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

Optimal. Leaf size=259 \[ \frac{3 d^2 (c+d x) \cos (a+b x)}{4 b^3}+\frac{d^2 (c+d x) \cos (3 a+3 b x)}{72 b^3}-\frac{3 d^2 (c+d x) \cos (5 a+5 b x)}{1000 b^3}+\frac{3 d (c+d x)^2 \sin (a+b x)}{8 b^2}+\frac{d (c+d x)^2 \sin (3 a+3 b x)}{48 b^2}-\frac{3 d (c+d x)^2 \sin (5 a+5 b x)}{400 b^2}-\frac{3 d^3 \sin (a+b x)}{4 b^4}-\frac{d^3 \sin (3 a+3 b x)}{216 b^4}+\frac{3 d^3 \sin (5 a+5 b x)}{5000 b^4}-\frac{(c+d x)^3 \cos (a+b x)}{8 b}-\frac{(c+d x)^3 \cos (3 a+3 b x)}{48 b}+\frac{(c+d x)^3 \cos (5 a+5 b x)}{80 b} \]

[Out]

(3*d^2*(c + d*x)*Cos[a + b*x])/(4*b^3) - ((c + d*x)^3*Cos[a + b*x])/(8*b) + (d^2*(c + d*x)*Cos[3*a + 3*b*x])/(
72*b^3) - ((c + d*x)^3*Cos[3*a + 3*b*x])/(48*b) - (3*d^2*(c + d*x)*Cos[5*a + 5*b*x])/(1000*b^3) + ((c + d*x)^3
*Cos[5*a + 5*b*x])/(80*b) - (3*d^3*Sin[a + b*x])/(4*b^4) + (3*d*(c + d*x)^2*Sin[a + b*x])/(8*b^2) - (d^3*Sin[3
*a + 3*b*x])/(216*b^4) + (d*(c + d*x)^2*Sin[3*a + 3*b*x])/(48*b^2) + (3*d^3*Sin[5*a + 5*b*x])/(5000*b^4) - (3*
d*(c + d*x)^2*Sin[5*a + 5*b*x])/(400*b^2)

