3.211 \(\int \cos ^2(c+d x) \sin (a+b x) \, dx\)

Optimal. Leaf size=62 \[ -\frac{\cos (a+x (b-2 d)-2 c)}{4 (b-2 d)}-\frac{\cos (a+x (b+2 d)+2 c)}{4 (b+2 d)}-\frac{\cos (a+b x)}{2 b} \]

[Out]

-Cos[a + b*x]/(2*b) - Cos[a - 2*c + (b - 2*d)*x]/(4*(b - 2*d)) - Cos[a + 2*c + (b + 2*d)*x]/(4*(b + 2*d))

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Rubi [A]  time = 0.0469182, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {4574, 2638} \[ -\frac{\cos (a+x (b-2 d)-2 c)}{4 (b-2 d)}-\frac{\cos (a+x (b+2 d)+2 c)}{4 (b+2 d)}-\frac{\cos (a+b x)}{2 b} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^2*Sin[a + b*x],x]

[Out]

-Cos[a + b*x]/(2*b) - Cos[a - 2*c + (b - 2*d)*x]/(4*(b - 2*d)) - Cos[a + 2*c + (b + 2*d)*x]/(4*(b + 2*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 ^2(c+d x) \sin (a+b x) \, dx &=\int \left (\frac{1}{2} \sin (a+b x)+\frac{1}{4} \sin (a-2 c+(b-2 d) x)+\frac{1}{4} \sin (a+2 c+(b+2 d) x)\right ) \, dx\\ &=\frac{1}{4} \int \sin (a-2 c+(b-2 d) x) \, dx+\frac{1}{4} \int \sin (a+2 c+(b+2 d) x) \, dx+\frac{1}{2} \int \sin (a+b x) \, dx\\ &=-\frac{\cos (a+b x)}{2 b}-\frac{\cos (a-2 c+(b-2 d) x)}{4 (b-2 d)}-\frac{\cos (a+2 c+(b+2 d) x)}{4 (b+2 d)}\\ \end{align*}

Mathematica [A]  time = 0.780621, size = 71, normalized size = 1.15 \[ \frac{1}{4} \left (-\frac{\cos (a+b x-2 c-2 d x)}{b-2 d}-\frac{\cos (a+b x+2 c+2 d x)}{b+2 d}+\frac{2 \sin (a) \sin (b x)}{b}-\frac{2 \cos (a) \cos (b x)}{b}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^2*Sin[a + b*x],x]

[Out]

((-2*Cos[a]*Cos[b*x])/b - Cos[a - 2*c + b*x - 2*d*x]/(b - 2*d) - Cos[a + 2*c + b*x + 2*d*x]/(b + 2*d) + (2*Sin
[a]*Sin[b*x])/b)/4

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Maple [A]  time = 0.016, size = 57, normalized size = 0.9 \begin{align*} -{\frac{\cos \left ( bx+a \right ) }{2\,b}}-{\frac{\cos \left ( a-2\,c+ \left ( b-2\,d \right ) x \right ) }{4\,b-8\,d}}-{\frac{\cos \left ( a+2\,c+ \left ( b+2\,d \right ) x \right ) }{4\,b+8\,d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^2*sin(b*x+a),x)

[Out]

-1/2*cos(b*x+a)/b-1/4*cos(a-2*c+(b-2*d)*x)/(b-2*d)-1/4*cos(a+2*c+(b+2*d)*x)/(b+2*d)

