3.1 \(\int (c+d x)^4 \cos (a+b x) \, dx\)

Optimal. Leaf size=91 \[ -\frac{12 d^2 (c+d x)^2 \sin (a+b x)}{b^3}-\frac{24 d^3 (c+d x) \cos (a+b x)}{b^4}+\frac{4 d (c+d x)^3 \cos (a+b x)}{b^2}+\frac{24 d^4 \sin (a+b x)}{b^5}+\frac{(c+d x)^4 \sin (a+b x)}{b} \]

[Out]

(-24*d^3*(c + d*x)*Cos[a + b*x])/b^4 + (4*d*(c + d*x)^3*Cos[a + b*x])/b^2 + (24*d^4*Sin[a + b*x])/b^5 - (12*d^
2*(c + d*x)^2*Sin[a + b*x])/b^3 + ((c + d*x)^4*Sin[a + b*x])/b

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Rubi [A]  time = 0.0932382, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3296, 2637} \[ -\frac{12 d^2 (c+d x)^2 \sin (a+b x)}{b^3}-\frac{24 d^3 (c+d x) \cos (a+b x)}{b^4}+\frac{4 d (c+d x)^3 \cos (a+b x)}{b^2}+\frac{24 d^4 \sin (a+b x)}{b^5}+\frac{(c+d x)^4 \sin (a+b x)}{b} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^4*Cos[a + b*x],x]

[Out]

(-24*d^3*(c + d*x)*Cos[a + b*x])/b^4 + (4*d*(c + d*x)^3*Cos[a + b*x])/b^2 + (24*d^4*Sin[a + b*x])/b^5 - (12*d^
2*(c + d*x)^2*Sin[a + b*x])/b^3 + ((c + d*x)^4*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 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 (c+d x)^4 \cos (a+b x) \, dx &=\frac{(c+d x)^4 \sin (a+b x)}{b}-\frac{(4 d) \int (c+d x)^3 \sin (a+b x) \, dx}{b}\\ &=\frac{4 d (c+d x)^3 \cos (a+b x)}{b^2}+\frac{(c+d x)^4 \sin (a+b x)}{b}-\frac{\left (12 d^2\right ) \int (c+d x)^2 \cos (a+b x) \, dx}{b^2}\\ &=\frac{4 d (c+d x)^3 \cos (a+b x)}{b^2}-\frac{12 d^2 (c+d x)^2 \sin (a+b x)}{b^3}+\frac{(c+d x)^4 \sin (a+b x)}{b}+\frac{\left (24 d^3\right ) \int (c+d x) \sin (a+b x) \, dx}{b^3}\\ &=-\frac{24 d^3 (c+d x) \cos (a+b x)}{b^4}+\frac{4 d (c+d x)^3 \cos (a+b x)}{b^2}-\frac{12 d^2 (c+d x)^2 \sin (a+b x)}{b^3}+\frac{(c+d x)^4 \sin (a+b x)}{b}+\frac{\left (24 d^4\right ) \int \cos (a+b x) \, dx}{b^4}\\ &=-\frac{24 d^3 (c+d x) \cos (a+b x)}{b^4}+\frac{4 d (c+d x)^3 \cos (a+b x)}{b^2}+\frac{24 d^4 \sin (a+b x)}{b^5}-\frac{12 d^2 (c+d x)^2 \sin (a+b x)}{b^3}+\frac{(c+d x)^4 \sin (a+b x)}{b}\\ \end{align*}

Mathematica [A]  time = 0.337115, size = 76, normalized size = 0.84 \[ \frac{\sin (a+b x) \left (-12 b^2 d^2 (c+d x)^2+b^4 (c+d x)^4+24 d^4\right )+4 b d (c+d x) \cos (a+b x) \left (b^2 (c+d x)^2-6 d^2\right )}{b^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^4*Cos[a + b*x],x]

[Out]

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

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Maple [B]  time = 0.029, size = 539, normalized size = 5.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^4*cos(b*x+a),x)

