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

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

[Out]

(-3*d^3*(c + d*x)*Cos[a + b*x])/b^4 + (d*(c + d*x)^3*Cos[a + b*x])/(2*b^2) + (d^3*(c + d*x)*Cos[3*a + 3*b*x])/
(54*b^4) - (d*(c + d*x)^3*Cos[3*a + 3*b*x])/(36*b^2) + (3*d^3*(c + d*x)*Cos[5*a + 5*b*x])/(1250*b^4) - (d*(c +
d*x)^3*Cos[5*a + 5*b*x])/(100*b^2) + (3*d^4*Sin[a + b*x])/b^5 - (3*d^2*(c + d*x)^2*Sin[a + b*x])/(2*b^3) + ((
c + d*x)^4*Sin[a + b*x])/(8*b) - (d^4*Sin[3*a + 3*b*x])/(162*b^5) + (d^2*(c + d*x)^2*Sin[3*a + 3*b*x])/(36*b^3
) - ((c + d*x)^4*Sin[3*a + 3*b*x])/(48*b) - (3*d^4*Sin[5*a + 5*b*x])/(6250*b^5) + (3*d^2*(c + d*x)^2*Sin[5*a +
5*b*x])/(500*b^3) - ((c + d*x)^4*Sin[5*a + 5*b*x])/(80*b)