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Rubi [A]  time = 0.279157, antiderivative size = 259, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {4406, 3296, 2637} \[ \frac{3 d^2 (c+d x) \cos (a+b x)}{4 b^3}+\frac{d^2 (c+d x) \cos (3 a+3 b x)}{72 b^3}-\frac{3 d^2 (c+d x) \cos (5 a+5 b x)}{1000 b^3}+\frac{3 d (c+d x)^2 \sin (a+b x)}{8 b^2}+\frac{d (c+d x)^2 \sin (3 a+3 b x)}{48 b^2}-\frac{3 d (c+d x)^2 \sin (5 a+5 b x)}{400 b^2}-\frac{3 d^3 \sin (a+b x)}{4 b^4}-\frac{d^3 \sin (3 a+3 b x)}{216 b^4}+\frac{3 d^3 \sin (5 a+5 b x)}{5000 b^4}-\frac{(c+d x)^3 \cos (a+b x)}{8 b}-\frac{(c+d x)^3 \cos (3 a+3 b x)}{48 b}+\frac{(c+d x)^3 \cos (5 a+5 b x)}{80 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*d^2*(c + d*x)*Cos[a + b*x])/(4*b^3) - ((c + d*x)^3*Cos[a + b*x])/(8*b) + (d^2*(c + d*x)*Cos[3*a + 3*b*x])/(
72*b^3) - ((c + d*x)^3*Cos[3*a + 3*b*x])/(48*b) - (3*d^2*(c + d*x)*Cos[5*a + 5*b*x])/(1000*b^3) + ((c + d*x)^3
*Cos[5*a + 5*b*x])/(80*b) - (3*d^3*Sin[a + b*x])/(4*b^4) + (3*d*(c + d*x)^2*Sin[a + b*x])/(8*b^2) - (d^3*Sin[3
*a + 3*b*x])/(216*b^4) + (d*(c + d*x)^2*Sin[3*a + 3*b*x])/(48*b^2) + (3*d^3*Sin[5*a + 5*b*x])/(5000*b^4) - (3*
d*(c + d*x)^2*Sin[5*a + 5*b*x])/(400*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 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)^3 \cos ^2(a+b x) \sin ^3(a+b x) \, dx &=\int \left (\frac{1}{8} (c+d x)^3 \sin (a+b x)+\frac{1}{16} (c+d x)^3 \sin (3 a+3 b x)-\frac{1}{16} (c+d x)^3 \sin (5 a+5 b x)\right ) \, dx\\ &=\frac{1}{16} \int (c+d x)^3 \sin (3 a+3 b x) \, dx-\frac{1}{16} \int (c+d x)^3 \sin (5 a+5 b x) \, dx+\frac{1}{8} \int (c+d x)^3 \sin (a+b x) \, dx\\ &=-\frac{(c+d x)^3 \cos (a+b x)}{8 b}-\frac{(c+d x)^3 \cos (3 a+3 b x)}{48 b}+\frac{(c+d x)^3 \cos (5 a+5 b x)}{80 b}-\frac{(3 d) \int (c+d x)^2 \cos (5 a+5 b x) \, dx}{80 b}+\frac{d \int (c+d x)^2 \cos (3 a+3 b x) \, dx}{16 b}+\frac{(3 d) \int (c+d x)^2 \cos (a+b x) \, dx}{8 b}\\ &=-\frac{(c+d x)^3 \cos (a+b x)}{8 b}-\frac{(c+d x)^3 \cos (3 a+3 b x)}{48 b}+\frac{(c+d x)^3 \cos (5 a+5 b x)}{80 b}+\frac{3 d (c+d x)^2 \sin (a+b x)}{8 b^2}+\frac{d (c+d x)^2 \sin (3 a+3 b x)}{48 b^2}-\frac{3 d (c+d x)^2 \sin (5 a+5 b x)}{400 b^2}+\frac{\left (3 d^2\right ) \int (c+d x) \sin (5 a+5 b x) \, dx}{200 b^2}-\frac{d^2 \int (c+d x) \sin (3 a+3 b x) \, dx}{24 b^2}-\frac{\left (3 d^2\right ) \int (c+d x) \sin (a+b x) \, dx}{4 b^2}\\ &=\frac{3 d^2 (c+d x) \cos (a+b x)}{4 b^3}-\frac{(c+d x)^3 \cos (a+b x)}{8 b}+\frac{d^2 (c+d x) \cos (3 a+3 b x)}{72 b^3}-\frac{(c+d x)^3 \cos (3 a+3 b x)}{48 b}-\frac{3 d^2 (c+d x) \cos (5 a+5 b x)}{1000 b^3}+\frac{(c+d x)^3 \cos (5 a+5 b x)}{80 b}+\frac{3 d (c+d x)^2 \sin (a+b x)}{8 b^2}+\frac{d (c+d x)^2 \sin (3 a+3 b x)}{48 b^2}-\frac{3 d (c+d x)^2 \sin (5 a+5 b x)}{400 b^2}+\frac{\left (3 d^3\right ) \int \cos (5 a+5 b x) \, dx}{1000 b^3}-\frac{d^3 \int \cos (3 a+3 b x) \, dx}{72 b^3}-\frac{\left (3 d^3\right ) \int \cos (a+b x) \, dx}{4 b^3}\\ &=\frac{3 d^2 (c+d x) \cos (a+b x)}{4 b^3}-\frac{(c+d x)^3 \cos (a+b x)}{8 b}+\frac{d^2 (c+d x) \cos (3 a+3 b x)}{72 b^3}-\frac{(c+d x)^3 \cos (3 a+3 b x)}{48 b}-\frac{3 d^2 (c+d x) \cos (5 a+5 b x)}{1000 b^3}+\frac{(c+d x)^3 \cos (5 a+5 b x)}{80 b}-\frac{3 d^3 \sin (a+b x)}{4 b^4}+\frac{3 d (c+d x)^2 \sin (a+b x)}{8 b^2}-\frac{d^3 \sin (3 a+3 b x)}{216 b^4}+\frac{d (c+d x)^2 \sin (3 a+3 b x)}{48 b^2}+\frac{3 d^3 \sin (5 a+5 b x)}{5000 b^4}-\frac{3 d (c+d x)^2 \sin (5 a+5 b x)}{400 b^2}\\ \end{align*}

