3.2 \(\int (c+d x)^3 \cos (a+b x) \, dx\)

Optimal. Leaf size=70 \[ -\frac{6 d^2 (c+d x) \sin (a+b x)}{b^3}+\frac{3 d (c+d x)^2 \cos (a+b x)}{b^2}-\frac{6 d^3 \cos (a+b x)}{b^4}+\frac{(c+d x)^3 \sin (a+b x)}{b} \]

[Out]

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

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Rubi [A]  time = 0.0655573, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3296, 2638} \[ -\frac{6 d^2 (c+d x) \sin (a+b x)}{b^3}+\frac{3 d (c+d x)^2 \cos (a+b x)}{b^2}-\frac{6 d^3 \cos (a+b x)}{b^4}+\frac{(c+d x)^3 \sin (a+b x)}{b} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^3*Cos[a + b*x],x]

[Out]

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

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 2638

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

Rubi steps

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

Mathematica [A]  time = 0.22181, size = 61, normalized size = 0.87 \[ \frac{b (c+d x) \sin (a+b x) \left (b^2 (c+d x)^2-6 d^2\right )+3 d \cos (a+b x) \left (b^2 (c+d x)^2-2 d^2\right )}{b^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^3*Cos[a + b*x],x]

[Out]

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

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Maple [B]  time = 0.029, size = 302, normalized size = 4.3 \begin{align*}{\frac{1}{b} \left ({\frac{{d}^{3} \left ( \left ( bx+a \right ) ^{3}\sin \left ( bx+a \right ) +3\, \left ( bx+a \right ) ^{2}\cos \left ( bx+a \right ) -6\,\cos \left ( bx+a \right ) -6\, \left ( bx+a \right ) \sin \left ( bx+a \right ) \right ) }{{b}^{3}}}-3\,{\frac{a{d}^{3} \left ( \left ( bx+a \right ) ^{2}\sin \left ( bx+a \right ) -2\,\sin \left ( bx+a \right ) +2\, \left ( bx+a \right ) \cos \left ( bx+a \right ) \right ) }{{b}^{3}}}+3\,{\frac{c{d}^{2} \left ( \left ( bx+a \right ) ^{2}\sin \left ( bx+a \right ) -2\,\sin \left ( bx+a \right ) +2\, \left ( bx+a \right ) \cos \left ( bx+a \right ) \right ) }{{b}^{2}}}+3\,{\frac{{a}^{2}{d}^{3} \left ( \cos \left ( bx+a \right ) + \left ( bx+a \right ) \sin \left ( bx+a \right ) \right ) }{{b}^{3}}}-6\,{\frac{ac{d}^{2} \left ( \cos \left ( bx+a \right ) + \left ( bx+a \right ) \sin \left ( bx+a \right ) \right ) }{{b}^{2}}}+3\,{\frac{{c}^{2}d \left ( \cos \left ( bx+a \right ) + \left ( bx+a \right ) \sin \left ( bx+a \right ) \right ) }{b}}-{\frac{{a}^{3}{d}^{3}\sin \left ( bx+a \right ) }{{b}^{3}}}+3\,{\frac{{a}^{2}c{d}^{2}\sin \left ( bx+a \right ) }{{b}^{2}}}-3\,{\frac{a{c}^{2}d\sin \left ( bx+a \right ) }{b}}+{c}^{3}\sin \left ( bx+a \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.01193, size = 375, normalized size = 5.36 \begin{align*} \frac{c^{3} \sin \left (b x + a\right ) - \frac{3 \, a c^{2} d \sin \left (b x + a\right )}{b} + \frac{3 \, a^{2} c d^{2} \sin \left (b x + a\right )}{b^{2}} - \frac{a^{3} d^{3} \sin \left (b x + a\right )}{b^{3}} + \frac{3 \,{\left ({\left (b x + a\right )} \sin \left (b x + a\right ) + \cos \left (b x + a\right )\right )} c^{2} d}{b} - \frac{6 \,{\left ({\left (b x + a\right )} \sin \left (b x + a\right ) + \cos \left (b x + a\right )\right )} a c d^{2}}{b^{2}} + \frac{3 \,{\left ({\left (b x + a\right )} \sin \left (b x + a\right ) + \cos \left (b x + a\right )\right )} a^{2} d^{3}}{b^{3}} + \frac{3 \,{\left (2 \,{\left (b x + a\right )} \cos \left (b x + a\right ) +{\left ({\left (b x + a\right )}^{2} - 2\right )} \sin \left (b x + a\right )\right )} c d^{2}}{b^{2}} - \frac{3 \,{\left (2 \,{\left (b x + a\right )} \cos \left (b x + a\right ) +{\left ({\left (b x + a\right )}^{2} - 2\right )} \sin \left (b x + a\right )\right )} a d^{3}}{b^{3}} + \frac{{\left (3 \,{\left ({\left (b x + a\right )}^{2} - 2\right )} \cos \left (b x + a\right ) +{\left ({\left (b x + a\right )}^{3} - 6 \, b x - 6 \, a\right )} \sin \left (b x + a\right )\right )} d^{3}}{b^{3}}}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(c^3*sin(b*x + a) - 3*a*c^2*d*sin(b*x + a)/b + 3*a^2*c*d^2*sin(b*x + a)/b^2 - a^3*d^3*sin(b*x + a)/b^3 + 3*((b
*x + a)*sin(b*x + a) + cos(b*x + a))*c^2*d/b - 6*((b*x + a)*sin(b*x + a) + cos(b*x + a))*a*c*d^2/b^2 + 3*((b*x
+ a)*sin(b*x + a) + cos(b*x + a))*a^2*d^3/b^3 + 3*(2*(b*x + a)*cos(b*x + a) + ((b*x + a)^2 - 2)*sin(b*x + a))
*c*d^2/b^2 - 3*(2*(b*x + a)*cos(b*x + a) + ((b*x + a)^2 - 2)*sin(b*x + a))*a*d^3/b^3 + (3*((b*x + a)^2 - 2)*co
s(b*x + a) + ((b*x + a)^3 - 6*b*x - 6*a)*sin(b*x + a))*d^3/b^3)/b

