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

Optimal. Leaf size=205 \[ -\frac{4 d^2 (c+d x)^2 \sin ^3(a+b x)}{9 b^3}-\frac{8 d^2 (c+d x)^2 \sin (a+b x)}{3 b^3}-\frac{160 d^3 (c+d x) \cos (a+b x)}{27 b^4}-\frac{8 d^3 (c+d x) \sin ^2(a+b x) \cos (a+b x)}{27 b^4}+\frac{8 d (c+d x)^3 \cos (a+b x)}{9 b^2}+\frac{4 d (c+d x)^3 \sin ^2(a+b x) \cos (a+b x)}{9 b^2}+\frac{8 d^4 \sin ^3(a+b x)}{81 b^5}+\frac{160 d^4 \sin (a+b x)}{27 b^5}+\frac{(c+d x)^4 \sin ^3(a+b x)}{3 b} \]

[Out]

(-160*d^3*(c + d*x)*Cos[a + b*x])/(27*b^4) + (8*d*(c + d*x)^3*Cos[a + b*x])/(9*b^2) + (160*d^4*Sin[a + b*x])/(
27*b^5) - (8*d^2*(c + d*x)^2*Sin[a + b*x])/(3*b^3) - (8*d^3*(c + d*x)*Cos[a + b*x]*Sin[a + b*x]^2)/(27*b^4) +
(4*d*(c + d*x)^3*Cos[a + b*x]*Sin[a + b*x]^2)/(9*b^2) + (8*d^4*Sin[a + b*x]^3)/(81*b^5) - (4*d^2*(c + d*x)^2*S
in[a + b*x]^3)/(9*b^3) + ((c + d*x)^4*Sin[a + b*x]^3)/(3*b)

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Rubi [A]  time = 0.199716, antiderivative size = 205, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {4404, 3311, 3296, 2637, 3310} \[ -\frac{4 d^2 (c+d x)^2 \sin ^3(a+b x)}{9 b^3}-\frac{8 d^2 (c+d x)^2 \sin (a+b x)}{3 b^3}-\frac{160 d^3 (c+d x) \cos (a+b x)}{27 b^4}-\frac{8 d^3 (c+d x) \sin ^2(a+b x) \cos (a+b x)}{27 b^4}+\frac{8 d (c+d x)^3 \cos (a+b x)}{9 b^2}+\frac{4 d (c+d x)^3 \sin ^2(a+b x) \cos (a+b x)}{9 b^2}+\frac{8 d^4 \sin ^3(a+b x)}{81 b^5}+\frac{160 d^4 \sin (a+b x)}{27 b^5}+\frac{(c+d x)^4 \sin ^3(a+b x)}{3 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-160*d^3*(c + d*x)*Cos[a + b*x])/(27*b^4) + (8*d*(c + d*x)^3*Cos[a + b*x])/(9*b^2) + (160*d^4*Sin[a + b*x])/(
27*b^5) - (8*d^2*(c + d*x)^2*Sin[a + b*x])/(3*b^3) - (8*d^3*(c + d*x)*Cos[a + b*x]*Sin[a + b*x]^2)/(27*b^4) +
(4*d*(c + d*x)^3*Cos[a + b*x]*Sin[a + b*x]^2)/(9*b^2) + (8*d^4*Sin[a + b*x]^3)/(81*b^5) - (4*d^2*(c + d*x)^2*S
in[a + b*x]^3)/(9*b^3) + ((c + d*x)^4*Sin[a + b*x]^3)/(3*b)

