3.89 \(\int (c+d x)^4 \cos ^2(a+b x) \sin ^3(a+b x) \, dx\)

Optimal. Leaf size=330 \[ -\frac{3 d^3 (c+d x) \sin (a+b x)}{b^4}-\frac{d^3 (c+d x) \sin (3 a+3 b x)}{54 b^4}+\frac{3 d^3 (c+d x) \sin (5 a+5 b x)}{1250 b^4}+\frac{3 d^2 (c+d x)^2 \cos (a+b x)}{2 b^3}+\frac{d^2 (c+d x)^2 \cos (3 a+3 b x)}{36 b^3}-\frac{3 d^2 (c+d x)^2 \cos (5 a+5 b x)}{500 b^3}+\frac{d (c+d x)^3 \sin (a+b x)}{2 b^2}+\frac{d (c+d x)^3 \sin (3 a+3 b x)}{36 b^2}-\frac{d (c+d x)^3 \sin (5 a+5 b x)}{100 b^2}-\frac{3 d^4 \cos (a+b x)}{b^5}-\frac{d^4 \cos (3 a+3 b x)}{162 b^5}+\frac{3 d^4 \cos (5 a+5 b x)}{6250 b^5}-\frac{(c+d x)^4 \cos (a+b x)}{8 b}-\frac{(c+d x)^4 \cos (3 a+3 b x)}{48 b}+\frac{(c+d x)^4 \cos (5 a+5 b x)}{80 b} \]

[Out]

(-3*d^4*Cos[a + b*x])/b^5 + (3*d^2*(c + d*x)^2*Cos[a + b*x])/(2*b^3) - ((c + d*x)^4*Cos[a + b*x])/(8*b) - (d^4
*Cos[3*a + 3*b*x])/(162*b^5) + (d^2*(c + d*x)^2*Cos[3*a + 3*b*x])/(36*b^3) - ((c + d*x)^4*Cos[3*a + 3*b*x])/(4
8*b) + (3*d^4*Cos[5*a + 5*b*x])/(6250*b^5) - (3*d^2*(c + d*x)^2*Cos[5*a + 5*b*x])/(500*b^3) + ((c + d*x)^4*Cos
[5*a + 5*b*x])/(80*b) - (3*d^3*(c + d*x)*Sin[a + b*x])/b^4 + (d*(c + d*x)^3*Sin[a + b*x])/(2*b^2) - (d^3*(c +
d*x)*Sin[3*a + 3*b*x])/(54*b^4) + (d*(c + d*x)^3*Sin[3*a + 3*b*x])/(36*b^2) + (3*d^3*(c + d*x)*Sin[5*a + 5*b*x
])/(1250*b^4) - (d*(c + d*x)^3*Sin[5*a + 5*b*x])/(100*b^2)

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Rubi [A]  time = 0.391063, antiderivative size = 330, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {4406, 3296, 2638} \[ -\frac{3 d^3 (c+d x) \sin (a+b x)}{b^4}-\frac{d^3 (c+d x) \sin (3 a+3 b x)}{54 b^4}+\frac{3 d^3 (c+d x) \sin (5 a+5 b x)}{1250 b^4}+\frac{3 d^2 (c+d x)^2 \cos (a+b x)}{2 b^3}+\frac{d^2 (c+d x)^2 \cos (3 a+3 b x)}{36 b^3}-\frac{3 d^2 (c+d x)^2 \cos (5 a+5 b x)}{500 b^3}+\frac{d (c+d x)^3 \sin (a+b x)}{2 b^2}+\frac{d (c+d x)^3 \sin (3 a+3 b x)}{36 b^2}-\frac{d (c+d x)^3 \sin (5 a+5 b x)}{100 b^2}-\frac{3 d^4 \cos (a+b x)}{b^5}-\frac{d^4 \cos (3 a+3 b x)}{162 b^5}+\frac{3 d^4 \cos (5 a+5 b x)}{6250 b^5}-\frac{(c+d x)^4 \cos (a+b x)}{8 b}-\frac{(c+d x)^4 \cos (3 a+3 b x)}{48 b}+\frac{(c+d x)^4 \cos (5 a+5 b x)}{80 b} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^4*Cos[a + b*x]^2*Sin[a + b*x]^3,x]

[Out]

