3.224 \(\int \cos (c+d x) \sin ^3(a+b x) \, dx\)
Optimal. Leaf size=97 \[ -\frac{3 \cos (a+x (b-d)-c)}{8 (b-d)}+\frac{\cos (3 a+x (3 b-d)-c)}{8 (3 b-d)}-\frac{3 \cos (a+x (b+d)+c)}{8 (b+d)}+\frac{\cos (3 a+x (3 b+d)+c)}{8 (3 b+d)} \]
[Out]
(-3*Cos[a - c + (b - d)*x])/(8*(b - d)) + Cos[3*a - c + (3*b - d)*x]/(8*(3*b - d)) - (3*Cos[a + c + (b + d)*x]
)/(8*(b + d)) + Cos[3*a + c + (3*b + d)*x]/(8*(3*b + d))
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Rubi [A] time = 0.0682744, antiderivative size = 97, normalized size of antiderivative = 1.,
number of steps used = 6, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used =
{4574, 2638} \[ -\frac{3 \cos (a+x (b-d)-c)}{8 (b-d)}+\frac{\cos (3 a+x (3 b-d)-c)}{8 (3 b-d)}-\frac{3 \cos (a+x (b+d)+c)}{8 (b+d)}+\frac{\cos (3 a+x (3 b+d)+c)}{8 (3 b+d)} \]
Antiderivative was successfully verified.
[In]
Int[Cos[c + d*x]*Sin[a + b*x]^3,x]
[Out]
(-3*Cos[a - c + (b - d)*x])/(8*(b - d)) + Cos[3*a - c + (3*b - d)*x]/(8*(3*b - d)) - (3*Cos[a + c + (b + d)*x]
)/(8*(b + d)) + Cos[3*a + c + (3*b + d)*x]/(8*(3*b + d))
Rule 4574
Int[Cos[w_]^(q_.)*Sin[v_]^(p_.), x_Symbol] :> Int[ExpandTrigReduce[Sin[v]^p*Cos[w]^q, x], x] /; IGtQ[p, 0] &&
IGtQ[q, 0] && ((PolynomialQ[v, x] && PolynomialQ[w, x]) || (BinomialQ[{v, w}, x] && IndependentQ[Cancel[v/w],
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 \cos (c+d x) \sin ^3(a+b x) \, dx &=\int \left (\frac{3}{8} \sin (a-c+(b-d) x)-\frac{1}{8} \sin (3 a-c+(3 b-d) x)+\frac{3}{8} \sin (a+c+(b+d) x)-\frac{1}{8} \sin (3 a+c+(3 b+d) x)\right ) \, dx\\ &=-\left (\frac{1}{8} \int \sin (3 a-c+(3 b-d) x) \, dx\right )-\frac{1}{8} \int \sin (3 a+c+(3 b+d) x) \, dx+\frac{3}{8} \int \sin (a-c+(b-d) x) \, dx+\frac{3}{8} \int \sin (a+c+(b+d) x) \, dx\\ &=-\frac{3 \cos (a-c+(b-d) x)}{8 (b-d)}+\frac{\cos (3 a-c+(3 b-d) x)}{8 (3 b-d)}-\frac{3 \cos (a+c+(b+d) x)}{8 (b+d)}+\frac{\cos (3 a+c+(3 b+d) x)}{8 (3 b+d)}\\ \end{align*}
Mathematica [A] time = 0.523228, size = 90, normalized size = 0.93 \[ \frac{1}{8} \left (-\frac{3 \cos (a+b x-c-d x)}{b-d}+\frac{\cos (3 a+3 b x-c-d x)}{3 b-d}+\frac{\cos (3 a+3 b x+c+d x)}{3 b+d}-\frac{3 \cos (a+x (b+d)+c)}{b+d}\right ) \]
Antiderivative was successfully verified.
