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

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

[Out]

Sin[x]^5/5 - Sin[x]^7/7

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Rubi [A]  time = 0.0366809, 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 \[ \frac{\sin ^5(x)}{5}-\frac{\sin ^7(x)}{7} \]

Antiderivative was successfully verified.

[In]  Int[Cos[x]^3*Sin[x]^4,x]

[Out]

Sin[x]^5/5 - Sin[x]^7/7

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Rubi in Sympy [A]  time = 2.34185, size = 12, normalized size = 0.71 \[ - \frac{\sin ^{7}{\left (x \right )}}{7} + \frac{\sin ^{5}{\left (x \right )}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sin(x)**7/7 + sin(x)**5/5

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Mathematica [A]  time = 0.0159662, size = 31, normalized size = 1.82 \[ \frac{3 \sin (x)}{64}-\frac{1}{64} \sin (3 x)-\frac{1}{320} \sin (5 x)+\frac{1}{448} \sin (7 x) \]

Antiderivative was successfully verified.

[In]  Integrate[Cos[x]^3*Sin[x]^4,x]

[Out]

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

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Maple [B]  time = 0.01, size = 30, normalized size = 1.8 \[ -{\frac{ \left ( \cos \left ( x \right ) \right ) ^{4} \left ( \sin \left ( x \right ) \right ) ^{3}}{7}}-{\frac{3\,\sin \left ( x \right ) \left ( \cos \left ( x \right ) \right ) ^{4}}{35}}+{\frac{ \left ( 2+ \left ( \cos \left ( x \right ) \right ) ^{2} \right ) \sin \left ( x \right ) }{35}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(cos(x)^3*sin(x)^4,x)

[Out]

-1/7*cos(x)^4*sin(x)^3-3/35*sin(x)*cos(x)^4+1/35*(2+cos(x)^2)*sin(x)

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Maxima [A]  time = 1.33705, size = 18, normalized size = 1.06 \[ -\frac{1}{7} \, \sin \left (x\right )^{7} + \frac{1}{5} \, \sin \left (x\right )^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/7*sin(x)^7 + 1/5*sin(x)^5

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Fricas [A]  time = 0.231487, size = 30, normalized size = 1.76 \[ \frac{1}{35} \,{\left (5 \, \cos \left (x\right )^{6} - 8 \, \cos \left (x\right )^{4} + \cos \left (x\right )^{2} + 2\right )} \sin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/35*(5*cos(x)^6 - 8*cos(x)^4 + cos(x)^2 + 2)*sin(x)

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Sympy [A]  time = 0.044802, size = 12, normalized size = 0.71 \[ - \frac{\sin ^{7}{\left (x \right )}}{7} + \frac{\sin ^{5}{\left (x \right )}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sin(x)**7/7 + sin(x)**5/5

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GIAC/XCAS [A]  time = 0.198859, size = 18, normalized size = 1.06 \[ -\frac{1}{7} \, \sin \left (x\right )^{7} + \frac{1}{5} \, \sin \left (x\right )^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/7*sin(x)^7 + 1/5*sin(x)^5