3.137 \(\int (c+d x)^4 \cos ^3(a+b x) \sin (a+b x) \, dx\)

Optimal. Leaf size=260 \[ \frac{3 d^2 (c+d x)^2 \cos ^4(a+b x)}{16 b^3}+\frac{9 d^2 (c+d x)^2 \cos ^2(a+b x)}{16 b^3}-\frac{3 d^3 (c+d x) \sin (a+b x) \cos ^3(a+b x)}{32 b^4}-\frac{45 d^3 (c+d x) \sin (a+b x) \cos (a+b x)}{64 b^4}+\frac{d (c+d x)^3 \sin (a+b x) \cos ^3(a+b x)}{4 b^2}+\frac{3 d (c+d x)^3 \sin (a+b x) \cos (a+b x)}{8 b^2}-\frac{3 d^4 \cos ^4(a+b x)}{128 b^5}-\frac{45 d^4 \cos ^2(a+b x)}{128 b^5}-\frac{(c+d x)^4 \cos ^4(a+b x)}{4 b}-\frac{45 c d^3 x}{64 b^3}-\frac{45 d^4 x^2}{128 b^3}+\frac{3 (c+d x)^4}{32 b} \]

[Out]

(-45*c*d^3*x)/(64*b^3) - (45*d^4*x^2)/(128*b^3) + (3*(c + d*x)^4)/(32*b) - (45*d^4*Cos[a + b*x]^2)/(128*b^5) +
(9*d^2*(c + d*x)^2*Cos[a + b*x]^2)/(16*b^3) - (3*d^4*Cos[a + b*x]^4)/(128*b^5) + (3*d^2*(c + d*x)^2*Cos[a + b
*x]^4)/(16*b^3) - ((c + d*x)^4*Cos[a + b*x]^4)/(4*b) - (45*d^3*(c + d*x)*Cos[a + b*x]*Sin[a + b*x])/(64*b^4) +
(3*d*(c + d*x)^3*Cos[a + b*x]*Sin[a + b*x])/(8*b^2) - (3*d^3*(c + d*x)*Cos[a + b*x]^3*Sin[a + b*x])/(32*b^4)
+ (d*(c + d*x)^3*Cos[a + b*x]^3*Sin[a + b*x])/(4*b^2)

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Rubi [A]  time = 0.234001, antiderivative size = 260, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {4405, 3311, 32, 3310} \[ \frac{3 d^2 (c+d x)^2 \cos ^4(a+b x)}{16 b^3}+\frac{9 d^2 (c+d x)^2 \cos ^2(a+b x)}{16 b^3}-\frac{3 d^3 (c+d x) \sin (a+b x) \cos ^3(a+b x)}{32 b^4}-\frac{45 d^3 (c+d x) \sin (a+b x) \cos (a+b x)}{64 b^4}+\frac{d (c+d x)^3 \sin (a+b x) \cos ^3(a+b x)}{4 b^2}+\frac{3 d (c+d x)^3 \sin (a+b x) \cos (a+b x)}{8 b^2}-\frac{3 d^4 \cos ^4(a+b x)}{128 b^5}-\frac{45 d^4 \cos ^2(a+b x)}{128 b^5}-\frac{(c+d x)^4 \cos ^4(a+b x)}{4 b}-\frac{45 c d^3 x}{64 b^3}-\frac{45 d^4 x^2}{128 b^3}+\frac{3 (c+d x)^4}{32 b} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^4*Cos[a + b*x]^3*Sin[a + b*x],x]

[Out]