Rule 4404

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

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 (a+b x) \sin ^2(a+b x) \, dx &=\frac{(c+d x)^4 \sin ^3(a+b x)}{3 b}-\frac{(4 d) \int (c+d x)^3 \sin ^3(a+b x) \, dx}{3 b}\\ &=\frac{4 d (c+d x)^3 \cos (a+b x) \sin ^2(a+b x)}{9 b^2}-\frac{4 d^2 (c+d x)^2 \sin ^3(a+b x)}{9 b^3}+\frac{(c+d x)^4 \sin ^3(a+b x)}{3 b}-\frac{(8 d) \int (c+d x)^3 \sin (a+b x) \, dx}{9 b}+\frac{\left (8 d^3\right ) \int (c+d x) \sin ^3(a+b x) \, dx}{9 b^3}\\ &=\frac{8 d (c+d x)^3 \cos (a+b x)}{9 b^2}-\frac{8 d^3 (c+d x) \cos (a+b x) \sin ^2(a+b x)}{27 b^4}+\frac{4 d (c+d x)^3 \cos (a+b x) \sin ^2(a+b x)}{9 b^2}+\frac{8 d^4 \sin ^3(a+b x)}{81 b^5}-\frac{4 d^2 (c+d x)^2 \sin ^3(a+b x)}{9 b^3}+\frac{(c+d x)^4 \sin ^3(a+b x)}{3 b}-\frac{\left (8 d^2\right ) \int (c+d x)^2 \cos (a+b x) \, dx}{3 b^2}+\frac{\left (16 d^3\right ) \int (c+d x) \sin (a+b x) \, dx}{27 b^3}\\ &=-\frac{16 d^3 (c+d x) \cos (a+b x)}{27 b^4}+\frac{8 d (c+d x)^3 \cos (a+b x)}{9 b^2}-\frac{8 d^2 (c+d x)^2 \sin (a+b x)}{3 b^3}-\frac{8 d^3 (c+d x) \cos (a+b x) \sin ^2(a+b x)}{27 b^4}+\frac{4 d (c+d x)^3 \cos (a+b x) \sin ^2(a+b x)}{9 b^2}+\frac{8 d^4 \sin ^3(a+b x)}{81 b^5}-\frac{4 d^2 (c+d x)^2 \sin ^3(a+b x)}{9 b^3}+\frac{(c+d x)^4 \sin ^3(a+b x)}{3 b}+\frac{\left (16 d^3\right ) \int (c+d x) \sin (a+b x) \, dx}{3 b^3}+\frac{\left (16 d^4\right ) \int \cos (a+b x) \, dx}{27 b^4}\\ &=-\frac{160 d^3 (c+d x) \cos (a+b x)}{27 b^4}+\frac{8 d (c+d x)^3 \cos (a+b x)}{9 b^2}+\frac{16 d^4 \sin (a+b x)}{27 b^5}-\frac{8 d^2 (c+d x)^2 \sin (a+b x)}{3 b^3}-\frac{8 d^3 (c+d x) \cos (a+b x) \sin ^2(a+b x)}{27 b^4}+\frac{4 d (c+d x)^3 \cos (a+b x) \sin ^2(a+b x)}{9 b^2}+\frac{8 d^4 \sin ^3(a+b x)}{81 b^5}-\frac{4 d^2 (c+d x)^2 \sin ^3(a+b x)}{9 b^3}+\frac{(c+d x)^4 \sin ^3(a+b x)}{3 b}+\frac{\left (16 d^4\right ) \int \cos (a+b x) \, dx}{3 b^4}\\ &=-\frac{160 d^3 (c+d x) \cos (a+b x)}{27 b^4}+\frac{8 d (c+d x)^3 \cos (a+b x)}{9 b^2}+\frac{160 d^4 \sin (a+b x)}{27 b^5}-\frac{8 d^2 (c+d x)^2 \sin (a+b x)}{3 b^3}-\frac{8 d^3 (c+d x) \cos (a+b x) \sin ^2(a+b x)}{27 b^4}+\frac{4 d (c+d x)^3 \cos (a+b x) \sin ^2(a+b x)}{9 b^2}+\frac{8 d^4 \sin ^3(a+b x)}{81 b^5}-\frac{4 d^2 (c+d x)^2 \sin ^3(a+b x)}{9 b^3}+\frac{(c+d x)^4 \sin ^3(a+b x)}{3 b}\\ \end{align*}

