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

Optimal. Leaf size=156 \[ \frac{3 a x^2 \sin (c+d x)}{d^2}-\frac{6 a \sin (c+d x)}{d^4}+\frac{6 a x \cos (c+d x)}{d^3}-\frac{a x^3 \cos (c+d x)}{d}+\frac{6 b x^5 \sin (c+d x)}{d^2}-\frac{120 b x^3 \sin (c+d x)}{d^4}+\frac{30 b x^4 \cos (c+d x)}{d^3}-\frac{360 b x^2 \cos (c+d x)}{d^5}+\frac{720 b x \sin (c+d x)}{d^6}+\frac{720 b \cos (c+d x)}{d^7}-\frac{b x^6 \cos (c+d x)}{d} \]

[Out]

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

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Rubi [A]  time = 0.248865, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {3339, 3296, 2637, 2638} \[ \frac{3 a x^2 \sin (c+d x)}{d^2}-\frac{6 a \sin (c+d x)}{d^4}+\frac{6 a x \cos (c+d x)}{d^3}-\frac{a x^3 \cos (c+d x)}{d}+\frac{6 b x^5 \sin (c+d x)}{d^2}-\frac{120 b x^3 \sin (c+d x)}{d^4}+\frac{30 b x^4 \cos (c+d x)}{d^3}-\frac{360 b x^2 \cos (c+d x)}{d^5}+\frac{720 b x \sin (c+d x)}{d^6}+\frac{720 b \cos (c+d x)}{d^7}-\frac{b x^6 \cos (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Int[x^3*(a + b*x^3)*Sin[c + d*x],x]

[Out]

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

Rule 3339

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

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]

Rule 2638

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

Rubi steps

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

Mathematica [A]  time = 0.207923, size = 101, normalized size = 0.65 \[ \frac{3 d \left (a d^2 \left (d^2 x^2-2\right )+2 b x \left (d^4 x^4-20 d^2 x^2+120\right )\right ) \sin (c+d x)-\left (a d^4 x \left (d^2 x^2-6\right )+b \left (d^6 x^6-30 d^4 x^4+360 d^2 x^2-720\right )\right ) \cos (c+d x)}{d^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-((a*d^4*x*(-6 + d^2*x^2) + b*(-720 + 360*d^2*x^2 - 30*d^4*x^4 + d^6*x^6))*Cos[c + d*x]) + 3*d*(a*d^2*(-2 + d
^2*x^2) + 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 = 556, normalized size = 3.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d^4*(1/d^3*b*(-(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^3*b*c*(-(d*x+c)^5*cos(d*x+c)+5*(d*x+c)^4*s
in(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^3*b*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))+a*(
-(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^3*b*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))-3*a*c*(-(d*x+c)^2*cos(d*x+c)+2*cos(d*x+c)+2*(d
*x+c)*sin(d*x+c))+15/d^3*b*c^4*(-(d*x+c)^2*cos(d*x+c)+2*cos(d*x+c)+2*(d*x+c)*sin(d*x+c))+3*a*c^2*(sin(d*x+c)-(
d*x+c)*cos(d*x+c))-6/d^3*b*c^5*(sin(d*x+c)-(d*x+c)*cos(d*x+c))+a*c^3*cos(d*x+c)-1/d^3*b*c^6*cos(d*x+c))

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Maxima [B]  time = 1.08813, size = 606, normalized size = 3.88 \begin{align*} \frac{a c^{3} \cos \left (d x + c\right ) - \frac{b c^{6} \cos \left (d x + c\right )}{d^{3}} - 3 \,{\left ({\left (d x + c\right )} \cos \left (d x + c\right ) - \sin \left (d x + c\right )\right )} a c^{2} + \frac{6 \,{\left ({\left (d x + c\right )} \cos \left (d x + c\right ) - \sin \left (d x + c\right )\right )} b c^{5}}{d^{3}} + 3 \,{\left ({\left ({\left (d x + c\right )}^{2} - 2\right )} \cos \left (d x + c\right ) - 2 \,{\left (d x + c\right )} \sin \left (d x + c\right )\right )} a c - \frac{15 \,{\left ({\left ({\left (d x + c\right )}^{2} - 2\right )} \cos \left (d x + c\right ) - 2 \,{\left (d x + c\right )} \sin \left (d x + c\right )\right )} b c^{4}}{d^{3}} -{\left ({\left ({\left (d x + c\right )}^{3} - 6 \, d x - 6 \, c\right )} \cos \left (d x + c\right ) - 3 \,{\left ({\left (d x + c\right )}^{2} - 2\right )} \sin \left (d x + c\right )\right )} a + \frac{20 \,{\left ({\left ({\left (d x + c\right )}^{3} - 6 \, d x - 6 \, c\right )} \cos \left (d x + c\right ) - 3 \,{\left ({\left (d x + c\right )}^{2} - 2\right )} \sin \left (d x + c\right )\right )} b c^{3}}{d^{3}} - \frac{15 \,{\left ({\left ({\left (d x + c\right )}^{4} - 12 \,{\left (d x + c\right )}^{2} + 24\right )} \cos \left (d x + c\right ) - 4 \,{\left ({\left (d x + c\right )}^{3} - 6 \, d x - 6 \, c\right )} \sin \left (d x + c\right )\right )} b c^{2}}{d^{3}} + \frac{6 \,{\left ({\left ({\left (d x + c\right )}^{5} - 20 \,{\left (d x + c\right )}^{3} + 120 \, d x + 120 \, c\right )} \cos \left (d x + c\right ) - 5 \,{\left ({\left (d x + c\right )}^{4} - 12 \,{\left (d x + c\right )}^{2} + 24\right )} \sin \left (d x + c\right )\right )} b c}{d^{3}} - \frac{{\left ({\left ({\left (d x + c\right )}^{6} - 30 \,{\left (d x + c\right )}^{4} + 360 \,{\left (d x + c\right )}^{2} - 720\right )} \cos \left (d x + c\right ) - 6 \,{\left ({\left (d x + c\right )}^{5} - 20 \,{\left (d x + c\right )}^{3} + 120 \, d x + 120 \, c\right )} \sin \left (d x + c\right )\right )} b}{d^{3}}}{d^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a*c^3*cos(d*x + c) - b*c^6*cos(d*x + c)/d^3 - 3*((d*x + c)*cos(d*x + c) - sin(d*x + c))*a*c^2 + 6*((d*x + c)*
cos(d*x + c) - sin(d*x + c))*b*c^5/d^3 + 3*(((d*x + c)^2 - 2)*cos(d*x + c) - 2*(d*x + c)*sin(d*x + c))*a*c - 1
5*(((d*x + c)^2 - 2)*cos(d*x + c) - 2*(d*x + c)*sin(d*x + c))*b*c^4/d^3 - (((d*x + c)^3 - 6*d*x - 6*c)*cos(d*x
+ c) - 3*((d*x + c)^2 - 2)*sin(d*x + c))*a + 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*c^3/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*c^2/d^3 + 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*c/d^3 - (((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/d^3)/d^4

