3.925 \(\int x^2 \cos ^7(a+b x^3) \sin (a+b x^3) \, dx\)

Optimal. Leaf size=17 \[ -\frac{\cos ^8\left (a+b x^3\right )}{24 b} \]

[Out]

-Cos[a + b*x^3]^8/(24*b)

________________________________________________________________________________________

Rubi [A]  time = 0.0241569, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {3442} \[ -\frac{\cos ^8\left (a+b x^3\right )}{24 b} \]

Antiderivative was successfully verified.

[In]

Int[x^2*Cos[a + b*x^3]^7*Sin[a + b*x^3],x]

[Out]

-Cos[a + b*x^3]^8/(24*b)

Rule 3442

Int[Cos[(a_.) + (b_.)*(x_)^(n_.)]^(p_.)*(x_)^(m_.)*Sin[(a_.) + (b_.)*(x_)^(n_.)], x_Symbol] :> -Simp[Cos[a + b
*x^n]^(p + 1)/(b*n*(p + 1)), x] /; FreeQ[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

\begin{align*} \int x^2 \cos ^7\left (a+b x^3\right ) \sin \left (a+b x^3\right ) \, dx &=-\frac{\cos ^8\left (a+b x^3\right )}{24 b}\\ \end{align*}

Mathematica [A]  time = 0.0211926, size = 17, normalized size = 1. \[ -\frac{\cos ^8\left (a+b x^3\right )}{24 b} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*Cos[a + b*x^3]^7*Sin[a + b*x^3],x]

[Out]

-Cos[a + b*x^3]^8/(24*b)

________________________________________________________________________________________

Maple [A]  time = 0.006, size = 16, normalized size = 0.9 \begin{align*} -{\frac{ \left ( \cos \left ( b{x}^{3}+a \right ) \right ) ^{8}}{24\,b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*cos(b*x^3+a)^7*sin(b*x^3+a),x)

[Out]

-1/24*cos(b*x^3+a)^8/b

________________________________________________________________________________________

Maxima [A]  time = 0.959524, size = 20, normalized size = 1.18 \begin{align*} -\frac{\cos \left (b x^{3} + a\right )^{8}}{24 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*cos(b*x^3+a)^7*sin(b*x^3+a),x, algorithm="maxima")

[Out]

-1/24*cos(b*x^3 + a)^8/b

________________________________________________________________________________________

Fricas [A]  time = 2.22122, size = 35, normalized size = 2.06 \begin{align*} -\frac{\cos \left (b x^{3} + a\right )^{8}}{24 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*cos(b*x^3+a)^7*sin(b*x^3+a),x, algorithm="fricas")

[Out]

-1/24*cos(b*x^3 + a)^8/b

________________________________________________________________________________________

Sympy [A]  time = 30.8132, size = 27, normalized size = 1.59 \begin{align*} \begin{cases} - \frac{\cos ^{8}{\left (a + b x^{3} \right )}}{24 b} & \text{for}\: b \neq 0 \\\frac{x^{3} \sin{\left (a \right )} \cos ^{7}{\left (a \right )}}{3} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*cos(b*x**3+a)**7*sin(b*x**3+a),x)

[Out]

Piecewise((-cos(a + b*x**3)**8/(24*b), Ne(b, 0)), (x**3*sin(a)*cos(a)**7/3, True))

________________________________________________________________________________________

Giac [A]  time = 1.1732, size = 20, normalized size = 1.18 \begin{align*} -\frac{\cos \left (b x^{3} + a\right )^{8}}{24 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*cos(b*x^3+a)^7*sin(b*x^3+a),x, algorithm="giac")

[Out]

-1/24*cos(b*x^3 + a)^8/b