3.210 \(\int \cos ^3(c+d x) \sin (a+b x) \, dx\)
Optimal. Leaf size=91 \[ -\frac {\cos (a+x (b-3 d)-3 c)}{8 (b-3 d)}-\frac {3 \cos (a+x (b-d)-c)}{8 (b-d)}-\frac {3 \cos (a+x (b+d)+c)}{8 (b+d)}-\frac {\cos (a+x (b+3 d)+3 c)}{8 (b+3 d)} \]
[Out]
-1/8*cos(a-3*c+(b-3*d)*x)/(b-3*d)-3/8*cos(a-c+(b-d)*x)/(b-d)-3/8*cos(a+c+(b+d)*x)/(b+d)-1/8*cos(a+3*c+(b+3*d)*
x)/(b+3*d)
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Rubi [A] time = 0.07, antiderivative size = 91, normalized size of antiderivative = 1.00,
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} \[ -\frac {\cos (a+x (b-3 d)-3 c)}{8 (b-3 d)}-\frac {3 \cos (a+x (b-d)-c)}{8 (b-d)}-\frac {3 \cos (a+x (b+d)+c)}{8 (b+d)}-\frac {\cos (a+x (b+3 d)+3 c)}{8 (b+3 d)} \]
Antiderivative was successfully verified.
[In]
Int[Cos[c + d*x]^3*Sin[a + b*x],x]
[Out]
-Cos[a - 3*c + (b - 3*d)*x]/(8*(b - 3*d)) - (3*Cos[a - c + (b - d)*x])/(8*(b - d)) - (3*Cos[a + c + (b + d)*x]
)/(8*(b + d)) - Cos[a + 3*c + (b + 3*d)*x]/(8*(b + 3*d))
Rule 2638
Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]
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]))
Rubi steps
\begin {align*} \int \cos ^3(c+d x) \sin (a+b x) \, dx &=\int \left (\frac {1}{8} \sin (a-3 c+(b-3 d) x)+\frac {3}{8} \sin (a-c+(b-d) x)+\frac {3}{8} \sin (a+c+(b+d) x)+\frac {1}{8} \sin (a+3 c+(b+3 d) x)\right ) \, dx\\ &=\frac {1}{8} \int \sin (a-3 c+(b-3 d) x) \, dx+\frac {1}{8} \int \sin (a+3 c+(b+3 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 {\cos (a-3 c+(b-3 d) x)}{8 (b-3 d)}-\frac {3 \cos (a-c+(b-d) x)}{8 (b-d)}-\frac {3 \cos (a+c+(b+d) x)}{8 (b+d)}-\frac {\cos (a+3 c+(b+3 d) x)}{8 (b+3 d)}\\ \end {align*}
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Mathematica [A] time = 0.50, size = 87, normalized size = 0.96 \[ \frac {1}{8} \left (-\frac {\cos (a+b x-3 c-3 d x)}{b-3 d}-\frac {3 \cos (a+b x-c-d x)}{b-d}-\frac {\cos (a+b x+3 c+3 d x)}{b+3 d}-\frac {3 \cos (a+x (b+d)+c)}{b+d}\right ) \]
Antiderivative was successfully verified.
[In]
Integrate[Cos[c + d*x]^3*Sin[a + b*x],x]
[Out]
(-(Cos[a - 3*c + b*x - 3*d*x]/(b - 3*d)) - (3*Cos[a - c + b*x - d*x])/(b - d) - Cos[a + 3*c + b*x + 3*d*x]/(b
+ 3*d) - (3*Cos[a + c + (b + d)*x])/(b + d))/8
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fricas [A] time = 0.42, size = 106, normalized size = 1.16 \[ \frac {6 \, b d^{2} \cos \left (b x + a\right ) \cos \left (d x + c\right ) - {\left (b^{3} - b d^{2}\right )} \cos \left (b x + a\right ) \cos \left (d x + c\right )^{3} + 3 \, {\left (2 \, d^{3} - {\left (b^{2} d - d^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (b x + a\right ) \sin \left (d x + c\right )}{b^{4} - 10 \, b^{2} d^{2} + 9 \, d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^3*sin(b*x+a),x, algorithm="fricas")
[Out]
(6*b*d^2*cos(b*x + a)*cos(d*x + c) - (b^3 - b*d^2)*cos(b*x + a)*cos(d*x + c)^3 + 3*(2*d^3 - (b^2*d - d^3)*cos(
d*x + c)^2)*sin(b*x + a)*sin(d*x + c))/(b^4 - 10*b^2*d^2 + 9*d^4)
