3.759 \(\int (\cos ^{12}(x) \sin ^{10}(x)-\cos ^{10}(x) \sin ^{12}(x)) \, dx\)

Optimal. Leaf size=12 \[ \frac{1}{11} \sin ^{11}(x) \cos ^{11}(x) \]

[Out]

(Cos[x]^11*Sin[x]^11)/11

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Rubi [B]  time = 0.3238, antiderivative size = 129, normalized size of antiderivative = 10.75, number of steps used = 25, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {2568, 2635, 8} \[ -\frac{1}{22} \sin ^9(x) \cos ^{13}(x)-\frac{9}{440} \sin ^7(x) \cos ^{13}(x)-\frac{7}{880} \sin ^5(x) \cos ^{13}(x)-\frac{7 \sin ^3(x) \cos ^{13}(x)}{2816}+\frac{1}{22} \sin ^{11}(x) \cos ^{11}(x)+\frac{1}{40} \sin ^9(x) \cos ^{11}(x)+\frac{1}{80} \sin ^7(x) \cos ^{11}(x)+\frac{7 \sin ^5(x) \cos ^{11}(x)}{1280}+\frac{1}{512} \sin ^3(x) \cos ^{11}(x)-\frac{3 \sin (x) \cos ^{13}(x)}{5632}+\frac{3 \sin (x) \cos ^{11}(x)}{5632} \]

Antiderivative was successfully verified.

[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 2568

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 2635

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] && Integer
Q[2*n]

Rule 8

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

Rubi steps

\begin{align*} \int \left (\cos ^{12}(x) \sin ^{10}(x)-\cos ^{10}(x) \sin ^{12}(x)\right ) \, dx &=\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]  time = 0.0262099, size = 49, normalized size = 4.08 \[ \frac{21 \sin (2 x)}{1048576}-\frac{15 \sin (6 x)}{1048576}+\frac{15 \sin (10 x)}{2097152}-\frac{5 \sin (14 x)}{2097152}+\frac{\sin (18 x)}{2097152}-\frac{\sin (22 x)}{23068672} \]

Antiderivative was successfully verified.

[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

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Maple [B]  time = 0.081, size = 176, normalized size = 14.7 \begin{align*} -{\frac{ \left ( \cos \left ( x \right ) \right ) ^{13} \left ( \sin \left ( x \right ) \right ) ^{9}}{22}}-{\frac{9\, \left ( \sin \left ( x \right ) \right ) ^{7} \left ( \cos \left ( x \right ) \right ) ^{13}}{440}}-{\frac{7\, \left ( \sin \left ( x \right ) \right ) ^{5} \left ( \cos \left ( x \right ) \right ) ^{13}}{880}}-{\frac{7\, \left ( \sin \left ( x \right ) \right ) ^{3} \left ( \cos \left ( x \right ) \right ) ^{13}}{2816}}-{\frac{3\,\sin \left ( x \right ) \left ( \cos \left ( x \right ) \right ) ^{13}}{5632}}+{\frac{\sin \left ( x \right ) }{22528} \left ( \left ( \cos \left ( x \right ) \right ) ^{11}+{\frac{11\, \left ( \cos \left ( x \right ) \right ) ^{9}}{10}}+{\frac{99\, \left ( \cos \left ( x \right ) \right ) ^{7}}{80}}+{\frac{231\, \left ( \cos \left ( x \right ) \right ) ^{5}}{160}}+{\frac{231\, \left ( \cos \left ( x \right ) \right ) ^{3}}{128}}+{\frac{693\,\cos \left ( x \right ) }{256}} \right ) }+{\frac{ \left ( \cos \left ( x \right ) \right ) ^{11} \left ( \sin \left ( x \right ) \right ) ^{11}}{22}}+{\frac{ \left ( \sin \left ( x \right ) \right ) ^{9} \left ( \cos \left ( x \right ) \right ) ^{11}}{40}}+{\frac{ \left ( \sin \left ( x \right ) \right ) ^{7} \left ( \cos \left ( x \right ) \right ) ^{11}}{80}}+{\frac{7\, \left ( \sin \left ( x \right ) \right ) ^{5} \left ( \cos \left ( x \right ) \right ) ^{11}}{1280}}+{\frac{ \left ( \sin \left ( x \right ) \right ) ^{3} \left ( \cos \left ( x \right ) \right ) ^{11}}{512}}+{\frac{\sin \left ( x \right ) \left ( \cos \left ( x \right ) \right ) ^{11}}{2048}}-{\frac{\sin \left ( x \right ) }{20480} \left ( \left ( \cos \left ( x \right ) \right ) ^{9}+{\frac{9\, \left ( \cos \left ( x \right ) \right ) ^{7}}{8}}+{\frac{21\, \left ( \cos \left ( x \right ) \right ) ^{5}}{16}}+{\frac{105\, \left ( \cos \left ( x \right ) \right ) ^{3}}{64}}+{\frac{315\,\cos \left ( x \right ) }{128}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/22*cos(x)^13*sin(x)^9-9/440*sin(x)^7*cos(x)^13-7/880*sin(x)^5*cos(x)^13-7/2816*sin(x)^3*cos(x)^13-3/5632*si
n(x)*cos(x)^13+1/22528*(cos(x)^11+11/10*cos(x)^9+99/80*cos(x)^7+231/160*cos(x)^5+231/128*cos(x)^3+693/256*cos(
x))*sin(x)+1/22*cos(x)^11*sin(x)^11+1/40*sin(x)^9*cos(x)^11+1/80*sin(x)^7*cos(x)^11+7/1280*sin(x)^5*cos(x)^11+
1/512*sin(x)^3*cos(x)^11+1/2048*sin(x)*cos(x)^11-1/20480*(cos(x)^9+9/8*cos(x)^7+21/16*cos(x)^5+105/64*cos(x)^3
+315/128*cos(x))*sin(x)

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Maxima [A]  time = 0.983177, size = 11, normalized size = 0.92 \begin{align*} \frac{1}{22528} \, \sin \left (2 \, x\right )^{11} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.45203, size = 130, normalized size = 10.83 \begin{align*} -\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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [B]  time = 0.089651, size = 236, normalized size = 19.67 \begin{align*} - \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Giac [B]  time = 1.11688, size = 50, normalized size = 4.17 \begin{align*} -\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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)