(-3*d^4*Cos[a + b*x])/b^5 + (3*d^2*(c + d*x)^2*Cos[a + b*x])/(2*b^3) - ((c + d*x)^4*Cos[a + b*x])/(8*b) - (d^4
*Cos[3*a + 3*b*x])/(162*b^5) + (d^2*(c + d*x)^2*Cos[3*a + 3*b*x])/(36*b^3) - ((c + d*x)^4*Cos[3*a + 3*b*x])/(4
8*b) + (3*d^4*Cos[5*a + 5*b*x])/(6250*b^5) - (3*d^2*(c + d*x)^2*Cos[5*a + 5*b*x])/(500*b^3) + ((c + d*x)^4*Cos
[5*a + 5*b*x])/(80*b) - (3*d^3*(c + d*x)*Sin[a + b*x])/b^4 + (d*(c + d*x)^3*Sin[a + b*x])/(2*b^2) - (d^3*(c +
d*x)*Sin[3*a + 3*b*x])/(54*b^4) + (d*(c + d*x)^3*Sin[3*a + 3*b*x])/(36*b^2) + (3*d^3*(c + d*x)*Sin[5*a + 5*b*x
])/(1250*b^4) - (d*(c + d*x)^3*Sin[5*a + 5*b*x])/(100*b^2)

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 2638

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

Rubi steps

\begin{align*} \int (c+d x)^4 \cos ^2(a+b x) \sin ^3(a+b x) \, dx &=\int \left (\frac{1}{8} (c+d x)^4 \sin (a+b x)+\frac{1}{16} (c+d x)^4 \sin (3 a+3 b x)-\frac{1}{16} (c+d x)^4 \sin (5 a+5 b x)\right ) \, dx\\ &=\frac{1}{16} \int (c+d x)^4 \sin (3 a+3 b x) \, dx-\frac{1}{16} \int (c+d x)^4 \sin (5 a+5 b x) \, dx+\frac{1}{8} \int (c+d x)^4 \sin (a+b x) \, dx\\ &=-\frac{(c+d x)^4 \cos (a+b x)}{8 b}-\frac{(c+d x)^4 \cos (3 a+3 b x)}{48 b}+\frac{(c+d x)^4 \cos (5 a+5 b x)}{80 b}-\frac{d \int (c+d x)^3 \cos (5 a+5 b x) \, dx}{20 b}+\frac{d \int (c+d x)^3 \cos (3 a+3 b x) \, dx}{12 b}+\frac{d \int (c+d x)^3 \cos (a+b x) \, dx}{2 b}\\ &=-\frac{(c+d x)^4 \cos (a+b x)}{8 b}-\frac{(c+d x)^4 \cos (3 a+3 b x)}{48 b}+\frac{(c+d x)^4 \cos (5 a+5 b x)}{80 b}+\frac{d (c+d x)^3 \sin (a+b x)}{2 b^2}+\frac{d (c+d x)^3 \sin (3 a+3 b x)}{36 b^2}-\frac{d (c+d x)^3 \sin (5 a+5 b x)}{100 b^2}+\frac{\left (3 d^2\right ) \int (c+d x)^2 \sin (5 a+5 b x) \, dx}{100 b^2}-\frac{d^2 \int (c+d x)^2 \sin (3 a+3 b x) \, dx}{12 b^2}-\frac{\left (3 d^2\right ) \int (c+d x)^2 \sin (a+b x) \, dx}{2 b^2}\\ &=\frac{3 d^2 (c+d x)^2 \cos (a+b x)}{2 b^3}-\frac{(c+d x)^4 \cos (a+b x)}{8 b}+\frac{d^2 (c+d x)^2 \cos (3 a+3 b x)}{36 b^3}-\frac{(c+d x)^4 \cos (3 a+3 b x)}{48 b}-\frac{3 d^2 (c+d x)^2 \cos (5 a+5 b x)}{500 b^3}+\frac{(c+d x)^4 \cos (5 a+5 b x)}{80 b}+\frac{d (c+d x)^3 \sin (a+b x)}{2 b^2}+\frac{d (c+d x)^3 \sin (3 a+3 b x)}{36 b^2}-\frac{d (c+d x)^3 \sin (5 a+5 b x)}{100 b^2}+\frac{\left (3 d^3\right ) \int (c+d x) \cos (5 a+5 b x) \, dx}{250 b^3}-\frac{d^3 \int (c+d x) \cos (3 a+3 b x) \, dx}{18 b^3}-\frac{\left (3 d^3\right ) \int (c+d x) \cos (a+b x) \, dx}{b^3}\\ &=\frac{3 d^2 (c+d x)^2 \cos (a+b x)}{2 b^3}-\frac{(c+d x)^4 \cos (a+b x)}{8 b}+\frac{d^2 (c+d x)^2 \cos (3 a+3 b x)}{36 b^3}-\frac{(c+d x)^4 \cos (3 a+3 b x)}{48 b}-\frac{3 d^2 (c+d x)^2 \cos (5 a+5 b x)}{500 b^3}+\frac{(c+d x)^4 \cos (5 a+5 b x)}{80 b}-\frac{3 d^3 (c+d x) \sin (a+b x)}{b^4}+\frac{d (c+d x)^3 \sin (a+b x)}{2 b^2}-\frac{d^3 (c+d x) \sin (3 a+3 b x)}{54 b^4}+\frac{d (c+d x)^3 \sin (3 a+3 b x)}{36 b^2}+\frac{3 d^3 (c+d x) \sin (5 a+5 b x)}{1250 b^4}-\frac{d (c+d x)^3 \sin (5 a+5 b x)}{100 b^2}-\frac{\left (3 d^4\right ) \int \sin (5 a+5 b x) \, dx}{1250 b^4}+\frac{d^4 \int \sin (3 a+3 b x) \, dx}{54 b^4}+\frac{\left (3 d^4\right ) \int \sin (a+b x) \, dx}{b^4}\\ &=-\frac{3 d^4 \cos (a+b x)}{b^5}+\frac{3 d^2 (c+d x)^2 \cos (a+b x)}{2 b^3}-\frac{(c+d x)^4 \cos (a+b x)}{8 b}-\frac{d^4 \cos (3 a+3 b x)}{162 b^5}+\frac{d^2 (c+d x)^2 \cos (3 a+3 b x)}{36 b^3}-\frac{(c+d x)^4 \cos (3 a+3 b x)}{48 b}+\frac{3 d^4 \cos (5 a+5 b x)}{6250 b^5}-\frac{3 d^2 (c+d x)^2 \cos (5 a+5 b x)}{500 b^3}+\frac{(c+d x)^4 \cos (5 a+5 b x)}{80 b}-\frac{3 d^3 (c+d x) \sin (a+b x)}{b^4}+\frac{d (c+d x)^3 \sin (a+b x)}{2 b^2}-\frac{d^3 (c+d x) \sin (3 a+3 b x)}{54 b^4}+\frac{d (c+d x)^3 \sin (3 a+3 b x)}{36 b^2}+\frac{3 d^3 (c+d x) \sin (5 a+5 b x)}{1250 b^4}-\frac{d (c+d x)^3 \sin (5 a+5 b x)}{100 b^2}\\ \end{align*}