(-45*c*d^3*x)/(64*b^3) - (45*d^4*x^2)/(128*b^3) + (3*(c + d*x)^4)/(32*b) - (45*d^4*Cos[a + b*x]^2)/(128*b^5) +
(9*d^2*(c + d*x)^2*Cos[a + b*x]^2)/(16*b^3) - (3*d^4*Cos[a + b*x]^4)/(128*b^5) + (3*d^2*(c + d*x)^2*Cos[a + b
*x]^4)/(16*b^3) - ((c + d*x)^4*Cos[a + b*x]^4)/(4*b) - (45*d^3*(c + d*x)*Cos[a + b*x]*Sin[a + b*x])/(64*b^4) +
(3*d*(c + d*x)^3*Cos[a + b*x]*Sin[a + b*x])/(8*b^2) - (3*d^3*(c + d*x)*Cos[a + b*x]^3*Sin[a + b*x])/(32*b^4)
+ (d*(c + d*x)^3*Cos[a + b*x]^3*Sin[a + b*x])/(4*b^2)

Rule 4405

Int[Cos[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)], x_Symbol] :> -Simp[((c +
d*x)^m*Cos[a + b*x]^(n + 1))/(b*(n + 1)), x] + Dist[(d*m)/(b*(n + 1)), Int[(c + d*x)^(m - 1)*Cos[a + b*x]^(n
+ 1), x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && NeQ[n, -1]

Rule 3311

Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(d*m*(c + d*x)^(m - 1)*(
b*Sin[e + f*x])^n)/(f^2*n^2), x] + (Dist[(b^2*(n - 1))/n, Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - D
ist[(d^2*m*(m - 1))/(f^2*n^2), Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x] - Simp[(b*(c + d*x)^m*Cos[e +
f*x]*(b*Sin[e + f*x])^(n - 1))/(f*n), x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 1]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 3310

Int[((c_.) + (d_.)*(x_))*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(d*(b*Sin[e + f*x])^n)/(f^2*n
^2), x] + (Dist[(b^2*(n - 1))/n, Int[(c + d*x)*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[(b*(c + d*x)*Cos[e + f*
x]*(b*Sin[e + f*x])^(n - 1))/(f*n), x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1]

Rubi steps

\begin{align*} \int (c+d x)^4 \cos ^3(a+b x) \sin (a+b x) \, dx &=-\frac{(c+d x)^4 \cos ^4(a+b x)}{4 b}+\frac{d \int (c+d x)^3 \cos ^4(a+b x) \, dx}{b}\\ &=\frac{3 d^2 (c+d x)^2 \cos ^4(a+b x)}{16 b^3}-\frac{(c+d x)^4 \cos ^4(a+b x)}{4 b}+\frac{d (c+d x)^3 \cos ^3(a+b x) \sin (a+b x)}{4 b^2}+\frac{(3 d) \int (c+d x)^3 \cos ^2(a+b x) \, dx}{4 b}-\frac{\left (3 d^3\right ) \int (c+d x) \cos ^4(a+b x) \, dx}{8 b^3}\\ &=\frac{9 d^2 (c+d x)^2 \cos ^2(a+b x)}{16 b^3}-\frac{3 d^4 \cos ^4(a+b x)}{128 b^5}+\frac{3 d^2 (c+d x)^2 \cos ^4(a+b x)}{16 b^3}-\frac{(c+d x)^4 \cos ^4(a+b x)}{4 b}+\frac{3 d (c+d x)^3 \cos (a+b x) \sin (a+b x)}{8 b^2}-\frac{3 d^3 (c+d x) \cos ^3(a+b x) \sin (a+b x)}{32 b^4}+\frac{d (c+d x)^3 \cos ^3(a+b x) \sin (a+b x)}{4 b^2}+\frac{(3 d) \int (c+d x)^3 \, dx}{8 b}-\frac{\left (9 d^3\right ) \int (c+d x) \cos ^2(a+b x) \, dx}{32 b^3}-\frac{\left (9 d^3\right ) \int (c+d x) \cos ^2(a+b x) \, dx}{8 b^3}\\ &=\frac{3 (c+d x)^4}{32 b}-\frac{45 d^4 \cos ^2(a+b x)}{128 b^5}+\frac{9 d^2 (c+d x)^2 \cos ^2(a+b x)}{16 b^3}-\frac{3 d^4 \cos ^4(a+b x)}{128 b^5}+\frac{3 d^2 (c+d x)^2 \cos ^4(a+b x)}{16 b^3}-\frac{(c+d x)^4 \cos ^4(a+b x)}{4 b}-\frac{45 d^3 (c+d x) \cos (a+b x) \sin (a+b x)}{64 b^4}+\frac{3 d (c+d x)^3 \cos (a+b x) \sin (a+b x)}{8 b^2}-\frac{3 d^3 (c+d x) \cos ^3(a+b x) \sin (a+b x)}{32 b^4}+\frac{d (c+d x)^3 \cos ^3(a+b x) \sin (a+b x)}{4 b^2}-\frac{\left (9 d^3\right ) \int (c+d x) \, dx}{64 b^3}-\frac{\left (9 d^3\right ) \int (c+d x) \, dx}{16 b^3}\\ &=-\frac{45 c d^3 x}{64 b^3}-\frac{45 d^4 x^2}{128 b^3}+\frac{3 (c+d x)^4}{32 b}-\frac{45 d^4 \cos ^2(a+b x)}{128 b^5}+\frac{9 d^2 (c+d x)^2 \cos ^2(a+b x)}{16 b^3}-\frac{3 d^4 \cos ^4(a+b x)}{128 b^5}+\frac{3 d^2 (c+d x)^2 \cos ^4(a+b x)}{16 b^3}-\frac{(c+d x)^4 \cos ^4(a+b x)}{4 b}-\frac{45 d^3 (c+d x) \cos (a+b x) \sin (a+b x)}{64 b^4}+\frac{3 d (c+d x)^3 \cos (a+b x) \sin (a+b x)}{8 b^2}-\frac{3 d^3 (c+d x) \cos ^3(a+b x) \sin (a+b x)}{32 b^4}+\frac{d (c+d x)^3 \cos ^3(a+b x) \sin (a+b x)}{4 b^2}\\ \end{align*}