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Fricas [A]  time = 1.06501, size = 228, normalized size = 3.26 \begin{align*} \frac{3 \,{\left (b^{2} d^{3} x^{2} + 2 \, b^{2} c d^{2} x + b^{2} c^{2} d - 2 \, d^{3}\right )} \cos \left (b x + a\right ) +{\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + b^{3} c^{3} - 6 \, b c d^{2} + 3 \,{\left (b^{3} c^{2} d - 2 \, b d^{3}\right )} x\right )} \sin \left (b x + a\right )}{b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 1.31311, size = 202, normalized size = 2.89 \begin{align*} \begin{cases} \frac{c^{3} \sin{\left (a + b x \right )}}{b} + \frac{3 c^{2} d x \sin{\left (a + b x \right )}}{b} + \frac{3 c d^{2} x^{2} \sin{\left (a + b x \right )}}{b} + \frac{d^{3} x^{3} \sin{\left (a + b x \right )}}{b} + \frac{3 c^{2} d \cos{\left (a + b x \right )}}{b^{2}} + \frac{6 c d^{2} x \cos{\left (a + b x \right )}}{b^{2}} + \frac{3 d^{3} x^{2} \cos{\left (a + b x \right )}}{b^{2}} - \frac{6 c d^{2} \sin{\left (a + b x \right )}}{b^{3}} - \frac{6 d^{3} x \sin{\left (a + b x \right )}}{b^{3}} - \frac{6 d^{3} \cos{\left (a + b x \right )}}{b^{4}} & \text{for}\: b \neq 0 \\\left (c^{3} x + \frac{3 c^{2} d x^{2}}{2} + c d^{2} x^{3} + \frac{d^{3} x^{4}}{4}\right ) \cos{\left (a \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.12317, size = 149, normalized size = 2.13 \begin{align*} \frac{3 \,{\left (b^{2} d^{3} x^{2} + 2 \, b^{2} c d^{2} x + b^{2} c^{2} d - 2 \, d^{3}\right )} \cos \left (b x + a\right )}{b^{4}} + \frac{{\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + b^{3} c^{3} - 6 \, b d^{3} x - 6 \, b c d^{2}\right )} \sin \left (b x + a\right )}{b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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