Mathematica [A]  time = 3.4162, size = 238, normalized size = 0.72 \[ \frac{120 b d (c+d x) \sin (a+b x) \left (16 \cos (2 (a+b x)) \left (75 b^2 (c+d x)^2-68 d^2\right )-27 \cos (4 (a+b x)) \left (25 b^2 (c+d x)^2-6 d^2\right )+17475 b^2 c^2+34950 b^2 c d x+17475 b^2 d^2 x^2-101794 d^2\right )-506250 \cos (a+b x) \left (-12 b^2 d^2 (c+d x)^2+b^4 (c+d x)^4+24 d^4\right )-3125 \cos (3 (a+b x)) \left (-36 b^2 d^2 (c+d x)^2+27 b^4 (c+d x)^4+8 d^4\right )+81 \cos (5 (a+b x)) \left (-300 b^2 d^2 (c+d x)^2+625 b^4 (c+d x)^4+24 d^4\right )}{4050000 b^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^4*Cos[a + b*x]^2*Sin[a + b*x]^3,x]

[Out]

(-506250*(24*d^4 - 12*b^2*d^2*(c + d*x)^2 + b^4*(c + d*x)^4)*Cos[a + b*x] - 3125*(8*d^4 - 36*b^2*d^2*(c + d*x)
^2 + 27*b^4*(c + d*x)^4)*Cos[3*(a + b*x)] + 81*(24*d^4 - 300*b^2*d^2*(c + d*x)^2 + 625*b^4*(c + d*x)^4)*Cos[5*
(a + b*x)] + 120*b*d*(c + d*x)*(17475*b^2*c^2 - 101794*d^2 + 34950*b^2*c*d*x + 17475*b^2*d^2*x^2 + 16*(-68*d^2
+ 75*b^2*(c + d*x)^2)*Cos[2*(a + b*x)] - 27*(-6*d^2 + 25*b^2*(c + d*x)^2)*Cos[4*(a + b*x)])*Sin[a + b*x])/(40
50000*b^5)

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Maple [B]  time = 0.087, size = 1812, normalized size = 5.5 \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)^3,x)

[Out]

