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

Optimal. Leaf size=185 \[ \frac{a^2 \sin (c+d x)}{d^2}-\frac{a^2 x \cos (c+d x)}{d}+\frac{6 a b x^2 \sin (c+d x)}{d^2}-\frac{12 a b \sin (c+d x)}{d^4}+\frac{12 a b x \cos (c+d x)}{d^3}-\frac{2 a b x^3 \cos (c+d x)}{d}+\frac{5 b^2 x^4 \sin (c+d x)}{d^2}-\frac{60 b^2 x^2 \sin (c+d x)}{d^4}+\frac{20 b^2 x^3 \cos (c+d x)}{d^3}+\frac{120 b^2 \sin (c+d x)}{d^6}-\frac{120 b^2 x \cos (c+d x)}{d^5}-\frac{b^2 x^5 \cos (c+d x)}{d} \]

[Out]

(-120*b^2*x*Cos[c + d*x])/d^5 + (12*a*b*x*Cos[c + d*x])/d^3 - (a^2*x*Cos[c + d*x])/d + (20*b^2*x^3*Cos[c + d*x
])/d^3 - (2*a*b*x^3*Cos[c + d*x])/d - (b^2*x^5*Cos[c + d*x])/d + (120*b^2*Sin[c + d*x])/d^6 - (12*a*b*Sin[c +
d*x])/d^4 + (a^2*Sin[c + d*x])/d^2 - (60*b^2*x^2*Sin[c + d*x])/d^4 + (6*a*b*x^2*Sin[c + d*x])/d^2 + (5*b^2*x^4
*Sin[c + d*x])/d^2

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Rubi [A]  time = 0.235175, antiderivative size = 185, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {3339, 3296, 2637} \[ \frac{a^2 \sin (c+d x)}{d^2}-\frac{a^2 x \cos (c+d x)}{d}+\frac{6 a b x^2 \sin (c+d x)}{d^2}-\frac{12 a b \sin (c+d x)}{d^4}+\frac{12 a b x \cos (c+d x)}{d^3}-\frac{2 a b x^3 \cos (c+d x)}{d}+\frac{5 b^2 x^4 \sin (c+d x)}{d^2}-\frac{60 b^2 x^2 \sin (c+d x)}{d^4}+\frac{20 b^2 x^3 \cos (c+d x)}{d^3}+\frac{120 b^2 \sin (c+d x)}{d^6}-\frac{120 b^2 x \cos (c+d x)}{d^5}-\frac{b^2 x^5 \cos (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*x^2)^2*Sin[c + d*x],x]

[Out]

(-120*b^2*x*Cos[c + d*x])/d^5 + (12*a*b*x*Cos[c + d*x])/d^3 - (a^2*x*Cos[c + d*x])/d + (20*b^2*x^3*Cos[c + d*x
])/d^3 - (2*a*b*x^3*Cos[c + d*x])/d - (b^2*x^5*Cos[c + d*x])/d + (120*b^2*Sin[c + d*x])/d^6 - (12*a*b*Sin[c +
d*x])/d^4 + (a^2*Sin[c + d*x])/d^2 - (60*b^2*x^2*Sin[c + d*x])/d^4 + (6*a*b*x^2*Sin[c + d*x])/d^2 + (5*b^2*x^4
*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]