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giac [A] time = 1.92, size = 84, normalized size = 0.92 \[ -\frac {\cos \left (b x + 3 \, d x + a + 3 \, c\right )}{8 \, {\left (b + 3 \, d\right )}} - \frac {3 \, \cos \left (b x + d x + a + c\right )}{8 \, {\left (b + d\right )}} - \frac {3 \, \cos \left (b x - d x + a - c\right )}{8 \, {\left (b - d\right )}} - \frac {\cos \left (b x - 3 \, d x + a - 3 \, c\right )}{8 \, {\left (b - 3 \, d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^3*sin(b*x+a),x, algorithm="giac")
[Out]
-1/8*cos(b*x + 3*d*x + a + 3*c)/(b + 3*d) - 3/8*cos(b*x + d*x + a + c)/(b + d) - 3/8*cos(b*x - d*x + a - c)/(b
- d) - 1/8*cos(b*x - 3*d*x + a - 3*c)/(b - 3*d)
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maple [A] time = 0.17, size = 84, normalized size = 0.92 \[ -\frac {\cos \left (a -3 c +\left (b -3 d \right ) x \right )}{8 \left (b -3 d \right )}-\frac {3 \cos \left (a -c +\left (b -d \right ) x \right )}{8 \left (b -d \right )}-\frac {3 \cos \left (a +c +\left (b +d \right ) x \right )}{8 \left (b +d \right )}-\frac {\cos \left (a +3 c +\left (b +3 d \right ) x \right )}{8 \left (b +3 d \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cos(d*x+c)^3*sin(b*x+a),x)
[Out]
-1/8*cos(a-3*c+(b-3*d)*x)/(b-3*d)-3/8*cos(a-c+(b-d)*x)/(b-d)-3/8*cos(a+c+(b+d)*x)/(b+d)-1/8*cos(a+3*c+(b+3*d)*
x)/(b+3*d)
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maxima [B] time = 0.39, size = 912, normalized size = 10.02 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^3*sin(b*x+a),x, algorithm="maxima")
[Out]
-1/16*((b^3*cos(3*c) - 3*b^2*d*cos(3*c) - b*d^2*cos(3*c) + 3*d^3*cos(3*c))*cos((b + 3*d)*x + a + 6*c) + (b^3*c
os(3*c) - 3*b^2*d*cos(3*c) - b*d^2*cos(3*c) + 3*d^3*cos(3*c))*cos((b + 3*d)*x + a) + 3*(b^3*cos(3*c) - b^2*d*c
os(3*c) - 9*b*d^2*cos(3*c) + 9*d^3*cos(3*c))*cos((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))*cos((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))*cos(-(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))*cos(-(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))*cos(-(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))*cos(-(b - 3*d)*x -
a) + (b^3*sin(3*c) - 3*b^2*d*sin(3*c) - b*d^2*sin(3*c) + 3*d^3*sin(3*c))*sin((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))*sin((b + 3*d)*x + a) + 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))*sin((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))*sin((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))*sin(-(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))*sin(-(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))*sin(-(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))*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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mupad [B] time = 1.61, size = 297, normalized size = 3.26 \[ -{\mathrm {e}}^{a\,1{}\mathrm {i}-c\,3{}\mathrm {i}+b\,x\,1{}\mathrm {i}-d\,x\,3{}\mathrm {i}}\,\left (\frac {b+3\,d}{16\,b^2-144\,d^2}+\frac {{\mathrm {e}}^{-a\,2{}\mathrm {i}-b\,x\,2{}\mathrm {i}}\,\left (b-3\,d\right )}{16\,b^2-144\,d^2}\right )-{\mathrm {e}}^{a\,1{}\mathrm {i}+c\,3{}\mathrm {i}+b\,x\,1{}\mathrm {i}+d\,x\,3{}\mathrm {i}}\,\left (\frac {b-3\,d}{16\,b^2-144\,d^2}+\frac {{\mathrm {e}}^{-a\,2{}\mathrm {i}-b\,x\,2{}\mathrm {i}}\,\left (b+3\,d\right )}{16\,b^2-144\,d^2}\right )-{\mathrm {e}}^{a\,1{}\mathrm {i}-c\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}\,\left (\frac {3\,b+3\,d}{16\,b^2-16\,d^2}+\frac {{\mathrm {e}}^{-a\,2{}\mathrm {i}-b\,x\,2{}\mathrm {i}}\,\left (3\,b-3\,d\right )}{16\,b^2-16\,d^2}\right )-{\mathrm {e}}^{a\,1{}\mathrm {i}+c\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,\left (\frac {3\,b-3\,d}{16\,b^2-16\,d^2}+\frac {{\mathrm {e}}^{-a\,2{}\mathrm {i}-b\,x\,2{}\mathrm {i}}\,\left (3\,b+3\,d\right )}{16\,b^2-16\,d^2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cos(c + d*x)^3*sin(a + b*x),x)
[Out]
- exp(a*1i - c*3i + b*x*1i - d*x*3i)*((b + 3*d)/(16*b^2 - 144*d^2) + (exp(- a*2i - b*x*2i)*(b - 3*d))/(16*b^2
- 144*d^2)) - exp(a*1i + c*3i + b*x*1i + d*x*3i)*((b - 3*d)/(16*b^2 - 144*d^2) + (exp(- a*2i - b*x*2i)*(b + 3*
d))/(16*b^2 - 144*d^2)) - exp(a*1i - c*1i + b*x*1i - d*x*1i)*((3*b + 3*d)/(16*b^2 - 16*d^2) + (exp(- a*2i - b*
x*2i)*(3*b - 3*d))/(16*b^2 - 16*d^2)) - exp(a*1i + c*1i + b*x*1i + d*x*1i)*((3*b - 3*d)/(16*b^2 - 16*d^2) + (e
xp(- a*2i - b*x*2i)*(3*b + 3*d))/(16*b^2 - 16*d^2))
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sympy [A] time = 32.16, size = 918, normalized size = 10.09 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)**3*sin(b*x+a),x)
[Out]
Piecewise((x*sin(a)*cos(c)**3, Eq(b, 0) & Eq(d, 0)), (-3*x*sin(a - 3*d*x)*sin(c + d*x)**2*cos(c + d*x)/8 + x*s
in(a - 3*d*x)*cos(c + d*x)**3/8 - x*sin(c + d*x)**3*cos(a - 3*d*x)/8 + 3*x*sin(c + d*x)*cos(a - 3*d*x)*cos(c +
d*x)**2/8 - sin(a - 3*d*x)*sin(c + d*x)**3/(24*d) - sin(a - 3*d*x)*sin(c + d*x)*cos(c + d*x)**2/(4*d) + 3*cos
(a - 3*d*x)*cos(c + d*x)**3/(8*d), Eq(b, -3*d)), (3*x*sin(a - d*x)*sin(c + d*x)**2*cos(c + d*x)/8 + 3*x*sin(a
- d*x)*cos(c + d*x)**3/8 + 3*x*sin(c + d*x)**3*cos(a - d*x)/8 + 3*x*sin(c + d*x)*cos(a - d*x)*cos(c + d*x)**2/
8 + 3*sin(a - d*x)*sin(c + d*x)**3/(8*d) + 3*sin(a - d*x)*sin(c + d*x)*cos(c + d*x)**2/(4*d) - cos(a - d*x)*co
s(c + d*x)**3/(8*d), Eq(b, -d)), (3*x*sin(a + d*x)*sin(c + d*x)**2*cos(c + d*x)/8 + 3*x*sin(a + d*x)*cos(c + d
*x)**3/8 - 3*x*sin(c + d*x)**3*cos(a + d*x)/8 - 3*x*sin(c + d*x)*cos(a + d*x)*cos(c + d*x)**2/8 + 3*sin(a + d*
x)*sin(c + d*x)**3/(8*d) + 3*sin(a + d*x)*sin(c + d*x)*cos(c + d*x)**2/(4*d) + cos(a + d*x)*cos(c + d*x)**3/(8
*d), Eq(b, d)), (-3*x*sin(a + 3*d*x)*sin(c + d*x)**2*cos(c + d*x)/8 + x*sin(a + 3*d*x)*cos(c + d*x)**3/8 + x*s
in(c + d*x)**3*cos(a + 3*d*x)/8 - 3*x*sin(c + d*x)*cos(a + 3*d*x)*cos(c + d*x)**2/8 - sin(a + 3*d*x)*sin(c + d
*x)**3/(24*d) - sin(a + 3*d*x)*sin(c + d*x)*cos(c + d*x)**2/(4*d) - 3*cos(a + 3*d*x)*cos(c + d*x)**3/(8*d), Eq
(b, 3*d)), (-b**3*cos(a + b*x)*cos(c + d*x)**3/(b**4 - 10*b**2*d**2 + 9*d**4) - 3*b**2*d*sin(a + b*x)*sin(c +
d*x)*cos(c + d*x)**2/(b**4 - 10*b**2*d**2 + 9*d**4) + 6*b*d**2*sin(c + d*x)**2*cos(a + b*x)*cos(c + d*x)/(b**4
- 10*b**2*d**2 + 9*d**4) + 7*b*d**2*cos(a + b*x)*cos(c + d*x)**3/(b**4 - 10*b**2*d**2 + 9*d**4) + 6*d**3*sin(
a + b*x)*sin(c + d*x)**3/(b**4 - 10*b**2*d**2 + 9*d**4) + 9*d**3*sin(a + b*x)*sin(c + d*x)*cos(c + d*x)**2/(b*
*4 - 10*b**2*d**2 + 9*d**4), True))
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