\(\int (c+d x)^4 \cos ^3(a+b x) \sin ^2(a+b x) \, dx\) [146]

Optimal result

Integrand size = 24, antiderivative size = 330 \[ \int (c+d x)^4 \cos ^3(a+b x) \sin ^2(a+b x) \, dx=-\frac {3 d^3 (c+d x) \cos (a+b x)}{b^4}+\frac {d (c+d x)^3 \cos (a+b x)}{2 b^2}+\frac {d^3 (c+d x) \cos (3 a+3 b x)}{54 b^4}-\frac {d (c+d x)^3 \cos (3 a+3 b x)}{36 b^2}+\frac {3 d^3 (c+d x) \cos (5 a+5 b x)}{1250 b^4}-\frac {d (c+d x)^3 \cos (5 a+5 b x)}{100 b^2}+\frac {3 d^4 \sin (a+b x)}{b^5}-\frac {3 d^2 (c+d x)^2 \sin (a+b x)}{2 b^3}+\frac {(c+d x)^4 \sin (a+b x)}{8 b}-\frac {d^4 \sin (3 a+3 b x)}{162 b^5}+\frac {d^2 (c+d x)^2 \sin (3 a+3 b x)}{36 b^3}-\frac {(c+d x)^4 \sin (3 a+3 b x)}{48 b}-\frac {3 d^4 \sin (5 a+5 b x)}{6250 b^5}+\frac {3 d^2 (c+d x)^2 \sin (5 a+5 b x)}{500 b^3}-\frac {(c+d x)^4 \sin (5 a+5 b x)}{80 b} \]

[Out]

-3*d^3*(d*x+c)*cos(b*x+a)/b^4+1/2*d*(d*x+c)^3*cos(b*x+a)/b^2+1/54*d^3*(d*x+c)*cos(3*b*x+3*a)/b^4-1/36*d*(d*x+c
)^3*cos(3*b*x+3*a)/b^2+3/1250*d^3*(d*x+c)*cos(5*b*x+5*a)/b^4-1/100*d*(d*x+c)^3*cos(5*b*x+5*a)/b^2+3*d^4*sin(b*
x+a)/b^5-3/2*d^2*(d*x+c)^2*sin(b*x+a)/b^3+1/8*(d*x+c)^4*sin(b*x+a)/b-1/162*d^4*sin(3*b*x+3*a)/b^5+1/36*d^2*(d*
x+c)^2*sin(3*b*x+3*a)/b^3-1/48*(d*x+c)^4*sin(3*b*x+3*a)/b-3/6250*d^4*sin(5*b*x+5*a)/b^5+3/500*d^2*(d*x+c)^2*si
n(5*b*x+5*a)/b^3-1/80*(d*x+c)^4*sin(5*b*x+5*a)/b

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 330, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {4491, 3377, 2717} \[ \int (c+d x)^4 \cos ^3(a+b x) \sin ^2(a+b x) \, dx=\frac {3 d^4 \sin (a+b x)}{b^5}-\frac {d^4 \sin (3 a+3 b x)}{162 b^5}-\frac {3 d^4 \sin (5 a+5 b x)}{6250 b^5}-\frac {3 d^3 (c+d x) \cos (a+b x)}{b^4}+\frac {d^3 (c+d x) \cos (3 a+3 b x)}{54 b^4}+\frac {3 d^3 (c+d x) \cos (5 a+5 b x)}{1250 b^4}-\frac {3 d^2 (c+d x)^2 \sin (a+b x)}{2 b^3}+\frac {d^2 (c+d x)^2 \sin (3 a+3 b x)}{36 b^3}+\frac {3 d^2 (c+d x)^2 \sin (5 a+5 b x)}{500 b^3}+\frac {d (c+d x)^3 \cos (a+b x)}{2 b^2}-\frac {d (c+d x)^3 \cos (3 a+3 b x)}{36 b^2}-\frac {d (c+d x)^3 \cos (5 a+5 b x)}{100 b^2}+\frac {(c+d x)^4 \sin (a+b x)}{8 b}-\frac {(c+d x)^4 \sin (3 a+3 b x)}{48 b}-\frac {(c+d x)^4 \sin (5 a+5 b x)}{80 b} \]

[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 2717

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3377

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 4491

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]

Rubi steps \begin{align*} \text {integral}& = \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] (verified)