[In]
Integrate[Cos[c + d*x]*Sin[a + b*x]^3,x]
[Out]
((-3*Cos[a - c + b*x - d*x])/(b - d) + Cos[3*a - c + 3*b*x - d*x]/(3*b - d) + Cos[3*a + c + 3*b*x + d*x]/(3*b
+ d) - (3*Cos[a + c + (b + d)*x])/(b + d))/8
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Maple [A] time = 0.02, size = 90, normalized size = 0.9 \begin{align*} -{\frac{3\,\cos \left ( a-c+ \left ( b-d \right ) x \right ) }{8\,b-8\,d}}+{\frac{\cos \left ( 3\,a-c+ \left ( 3\,b-d \right ) x \right ) }{24\,b-8\,d}}-{\frac{3\,\cos \left ( a+c+ \left ( b+d \right ) x \right ) }{8\,b+8\,d}}+{\frac{\cos \left ( 3\,a+c+ \left ( 3\,b+d \right ) x \right ) }{24\,b+8\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cos(d*x+c)*sin(b*x+a)^3,x)
[Out]
-3/8*cos(a-c+(b-d)*x)/(b-d)+1/8*cos(3*a-c+(3*b-d)*x)/(3*b-d)-3/8*cos(a+c+(b+d)*x)/(b+d)+1/8*cos(3*a+c+(3*b+d)*
x)/(3*b+d)
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Maxima [B] time = 1.26686, size = 1060, normalized size = 10.93 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)*sin(b*x+a)^3,x, algorithm="maxima")
[Out]
1/16*((3*b^3*cos(c) - b^2*d*cos(c) - 3*b*d^2*cos(c) + d^3*cos(c))*cos((3*b + d)*x + 3*a + 2*c) + (3*b^3*cos(c)
- b^2*d*cos(c) - 3*b*d^2*cos(c) + d^3*cos(c))*cos((3*b + d)*x + 3*a) + (3*b^3*cos(c) + b^2*d*cos(c) - 3*b*d^2
*cos(c) - d^3*cos(c))*cos(-(3*b - d)*x - 3*a + 2*c) + (3*b^3*cos(c) + b^2*d*cos(c) - 3*b*d^2*cos(c) - d^3*cos(
c))*cos(-(3*b - d)*x - 3*a) - 3*(9*b^3*cos(c) - 9*b^2*d*cos(c) - b*d^2*cos(c) + d^3*cos(c))*cos((b + d)*x + a
+ 2*c) - 3*(9*b^3*cos(c) - 9*b^2*d*cos(c) - b*d^2*cos(c) + d^3*cos(c))*cos((b + d)*x + a) - 3*(9*b^3*cos(c) +
9*b^2*d*cos(c) - b*d^2*cos(c) - d^3*cos(c))*cos(-(b - d)*x - a + 2*c) - 3*(9*b^3*cos(c) + 9*b^2*d*cos(c) - b*d
^2*cos(c) - d^3*cos(c))*cos(-(b - d)*x - a) + (3*b^3*sin(c) - b^2*d*sin(c) - 3*b*d^2*sin(c) + d^3*sin(c))*sin(
(3*b + d)*x + 3*a + 2*c) - (3*b^3*sin(c) - b^2*d*sin(c) - 3*b*d^2*sin(c) + d^3*sin(c))*sin((3*b + d)*x + 3*a)
+ (3*b^3*sin(c) + b^2*d*sin(c) - 3*b*d^2*sin(c) - d^3*sin(c))*sin(-(3*b - d)*x - 3*a + 2*c) - (3*b^3*sin(c) +
b^2*d*sin(c) - 3*b*d^2*sin(c) - d^3*sin(c))*sin(-(3*b - d)*x - 3*a) - 3*(9*b^3*sin(c) - 9*b^2*d*sin(c) - b*d^2
*sin(c) + d^3*sin(c))*sin((b + d)*x + a + 2*c) + 3*(9*b^3*sin(c) - 9*b^2*d*sin(c) - b*d^2*sin(c) + d^3*sin(c))
*sin((b + d)*x + a) - 3*(9*b^3*sin(c) + 9*b^2*d*sin(c) - b*d^2*sin(c) - d^3*sin(c))*sin(-(b - d)*x - a + 2*c)
+ 3*(9*b^3*sin(c) + 9*b^2*d*sin(c) - b*d^2*sin(c) - d^3*sin(c))*sin(-(b - d)*x - a))/(9*b^4*cos(c)^2 + 9*b^4*s
in(c)^2 + (cos(c)^2 + sin(c)^2)*d^4 - 10*(b^2*cos(c)^2 + b^2*sin(c)^2)*d^2)
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Fricas [A] time = 0.504789, size = 247, normalized size = 2.55 \begin{align*} -\frac{{\left (7 \, b^{2} d - d^{3} -{\left (b^{2} d - d^{3}\right )} \cos \left (b x + a\right )^{2}\right )} \sin \left (b x + a\right ) \sin \left (d x + c\right ) - 3 \,{\left ({\left (b^{3} - b d^{2}\right )} \cos \left (b x + a\right )^{3} -{\left (3 \, b^{3} - b d^{2}\right )} \cos \left (b x + a\right )\right )} \cos \left (d x + c\right )}{9 \, b^{4} - 10 \, b^{2} d^{2} + d^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)*sin(b*x+a)^3,x, algorithm="fricas")
