3.206 \(\int \sin ^3(a+b x) \sin (c+d x) \, dx\)

Optimal. Leaf size=97 \[ \frac{3 \sin (a+x (b-d)-c)}{8 (b-d)}-\frac{\sin (3 a+x (3 b-d)-c)}{8 (3 b-d)}-\frac{3 \sin (a+x (b+d)+c)}{8 (b+d)}+\frac{\sin (3 a+x (3 b+d)+c)}{8 (3 b+d)} \]

[Out]

(3*Sin[a - c + (b - d)*x])/(8*(b - d)) - Sin[3*a - c + (3*b - d)*x]/(8*(3*b - d)) - (3*Sin[a + c + (b + d)*x])
/(8*(b + d)) + Sin[3*a + c + (3*b + d)*x]/(8*(3*b + d))

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Rubi [A]  time = 0.074767, 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 = {4569, 2637} \[ \frac{3 \sin (a+x (b-d)-c)}{8 (b-d)}-\frac{\sin (3 a+x (3 b-d)-c)}{8 (3 b-d)}-\frac{3 \sin (a+x (b+d)+c)}{8 (b+d)}+\frac{\sin (3 a+x (3 b+d)+c)}{8 (3 b+d)} \]

Antiderivative was successfully verified.

[In]

Int[Sin[a + b*x]^3*Sin[c + d*x],x]

[Out]

(3*Sin[a - c + (b - d)*x])/(8*(b - d)) - Sin[3*a - c + (3*b - d)*x]/(8*(3*b - d)) - (3*Sin[a + c + (b + d)*x])
/(8*(b + d)) + Sin[3*a + c + (3*b + d)*x]/(8*(3*b + 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 ^3(a+b x) \sin (c+d x) \, dx &=\int \left (\frac{3}{8} \cos (a-c+(b-d) x)-\frac{1}{8} \cos (3 a-c+(3 b-d) x)-\frac{3}{8} \cos (a+c+(b+d) x)+\frac{1}{8} \cos (3 a+c+(3 b+d) x)\right ) \, dx\\ &=-\left (\frac{1}{8} \int \cos (3 a-c+(3 b-d) x) \, dx\right )+\frac{1}{8} \int \cos (3 a+c+(3 b+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{3 \sin (a-c+(b-d) x)}{8 (b-d)}-\frac{\sin (3 a-c+(3 b-d) x)}{8 (3 b-d)}-\frac{3 \sin (a+c+(b+d) x)}{8 (b+d)}+\frac{\sin (3 a+c+(3 b+d) x)}{8 (3 b+d)}\\ \end{align*}

Mathematica [A]  time = 0.51821, size = 91, normalized size = 0.94 \[ \frac{1}{8} \left (\frac{3 \sin (a+b x-c-d x)}{b-d}-\frac{\sin (3 a+3 b x-c-d x)}{3 b-d}+\frac{\sin (3 a+3 b x+c+d x)}{3 b+d}-\frac{3 \sin (a+x (b+d)+c)}{b+d}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[a + b*x]^3*Sin[c + d*x],x]

[Out]

((3*Sin[a - c + b*x - d*x])/(b - d) - Sin[3*a - c + 3*b*x - d*x]/(3*b - d) + Sin[3*a + c + 3*b*x + d*x]/(3*b +
 d) - (3*Sin[a + c + (b + d)*x])/(b + d))/8

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Maple [A]  time = 0.024, size = 90, normalized size = 0.9 \begin{align*}{\frac{3\,\sin \left ( a-c+ \left ( b-d \right ) x \right ) }{8\,b-8\,d}}-{\frac{\sin \left ( 3\,a-c+ \left ( 3\,b-d \right ) x \right ) }{24\,b-8\,d}}-{\frac{3\,\sin \left ( a+c+ \left ( b+d \right ) x \right ) }{8\,b+8\,d}}+{\frac{\sin \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(sin(b*x+a)^3*sin(d*x+c),x)

[Out]

3/8*sin(a-c+(b-d)*x)/(b-d)-1/8*sin(3*a-c+(3*b-d)*x)/(3*b-d)-3/8*sin(a+c+(b+d)*x)/(b+d)+1/8*sin(3*a+c+(3*b+d)*x
)/(3*b+d)

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Maxima [B]  time = 1.38801, size = 1065, normalized size = 10.98 \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)^3*sin(d*x+c),x, algorithm="maxima")

[Out]

-1/16*((3*b^3*sin(c) - b^2*d*sin(c) - 3*b*d^2*sin(c) + d^3*sin(c))*cos((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))*cos((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))*cos(-(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))*cos(-(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))*cos((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))*cos((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))*cos(-(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))*cos(-(b - d)*x - a) - (3*b^3*cos(c) - b^2*d*cos(c) - 3*b*d^2*cos(c) + d^3*cos(c))*sin
((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))*sin((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))*sin(-(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))*sin(-(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))*sin((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)
)*sin((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))*sin(-(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))*sin(-(b - d)*x - a))/(9*b^4*cos(c)^2 + 9*b^4*
sin(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 = 0.49894, size = 246, normalized size = 2.54 \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 )} \cos \left (d x + c\right ) \sin \left (b x + a\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 )} \sin \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(sin(b*x+a)^3*sin(d*x+c),x, algorithm="fricas")