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Fricas [A]  time = 1.59165, size = 238, normalized size = 1.53 \begin{align*} -\frac{{\left (b d^{6} x^{6} + a d^{6} x^{3} - 30 \, b d^{4} x^{4} - 6 \, a d^{4} x + 360 \, b d^{2} x^{2} - 720 \, b\right )} \cos \left (d x + c\right ) - 3 \,{\left (2 \, b d^{5} x^{5} + a d^{5} x^{2} - 40 \, b d^{3} x^{3} - 2 \, a d^{3} + 240 \, b 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(x^3*(b*x^3+a)*sin(d*x+c),x, algorithm="fricas")

[Out]

-((b*d^6*x^6 + a*d^6*x^3 - 30*b*d^4*x^4 - 6*a*d^4*x + 360*b*d^2*x^2 - 720*b)*cos(d*x + c) - 3*(2*b*d^5*x^5 + a
*d^5*x^2 - 40*b*d^3*x^3 - 2*a*d^3 + 240*b*d*x)*sin(d*x + c))/d^7

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Sympy [A]  time = 7.1842, size = 185, normalized size = 1.19 \begin{align*} \begin{cases} - \frac{a x^{3} \cos{\left (c + d x \right )}}{d} + \frac{3 a x^{2} \sin{\left (c + d x \right )}}{d^{2}} + \frac{6 a x \cos{\left (c + d x \right )}}{d^{3}} - \frac{6 a \sin{\left (c + d x \right )}}{d^{4}} - \frac{b x^{6} \cos{\left (c + d x \right )}}{d} + \frac{6 b x^{5} \sin{\left (c + d x \right )}}{d^{2}} + \frac{30 b x^{4} \cos{\left (c + d x \right )}}{d^{3}} - \frac{120 b x^{3} \sin{\left (c + d x \right )}}{d^{4}} - \frac{360 b x^{2} \cos{\left (c + d x \right )}}{d^{5}} + \frac{720 b x \sin{\left (c + d x \right )}}{d^{6}} + \frac{720 b \cos{\left (c + d x \right )}}{d^{7}} & \text{for}\: d \neq 0 \\\left (\frac{a x^{4}}{4} + \frac{b 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(x**3*(b*x**3+a)*sin(d*x+c),x)

[Out]

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

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Giac [A]  time = 1.10401, size = 143, normalized size = 0.92 \begin{align*} -\frac{{\left (b d^{6} x^{6} + a d^{6} x^{3} - 30 \, b d^{4} x^{4} - 6 \, a d^{4} x + 360 \, b d^{2} x^{2} - 720 \, b\right )} \cos \left (d x + c\right )}{d^{7}} + \frac{3 \,{\left (2 \, b d^{5} x^{5} + a d^{5} x^{2} - 40 \, b d^{3} x^{3} - 2 \, a d^{3} + 240 \, b 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(x^3*(b*x^3+a)*sin(d*x+c),x, algorithm="giac")

[Out]

-(b*d^6*x^6 + a*d^6*x^3 - 30*b*d^4*x^4 - 6*a*d^4*x + 360*b*d^2*x^2 - 720*b)*cos(d*x + c)/d^7 + 3*(2*b*d^5*x^5
+ a*d^5*x^2 - 40*b*d^3*x^3 - 2*a*d^3 + 240*b*d*x)*sin(d*x + c)/d^7