3.88 \(\int (a+b x^3)^2 \sin (c+d x) \, dx\)

Optimal. Leaf size=188 \[ -\frac{a^2 \cos (c+d x)}{d}+\frac{6 a b x^2 \sin (c+d x)}{d^2}-\frac{12 a b \sin (c+d x)}{d^4}+\frac{12 a b x \cos (c+d x)}{d^3}-\frac{2 a b x^3 \cos (c+d x)}{d}+\frac{6 b^2 x^5 \sin (c+d x)}{d^2}-\frac{120 b^2 x^3 \sin (c+d x)}{d^4}+\frac{30 b^2 x^4 \cos (c+d x)}{d^3}-\frac{360 b^2 x^2 \cos (c+d x)}{d^5}+\frac{720 b^2 x \sin (c+d x)}{d^6}+\frac{720 b^2 \cos (c+d x)}{d^7}-\frac{b^2 x^6 \cos (c+d x)}{d} \]

[Out]

(720*b^2*Cos[c + d*x])/d^7 - (a^2*Cos[c + d*x])/d + (12*a*b*x*Cos[c + d*x])/d^3 - (360*b^2*x^2*Cos[c + d*x])/d
^5 - (2*a*b*x^3*Cos[c + d*x])/d + (30*b^2*x^4*Cos[c + d*x])/d^3 - (b^2*x^6*Cos[c + d*x])/d - (12*a*b*Sin[c + d
*x])/d^4 + (720*b^2*x*Sin[c + d*x])/d^6 + (6*a*b*x^2*Sin[c + d*x])/d^2 - (120*b^2*x^3*Sin[c + d*x])/d^4 + (6*b
^2*x^5*Sin[c + d*x])/d^2

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Rubi [A]  time = 0.242373, antiderivative size = 188, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3329, 2638, 3296, 2637} \[ -\frac{a^2 \cos (c+d x)}{d}+\frac{6 a b x^2 \sin (c+d x)}{d^2}-\frac{12 a b \sin (c+d x)}{d^4}+\frac{12 a b x \cos (c+d x)}{d^3}-\frac{2 a b x^3 \cos (c+d x)}{d}+\frac{6 b^2 x^5 \sin (c+d x)}{d^2}-\frac{120 b^2 x^3 \sin (c+d x)}{d^4}+\frac{30 b^2 x^4 \cos (c+d x)}{d^3}-\frac{360 b^2 x^2 \cos (c+d x)}{d^5}+\frac{720 b^2 x \sin (c+d x)}{d^6}+\frac{720 b^2 \cos (c+d x)}{d^7}-\frac{b^2 x^6 \cos (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^2*Sin[c + d*x],x]

[Out]

(720*b^2*Cos[c + d*x])/d^7 - (a^2*Cos[c + d*x])/d + (12*a*b*x*Cos[c + d*x])/d^3 - (360*b^2*x^2*Cos[c + d*x])/d
^5 - (2*a*b*x^3*Cos[c + d*x])/d + (30*b^2*x^4*Cos[c + d*x])/d^3 - (b^2*x^6*Cos[c + d*x])/d - (12*a*b*Sin[c + d
*x])/d^4 + (720*b^2*x*Sin[c + d*x])/d^6 + (6*a*b*x^2*Sin[c + d*x])/d^2 - (120*b^2*x^3*Sin[c + d*x])/d^4 + (6*b
^2*x^5*Sin[c + d*x])/d^2

Rule 3329

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*Sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Int[ExpandIntegrand[Sin[c + d*x], (a
+ b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[p, 0]

