3.62 \(\int \cos ^4(x) \sin ^3(x) \, dx\)

Optimal. Leaf size=17 \[ \frac{\cos ^7(x)}{7}-\frac{\cos ^5(x)}{5} \]

[Out]

-Cos[x]^5/5 + Cos[x]^7/7

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Rubi [A]  time = 0.0360282, antiderivative size = 17, 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 \[ \frac{\cos ^7(x)}{7}-\frac{\cos ^5(x)}{5} \]

Antiderivative was successfully verified.

[In]  Int[Cos[x]^4*Sin[x]^3,x]

[Out]

-Cos[x]^5/5 + Cos[x]^7/7

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Rubi in Sympy [A]  time = 2.33001, size = 12, normalized size = 0.71 \[ \frac{\cos ^{7}{\left (x \right )}}{7} - \frac{\cos ^{5}{\left (x \right )}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(cos(x)**4*sin(x)**3,x)

[Out]

cos(x)**7/7 - cos(x)**5/5

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Mathematica [A]  time = 0.0157803, size = 31, normalized size = 1.82 \[ -\frac{3 \cos (x)}{64}-\frac{1}{64} \cos (3 x)+\frac{1}{320} \cos (5 x)+\frac{1}{448} \cos (7 x) \]

Antiderivative was successfully verified.

[In]  Integrate[Cos[x]^4*Sin[x]^3,x]

[Out]

(-3*Cos[x])/64 - Cos[3*x]/64 + Cos[5*x]/320 + Cos[7*x]/448

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Maple [A]  time = 0.01, size = 18, normalized size = 1.1 \[ -{\frac{ \left ( \cos \left ( x \right ) \right ) ^{5} \left ( \sin \left ( x \right ) \right ) ^{2}}{7}}-{\frac{2\, \left ( \cos \left ( x \right ) \right ) ^{5}}{35}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(cos(x)^4*sin(x)^3,x)

[Out]

-1/7*cos(x)^5*sin(x)^2-2/35*cos(x)^5

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Maxima [A]  time = 1.32518, size = 18, normalized size = 1.06 \[ \frac{1}{7} \, \cos \left (x\right )^{7} - \frac{1}{5} \, \cos \left (x\right )^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(x)^4*sin(x)^3,x, algorithm="maxima")

[Out]

1/7*cos(x)^7 - 1/5*cos(x)^5

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Fricas [A]  time = 0.221402, size = 18, normalized size = 1.06 \[ \frac{1}{7} \, \cos \left (x\right )^{7} - \frac{1}{5} \, \cos \left (x\right )^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(x)^4*sin(x)^3,x, algorithm="fricas")

[Out]

1/7*cos(x)^7 - 1/5*cos(x)^5

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Sympy [A]  time = 0.05273, size = 12, normalized size = 0.71 \[ \frac{\cos ^{7}{\left (x \right )}}{7} - \frac{\cos ^{5}{\left (x \right )}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(x)**4*sin(x)**3,x)

[Out]

cos(x)**7/7 - cos(x)**5/5

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GIAC/XCAS [A]  time = 0.201719, size = 18, normalized size = 1.06 \[ \frac{1}{7} \, \cos \left (x\right )^{7} - \frac{1}{5} \, \cos \left (x\right )^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(cos(x)^4*sin(x)^3,x, algorithm="giac")

[Out]

1/7*cos(x)^7 - 1/5*cos(x)^5