\(\int \cos (c+d x) \sin ^3(a+b x) \, dx\) [224]

Optimal result

Integrand size = 15, antiderivative size = 97 \[ \int \cos (c+d x) \sin ^3(a+b x) \, dx=-\frac {3 \cos (a-c+(b-d) x)}{8 (b-d)}+\frac {\cos (3 a-c+(3 b-d) x)}{8 (3 b-d)}-\frac {3 \cos (a+c+(b+d) x)}{8 (b+d)}+\frac {\cos (3 a+c+(3 b+d) x)}{8 (3 b+d)} \]

[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)

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {4670, 2718} \[ \int \cos (c+d x) \sin ^3(a+b x) \, dx=-\frac {3 \cos (a+x (b-d)-c)}{8 (b-d)}+\frac {\cos (3 a+x (3 b-d)-c)}{8 (3 b-d)}-\frac {3 \cos (a+x (b+d)+c)}{8 (b+d)}+\frac {\cos (3 a+x (3 b+d)+c)}{8 (3 b+d)} \]

[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 2718

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 4670

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*} \text {integral}& = \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] (verified)

Time = 0.56 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.93 \[ \int \cos (c+d x) \sin ^3(a+b x) \, dx=\frac {1}{8} \left (-\frac {3 \cos (a-c+b x-d x)}{b-d}+\frac {\cos (3 a-c+3 b x-d x)}{3 b-d}+\frac {\cos (3 a+c+3 b x+d x)}{3 b+d}-\frac {3 \cos (a+c+(b+d) x)}{b+d}\right ) \]

[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

Maple [A] (verified)

Time = 0.72 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.93

method	result	size
default	\(-\frac {3 \cos \left (a -c +\left (b -d \right ) x \right )}{8 \left (b -d \right )}+\frac {\cos \left (3 a -c +\left (3 b -d \right ) x \right )}{24 b -8 d}-\frac {3 \cos \left (a +c +\left (b +d \right ) x \right )}{8 \left (b +d \right )}+\frac {\cos \left (3 a +c +\left (3 b +d \right ) x \right )}{24 b +8 d}\)	\(90\)
parallelrisch	\(\frac {-12 \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{6} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b^{3}-24 b^{2} d \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{5}+12 \left (\left (-3 b^{3}+b \,d^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-b \,d^{2}\right ) \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{4}+16 \left (-4 b^{2} d +d^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{3}+12 \left (-\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b \,d^{2}-3 b^{3}+b \,d^{2}\right ) \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}-24 \tan \left (\frac {a}{2}+\frac {x b}{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{2} d -12 b^{3}}{\left (b +d \right ) \left (1+\tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}\right )^{3} \left (3 b +d \right ) \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right ) \left (3 b -d \right ) \left (b -d \right )}\)	\(260\)
risch	\(-\frac {27 \cos \left (x b -d x +a -c \right ) b^{3}}{8 \left (b +d \right ) \left (3 b +d \right ) \left (-b +d \right ) \left (-3 b +d \right )}-\frac {27 \cos \left (x b -d x +a -c \right ) b^{2} d}{8 \left (b +d \right ) \left (3 b +d \right ) \left (-b +d \right ) \left (-3 b +d \right )}+\frac {3 \cos \left (x b -d x +a -c \right ) b \,d^{2}}{8 \left (b +d \right ) \left (3 b +d \right ) \left (-b +d \right ) \left (-3 b +d \right )}+\frac {3 \cos \left (x b -d x +a -c \right ) d^{3}}{8 \left (b +d \right ) \left (3 b +d \right ) \left (-b +d \right ) \left (-3 b +d \right )}-\frac {27 \cos \left (x b +d x +a +c \right ) b^{3}}{8 \left (b +d \right ) \left (3 b +d \right ) \left (b -d \right ) \left (3 b -d \right )}+\frac {27 \cos \left (x b +d x +a +c \right ) b^{2} d}{8 \left (b +d \right ) \left (3 b +d \right ) \left (b -d \right ) \left (3 b -d \right )}+\frac {3 \cos \left (x b +d x +a +c \right ) b \,d^{2}}{8 \left (b +d \right ) \left (3 b +d \right ) \left (b -d \right ) \left (3 b -d \right )}-\frac {3 \cos \left (x b +d x +a +c \right ) d^{3}}{8 \left (b +d \right ) \left (3 b +d \right ) \left (b -d \right ) \left (3 b -d \right )}+\frac {3 \cos \left (3 x b -d x +3 a -c \right ) b^{3}}{8 \left (b +d \right ) \left (3 b +d \right ) \left (-b +d \right ) \left (-3 b +d \right )}+\frac {\cos \left (3 x b -d x +3 a -c \right ) b^{2} d}{8 \left (b +d \right ) \left (3 b +d \right ) \left (-b +d \right ) \left (-3 b +d \right )}-\frac {3 \cos \left (3 x b -d x +3 a -c \right ) b \,d^{2}}{8 \left (b +d \right ) \left (3 b +d \right ) \left (-b +d \right ) \left (-3 b +d \right )}-\frac {\cos \left (3 x b -d x +3 a -c \right ) d^{3}}{8 \left (b +d \right ) \left (3 b +d \right ) \left (-b +d \right ) \left (-3 b +d \right )}+\frac {3 \cos \left (3 x b +d x +3 a +c \right ) b^{3}}{8 \left (b +d \right ) \left (3 b +d \right ) \left (b -d \right ) \left (3 b -d \right )}-\frac {\cos \left (3 x b +d x +3 a +c \right ) b^{2} d}{8 \left (b +d \right ) \left (3 b +d \right ) \left (b -d \right ) \left (3 b -d \right )}-\frac {3 \cos \left (3 x b +d x +3 a +c \right ) b \,d^{2}}{8 \left (b +d \right ) \left (3 b +d \right ) \left (b -d \right ) \left (3 b -d \right )}+\frac {\cos \left (3 x b +d x +3 a +c \right ) d^{3}}{8 \left (b +d \right ) \left (3 b +d \right ) \left (b -d \right ) \left (3 b -d \right )}\)	\(730\)