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Maxima [B]  time = 1.21244, size = 559, normalized size = 9.02 \begin{align*} -\frac{{\left (b^{2} \cos \left (2 \, c\right ) - 2 \, b d \cos \left (2 \, c\right )\right )} \cos \left ({\left (b + 2 \, d\right )} x + a + 4 \, c\right ) +{\left (b^{2} \cos \left (2 \, c\right ) - 2 \, b d \cos \left (2 \, c\right )\right )} \cos \left ({\left (b + 2 \, d\right )} x + a\right ) +{\left (b^{2} \cos \left (2 \, c\right ) + 2 \, b d \cos \left (2 \, c\right )\right )} \cos \left (-{\left (b - 2 \, d\right )} x - a + 4 \, c\right ) +{\left (b^{2} \cos \left (2 \, c\right ) + 2 \, b d \cos \left (2 \, c\right )\right )} \cos \left (-{\left (b - 2 \, d\right )} x - a\right ) + 2 \,{\left (b^{2} \cos \left (2 \, c\right ) - 4 \, d^{2} \cos \left (2 \, c\right )\right )} \cos \left (b x + a + 2 \, c\right ) + 2 \,{\left (b^{2} \cos \left (2 \, c\right ) - 4 \, d^{2} \cos \left (2 \, c\right )\right )} \cos \left (b x + a - 2 \, c\right ) +{\left (b^{2} \sin \left (2 \, c\right ) - 2 \, b d \sin \left (2 \, c\right )\right )} \sin \left ({\left (b + 2 \, d\right )} x + a + 4 \, c\right ) -{\left (b^{2} \sin \left (2 \, c\right ) - 2 \, b d \sin \left (2 \, c\right )\right )} \sin \left ({\left (b + 2 \, d\right )} x + a\right ) +{\left (b^{2} \sin \left (2 \, c\right ) + 2 \, b d \sin \left (2 \, c\right )\right )} \sin \left (-{\left (b - 2 \, d\right )} x - a + 4 \, c\right ) -{\left (b^{2} \sin \left (2 \, c\right ) + 2 \, b d \sin \left (2 \, c\right )\right )} \sin \left (-{\left (b - 2 \, d\right )} x - a\right ) + 2 \,{\left (b^{2} \sin \left (2 \, c\right ) - 4 \, d^{2} \sin \left (2 \, c\right )\right )} \sin \left (b x + a + 2 \, c\right ) - 2 \,{\left (b^{2} \sin \left (2 \, c\right ) - 4 \, d^{2} \sin \left (2 \, c\right )\right )} \sin \left (b x + a - 2 \, c\right )}{8 \,{\left (b^{3} \cos \left (2 \, c\right )^{2} + b^{3} \sin \left (2 \, c\right )^{2} - 4 \,{\left (b \cos \left (2 \, c\right )^{2} + b \sin \left (2 \, c\right )^{2}\right )} d^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^2*sin(b*x+a),x, algorithm="maxima")

[Out]

-1/8*((b^2*cos(2*c) - 2*b*d*cos(2*c))*cos((b + 2*d)*x + a + 4*c) + (b^2*cos(2*c) - 2*b*d*cos(2*c))*cos((b + 2*
d)*x + a) + (b^2*cos(2*c) + 2*b*d*cos(2*c))*cos(-(b - 2*d)*x - a + 4*c) + (b^2*cos(2*c) + 2*b*d*cos(2*c))*cos(
-(b - 2*d)*x - a) + 2*(b^2*cos(2*c) - 4*d^2*cos(2*c))*cos(b*x + a + 2*c) + 2*(b^2*cos(2*c) - 4*d^2*cos(2*c))*c
os(b*x + a - 2*c) + (b^2*sin(2*c) - 2*b*d*sin(2*c))*sin((b + 2*d)*x + a + 4*c) - (b^2*sin(2*c) - 2*b*d*sin(2*c
))*sin((b + 2*d)*x + a) + (b^2*sin(2*c) + 2*b*d*sin(2*c))*sin(-(b - 2*d)*x - a + 4*c) - (b^2*sin(2*c) + 2*b*d*
sin(2*c))*sin(-(b - 2*d)*x - a) + 2*(b^2*sin(2*c) - 4*d^2*sin(2*c))*sin(b*x + a + 2*c) - 2*(b^2*sin(2*c) - 4*d
^2*sin(2*c))*sin(b*x + a - 2*c))/(b^3*cos(2*c)^2 + b^3*sin(2*c)^2 - 4*(b*cos(2*c)^2 + b*sin(2*c)^2)*d^2)