Mathematica [A]  time = 1.59901, size = 369, normalized size = 1.42 \[ \frac{101250 b^2 c^2 d \sin (a+b x)+5625 b^2 c^2 d \sin (3 (a+b x))-2025 b^2 c^2 d \sin (5 (a+b x))+10125 b^3 c^2 d x \cos (5 (a+b x))+3375 b^3 c^3 \cos (5 (a+b x))+10125 b^3 c d^2 x^2 \cos (5 (a+b x))+202500 b^2 c d^2 x \sin (a+b x)+11250 b^2 c d^2 x \sin (3 (a+b x))-4050 b^2 c d^2 x \sin (5 (a+b x))-33750 b (c+d x) \cos (a+b x) \left (b^2 (c+d x)^2-6 d^2\right )-1875 b (c+d x) \cos (3 (a+b x)) \left (3 b^2 (c+d x)^2-2 d^2\right )+101250 b^2 d^3 x^2 \sin (a+b x)+5625 b^2 d^3 x^2 \sin (3 (a+b x))-2025 b^2 d^3 x^2 \sin (5 (a+b x))+3375 b^3 d^3 x^3 \cos (5 (a+b x))-810 b c d^2 \cos (5 (a+b x))-202500 d^3 \sin (a+b x)-1250 d^3 \sin (3 (a+b x))+162 d^3 \sin (5 (a+b x))-810 b d^3 x \cos (5 (a+b x))}{270000 b^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-33750*b*(c + d*x)*(-6*d^2 + b^2*(c + d*x)^2)*Cos[a + b*x] - 1875*b*(c + d*x)*(-2*d^2 + 3*b^2*(c + d*x)^2)*Co
s[3*(a + b*x)] + 3375*b^3*c^3*Cos[5*(a + b*x)] - 810*b*c*d^2*Cos[5*(a + b*x)] + 10125*b^3*c^2*d*x*Cos[5*(a + b
*x)] - 810*b*d^3*x*Cos[5*(a + b*x)] + 10125*b^3*c*d^2*x^2*Cos[5*(a + b*x)] + 3375*b^3*d^3*x^3*Cos[5*(a + b*x)]
+ 101250*b^2*c^2*d*Sin[a + b*x] - 202500*d^3*Sin[a + b*x] + 202500*b^2*c*d^2*x*Sin[a + b*x] + 101250*b^2*d^3*
x^2*Sin[a + b*x] + 5625*b^2*c^2*d*Sin[3*(a + b*x)] - 1250*d^3*Sin[3*(a + b*x)] + 11250*b^2*c*d^2*x*Sin[3*(a +
b*x)] + 5625*b^2*d^3*x^2*Sin[3*(a + b*x)] - 2025*b^2*c^2*d*Sin[5*(a + b*x)] + 162*d^3*Sin[5*(a + b*x)] - 4050*
b^2*c*d^2*x*Sin[5*(a + b*x)] - 2025*b^2*d^3*x^2*Sin[5*(a + b*x)])/(270000*b^4)

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Maple [B]  time = 0.024, size = 992, normalized size = 3.8 \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)^3*cos(b*x+a)^2*sin(b*x+a)^3,x)

[Out]

1/b*(1/b^3*d^3*(-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)-3/b^3*a*d^3*(-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*si
n(b*x+a)^2)*cos(b*x+a))+3/b^2*c*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)*s
in(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))+3/b^3*a^2
*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)-6/b^2*a*c*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)+3/b
*c^2*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^3*a^3*d^3*(-1/5*sin(b*x+a)^2*cos(b*x+a)^3-2/15*cos(b*x+
a)^3)+3/b^2*a^2*c*d^2*(-1/5*sin(b*x+a)^2*cos(b*x+a)^3-2/15*cos(b*x+a)^3)-3/b*a*c^2*d*(-1/5*sin(b*x+a)^2*cos(b*
x+a)^3-2/15*cos(b*x+a)^3)+c^3*(-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 = 1.38457, size = 1034, normalized size = 3.99 \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)^3*cos(b*x+a)^2*sin(b*x+a)^3,x, algorithm="maxima")

[Out]

1/270000*(18000*(3*cos(b*x + a)^5 - 5*cos(b*x + a)^3)*c^3 - 54000*(3*cos(b*x + a)^5 - 5*cos(b*x + a)^3)*a*c^2*
d/b + 54000*(3*cos(b*x + a)^5 - 5*cos(b*x + a)^3)*a^2*c*d^2/b^2 - 18000*(3*cos(b*x + a)^5 - 5*cos(b*x + a)^3)*
a^3*d^3/b^3 + 225*(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^2*d/b - 450*(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*si
n(b*x + a))*a*c*d^2/b^2 + 225*(45*(b*x + a)*cos(5*b*x + 5*a) - 75*(b*x + a)*cos(3*b*x + 3*a) - 450*(b*x + a)*c
os(b*x + a) - 9*sin(5*b*x + 5*a) + 25*sin(3*b*x + 3*a) + 450*sin(b*x + a))*a^2*d^3/b^3 + 15*(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*d^2/b^2 - 15*(2
7*(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)*co
s(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
*d^3/b^3 + (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))*d^3/b^3)/b