[In]

int(cos(d*x+c)*sin(b*x+a)^3,x,method=_RETURNVERBOSE)

[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)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.20 \[ \int \cos (c+d x) \sin ^3(a+b x) \, dx=-\frac {{\left (7 \, b^{2} d - d^{3} - {\left (b^{2} d - d^{3}\right )} \cos \left (b x + a\right )^{2}\right )} \sin \left (b x + a\right ) \sin \left (d x + c\right ) - 3 \, {\left ({\left (b^{3} - b d^{2}\right )} \cos \left (b x + a\right )^{3} - {\left (3 \, b^{3} - b d^{2}\right )} \cos \left (b x + a\right )\right )} \cos \left (d x + c\right )}{9 \, b^{4} - 10 \, b^{2} d^{2} + d^{4}} \]

[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)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 932 vs. \(2 (76) = 152\).

Time = 2.04 (sec) , antiderivative size = 932, normalized size of antiderivative = 9.61 \[ \int \cos (c+d x) \sin ^3(a+b x) \, dx=\text {Too large to display} \]

[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 - sin(a - d*x)**3*sin(c + d*x)/(8*d) + 3*sin(a - d*x)**2*cos(a - d*x)*cos(c + d*x)/(4*d) + 3*cos(a - d*x
)**3*cos(c + d*x)/(8*d), Eq(b, -d)), (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 +
9*sin(a - d*x/3)**3*sin(c + d*x)/(8*d) - 3*sin(a - d*x/3)**2*cos(a - d*x/3)*cos(c + d*x)/(4*d) - cos(a - d*x/3
)**3*cos(c + d*x)/(8*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
+ 9*sin(a + d*x/3)**3*sin(c + d*x)/(8*d) + 3*sin(a + d*x/3)**2*cos(a + d*x/3)*cos(c + d*x)/(4*d) + cos(a + d*x
/3)**3*cos(c + d*x)/(8*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)/(8*d) - 3*sin(a + d*x)**2*cos(a + d*x)*cos(c + d*x)/(4*d) - 3*cos(a + d*x)**3*cos(c +
d*x)/(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))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 785 vs. \(2 (89) = 178\).

Time = 0.28 (sec) , antiderivative size = 785, normalized size of antiderivative = 8.09 \[ \int \cos (c+d x) \sin ^3(a+b x) \, dx=\text {Too large to display} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.92 \[ \int \cos (c+d x) \sin ^3(a+b x) \, dx=\frac {\cos \left (3 \, b x + d x + 3 \, a + c\right )}{8 \, {\left (3 \, b + d\right )}} + \frac {\cos \left (3 \, b x - d x + 3 \, a - c\right )}{8 \, {\left (3 \, b - d\right )}} - \frac {3 \, \cos \left (b x + d x + a + c\right )}{8 \, {\left (b + d\right )}} - \frac {3 \, \cos \left (b x - d x + a - c\right )}{8 \, {\left (b - d\right )}} \]