1/b*(1/b^4*d^4*(-1/3*(b*x+a)^4*(2+sin(b*x+a)^2)*cos(b*x+a)+8/15*(b*x+a)^3*sin(b*x+a)+8/5*(b*x+a)^2*cos(b*x+a)-
3424/1125*cos(b*x+a)-3424/1125*(b*x+a)*sin(b*x+a)+4/45*(b*x+a)^3*sin(b*x+a)^3+4/45*(b*x+a)^2*(2+sin(b*x+a)^2)*
cos(b*x+a)+88/3375*(b*x+a)*sin(b*x+a)^3+88/10125*(2+sin(b*x+a)^2)*cos(b*x+a)+1/5*(b*x+a)^4*(8/3+sin(b*x+a)^4+4
/3*sin(b*x+a)^2)*cos(b*x+a)-4/25*(b*x+a)^3*sin(b*x+a)^5-12/125*(b*x+a)^2*(8/3+sin(b*x+a)^4+4/3*sin(b*x+a)^2)*c
os(b*x+a)+24/625*(b*x+a)*sin(b*x+a)^5+24/3125*(8/3+sin(b*x+a)^4+4/3*sin(b*x+a)^2)*cos(b*x+a))-4/b^4*a*d^4*(-1/
3*(b*x+a)^3*(2+sin(b*x+a)^2)*cos(b*x+a)+2/5*(b*x+a)^2*sin(b*x+a)-856/1125*sin(b*x+a)+4/5*(b*x+a)*cos(b*x+a)+1/
15*(b*x+a)^2*sin(b*x+a)^3+2/45*(b*x+a)*(2+sin(b*x+a)^2)*cos(b*x+a)+22/3375*sin(b*x+a)^3+1/5*(b*x+a)^3*(8/3+sin
(b*x+a)^4+4/3*sin(b*x+a)^2)*cos(b*x+a)-3/25*(b*x+a)^2*sin(b*x+a)^5-6/125*(b*x+a)*(8/3+sin(b*x+a)^4+4/3*sin(b*x
+a)^2)*cos(b*x+a)+6/625*sin(b*x+a)^5)+4/b^3*c*d^3*(-1/3*(b*x+a)^3*(2+sin(b*x+a)^2)*cos(b*x+a)+2/5*(b*x+a)^2*si
n(b*x+a)-856/1125*sin(b*x+a)+4/5*(b*x+a)*cos(b*x+a)+1/15*(b*x+a)^2*sin(b*x+a)^3+2/45*(b*x+a)*(2+sin(b*x+a)^2)*
cos(b*x+a)+22/3375*sin(b*x+a)^3+1/5*(b*x+a)^3*(8/3+sin(b*x+a)^4+4/3*sin(b*x+a)^2)*cos(b*x+a)-3/25*(b*x+a)^2*si
n(b*x+a)^5-6/125*(b*x+a)*(8/3+sin(b*x+a)^4+4/3*sin(b*x+a)^2)*cos(b*x+a)+6/625*sin(b*x+a)^5)+6/b^4*a^2*d^4*(-1/
3*(b*x+a)^2*(2+sin(b*x+a)^2)*cos(b*x+a)+4/15*cos(b*x+a)+4/15*(b*x+a)*sin(b*x+a)+2/45*(b*x+a)*sin(b*x+a)^3+2/13
5*(2+sin(b*x+a)^2)*cos(b*x+a)+1/5*(b*x+a)^2*(8/3+sin(b*x+a)^4+4/3*sin(b*x+a)^2)*cos(b*x+a)-2/25*(b*x+a)*sin(b*
x+a)^5-2/125*(8/3+sin(b*x+a)^4+4/3*sin(b*x+a)^2)*cos(b*x+a))-12/b^3*a*c*d^3*(-1/3*(b*x+a)^2*(2+sin(b*x+a)^2)*c
os(b*x+a)+4/15*cos(b*x+a)+4/15*(b*x+a)*sin(b*x+a)+2/45*(b*x+a)*sin(b*x+a)^3+2/135*(2+sin(b*x+a)^2)*cos(b*x+a)+
1/5*(b*x+a)^2*(8/3+sin(b*x+a)^4+4/3*sin(b*x+a)^2)*cos(b*x+a)-2/25*(b*x+a)*sin(b*x+a)^5-2/125*(8/3+sin(b*x+a)^4
+4/3*sin(b*x+a)^2)*cos(b*x+a))+6/b^2*c^2*d^2*(-1/3*(b*x+a)^2*(2+sin(b*x+a)^2)*cos(b*x+a)+4/15*cos(b*x+a)+4/15*
(b*x+a)*sin(b*x+a)+2/45*(b*x+a)*sin(b*x+a)^3+2/135*(2+sin(b*x+a)^2)*cos(b*x+a)+1/5*(b*x+a)^2*(8/3+sin(b*x+a)^4
+4/3*sin(b*x+a)^2)*cos(b*x+a)-2/25*(b*x+a)*sin(b*x+a)^5-2/125*(8/3+sin(b*x+a)^4+4/3*sin(b*x+a)^2)*cos(b*x+a))-
4/b^4*a^3*d^4*(-1/3*(b*x+a)*(2+sin(b*x+a)^2)*cos(b*x+a)+1/45*sin(b*x+a)^3+2/15*sin(b*x+a)+1/5*(b*x+a)*(8/3+sin
(b*x+a)^4+4/3*sin(b*x+a)^2)*cos(b*x+a)-1/25*sin(b*x+a)^5)+12/b^3*a^2*c*d^3*(-1/3*(b*x+a)*(2+sin(b*x+a)^2)*cos(
b*x+a)+1/45*sin(b*x+a)^3+2/15*sin(b*x+a)+1/5*(b*x+a)*(8/3+sin(b*x+a)^4+4/3*sin(b*x+a)^2)*cos(b*x+a)-1/25*sin(b
*x+a)^5)-12/b^2*a*c^2*d^2*(-1/3*(b*x+a)*(2+sin(b*x+a)^2)*cos(b*x+a)+1/45*sin(b*x+a)^3+2/15*sin(b*x+a)+1/5*(b*x
+a)*(8/3+sin(b*x+a)^4+4/3*sin(b*x+a)^2)*cos(b*x+a)-1/25*sin(b*x+a)^5)+4/b*c^3*d*(-1/3*(b*x+a)*(2+sin(b*x+a)^2)
*cos(b*x+a)+1/45*sin(b*x+a)^3+2/15*sin(b*x+a)+1/5*(b*x+a)*(8/3+sin(b*x+a)^4+4/3*sin(b*x+a)^2)*cos(b*x+a)-1/25*
sin(b*x+a)^5)+1/b^4*a^4*d^4*(-1/5*sin(b*x+a)^2*cos(b*x+a)^3-2/15*cos(b*x+a)^3)-4/b^3*a^3*c*d^3*(-1/5*sin(b*x+a
)^2*cos(b*x+a)^3-2/15*cos(b*x+a)^3)+6/b^2*a^2*c^2*d^2*(-1/5*sin(b*x+a)^2*cos(b*x+a)^3-2/15*cos(b*x+a)^3)-4/b*a
*c^3*d*(-1/5*sin(b*x+a)^2*cos(b*x+a)^3-2/15*cos(b*x+a)^3)+c^4*(-1/5*sin(b*x+a)^2*cos(b*x+a)^3-2/15*cos(b*x+a)^
3))

