\(\int (c+d x)^4 \cos ^2(a+b x) \sin ^2(a+b x) \, dx\) [80]

Optimal result

Integrand size = 24, antiderivative size = 131 \[ \int (c+d x)^4 \cos ^2(a+b x) \sin ^2(a+b x) \, dx=\frac {(c+d x)^5}{40 d}+\frac {3 d^3 (c+d x) \cos (4 a+4 b x)}{256 b^4}-\frac {d (c+d x)^3 \cos (4 a+4 b x)}{32 b^2}-\frac {3 d^4 \sin (4 a+4 b x)}{1024 b^5}+\frac {3 d^2 (c+d x)^2 \sin (4 a+4 b x)}{128 b^3}-\frac {(c+d x)^4 \sin (4 a+4 b x)}{32 b} \]

[Out]

1/40*(d*x+c)^5/d+3/256*d^3*(d*x+c)*cos(4*b*x+4*a)/b^4-1/32*d*(d*x+c)^3*cos(4*b*x+4*a)/b^2-3/1024*d^4*sin(4*b*x
+4*a)/b^5+3/128*d^2*(d*x+c)^2*sin(4*b*x+4*a)/b^3-1/32*(d*x+c)^4*sin(4*b*x+4*a)/b

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {4491, 3377, 2717} \[ \int (c+d x)^4 \cos ^2(a+b x) \sin ^2(a+b x) \, dx=-\frac {3 d^4 \sin (4 a+4 b x)}{1024 b^5}+\frac {3 d^3 (c+d x) \cos (4 a+4 b x)}{256 b^4}+\frac {3 d^2 (c+d x)^2 \sin (4 a+4 b x)}{128 b^3}-\frac {d (c+d x)^3 \cos (4 a+4 b x)}{32 b^2}-\frac {(c+d x)^4 \sin (4 a+4 b x)}{32 b}+\frac {(c+d x)^5}{40 d} \]

[In]

Int[(c + d*x)^4*Cos[a + b*x]^2*Sin[a + b*x]^2,x]

[Out]

(c + d*x)^5/(40*d) + (3*d^3*(c + d*x)*Cos[4*a + 4*b*x])/(256*b^4) - (d*(c + d*x)^3*Cos[4*a + 4*b*x])/(32*b^2)
- (3*d^4*Sin[4*a + 4*b*x])/(1024*b^5) + (3*d^2*(c + d*x)^2*Sin[4*a + 4*b*x])/(128*b^3) - ((c + d*x)^4*Sin[4*a
+ 4*b*x])/(32*b)

Rule 2717

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3377

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(-(c + d*x)^m)*(Cos[e + f*x]/f), x]
+ Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 4491