Mathematica [A]  time = 1.5138, size = 385, normalized size = 1.88 \[ \frac{486 b^4 c^2 d^2 x^2 \sin (a+b x)-162 b^4 c^2 d^2 x^2 \sin (3 (a+b x))-972 b^2 c^2 d^2 \sin (a+b x)+36 b^2 c^2 d^2 \sin (3 (a+b x))+324 b^4 c^3 d x \sin (a+b x)-108 b^4 c^3 d x \sin (3 (a+b x))+81 b^4 c^4 \sin (a+b x)-27 b^4 c^4 \sin (3 (a+b x))+324 b^4 c d^3 x^3 \sin (a+b x)-108 b^4 c d^3 x^3 \sin (3 (a+b x))-1944 b^2 c d^3 x \sin (a+b x)+72 b^2 c d^3 x \sin (3 (a+b x))+324 b d (c+d x) \cos (a+b x) \left (b^2 (c+d x)^2-6 d^2\right )-12 b d (c+d x) \cos (3 (a+b x)) \left (3 b^2 (c+d x)^2-2 d^2\right )+81 b^4 d^4 x^4 \sin (a+b x)-27 b^4 d^4 x^4 \sin (3 (a+b x))-972 b^2 d^4 x^2 \sin (a+b x)+36 b^2 d^4 x^2 \sin (3 (a+b x))+1944 d^4 \sin (a+b x)-8 d^4 \sin (3 (a+b x))}{324 b^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(324*b*d*(c + d*x)*(-6*d^2 + b^2*(c + d*x)^2)*Cos[a + b*x] - 12*b*d*(c + d*x)*(-2*d^2 + 3*b^2*(c + d*x)^2)*Cos
[3*(a + b*x)] + 81*b^4*c^4*Sin[a + b*x] - 972*b^2*c^2*d^2*Sin[a + b*x] + 1944*d^4*Sin[a + b*x] + 324*b^4*c^3*d
*x*Sin[a + b*x] - 1944*b^2*c*d^3*x*Sin[a + b*x] + 486*b^4*c^2*d^2*x^2*Sin[a + b*x] - 972*b^2*d^4*x^2*Sin[a + b
*x] + 324*b^4*c*d^3*x^3*Sin[a + b*x] + 81*b^4*d^4*x^4*Sin[a + b*x] - 27*b^4*c^4*Sin[3*(a + b*x)] + 36*b^2*c^2*
d^2*Sin[3*(a + b*x)] - 8*d^4*Sin[3*(a + b*x)] - 108*b^4*c^3*d*x*Sin[3*(a + b*x)] + 72*b^2*c*d^3*x*Sin[3*(a + b
*x)] - 162*b^4*c^2*d^2*x^2*Sin[3*(a + b*x)] + 36*b^2*d^4*x^2*Sin[3*(a + b*x)] - 108*b^4*c*d^3*x^3*Sin[3*(a + b
*x)] - 27*b^4*d^4*x^4*Sin[3*(a + b*x)])/(324*b^5)

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Maple [B]  time = 0.058, size = 835, normalized size = 4.1 \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)*sin(b*x+a)^2,x)

[Out]

1/b*(1/b^4*d^4*(1/3*(b*x+a)^4*sin(b*x+a)^3+4/9*(b*x+a)^3*(2+sin(b*x+a)^2)*cos(b*x+a)-8/3*(b*x+a)^2*sin(b*x+a)+
160/27*sin(b*x+a)-16/3*(b*x+a)*cos(b*x+a)-4/9*(b*x+a)^2*sin(b*x+a)^3-8/27*(b*x+a)*(2+sin(b*x+a)^2)*cos(b*x+a)+
8/81*sin(b*x+a)^3)-4/b^4*a*d^4*(1/3*(b*x+a)^3*sin(b*x+a)^3+1/3*(b*x+a)^2*(2+sin(b*x+a)^2)*cos(b*x+a)-4/3*cos(b
*x+a)-4/3*(b*x+a)*sin(b*x+a)-2/9*(b*x+a)*sin(b*x+a)^3-2/27*(2+sin(b*x+a)^2)*cos(b*x+a))+4/b^3*c*d^3*(1/3*(b*x+
a)^3*sin(b*x+a)^3+1/3*(b*x+a)^2*(2+sin(b*x+a)^2)*cos(b*x+a)-4/3*cos(b*x+a)-4/3*(b*x+a)*sin(b*x+a)-2/9*(b*x+a)*
sin(b*x+a)^3-2/27*(2+sin(b*x+a)^2)*cos(b*x+a))+6/b^4*a^2*d^4*(1/3*(b*x+a)^2*sin(b*x+a)^3+2/9*(b*x+a)*(2+sin(b*
x+a)^2)*cos(b*x+a)-2/27*sin(b*x+a)^3-4/9*sin(b*x+a))-12/b^3*a*c*d^3*(1/3*(b*x+a)^2*sin(b*x+a)^3+2/9*(b*x+a)*(2
+sin(b*x+a)^2)*cos(b*x+a)-2/27*sin(b*x+a)^3-4/9*sin(b*x+a))+6/b^2*c^2*d^2*(1/3*(b*x+a)^2*sin(b*x+a)^3+2/9*(b*x
+a)*(2+sin(b*x+a)^2)*cos(b*x+a)-2/27*sin(b*x+a)^3-4/9*sin(b*x+a))-4/b^4*a^3*d^4*(1/3*(b*x+a)*sin(b*x+a)^3+1/9*
(2+sin(b*x+a)^2)*cos(b*x+a))+12/b^3*a^2*c*d^3*(1/3*(b*x+a)*sin(b*x+a)^3+1/9*(2+sin(b*x+a)^2)*cos(b*x+a))-12/b^
2*a*c^2*d^2*(1/3*(b*x+a)*sin(b*x+a)^3+1/9*(2+sin(b*x+a)^2)*cos(b*x+a))+4/b*c^3*d*(1/3*(b*x+a)*sin(b*x+a)^3+1/9
*(2+sin(b*x+a)^2)*cos(b*x+a))+1/3/b^4*a^4*d^4*sin(b*x+a)^3-4/3/b^3*a^3*c*d^3*sin(b*x+a)^3+2/b^2*a^2*c^2*d^2*si
n(b*x+a)^3-4/3/b*a*c^3*d*sin(b*x+a)^3+1/3*c^4*sin(b*x+a)^3)

