos {\left (x \right )}}{327680} + \frac {21 \sin ^{3}{\left (x \right )} \cos {\left (x \right )}}{262144} - \frac {\sin {\left (x \right )} \cos ^{21}{\left (x \right )}}{22} + \frac {89 \sin {\left (x \right )} \cos ^{19}{\left (x \right )}}{440} - \frac {301 \sin {\left (x \right )} \cos ^{17}{\left (x \right )}}{880} + \frac {3683 \sin {\left (x \right )} \cos ^{15}{\left (x \right )}}{14080} - \frac {433 \sin {\left (x \right )} \cos ^{13}{\left (x \right )}}{5632} + \frac {\sin {\left (x \right )} \cos ^{11}{\left (x \right )}}{22528} + \frac {\sin {\left (x \right )} \cos ^{9}{\left (x \right )}}{20480} + \frac {9 \sin {\left (x \right )} \cos ^{7}{\left (x \right )}}{163840} + \frac {21 \sin {\left (x \right )} \cos ^{5}{\left (x \right )}}{327680} + \frac {21 \sin {\left (x \right )} \cos ^{3}{\left (x \right )}}{262144} + \frac {63 \sin {\left (x \right )} \cos {\left (x \right )}}{262144} \]

[In]

integrate(cos(x)**12*sin(x)**10-cos(x)**10*sin(x)**12,x)

[Out]

-sin(x)**21*cos(x)/22 + 89*sin(x)**19*cos(x)/440 - 301*sin(x)**17*cos(x)/880 + 3683*sin(x)**15*cos(x)/14080 -
433*sin(x)**13*cos(x)/5632 + sin(x)**11*cos(x)/22528 + sin(x)**9*cos(x)/20480 + 9*sin(x)**7*cos(x)/163840 + 21
*sin(x)**5*cos(x)/327680 + 21*sin(x)**3*cos(x)/262144 - sin(x)*cos(x)**21/22 + 89*sin(x)*cos(x)**19/440 - 301*
sin(x)*cos(x)**17/880 + 3683*sin(x)*cos(x)**15/14080 - 433*sin(x)*cos(x)**13/5632 + sin(x)*cos(x)**11/22528 +
sin(x)*cos(x)**9/20480 + 9*sin(x)*cos(x)**7/163840 + 21*sin(x)*cos(x)**5/327680 + 21*sin(x)*cos(x)**3/262144 +
63*sin(x)*cos(x)/262144

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.67 \[ \int \left (\cos ^{12}(x) \sin ^{10}(x)-\cos ^{10}(x) \sin ^{12}(x)\right ) \, dx=\frac {1}{22528} \, \sin \left (2 \, x\right )^{11} \]

[In]

integrate(cos(x)^12*sin(x)^10-cos(x)^10*sin(x)^12,x, algorithm="maxima")

[Out]

1/22528*sin(2*x)^11

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 37 vs. \(2 (10) = 20\).

Time = 0.28 (sec) , antiderivative size = 37, normalized size of antiderivative = 3.08 \[ \int \left (\cos ^{12}(x) \sin ^{10}(x)-\cos ^{10}(x) \sin ^{12}(x)\right ) \, dx=-\frac {1}{23068672} \, \sin \left (22 \, x\right ) + \frac {1}{2097152} \, \sin \left (18 \, x\right ) - \frac {5}{2097152} \, \sin \left (14 \, x\right ) + \frac {15}{2097152} \, \sin \left (10 \, x\right ) - \frac {15}{1048576} \, \sin \left (6 \, x\right ) + \frac {21}{1048576} \, \sin \left (2 \, x\right ) \]

[In]

integrate(cos(x)^12*sin(x)^10-cos(x)^10*sin(x)^12,x, algorithm="giac")

[Out]

-1/23068672*sin(22*x) + 1/2097152*sin(18*x) - 5/2097152*sin(14*x) + 15/2097152*sin(10*x) - 15/1048576*sin(6*x)
+ 21/1048576*sin(2*x)

Mupad [B] (verification not implemented)

Time = 26.58 (sec) , antiderivative size = 49, normalized size of antiderivative = 4.08 \[ \int \left (\cos ^{12}(x) \sin ^{10}(x)-\cos ^{10}(x) \sin ^{12}(x)\right ) \, dx=-\frac {\sin \left (x\right )\,{\cos \left (x\right )}^{21}}{11}+\frac {5\,\sin \left (x\right )\,{\cos \left (x\right )}^{19}}{11}-\frac {10\,\sin \left (x\right )\,{\cos \left (x\right )}^{17}}{11}+\frac {10\,\sin \left (x\right )\,{\cos \left (x\right )}^{15}}{11}-\frac {5\,\sin \left (x\right )\,{\cos \left (x\right )}^{13}}{11}+\frac {\sin \left (x\right )\,{\cos \left (x\right )}^{11}}{11} \]

[In]

int(cos(x)^12*sin(x)^10 - cos(x)^10*sin(x)^12,x)

[Out]

(cos(x)^11*sin(x))/11 - (5*cos(x)^13*sin(x))/11 + (10*cos(x)^15*sin(x))/11 - (10*cos(x)^17*sin(x))/11 + (5*cos
(x)^19*sin(x))/11 - (cos(x)^21*sin(x))/11