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[E
xpandTrigReduce[(c + d*x)^m, Sin[a + b*x]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0]
&& IGtQ[p, 0]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{8} (c+d x)^4-\frac {1}{8} (c+d x)^4 \cos (4 a+4 b x)\right ) \, dx \\ & = \frac {(c+d x)^5}{40 d}-\frac {1}{8} \int (c+d x)^4 \cos (4 a+4 b x) \, dx \\ & = \frac {(c+d x)^5}{40 d}-\frac {(c+d x)^4 \sin (4 a+4 b x)}{32 b}+\frac {d \int (c+d x)^3 \sin (4 a+4 b x) \, dx}{8 b} \\ & = \frac {(c+d x)^5}{40 d}-\frac {d (c+d x)^3 \cos (4 a+4 b x)}{32 b^2}-\frac {(c+d x)^4 \sin (4 a+4 b x)}{32 b}+\frac {\left (3 d^2\right ) \int (c+d x)^2 \cos (4 a+4 b x) \, dx}{32 b^2} \\ & = \frac {(c+d x)^5}{40 d}-\frac {d (c+d x)^3 \cos (4 a+4 b x)}{32 b^2}+\frac {3 d^2 (c+d x)^2 \sin (4 a+4 b x)}{128 b^3}-\frac {(c+d x)^4 \sin (4 a+4 b x)}{32 b}-\frac {\left (3 d^3\right ) \int (c+d x) \sin (4 a+4 b x) \, dx}{64 b^3} \\ & = \frac {(c+d x)^5}{40 d}+\frac {3 d^3 (c+d x) \cos (4 a+4 b x)}{256 b^4}-\frac {d (c+d x)^3 \cos (4 a+4 b x)}{32 b^2}+\frac {3 d^2 (c+d x)^2 \sin (4 a+4 b x)}{128 b^3}-\frac {(c+d x)^4 \sin (4 a+4 b x)}{32 b}-\frac {\left (3 d^4\right ) \int \cos (4 a+4 b x) \, dx}{256 b^4} \\ & = \frac {(c+d x)^5}{40 d}+\frac {3 d^3 (c+d x) \cos (4 a+4 b x)}{256 b^4}-\frac {d (c+d x)^3 \cos (4 a+4 b x)}{32 b^2}-\frac {3 d^4 \sin (4 a+4 b x)}{1024 b^5}+\frac {3 d^2 (c+d x)^2 \sin (4 a+4 b x)}{128 b^3}-\frac {(c+d x)^4 \sin (4 a+4 b x)}{32 b} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.25 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.01 \[ \int (c+d x)^4 \cos ^2(a+b x) \sin ^2(a+b x) \, dx=\frac {128 b^5 x \left (5 c^4+10 c^3 d x+10 c^2 d^2 x^2+5 c d^3 x^3+d^4 x^4\right )+20 b d (c+d x) \left (3 d^2-8 b^2 (c+d x)^2\right ) \cos (4 (a+b x))-5 \left (3 d^4-24 b^2 d^2 (c+d x)^2+32 b^4 (c+d x)^4\right ) \sin (4 (a+b x))}{5120 b^5} \]

[In]

Integrate[(c + d*x)^4*Cos[a + b*x]^2*Sin[a + b*x]^2,x]

[Out]

(128*b^5*x*(5*c^4 + 10*c^3*d*x + 10*c^2*d^2*x^2 + 5*c*d^3*x^3 + d^4*x^4) + 20*b*d*(c + d*x)*(3*d^2 - 8*b^2*(c
+ d*x)^2)*Cos[4*(a + b*x)] - 5*(3*d^4 - 24*b^2*d^2*(c + d*x)^2 + 32*b^4*(c + d*x)^4)*Sin[4*(a + b*x)])/(5120*b
^5)

Maple [A] (verified)

Time = 2.63 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.11

method	result	size
parallelrisch	\(\frac {\left (-32 b^{4} \left (d x +c \right )^{4}+24 d^{2} \left (d x +c \right )^{2} b^{2}-3 d^{4}\right ) \sin \left (4 x b +4 a \right )+128 b \left (-\frac {\left (\left (d x +c \right )^{2} b^{2}-\frac {3 d^{2}}{8}\right ) d \left (d x +c \right ) \cos \left (4 x b +4 a \right )}{4}+x \left (\frac {1}{5} d^{4} x^{4}+c \,d^{3} x^{3}+2 c^{2} d^{2} x^{2}+2 c^{3} d x +c^{4}\right ) b^{4}+\frac {b^{2} c^{3} d}{4}-\frac {3 d^{3} c}{32}\right )}{1024 b^{5}}\)	\(146\)
risch	\(\frac {d^{4} x^{5}}{40}+\frac {d^{3} c \,x^{4}}{8}+\frac {d^{2} c^{2} x^{3}}{4}+\frac {d \,c^{3} x^{2}}{4}+\frac {c^{4} x}{8}+\frac {c^{5}}{40 d}-\frac {d \left (8 b^{2} d^{3} x^{3}+24 b^{2} c \,d^{2} x^{2}+24 b^{2} c^{2} d x +8 b^{2} c^{3}-3 d^{3} x -3 c \,d^{2}\right ) \cos \left (4 x b +4 a \right )}{256 b^{4}}-\frac {\left (32 d^{4} x^{4} b^{4}+128 b^{4} c \,d^{3} x^{3}+192 b^{4} c^{2} d^{2} x^{2}+128 b^{4} c^{3} d x +32 b^{4} c^{4}-24 b^{2} d^{4} x^{2}-48 b^{2} c \,d^{3} x -24 b^{2} c^{2} d^{2}+3 d^{4}\right ) \sin \left (4 x b +4 a \right )}{1024 b^{5}}\)	\(229\)
derivativedivides	\(\text {Expression too large to display}\)	\(1951\)
default	\(\text {Expression too large to display}\)	\(1951\)