Rubi steps

\begin{align*} \int x \left (a+b x^2\right )^2 \sin (c+d x) \, dx &=\int \left (a^2 x \sin (c+d x)+2 a b x^3 \sin (c+d x)+b^2 x^5 \sin (c+d x)\right ) \, dx\\ &=a^2 \int x \sin (c+d x) \, dx+(2 a b) \int x^3 \sin (c+d x) \, dx+b^2 \int x^5 \sin (c+d x) \, dx\\ &=-\frac{a^2 x \cos (c+d x)}{d}-\frac{2 a b x^3 \cos (c+d x)}{d}-\frac{b^2 x^5 \cos (c+d x)}{d}+\frac{a^2 \int \cos (c+d x) \, dx}{d}+\frac{(6 a b) \int x^2 \cos (c+d x) \, dx}{d}+\frac{\left (5 b^2\right ) \int x^4 \cos (c+d x) \, dx}{d}\\ &=-\frac{a^2 x \cos (c+d x)}{d}-\frac{2 a b x^3 \cos (c+d x)}{d}-\frac{b^2 x^5 \cos (c+d x)}{d}+\frac{a^2 \sin (c+d x)}{d^2}+\frac{6 a b x^2 \sin (c+d x)}{d^2}+\frac{5 b^2 x^4 \sin (c+d x)}{d^2}-\frac{(12 a b) \int x \sin (c+d x) \, dx}{d^2}-\frac{\left (20 b^2\right ) \int x^3 \sin (c+d x) \, dx}{d^2}\\ &=\frac{12 a b x \cos (c+d x)}{d^3}-\frac{a^2 x \cos (c+d x)}{d}+\frac{20 b^2 x^3 \cos (c+d x)}{d^3}-\frac{2 a b x^3 \cos (c+d x)}{d}-\frac{b^2 x^5 \cos (c+d x)}{d}+\frac{a^2 \sin (c+d x)}{d^2}+\frac{6 a b x^2 \sin (c+d x)}{d^2}+\frac{5 b^2 x^4 \sin (c+d x)}{d^2}-\frac{(12 a b) \int \cos (c+d x) \, dx}{d^3}-\frac{\left (60 b^2\right ) \int x^2 \cos (c+d x) \, dx}{d^3}\\ &=\frac{12 a b x \cos (c+d x)}{d^3}-\frac{a^2 x \cos (c+d x)}{d}+\frac{20 b^2 x^3 \cos (c+d x)}{d^3}-\frac{2 a b x^3 \cos (c+d x)}{d}-\frac{b^2 x^5 \cos (c+d x)}{d}-\frac{12 a b \sin (c+d x)}{d^4}+\frac{a^2 \sin (c+d x)}{d^2}-\frac{60 b^2 x^2 \sin (c+d x)}{d^4}+\frac{6 a b x^2 \sin (c+d x)}{d^2}+\frac{5 b^2 x^4 \sin (c+d x)}{d^2}+\frac{\left (120 b^2\right ) \int x \sin (c+d x) \, dx}{d^4}\\ &=-\frac{120 b^2 x \cos (c+d x)}{d^5}+\frac{12 a b x \cos (c+d x)}{d^3}-\frac{a^2 x \cos (c+d x)}{d}+\frac{20 b^2 x^3 \cos (c+d x)}{d^3}-\frac{2 a b x^3 \cos (c+d x)}{d}-\frac{b^2 x^5 \cos (c+d x)}{d}-\frac{12 a b \sin (c+d x)}{d^4}+\frac{a^2 \sin (c+d x)}{d^2}-\frac{60 b^2 x^2 \sin (c+d x)}{d^4}+\frac{6 a b x^2 \sin (c+d x)}{d^2}+\frac{5 b^2 x^4 \sin (c+d x)}{d^2}+\frac{\left (120 b^2\right ) \int \cos (c+d x) \, dx}{d^5}\\ &=-\frac{120 b^2 x \cos (c+d x)}{d^5}+\frac{12 a b x \cos (c+d x)}{d^3}-\frac{a^2 x \cos (c+d x)}{d}+\frac{20 b^2 x^3 \cos (c+d x)}{d^3}-\frac{2 a b x^3 \cos (c+d x)}{d}-\frac{b^2 x^5 \cos (c+d x)}{d}+\frac{120 b^2 \sin (c+d x)}{d^6}-\frac{12 a b \sin (c+d x)}{d^4}+\frac{a^2 \sin (c+d x)}{d^2}-\frac{60 b^2 x^2 \sin (c+d x)}{d^4}+\frac{6 a b x^2 \sin (c+d x)}{d^2}+\frac{5 b^2 x^4 \sin (c+d x)}{d^2}\\ \end{align*}

