3.1 \(\int x^3 \cos (a+b x^2) \, dx\)

Optimal. Leaf size=34 \[ \frac {\cos \left (a+b x^2\right )}{2 b^2}+\frac {x^2 \sin \left (a+b x^2\right )}{2 b} \]

[Out]

1/2*cos(b*x^2+a)/b^2+1/2*x^2*sin(b*x^2+a)/b

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Rubi [A]  time = 0.03, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3380, 3296, 2638} \[ \frac {\cos \left (a+b x^2\right )}{2 b^2}+\frac {x^2 \sin \left (a+b x^2\right )}{2 b} \]

Antiderivative was successfully verified.

[In]

Int[x^3*Cos[a + b*x^2],x]

[Out]

Cos[a + b*x^2]/(2*b^2) + (x^2*Sin[a + b*x^2])/(2*b)

Rule 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3296

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 3380

Int[((a_.) + Cos[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*Cos[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IntegerQ[Simpl
ify[(m + 1)/n]] && (EqQ[p, 1] || EqQ[m, n - 1] || (IntegerQ[p] && GtQ[Simplify[(m + 1)/n], 0]))

Rubi steps

\begin {align*} \int x^3 \cos \left (a+b x^2\right ) \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int x \cos (a+b x) \, dx,x,x^2\right )\\ &=\frac {x^2 \sin \left (a+b x^2\right )}{2 b}-\frac {\operatorname {Subst}\left (\int \sin (a+b x) \, dx,x,x^2\right )}{2 b}\\ &=\frac {\cos \left (a+b x^2\right )}{2 b^2}+\frac {x^2 \sin \left (a+b x^2\right )}{2 b}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 29, normalized size = 0.85 \[ \frac {b x^2 \sin \left (a+b x^2\right )+\cos \left (a+b x^2\right )}{2 b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*Cos[a + b*x^2],x]

[Out]

(Cos[a + b*x^2] + b*x^2*Sin[a + b*x^2])/(2*b^2)

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fricas [A]  time = 0.57, size = 27, normalized size = 0.79 \[ \frac {b x^{2} \sin \left (b x^{2} + a\right ) + \cos \left (b x^{2} + a\right )}{2 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*cos(b*x^2+a),x, algorithm="fricas")

[Out]

1/2*(b*x^2*sin(b*x^2 + a) + cos(b*x^2 + a))/b^2

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giac [A]  time = 0.38, size = 27, normalized size = 0.79 \[ \frac {b x^{2} \sin \left (b x^{2} + a\right ) + \cos \left (b x^{2} + a\right )}{2 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*cos(b*x^2+a),x, algorithm="giac")

[Out]

1/2*(b*x^2*sin(b*x^2 + a) + cos(b*x^2 + a))/b^2

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maple [A]  time = 0.02, size = 31, normalized size = 0.91 \[ \frac {\cos \left (b \,x^{2}+a \right )}{2 b^{2}}+\frac {x^{2} \sin \left (b \,x^{2}+a \right )}{2 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*cos(b*x^2+a),x)

[Out]

1/2*cos(b*x^2+a)/b^2+1/2*x^2*sin(b*x^2+a)/b

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maxima [A]  time = 1.11, size = 27, normalized size = 0.79 \[ \frac {b x^{2} \sin \left (b x^{2} + a\right ) + \cos \left (b x^{2} + a\right )}{2 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*cos(b*x^2+a),x, algorithm="maxima")

[Out]

1/2*(b*x^2*sin(b*x^2 + a) + cos(b*x^2 + a))/b^2

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mupad [B]  time = 0.10, size = 27, normalized size = 0.79 \[ \frac {\cos \left (b\,x^2+a\right )+b\,x^2\,\sin \left (b\,x^2+a\right )}{2\,b^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*cos(a + b*x^2),x)

[Out]

(cos(a + b*x^2) + b*x^2*sin(a + b*x^2))/(2*b^2)

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sympy [A]  time = 0.77, size = 36, normalized size = 1.06 \[ \begin {cases} \frac {x^{2} \sin {\left (a + b x^{2} \right )}}{2 b} + \frac {\cos {\left (a + b x^{2} \right )}}{2 b^{2}} & \text {for}\: b \neq 0 \\\frac {x^{4} \cos {\relax (a )}}{4} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*cos(b*x**2+a),x)

[Out]

Piecewise((x**2*sin(a + b*x**2)/(2*b) + cos(a + b*x**2)/(2*b**2), Ne(b, 0)), (x**4*cos(a)/4, True))

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