\(\int x^2 \cos ^7(a+b x^3) \sin (a+b x^3) \, dx\) [925]

Optimal result

Integrand size = 22, antiderivative size = 17 \[ \int x^2 \cos ^7\left (a+b x^3\right ) \sin \left (a+b x^3\right ) \, dx=-\frac {\cos ^8\left (a+b x^3\right )}{24 b} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {3523} \[ \int x^2 \cos ^7\left (a+b x^3\right ) \sin \left (a+b x^3\right ) \, dx=-\frac {\cos ^8\left (a+b x^3\right )}{24 b} \]

[In]

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

[Out]

-1/24*Cos[a + b*x^3]^8/b

Rule 3523

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*} \text {integral}& = -\frac {\cos ^8\left (a+b x^3\right )}{24 b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int x^2 \cos ^7\left (a+b x^3\right ) \sin \left (a+b x^3\right ) \, dx=-\frac {\cos ^8\left (a+b x^3\right )}{24 b} \]

[In]

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

[Out]

-1/24*Cos[a + b*x^3]^8/b

Maple [A] (verified)

Time = 0.55 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.94

method	result	size
derivativedivides	\(-\frac {\cos \left (b \,x^{3}+a \right )^{8}}{24 b}\)	\(16\)
default	\(-\frac {\cos \left (b \,x^{3}+a \right )^{8}}{24 b}\)	\(16\)
risch	\(-\frac {\cos \left (8 b \,x^{3}+8 a \right )}{3072 b}-\frac {\cos \left (6 b \,x^{3}+6 a \right )}{384 b}-\frac {7 \cos \left (4 b \,x^{3}+4 a \right )}{768 b}-\frac {7 \cos \left (2 b \,x^{3}+2 a \right )}{384 b}\)	\(66\)
parallelrisch	\(\frac {\frac {2 \tan \left (\frac {a}{2}+\frac {b \,x^{3}}{2}\right )^{14}}{3}+\frac {14 \tan \left (\frac {a}{2}+\frac {b \,x^{3}}{2}\right )^{10}}{3}+\frac {14 \tan \left (\frac {a}{2}+\frac {b \,x^{3}}{2}\right )^{6}}{3}+\frac {2 \tan \left (\frac {a}{2}+\frac {b \,x^{3}}{2}\right )^{2}}{3}}{b {\left (1+\tan \left (\frac {a}{2}+\frac {b \,x^{3}}{2}\right )^{2}\right )}^{8}}\)	\(80\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int x^2 \cos ^7\left (a+b x^3\right ) \sin \left (a+b x^3\right ) \, dx=-\frac {\cos \left (b x^{3} + a\right )^{8}}{24 \, b} \]

[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] (verification not implemented)

Time = 1.23 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.59 \[ \int x^2 \cos ^7\left (a+b x^3\right ) \sin \left (a+b x^3\right ) \, dx=\begin {cases} - \frac {\cos ^{8}{\left (a + b x^{3} \right )}}{24 b} & \text {for}\: b \neq 0 \\\frac {x^{3} \sin {\left (a \right )} \cos ^{7}{\left (a \right )}}{3} & \text {otherwise} \end {cases} \]

[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))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int x^2 \cos ^7\left (a+b x^3\right ) \sin \left (a+b x^3\right ) \, dx=-\frac {\cos \left (b x^{3} + a\right )^{8}}{24 \, b} \]

[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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int x^2 \cos ^7\left (a+b x^3\right ) \sin \left (a+b x^3\right ) \, dx=-\frac {\cos \left (b x^{3} + a\right )^{8}}{24 \, b} \]

[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

Mupad [B] (verification not implemented)

Time = 26.99 (sec) , antiderivative size = 56, normalized size of antiderivative = 3.29 \[ \int x^2 \cos ^7\left (a+b x^3\right ) \sin \left (a+b x^3\right ) \, dx=-\frac {56\,\cos \left (2\,b\,x^3+2\,a\right )+28\,\cos \left (4\,b\,x^3+4\,a\right )+8\,\cos \left (6\,b\,x^3+6\,a\right )+\cos \left (8\,b\,x^3+8\,a\right )}{3072\,b} \]

[In]

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

[Out]

-(56*cos(2*a + 2*b*x^3) + 28*cos(4*a + 4*b*x^3) + 8*cos(6*a + 6*b*x^3) + cos(8*a + 8*b*x^3))/(3072*b)