[In]

int((d*x+c)^4*cos(b*x+a)^2*sin(b*x+a)^2,x,method=_RETURNVERBOSE)

[Out]

1/1024*((-32*b^4*(d*x+c)^4+24*d^2*(d*x+c)^2*b^2-3*d^4)*sin(4*b*x+4*a)+128*b*(-1/4*((d*x+c)^2*b^2-3/8*d^2)*d*(d
*x+c)*cos(4*b*x+4*a)+x*(1/5*d^4*x^4+c*d^3*x^3+2*c^2*d^2*x^2+2*c^3*d*x+c^4)*b^4+1/4*b^2*c^3*d-3/32*d^3*c))/b^5

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 466 vs. \(2 (119) = 238\).

Time = 0.26 (sec) , antiderivative size = 466, normalized size of antiderivative = 3.56 \[ \int (c+d x)^4 \cos ^2(a+b x) \sin ^2(a+b x) \, dx=\frac {32 \, b^{5} d^{4} x^{5} + 160 \, b^{5} c d^{3} x^{4} - 40 \, {\left (8 \, b^{3} d^{4} x^{3} + 24 \, b^{3} c d^{3} x^{2} + 8 \, b^{3} c^{3} d - 3 \, b c d^{3} + 3 \, {\left (8 \, b^{3} c^{2} d^{2} - b d^{4}\right )} x\right )} \cos \left (b x + a\right )^{4} + 40 \, {\left (8 \, b^{5} c^{2} d^{2} - b^{3} d^{4}\right )} x^{3} + 40 \, {\left (8 \, b^{5} c^{3} d - 3 \, b^{3} c d^{3}\right )} x^{2} + 40 \, {\left (8 \, b^{3} d^{4} x^{3} + 24 \, b^{3} c d^{3} x^{2} + 8 \, b^{3} c^{3} d - 3 \, b c d^{3} + 3 \, {\left (8 \, b^{3} c^{2} d^{2} - b d^{4}\right )} x\right )} \cos \left (b x + a\right )^{2} + 5 \, {\left (32 \, b^{5} c^{4} - 24 \, b^{3} c^{2} d^{2} + 3 \, b d^{4}\right )} x - 5 \, {\left (2 \, {\left (32 \, b^{4} d^{4} x^{4} + 128 \, b^{4} c d^{3} x^{3} + 32 \, b^{4} c^{4} - 24 \, b^{2} c^{2} d^{2} + 3 \, d^{4} + 24 \, {\left (8 \, b^{4} c^{2} d^{2} - b^{2} d^{4}\right )} x^{2} + 16 \, {\left (8 \, b^{4} c^{3} d - 3 \, b^{2} c d^{3}\right )} x\right )} \cos \left (b x + a\right )^{3} - {\left (32 \, b^{4} d^{4} x^{4} + 128 \, b^{4} c d^{3} x^{3} + 32 \, b^{4} c^{4} - 24 \, b^{2} c^{2} d^{2} + 3 \, d^{4} + 24 \, {\left (8 \, b^{4} c^{2} d^{2} - b^{2} d^{4}\right )} x^{2} + 16 \, {\left (8 \, b^{4} c^{3} d - 3 \, b^{2} c d^{3}\right )} x\right )} \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )}{1280 \, b^{5}} \]

[In]

integrate((d*x+c)^4*cos(b*x+a)^2*sin(b*x+a)^2,x, algorithm="fricas")

[Out]