Mathematica [A]  time = 1.8831, size = 158, normalized size = 0.61 \[ -\frac{64 \cos (2 (a+b x)) \left (-6 b^2 d^2 (c+d x)^2+2 b^4 (c+d x)^4+3 d^4\right )+\cos (4 (a+b x)) \left (-24 b^2 d^2 (c+d x)^2+32 b^4 (c+d x)^4+3 d^4\right )-8 b d (c+d x) \sin (2 (a+b x)) \left (\cos (2 (a+b x)) \left (8 b^2 (c+d x)^2-3 d^2\right )+16 \left (2 b^2 (c+d x)^2-3 d^2\right )\right )}{1024 b^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^4*Cos[a + b*x]^3*Sin[a + b*x],x]

[Out]

-(64*(3*d^4 - 6*b^2*d^2*(c + d*x)^2 + 2*b^4*(c + d*x)^4)*Cos[2*(a + b*x)] + (3*d^4 - 24*b^2*d^2*(c + d*x)^2 +
32*b^4*(c + d*x)^4)*Cos[4*(a + b*x)] - 8*b*d*(c + d*x)*(16*(-3*d^2 + 2*b^2*(c + d*x)^2) + (-3*d^2 + 8*b^2*(c +
d*x)^2)*Cos[2*(a + b*x)])*Sin[2*(a + b*x)])/(1024*b^5)

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Maple [B]  time = 0.049, size = 1150, normalized size = 4.4 \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)^3*sin(b*x+a),x)

[Out]