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Maxima [B]  time = 1.35791, size = 1188, normalized size = 5.8 \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)*sin(b*x+a)^2,x, algorithm="maxima")

[Out]

1/324*(108*c^4*sin(b*x + a)^3 - 432*a*c^3*d*sin(b*x + a)^3/b + 648*a^2*c^2*d^2*sin(b*x + a)^3/b^2 - 432*a^3*c*
d^3*sin(b*x + a)^3/b^3 + 108*a^4*d^4*sin(b*x + a)^3/b^4 - 36*(3*(b*x + a)*sin(3*b*x + 3*a) - 9*(b*x + a)*sin(b
*x + a) + cos(3*b*x + 3*a) - 9*cos(b*x + a))*c^3*d/b + 108*(3*(b*x + a)*sin(3*b*x + 3*a) - 9*(b*x + a)*sin(b*x
+ a) + cos(3*b*x + 3*a) - 9*cos(b*x + a))*a*c^2*d^2/b^2 - 108*(3*(b*x + a)*sin(3*b*x + 3*a) - 9*(b*x + a)*sin
(b*x + a) + cos(3*b*x + 3*a) - 9*cos(b*x + a))*a^2*c*d^3/b^3 + 36*(3*(b*x + a)*sin(3*b*x + 3*a) - 9*(b*x + a)*
sin(b*x + a) + cos(3*b*x + 3*a) - 9*cos(b*x + a))*a^3*d^4/b^4 - 18*(6*(b*x + a)*cos(3*b*x + 3*a) - 54*(b*x + a
)*cos(b*x + a) + (9*(b*x + a)^2 - 2)*sin(3*b*x + 3*a) - 27*((b*x + a)^2 - 2)*sin(b*x + a))*c^2*d^2/b^2 + 36*(6
*(b*x + a)*cos(3*b*x + 3*a) - 54*(b*x + a)*cos(b*x + a) + (9*(b*x + a)^2 - 2)*sin(3*b*x + 3*a) - 27*((b*x + a)
^2 - 2)*sin(b*x + a))*a*c*d^3/b^3 - 18*(6*(b*x + a)*cos(3*b*x + 3*a) - 54*(b*x + a)*cos(b*x + a) + (9*(b*x + a
)^2 - 2)*sin(3*b*x + 3*a) - 27*((b*x + a)^2 - 2)*sin(b*x + a))*a^2*d^4/b^4 - 12*((9*(b*x + a)^2 - 2)*cos(3*b*x
+ 3*a) - 81*((b*x + a)^2 - 2)*cos(b*x + a) + 3*(3*(b*x + a)^3 - 2*b*x - 2*a)*sin(3*b*x + 3*a) - 27*((b*x + a)
^3 - 6*b*x - 6*a)*sin(b*x + a))*c*d^3/b^3 + 12*((9*(b*x + a)^2 - 2)*cos(3*b*x + 3*a) - 81*((b*x + a)^2 - 2)*co
s(b*x + a) + 3*(3*(b*x + a)^3 - 2*b*x - 2*a)*sin(3*b*x + 3*a) - 27*((b*x + a)^3 - 6*b*x - 6*a)*sin(b*x + a))*a
*d^4/b^4 - (12*(3*(b*x + a)^3 - 2*b*x - 2*a)*cos(3*b*x + 3*a) - 324*((b*x + a)^3 - 6*b*x - 6*a)*cos(b*x + a) +
(27*(b*x + a)^4 - 36*(b*x + a)^2 + 8)*sin(3*b*x + 3*a) - 81*((b*x + a)^4 - 12*(b*x + a)^2 + 24)*sin(b*x + a))
*d^4/b^4)/b