1/1280*(32*b^5*d^4*x^5 + 160*b^5*c*d^3*x^4 - 40*(8*b^3*d^4*x^3 + 24*b^3*c*d^3*x^2 + 8*b^3*c^3*d - 3*b*c*d^3 +
3*(8*b^3*c^2*d^2 - b*d^4)*x)*cos(b*x + a)^4 + 40*(8*b^5*c^2*d^2 - b^3*d^4)*x^3 + 40*(8*b^5*c^3*d - 3*b^3*c*d^3
)*x^2 + 40*(8*b^3*d^4*x^3 + 24*b^3*c*d^3*x^2 + 8*b^3*c^3*d - 3*b*c*d^3 + 3*(8*b^3*c^2*d^2 - b*d^4)*x)*cos(b*x
+ a)^2 + 5*(32*b^5*c^4 - 24*b^3*c^2*d^2 + 3*b*d^4)*x - 5*(2*(32*b^4*d^4*x^4 + 128*b^4*c*d^3*x^3 + 32*b^4*c^4 -
24*b^2*c^2*d^2 + 3*d^4 + 24*(8*b^4*c^2*d^2 - b^2*d^4)*x^2 + 16*(8*b^4*c^3*d - 3*b^2*c*d^3)*x)*cos(b*x + a)^3
- (32*b^4*d^4*x^4 + 128*b^4*c*d^3*x^3 + 32*b^4*c^4 - 24*b^2*c^2*d^2 + 3*d^4 + 24*(8*b^4*c^2*d^2 - b^2*d^4)*x^2
+ 16*(8*b^4*c^3*d - 3*b^2*c*d^3)*x)*cos(b*x + a))*sin(b*x + a))/b^5

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1231 vs. \(2 (126) = 252\).

Time = 0.91 (sec) , antiderivative size = 1231, normalized size of antiderivative = 9.40 \[ \int (c+d x)^4 \cos ^2(a+b x) \sin ^2(a+b x) \, dx=\text {Too large to display} \]

[In]

integrate((d*x+c)**4*cos(b*x+a)**2*sin(b*x+a)**2,x)

[Out]