1/b*(1/b^4*d^4*(-1/4*(b*x+a)^4*cos(b*x+a)^4+(b*x+a)^3*(1/4*(cos(b*x+a)^3+3/2*cos(b*x+a))*sin(b*x+a)+3/8*b*x+3/
8*a)+3/16*(b*x+a)^2*cos(b*x+a)^4-3/8*(b*x+a)*(1/4*(cos(b*x+a)^3+3/2*cos(b*x+a))*sin(b*x+a)+3/8*b*x+3/8*a)+45/1
28*(b*x+a)^2-3/128*cos(b*x+a)^4-9/128*cos(b*x+a)^2+9/16*(b*x+a)^2*cos(b*x+a)^2-9/8*(b*x+a)*(1/2*cos(b*x+a)*sin
(b*x+a)+1/2*b*x+1/2*a)+9/32*sin(b*x+a)^2-9/32*(b*x+a)^4)-4/b^4*a*d^4*(-1/4*(b*x+a)^3*cos(b*x+a)^4+3/4*(b*x+a)^
2*(1/4*(cos(b*x+a)^3+3/2*cos(b*x+a))*sin(b*x+a)+3/8*b*x+3/8*a)+3/32*(b*x+a)*cos(b*x+a)^4-3/128*(cos(b*x+a)^3+3
/2*cos(b*x+a))*sin(b*x+a)-45/256*b*x-45/256*a+9/32*(b*x+a)*cos(b*x+a)^2-9/64*cos(b*x+a)*sin(b*x+a)-3/16*(b*x+a
)^3)+4/b^3*c*d^3*(-1/4*(b*x+a)^3*cos(b*x+a)^4+3/4*(b*x+a)^2*(1/4*(cos(b*x+a)^3+3/2*cos(b*x+a))*sin(b*x+a)+3/8*
b*x+3/8*a)+3/32*(b*x+a)*cos(b*x+a)^4-3/128*(cos(b*x+a)^3+3/2*cos(b*x+a))*sin(b*x+a)-45/256*b*x-45/256*a+9/32*(
b*x+a)*cos(b*x+a)^2-9/64*cos(b*x+a)*sin(b*x+a)-3/16*(b*x+a)^3)+6/b^4*a^2*d^4*(-1/4*(b*x+a)^2*cos(b*x+a)^4+1/2*
(b*x+a)*(1/4*(cos(b*x+a)^3+3/2*cos(b*x+a))*sin(b*x+a)+3/8*b*x+3/8*a)-3/32*(b*x+a)^2+1/32*cos(b*x+a)^4+3/32*cos
(b*x+a)^2)-12/b^3*a*c*d^3*(-1/4*(b*x+a)^2*cos(b*x+a)^4+1/2*(b*x+a)*(1/4*(cos(b*x+a)^3+3/2*cos(b*x+a))*sin(b*x+
a)+3/8*b*x+3/8*a)-3/32*(b*x+a)^2+1/32*cos(b*x+a)^4+3/32*cos(b*x+a)^2)+6/b^2*c^2*d^2*(-1/4*(b*x+a)^2*cos(b*x+a)
^4+1/2*(b*x+a)*(1/4*(cos(b*x+a)^3+3/2*cos(b*x+a))*sin(b*x+a)+3/8*b*x+3/8*a)-3/32*(b*x+a)^2+1/32*cos(b*x+a)^4+3
/32*cos(b*x+a)^2)-4/b^4*a^3*d^4*(-1/4*(b*x+a)*cos(b*x+a)^4+1/16*(cos(b*x+a)^3+3/2*cos(b*x+a))*sin(b*x+a)+3/32*
b*x+3/32*a)+12/b^3*a^2*c*d^3*(-1/4*(b*x+a)*cos(b*x+a)^4+1/16*(cos(b*x+a)^3+3/2*cos(b*x+a))*sin(b*x+a)+3/32*b*x
+3/32*a)-12/b^2*a*c^2*d^2*(-1/4*(b*x+a)*cos(b*x+a)^4+1/16*(cos(b*x+a)^3+3/2*cos(b*x+a))*sin(b*x+a)+3/32*b*x+3/
32*a)+4/b*c^3*d*(-1/4*(b*x+a)*cos(b*x+a)^4+1/16*(cos(b*x+a)^3+3/2*cos(b*x+a))*sin(b*x+a)+3/32*b*x+3/32*a)-1/4/
b^4*a^4*d^4*cos(b*x+a)^4+1/b^3*a^3*c*d^3*cos(b*x+a)^4-3/2/b^2*a^2*c^2*d^2*cos(b*x+a)^4+1/b*a*c^3*d*cos(b*x+a)^
4-1/4*c^4*cos(b*x+a)^4)