[Out]

1/b*(1/b^4*d^4*((b*x+a)^4*sin(b*x+a)+4*(b*x+a)^3*cos(b*x+a)-12*(b*x+a)^2*sin(b*x+a)+24*sin(b*x+a)-24*(b*x+a)*c
os(b*x+a))-4/b^4*a*d^4*((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))+4/b^3*c
*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))+6/b^4*a^2*d^4*((b*x+a)^2*
sin(b*x+a)-2*sin(b*x+a)+2*(b*x+a)*cos(b*x+a))-12/b^3*a*c*d^3*((b*x+a)^2*sin(b*x+a)-2*sin(b*x+a)+2*(b*x+a)*cos(
b*x+a))+6/b^2*c^2*d^2*((b*x+a)^2*sin(b*x+a)-2*sin(b*x+a)+2*(b*x+a)*cos(b*x+a))-4/b^4*a^3*d^4*(cos(b*x+a)+(b*x+
a)*sin(b*x+a))+12/b^3*a^2*c*d^3*(cos(b*x+a)+(b*x+a)*sin(b*x+a))-12/b^2*a*c^2*d^2*(cos(b*x+a)+(b*x+a)*sin(b*x+a
))+4/b*c^3*d*(cos(b*x+a)+(b*x+a)*sin(b*x+a))+1/b^4*a^4*d^4*sin(b*x+a)-4/b^3*a^3*c*d^3*sin(b*x+a)+6/b^2*a^2*c^2
*d^2*sin(b*x+a)-4/b*a*c^3*d*sin(b*x+a)+c^4*sin(b*x+a))

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Maxima [B]  time = 1.1163, size = 649, normalized size = 7.13 \begin{align*} \frac{c^{4} \sin \left (b x + a\right ) - \frac{4 \, a c^{3} d \sin \left (b x + a\right )}{b} + \frac{6 \, a^{2} c^{2} d^{2} \sin \left (b x + a\right )}{b^{2}} - \frac{4 \, a^{3} c d^{3} \sin \left (b x + a\right )}{b^{3}} + \frac{a^{4} d^{4} \sin \left (b x + a\right )}{b^{4}} + \frac{4 \,{\left ({\left (b x + a\right )} \sin \left (b x + a\right ) + \cos \left (b x + a\right )\right )} c^{3} d}{b} - \frac{12 \,{\left ({\left (b x + a\right )} \sin \left (b x + a\right ) + \cos \left (b x + a\right )\right )} a c^{2} d^{2}}{b^{2}} + \frac{12 \,{\left ({\left (b x + a\right )} \sin \left (b x + a\right ) + \cos \left (b x + a\right )\right )} a^{2} c d^{3}}{b^{3}} - \frac{4 \,{\left ({\left (b x + a\right )} \sin \left (b x + a\right ) + \cos \left (b x + a\right )\right )} a^{3} d^{4}}{b^{4}} + \frac{6 \,{\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^{2} d^{2}}{b^{2}} - \frac{12 \,{\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 c d^{3}}{b^{3}} + \frac{6 \,{\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^{2} d^{4}}{b^{4}} + \frac{4 \,{\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 )} c d^{3}}{b^{3}} - \frac{4 \,{\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 )} a d^{4}}{b^{4}} + \frac{{\left (4 \,{\left ({\left (b x + a\right )}^{3} - 6 \, b x - 6 \, a\right )} \cos \left (b x + a\right ) +{\left ({\left (b x + a\right )}^{4} - 12 \,{\left (b x + a\right )}^{2} + 24\right )} \sin \left (b x + a\right )\right )} d^{4}}{b^{4}}}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^4*cos(b*x+a),x, algorithm="maxima")

[Out]

