3.192 \(\int \sin (a+b x) \sin ^3(c+d x) \, dx\)
Optimal. Leaf size=91 \[ -\frac{\sin (a+x (b-3 d)-3 c)}{8 (b-3 d)}+\frac{3 \sin (a+x (b-d)-c)}{8 (b-d)}-\frac{3 \sin (a+x (b+d)+c)}{8 (b+d)}+\frac{\sin (a+x (b+3 d)+3 c)}{8 (b+3 d)} \]
[Out]
-Sin[a - 3*c + (b - 3*d)*x]/(8*(b - 3*d)) + (3*Sin[a - c + (b - d)*x])/(8*(b - d)) - (3*Sin[a + c + (b + d)*x]
)/(8*(b + d)) + Sin[a + 3*c + (b + 3*d)*x]/(8*(b + 3*d))
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Rubi [A] time = 0.0764363, antiderivative size = 91, 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 =
{4569, 2637} \[ -\frac{\sin (a+x (b-3 d)-3 c)}{8 (b-3 d)}+\frac{3 \sin (a+x (b-d)-c)}{8 (b-d)}-\frac{3 \sin (a+x (b+d)+c)}{8 (b+d)}+\frac{\sin (a+x (b+3 d)+3 c)}{8 (b+3 d)} \]
Antiderivative was successfully verified.
[In]
Int[Sin[a + b*x]*Sin[c + d*x]^3,x]
[Out]
-Sin[a - 3*c + (b - 3*d)*x]/(8*(b - 3*d)) + (3*Sin[a - c + (b - d)*x])/(8*(b - d)) - (3*Sin[a + c + (b + d)*x]
)/(8*(b + d)) + Sin[a + 3*c + (b + 3*d)*x]/(8*(b + 3*d))
Rule 4569
Int[Sin[v_]^(p_.)*Sin[w_]^(q_.), x_Symbol] :> Int[ExpandTrigReduce[Sin[v]^p*Sin[w]^q, x], x] /; ((PolynomialQ[
v, x] && PolynomialQ[w, x]) || (BinomialQ[{v, w}, x] && IndependentQ[Cancel[v/w], x])) && IGtQ[p, 0] && IGtQ[q
, 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 \sin (a+b x) \sin ^3(c+d x) \, dx &=\int \left (-\frac{1}{8} \cos (a-3 c+(b-3 d) x)+\frac{3}{8} \cos (a-c+(b-d) x)-\frac{3}{8} \cos (a+c+(b+d) x)+\frac{1}{8} \cos (a+3 c+(b+3 d) x)\right ) \, dx\\ &=-\left (\frac{1}{8} \int \cos (a-3 c+(b-3 d) x) \, dx\right )+\frac{1}{8} \int \cos (a+3 c+(b+3 d) x) \, dx+\frac{3}{8} \int \cos (a-c+(b-d) x) \, dx-\frac{3}{8} \int \cos (a+c+(b+d) x) \, dx\\ &=-\frac{\sin (a-3 c+(b-3 d) x)}{8 (b-3 d)}+\frac{3 \sin (a-c+(b-d) x)}{8 (b-d)}-\frac{3 \sin (a+c+(b+d) x)}{8 (b+d)}+\frac{\sin (a+3 c+(b+3 d) x)}{8 (b+3 d)}\\ \end{align*}
Mathematica [A] time = 0.511428, size = 86, normalized size = 0.95 \[ \frac{1}{8} \left (-\frac{\sin (a+b x-3 c-3 d x)}{b-3 d}+\frac{3 \sin (a+b x-c-d x)}{b-d}+\frac{\sin (a+b x+3 c+3 d x)}{b+3 d}-\frac{3 \sin (a+x (b+d)+c)}{b+d}\right ) \]
Antiderivative was successfully verified.