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Maxima [B]  time = 1.394, size = 1305, normalized size = 5.02 \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)^3*sin(b*x+a),x, algorithm="maxima")

[Out]

-1/1024*(256*c^4*cos(b*x + a)^4 - 1024*a*c^3*d*cos(b*x + a)^4/b + 1536*a^2*c^2*d^2*cos(b*x + a)^4/b^2 - 1024*a
^3*c*d^3*cos(b*x + a)^4/b^3 + 256*a^4*d^4*cos(b*x + a)^4/b^4 + 32*(4*(b*x + a)*cos(4*b*x + 4*a) + 16*(b*x + a)
*cos(2*b*x + 2*a) - sin(4*b*x + 4*a) - 8*sin(2*b*x + 2*a))*c^3*d/b - 96*(4*(b*x + a)*cos(4*b*x + 4*a) + 16*(b*
x + a)*cos(2*b*x + 2*a) - sin(4*b*x + 4*a) - 8*sin(2*b*x + 2*a))*a*c^2*d^2/b^2 + 96*(4*(b*x + a)*cos(4*b*x + 4
*a) + 16*(b*x + a)*cos(2*b*x + 2*a) - sin(4*b*x + 4*a) - 8*sin(2*b*x + 2*a))*a^2*c*d^3/b^3 - 32*(4*(b*x + a)*c
os(4*b*x + 4*a) + 16*(b*x + a)*cos(2*b*x + 2*a) - sin(4*b*x + 4*a) - 8*sin(2*b*x + 2*a))*a^3*d^4/b^4 + 24*((8*
(b*x + a)^2 - 1)*cos(4*b*x + 4*a) + 16*(2*(b*x + a)^2 - 1)*cos(2*b*x + 2*a) - 4*(b*x + a)*sin(4*b*x + 4*a) - 3
2*(b*x + a)*sin(2*b*x + 2*a))*c^2*d^2/b^2 - 48*((8*(b*x + a)^2 - 1)*cos(4*b*x + 4*a) + 16*(2*(b*x + a)^2 - 1)*
cos(2*b*x + 2*a) - 4*(b*x + a)*sin(4*b*x + 4*a) - 32*(b*x + a)*sin(2*b*x + 2*a))*a*c*d^3/b^3 + 24*((8*(b*x + a
)^2 - 1)*cos(4*b*x + 4*a) + 16*(2*(b*x + a)^2 - 1)*cos(2*b*x + 2*a) - 4*(b*x + a)*sin(4*b*x + 4*a) - 32*(b*x +
a)*sin(2*b*x + 2*a))*a^2*d^4/b^4 + 4*(4*(8*(b*x + a)^3 - 3*b*x - 3*a)*cos(4*b*x + 4*a) + 64*(2*(b*x + a)^3 -
3*b*x - 3*a)*cos(2*b*x + 2*a) - 3*(8*(b*x + a)^2 - 1)*sin(4*b*x + 4*a) - 96*(2*(b*x + a)^2 - 1)*sin(2*b*x + 2*
a))*c*d^3/b^3 - 4*(4*(8*(b*x + a)^3 - 3*b*x - 3*a)*cos(4*b*x + 4*a) + 64*(2*(b*x + a)^3 - 3*b*x - 3*a)*cos(2*b
*x + 2*a) - 3*(8*(b*x + a)^2 - 1)*sin(4*b*x + 4*a) - 96*(2*(b*x + a)^2 - 1)*sin(2*b*x + 2*a))*a*d^4/b^4 + ((32
*(b*x + a)^4 - 24*(b*x + a)^2 + 3)*cos(4*b*x + 4*a) + 64*(2*(b*x + a)^4 - 6*(b*x + a)^2 + 3)*cos(2*b*x + 2*a)
- 4*(8*(b*x + a)^3 - 3*b*x - 3*a)*sin(4*b*x + 4*a) - 128*(2*(b*x + a)^3 - 3*b*x - 3*a)*sin(2*b*x + 2*a))*d^4/b
^4)/b

