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

Optimal. Leaf size=131 \[ \frac{3 d^2 (c+d x)^2 \sin (4 a+4 b x)}{128 b^3}+\frac{3 d^3 (c+d x) \cos (4 a+4 b x)}{256 b^4}-\frac{d (c+d x)^3 \cos (4 a+4 b x)}{32 b^2}-\frac{3 d^4 \sin (4 a+4 b x)}{1024 b^5}-\frac{(c+d x)^4 \sin (4 a+4 b x)}{32 b}+\frac{(c+d x)^5}{40 d} \]

[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)

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Rubi [A]  time = 0.1644, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {4406, 3296, 2637} \[ \frac{3 d^2 (c+d x)^2 \sin (4 a+4 b x)}{128 b^3}+\frac{3 d^3 (c+d x) \cos (4 a+4 b x)}{256 b^4}-\frac{d (c+d x)^3 \cos (4 a+4 b x)}{32 b^2}-\frac{3 d^4 \sin (4 a+4 b x)}{1024 b^5}-\frac{(c+d x)^4 \sin (4 a+4 b x)}{32 b}+\frac{(c+d x)^5}{40 d} \]

Antiderivative was successfully verified.

[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 4406

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]

Rule 3296

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 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 (c+d x)^4 \cos ^2(a+b x) \sin ^2(a+b x) \, dx &=\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]  time = 1.36655, size = 132, normalized size = 1.01 \[ \frac{-5 \sin (4 (a+b x)) \left (-24 b^2 d^2 (c+d x)^2+32 b^4 (c+d x)^4+3 d^4\right )+20 b d (c+d x) \cos (4 (a+b x)) \left (3 d^2-8 b^2 (c+d x)^2\right )+128 b^5 x \left (10 c^2 d^2 x^2+10 c^3 d x+5 c^4+5 c d^3 x^3+d^4 x^4\right )}{5120 b^5} \]

Antiderivative was successfully verified.

[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)

