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

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

[Out]

Sin[x] + Sin[x]^4/4

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Rubi [A]  time = 0.0120964, antiderivative size = 11, normalized size of antiderivative = 1., 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*}

Mathematica [A]  time = 0.0029038, size = 11, normalized size = 1. \[ \frac{\sin ^4(x)}{4}+\sin (x) \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[x]*(1 + Sin[x]^3),x]

[Out]

Sin[x] + Sin[x]^4/4

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Maple [A]  time = 0.013, size = 10, normalized size = 0.9 \begin{align*} \sin \left ( x \right ) +{\frac{ \left ( \sin \left ( x \right ) \right ) ^{4}}{4}} \end{align*}

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 = 0.916528, size = 12, normalized size = 1.09 \begin{align*} \frac{1}{4} \, \sin \left (x\right )^{4} + \sin \left (x\right ) \end{align*}

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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Fricas [A]  time = 2.01435, size = 51, normalized size = 4.64 \begin{align*} \frac{1}{4} \, \cos \left (x\right )^{4} - \frac{1}{2} \, \cos \left (x\right )^{2} + \sin \left (x\right ) \end{align*}

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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Sympy [A]  time = 0.536143, size = 8, normalized size = 0.73 \begin{align*} \frac{\sin ^{4}{\left (x \right )}}{4} + \sin{\left (x \right )} \end{align*}

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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Giac [A]  time = 1.04765, size = 12, normalized size = 1.09 \begin{align*} \frac{1}{4} \, \sin \left (x\right )^{4} + \sin \left (x\right ) \end{align*}

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)