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Maxima [B]  time = 1.42847, size = 1808, normalized size = 5.48 \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)^3,x, algorithm="maxima")

[Out]

1/4050000*(270000*(3*cos(b*x + a)^5 - 5*cos(b*x + a)^3)*c^4 - 1080000*(3*cos(b*x + a)^5 - 5*cos(b*x + a)^3)*a*
c^3*d/b + 1620000*(3*cos(b*x + a)^5 - 5*cos(b*x + a)^3)*a^2*c^2*d^2/b^2 - 1080000*(3*cos(b*x + a)^5 - 5*cos(b*
x + a)^3)*a^3*c*d^3/b^3 + 270000*(3*cos(b*x + a)^5 - 5*cos(b*x + a)^3)*a^4*d^4/b^4 + 4500*(45*(b*x + a)*cos(5*
b*x + 5*a) - 75*(b*x + a)*cos(3*b*x + 3*a) - 450*(b*x + a)*cos(b*x + a) - 9*sin(5*b*x + 5*a) + 25*sin(3*b*x +
3*a) + 450*sin(b*x + a))*c^3*d/b - 13500*(45*(b*x + a)*cos(5*b*x + 5*a) - 75*(b*x + a)*cos(3*b*x + 3*a) - 450*
(b*x + a)*cos(b*x + a) - 9*sin(5*b*x + 5*a) + 25*sin(3*b*x + 3*a) + 450*sin(b*x + a))*a*c^2*d^2/b^2 + 13500*(4
5*(b*x + a)*cos(5*b*x + 5*a) - 75*(b*x + a)*cos(3*b*x + 3*a) - 450*(b*x + a)*cos(b*x + a) - 9*sin(5*b*x + 5*a)
+ 25*sin(3*b*x + 3*a) + 450*sin(b*x + a))*a^2*c*d^3/b^3 - 4500*(45*(b*x + a)*cos(5*b*x + 5*a) - 75*(b*x + a)*
cos(3*b*x + 3*a) - 450*(b*x + a)*cos(b*x + a) - 9*sin(5*b*x + 5*a) + 25*sin(3*b*x + 3*a) + 450*sin(b*x + a))*a
^3*d^4/b^4 + 450*(27*(25*(b*x + a)^2 - 2)*cos(5*b*x + 5*a) - 125*(9*(b*x + a)^2 - 2)*cos(3*b*x + 3*a) - 6750*(
(b*x + a)^2 - 2)*cos(b*x + a) - 270*(b*x + a)*sin(5*b*x + 5*a) + 750*(b*x + a)*sin(3*b*x + 3*a) + 13500*(b*x +
a)*sin(b*x + a))*c^2*d^2/b^2 - 900*(27*(25*(b*x + a)^2 - 2)*cos(5*b*x + 5*a) - 125*(9*(b*x + a)^2 - 2)*cos(3*
b*x + 3*a) - 6750*((b*x + a)^2 - 2)*cos(b*x + a) - 270*(b*x + a)*sin(5*b*x + 5*a) + 750*(b*x + a)*sin(3*b*x +
3*a) + 13500*(b*x + a)*sin(b*x + a))*a*c*d^3/b^3 + 450*(27*(25*(b*x + a)^2 - 2)*cos(5*b*x + 5*a) - 125*(9*(b*x
+ a)^2 - 2)*cos(3*b*x + 3*a) - 6750*((b*x + a)^2 - 2)*cos(b*x + a) - 270*(b*x + a)*sin(5*b*x + 5*a) + 750*(b*
x + a)*sin(3*b*x + 3*a) + 13500*(b*x + a)*sin(b*x + a))*a^2*d^4/b^4 + 60*(135*(25*(b*x + a)^3 - 6*b*x - 6*a)*c
os(5*b*x + 5*a) - 1875*(3*(b*x + a)^3 - 2*b*x - 2*a)*cos(3*b*x + 3*a) - 33750*((b*x + a)^3 - 6*b*x - 6*a)*cos(
b*x + a) - 81*(25*(b*x + a)^2 - 2)*sin(5*b*x + 5*a) + 625*(9*(b*x + a)^2 - 2)*sin(3*b*x + 3*a) + 101250*((b*x
+ a)^2 - 2)*sin(b*x + a))*c*d^3/b^3 - 60*(135*(25*(b*x + a)^3 - 6*b*x - 6*a)*cos(5*b*x + 5*a) - 1875*(3*(b*x +