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Fricas [A]  time = 0.536091, size = 802, normalized size = 3.08 \begin{align*} \frac{12 \, b^{4} d^{4} x^{4} + 48 \, b^{4} c d^{3} x^{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 )^{4} + 9 \,{\left (8 \, b^{4} c^{2} d^{2} - 5 \, b^{2} d^{4}\right )} x^{2} + 9 \,{\left (8 \, b^{2} d^{4} x^{2} + 16 \, b^{2} c d^{3} x + 8 \, b^{2} c^{2} d^{2} - 5 \, d^{4}\right )} \cos \left (b x + a\right )^{2} + 6 \,{\left (8 \, b^{4} c^{3} d - 15 \, b^{2} c d^{3}\right )} x + 2 \,{\left (2 \,{\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 )^{3} + 3 \,{\left (8 \, b^{3} d^{4} x^{3} + 24 \, b^{3} c d^{3} x^{2} + 8 \, b^{3} c^{3} d - 15 \, b c d^{3} + 3 \,{\left (8 \, b^{3} c^{2} d^{2} - 5 \, b d^{4}\right )} x\right )} \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )}{128 \, b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^4*cos(b*x+a)^3*sin(b*x+a),x, algorithm="fricas")

[Out]

1/128*(12*b^4*d^4*x^4 + 48*b^4*c*d^3*x^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)^4 + 9*(8*b^4*c^2*d^
2 - 5*b^2*d^4)*x^2 + 9*(8*b^2*d^4*x^2 + 16*b^2*c*d^3*x + 8*b^2*c^2*d^2 - 5*d^4)*cos(b*x + a)^2 + 6*(8*b^4*c^3*
d - 15*b^2*c*d^3)*x + 2*(2*(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)^3 + 3*(8*b^3*d^4*x^3 + 24*b^3*c*d^3*x^2 + 8*b^3*c^3*d - 15*b*c*d^3 + 3*(8*b^3*c^2*d^2 - 5
*b*d^4)*x)*cos(b*x + a))*sin(b*x + a))/b^5

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Sympy [A]  time = 21.9182, size = 976, normalized size = 3.75 \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)**3*sin(b*x+a),x)

[Out]