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Fricas [A]  time = 0.490526, size = 162, normalized size = 2.61 \begin{align*} -\frac{b^{2} \cos \left (b x + a\right ) \cos \left (d x + c\right )^{2} + 2 \, b d \cos \left (d x + c\right ) \sin \left (b x + a\right ) \sin \left (d x + c\right ) - 2 \, d^{2} \cos \left (b x + a\right )}{b^{3} - 4 \, b d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^2*sin(b*x+a),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 12.1257, size = 401, normalized size = 6.47 \begin{align*} \begin{cases} x \sin{\left (a \right )} \cos ^{2}{\left (c \right )} & \text{for}\: b = 0 \wedge d = 0 \\\left (\frac{x \sin ^{2}{\left (c + d x \right )}}{2} + \frac{x \cos ^{2}{\left (c + d x \right )}}{2} + \frac{\sin{\left (c + d x \right )} \cos{\left (c + d x \right )}}{2 d}\right ) \sin{\left (a \right )} & \text{for}\: b = 0 \\- \frac{x \sin{\left (a - 2 d x \right )} \sin ^{2}{\left (c + d x \right )}}{4} + \frac{x \sin{\left (a - 2 d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac{x \sin{\left (c + d x \right )} \cos{\left (a - 2 d x \right )} \cos{\left (c + d x \right )}}{2} + \frac{\sin ^{2}{\left (c + d x \right )} \cos{\left (a - 2 d x \right )}}{8 d} + \frac{3 \cos{\left (a - 2 d x \right )} \cos ^{2}{\left (c + d x \right )}}{8 d} & \text{for}\: b = - 2 d \\- \frac{x \sin{\left (a + 2 d x \right )} \sin ^{2}{\left (c + d x \right )}}{4} + \frac{x \sin{\left (a + 2 d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} - \frac{x \sin{\left (c + d x \right )} \cos{\left (a + 2 d x \right )} \cos{\left (c + d x \right )}}{2} - \frac{\sin{\left (a + 2 d x \right )} \sin{\left (c + d x \right )} \cos{\left (c + d x \right )}}{4 d} - \frac{\cos{\left (a + 2 d x \right )} \cos ^{2}{\left (c + d x \right )}}{2 d} & \text{for}\: b = 2 d \\- \frac{b^{2} \cos{\left (a + b x \right )} \cos ^{2}{\left (c + d x \right )}}{b^{3} - 4 b d^{2}} - \frac{2 b d \sin{\left (a + b x \right )} \sin{\left (c + d x \right )} \cos{\left (c + d x \right )}}{b^{3} - 4 b d^{2}} + \frac{2 d^{2} \sin ^{2}{\left (c + d x \right )} \cos{\left (a + b x \right )}}{b^{3} - 4 b d^{2}} + \frac{2 d^{2} \cos{\left (a + b x \right )} \cos ^{2}{\left (c + d x \right )}}{b^{3} - 4 b d^{2}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**2*sin(b*x+a),x)

[Out]

Piecewise((x*sin(a)*cos(c)**2, Eq(b, 0) & Eq(d, 0)), ((x*sin(c + d*x)**2/2 + x*cos(c + d*x)**2/2 + sin(c + d*x
)*cos(c + d*x)/(2*d))*sin(a), Eq(b, 0)), (-x*sin(a - 2*d*x)*sin(c + d*x)**2/4 + x*sin(a - 2*d*x)*cos(c + d*x)*
*2/4 + x*sin(c + d*x)*cos(a - 2*d*x)*cos(c + d*x)/2 + sin(c + d*x)**2*cos(a - 2*d*x)/(8*d) + 3*cos(a - 2*d*x)*
cos(c + d*x)**2/(8*d), Eq(b, -2*d)), (-x*sin(a + 2*d*x)*sin(c + d*x)**2/4 + x*sin(a + 2*d*x)*cos(c + d*x)**2/4
- x*sin(c + d*x)*cos(a + 2*d*x)*cos(c + d*x)/2 - sin(a + 2*d*x)*sin(c + d*x)*cos(c + d*x)/(4*d) - cos(a + 2*d
*x)*cos(c + d*x)**2/(2*d), Eq(b, 2*d)), (-b**2*cos(a + b*x)*cos(c + d*x)**2/(b**3 - 4*b*d**2) - 2*b*d*sin(a +
b*x)*sin(c + d*x)*cos(c + d*x)/(b**3 - 4*b*d**2) + 2*d**2*sin(c + d*x)**2*cos(a + b*x)/(b**3 - 4*b*d**2) + 2*d
**2*cos(a + b*x)*cos(c + d*x)**2/(b**3 - 4*b*d**2), True))

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Giac [A]  time = 1.13856, size = 76, normalized size = 1.23 \begin{align*} -\frac{\cos \left (b x + 2 \, d x + a + 2 \, c\right )}{4 \,{\left (b + 2 \, d\right )}} - \frac{\cos \left (b x - 2 \, d x + a - 2 \, c\right )}{4 \,{\left (b - 2 \, d\right )}} - \frac{\cos \left (b x + a\right )}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^2*sin(b*x+a),x, algorithm="giac")

[Out]

-1/4*cos(b*x + 2*d*x + a + 2*c)/(b + 2*d) - 1/4*cos(b*x - 2*d*x + a - 2*c)/(b - 2*d) - 1/2*cos(b*x + a)/b