a)^3 - 2*b*x - 2*a)*cos(3*b*x + 3*a) - 33750*((b*x + a)^3 - 6*b*x - 6*a)*cos(b*x + a) - 81*(25*(b*x + a)^2 -
2)*sin(5*b*x + 5*a) + 625*(9*(b*x + a)^2 - 2)*sin(3*b*x + 3*a) + 101250*((b*x + a)^2 - 2)*sin(b*x + a))*a*d^4/
b^4 + (81*(625*(b*x + a)^4 - 300*(b*x + a)^2 + 24)*cos(5*b*x + 5*a) - 3125*(27*(b*x + a)^4 - 36*(b*x + a)^2 +
8)*cos(3*b*x + 3*a) - 506250*((b*x + a)^4 - 12*(b*x + a)^2 + 24)*cos(b*x + a) - 1620*(25*(b*x + a)^3 - 6*b*x -
6*a)*sin(5*b*x + 5*a) + 37500*(3*(b*x + a)^3 - 2*b*x - 2*a)*sin(3*b*x + 3*a) + 2025000*((b*x + a)^3 - 6*b*x -
6*a)*sin(b*x + a))*d^4/b^4)/b

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Fricas [A]  time = 0.610837, size = 1121, normalized size = 3.4 \begin{align*} \frac{81 \,{\left (625 \, b^{4} d^{4} x^{4} + 2500 \, b^{4} c d^{3} x^{3} + 625 \, b^{4} c^{4} - 300 \, b^{2} c^{2} d^{2} + 24 \, d^{4} + 150 \,{\left (25 \, b^{4} c^{2} d^{2} - 2 \, b^{2} d^{4}\right )} x^{2} + 100 \,{\left (25 \, b^{4} c^{3} d - 6 \, b^{2} c d^{3}\right )} x\right )} \cos \left (b x + a\right )^{5} - 5 \,{\left (16875 \, b^{4} d^{4} x^{4} + 67500 \, b^{4} c d^{3} x^{3} + 16875 \, b^{4} c^{4} - 11700 \, b^{2} c^{2} d^{2} + 1736 \, d^{4} + 450 \,{\left (225 \, b^{4} c^{2} d^{2} - 26 \, b^{2} d^{4}\right )} x^{2} + 900 \,{\left (75 \, b^{4} c^{3} d - 26 \, b^{2} c d^{3}\right )} x\right )} \cos \left (b x + a\right )^{3} + 120 \,{\left (2925 \, b^{2} d^{4} x^{2} + 5850 \, b^{2} c d^{3} x + 2925 \, b^{2} c^{2} d^{2} - 6284 \, d^{4}\right )} \cos \left (b x + a\right ) + 60 \,{\left (1950 \, b^{3} d^{4} x^{3} + 5850 \, b^{3} c d^{3} x^{2} + 1950 \, b^{3} c^{3} d - 12568 \, b c d^{3} - 27 \,{\left (25 \, b^{3} d^{4} x^{3} + 75 \, b^{3} c d^{3} x^{2} + 25 \, b^{3} c^{3} d - 6 \, b c d^{3} + 3 \,{\left (25 \, b^{3} c^{2} d^{2} - 2 \, b d^{4}\right )} x\right )} \cos \left (b x + a\right )^{4} +{\left (975 \, b^{3} d^{4} x^{3} + 2925 \, b^{3} c d^{3} x^{2} + 975 \, b^{3} c^{3} d - 434 \, b c d^{3} +{\left (2925 \, b^{3} c^{2} d^{2} - 434 \, b d^{4}\right )} x\right )} \cos \left (b x + a\right )^{2} + 2 \,{\left (2925 \, b^{3} c^{2} d^{2} - 6284 \, b d^{4}\right )} x\right )} \sin \left (b x + a\right )}{253125 \, 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)^3,x, algorithm="fricas")

