3.1.65 \(\int \cos (x) (1+\sin ^3(x)) \, dx\) [65]

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

[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 = {3302} \begin {gather*} \frac {\sin ^4(x)}{4}+\sin (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cos[x]*(1 + Sin[x]^3),x]

[Out]

Sin[x] + Sin[x]^4/4

Rule 3302

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 &=\text {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 \begin {gather*} \sin (x)+\frac {\sin ^4(x)}{4} \end {gather*}

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 = 0.04, size = 10, normalized size = 0.91

method	result	size
derivativedivides	\(\sin \left (x \right )+\frac {\left (\sin ^{4}\left (x \right )\right )}{4}\)	\(10\)
default	\(\sin \left (x \right )+\frac {\left (\sin ^{4}\left (x \right )\right )}{4}\)	\(10\)
risch	\(\sin \left (x \right )+\frac {\cos \left (4 x \right )}{32}-\frac {\cos \left (2 x \right )}{8}\)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 1.30, size = 9, normalized size = 0.82 \begin {gather*} \frac {1}{4} \, \sin \left (x\right )^{4} + \sin \left (x\right ) \end {gather*}

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 = 1.02, size = 15, normalized size = 1.36 \begin {gather*} \frac {1}{4} \, \cos \left (x\right )^{4} - \frac {1}{2} \, \cos \left (x\right )^{2} + \sin \left (x\right ) \end {gather*}

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 = 0.13, size = 8, normalized size = 0.73 \begin {gather*} \frac {\sin ^{4}{\left (x \right )}}{4} + \sin {\left (x \right )} \end {gather*}

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 = 0.43, size = 9, normalized size = 0.82 \begin {gather*} \frac {1}{4} \, \sin \left (x\right )^{4} + \sin \left (x\right ) \end {gather*}

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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Mupad [B]
time = 0.04, size = 9, normalized size = 0.82 \begin {gather*} \frac {{\sin \left (x\right )}^4}{4}+\sin \left (x\right ) \end {gather*}

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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