Piecewise((c**4*sin(a + b*x)**4/(4*b) + c**4*sin(a + b*x)**2*cos(a + b*x)**2/(2*b) + 3*c**3*d*x*sin(a + b*x)**
4/(8*b) + 3*c**3*d*x*sin(a + b*x)**2*cos(a + b*x)**2/(4*b) - 5*c**3*d*x*cos(a + b*x)**4/(8*b) + 9*c**2*d**2*x*
*2*sin(a + b*x)**4/(16*b) + 9*c**2*d**2*x**2*sin(a + b*x)**2*cos(a + b*x)**2/(8*b) - 15*c**2*d**2*x**2*cos(a +
b*x)**4/(16*b) + 3*c*d**3*x**3*sin(a + b*x)**4/(8*b) + 3*c*d**3*x**3*sin(a + b*x)**2*cos(a + b*x)**2/(4*b) -
5*c*d**3*x**3*cos(a + b*x)**4/(8*b) + 3*d**4*x**4*sin(a + b*x)**4/(32*b) + 3*d**4*x**4*sin(a + b*x)**2*cos(a +
b*x)**2/(16*b) - 5*d**4*x**4*cos(a + b*x)**4/(32*b) + 3*c**3*d*sin(a + b*x)**3*cos(a + b*x)/(8*b**2) + 5*c**3
*d*sin(a + b*x)*cos(a + b*x)**3/(8*b**2) + 9*c**2*d**2*x*sin(a + b*x)**3*cos(a + b*x)/(8*b**2) + 15*c**2*d**2*
x*sin(a + b*x)*cos(a + b*x)**3/(8*b**2) + 9*c*d**3*x**2*sin(a + b*x)**3*cos(a + b*x)/(8*b**2) + 15*c*d**3*x**2
*sin(a + b*x)*cos(a + b*x)**3/(8*b**2) + 3*d**4*x**3*sin(a + b*x)**3*cos(a + b*x)/(8*b**2) + 5*d**4*x**3*sin(a
+ b*x)*cos(a + b*x)**3/(8*b**2) - 3*c**2*d**2*sin(a + b*x)**4/(4*b**3) - 15*c**2*d**2*sin(a + b*x)**2*cos(a +
b*x)**2/(16*b**3) - 45*c*d**3*x*sin(a + b*x)**4/(64*b**3) - 9*c*d**3*x*sin(a + b*x)**2*cos(a + b*x)**2/(32*b*
*3) + 51*c*d**3*x*cos(a + b*x)**4/(64*b**3) - 45*d**4*x**2*sin(a + b*x)**4/(128*b**3) - 9*d**4*x**2*sin(a + b*
x)**2*cos(a + b*x)**2/(64*b**3) + 51*d**4*x**2*cos(a + b*x)**4/(128*b**3) - 45*c*d**3*sin(a + b*x)**3*cos(a +
b*x)/(64*b**4) - 51*c*d**3*sin(a + b*x)*cos(a + b*x)**3/(64*b**4) - 45*d**4*x*sin(a + b*x)**3*cos(a + b*x)/(64
*b**4) - 51*d**4*x*sin(a + b*x)*cos(a + b*x)**3/(64*b**4) + 3*d**4*sin(a + b*x)**4/(8*b**5) + 51*d**4*sin(a +
b*x)**2*cos(a + b*x)**2/(128*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)*cos(a)**3, True))

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Giac [A]  time = 1.10955, size = 487, normalized size = 1.87 \begin{align*} -\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 )} \cos \left (4 \, b x + 4 \, a\right )}{1024 \, b^{5}} - \frac{{\left (2 \, b^{4} d^{4} x^{4} + 8 \, b^{4} c d^{3} x^{3} + 12 \, b^{4} c^{2} d^{2} x^{2} + 8 \, b^{4} c^{3} d x + 2 \, b^{4} c^{4} - 6 \, b^{2} d^{4} x^{2} - 12 \, b^{2} c d^{3} x - 6 \, b^{2} c^{2} d^{2} + 3 \, d^{4}\right )} \cos \left (2 \, b x + 2 \, a\right )}{16 \, b^{5}} + \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 )} \sin \left (4 \, b x + 4 \, a\right )}{256 \, b^{5}} + \frac{{\left (2 \, b^{3} d^{4} x^{3} + 6 \, b^{3} c d^{3} x^{2} + 6 \, b^{3} c^{2} d^{2} x + 2 \, b^{3} c^{3} d - 3 \, b d^{4} x - 3 \, b c d^{3}\right )} \sin \left (2 \, b x + 2 \, a\right )}{8 \, b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^4*cos(b*x+a)^3*sin(b*x+a),x, algorithm="giac")

[Out]

-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)*cos(4*b*x + 4*a)/b^5 - 1/16*(2*b^4*d^4*x^4 + 8*b^4*c*d^3*x^3 +
12*b^4*c^2*d^2*x^2 + 8*b^4*c^3*d*x + 2*b^4*c^4 - 6*b^2*d^4*x^2 - 12*b^2*c*d^3*x - 6*b^2*c^2*d^2 + 3*d^4)*cos(2
*b*x + 2*a)/b^5 + 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)*sin(4*b*x + 4*a)/b^5 + 1/8*(2*b^3*d^4*x^3 + 6*b^3*c*d^3*x^2 + 6*b^3*c^2*d^2*x + 2*b^3*c^3*d - 3*b*d^4*x
- 3*b*c*d^3)*sin(2*b*x + 2*a)/b^5