[Out]

1/253125*(81*(625*b^4*d^4*x^4 + 2500*b^4*c*d^3*x^3 + 625*b^4*c^4 - 300*b^2*c^2*d^2 + 24*d^4 + 150*(25*b^4*c^2*
d^2 - 2*b^2*d^4)*x^2 + 100*(25*b^4*c^3*d - 6*b^2*c*d^3)*x)*cos(b*x + a)^5 - 5*(16875*b^4*d^4*x^4 + 67500*b^4*c
*d^3*x^3 + 16875*b^4*c^4 - 11700*b^2*c^2*d^2 + 1736*d^4 + 450*(225*b^4*c^2*d^2 - 26*b^2*d^4)*x^2 + 900*(75*b^4
*c^3*d - 26*b^2*c*d^3)*x)*cos(b*x + a)^3 + 120*(2925*b^2*d^4*x^2 + 5850*b^2*c*d^3*x + 2925*b^2*c^2*d^2 - 6284*
d^4)*cos(b*x + a) + 60*(1950*b^3*d^4*x^3 + 5850*b^3*c*d^3*x^2 + 1950*b^3*c^3*d - 12568*b*c*d^3 - 27*(25*b^3*d^
4*x^3 + 75*b^3*c*d^3*x^2 + 25*b^3*c^3*d - 6*b*c*d^3 + 3*(25*b^3*c^2*d^2 - 2*b*d^4)*x)*cos(b*x + a)^4 + (975*b^
3*d^4*x^3 + 2925*b^3*c*d^3*x^2 + 975*b^3*c^3*d - 434*b*c*d^3 + (2925*b^3*c^2*d^2 - 434*b*d^4)*x)*cos(b*x + a)^
2 + 2*(2925*b^3*c^2*d^2 - 6284*b*d^4)*x)*sin(b*x + a))/b^5

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Sympy [A]  time = 79.2152, size = 1098, normalized size = 3.33 \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)**3,x)

[Out]