________________________________________________________________________________________

Rubi [A]  time = 0.368317, 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, 2637} \[ -\frac{3 d^2 (c+d x)^2 \sin (a+b x)}{2 b^3}+\frac{d^2 (c+d x)^2 \sin (3 a+3 b x)}{36 b^3}+\frac{3 d^2 (c+d x)^2 \sin (5 a+5 b x)}{500 b^3}-\frac{3 d^3 (c+d x) \cos (a+b x)}{b^4}+\frac{d^3 (c+d x) \cos (3 a+3 b x)}{54 b^4}+\frac{3 d^3 (c+d x) \cos (5 a+5 b x)}{1250 b^4}+\frac{d (c+d x)^3 \cos (a+b x)}{2 b^2}-\frac{d (c+d x)^3 \cos (3 a+3 b x)}{36 b^2}-\frac{d (c+d x)^3 \cos (5 a+5 b x)}{100 b^2}+\frac{3 d^4 \sin (a+b x)}{b^5}-\frac{d^4 \sin (3 a+3 b x)}{162 b^5}-\frac{3 d^4 \sin (5 a+5 b x)}{6250 b^5}+\frac{(c+d x)^4 \sin (a+b x)}{8 b}-\frac{(c+d x)^4 \sin (3 a+3 b x)}{48 b}-\frac{(c+d x)^4 \sin (5 a+5 b x)}{80 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 3.75681, size = 563, normalized size = 1.71 \[ -\frac{-3037500 b^2 c^2 d^2 \left (\left (b^2 x^2-2\right ) \sin (a+b x)+2 b x \cos (a+b x)\right )+56250 b^2 c^2 d^2 \left (\left (9 b^2 x^2-2\right ) \sin (3 (a+b x))+6 b x \cos (3 (a+b x))\right )+12150 b^2 c^2 d^2 \left (\left (25 b^2 x^2-2\right ) \sin (5 (a+b x))+10 b x \cos (5 (a+b x))\right )-2025000 b^3 c^3 d (b x \sin (a+b x)+\cos (a+b x))+112500 b^3 c^3 d (3 b x \sin (3 (a+b x))+\cos (3 (a+b x)))+40500 b^3 c^3 d (5 b x \sin (5 (a+b x))+\cos (5 (a+b x)))-506250 b^4 c^4 \sin (a+b x)+84375 b^4 c^4 \sin (3 (a+b x))+50625 b^4 c^4 \sin (5 (a+b x))-2025000 b c d^3 \left (b x \left (b^2 x^2-6\right ) \sin (a+b x)+3 \left (b^2 x^2-2\right ) \cos (a+b x)\right )+37500 b c d^3 \left (3 b x \left (3 b^2 x^2-2\right ) \sin (3 (a+b x))+\left (9 b^2 x^2-2\right ) \cos (3 (a+b x))\right )+1620 b c d^3 \left (5 b x \left (25 b^2 x^2-6\right ) \sin (5 (a+b x))+\left (75 b^2 x^2-6\right ) \cos (5 (a+b x))\right )-506250 d^4 \left (\left (b^4 x^4-12 b^2 x^2+24\right ) \sin (a+b x)+4 b x \left (b^2 x^2-6\right ) \cos (a+b x)\right )+3125 d^4 \left (\left (27 b^4 x^4-36 b^2 x^2+8\right ) \sin (3 (a+b x))+12 b x \left (3 b^2 x^2-2\right ) \cos (3 (a+b x))\right )+81 d^4 \left (\left (625 b^4 x^4-300 b^2 x^2+24\right ) \sin (5 (a+b x))+20 b x \left (25 b^2 x^2-6\right ) \cos (5 (a+b x))\right )}{4050000 b^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(-506250*b^4*c^4*Sin[a + b*x] - 2025000*b^3*c^3*d*(Cos[a + b*x] + b*x*Sin[a + b*x]) - 2025000*b*c*d^3*(3*(-2
+ b^2*x^2)*Cos[a + b*x] + b*x*(-6 + b^2*x^2)*Sin[a + b*x]) - 3037500*b^2*c^2*d^2*(2*b*x*Cos[a + b*x] + (-2 + b
^2*x^2)*Sin[a + b*x]) - 506250*d^4*(4*b*x*(-6 + b^2*x^2)*Cos[a + b*x] + (24 - 12*b^2*x^2 + b^4*x^4)*Sin[a + b*
x]) + 84375*b^4*c^4*Sin[3*(a + b*x)] + 112500*b^3*c^3*d*(Cos[3*(a + b*x)] + 3*b*x*Sin[3*(a + b*x)]) + 37500*b*
c*d^3*((-2 + 9*b^2*x^2)*Cos[3*(a + b*x)] + 3*b*x*(-2 + 3*b^2*x^2)*Sin[3*(a + b*x)]) + 56250*b^2*c^2*d^2*(6*b*x
*Cos[3*(a + b*x)] + (-2 + 9*b^2*x^2)*Sin[3*(a + b*x)]) + 3125*d^4*(12*b*x*(-2 + 3*b^2*x^2)*Cos[3*(a + b*x)] +
(8 - 36*b^2*x^2 + 27*b^4*x^4)*Sin[3*(a + b*x)]) + 50625*b^4*c^4*Sin[5*(a + b*x)] + 40500*b^3*c^3*d*(Cos[5*(a +
b*x)] + 5*b*x*Sin[5*(a + b*x)]) + 1620*b*c*d^3*((-6 + 75*b^2*x^2)*Cos[5*(a + b*x)] + 5*b*x*(-6 + 25*b^2*x^2)*
Sin[5*(a + b*x)]) + 12150*b^2*c^2*d^2*(10*b*x*Cos[5*(a + b*x)] + (-2 + 25*b^2*x^2)*Sin[5*(a + b*x)]) + 81*d^4*
(20*b*x*(-6 + 25*b^2*x^2)*Cos[5*(a + b*x)] + (24 - 300*b^2*x^2 + 625*b^4*x^4)*Sin[5*(a + b*x)]))/(4050000*b^5)

________________________________________________________________________________________

Maple [B]  time = 0.052, size = 1842, normalized size = 5.6 \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)^3*sin(b*x+a)^2,x)

[Out]