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 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int \left (a+b x^3\right )^2 \sin (c+d x) \, dx &=\int \left (a^2 \sin (c+d x)+2 a b x^3 \sin (c+d x)+b^2 x^6 \sin (c+d x)\right ) \, dx\\ &=a^2 \int \sin (c+d x) \, dx+(2 a b) \int x^3 \sin (c+d x) \, dx+b^2 \int x^6 \sin (c+d x) \, dx\\ &=-\frac{a^2 \cos (c+d x)}{d}-\frac{2 a b x^3 \cos (c+d x)}{d}-\frac{b^2 x^6 \cos (c+d x)}{d}+\frac{(6 a b) \int x^2 \cos (c+d x) \, dx}{d}+\frac{\left (6 b^2\right ) \int x^5 \cos (c+d x) \, dx}{d}\\ &=-\frac{a^2 \cos (c+d x)}{d}-\frac{2 a b x^3 \cos (c+d x)}{d}-\frac{b^2 x^6 \cos (c+d x)}{d}+\frac{6 a b x^2 \sin (c+d x)}{d^2}+\frac{6 b^2 x^5 \sin (c+d x)}{d^2}-\frac{(12 a b) \int x \sin (c+d x) \, dx}{d^2}-\frac{\left (30 b^2\right ) \int x^4 \sin (c+d x) \, dx}{d^2}\\ &=-\frac{a^2 \cos (c+d x)}{d}+\frac{12 a b x \cos (c+d x)}{d^3}-\frac{2 a b x^3 \cos (c+d x)}{d}+\frac{30 b^2 x^4 \cos (c+d x)}{d^3}-\frac{b^2 x^6 \cos (c+d x)}{d}+\frac{6 a b x^2 \sin (c+d x)}{d^2}+\frac{6 b^2 x^5 \sin (c+d x)}{d^2}-\frac{(12 a b) \int \cos (c+d x) \, dx}{d^3}-\frac{\left (120 b^2\right ) \int x^3 \cos (c+d x) \, dx}{d^3}\\ &=-\frac{a^2 \cos (c+d x)}{d}+\frac{12 a b x \cos (c+d x)}{d^3}-\frac{2 a b x^3 \cos (c+d x)}{d}+\frac{30 b^2 x^4 \cos (c+d x)}{d^3}-\frac{b^2 x^6 \cos (c+d x)}{d}-\frac{12 a b \sin (c+d x)}{d^4}+\frac{6 a b x^2 \sin (c+d x)}{d^2}-\frac{120 b^2 x^3 \sin (c+d x)}{d^4}+\frac{6 b^2 x^5 \sin (c+d x)}{d^2}+\frac{\left (360 b^2\right ) \int x^2 \sin (c+d x) \, dx}{d^4}\\ &=-\frac{a^2 \cos (c+d x)}{d}+\frac{12 a b x \cos (c+d x)}{d^3}-\frac{360 b^2 x^2 \cos (c+d x)}{d^5}-\frac{2 a b x^3 \cos (c+d x)}{d}+\frac{30 b^2 x^4 \cos (c+d x)}{d^3}-\frac{b^2 x^6 \cos (c+d x)}{d}-\frac{12 a b \sin (c+d x)}{d^4}+\frac{6 a b x^2 \sin (c+d x)}{d^2}-\frac{120 b^2 x^3 \sin (c+d x)}{d^4}+\frac{6 b^2 x^5 \sin (c+d x)}{d^2}+\frac{\left (720 b^2\right ) \int x \cos (c+d x) \, dx}{d^5}\\ &=-\frac{a^2 \cos (c+d x)}{d}+\frac{12 a b x \cos (c+d x)}{d^3}-\frac{360 b^2 x^2 \cos (c+d x)}{d^5}-\frac{2 a b x^3 \cos (c+d x)}{d}+\frac{30 b^2 x^4 \cos (c+d x)}{d^3}-\frac{b^2 x^6 \cos (c+d x)}{d}-\frac{12 a b \sin (c+d x)}{d^4}+\frac{720 b^2 x \sin (c+d x)}{d^6}+\frac{6 a b x^2 \sin (c+d x)}{d^2}-\frac{120 b^2 x^3 \sin (c+d x)}{d^4}+\frac{6 b^2 x^5 \sin (c+d x)}{d^2}-\frac{\left (720 b^2\right ) \int \sin (c+d x) \, dx}{d^6}\\ &=\frac{720 b^2 \cos (c+d x)}{d^7}-\frac{a^2 \cos (c+d x)}{d}+\frac{12 a b x \cos (c+d x)}{d^3}-\frac{360 b^2 x^2 \cos (c+d x)}{d^5}-\frac{2 a b x^3 \cos (c+d x)}{d}+\frac{30 b^2 x^4 \cos (c+d x)}{d^3}-\frac{b^2 x^6 \cos (c+d x)}{d}-\frac{12 a b \sin (c+d x)}{d^4}+\frac{720 b^2 x \sin (c+d x)}{d^6}+\frac{6 a b x^2 \sin (c+d x)}{d^2}-\frac{120 b^2 x^3 \sin (c+d x)}{d^4}+\frac{6 b^2 x^5 \sin (c+d x)}{d^2}\\ \end{align*}

Mathematica [A]  time = 0.315224, size = 112, normalized size = 0.6 \[ \frac{6 b d \left (a d^2 \left (d^2 x^2-2\right )+b x \left (d^4 x^4-20 d^2 x^2+120\right )\right ) \sin (c+d x)-\left (a^2 d^6+2 a b d^4 x \left (d^2 x^2-6\right )+b^2 \left (d^6 x^6-30 d^4 x^4+360 d^2 x^2-720\right )\right ) \cos (c+d x)}{d^7} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^2*Sin[c + d*x],x]

[Out]