Mathematica [A]  time = 0.250821, size = 113, normalized size = 0.61 \[ \frac{\left (a^2 d^4+6 a b d^2 \left (d^2 x^2-2\right )+5 b^2 \left (d^4 x^4-12 d^2 x^2+24\right )\right ) \sin (c+d x)-d x \left (a^2 d^4+2 a b d^2 \left (d^2 x^2-6\right )+b^2 \left (d^4 x^4-20 d^2 x^2+120\right )\right ) \cos (c+d x)}{d^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-(d*x*(a^2*d^4 + 2*a*b*d^2*(-6 + d^2*x^2) + b^2*(120 - 20*d^2*x^2 + d^4*x^4))*Cos[c + d*x]) + (a^2*d^4 + 6*a*
b*d^2*(-2 + d^2*x^2) + 5*b^2*(24 - 12*d^2*x^2 + d^4*x^4))*Sin[c + d*x])/d^6

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Maple [B]  time = 0.007, size = 514, normalized size = 2.8 \begin{align*}{\frac{1}{{d}^{2}} \left ({\frac{{b}^{2} \left ( - \left ( dx+c \right ) ^{5}\cos \left ( dx+c \right ) +5\, \left ( dx+c \right ) ^{4}\sin \left ( dx+c \right ) +20\, \left ( dx+c \right ) ^{3}\cos \left ( dx+c \right ) -60\, \left ( dx+c \right ) ^{2}\sin \left ( dx+c \right ) +120\,\sin \left ( dx+c \right ) -120\, \left ( dx+c \right ) \cos \left ( dx+c \right ) \right ) }{{d}^{4}}}-5\,{\frac{{b}^{2}c \left ( - \left ( dx+c \right ) ^{4}\cos \left ( dx+c \right ) +4\, \left ( dx+c \right ) ^{3}\sin \left ( dx+c \right ) +12\, \left ( dx+c \right ) ^{2}\cos \left ( dx+c \right ) -24\,\cos \left ( dx+c \right ) -24\, \left ( dx+c \right ) \sin \left ( dx+c \right ) \right ) }{{d}^{4}}}+2\,{\frac{ab \left ( - \left ( dx+c \right ) ^{3}\cos \left ( dx+c \right ) +3\, \left ( dx+c \right ) ^{2}\sin \left ( dx+c \right ) -6\,\sin \left ( dx+c \right ) +6\, \left ( dx+c \right ) \cos \left ( dx+c \right ) \right ) }{{d}^{2}}}+10\,{\frac{{b}^{2}{c}^{2} \left ( - \left ( dx+c \right ) ^{3}\cos \left ( dx+c \right ) +3\, \left ( dx+c \right ) ^{2}\sin \left ( dx+c \right ) -6\,\sin \left ( dx+c \right ) +6\, \left ( dx+c \right ) \cos \left ( dx+c \right ) \right ) }{{d}^{4}}}-6\,{\frac{abc \left ( - \left ( dx+c \right ) ^{2}\cos \left ( dx+c \right ) +2\,\cos \left ( dx+c \right ) +2\, \left ( dx+c \right ) \sin \left ( dx+c \right ) \right ) }{{d}^{2}}}-10\,{\frac{{b}^{2}{c}^{3} \left ( - \left ( dx+c \right ) ^{2}\cos \left ( dx+c \right ) +2\,\cos \left ( dx+c \right ) +2\, \left ( dx+c \right ) \sin \left ( dx+c \right ) \right ) }{{d}^{4}}}+{a}^{2} \left ( \sin \left ( dx+c \right ) - \left ( dx+c \right ) \cos \left ( dx+c \right ) \right ) +6\,{\frac{ab{c}^{2} \left ( \sin \left ( dx+c \right ) - \left ( dx+c \right ) \cos \left ( dx+c \right ) \right ) }{{d}^{2}}}+5\,{\frac{{b}^{2}{c}^{4} \left ( \sin \left ( dx+c \right ) - \left ( dx+c \right ) \cos \left ( dx+c \right ) \right ) }{{d}^{4}}}+{a}^{2}c\cos \left ( dx+c \right ) +2\,{\frac{ab{c}^{3}\cos \left ( dx+c \right ) }{{d}^{2}}}+{\frac{{b}^{2}{c}^{5}\cos \left ( dx+c \right ) }{{d}^{4}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d^2*(1/d^4*b^2*(-(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))-5/d^4*b^2*c*(-(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^2*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))+10/d^4*b^2*c^2*(-(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^2*a*b*c*(-(d*x+c)^2*cos(d*x+c)+2*cos(d*x+c)+2*(d*x+c)*sin(d*x+c))-10/d^4*b^2*c^3*(-(d*x+c)
^2*cos(d*x+c)+2*cos(d*x+c)+2*(d*x+c)*sin(d*x+c))+a^2*(sin(d*x+c)-(d*x+c)*cos(d*x+c))+6/d^2*a*b*c^2*(sin(d*x+c)
-(d*x+c)*cos(d*x+c))+5/d^4*b^2*c^4*(sin(d*x+c)-(d*x+c)*cos(d*x+c))+a^2*c*cos(d*x+c)+2/d^2*a*b*c^3*cos(d*x+c)+1
/d^4*b^2*c^5*cos(d*x+c))