(c^4*sin(b*x + a) - 4*a*c^3*d*sin(b*x + a)/b + 6*a^2*c^2*d^2*sin(b*x + a)/b^2 - 4*a^3*c*d^3*sin(b*x + a)/b^3 +
a^4*d^4*sin(b*x + a)/b^4 + 4*((b*x + a)*sin(b*x + a) + cos(b*x + a))*c^3*d/b - 12*((b*x + a)*sin(b*x + a) + c
os(b*x + a))*a*c^2*d^2/b^2 + 12*((b*x + a)*sin(b*x + a) + cos(b*x + a))*a^2*c*d^3/b^3 - 4*((b*x + a)*sin(b*x +
a) + cos(b*x + a))*a^3*d^4/b^4 + 6*(2*(b*x + a)*cos(b*x + a) + ((b*x + a)^2 - 2)*sin(b*x + a))*c^2*d^2/b^2 -
12*(2*(b*x + a)*cos(b*x + a) + ((b*x + a)^2 - 2)*sin(b*x + a))*a*c*d^3/b^3 + 6*(2*(b*x + a)*cos(b*x + a) + ((b
*x + a)^2 - 2)*sin(b*x + a))*a^2*d^4/b^4 + 4*(3*((b*x + a)^2 - 2)*cos(b*x + a) + ((b*x + a)^3 - 6*b*x - 6*a)*s
in(b*x + a))*c*d^3/b^3 - 4*(3*((b*x + a)^2 - 2)*cos(b*x + a) + ((b*x + a)^3 - 6*b*x - 6*a)*sin(b*x + a))*a*d^4
/b^4 + (4*((b*x + a)^3 - 6*b*x - 6*a)*cos(b*x + a) + ((b*x + a)^4 - 12*(b*x + a)^2 + 24)*sin(b*x + a))*d^4/b^4
)/b

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Fricas [A]  time = 1.02636, size = 347, normalized size = 3.81 \begin{align*} \frac{4 \,{\left (b^{3} d^{4} x^{3} + 3 \, b^{3} c d^{3} x^{2} + b^{3} c^{3} d - 6 \, b c d^{3} + 3 \,{\left (b^{3} c^{2} d^{2} - 2 \, b d^{4}\right )} x\right )} \cos \left (b x + a\right ) +{\left (b^{4} d^{4} x^{4} + 4 \, b^{4} c d^{3} x^{3} + b^{4} c^{4} - 12 \, b^{2} c^{2} d^{2} + 24 \, d^{4} + 6 \,{\left (b^{4} c^{2} d^{2} - 2 \, b^{2} d^{4}\right )} x^{2} + 4 \,{\left (b^{4} c^{3} d - 6 \, b^{2} c d^{3}\right )} x\right )} \sin \left (b x + a\right )}{b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^4*cos(b*x+a),x, algorithm="fricas")

[Out]

(4*(b^3*d^4*x^3 + 3*b^3*c*d^3*x^2 + b^3*c^3*d - 6*b*c*d^3 + 3*(b^3*c^2*d^2 - 2*b*d^4)*x)*cos(b*x + a) + (b^4*d
^4*x^4 + 4*b^4*c*d^3*x^3 + b^4*c^4 - 12*b^2*c^2*d^2 + 24*d^4 + 6*(b^4*c^2*d^2 - 2*b^2*d^4)*x^2 + 4*(b^4*c^3*d
- 6*b^2*c*d^3)*x)*sin(b*x + a))/b^5