[In]
Integrate[Sin[a + b*x]*Sin[c + d*x]^3,x]
[Out]
(-(Sin[a - 3*c + b*x - 3*d*x]/(b - 3*d)) + (3*Sin[a - c + b*x - d*x])/(b - d) + Sin[a + 3*c + b*x + 3*d*x]/(b
+ 3*d) - (3*Sin[a + c + (b + d)*x])/(b + d))/8
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Maple [A] time = 0.027, size = 84, normalized size = 0.9 \begin{align*} -{\frac{\sin \left ( a-3\,c+ \left ( b-3\,d \right ) x \right ) }{8\,b-24\,d}}+{\frac{3\,\sin \left ( a-c+ \left ( b-d \right ) x \right ) }{8\,b-8\,d}}-{\frac{3\,\sin \left ( a+c+ \left ( b+d \right ) x \right ) }{8\,b+8\,d}}+{\frac{\sin \left ( a+3\,c+ \left ( b+3\,d \right ) x \right ) }{8\,b+24\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(sin(b*x+a)*sin(d*x+c)^3,x)
[Out]
-1/8*sin(a-3*c+(b-3*d)*x)/(b-3*d)+3/8*sin(a-c+(b-d)*x)/(b-d)-3/8*sin(a+c+(b+d)*x)/(b+d)+1/8*sin(a+3*c+(b+3*d)*
x)/(b+3*d)
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Maxima [B] time = 1.60639, size = 1237, normalized size = 13.59 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sin(b*x+a)*sin(d*x+c)^3,x, algorithm="maxima")
[Out]
-1/16*((b^3*sin(3*c) - 3*b^2*d*sin(3*c) - b*d^2*sin(3*c) + 3*d^3*sin(3*c))*cos((b + 3*d)*x + a + 6*c) - (b^3*s
in(3*c) - 3*b^2*d*sin(3*c) - b*d^2*sin(3*c) + 3*d^3*sin(3*c))*cos((b + 3*d)*x + a) - 3*(b^3*sin(3*c) - b^2*d*s
in(3*c) - 9*b*d^2*sin(3*c) + 9*d^3*sin(3*c))*cos((b + d)*x + a + 4*c) + 3*(b^3*sin(3*c) - b^2*d*sin(3*c) - 9*b
*d^2*sin(3*c) + 9*d^3*sin(3*c))*cos((b + d)*x + a - 2*c) - 3*(b^3*sin(3*c) + b^2*d*sin(3*c) - 9*b*d^2*sin(3*c)
- 9*d^3*sin(3*c))*cos(-(b - d)*x - a + 4*c) + 3*(b^3*sin(3*c) + b^2*d*sin(3*c) - 9*b*d^2*sin(3*c) - 9*d^3*sin
(3*c))*cos(-(b - d)*x - a - 2*c) + (b^3*sin(3*c) + 3*b^2*d*sin(3*c) - b*d^2*sin(3*c) - 3*d^3*sin(3*c))*cos(-(b
- 3*d)*x - a + 6*c) - (b^3*sin(3*c) + 3*b^2*d*sin(3*c) - b*d^2*sin(3*c) - 3*d^3*sin(3*c))*cos(-(b - 3*d)*x -
a) - (b^3*cos(3*c) - 3*b^2*d*cos(3*c) - b*d^2*cos(3*c) + 3*d^3*cos(3*c))*sin((b + 3*d)*x + a + 6*c) - (b^3*cos
(3*c) - 3*b^2*d*cos(3*c) - b*d^2*cos(3*c) + 3*d^3*cos(3*c))*sin((b + 3*d)*x + a) + 3*(b^3*cos(3*c) - b^2*d*cos
(3*c) - 9*b*d^2*cos(3*c) + 9*d^3*cos(3*c))*sin((b + d)*x + a + 4*c) + 3*(b^3*cos(3*c) - b^2*d*cos(3*c) - 9*b*d
^2*cos(3*c) + 9*d^3*cos(3*c))*sin((b + d)*x + a - 2*c) + 3*(b^3*cos(3*c) + b^2*d*cos(3*c) - 9*b*d^2*cos(3*c) -
9*d^3*cos(3*c))*sin(-(b - d)*x - a + 4*c) + 3*(b^3*cos(3*c) + b^2*d*cos(3*c) - 9*b*d^2*cos(3*c) - 9*d^3*cos(3
*c))*sin(-(b - d)*x - a - 2*c) - (b^3*cos(3*c) + 3*b^2*d*cos(3*c) - b*d^2*cos(3*c) - 3*d^3*cos(3*c))*sin(-(b -
3*d)*x - a + 6*c) - (b^3*cos(3*c) + 3*b^2*d*cos(3*c) - b*d^2*cos(3*c) - 3*d^3*cos(3*c))*sin(-(b - 3*d)*x - a)
)/(b^4*cos(3*c)^2 + b^4*sin(3*c)^2 + 9*(cos(3*c)^2 + sin(3*c)^2)*d^4 - 10*(b^2*cos(3*c)^2 + b^2*sin(3*c)^2)*d^
2)
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Fricas [A] time = 0.507553, size = 267, normalized size = 2.93 \begin{align*} -\frac{3 \,{\left ({\left (b^{2} d - d^{3}\right )} \cos \left (d x + c\right )^{3} -{\left (b^{2} d - 3 \, d^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (b x + a\right ) -{\left ({\left (b^{3} - b d^{2}\right )} \cos \left (b x + a\right ) \cos \left (d x + c\right )^{2} -{\left (b^{3} - 7 \, b d^{2}\right )} \cos \left (b x + a\right )\right )} \sin \left (d x + c\right )}{b^{4} - 10 \, b^{2} d^{2} + 9 \, d^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sin(b*x+a)*sin(d*x+c)^3,x, algorithm="fricas")