[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)

Mupad [B] (verification not implemented)

Time = 21.98 (sec) , antiderivative size = 471, normalized size of antiderivative = 4.86 \[ \int \cos (c+d x) \sin ^3(a+b x) \, dx=-{\mathrm {e}}^{a\,3{}\mathrm {i}-c\,1{}\mathrm {i}+b\,x\,3{}\mathrm {i}-d\,x\,1{}\mathrm {i}}\,\left (\frac {-3\,b^3-b^2\,d+3\,b\,d^2+d^3}{144\,b^4-160\,b^2\,d^2+16\,d^4}+\frac {{\mathrm {e}}^{-a\,6{}\mathrm {i}-b\,x\,6{}\mathrm {i}}\,\left (-3\,b^3+b^2\,d+3\,b\,d^2-d^3\right )}{144\,b^4-160\,b^2\,d^2+16\,d^4}-\frac {{\mathrm {e}}^{-a\,2{}\mathrm {i}-b\,x\,2{}\mathrm {i}}\,\left (-27\,b^3-27\,b^2\,d+3\,b\,d^2+3\,d^3\right )}{144\,b^4-160\,b^2\,d^2+16\,d^4}-\frac {{\mathrm {e}}^{-a\,4{}\mathrm {i}-b\,x\,4{}\mathrm {i}}\,\left (-27\,b^3+27\,b^2\,d+3\,b\,d^2-3\,d^3\right )}{144\,b^4-160\,b^2\,d^2+16\,d^4}\right )-{\mathrm {e}}^{a\,3{}\mathrm {i}+c\,1{}\mathrm {i}+b\,x\,3{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,\left (\frac {-3\,b^3+b^2\,d+3\,b\,d^2-d^3}{144\,b^4-160\,b^2\,d^2+16\,d^4}+\frac {{\mathrm {e}}^{-a\,6{}\mathrm {i}-b\,x\,6{}\mathrm {i}}\,\left (-3\,b^3-b^2\,d+3\,b\,d^2+d^3\right )}{144\,b^4-160\,b^2\,d^2+16\,d^4}-\frac {{\mathrm {e}}^{-a\,2{}\mathrm {i}-b\,x\,2{}\mathrm {i}}\,\left (-27\,b^3+27\,b^2\,d+3\,b\,d^2-3\,d^3\right )}{144\,b^4-160\,b^2\,d^2+16\,d^4}-\frac {{\mathrm {e}}^{-a\,4{}\mathrm {i}-b\,x\,4{}\mathrm {i}}\,\left (-27\,b^3-27\,b^2\,d+3\,b\,d^2+3\,d^3\right )}{144\,b^4-160\,b^2\,d^2+16\,d^4}\right ) \]

[In]

int(cos(c + d*x)*sin(a + b*x)^3,x)

[Out]

- exp(a*3i - c*1i + b*x*3i - d*x*1i)*((3*b*d^2 - b^2*d - 3*b^3 + d^3)/(144*b^4 + 16*d^4 - 160*b^2*d^2) + (exp(
- a*6i - b*x*6i)*(3*b*d^2 + b^2*d - 3*b^3 - d^3))/(144*b^4 + 16*d^4 - 160*b^2*d^2) - (exp(- a*2i - b*x*2i)*(3*
b*d^2 - 27*b^2*d - 27*b^3 + 3*d^3))/(144*b^4 + 16*d^4 - 160*b^2*d^2) - (exp(- a*4i - b*x*4i)*(3*b*d^2 + 27*b^2
*d - 27*b^3 - 3*d^3))/(144*b^4 + 16*d^4 - 160*b^2*d^2)) - exp(a*3i + c*1i + b*x*3i + d*x*1i)*((3*b*d^2 + b^2*d
- 3*b^3 - d^3)/(144*b^4 + 16*d^4 - 160*b^2*d^2) + (exp(- a*6i - b*x*6i)*(3*b*d^2 - b^2*d - 3*b^3 + d^3))/(144
*b^4 + 16*d^4 - 160*b^2*d^2) - (exp(- a*2i - b*x*2i)*(3*b*d^2 + 27*b^2*d - 27*b^3 - 3*d^3))/(144*b^4 + 16*d^4
- 160*b^2*d^2) - (exp(- a*4i - b*x*4i)*(3*b*d^2 - 27*b^2*d - 27*b^3 + 3*d^3))/(144*b^4 + 16*d^4 - 160*b^2*d^2)
)