3.1.2 \(\int (c+d x)^3 \cos (a+b x) \, dx\) [2]

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

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 70, normalized size of antiderivative = 1.00, 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 = {3377, 2718} \begin {gather*} -\frac {6 d^3 \cos (a+b x)}{b^4}-\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 {(c+d x)^3 \sin (a+b x)}{b} \end {gather*}

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 2718

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

Rule 3377

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]

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*}

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Mathematica [A]
time = 0.12, size = 61, normalized size = 0.87 \begin {gather*} \frac {3 d \left (-2 d^2+b^2 (c+d x)^2\right ) \cos (a+b x)+b (c+d x) \left (-6 d^2+b^2 (c+d x)^2\right ) \sin (a+b x)}{b^4} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. \(301\) vs. \(2(70)=140\).
time = 0.09, size = 302, normalized size = 4.31

method	result	size
risch	\(\frac {3 d \left (d^{2} x^{2} b^{2}+2 b^{2} c d x +b^{2} c^{2}-2 d^{2}\right ) \cos \left (b x +a \right )}{b^{4}}+\frac {\left (b^{2} d^{3} x^{3}+3 b^{2} c \,d^{2} x^{2}+3 b^{2} c^{2} d x +b^{2} c^{3}-6 d^{3} x -6 c \,d^{2}\right ) \sin \left (b x +a \right )}{b^{3}}\)	\(107\)
norman	\(\frac {\frac {\left (-6 b^{2} c^{2} d +12 d^{3}\right ) \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b^{4}}+\frac {3 d^{3} x^{2}}{b^{2}}+\frac {6 c \,d^{2} x}{b^{2}}+\frac {2 c \left (b^{2} c^{2}-6 d^{2}\right ) \tan \left (\frac {b x}{2}+\frac {a}{2}\right )}{b^{3}}+\frac {2 d^{3} x^{3} \tan \left (\frac {b x}{2}+\frac {a}{2}\right )}{b}-\frac {3 d^{3} x^{2} \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b^{2}}+\frac {6 c \,d^{2} x^{2} \tan \left (\frac {b x}{2}+\frac {a}{2}\right )}{b}-\frac {6 c \,d^{2} x \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b^{2}}+\frac {6 d \left (b^{2} c^{2}-2 d^{2}\right ) x \tan \left (\frac {b x}{2}+\frac {a}{2}\right )}{b^{3}}}{1+\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )}\)	\(210\)
derivativedivides	\(\frac {-\frac {a^{3} d^{3} \sin \left (b x +a \right )}{b^{3}}+\frac {3 a^{2} c \,d^{2} \sin \left (b x +a \right )}{b^{2}}+\frac {3 a^{2} d^{3} \left (\cos \left (b x +a \right )+\left (b x +a \right ) \sin \left (b x +a \right )\right )}{b^{3}}-\frac {3 a \,c^{2} d \sin \left (b x +a \right )}{b}-\frac {6 a c \,d^{2} \left (\cos \left (b x +a \right )+\left (b x +a \right ) \sin \left (b x +a \right )\right )}{b^{2}}-\frac {3 a \,d^{3} \left (\left (b x +a \right )^{2} \sin \left (b x +a \right )-2 \sin \left (b x +a \right )+2 \left (b x +a \right ) \cos \left (b x +a \right )\right )}{b^{3}}+c^{3} \sin \left (b x +a \right )+\frac {3 c^{2} d \left (\cos \left (b x +a \right )+\left (b x +a \right ) \sin \left (b x +a \right )\right )}{b}+\frac {3 c \,d^{2} \left (\left (b x +a \right )^{2} \sin \left (b x +a \right )-2 \sin \left (b x +a \right )+2 \left (b x +a \right ) \cos \left (b x +a \right )\right )}{b^{2}}+\frac {d^{3} \left (\left (b x +a \right )^{3} \sin \left (b x +a \right )+3 \left (b x +a \right )^{2} \cos \left (b x +a \right )-6 \cos \left (b x +a \right )-6 \left (b x +a \right ) \sin \left (b x +a \right )\right )}{b^{3}}}{b}\)	\(302\)
default	\(\frac {-\frac {a^{3} d^{3} \sin \left (b x +a \right )}{b^{3}}+\frac {3 a^{2} c \,d^{2} \sin \left (b x +a \right )}{b^{2}}+\frac {3 a^{2} d^{3} \left (\cos \left (b x +a \right )+\left (b x +a \right ) \sin \left (b x +a \right )\right )}{b^{3}}-\frac {3 a \,c^{2} d \sin \left (b x +a \right )}{b}-\frac {6 a c \,d^{2} \left (\cos \left (b x +a \right )+\left (b x +a \right ) \sin \left (b x +a \right )\right )}{b^{2}}-\frac {3 a \,d^{3} \left (\left (b x +a \right )^{2} \sin \left (b x +a \right )-2 \sin \left (b x +a \right )+2 \left (b x +a \right ) \cos \left (b x +a \right )\right )}{b^{3}}+c^{3} \sin \left (b x +a \right )+\frac {3 c^{2} d \left (\cos \left (b x +a \right )+\left (b x +a \right ) \sin \left (b x +a \right )\right )}{b}+\frac {3 c \,d^{2} \left (\left (b x +a \right )^{2} \sin \left (b x +a \right )-2 \sin \left (b x +a \right )+2 \left (b x +a \right ) \cos \left (b x +a \right )\right )}{b^{2}}+\frac {d^{3} \left (\left (b x +a \right )^{3} \sin \left (b x +a \right )+3 \left (b x +a \right )^{2} \cos \left (b x +a \right )-6 \cos \left (b x +a \right )-6 \left (b x +a \right ) \sin \left (b x +a \right )\right )}{b^{3}}}{b}\)	\(302\)
meijerg	\(\frac {8 d^{3} \cos \left (a \right ) \sqrt {\pi }\, \left (\frac {3}{4 \sqrt {\pi }}-\frac {\left (-\frac {3 x^{2} b^{2}}{2}+3\right ) \cos \left (b x \right )}{4 \sqrt {\pi }}-\frac {x b \left (-\frac {x^{2} b^{2}}{2}+3\right ) \sin \left (b x \right )}{4 \sqrt {\pi }}\right )}{b^{4}}-\frac {8 d^{3} \sin \left (a \right ) \sqrt {\pi }\, \left (\frac {x b \left (-\frac {5 x^{2} b^{2}}{2}+15\right ) \cos \left (b x \right )}{20 \sqrt {\pi }}-\frac {\left (-\frac {15 x^{2} b^{2}}{2}+15\right ) \sin \left (b x \right )}{20 \sqrt {\pi }}\right )}{b^{4}}+\frac {12 c \,d^{2} \cos \left (a \right ) \sqrt {\pi }\, \left (\frac {x \left (b^{2}\right )^{\frac {3}{2}} \cos \left (b x \right )}{2 \sqrt {\pi }\, b^{2}}-\frac {\left (b^{2}\right )^{\frac {3}{2}} \left (-\frac {3 x^{2} b^{2}}{2}+3\right ) \sin \left (b x \right )}{6 \sqrt {\pi }\, b^{3}}\right )}{b^{2} \sqrt {b^{2}}}-\frac {12 c \,d^{2} \sin \left (a \right ) \sqrt {\pi }\, \left (-\frac {1}{2 \sqrt {\pi }}+\frac {\left (-\frac {x^{2} b^{2}}{2}+1\right ) \cos \left (b x \right )}{2 \sqrt {\pi }}+\frac {x b \sin \left (b x \right )}{2 \sqrt {\pi }}\right )}{b^{3}}+\frac {6 d \,c^{2} \cos \left (a \right ) \sqrt {\pi }\, \left (-\frac {1}{2 \sqrt {\pi }}+\frac {\cos \left (b x \right )}{2 \sqrt {\pi }}+\frac {x b \sin \left (b x \right )}{2 \sqrt {\pi }}\right )}{b^{2}}-\frac {6 d \,c^{2} \sin \left (a \right ) \sqrt {\pi }\, \left (-\frac {x b \cos \left (b x \right )}{2 \sqrt {\pi }}+\frac {\sin \left (b x \right )}{2 \sqrt {\pi }}\right )}{b^{2}}+\frac {c^{3} \cos \left (a \right ) \sin \left (b x \right )}{b}-\frac {c^{3} \sin \left (a \right ) \sqrt {\pi }\, \left (\frac {1}{\sqrt {\pi }}-\frac {\cos \left (b x \right )}{\sqrt {\pi }}\right )}{b}\)	\(344\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 278 vs. \(2 (70) = 140\).
time = 0.33, size = 278, normalized size = 3.97 \begin {gather*} \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 {gather*}