1/b*(1/b^4*d^4*(1/3*(b*x+a)^4*(2+cos(b*x+a)^2)*sin(b*x+a)+8/15*(b*x+a)^3*cos(b*x+a)-8/5*(b*x+a)^2*sin(b*x+a)+3
424/1125*sin(b*x+a)-3424/1125*(b*x+a)*cos(b*x+a)+4/45*(b*x+a)^3*cos(b*x+a)^3-4/45*(b*x+a)^2*(2+cos(b*x+a)^2)*s
in(b*x+a)+88/3375*(b*x+a)*cos(b*x+a)^3-88/10125*(2+cos(b*x+a)^2)*sin(b*x+a)-1/5*(b*x+a)^4*(8/3+cos(b*x+a)^4+4/
3*cos(b*x+a)^2)*sin(b*x+a)-4/25*(b*x+a)^3*cos(b*x+a)^5+12/125*(b*x+a)^2*(8/3+cos(b*x+a)^4+4/3*cos(b*x+a)^2)*si
n(b*x+a)+24/625*(b*x+a)*cos(b*x+a)^5-24/3125*(8/3+cos(b*x+a)^4+4/3*cos(b*x+a)^2)*sin(b*x+a))-4/b^4*a*d^4*(1/3*
(b*x+a)^3*(2+cos(b*x+a)^2)*sin(b*x+a)+2/5*(b*x+a)^2*cos(b*x+a)-856/1125*cos(b*x+a)-4/5*(b*x+a)*sin(b*x+a)+1/15
*(b*x+a)^2*cos(b*x+a)^3-2/45*(b*x+a)*(2+cos(b*x+a)^2)*sin(b*x+a)+22/3375*cos(b*x+a)^3-1/5*(b*x+a)^3*(8/3+cos(b
*x+a)^4+4/3*cos(b*x+a)^2)*sin(b*x+a)-3/25*(b*x+a)^2*cos(b*x+a)^5+6/125*(b*x+a)*(8/3+cos(b*x+a)^4+4/3*cos(b*x+a
)^2)*sin(b*x+a)+6/625*cos(b*x+a)^5)+4/b^3*c*d^3*(1/3*(b*x+a)^3*(2+cos(b*x+a)^2)*sin(b*x+a)+2/5*(b*x+a)^2*cos(b
*x+a)-856/1125*cos(b*x+a)-4/5*(b*x+a)*sin(b*x+a)+1/15*(b*x+a)^2*cos(b*x+a)^3-2/45*(b*x+a)*(2+cos(b*x+a)^2)*sin
(b*x+a)+22/3375*cos(b*x+a)^3-1/5*(b*x+a)^3*(8/3+cos(b*x+a)^4+4/3*cos(b*x+a)^2)*sin(b*x+a)-3/25*(b*x+a)^2*cos(b
*x+a)^5+6/125*(b*x+a)*(8/3+cos(b*x+a)^4+4/3*cos(b*x+a)^2)*sin(b*x+a)+6/625*cos(b*x+a)^5)+6/b^4*a^2*d^4*(1/3*(b
*x+a)^2*(2+cos(b*x+a)^2)*sin(b*x+a)-4/15*sin(b*x+a)+4/15*(b*x+a)*cos(b*x+a)+2/45*(b*x+a)*cos(b*x+a)^3-2/135*(2
+cos(b*x+a)^2)*sin(b*x+a)-1/5*(b*x+a)^2*(8/3+cos(b*x+a)^4+4/3*cos(b*x+a)^2)*sin(b*x+a)-2/25*(b*x+a)*cos(b*x+a)
^5+2/125*(8/3+cos(b*x+a)^4+4/3*cos(b*x+a)^2)*sin(b*x+a))-12/b^3*a*c*d^3*(1/3*(b*x+a)^2*(2+cos(b*x+a)^2)*sin(b*
