3.4.42 \(\int \cos ^2(x) \sin ^5(x) \, dx\) [342]

Optimal. Leaf size=25 \[ -\frac {1}{3} \cos ^3(x)+\frac {2 \cos ^5(x)}{5}-\frac {\cos ^7(x)}{7} \]

[Out]

-1/3*cos(x)^3+2/5*cos(x)^5-1/7*cos(x)^7

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Rubi [A]
time = 0.02, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2645, 276} \begin {gather*} -\frac {1}{7} \cos ^7(x)+\frac {2 \cos ^5(x)}{5}-\frac {\cos ^3(x)}{3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cos[x]^2*Sin[x]^5,x]

[Out]

-1/3*Cos[x]^3 + (2*Cos[x]^5)/5 - Cos[x]^7/7

Rule 276

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rule 2645

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

Rubi steps

\begin {align*} \int \cos ^2(x) \sin ^5(x) \, dx &=-\text {Subst}\left (\int x^2 \left (1-x^2\right )^2 \, dx,x,\cos (x)\right )\\ &=-\text {Subst}\left (\int \left (x^2-2 x^4+x^6\right ) \, dx,x,\cos (x)\right )\\ &=-\frac {1}{3} \cos ^3(x)+\frac {2 \cos ^5(x)}{5}-\frac {\cos ^7(x)}{7}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 31, normalized size = 1.24 \begin {gather*} -\frac {5 \cos (x)}{64}-\frac {1}{192} \cos (3 x)+\frac {3}{320} \cos (5 x)-\frac {1}{448} \cos (7 x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cos[x]^2*Sin[x]^5,x]

[Out]

(-5*Cos[x])/64 - Cos[3*x]/192 + (3*Cos[5*x])/320 - Cos[7*x]/448

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Mathics [A]
time = 1.80, size = 20, normalized size = 0.80 \begin {gather*} -\frac {\left (8+12 \text {Sin}\left [x\right ]^2+15 \text {Sin}\left [x\right ]^4\right ) \text {Cos}\left [x\right ]^3}{105} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[Cos[x]^2*Sin[x]^5,x]')

[Out]

-(8 + 12 Sin[x] ^ 2 + 15 Sin[x] ^ 4) Cos[x] ^ 3 / 105

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Maple [A]
time = 0.03, size = 28, normalized size = 1.12

method	result	size
risch	\(-\frac {5 \cos \left (x \right )}{64}-\frac {\cos \left (7 x \right )}{448}+\frac {3 \cos \left (5 x \right )}{320}-\frac {\cos \left (3 x \right )}{192}\)	\(24\)
default	\(-\frac {\left (\cos ^{3}\left (x \right )\right ) \left (\sin ^{4}\left (x \right )\right )}{7}-\frac {4 \left (\cos ^{3}\left (x \right )\right ) \left (\sin ^{2}\left (x \right )\right )}{35}-\frac {8 \left (\cos ^{3}\left (x \right )\right )}{105}\)	\(28\)
norman	\(\frac {16 \left (\tan ^{6}\left (\frac {x}{2}\right )\right )+\frac {16 \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{5}+\frac {16 \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{15}+\frac {32 \left (\tan ^{10}\left (\frac {x}{2}\right )\right )}{5}+\frac {32 \left (\tan ^{12}\left (\frac {x}{2}\right )\right )}{15}+\frac {32 \left (\tan ^{14}\left (\frac {x}{2}\right )\right )}{105}+\frac {16}{105}}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )^{7}}\)	\(62\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)^2*sin(x)^5,x,method=_RETURNVERBOSE)

[Out]

-1/7*cos(x)^3*sin(x)^4-4/35*cos(x)^3*sin(x)^2-8/105*cos(x)^3

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Maxima [A]
time = 0.27, size = 19, normalized size = 0.76 \begin {gather*} -\frac {1}{7} \, \cos \left (x\right )^{7} + \frac {2}{5} \, \cos \left (x\right )^{5} - \frac {1}{3} \, \cos \left (x\right )^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^2*sin(x)^5,x, algorithm="maxima")

[Out]

-1/7*cos(x)^7 + 2/5*cos(x)^5 - 1/3*cos(x)^3

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Fricas [A]
time = 0.37, size = 19, normalized size = 0.76 \begin {gather*} -\frac {1}{7} \, \cos \left (x\right )^{7} + \frac {2}{5} \, \cos \left (x\right )^{5} - \frac {1}{3} \, \cos \left (x\right )^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^2*sin(x)^5,x, algorithm="fricas")

[Out]

-1/7*cos(x)^7 + 2/5*cos(x)^5 - 1/3*cos(x)^3

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Sympy [A]
time = 0.03, size = 20, normalized size = 0.80 \begin {gather*} - \frac {\cos ^{7}{\left (x \right )}}{7} + \frac {2 \cos ^{5}{\left (x \right )}}{5} - \frac {\cos ^{3}{\left (x \right )}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)**2*sin(x)**5,x)

[Out]

-cos(x)**7/7 + 2*cos(x)**5/5 - cos(x)**3/3

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Giac [A]
time = 0.00, size = 25, normalized size = 1.00 \begin {gather*} -\frac {\cos ^{7}x}{7}+\frac {2}{5} \cos ^{5}x-\frac {\cos ^{3}x}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^2*sin(x)^5,x)

[Out]

-1/7*cos(x)^7 + 2/5*cos(x)^5 - 1/3*cos(x)^3

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Mupad [B]
time = 0.17, size = 19, normalized size = 0.76 \begin {gather*} -\frac {{\cos \left (x\right )}^7}{7}+\frac {2\,{\cos \left (x\right )}^5}{5}-\frac {{\cos \left (x\right )}^3}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)^2*sin(x)^5,x)

[Out]

(2*cos(x)^5)/5 - cos(x)^3/3 - cos(x)^7/7

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