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Fricas [A]  time = 0.520872, size = 759, normalized size = 3.7 \begin{align*} -\frac{12 \,{\left (3 \, b^{3} d^{4} x^{3} + 9 \, b^{3} c d^{3} x^{2} + 3 \, b^{3} c^{3} d - 2 \, b c d^{3} +{\left (9 \, b^{3} c^{2} d^{2} - 2 \, b d^{4}\right )} x\right )} \cos \left (b x + a\right )^{3} - 36 \,{\left (3 \, b^{3} d^{4} x^{3} + 9 \, b^{3} c d^{3} x^{2} + 3 \, b^{3} c^{3} d - 14 \, b c d^{3} +{\left (9 \, b^{3} c^{2} d^{2} - 14 \, b d^{4}\right )} x\right )} \cos \left (b x + a\right ) -{\left (27 \, b^{4} d^{4} x^{4} + 108 \, b^{4} c d^{3} x^{3} + 27 \, b^{4} c^{4} - 252 \, b^{2} c^{2} d^{2} + 488 \, d^{4} + 18 \,{\left (9 \, b^{4} c^{2} d^{2} - 14 \, b^{2} d^{4}\right )} x^{2} -{\left (27 \, b^{4} d^{4} x^{4} + 108 \, b^{4} c d^{3} x^{3} + 27 \, b^{4} c^{4} - 36 \, b^{2} c^{2} d^{2} + 8 \, d^{4} + 18 \,{\left (9 \, b^{4} c^{2} d^{2} - 2 \, b^{2} d^{4}\right )} x^{2} + 36 \,{\left (3 \, b^{4} c^{3} d - 2 \, b^{2} c d^{3}\right )} x\right )} \cos \left (b x + a\right )^{2} + 36 \,{\left (3 \, b^{4} c^{3} d - 14 \, b^{2} c d^{3}\right )} x\right )} \sin \left (b x + a\right )}{81 \, b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/81*(12*(3*b^3*d^4*x^3 + 9*b^3*c*d^3*x^2 + 3*b^3*c^3*d - 2*b*c*d^3 + (9*b^3*c^2*d^2 - 2*b*d^4)*x)*cos(b*x +
a)^3 - 36*(3*b^3*d^4*x^3 + 9*b^3*c*d^3*x^2 + 3*b^3*c^3*d - 14*b*c*d^3 + (9*b^3*c^2*d^2 - 14*b*d^4)*x)*cos(b*x
+ a) - (27*b^4*d^4*x^4 + 108*b^4*c*d^3*x^3 + 27*b^4*c^4 - 252*b^2*c^2*d^2 + 488*d^4 + 18*(9*b^4*c^2*d^2 - 14*b
^2*d^4)*x^2 - (27*b^4*d^4*x^4 + 108*b^4*c*d^3*x^3 + 27*b^4*c^4 - 36*b^2*c^2*d^2 + 8*d^4 + 18*(9*b^4*c^2*d^2 -
2*b^2*d^4)*x^2 + 36*(3*b^4*c^3*d - 2*b^2*c*d^3)*x)*cos(b*x + a)^2 + 36*(3*b^4*c^3*d - 14*b^2*c*d^3)*x)*sin(b*x
+ a))/b^5

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Sympy [A]  time = 11.3384, size = 646, normalized size = 3.15 \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)*sin(b*x+a)**2,x)

[Out]