x+a)-4/15*sin(b*x+a)+4/15*(b*x+a)*cos(b*x+a)+2/45*(b*x+a)*cos(b*x+a)^3-2/135*(2+cos(b*x+a)^2)*sin(b*x+a)-1/5*(
b*x+a)^2*(8/3+cos(b*x+a)^4+4/3*cos(b*x+a)^2)*sin(b*x+a)-2/25*(b*x+a)*cos(b*x+a)^5+2/125*(8/3+cos(b*x+a)^4+4/3*
cos(b*x+a)^2)*sin(b*x+a))+6/b^2*c^2*d^2*(1/3*(b*x+a)^2*(2+cos(b*x+a)^2)*sin(b*x+a)-4/15*sin(b*x+a)+4/15*(b*x+a
)*cos(b*x+a)+2/45*(b*x+a)*cos(b*x+a)^3-2/135*(2+cos(b*x+a)^2)*sin(b*x+a)-1/5*(b*x+a)^2*(8/3+cos(b*x+a)^4+4/3*c
os(b*x+a)^2)*sin(b*x+a)-2/25*(b*x+a)*cos(b*x+a)^5+2/125*(8/3+cos(b*x+a)^4+4/3*cos(b*x+a)^2)*sin(b*x+a))-4/b^4*
a^3*d^4*(1/3*(b*x+a)*(2+cos(b*x+a)^2)*sin(b*x+a)+1/45*cos(b*x+a)^3+2/15*cos(b*x+a)-1/5*(b*x+a)*(8/3+cos(b*x+a)
^4+4/3*cos(b*x+a)^2)*sin(b*x+a)-1/25*cos(b*x+a)^5)+12/b^3*a^2*c*d^3*(1/3*(b*x+a)*(2+cos(b*x+a)^2)*sin(b*x+a)+1
/45*cos(b*x+a)^3+2/15*cos(b*x+a)-1/5*(b*x+a)*(8/3+cos(b*x+a)^4+4/3*cos(b*x+a)^2)*sin(b*x+a)-1/25*cos(b*x+a)^5)
-12/b^2*a*c^2*d^2*(1/3*(b*x+a)*(2+cos(b*x+a)^2)*sin(b*x+a)+1/45*cos(b*x+a)^3+2/15*cos(b*x+a)-1/5*(b*x+a)*(8/3+
cos(b*x+a)^4+4/3*cos(b*x+a)^2)*sin(b*x+a)-1/25*cos(b*x+a)^5)+4/b*c^3*d*(1/3*(b*x+a)*(2+cos(b*x+a)^2)*sin(b*x+a
)+1/45*cos(b*x+a)^3+2/15*cos(b*x+a)-1/5*(b*x+a)*(8/3+cos(b*x+a)^4+4/3*cos(b*x+a)^2)*sin(b*x+a)-1/25*cos(b*x+a)
^5)+1/b^4*a^4*d^4*(-1/5*sin(b*x+a)*cos(b*x+a)^4+1/15*(2+cos(b*x+a)^2)*sin(b*x+a))-4/b^3*a^3*c*d^3*(-1/5*sin(b*
x+a)*cos(b*x+a)^4+1/15*(2+cos(b*x+a)^2)*sin(b*x+a))+6/b^2*a^2*c^2*d^2*(-1/5*sin(b*x+a)*cos(b*x+a)^4+1/15*(2+co
s(b*x+a)^2)*sin(b*x+a))-4/b*a*c^3*d*(-1/5*sin(b*x+a)*cos(b*x+a)^4+1/15*(2+cos(b*x+a)^2)*sin(b*x+a))+c^4*(-1/5*
sin(b*x+a)*cos(b*x+a)^4+1/15*(2+cos(b*x+a)^2)*sin(b*x+a)))