Piecewise((c**4*x*sin(a + b*x)**4/8 + c**4*x*sin(a + b*x)**2*cos(a + b*x)**2/4 + c**4*x*cos(a + b*x)**4/8 + c*
*3*d*x**2*sin(a + b*x)**4/4 + c**3*d*x**2*sin(a + b*x)**2*cos(a + b*x)**2/2 + c**3*d*x**2*cos(a + b*x)**4/4 +
c**2*d**2*x**3*sin(a + b*x)**4/4 + c**2*d**2*x**3*sin(a + b*x)**2*cos(a + b*x)**2/2 + c**2*d**2*x**3*cos(a + b
*x)**4/4 + c*d**3*x**4*sin(a + b*x)**4/8 + c*d**3*x**4*sin(a + b*x)**2*cos(a + b*x)**2/4 + c*d**3*x**4*cos(a +
b*x)**4/8 + d**4*x**5*sin(a + b*x)**4/40 + d**4*x**5*sin(a + b*x)**2*cos(a + b*x)**2/20 + d**4*x**5*cos(a + b
*x)**4/40 + c**4*sin(a + b*x)**3*cos(a + b*x)/(8*b) - c**4*sin(a + b*x)*cos(a + b*x)**3/(8*b) + c**3*d*x*sin(a
+ b*x)**3*cos(a + b*x)/(2*b) - c**3*d*x*sin(a + b*x)*cos(a + b*x)**3/(2*b) + 3*c**2*d**2*x**2*sin(a + b*x)**3
*cos(a + b*x)/(4*b) - 3*c**2*d**2*x**2*sin(a + b*x)*cos(a + b*x)**3/(4*b) + c*d**3*x**3*sin(a + b*x)**3*cos(a
+ b*x)/(2*b) - c*d**3*x**3*sin(a + b*x)*cos(a + b*x)**3/(2*b) + d**4*x**4*sin(a + b*x)**3*cos(a + b*x)/(8*b) -
d**4*x**4*sin(a + b*x)*cos(a + b*x)**3/(8*b) - c**3*d*sin(a + b*x)**4/(8*b**2) - c**3*d*cos(a + b*x)**4/(8*b*
*2) - 3*c**2*d**2*x*sin(a + b*x)**4/(32*b**2) + 9*c**2*d**2*x*sin(a + b*x)**2*cos(a + b*x)**2/(16*b**2) - 3*c*
*2*d**2*x*cos(a + b*x)**4/(32*b**2) - 3*c*d**3*x**2*sin(a + b*x)**4/(32*b**2) + 9*c*d**3*x**2*sin(a + b*x)**2*
cos(a + b*x)**2/(16*b**2) - 3*c*d**3*x**2*cos(a + b*x)**4/(32*b**2) - d**4*x**3*sin(a + b*x)**4/(32*b**2) + 3*
d**4*x**3*sin(a + b*x)**2*cos(a + b*x)**2/(16*b**2) - d**4*x**3*cos(a + b*x)**4/(32*b**2) - 3*c**2*d**2*sin(a
+ b*x)**3*cos(a + b*x)/(32*b**3) + 3*c**2*d**2*sin(a + b*x)*cos(a + b*x)**3/(32*b**3) - 3*c*d**3*x*sin(a + b*x
)**3*cos(a + b*x)/(16*b**3) + 3*c*d**3*x*sin(a + b*x)*cos(a + b*x)**3/(16*b**3) - 3*d**4*x**2*sin(a + b*x)**3*
cos(a + b*x)/(32*b**3) + 3*d**4*x**2*sin(a + b*x)*cos(a + b*x)**3/(32*b**3) + 3*c*d**3*sin(a + b*x)**4/(64*b**
4) + 3*c*d**3*cos(a + b*x)**4/(64*b**4) + 3*d**4*x*sin(a + b*x)**4/(256*b**4) - 9*d**4*x*sin(a + b*x)**2*cos(a
+ b*x)**2/(128*b**4) + 3*d**4*x*cos(a + b*x)**4/(256*b**4) + 3*d**4*sin(a + b*x)**3*cos(a + b*x)/(256*b**5) -
3*d**4*sin(a + b*x)*cos(a + b*x)**3/(256*b**5), Ne(b, 0)), ((c**4*x + 2*c**3*d*x**2 + 2*c**2*d**2*x**3 + c*d*
*3*x**4 + d**4*x**5/5)*sin(a)**2*cos(a)**2, True))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 735 vs. \(2 (119) = 238\).

