3.65 \(\int \cos (x) (1+\sin ^3(x)) \, dx\)

Optimal. Leaf size=11 \[ \frac {\sin ^4(x)}{4}+\sin (x) \]

[Out]

sin(x)+1/4*sin(x)^4

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Rubi [A]  time = 0.01, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {3223} \[ \frac {\sin ^4(x)}{4}+\sin (x) \]

Antiderivative was successfully verified.

[In]

Int[Cos[x]*(1 + Sin[x]^3),x]

[Out]

Sin[x] + Sin[x]^4/4

Rule 3223

Int[cos[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*((c_.)*sin[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol] :> With
[{ff = FreeFactors[Sin[e + f*x], x]}, Dist[ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b*(c*ff*x)^n)^p, x]
, x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && IntegerQ[(m - 1)/2] && (EqQ[n, 4] || GtQ[m, 0
] || IGtQ[p, 0] || IntegersQ[m, p])

Rubi steps

\begin {align*} \int \cos (x) \left (1+\sin ^3(x)\right ) \, dx &=\operatorname {Subst}\left (\int \left (1+x^3\right ) \, dx,x,\sin (x)\right )\\ &=\sin (x)+\frac {\sin ^4(x)}{4}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 11, normalized size = 1.00 \[ \frac {\sin ^4(x)}{4}+\sin (x) \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[x]*(1 + Sin[x]^3),x]

[Out]

Sin[x] + Sin[x]^4/4

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fricas [A]  time = 0.45, size = 15, normalized size = 1.36 \[ \frac {1}{4} \, \cos \relax (x)^{4} - \frac {1}{2} \, \cos \relax (x)^{2} + \sin \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*(1+sin(x)^3),x, algorithm="fricas")

[Out]

1/4*cos(x)^4 - 1/2*cos(x)^2 + sin(x)

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giac [A]  time = 0.96, size = 9, normalized size = 0.82 \[ \frac {1}{4} \, \sin \relax (x)^{4} + \sin \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*(1+sin(x)^3),x, algorithm="giac")

[Out]

1/4*sin(x)^4 + sin(x)

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maple [A]  time = 0.03, size = 10, normalized size = 0.91 \[ \frac {\left (\sin ^{4}\relax (x )\right )}{4}+\sin \relax (x ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)*(1+sin(x)^3),x)

[Out]

sin(x)+1/4*sin(x)^4

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maxima [A]  time = 0.53, size = 9, normalized size = 0.82 \[ \frac {1}{4} \, \sin \relax (x)^{4} + \sin \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*(1+sin(x)^3),x, algorithm="maxima")

[Out]

1/4*sin(x)^4 + sin(x)

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mupad [B]  time = 0.04, size = 9, normalized size = 0.82 \[ \frac {{\sin \relax (x)}^4}{4}+\sin \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)*(sin(x)^3 + 1),x)

[Out]

sin(x) + sin(x)^4/4

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sympy [A]  time = 0.57, size = 8, normalized size = 0.73 \[ \frac {\sin ^{4}{\relax (x )}}{4} + \sin {\relax (x )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*(1+sin(x)**3),x)

[Out]

sin(x)**4/4 + sin(x)

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