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Maxima [B]  time = 1.48933, 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)^3*sin(b*x+a)^2,x, algorithm="maxima")

[Out]

-1/4050000*(270000*(3*sin(b*x + a)^5 - 5*sin(b*x + a)^3)*c^4 - 1080000*(3*sin(b*x + a)^5 - 5*sin(b*x + a)^3)*a
*c^3*d/b + 1620000*(3*sin(b*x + a)^5 - 5*sin(b*x + a)^3)*a^2*c^2*d^2/b^2 - 1080000*(3*sin(b*x + a)^5 - 5*sin(b
*x + a)^3)*a^3*c*d^3/b^3 + 270000*(3*sin(b*x + a)^5 - 5*sin(b*x + a)^3)*a^4*d^4/b^4 + 4500*(45*(b*x + a)*sin(5
*b*x + 5*a) + 75*(b*x + a)*sin(3*b*x + 3*a) - 450*(b*x + a)*sin(b*x + a) + 9*cos(5*b*x + 5*a) + 25*cos(3*b*x +
3*a) - 450*cos(b*x + a))*c^3*d/b - 13500*(45*(b*x + a)*sin(5*b*x + 5*a) + 75*(b*x + a)*sin(3*b*x + 3*a) - 450
*(b*x + a)*sin(b*x + a) + 9*cos(5*b*x + 5*a) + 25*cos(3*b*x + 3*a) - 450*cos(b*x + a))*a*c^2*d^2/b^2 + 13500*(
45*(b*x + a)*sin(5*b*x + 5*a) + 75*(b*x + a)*sin(3*b*x + 3*a) - 450*(b*x + a)*sin(b*x + a) + 9*cos(5*b*x + 5*a
) + 25*cos(3*b*x + 3*a) - 450*cos(b*x + a))*a^2*c*d^3/b^3 - 4500*(45*(b*x + a)*sin(5*b*x + 5*a) + 75*(b*x + a)
*sin(3*b*x + 3*a) - 450*(b*x + a)*sin(b*x + a) + 9*cos(5*b*x + 5*a) + 25*cos(3*b*x + 3*a) - 450*cos(b*x + a))*
a^3*d^4/b^4 + 450*(270*(b*x + a)*cos(5*b*x + 5*a) + 750*(b*x + a)*cos(3*b*x + 3*a) - 13500*(b*x + a)*cos(b*x +
a) + 27*(25*(b*x + a)^2 - 2)*sin(5*b*x + 5*a) + 125*(9*(b*x + a)^2 - 2)*sin(3*b*x + 3*a) - 6750*((b*x + a)^2
- 2)*sin(b*x + a))*c^2*d^2/b^2 - 900*(270*(b*x + a)*cos(5*b*x + 5*a) + 750*(b*x + a)*cos(3*b*x + 3*a) - 13500*
(b*x + a)*cos(b*x + a) + 27*(25*(b*x + a)^2 - 2)*sin(5*b*x + 5*a) + 125*(9*(b*x + a)^2 - 2)*sin(3*b*x + 3*a) -
6750*((b*x + a)^2 - 2)*sin(b*x + a))*a*c*d^3/b^3 + 450*(270*(b*x + a)*cos(5*b*x + 5*a) + 750*(b*x + a)*cos(3*
b*x + 3*a) - 13500*(b*x + a)*cos(b*x + a) + 27*(25*(b*x + a)^2 - 2)*sin(5*b*x + 5*a) + 125*(9*(b*x + a)^2 - 2)
*sin(3*b*x + 3*a) - 6750*((b*x + a)^2 - 2)*sin(b*x + a))*a^2*d^4/b^4 + 60*(81*(25*(b*x + a)^2 - 2)*cos(5*b*x +
5*a) + 625*(9*(b*x + a)^2 - 2)*cos(3*b*x + 3*a) - 101250*((b*x + a)^2 - 2)*cos(b*x + a) + 135*(25*(b*x + a)^3
- 6*b*x - 6*a)*sin(5*b*x + 5*a) + 1875*(3*(b*x + a)^3 - 2*b*x - 2*a)*sin(3*b*x + 3*a) - 33750*((b*x + a)^3 -
6*b*x - 6*a)*sin(b*x + a))*c*d^3/b^3 - 60*(81*(25*(b*x + a)^2 - 2)*cos(5*b*x + 5*a) + 625*(9*(b*x + a)^2 - 2)*
cos(3*b*x + 3*a) - 101250*((b*x + a)^2 - 2)*cos(b*x + a) + 135*(25*(b*x + a)^3 - 6*b*x - 6*a)*sin(5*b*x + 5*a)
+ 1875*(3*(b*x + a)^3 - 2*b*x - 2*a)*sin(3*b*x + 3*a) - 33750*((b*x + a)^3 - 6*b*x - 6*a)*sin(b*x + a))*a*d^4
/b^4 + (1620*(25*(b*x + a)^3 - 6*b*x - 6*a)*cos(5*b*x + 5*a) + 37500*(3*(b*x + a)^3 - 2*b*x - 2*a)*cos(3*b*x +
3*a) - 2025000*((b*x + a)^3 - 6*b*x - 6*a)*cos(b*x + a) + 81*(625*(b*x + a)^4 - 300*(b*x + a)^2 + 24)*sin(5*b
*x + 5*a) + 3125*(27*(b*x + a)^4 - 36*(b*x + a)^2 + 8)*sin(3*b*x + 3*a) - 506250*((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 = 0.568267, size = 1245, normalized size = 3.77 \begin{align*} -\frac{1620 \,{\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 )^{5} - 300 \,{\left (75 \, b^{3} d^{4} x^{3} + 225 \, b^{3} c d^{3} x^{2} + 75 \, b^{3} c^{3} d + 22 \, b c d^{3} +{\left (225 \, b^{3} c^{2} d^{2} + 22 \, b d^{4}\right )} x\right )} \cos \left (b x + a\right )^{3} - 1800 \,{\left (75 \, b^{3} d^{4} x^{3} + 225 \, b^{3} c d^{3} x^{2} + 75 \, b^{3} c^{3} d - 428 \, b c d^{3} +{\left (225 \, b^{3} c^{2} d^{2} - 428 \, b d^{4}\right )} x\right )} \cos \left (b x + a\right ) -{\left (33750 \, b^{4} d^{4} x^{4} + 135000 \, b^{4} c d^{3} x^{3} + 33750 \, b^{4} c^{4} - 385200 \, b^{2} c^{2} d^{2} - 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 )^{4} + 760816 \, d^{4} + 900 \,{\left (225 \, b^{4} c^{2} d^{2} - 428 \, b^{2} d^{4}\right )} x^{2} +{\left (16875 \, b^{4} d^{4} x^{4} + 67500 \, b^{4} c d^{3} x^{3} + 16875 \, b^{4} c^{4} + 9900 \, b^{2} c^{2} d^{2} - 4792 \, d^{4} + 450 \,{\left (225 \, b^{4} c^{2} d^{2} + 22 \, b^{2} d^{4}\right )} x^{2} + 900 \,{\left (75 \, b^{4} c^{3} d + 22 \, b^{2} c d^{3}\right )} x\right )} \cos \left (b x + a\right )^{2} + 1800 \,{\left (75 \, b^{4} c^{3} d - 428 \, b^{2} c d^{3}\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)^3*sin(b*x+a)^2,x, algorithm="fricas")