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Fricas [A]  time = 0.538011, size = 706, normalized size = 2.73 \begin{align*} \frac{135 \,{\left (25 \, b^{3} d^{3} x^{3} + 75 \, b^{3} c d^{2} x^{2} + 25 \, b^{3} c^{3} - 6 \, b c d^{2} + 3 \,{\left (25 \, b^{3} c^{2} d - 2 \, b d^{3}\right )} x\right )} \cos \left (b x + a\right )^{5} - 75 \,{\left (75 \, b^{3} d^{3} x^{3} + 225 \, b^{3} c d^{2} x^{2} + 75 \, b^{3} c^{3} - 26 \, b c d^{2} +{\left (225 \, b^{3} c^{2} d - 26 \, b d^{3}\right )} x\right )} \cos \left (b x + a\right )^{3} + 11700 \,{\left (b d^{3} x + b c d^{2}\right )} \cos \left (b x + a\right ) +{\left (5850 \, b^{2} d^{3} x^{2} + 11700 \, b^{2} c d^{2} x + 5850 \, b^{2} c^{2} d - 81 \,{\left (25 \, b^{2} d^{3} x^{2} + 50 \, b^{2} c d^{2} x + 25 \, b^{2} c^{2} d - 2 \, d^{3}\right )} \cos \left (b x + a\right )^{4} - 12568 \, d^{3} +{\left (2925 \, b^{2} d^{3} x^{2} + 5850 \, b^{2} c d^{2} x + 2925 \, b^{2} c^{2} d - 434 \, d^{3}\right )} \cos \left (b x + a\right )^{2}\right )} \sin \left (b x + a\right )}{16875 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3*cos(b*x+a)^2*sin(b*x+a)^3,x, algorithm="fricas")

[Out]

1/16875*(135*(25*b^3*d^3*x^3 + 75*b^3*c*d^2*x^2 + 25*b^3*c^3 - 6*b*c*d^2 + 3*(25*b^3*c^2*d - 2*b*d^3)*x)*cos(b
*x + a)^5 - 75*(75*b^3*d^3*x^3 + 225*b^3*c*d^2*x^2 + 75*b^3*c^3 - 26*b*c*d^2 + (225*b^3*c^2*d - 26*b*d^3)*x)*c
os(b*x + a)^3 + 11700*(b*d^3*x + b*c*d^2)*cos(b*x + a) + (5850*b^2*d^3*x^2 + 11700*b^2*c*d^2*x + 5850*b^2*c^2*
d - 81*(25*b^2*d^3*x^2 + 50*b^2*c*d^2*x + 25*b^2*c^2*d - 2*d^3)*cos(b*x + a)^4 - 12568*d^3 + (2925*b^2*d^3*x^2
+ 5850*b^2*c*d^2*x + 2925*b^2*c^2*d - 434*d^3)*cos(b*x + a)^2)*sin(b*x + a))/b^4

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Sympy [A]  time = 23.6664, size = 690, normalized size = 2.66 \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)**3*cos(b*x+a)**2*sin(b*x+a)**3,x)

[Out]

Piecewise((-c**3*sin(a + b*x)**2*cos(a + b*x)**3/(3*b) - 2*c**3*cos(a + b*x)**5/(15*b) - c**2*d*x*sin(a + b*x)
**2*cos(a + b*x)**3/b - 2*c**2*d*x*cos(a + b*x)**5/(5*b) - c*d**2*x**2*sin(a + b*x)**2*cos(a + b*x)**3/b - 2*c
*d**2*x**2*cos(a + b*x)**5/(5*b) - d**3*x**3*sin(a + b*x)**2*cos(a + b*x)**3/(3*b) - 2*d**3*x**3*cos(a + b*x)*
*5/(15*b) + 26*c**2*d*sin(a + b*x)**5/(75*b**2) + 13*c**2*d*sin(a + b*x)**3*cos(a + b*x)**2/(15*b**2) + 2*c**2
*d*sin(a + b*x)*cos(a + b*x)**4/(5*b**2) + 52*c*d**2*x*sin(a + b*x)**5/(75*b**2) + 26*c*d**2*x*sin(a + b*x)**3
*cos(a + b*x)**2/(15*b**2) + 4*c*d**2*x*sin(a + b*x)*cos(a + b*x)**4/(5*b**2) + 26*d**3*x**2*sin(a + b*x)**5/(
75*b**2) + 13*d**3*x**2*sin(a + b*x)**3*cos(a + b*x)**2/(15*b**2) + 2*d**3*x**2*sin(a + b*x)*cos(a + b*x)**4/(
5*b**2) + 52*c*d**2*sin(a + b*x)**4*cos(a + b*x)/(75*b**3) + 338*c*d**2*sin(a + b*x)**2*cos(a + b*x)**3/(225*b
**3) + 856*c*d**2*cos(a + b*x)**5/(1125*b**3) + 52*d**3*x*sin(a + b*x)**4*cos(a + b*x)/(75*b**3) + 338*d**3*x*
sin(a + b*x)**2*cos(a + b*x)**3/(225*b**3) + 856*d**3*x*cos(a + b*x)**5/(1125*b**3) - 12568*d**3*sin(a + b*x)*
*5/(16875*b**4) - 5114*d**3*sin(a + b*x)**3*cos(a + b*x)**2/(3375*b**4) - 856*d**3*sin(a + b*x)*cos(a + b*x)**
4/(1125*b**4), Ne(b, 0)), ((c**3*x + 3*c**2*d*x**2/2 + c*d**2*x**3 + d**3*x**4/4)*sin(a)**3*cos(a)**2, True))

