3.345 \(\int \cos ^6(x) \sin ^7(x) \, dx\)

Optimal. Leaf size=33 \[ \frac{\cos ^{13}(x)}{13}-\frac{3 \cos ^{11}(x)}{11}+\frac{\cos ^9(x)}{3}-\frac{\cos ^7(x)}{7} \]

[Out]

-Cos[x]^7/7 + Cos[x]^9/3 - (3*Cos[x]^11)/11 + Cos[x]^13/13

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Rubi [A]  time = 0.0555727, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{\cos ^{13}(x)}{13}-\frac{3 \cos ^{11}(x)}{11}+\frac{\cos ^9(x)}{3}-\frac{\cos ^7(x)}{7} \]

Antiderivative was successfully verified.

[In]  Int[Cos[x]^6*Sin[x]^7,x]

[Out]

-Cos[x]^7/7 + Cos[x]^9/3 - (3*Cos[x]^11)/11 + Cos[x]^13/13

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Rubi in Sympy [A]  time = 3.256, size = 27, normalized size = 0.82 \[ \frac{\cos ^{13}{\left (x \right )}}{13} - \frac{3 \cos ^{11}{\left (x \right )}}{11} + \frac{\cos ^{9}{\left (x \right )}}{3} - \frac{\cos ^{7}{\left (x \right )}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(cos(x)**6*sin(x)**7,x)

[Out]

cos(x)**13/13 - 3*cos(x)**11/11 + cos(x)**9/3 - cos(x)**7/7

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Mathematica [A]  time = 0.034187, size = 55, normalized size = 1.67 \[ -\frac{5 \cos (x)}{1024}-\frac{5 \cos (3 x)}{4096}+\frac{3 \cos (5 x)}{4096}+\frac{3 \cos (7 x)}{14336}-\frac{\cos (9 x)}{6144}-\frac{\cos (11 x)}{45056}+\frac{\cos (13 x)}{53248} \]

Antiderivative was successfully verified.

[In]  Integrate[Cos[x]^6*Sin[x]^7,x]

[Out]

(-5*Cos[x])/1024 - (5*Cos[3*x])/4096 + (3*Cos[5*x])/4096 + (3*Cos[7*x])/14336 -
Cos[9*x]/6144 - Cos[11*x]/45056 + Cos[13*x]/53248

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Maple [A]  time = 0.012, size = 38, normalized size = 1.2 \[ -{\frac{ \left ( \cos \left ( x \right ) \right ) ^{7} \left ( \sin \left ( x \right ) \right ) ^{6}}{13}}-{\frac{6\, \left ( \sin \left ( x \right ) \right ) ^{4} \left ( \cos \left ( x \right ) \right ) ^{7}}{143}}-{\frac{8\, \left ( \sin \left ( x \right ) \right ) ^{2} \left ( \cos \left ( x \right ) \right ) ^{7}}{429}}-{\frac{16\, \left ( \cos \left ( x \right ) \right ) ^{7}}{3003}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(cos(x)^6*sin(x)^7,x)

[Out]

-1/13*cos(x)^7*sin(x)^6-6/143*sin(x)^4*cos(x)^7-8/429*sin(x)^2*cos(x)^7-16/3003*
cos(x)^7

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Maxima [A]  time = 1.42244, size = 34, normalized size = 1.03 \[ \frac{1}{13} \, \cos \left (x\right )^{13} - \frac{3}{11} \, \cos \left (x\right )^{11} + \frac{1}{3} \, \cos \left (x\right )^{9} - \frac{1}{7} \, \cos \left (x\right )^{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(x)^6*sin(x)^7,x, algorithm="maxima")

[Out]

1/13*cos(x)^13 - 3/11*cos(x)^11 + 1/3*cos(x)^9 - 1/7*cos(x)^7

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Fricas [A]  time = 0.234401, size = 34, normalized size = 1.03 \[ \frac{1}{13} \, \cos \left (x\right )^{13} - \frac{3}{11} \, \cos \left (x\right )^{11} + \frac{1}{3} \, \cos \left (x\right )^{9} - \frac{1}{7} \, \cos \left (x\right )^{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(x)^6*sin(x)^7,x, algorithm="fricas")

[Out]

1/13*cos(x)^13 - 3/11*cos(x)^11 + 1/3*cos(x)^9 - 1/7*cos(x)^7

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Sympy [A]  time = 0.052812, size = 27, normalized size = 0.82 \[ \frac{\cos ^{13}{\left (x \right )}}{13} - \frac{3 \cos ^{11}{\left (x \right )}}{11} + \frac{\cos ^{9}{\left (x \right )}}{3} - \frac{\cos ^{7}{\left (x \right )}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(x)**6*sin(x)**7,x)

[Out]

cos(x)**13/13 - 3*cos(x)**11/11 + cos(x)**9/3 - cos(x)**7/7

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GIAC/XCAS [A]  time = 0.202166, size = 34, normalized size = 1.03 \[ \frac{1}{13} \, \cos \left (x\right )^{13} - \frac{3}{11} \, \cos \left (x\right )^{11} + \frac{1}{3} \, \cos \left (x\right )^{9} - \frac{1}{7} \, \cos \left (x\right )^{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(x)^6*sin(x)^7,x, algorithm="giac")

[Out]

1/13*cos(x)^13 - 3/11*cos(x)^11 + 1/3*cos(x)^9 - 1/7*cos(x)^7