[Out]

((7*b^2*d - d^3 - (b^2*d - d^3)*cos(b*x + a)^2)*cos(d*x + c)*sin(b*x + a) + 3*((b^3 - b*d^2)*cos(b*x + a)^3 -
(3*b^3 - b*d^2)*cos(b*x + a))*sin(d*x + c))/(9*b^4 - 10*b^2*d^2 + d^4)

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Sympy [A]  time = 88.9814, size = 957, normalized size = 9.87 \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)**3*sin(d*x+c),x)

[Out]

Piecewise((x*sin(a)**3*sin(c), Eq(b, 0) & Eq(d, 0)), (3*x*sin(a - d*x)**3*sin(c + d*x)/8 - 3*x*sin(a - d*x)**2
*cos(a - d*x)*cos(c + d*x)/8 + 3*x*sin(a - d*x)*sin(c + d*x)*cos(a - d*x)**2/8 - 3*x*cos(a - d*x)**3*cos(c + d
*x)/8 + 5*sin(a - d*x)**2*sin(c + d*x)*cos(a - d*x)/(8*d) - sin(a - d*x)*cos(a - d*x)**2*cos(c + d*x)/(8*d) +
sin(c + d*x)*cos(a - d*x)**3/(4*d), Eq(b, -d)), (x*sin(a - d*x/3)**3*sin(c + d*x)/8 - 3*x*sin(a - d*x/3)**2*co
s(a - d*x/3)*cos(c + d*x)/8 - 3*x*sin(a - d*x/3)*sin(c + d*x)*cos(a - d*x/3)**2/8 + x*cos(a - d*x/3)**3*cos(c
+ d*x)/8 + 21*sin(a - d*x/3)**2*sin(c + d*x)*cos(a - d*x/3)/(8*d) - 27*sin(a - d*x/3)*cos(a - d*x/3)**2*cos(c
+ d*x)/(8*d) - 5*sin(c + d*x)*cos(a - d*x/3)**3/(4*d), Eq(b, -d/3)), (x*sin(a + d*x/3)**3*sin(c + d*x)/8 + 3*x
*sin(a + d*x/3)**2*cos(a + d*x/3)*cos(c + d*x)/8 - 3*x*sin(a + d*x/3)*sin(c + d*x)*cos(a + d*x/3)**2/8 - x*cos
(a + d*x/3)**3*cos(c + d*x)/8 - 21*sin(a + d*x/3)**2*sin(c + d*x)*cos(a + d*x/3)/(8*d) - 27*sin(a + d*x/3)*cos
(a + d*x/3)**2*cos(c + d*x)/(8*d) + 5*sin(c + d*x)*cos(a + d*x/3)**3/(4*d), Eq(b, d/3)), (3*x*sin(a + d*x)**3*
sin(c + d*x)/8 + 3*x*sin(a + d*x)**2*cos(a + d*x)*cos(c + d*x)/8 + 3*x*sin(a + d*x)*sin(c + d*x)*cos(a + d*x)*
*2/8 + 3*x*cos(a + d*x)**3*cos(c + d*x)/8 + sin(a + d*x)**3*cos(c + d*x)/(8*d) - 3*sin(a + d*x)**2*sin(c + d*x
)*cos(a + d*x)/(4*d) - 3*sin(c + d*x)*cos(a + d*x)**3/(8*d), Eq(b, d)), (-9*b**3*sin(a + b*x)**2*sin(c + d*x)*
cos(a + b*x)/(9*b**4 - 10*b**2*d**2 + d**4) - 6*b**3*sin(c + d*x)*cos(a + b*x)**3/(9*b**4 - 10*b**2*d**2 + d**
4) + 7*b**2*d*sin(a + b*x)**3*cos(c + d*x)/(9*b**4 - 10*b**2*d**2 + d**4) + 6*b**2*d*sin(a + b*x)*cos(a + b*x)
**2*cos(c + d*x)/(9*b**4 - 10*b**2*d**2 + d**4) + 3*b*d**2*sin(a + b*x)**2*sin(c + d*x)*cos(a + b*x)/(9*b**4 -
 10*b**2*d**2 + d**4) - d**3*sin(a + b*x)**3*cos(c + d*x)/(9*b**4 - 10*b**2*d**2 + d**4), True))

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Giac [A]  time = 1.1318, size = 120, normalized size = 1.24 \begin{align*} \frac{\sin \left (3 \, b x + d x + 3 \, a + c\right )}{8 \,{\left (3 \, b + d\right )}} - \frac{\sin \left (3 \, b x - d x + 3 \, a - c\right )}{8 \,{\left (3 \, b - 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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*sin(3*b*x + d*x + 3*a + c)/(3*b + d) - 1/8*sin(3*b*x - d*x + 3*a - c)/(3*b - d) - 3/8*sin(b*x + d*x + a +
c)/(b + d) + 3/8*sin(b*x - d*x + a - c)/(b - d)