(-((a^2*d^6 + 2*a*b*d^4*x*(-6 + d^2*x^2) + b^2*(-720 + 360*d^2*x^2 - 30*d^4*x^4 + d^6*x^6))*Cos[c + d*x]) + 6*
b*d*(a*d^2*(-2 + d^2*x^2) + b*x*(120 - 20*d^2*x^2 + d^4*x^4))*Sin[c + d*x])/d^7

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Maple [B]  time = 0.007, size = 599, normalized size = 3.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^2*sin(d*x+c),x)

[Out]

1/d*(1/d^6*b^2*(-(d*x+c)^6*cos(d*x+c)+6*(d*x+c)^5*sin(d*x+c)+30*(d*x+c)^4*cos(d*x+c)-120*(d*x+c)^3*sin(d*x+c)-
360*(d*x+c)^2*cos(d*x+c)+720*cos(d*x+c)+720*(d*x+c)*sin(d*x+c))-6/d^6*b^2*c*(-(d*x+c)^5*cos(d*x+c)+5*(d*x+c)^4
*sin(d*x+c)+20*(d*x+c)^3*cos(d*x+c)-60*(d*x+c)^2*sin(d*x+c)+120*sin(d*x+c)-120*(d*x+c)*cos(d*x+c))+15/d^6*b^2*
c^2*(-(d*x+c)^4*cos(d*x+c)+4*(d*x+c)^3*sin(d*x+c)+12*(d*x+c)^2*cos(d*x+c)-24*cos(d*x+c)-24*(d*x+c)*sin(d*x+c))
+2/d^3*a*b*(-(d*x+c)^3*cos(d*x+c)+3*(d*x+c)^2*sin(d*x+c)-6*sin(d*x+c)+6*(d*x+c)*cos(d*x+c))-20/d^6*b^2*c^3*(-(
d*x+c)^3*cos(d*x+c)+3*(d*x+c)^2*sin(d*x+c)-6*sin(d*x+c)+6*(d*x+c)*cos(d*x+c))-6/d^3*a*b*c*(-(d*x+c)^2*cos(d*x+
c)+2*cos(d*x+c)+2*(d*x+c)*sin(d*x+c))+15/d^6*b^2*c^4*(-(d*x+c)^2*cos(d*x+c)+2*cos(d*x+c)+2*(d*x+c)*sin(d*x+c))
+6/d^3*a*b*c^2*(sin(d*x+c)-(d*x+c)*cos(d*x+c))-6/d^6*b^2*c^5*(sin(d*x+c)-(d*x+c)*cos(d*x+c))-a^2*cos(d*x+c)+2/
d^3*a*b*c^3*cos(d*x+c)-1/d^6*b^2*c^6*cos(d*x+c))

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Maxima [B]  time = 1.07572, size = 660, normalized size = 3.51 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(a^2*cos(d*x + c) + b^2*c^6*cos(d*x + c)/d^6 - 2*a*b*c^3*cos(d*x + c)/d^3 - 6*((d*x + c)*cos(d*x + c) - sin(d
*x + c))*b^2*c^5/d^6 + 6*((d*x + c)*cos(d*x + c) - sin(d*x + c))*a*b*c^2/d^3 + 15*(((d*x + c)^2 - 2)*cos(d*x +
c) - 2*(d*x + c)*sin(d*x + c))*b^2*c^4/d^6 - 6*(((d*x + c)^2 - 2)*cos(d*x + c) - 2*(d*x + c)*sin(d*x + c))*a*
b*c/d^3 - 20*(((d*x + c)^3 - 6*d*x - 6*c)*cos(d*x + c) - 3*((d*x + c)^2 - 2)*sin(d*x + c))*b^2*c^3/d^6 + 2*(((
d*x + c)^3 - 6*d*x - 6*c)*cos(d*x + c) - 3*((d*x + c)^2 - 2)*sin(d*x + c))*a*b/d^3 + 15*(((d*x + c)^4 - 12*(d*
x + c)^2 + 24)*cos(d*x + c) - 4*((d*x + c)^3 - 6*d*x - 6*c)*sin(d*x + c))*b^2*c^2/d^6 - 6*(((d*x + c)^5 - 20*(
d*x + c)^3 + 120*d*x + 120*c)*cos(d*x + c) - 5*((d*x + c)^4 - 12*(d*x + c)^2 + 24)*sin(d*x + c))*b^2*c/d^6 + (
((d*x + c)^6 - 30*(d*x + c)^4 + 360*(d*x + c)^2 - 720)*cos(d*x + c) - 6*((d*x + c)^5 - 20*(d*x + c)^3 + 120*d*
x + 120*c)*sin(d*x + c))*b^2/d^6)/d