Time = 0.27 (sec) , antiderivative size = 735, normalized size of antiderivative = 5.61 \[ \int (c+d x)^4 \cos ^2(a+b x) \sin ^2(a+b x) \, dx=\frac {160 \, {\left (4 \, b x + 4 \, a - \sin \left (4 \, b x + 4 \, a\right )\right )} c^{4} - \frac {640 \, {\left (4 \, b x + 4 \, a - \sin \left (4 \, b x + 4 \, a\right )\right )} a c^{3} d}{b} + \frac {960 \, {\left (4 \, b x + 4 \, a - \sin \left (4 \, b x + 4 \, a\right )\right )} a^{2} c^{2} d^{2}}{b^{2}} - \frac {640 \, {\left (4 \, b x + 4 \, a - \sin \left (4 \, b x + 4 \, a\right )\right )} a^{3} c d^{3}}{b^{3}} + \frac {160 \, {\left (4 \, b x + 4 \, a - \sin \left (4 \, b x + 4 \, a\right )\right )} a^{4} d^{4}}{b^{4}} + \frac {160 \, {\left (8 \, {\left (b x + a\right )}^{2} - 4 \, {\left (b x + a\right )} \sin \left (4 \, b x + 4 \, a\right ) - \cos \left (4 \, b x + 4 \, a\right )\right )} c^{3} d}{b} - \frac {480 \, {\left (8 \, {\left (b x + a\right )}^{2} - 4 \, {\left (b x + a\right )} \sin \left (4 \, b x + 4 \, a\right ) - \cos \left (4 \, b x + 4 \, a\right )\right )} a c^{2} d^{2}}{b^{2}} + \frac {480 \, {\left (8 \, {\left (b x + a\right )}^{2} - 4 \, {\left (b x + a\right )} \sin \left (4 \, b x + 4 \, a\right ) - \cos \left (4 \, b x + 4 \, a\right )\right )} a^{2} c d^{3}}{b^{3}} - \frac {160 \, {\left (8 \, {\left (b x + a\right )}^{2} - 4 \, {\left (b x + a\right )} \sin \left (4 \, b x + 4 \, a\right ) - \cos \left (4 \, b x + 4 \, a\right )\right )} a^{3} d^{4}}{b^{4}} + \frac {40 \, {\left (32 \, {\left (b x + a\right )}^{3} - 12 \, {\left (b x + a\right )} \cos \left (4 \, b x + 4 \, a\right ) - 3 \, {\left (8 \, {\left (b x + a\right )}^{2} - 1\right )} \sin \left (4 \, b x + 4 \, a\right )\right )} c^{2} d^{2}}{b^{2}} - \frac {80 \, {\left (32 \, {\left (b x + a\right )}^{3} - 12 \, {\left (b x + a\right )} \cos \left (4 \, b x + 4 \, a\right ) - 3 \, {\left (8 \, {\left (b x + a\right )}^{2} - 1\right )} \sin \left (4 \, b x + 4 \, a\right )\right )} a c d^{3}}{b^{3}} + \frac {40 \, {\left (32 \, {\left (b x + a\right )}^{3} - 12 \, {\left (b x + a\right )} \cos \left (4 \, b x + 4 \, a\right ) - 3 \, {\left (8 \, {\left (b x + a\right )}^{2} - 1\right )} \sin \left (4 \, b x + 4 \, a\right )\right )} a^{2} d^{4}}{b^{4}} + \frac {20 \, {\left (32 \, {\left (b x + a\right )}^{4} - 3 \, {\left (8 \, {\left (b x + a\right )}^{2} - 1\right )} \cos \left (4 \, b x + 4 \, a\right ) - 4 \, {\left (8 \, {\left (b x + a\right )}^{3} - 3 \, b x - 3 \, a\right )} \sin \left (4 \, b x + 4 \, a\right )\right )} c d^{3}}{b^{3}} - \frac {20 \, {\left (32 \, {\left (b x + a\right )}^{4} - 3 \, {\left (8 \, {\left (b x + a\right )}^{2} - 1\right )} \cos \left (4 \, b x + 4 \, a\right ) - 4 \, {\left (8 \, {\left (b x + a\right )}^{3} - 3 \, b x - 3 \, a\right )} \sin \left (4 \, b x + 4 \, a\right )\right )} a d^{4}}{b^{4}} + \frac {{\left (128 \, {\left (b x + a\right )}^{5} - 20 \, {\left (8 \, {\left (b x + a\right )}^{3} - 3 \, b x - 3 \, a\right )} \cos \left (4 \, b x + 4 \, a\right ) - 5 \, {\left (32 \, {\left (b x + a\right )}^{4} - 24 \, {\left (b x + a\right )}^{2} + 3\right )} \sin \left (4 \, b x + 4 \, a\right )\right )} d^{4}}{b^{4}}}{5120 \, b} \]

[In]

integrate((d*x+c)^4*cos(b*x+a)^2*sin(b*x+a)^2,x, algorithm="maxima")

[Out]