Piecewise((c**4*sin(a + b*x)**3/(3*b) + 4*c**3*d*x*sin(a + b*x)**3/(3*b) + 2*c**2*d**2*x**2*sin(a + b*x)**3/b
+ 4*c*d**3*x**3*sin(a + b*x)**3/(3*b) + d**4*x**4*sin(a + b*x)**3/(3*b) + 4*c**3*d*sin(a + b*x)**2*cos(a + b*x
)/(3*b**2) + 8*c**3*d*cos(a + b*x)**3/(9*b**2) + 4*c**2*d**2*x*sin(a + b*x)**2*cos(a + b*x)/b**2 + 8*c**2*d**2
*x*cos(a + b*x)**3/(3*b**2) + 4*c*d**3*x**2*sin(a + b*x)**2*cos(a + b*x)/b**2 + 8*c*d**3*x**2*cos(a + b*x)**3/
(3*b**2) + 4*d**4*x**3*sin(a + b*x)**2*cos(a + b*x)/(3*b**2) + 8*d**4*x**3*cos(a + b*x)**3/(9*b**2) - 28*c**2*
d**2*sin(a + b*x)**3/(9*b**3) - 8*c**2*d**2*sin(a + b*x)*cos(a + b*x)**2/(3*b**3) - 56*c*d**3*x*sin(a + b*x)**
3/(9*b**3) - 16*c*d**3*x*sin(a + b*x)*cos(a + b*x)**2/(3*b**3) - 28*d**4*x**2*sin(a + b*x)**3/(9*b**3) - 8*d**
4*x**2*sin(a + b*x)*cos(a + b*x)**2/(3*b**3) - 56*c*d**3*sin(a + b*x)**2*cos(a + b*x)/(9*b**4) - 160*c*d**3*co
s(a + b*x)**3/(27*b**4) - 56*d**4*x*sin(a + b*x)**2*cos(a + b*x)/(9*b**4) - 160*d**4*x*cos(a + b*x)**3/(27*b**
4) + 488*d**4*sin(a + b*x)**3/(81*b**5) + 160*d**4*sin(a + b*x)*cos(a + b*x)**2/(27*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), True))

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Giac [A]  time = 1.12878, size = 473, normalized size = 2.31 \begin{align*} -\frac{{\left (3 \, b^{3} d^{4} x^{3} + 9 \, b^{3} c d^{3} x^{2} + 9 \, b^{3} c^{2} d^{2} x + 3 \, b^{3} c^{3} d - 2 \, b d^{4} x - 2 \, b c d^{3}\right )} \cos \left (3 \, b x + 3 \, a\right )}{27 \, b^{5}} + \frac{{\left (b^{3} d^{4} x^{3} + 3 \, b^{3} c d^{3} x^{2} + 3 \, b^{3} c^{2} d^{2} x + b^{3} c^{3} d - 6 \, b d^{4} x - 6 \, b c d^{3}\right )} \cos \left (b x + a\right )}{b^{5}} - \frac{{\left (27 \, b^{4} d^{4} x^{4} + 108 \, b^{4} c d^{3} x^{3} + 162 \, b^{4} c^{2} d^{2} x^{2} + 108 \, b^{4} c^{3} d x + 27 \, b^{4} c^{4} - 36 \, b^{2} d^{4} x^{2} - 72 \, b^{2} c d^{3} x - 36 \, b^{2} c^{2} d^{2} + 8 \, d^{4}\right )} \sin \left (3 \, b x + 3 \, a\right )}{324 \, b^{5}} + \frac{{\left (b^{4} d^{4} x^{4} + 4 \, b^{4} c d^{3} x^{3} + 6 \, b^{4} c^{2} d^{2} x^{2} + 4 \, b^{4} c^{3} d x + b^{4} c^{4} - 12 \, b^{2} d^{4} x^{2} - 24 \, b^{2} c d^{3} x - 12 \, b^{2} c^{2} d^{2} + 24 \, d^{4}\right )} \sin \left (b x + a\right )}{4 \, b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/27*(3*b^3*d^4*x^3 + 9*b^3*c*d^3*x^2 + 9*b^3*c^2*d^2*x + 3*b^3*c^3*d - 2*b*d^4*x - 2*b*c*d^3)*cos(3*b*x + 3*
a)/b^5 + (b^3*d^4*x^3 + 3*b^3*c*d^3*x^2 + 3*b^3*c^2*d^2*x + b^3*c^3*d - 6*b*d^4*x - 6*b*c*d^3)*cos(b*x + a)/b^
5 - 1/324*(27*b^4*d^4*x^4 + 108*b^4*c*d^3*x^3 + 162*b^4*c^2*d^2*x^2 + 108*b^4*c^3*d*x + 27*b^4*c^4 - 36*b^2*d^
4*x^2 - 72*b^2*c*d^3*x - 36*b^2*c^2*d^2 + 8*d^4)*sin(3*b*x + 3*a)/b^5 + 1/4*(b^4*d^4*x^4 + 4*b^4*c*d^3*x^3 + 6
*b^4*c^2*d^2*x^2 + 4*b^4*c^3*d*x + b^4*c^4 - 12*b^2*d^4*x^2 - 24*b^2*c*d^3*x - 12*b^2*c^2*d^2 + 24*d^4)*sin(b*
x + a)/b^5