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 = 0.37, size = 109, normalized size = 1.56 \begin {gather*} \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 {gather*}

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 [B] Leaf count of result is larger than twice the leaf count of optimal. 202 vs. \(2 (70) = 140\).
time = 0.22, size = 202, normalized size = 2.89 \begin {gather*} \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 {gather*}

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 = 0.43, size = 110, normalized size = 1.57 \begin {gather*} \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 {gather*}

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

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Mupad [B]
time = 0.29, size = 147, normalized size = 2.10 \begin {gather*} \frac {3\,d^3\,x^2\,\cos \left (a+b\,x\right )}{b^2}-\frac {\sin \left (a+b\,x\right )\,\left (6\,c\,d^2-b^2\,c^3\right )}{b^3}-\frac {3\,\cos \left (a+b\,x\right )\,\left (2\,d^3-b^2\,c^2\,d\right )}{b^4}+\frac {d^3\,x^3\,\sin \left (a+b\,x\right )}{b}-\frac {3\,x\,\sin \left (a+b\,x\right )\,\left (2\,d^3-b^2\,c^2\,d\right )}{b^3}+\frac {6\,c\,d^2\,x\,\cos \left (a+b\,x\right )}{b^2}+\frac {3\,c\,d^2\,x^2\,\sin \left (a+b\,x\right )}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(a + b*x)*(c + d*x)^3,x)

[Out]

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

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