1/5120*(160*(4*b*x + 4*a - sin(4*b*x + 4*a))*c^4 - 640*(4*b*x + 4*a - sin(4*b*x + 4*a))*a*c^3*d/b + 960*(4*b*x
+ 4*a - sin(4*b*x + 4*a))*a^2*c^2*d^2/b^2 - 640*(4*b*x + 4*a - sin(4*b*x + 4*a))*a^3*c*d^3/b^3 + 160*(4*b*x +
4*a - sin(4*b*x + 4*a))*a^4*d^4/b^4 + 160*(8*(b*x + a)^2 - 4*(b*x + a)*sin(4*b*x + 4*a) - cos(4*b*x + 4*a))*c
^3*d/b - 480*(8*(b*x + a)^2 - 4*(b*x + a)*sin(4*b*x + 4*a) - cos(4*b*x + 4*a))*a*c^2*d^2/b^2 + 480*(8*(b*x + a
)^2 - 4*(b*x + a)*sin(4*b*x + 4*a) - cos(4*b*x + 4*a))*a^2*c*d^3/b^3 - 160*(8*(b*x + a)^2 - 4*(b*x + a)*sin(4*
b*x + 4*a) - cos(4*b*x + 4*a))*a^3*d^4/b^4 + 40*(32*(b*x + a)^3 - 12*(b*x + a)*cos(4*b*x + 4*a) - 3*(8*(b*x +
a)^2 - 1)*sin(4*b*x + 4*a))*c^2*d^2/b^2 - 80*(32*(b*x + a)^3 - 12*(b*x + a)*cos(4*b*x + 4*a) - 3*(8*(b*x + a)^
2 - 1)*sin(4*b*x + 4*a))*a*c*d^3/b^3 + 40*(32*(b*x + a)^3 - 12*(b*x + a)*cos(4*b*x + 4*a) - 3*(8*(b*x + a)^2 -
1)*sin(4*b*x + 4*a))*a^2*d^4/b^4 + 20*(32*(b*x + a)^4 - 3*(8*(b*x + a)^2 - 1)*cos(4*b*x + 4*a) - 4*(8*(b*x +
a)^3 - 3*b*x - 3*a)*sin(4*b*x + 4*a))*c*d^3/b^3 - 20*(32*(b*x + a)^4 - 3*(8*(b*x + a)^2 - 1)*cos(4*b*x + 4*a)
- 4*(8*(b*x + a)^3 - 3*b*x - 3*a)*sin(4*b*x + 4*a))*a*d^4/b^4 + (128*(b*x + a)^5 - 20*(8*(b*x + a)^3 - 3*b*x -
3*a)*cos(4*b*x + 4*a) - 5*(32*(b*x + a)^4 - 24*(b*x + a)^2 + 3)*sin(4*b*x + 4*a))*d^4/b^4)/b

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.71 \[ \int (c+d x)^4 \cos ^2(a+b x) \sin ^2(a+b x) \, dx=\frac {1}{40} \, d^{4} x^{5} + \frac {1}{8} \, c d^{3} x^{4} + \frac {1}{4} \, c^{2} d^{2} x^{3} + \frac {1}{4} \, c^{3} d x^{2} + \frac {1}{8} \, c^{4} x - \frac {{\left (8 \, b^{3} d^{4} x^{3} + 24 \, b^{3} c d^{3} x^{2} + 24 \, b^{3} c^{2} d^{2} x + 8 \, b^{3} c^{3} d - 3 \, b d^{4} x - 3 \, b c d^{3}\right )} \cos \left (4 \, b x + 4 \, a\right )}{256 \, b^{5}} - \frac {{\left (32 \, b^{4} d^{4} x^{4} + 128 \, b^{4} c d^{3} x^{3} + 192 \, b^{4} c^{2} d^{2} x^{2} + 128 \, b^{4} c^{3} d x + 32 \, b^{4} c^{4} - 24 \, b^{2} d^{4} x^{2} - 48 \, b^{2} c d^{3} x - 24 \, b^{2} c^{2} d^{2} + 3 \, d^{4}\right )} \sin \left (4 \, b x + 4 \, a\right )}{1024 \, b^{5}} \]

[In]

integrate((d*x+c)^4*cos(b*x+a)^2*sin(b*x+a)^2,x, algorithm="giac")

[Out]