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Maxima [B]  time = 1.11577, size = 591, normalized size = 3.19 \begin{align*} \frac{a^{2} c \cos \left (d x + c\right ) + \frac{b^{2} c^{5} \cos \left (d x + c\right )}{d^{4}} + \frac{2 \, a b c^{3} \cos \left (d x + c\right )}{d^{2}} -{\left ({\left (d x + c\right )} \cos \left (d x + c\right ) - \sin \left (d x + c\right )\right )} a^{2} - \frac{5 \,{\left ({\left (d x + c\right )} \cos \left (d x + c\right ) - \sin \left (d x + c\right )\right )} b^{2} c^{4}}{d^{4}} - \frac{6 \,{\left ({\left (d x + c\right )} \cos \left (d x + c\right ) - \sin \left (d x + c\right )\right )} a b c^{2}}{d^{2}} + \frac{10 \,{\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^{2} c^{3}}{d^{4}} + \frac{6 \,{\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 b c}{d^{2}} - \frac{10 \,{\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^{2} c^{2}}{d^{4}} - \frac{2 \,{\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 b}{d^{2}} + \frac{5 \,{\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^{2} c}{d^{4}} - \frac{{\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^{2}}{d^{4}}}{d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a^2*c*cos(d*x + c) + b^2*c^5*cos(d*x + c)/d^4 + 2*a*b*c^3*cos(d*x + c)/d^2 - ((d*x + c)*cos(d*x + c) - sin(d*
x + c))*a^2 - 5*((d*x + c)*cos(d*x + c) - sin(d*x + c))*b^2*c^4/d^4 - 6*((d*x + c)*cos(d*x + c) - sin(d*x + c)
)*a*b*c^2/d^2 + 10*(((d*x + c)^2 - 2)*cos(d*x + c) - 2*(d*x + c)*sin(d*x + c))*b^2*c^3/d^4 + 6*(((d*x + c)^2 -
2)*cos(d*x + c) - 2*(d*x + c)*sin(d*x + c))*a*b*c/d^2 - 10*(((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^2/d^4 - 2*(((d*x + c)^3 - 6*d*x - 6*c)*cos(d*x + c) - 3*((d*x + c)^2 - 2)*si
n(d*x + c))*a*b/d^2 + 5*(((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/d^4 - (((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/d^4)/d^2