[Out]
-((7*b^2*d - d^3 - (b^2*d - d^3)*cos(b*x + a)^2)*sin(b*x + a)*sin(d*x + c) - 3*((b^3 - b*d^2)*cos(b*x + a)^3 -
(3*b^3 - b*d^2)*cos(b*x + a))*cos(d*x + c))/(9*b^4 - 10*b^2*d^2 + d^4)
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Sympy [A] time = 88.4002, size = 964, normalized size = 9.94 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)*sin(b*x+a)**3,x)
[Out]
Piecewise((x*sin(a)**3*cos(c), Eq(b, 0) & Eq(d, 0)), (3*x*sin(a - d*x)**3*cos(c + d*x)/8 + 3*x*sin(a - d*x)**2
*sin(c + d*x)*cos(a - d*x)/8 + 3*x*sin(a - d*x)*cos(a - d*x)**2*cos(c + d*x)/8 + 3*x*sin(c + d*x)*cos(a - d*x)
**3/8 + 5*sin(a - d*x)**2*cos(a - d*x)*cos(c + d*x)/(8*d) + sin(a - d*x)*sin(c + d*x)*cos(a - d*x)**2/(8*d) +
cos(a - d*x)**3*cos(c + d*x)/(4*d), Eq(b, -d)), (x*sin(a - d*x/3)**3*cos(c + d*x)/8 + 3*x*sin(a - d*x/3)**2*si
n(c + d*x)*cos(a - d*x/3)/8 - 3*x*sin(a - d*x/3)*cos(a - d*x/3)**2*cos(c + d*x)/8 - x*sin(c + d*x)*cos(a - d*x
/3)**3/8 + 21*sin(a - d*x/3)**2*cos(a - d*x/3)*cos(c + d*x)/(8*d) + 27*sin(a - d*x/3)*sin(c + d*x)*cos(a - d*x
/3)**2/(8*d) - 5*cos(a - d*x/3)**3*cos(c + d*x)/(4*d), Eq(b, -d/3)), (x*sin(a + d*x/3)**3*cos(c + d*x)/8 - 3*x
*sin(a + d*x/3)**2*sin(c + d*x)*cos(a + d*x/3)/8 - 3*x*sin(a + d*x/3)*cos(a + d*x/3)**2*cos(c + d*x)/8 + x*sin
(c + d*x)*cos(a + d*x/3)**3/8 - 21*sin(a + d*x/3)**2*cos(a + d*x/3)*cos(c + d*x)/(8*d) + 27*sin(a + d*x/3)*sin
(c + d*x)*cos(a + d*x/3)**2/(8*d) + 5*cos(a + d*x/3)**3*cos(c + d*x)/(4*d), Eq(b, d/3)), (3*x*sin(a + d*x)**3*
cos(c + d*x)/8 - 3*x*sin(a + d*x)**2*sin(c + d*x)*cos(a + d*x)/8 + 3*x*sin(a + d*x)*cos(a + d*x)**2*cos(c + d*
x)/8 - 3*x*sin(c + d*x)*cos(a + d*x)**3/8 + sin(a + d*x)**3*sin(c + d*x)/(4*d) - 3*sin(a + d*x)**2*cos(a + d*x
)*cos(c + d*x)/(8*d) + 3*sin(a + d*x)*sin(c + d*x)*cos(a + d*x)**2/(8*d), Eq(b, d)), (-9*b**3*sin(a + b*x)**2*
cos(a + b*x)*cos(c + d*x)/(9*b**4 - 10*b**2*d**2 + d**4) - 6*b**3*cos(a + b*x)**3*cos(c + d*x)/(9*b**4 - 10*b*
*2*d**2 + d**4) - 7*b**2*d*sin(a + b*x)**3*sin(c + d*x)/(9*b**4 - 10*b**2*d**2 + d**4) - 6*b**2*d*sin(a + b*x)
*sin(c + d*x)*cos(a + b*x)**2/(9*b**4 - 10*b**2*d**2 + d**4) + 3*b*d**2*sin(a + b*x)**2*cos(a + b*x)*cos(c + d
*x)/(9*b**4 - 10*b**2*d**2 + d**4) + d**3*sin(a + b*x)**3*sin(c + d*x)/(9*b**4 - 10*b**2*d**2 + d**4), True))
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Giac [A] time = 1.127, size = 120, normalized size = 1.24 \begin{align*} \frac{\cos \left (3 \, b x + d x + 3 \, a + c\right )}{8 \,{\left (3 \, b + d\right )}} + \frac{\cos \left (3 \, b x - d x + 3 \, a - c\right )}{8 \,{\left (3 \, b - d\right )}} - \frac{3 \, \cos \left (b x + d x + a + c\right )}{8 \,{\left (b + d\right )}} - \frac{3 \, \cos \left (b x - d x + a - c\right )}{8 \,{\left (b - d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)*sin(b*x+a)^3,x, algorithm="giac")
[Out]
1/8*cos(3*b*x + d*x + 3*a + c)/(3*b + d) + 1/8*cos(3*b*x - d*x + 3*a - c)/(3*b - d) - 3/8*cos(b*x + d*x + a +
c)/(b + d) - 3/8*cos(b*x - d*x + a - c)/(b - d)