[Out]

-1/253125*(1620*(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)^5 - 300*(75*b^3*d^4*x^3 + 225*b^3*c*d^3*x^2 + 75*b^3*c^3*d + 22*b*c*d^3 + (225*b^3*c^2*d^2 + 22
*b*d^4)*x)*cos(b*x + a)^3 - 1800*(75*b^3*d^4*x^3 + 225*b^3*c*d^3*x^2 + 75*b^3*c^3*d - 428*b*c*d^3 + (225*b^3*c
^2*d^2 - 428*b*d^4)*x)*cos(b*x + a) - (33750*b^4*d^4*x^4 + 135000*b^4*c*d^3*x^3 + 33750*b^4*c^4 - 385200*b^2*c
^2*d^2 - 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)^4 + 760816*d^4 + 900*(225*b^4*c^2*d^2 -
428*b^2*d^4)*x^2 + (16875*b^4*d^4*x^4 + 67500*b^4*c*d^3*x^3 + 16875*b^4*c^4 + 9900*b^2*c^2*d^2 - 4792*d^4 + 4
50*(225*b^4*c^2*d^2 + 22*b^2*d^4)*x^2 + 900*(75*b^4*c^3*d + 22*b^2*c*d^3)*x)*cos(b*x + a)^2 + 1800*(75*b^4*c^3
*d - 428*b^2*c*d^3)*x)*sin(b*x + a))/b^5

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Sympy [A]  time = 48.0104, 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)**3*sin(b*x+a)**2,x)

[Out]