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Fricas [A]  time = 1.7336, size = 270, normalized size = 1.46 \begin{align*} -\frac{{\left (b^{2} d^{5} x^{5} + 2 \,{\left (a b d^{5} - 10 \, b^{2} d^{3}\right )} x^{3} +{\left (a^{2} d^{5} - 12 \, a b d^{3} + 120 \, b^{2} d\right )} x\right )} \cos \left (d x + c\right ) -{\left (5 \, b^{2} d^{4} x^{4} + a^{2} d^{4} - 12 \, a b d^{2} + 6 \,{\left (a b d^{4} - 10 \, b^{2} d^{2}\right )} x^{2} + 120 \, b^{2}\right )} \sin \left (d x + c\right )}{d^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((b^2*d^5*x^5 + 2*(a*b*d^5 - 10*b^2*d^3)*x^3 + (a^2*d^5 - 12*a*b*d^3 + 120*b^2*d)*x)*cos(d*x + c) - (5*b^2*d^
4*x^4 + a^2*d^4 - 12*a*b*d^2 + 6*(a*b*d^4 - 10*b^2*d^2)*x^2 + 120*b^2)*sin(d*x + c))/d^6

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Sympy [A]  time = 5.39049, size = 226, normalized size = 1.22 \begin{align*} \begin{cases} - \frac{a^{2} x \cos{\left (c + d x \right )}}{d} + \frac{a^{2} \sin{\left (c + d x \right )}}{d^{2}} - \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^{5} \cos{\left (c + d x \right )}}{d} + \frac{5 b^{2} x^{4} \sin{\left (c + d x \right )}}{d^{2}} + \frac{20 b^{2} x^{3} \cos{\left (c + d x \right )}}{d^{3}} - \frac{60 b^{2} x^{2} \sin{\left (c + d x \right )}}{d^{4}} - \frac{120 b^{2} x \cos{\left (c + d x \right )}}{d^{5}} + \frac{120 b^{2} \sin{\left (c + d x \right )}}{d^{6}} & \text{for}\: d \neq 0 \\\left (\frac{a^{2} x^{2}}{2} + \frac{a b x^{4}}{2} + \frac{b^{2} x^{6}}{6}\right ) \sin{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-a**2*x*cos(c + d*x)/d + a**2*sin(c + d*x)/d**2 - 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**5*cos(c + d*x)/d + 5*b**2*x**4*sin(c
+ d*x)/d**2 + 20*b**2*x**3*cos(c + d*x)/d**3 - 60*b**2*x**2*sin(c + d*x)/d**4 - 120*b**2*x*cos(c + d*x)/d**5
+ 120*b**2*sin(c + d*x)/d**6, Ne(d, 0)), ((a**2*x**2/2 + a*b*x**4/2 + b**2*x**6/6)*sin(c), True))

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Giac [A]  time = 1.10488, size = 174, normalized size = 0.94 \begin{align*} -\frac{{\left (b^{2} d^{5} x^{5} + 2 \, a b d^{5} x^{3} + a^{2} d^{5} x - 20 \, b^{2} d^{3} x^{3} - 12 \, a b d^{3} x + 120 \, b^{2} d x\right )} \cos \left (d x + c\right )}{d^{6}} + \frac{{\left (5 \, b^{2} d^{4} x^{4} + 6 \, a b d^{4} x^{2} + a^{2} d^{4} - 60 \, b^{2} d^{2} x^{2} - 12 \, a b d^{2} + 120 \, b^{2}\right )} \sin \left (d x + c\right )}{d^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(b^2*d^5*x^5 + 2*a*b*d^5*x^3 + a^2*d^5*x - 20*b^2*d^3*x^3 - 12*a*b*d^3*x + 120*b^2*d*x)*cos(d*x + c)/d^6 + (5
*b^2*d^4*x^4 + 6*a*b*d^4*x^2 + a^2*d^4 - 60*b^2*d^2*x^2 - 12*a*b*d^2 + 120*b^2)*sin(d*x + c)/d^6