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Fricas [A]  time = 1.70757, size = 282, normalized size = 1.5 \begin{align*} -\frac{{\left (b^{2} d^{6} x^{6} + 2 \, a b d^{6} x^{3} - 30 \, b^{2} d^{4} x^{4} + a^{2} d^{6} - 12 \, a b d^{4} x + 360 \, b^{2} d^{2} x^{2} - 720 \, b^{2}\right )} \cos \left (d x + c\right ) - 6 \,{\left (b^{2} d^{5} x^{5} + a b d^{5} x^{2} - 20 \, b^{2} d^{3} x^{3} - 2 \, a b d^{3} + 120 \, b^{2} d x\right )} \sin \left (d x + c\right )}{d^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((b^2*d^6*x^6 + 2*a*b*d^6*x^3 - 30*b^2*d^4*x^4 + a^2*d^6 - 12*a*b*d^4*x + 360*b^2*d^2*x^2 - 720*b^2)*cos(d*x
+ c) - 6*(b^2*d^5*x^5 + a*b*d^5*x^2 - 20*b^2*d^3*x^3 - 2*a*b*d^3 + 120*b^2*d*x)*sin(d*x + c))/d^7

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Sympy [A]  time = 7.54425, size = 226, normalized size = 1.2 \begin{align*} \begin{cases} - \frac{a^{2} \cos{\left (c + d x \right )}}{d} - \frac{2 a b x^{3} \cos{\left (c + d x \right )}}{d} + \frac{6 a b x^{2} \sin{\left (c + d x \right )}}{d^{2}} + \frac{12 a b x \cos{\left (c + d x \right )}}{d^{3}} - \frac{12 a b \sin{\left (c + d x \right )}}{d^{4}} - \frac{b^{2} x^{6} \cos{\left (c + d x \right )}}{d} + \frac{6 b^{2} x^{5} \sin{\left (c + d x \right )}}{d^{2}} + \frac{30 b^{2} x^{4} \cos{\left (c + d x \right )}}{d^{3}} - \frac{120 b^{2} x^{3} \sin{\left (c + d x \right )}}{d^{4}} - \frac{360 b^{2} x^{2} \cos{\left (c + d x \right )}}{d^{5}} + \frac{720 b^{2} x \sin{\left (c + d x \right )}}{d^{6}} + \frac{720 b^{2} \cos{\left (c + d x \right )}}{d^{7}} & \text{for}\: d \neq 0 \\\left (a^{2} x + \frac{a b x^{4}}{2} + \frac{b^{2} x^{7}}{7}\right ) \sin{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**2*sin(d*x+c),x)

[Out]

Piecewise((-a**2*cos(c + d*x)/d - 2*a*b*x**3*cos(c + d*x)/d + 6*a*b*x**2*sin(c + d*x)/d**2 + 12*a*b*x*cos(c +
d*x)/d**3 - 12*a*b*sin(c + d*x)/d**4 - b**2*x**6*cos(c + d*x)/d + 6*b**2*x**5*sin(c + d*x)/d**2 + 30*b**2*x**4
*cos(c + d*x)/d**3 - 120*b**2*x**3*sin(c + d*x)/d**4 - 360*b**2*x**2*cos(c + d*x)/d**5 + 720*b**2*x*sin(c + d*
x)/d**6 + 720*b**2*cos(c + d*x)/d**7, Ne(d, 0)), ((a**2*x + a*b*x**4/2 + b**2*x**7/7)*sin(c), True))

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Giac [A]  time = 1.15032, size = 177, normalized size = 0.94 \begin{align*} -\frac{{\left (b^{2} d^{6} x^{6} + 2 \, a b d^{6} x^{3} - 30 \, b^{2} d^{4} x^{4} + a^{2} d^{6} - 12 \, a b d^{4} x + 360 \, b^{2} d^{2} x^{2} - 720 \, b^{2}\right )} \cos \left (d x + c\right )}{d^{7}} + \frac{6 \,{\left (b^{2} d^{5} x^{5} + a b d^{5} x^{2} - 20 \, b^{2} d^{3} x^{3} - 2 \, a b d^{3} + 120 \, b^{2} d x\right )} \sin \left (d x + c\right )}{d^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(b^2*d^6*x^6 + 2*a*b*d^6*x^3 - 30*b^2*d^4*x^4 + a^2*d^6 - 12*a*b*d^4*x + 360*b^2*d^2*x^2 - 720*b^2)*cos(d*x +
c)/d^7 + 6*(b^2*d^5*x^5 + a*b*d^5*x^2 - 20*b^2*d^3*x^3 - 2*a*b*d^3 + 120*b^2*d*x)*sin(d*x + c)/d^7