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Maple [B]  time = 0.076, size = 1915, normalized size = 14.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/b*(1/b^4*d^4*((b*x+a)^4*(-1/2*cos(b*x+a)*sin(b*x+a)+1/2*b*x+1/2*a)-1/4*(b*x+a)^3*cos(b*x+a)^2+3/4*(b*x+a)^2*
(1/2*cos(b*x+a)*sin(b*x+a)+1/2*b*x+1/2*a)+3/32*(b*x+a)*cos(b*x+a)^2-3/64*cos(b*x+a)*sin(b*x+a)-21/256*b*x-21/2
56*a-7/16*(b*x+a)^3-1/10*(b*x+a)^5-(b*x+a)^4*(-1/4*(sin(b*x+a)^3+3/2*sin(b*x+a))*cos(b*x+a)+3/8*b*x+3/8*a)-1/4
*(b*x+a)^3*sin(b*x+a)^4+3/4*(b*x+a)^2*(-1/4*(sin(b*x+a)^3+3/2*sin(b*x+a))*cos(b*x+a)+3/8*b*x+3/8*a)+3/32*(b*x+
a)*sin(b*x+a)^4+3/128*(sin(b*x+a)^3+3/2*sin(b*x+a))*cos(b*x+a))-4/b^4*a*d^4*((b*x+a)^3*(-1/2*cos(b*x+a)*sin(b*
x+a)+1/2*b*x+1/2*a)-3/16*(b*x+a)^2*cos(b*x+a)^2+3/8*(b*x+a)*(1/2*cos(b*x+a)*sin(b*x+a)+1/2*b*x+1/2*a)-21/128*(
b*x+a)^2-3/128*sin(b*x+a)^2-3/32*(b*x+a)^4-(b*x+a)^3*(-1/4*(sin(b*x+a)^3+3/2*sin(b*x+a))*cos(b*x+a)+3/8*b*x+3/
8*a)-3/16*(b*x+a)^2*sin(b*x+a)^4+3/8*(b*x+a)*(-1/4*(sin(b*x+a)^3+3/2*sin(b*x+a))*cos(b*x+a)+3/8*b*x+3/8*a)+3/1
28*sin(b*x+a)^4)+4/b^3*c*d^3*((b*x+a)^3*(-1/2*cos(b*x+a)*sin(b*x+a)+1/2*b*x+1/2*a)-3/16*(b*x+a)^2*cos(b*x+a)^2
+3/8*(b*x+a)*(1/2*cos(b*x+a)*sin(b*x+a)+1/2*b*x+1/2*a)-21/128*(b*x+a)^2-3/128*sin(b*x+a)^2-3/32*(b*x+a)^4-(b*x
+a)^3*(-1/4*(sin(b*x+a)^3+3/2*sin(b*x+a))*cos(b*x+a)+3/8*b*x+3/8*a)-3/16*(b*x+a)^2*sin(b*x+a)^4+3/8*(b*x+a)*(-
1/4*(sin(b*x+a)^3+3/2*sin(b*x+a))*cos(b*x+a)+3/8*b*x+3/8*a)+3/128*sin(b*x+a)^4)+6/b^4*a^2*d^4*((b*x+a)^2*(-1/2
*cos(b*x+a)*sin(b*x+a)+1/2*b*x+1/2*a)-1/8*(b*x+a)*cos(b*x+a)^2+1/16*cos(b*x+a)*sin(b*x+a)+7/64*b*x+7/64*a-1/12
*(b*x+a)^3-(b*x+a)^2*(-1/4*(sin(b*x+a)^3+3/2*sin(b*x+a))*cos(b*x+a)+3/8*b*x+3/8*a)-1/8*(b*x+a)*sin(b*x+a)^4-1/
32*(sin(b*x+a)^3+3/2*sin(b*x+a))*cos(b*x+a))-12/b^3*a*c*d^3*((b*x+a)^2*(-1/2*cos(b*x+a)*sin(b*x+a)+1/2*b*x+1/2
*a)-1/8*(b*x+a)*cos(b*x+a)^2+1/16*cos(b*x+a)*sin(b*x+a)+7/64*b*x+7/64*a-1/12*(b*x+a)^3-(b*x+a)^2*(-1/4*(sin(b*
x+a)^3+3/2*sin(b*x+a))*cos(b*x+a)+3/8*b*x+3/8*a)-1/8*(b*x+a)*sin(b*x+a)^4-1/32*(sin(b*x+a)^3+3/2*sin(b*x+a))*c
os(b*x+a))+6/b^2*c^2*d^2*((b*x+a)^2*(-1/2*cos(b*x+a)*sin(b*x+a)+1/2*b*x+1/2*a)-1/8*(b*x+a)*cos(b*x+a)^2+1/16*c
os(b*x+a)*sin(b*x+a)+7/64*b*x+7/64*a-1/12*(b*x+a)^3-(b*x+a)^2*(-1/4*(sin(b*x+a)^3+3/2*sin(b*x+a))*cos(b*x+a)+3
/8*b*x+3/8*a)-1/8*(b*x+a)*sin(b*x+a)^4-1/32*(sin(b*x+a)^3+3/2*sin(b*x+a))*cos(b*x+a))-4/b^4*a^3*d^4*((b*x+a)*(
-1/2*cos(b*x+a)*sin(b*x+a)+1/2*b*x+1/2*a)-1/16*(b*x+a)^2+1/16*sin(b*x+a)^2-(b*x+a)*(-1/4*(sin(b*x+a)^3+3/2*sin
(b*x+a))*cos(b*x+a)+3/8*b*x+3/8*a)-1/16*sin(b*x+a)^4)+12/b^3*a^2*c*d^3*((b*x+a)*(-1/2*cos(b*x+a)*sin(b*x+a)+1/
2*b*x+1/2*a)-1/16*(b*x+a)^2+1/16*sin(b*x+a)^2-(b*x+a)*(-1/4*(sin(b*x+a)^3+3/2*sin(b*x+a))*cos(b*x+a)+3/8*b*x+3
/8*a)-1/16*sin(b*x+a)^4)-12/b^2*a*c^2*d^2*((b*x+a)*(-1/2*cos(b*x+a)*sin(b*x+a)+1/2*b*x+1/2*a)-1/16*(b*x+a)^2+1
/16*sin(b*x+a)^2-(b*x+a)*(-1/4*(sin(b*x+a)^3+3/2*sin(b*x+a))*cos(b*x+a)+3/8*b*x+3/8*a)-1/16*sin(b*x+a)^4)+4/b*
c^3*d*((b*x+a)*(-1/2*cos(b*x+a)*sin(b*x+a)+1/2*b*x+1/2*a)-1/16*(b*x+a)^2+1/16*sin(b*x+a)^2-(b*x+a)*(-1/4*(sin(
b*x+a)^3+3/2*sin(b*x+a))*cos(b*x+a)+3/8*b*x+3/8*a)-1/16*sin(b*x+a)^4)+1/b^4*a^4*d^4*(-1/4*sin(b*x+a)*cos(b*x+a
)^3+1/8*cos(b*x+a)*sin(b*x+a)+1/8*b*x+1/8*a)-4/b^3*a^3*c*d^3*(-1/4*sin(b*x+a)*cos(b*x+a)^3+1/8*cos(b*x+a)*sin(
b*x+a)+1/8*b*x+1/8*a)+6/b^2*a^2*c^2*d^2*(-1/4*sin(b*x+a)*cos(b*x+a)^3+1/8*cos(b*x+a)*sin(b*x+a)+1/8*b*x+1/8*a)
-4/b*a*c^3*d*(-1/4*sin(b*x+a)*cos(b*x+a)^3+1/8*cos(b*x+a)*sin(b*x+a)+1/8*b*x+1/8*a)+c^4*(-1/4*sin(b*x+a)*cos(b
*x+a)^3+1/8*cos(b*x+a)*sin(b*x+a)+1/8*b*x+1/8*a))

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Maxima [B]  time = 1.37141, size = 992, normalized size = 7.57 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [B]  time = 0.531492, size = 986, normalized size = 7.53 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Sympy [A]  time = 17.6012, size = 1209, normalized size = 9.23 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)**2*cos(a + b*x)**2/(4*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/(1
6*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)**2*cos(a + b*x)**2/(32*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/(2
56*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))

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Giac [A]  time = 1.16027, size = 302, normalized size = 2.31 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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