Piecewise((2*c**4*sin(a + b*x)**5/(15*b) + c**4*sin(a + b*x)**3*cos(a + b*x)**2/(3*b) + 8*c**3*d*x*sin(a + b*x
)**5/(15*b) + 4*c**3*d*x*sin(a + b*x)**3*cos(a + b*x)**2/(3*b) + 4*c**2*d**2*x**2*sin(a + b*x)**5/(5*b) + 2*c*
*2*d**2*x**2*sin(a + b*x)**3*cos(a + b*x)**2/b + 8*c*d**3*x**3*sin(a + b*x)**5/(15*b) + 4*c*d**3*x**3*sin(a +
b*x)**3*cos(a + b*x)**2/(3*b) + 2*d**4*x**4*sin(a + b*x)**5/(15*b) + d**4*x**4*sin(a + b*x)**3*cos(a + b*x)**2
/(3*b) + 8*c**3*d*sin(a + b*x)**4*cos(a + b*x)/(15*b**2) + 52*c**3*d*sin(a + b*x)**2*cos(a + b*x)**3/(45*b**2)
+ 104*c**3*d*cos(a + b*x)**5/(225*b**2) + 8*c**2*d**2*x*sin(a + b*x)**4*cos(a + b*x)/(5*b**2) + 52*c**2*d**2*
x*sin(a + b*x)**2*cos(a + b*x)**3/(15*b**2) + 104*c**2*d**2*x*cos(a + b*x)**5/(75*b**2) + 8*c*d**3*x**2*sin(a
+ b*x)**4*cos(a + b*x)/(5*b**2) + 52*c*d**3*x**2*sin(a + b*x)**2*cos(a + b*x)**3/(15*b**2) + 104*c*d**3*x**2*c
os(a + b*x)**5/(75*b**2) + 8*d**4*x**3*sin(a + b*x)**4*cos(a + b*x)/(15*b**2) + 52*d**4*x**3*sin(a + b*x)**2*c
os(a + b*x)**3/(45*b**2) + 104*d**4*x**3*cos(a + b*x)**5/(225*b**2) - 1712*c**2*d**2*sin(a + b*x)**5/(1125*b**
3) - 676*c**2*d**2*sin(a + b*x)**3*cos(a + b*x)**2/(225*b**3) - 104*c**2*d**2*sin(a + b*x)*cos(a + b*x)**4/(75
*b**3) - 3424*c*d**3*x*sin(a + b*x)**5/(1125*b**3) - 1352*c*d**3*x*sin(a + b*x)**3*cos(a + b*x)**2/(225*b**3)
- 208*c*d**3*x*sin(a + b*x)*cos(a + b*x)**4/(75*b**3) - 1712*d**4*x**2*sin(a + b*x)**5/(1125*b**3) - 676*d**4*
x**2*sin(a + b*x)**3*cos(a + b*x)**2/(225*b**3) - 104*d**4*x**2*sin(a + b*x)*cos(a + b*x)**4/(75*b**3) - 3424*
c*d**3*sin(a + b*x)**4*cos(a + b*x)/(1125*b**4) - 20456*c*d**3*sin(a + b*x)**2*cos(a + b*x)**3/(3375*b**4) - 5
0272*c*d**3*cos(a + b*x)**5/(16875*b**4) - 3424*d**4*x*sin(a + b*x)**4*cos(a + b*x)/(1125*b**4) - 20456*d**4*x
*sin(a + b*x)**2*cos(a + b*x)**3/(3375*b**4) - 50272*d**4*x*cos(a + b*x)**5/(16875*b**4) + 760816*d**4*sin(a +
b*x)**5/(253125*b**5) + 303368*d**4*sin(a + b*x)**3*cos(a + b*x)**2/(50625*b**5) + 50272*d**4*sin(a + b*x)*co
s(a + b*x)**4/(16875*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)**3, True))

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Giac [A]  time = 1.14666, size = 717, normalized size = 2.17 \begin{align*} -\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 )} \cos \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 )} \cos \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 )} \cos \left (b x + a\right )}{2 \, b^{5}} - \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 )} \sin \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 )} \sin \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 )} \sin \left (b x + 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)^2,x, algorithm="giac")

[Out]

-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)*cos(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)
*cos(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)*cos(b*x + a)/b^5 - 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)*sin(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)*sin(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)*sin(b*x + a)/b^5