[Out]
-(3*((b^2*d - d^3)*cos(d*x + c)^3 - (b^2*d - 3*d^3)*cos(d*x + c))*sin(b*x + a) - ((b^3 - b*d^2)*cos(b*x + a)*c
os(d*x + c)^2 - (b^3 - 7*b*d^2)*cos(b*x + a))*sin(d*x + c))/(b^4 - 10*b^2*d^2 + 9*d^4)
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Sympy [A] time = 133.383, size = 921, normalized size = 10.12 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sin(b*x+a)*sin(d*x+c)**3,x)
[Out]
Piecewise((x*sin(a)*sin(c)**3, Eq(b, 0) & Eq(d, 0)), (x*sin(a - 3*d*x)*sin(c + d*x)**3/8 - 3*x*sin(a - 3*d*x)*
sin(c + d*x)*cos(c + d*x)**2/8 - 3*x*sin(c + d*x)**2*cos(a - 3*d*x)*cos(c + d*x)/8 + x*cos(a - 3*d*x)*cos(c +
d*x)**3/8 + sin(a - 3*d*x)*cos(c + d*x)**3/(8*d) + 7*sin(c + d*x)**3*cos(a - 3*d*x)/(24*d) + sin(c + d*x)*cos(
a - 3*d*x)*cos(c + d*x)**2/(4*d), Eq(b, -3*d)), (3*x*sin(a - d*x)*sin(c + d*x)**3/8 + 3*x*sin(a - d*x)*sin(c +
d*x)*cos(c + d*x)**2/8 - 3*x*sin(c + d*x)**2*cos(a - d*x)*cos(c + d*x)/8 - 3*x*cos(a - d*x)*cos(c + d*x)**3/8
+ 3*sin(a - d*x)*cos(c + d*x)**3/(8*d) + 5*sin(c + d*x)**3*cos(a - d*x)/(8*d) + 3*sin(c + d*x)*cos(a - d*x)*c
os(c + d*x)**2/(4*d), Eq(b, -d)), (3*x*sin(a + d*x)*sin(c + d*x)**3/8 + 3*x*sin(a + d*x)*sin(c + d*x)*cos(c +
d*x)**2/8 + 3*x*sin(c + d*x)**2*cos(a + d*x)*cos(c + d*x)/8 + 3*x*cos(a + d*x)*cos(c + d*x)**3/8 + 3*sin(a + d
*x)*cos(c + d*x)**3/(8*d) - 5*sin(c + d*x)**3*cos(a + d*x)/(8*d) - 3*sin(c + d*x)*cos(a + d*x)*cos(c + d*x)**2
/(4*d), Eq(b, d)), (x*sin(a + 3*d*x)*sin(c + d*x)**3/8 - 3*x*sin(a + 3*d*x)*sin(c + d*x)*cos(c + d*x)**2/8 + 3
*x*sin(c + d*x)**2*cos(a + 3*d*x)*cos(c + d*x)/8 - x*cos(a + 3*d*x)*cos(c + d*x)**3/8 + sin(a + 3*d*x)*cos(c +
d*x)**3/(8*d) - 7*sin(c + d*x)**3*cos(a + 3*d*x)/(24*d) - sin(c + d*x)*cos(a + 3*d*x)*cos(c + d*x)**2/(4*d),
Eq(b, 3*d)), (-b**3*sin(c + d*x)**3*cos(a + b*x)/(b**4 - 10*b**2*d**2 + 9*d**4) + 3*b**2*d*sin(a + b*x)*sin(c
+ d*x)**2*cos(c + d*x)/(b**4 - 10*b**2*d**2 + 9*d**4) + 7*b*d**2*sin(c + d*x)**3*cos(a + b*x)/(b**4 - 10*b**2*
d**2 + 9*d**4) + 6*b*d**2*sin(c + d*x)*cos(a + b*x)*cos(c + d*x)**2/(b**4 - 10*b**2*d**2 + 9*d**4) - 9*d**3*si
n(a + b*x)*sin(c + d*x)**2*cos(c + d*x)/(b**4 - 10*b**2*d**2 + 9*d**4) - 6*d**3*sin(a + b*x)*cos(c + d*x)**3/(
b**4 - 10*b**2*d**2 + 9*d**4), True))
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Giac [A] time = 1.13697, size = 113, normalized size = 1.24 \begin{align*} \frac{\sin \left (b x + 3 \, d x + a + 3 \, c\right )}{8 \,{\left (b + 3 \, d\right )}} - \frac{3 \, \sin \left (b x + d x + a + c\right )}{8 \,{\left (b + d\right )}} + \frac{3 \, \sin \left (b x - d x + a - c\right )}{8 \,{\left (b - d\right )}} - \frac{\sin \left (b x - 3 \, d x + a - 3 \, c\right )}{8 \,{\left (b - 3 \, d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sin(b*x+a)*sin(d*x+c)^3,x, algorithm="giac")
[Out]
1/8*sin(b*x + 3*d*x + a + 3*c)/(b + 3*d) - 3/8*sin(b*x + d*x + a + c)/(b + d) + 3/8*sin(b*x - d*x + a - c)/(b
- d) - 1/8*sin(b*x - 3*d*x + a - 3*c)/(b - 3*d)