1/40*d^4*x^5 + 1/8*c*d^3*x^4 + 1/4*c^2*d^2*x^3 + 1/4*c^3*d*x^2 + 1/8*c^4*x - 1/256*(8*b^3*d^4*x^3 + 24*b^3*c*d
^3*x^2 + 24*b^3*c^2*d^2*x + 8*b^3*c^3*d - 3*b*d^4*x - 3*b*c*d^3)*cos(4*b*x + 4*a)/b^5 - 1/1024*(32*b^4*d^4*x^4
+ 128*b^4*c*d^3*x^3 + 192*b^4*c^2*d^2*x^2 + 128*b^4*c^3*d*x + 32*b^4*c^4 - 24*b^2*d^4*x^2 - 48*b^2*c*d^3*x -
24*b^2*c^2*d^2 + 3*d^4)*sin(4*b*x + 4*a)/b^5

Mupad [B] (verification not implemented)

Time = 23.66 (sec) , antiderivative size = 349, normalized size of antiderivative = 2.66 \[ \int (c+d x)^4 \cos ^2(a+b x) \sin ^2(a+b x) \, dx=-\frac {15\,d^4\,\sin \left (4\,a+4\,b\,x\right )-640\,b^5\,c^4\,x+160\,b^4\,c^4\,\sin \left (4\,a+4\,b\,x\right )-128\,b^5\,d^4\,x^5+160\,b^3\,c^3\,d\,\cos \left (4\,a+4\,b\,x\right )-1280\,b^5\,c^3\,d\,x^2-640\,b^5\,c\,d^3\,x^4-120\,b^2\,c^2\,d^2\,\sin \left (4\,a+4\,b\,x\right )+160\,b^3\,d^4\,x^3\,\cos \left (4\,a+4\,b\,x\right )-1280\,b^5\,c^2\,d^2\,x^3-120\,b^2\,d^4\,x^2\,\sin \left (4\,a+4\,b\,x\right )+160\,b^4\,d^4\,x^4\,\sin \left (4\,a+4\,b\,x\right )-60\,b\,c\,d^3\,\cos \left (4\,a+4\,b\,x\right )-60\,b\,d^4\,x\,\cos \left (4\,a+4\,b\,x\right )+960\,b^4\,c^2\,d^2\,x^2\,\sin \left (4\,a+4\,b\,x\right )-240\,b^2\,c\,d^3\,x\,\sin \left (4\,a+4\,b\,x\right )+640\,b^4\,c^3\,d\,x\,\sin \left (4\,a+4\,b\,x\right )+480\,b^3\,c^2\,d^2\,x\,\cos \left (4\,a+4\,b\,x\right )+480\,b^3\,c\,d^3\,x^2\,\cos \left (4\,a+4\,b\,x\right )+640\,b^4\,c\,d^3\,x^3\,\sin \left (4\,a+4\,b\,x\right )}{5120\,b^5} \]

[In]

int(cos(a + b*x)^2*sin(a + b*x)^2*(c + d*x)^4,x)

[Out]

-(15*d^4*sin(4*a + 4*b*x) - 640*b^5*c^4*x + 160*b^4*c^4*sin(4*a + 4*b*x) - 128*b^5*d^4*x^5 + 160*b^3*c^3*d*cos
(4*a + 4*b*x) - 1280*b^5*c^3*d*x^2 - 640*b^5*c*d^3*x^4 - 120*b^2*c^2*d^2*sin(4*a + 4*b*x) + 160*b^3*d^4*x^3*co
s(4*a + 4*b*x) - 1280*b^5*c^2*d^2*x^3 - 120*b^2*d^4*x^2*sin(4*a + 4*b*x) + 160*b^4*d^4*x^4*sin(4*a + 4*b*x) -
60*b*c*d^3*cos(4*a + 4*b*x) - 60*b*d^4*x*cos(4*a + 4*b*x) + 960*b^4*c^2*d^2*x^2*sin(4*a + 4*b*x) - 240*b^2*c*d
^3*x*sin(4*a + 4*b*x) + 640*b^4*c^3*d*x*sin(4*a + 4*b*x) + 480*b^3*c^2*d^2*x*cos(4*a + 4*b*x) + 480*b^3*c*d^3*
x^2*cos(4*a + 4*b*x) + 640*b^4*c*d^3*x^3*sin(4*a + 4*b*x))/(5120*b^5)