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Sympy [A]  time = 2.87839, size = 311, normalized size = 3.42 \begin{align*} \begin{cases} \frac{c^{4} \sin{\left (a + b x \right )}}{b} + \frac{4 c^{3} d x \sin{\left (a + b x \right )}}{b} + \frac{6 c^{2} d^{2} x^{2} \sin{\left (a + b x \right )}}{b} + \frac{4 c d^{3} x^{3} \sin{\left (a + b x \right )}}{b} + \frac{d^{4} x^{4} \sin{\left (a + b x \right )}}{b} + \frac{4 c^{3} d \cos{\left (a + b x \right )}}{b^{2}} + \frac{12 c^{2} d^{2} x \cos{\left (a + b x \right )}}{b^{2}} + \frac{12 c d^{3} x^{2} \cos{\left (a + b x \right )}}{b^{2}} + \frac{4 d^{4} x^{3} \cos{\left (a + b x \right )}}{b^{2}} - \frac{12 c^{2} d^{2} \sin{\left (a + b x \right )}}{b^{3}} - \frac{24 c d^{3} x \sin{\left (a + b x \right )}}{b^{3}} - \frac{12 d^{4} x^{2} \sin{\left (a + b x \right )}}{b^{3}} - \frac{24 c d^{3} \cos{\left (a + b x \right )}}{b^{4}} - \frac{24 d^{4} x \cos{\left (a + b x \right )}}{b^{4}} + \frac{24 d^{4} \sin{\left (a + b x \right )}}{b^{5}} & \text{for}\: b \neq 0 \\\left (c^{4} x + 2 c^{3} d x^{2} + 2 c^{2} d^{2} x^{3} + c d^{3} x^{4} + \frac{d^{4} x^{5}}{5}\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)**4*cos(b*x+a),x)

[Out]

Piecewise((c**4*sin(a + b*x)/b + 4*c**3*d*x*sin(a + b*x)/b + 6*c**2*d**2*x**2*sin(a + b*x)/b + 4*c*d**3*x**3*s
in(a + b*x)/b + d**4*x**4*sin(a + b*x)/b + 4*c**3*d*cos(a + b*x)/b**2 + 12*c**2*d**2*x*cos(a + b*x)/b**2 + 12*
c*d**3*x**2*cos(a + b*x)/b**2 + 4*d**4*x**3*cos(a + b*x)/b**2 - 12*c**2*d**2*sin(a + b*x)/b**3 - 24*c*d**3*x*s
in(a + b*x)/b**3 - 12*d**4*x**2*sin(a + b*x)/b**3 - 24*c*d**3*cos(a + b*x)/b**4 - 24*d**4*x*cos(a + b*x)/b**4
+ 24*d**4*sin(a + b*x)/b**5, Ne(b, 0)), ((c**4*x + 2*c**3*d*x**2 + 2*c**2*d**2*x**3 + c*d**3*x**4 + d**4*x**5/
5)*cos(a), True))

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Giac [A]  time = 1.13352, size = 230, normalized size = 2.53 \begin{align*} \frac{4 \,{\left (b^{3} d^{4} x^{3} + 3 \, b^{3} c d^{3} x^{2} + 3 \, b^{3} c^{2} d^{2} x + b^{3} c^{3} d - 6 \, b d^{4} x - 6 \, b c d^{3}\right )} \cos \left (b x + a\right )}{b^{5}} + \frac{{\left (b^{4} d^{4} x^{4} + 4 \, b^{4} c d^{3} x^{3} + 6 \, b^{4} c^{2} d^{2} x^{2} + 4 \, b^{4} c^{3} d x + b^{4} c^{4} - 12 \, b^{2} d^{4} x^{2} - 24 \, b^{2} c d^{3} x - 12 \, b^{2} c^{2} d^{2} + 24 \, d^{4}\right )} \sin \left (b x + a\right )}{b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^4*cos(b*x+a),x, algorithm="giac")

[Out]

4*(b^3*d^4*x^3 + 3*b^3*c*d^3*x^2 + 3*b^3*c^2*d^2*x + b^3*c^3*d - 6*b*d^4*x - 6*b*c*d^3)*cos(b*x + a)/b^5 + (b^
4*d^4*x^4 + 4*b^4*c*d^3*x^3 + 6*b^4*c^2*d^2*x^2 + 4*b^4*c^3*d*x + b^4*c^4 - 12*b^2*d^4*x^2 - 24*b^2*c*d^3*x -
12*b^2*c^2*d^2 + 24*d^4)*sin(b*x + a)/b^5