Piecewise((-c**4*sin(a + b*x)**2*cos(a + b*x)**3/(3*b) - 2*c**4*cos(a + b*x)**5/(15*b) - 4*c**3*d*x*sin(a + b*
x)**2*cos(a + b*x)**3/(3*b) - 8*c**3*d*x*cos(a + b*x)**5/(15*b) - 2*c**2*d**2*x**2*sin(a + b*x)**2*cos(a + b*x
)**3/b - 4*c**2*d**2*x**2*cos(a + b*x)**5/(5*b) - 4*c*d**3*x**3*sin(a + b*x)**2*cos(a + b*x)**3/(3*b) - 8*c*d*
*3*x**3*cos(a + b*x)**5/(15*b) - d**4*x**4*sin(a + b*x)**2*cos(a + b*x)**3/(3*b) - 2*d**4*x**4*cos(a + b*x)**5
/(15*b) + 104*c**3*d*sin(a + b*x)**5/(225*b**2) + 52*c**3*d*sin(a + b*x)**3*cos(a + b*x)**2/(45*b**2) + 8*c**3
*d*sin(a + b*x)*cos(a + b*x)**4/(15*b**2) + 104*c**2*d**2*x*sin(a + b*x)**5/(75*b**2) + 52*c**2*d**2*x*sin(a +
b*x)**3*cos(a + b*x)**2/(15*b**2) + 8*c**2*d**2*x*sin(a + b*x)*cos(a + b*x)**4/(5*b**2) + 104*c*d**3*x**2*sin
(a + b*x)**5/(75*b**2) + 52*c*d**3*x**2*sin(a + b*x)**3*cos(a + b*x)**2/(15*b**2) + 8*c*d**3*x**2*sin(a + b*x)
*cos(a + b*x)**4/(5*b**2) + 104*d**4*x**3*sin(a + b*x)**5/(225*b**2) + 52*d**4*x**3*sin(a + b*x)**3*cos(a + b*
x)**2/(45*b**2) + 8*d**4*x**3*sin(a + b*x)*cos(a + b*x)**4/(15*b**2) + 104*c**2*d**2*sin(a + b*x)**4*cos(a + b
*x)/(75*b**3) + 676*c**2*d**2*sin(a + b*x)**2*cos(a + b*x)**3/(225*b**3) + 1712*c**2*d**2*cos(a + b*x)**5/(112
5*b**3) + 208*c*d**3*x*sin(a + b*x)**4*cos(a + b*x)/(75*b**3) + 1352*c*d**3*x*sin(a + b*x)**2*cos(a + b*x)**3/
(225*b**3) + 3424*c*d**3*x*cos(a + b*x)**5/(1125*b**3) + 104*d**4*x**2*sin(a + b*x)**4*cos(a + b*x)/(75*b**3)
+ 676*d**4*x**2*sin(a + b*x)**2*cos(a + b*x)**3/(225*b**3) + 1712*d**4*x**2*cos(a + b*x)**5/(1125*b**3) - 5027
2*c*d**3*sin(a + b*x)**5/(16875*b**4) - 20456*c*d**3*sin(a + b*x)**3*cos(a + b*x)**2/(3375*b**4) - 3424*c*d**3
*sin(a + b*x)*cos(a + b*x)**4/(1125*b**4) - 50272*d**4*x*sin(a + b*x)**5/(16875*b**4) - 20456*d**4*x*sin(a + b
*x)**3*cos(a + b*x)**2/(3375*b**4) - 3424*d**4*x*sin(a + b*x)*cos(a + b*x)**4/(1125*b**4) - 50272*d**4*sin(a +
b*x)**4*cos(a + b*x)/(16875*b**5) - 303368*d**4*sin(a + b*x)**2*cos(a + b*x)**3/(50625*b**5) - 760816*d**4*co
s(a + b*x)**5/(253125*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)**3*cos(a)**2, True))

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Giac [A]  time = 1.13131, size = 717, normalized size = 2.17 \begin{align*} \frac{{\left (625 \, b^{4} d^{4} x^{4} + 2500 \, b^{4} c d^{3} x^{3} + 3750 \, b^{4} c^{2} d^{2} x^{2} + 2500 \, b^{4} c^{3} d x + 625 \, b^{4} c^{4} - 300 \, b^{2} d^{4} x^{2} - 600 \, b^{2} c d^{3} x - 300 \, b^{2} c^{2} d^{2} + 24 \, d^{4}\right )} \cos \left (5 \, b x + 5 \, a\right )}{50000 \, 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 )} \cos \left (3 \, b x + 3 \, a\right )}{1296 \, 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 )} \cos \left (b x + a\right )}{8 \, b^{5}} - \frac{{\left (25 \, b^{3} d^{4} x^{3} + 75 \, b^{3} c d^{3} x^{2} + 75 \, b^{3} c^{2} d^{2} x + 25 \, b^{3} c^{3} d - 6 \, b d^{4} x - 6 \, b c d^{3}\right )} \sin \left (5 \, b x + 5 \, a\right )}{2500 \, b^{5}} + \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 )} \sin \left (3 \, b x + 3 \, a\right )}{108 \, 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 )} \sin \left (b x + a\right )}{2 \, 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)^3,x, algorithm="giac")

[Out]

1/50000*(625*b^4*d^4*x^4 + 2500*b^4*c*d^3*x^3 + 3750*b^4*c^2*d^2*x^2 + 2500*b^4*c^3*d*x + 625*b^4*c^4 - 300*b^
2*d^4*x^2 - 600*b^2*c*d^3*x - 300*b^2*c^2*d^2 + 24*d^4)*cos(5*b*x + 5*a)/b^5 - 1/1296*(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)*cos(3*b*x + 3*a)/b^5 - 1/8*(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)*cos(b*x + a)/b^5 - 1/2500*(25*b^3*d^4*x^
3 + 75*b^3*c*d^3*x^2 + 75*b^3*c^2*d^2*x + 25*b^3*c^3*d - 6*b*d^4*x - 6*b*c*d^3)*sin(5*b*x + 5*a)/b^5 + 1/108*(
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)*sin(3*b*x + 3*a)/b^5
+ 1/2*(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)*sin(b*x + a)/b^5