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Giac [A]  time = 1.11185, size = 474, normalized size = 1.83 \begin{align*} \frac{{\left (25 \, b^{3} d^{3} x^{3} + 75 \, b^{3} c d^{2} x^{2} + 75 \, b^{3} c^{2} d x + 25 \, b^{3} c^{3} - 6 \, b d^{3} x - 6 \, b c d^{2}\right )} \cos \left (5 \, b x + 5 \, a\right )}{2000 \, b^{4}} - \frac{{\left (3 \, b^{3} d^{3} x^{3} + 9 \, b^{3} c d^{2} x^{2} + 9 \, b^{3} c^{2} d x + 3 \, b^{3} c^{3} - 2 \, b d^{3} x - 2 \, b c d^{2}\right )} \cos \left (3 \, b x + 3 \, a\right )}{144 \, b^{4}} - \frac{{\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + b^{3} c^{3} - 6 \, b d^{3} x - 6 \, b c d^{2}\right )} \cos \left (b x + a\right )}{8 \, b^{4}} - \frac{3 \,{\left (25 \, b^{2} d^{3} x^{2} + 50 \, b^{2} c d^{2} x + 25 \, b^{2} c^{2} d - 2 \, d^{3}\right )} \sin \left (5 \, b x + 5 \, a\right )}{10000 \, b^{4}} + \frac{{\left (9 \, b^{2} d^{3} x^{2} + 18 \, b^{2} c d^{2} x + 9 \, b^{2} c^{2} d - 2 \, d^{3}\right )} \sin \left (3 \, b x + 3 \, a\right )}{432 \, b^{4}} + \frac{3 \,{\left (b^{2} d^{3} x^{2} + 2 \, b^{2} c d^{2} x + b^{2} c^{2} d - 2 \, d^{3}\right )} \sin \left (b x + a\right )}{8 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3*cos(b*x+a)^2*sin(b*x+a)^3,x, algorithm="giac")

[Out]

1/2000*(25*b^3*d^3*x^3 + 75*b^3*c*d^2*x^2 + 75*b^3*c^2*d*x + 25*b^3*c^3 - 6*b*d^3*x - 6*b*c*d^2)*cos(5*b*x + 5
*a)/b^4 - 1/144*(3*b^3*d^3*x^3 + 9*b^3*c*d^2*x^2 + 9*b^3*c^2*d*x + 3*b^3*c^3 - 2*b*d^3*x - 2*b*c*d^2)*cos(3*b*
x + 3*a)/b^4 - 1/8*(b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 + 3*b^3*c^2*d*x + b^3*c^3 - 6*b*d^3*x - 6*b*c*d^2)*cos(b*x +
a)/b^4 - 3/10000*(25*b^2*d^3*x^2 + 50*b^2*c*d^2*x + 25*b^2*c^2*d - 2*d^3)*sin(5*b*x + 5*a)/b^4 + 1/432*(9*b^2
*d^3*x^2 + 18*b^2*c*d^2*x + 9*b^2*c^2*d - 2*d^3)*sin(3*b*x + 3*a)/b^4 + 3/8*(b^2*d^3*x^2 + 2*b^2*c*d^2*x + b^2
*c^2*d - 2*d^3)*sin(b*x + a)/b^4