Time = 3.71 (sec) , antiderivative size = 563, normalized size of antiderivative = 1.71 \[ \int (c+d x)^4 \cos ^3(a+b x) \sin ^2(a+b x) \, dx=-\frac {-506250 b^4 c^4 \sin (a+b x)-2025000 b^3 c^3 d (\cos (a+b x)+b x \sin (a+b x))-2025000 b c d^3 \left (3 \left (-2+b^2 x^2\right ) \cos (a+b x)+b x \left (-6+b^2 x^2\right ) \sin (a+b x)\right )-3037500 b^2 c^2 d^2 \left (2 b x \cos (a+b x)+\left (-2+b^2 x^2\right ) \sin (a+b x)\right )-506250 d^4 \left (4 b x \left (-6+b^2 x^2\right ) \cos (a+b x)+\left (24-12 b^2 x^2+b^4 x^4\right ) \sin (a+b x)\right )+84375 b^4 c^4 \sin (3 (a+b x))+112500 b^3 c^3 d (\cos (3 (a+b x))+3 b x \sin (3 (a+b x)))+37500 b c d^3 \left (\left (-2+9 b^2 x^2\right ) \cos (3 (a+b x))+3 b x \left (-2+3 b^2 x^2\right ) \sin (3 (a+b x))\right )+56250 b^2 c^2 d^2 \left (6 b x \cos (3 (a+b x))+\left (-2+9 b^2 x^2\right ) \sin (3 (a+b x))\right )+3125 d^4 \left (12 b x \left (-2+3 b^2 x^2\right ) \cos (3 (a+b x))+\left (8-36 b^2 x^2+27 b^4 x^4\right ) \sin (3 (a+b x))\right )+50625 b^4 c^4 \sin (5 (a+b x))+40500 b^3 c^3 d (\cos (5 (a+b x))+5 b x \sin (5 (a+b x)))+1620 b c d^3 \left (\left (-6+75 b^2 x^2\right ) \cos (5 (a+b x))+5 b x \left (-6+25 b^2 x^2\right ) \sin (5 (a+b x))\right )+12150 b^2 c^2 d^2 \left (10 b x \cos (5 (a+b x))+\left (-2+25 b^2 x^2\right ) \sin (5 (a+b x))\right )+81 d^4 \left (20 b x \left (-6+25 b^2 x^2\right ) \cos (5 (a+b x))+\left (24-300 b^2 x^2+625 b^4 x^4\right ) \sin (5 (a+b x))\right )}{4050000 b^5} \]

[In]

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

[Out]

-1/4050000*(-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)]))/b^5

Maple [A] (verified)

Time = 3.18 (sec) , antiderivative size = 253, normalized size of antiderivative = 0.77

method	result	size
parallelrisch	\(\frac {\left (-84375 b^{4} \left (d x +c \right )^{4}+112500 d^{2} \left (d x +c \right )^{2} b^{2}-25000 d^{4}\right ) \sin \left (3 x b +3 a \right )+\left (-50625 b^{4} \left (d x +c \right )^{4}+24300 d^{2} \left (d x +c \right )^{2} b^{2}-1944 d^{4}\right ) \sin \left (5 x b +5 a \right )-112500 b \left (\left (d x +c \right )^{2} b^{2}-\frac {2 d^{2}}{3}\right ) d \left (d x +c \right ) \cos \left (3 x b +3 a \right )-40500 b \left (\left (d x +c \right )^{2} b^{2}-\frac {6 d^{2}}{25}\right ) d \left (d x +c \right ) \cos \left (5 x b +5 a \right )+\left (506250 b^{4} \left (d x +c \right )^{4}-6075000 d^{2} \left (d x +c \right )^{2} b^{2}[phone] d^{4}\right ) \sin \left (x b +a \right )+2025000 b d \left (\left (\left (d x +c \right )^{2} b^{2}-6 d^{2}\right ) \left (d x +c \right ) \cos \left (x b +a \right )-\frac {208 c \left (b^{2} c^{2}-\frac {6284 d^{2}}{975}\right )}{225}\right )}{4050000 b^{5}}\)	\(253\)
risch	\(\frac {d \left (b^{2} d^{3} x^{3}+3 b^{2} c \,d^{2} x^{2}+3 b^{2} c^{2} d x +b^{2} c^{3}-6 d^{3} x -6 c \,d^{2}\right ) \cos \left (x b +a \right )}{2 b^{4}}+\frac {\left (d^{4} x^{4} b^{4}+4 b^{4} c \,d^{3} x^{3}+6 b^{4} c^{2} d^{2} x^{2}+4 b^{4} c^{3} d x +b^{4} c^{4}-12 b^{2} d^{4} x^{2}-24 b^{2} c \,d^{3} x -12 b^{2} c^{2} d^{2}+24 d^{4}\right ) \sin \left (x b +a \right )}{8 b^{5}}-\frac {d \left (25 b^{2} d^{3} x^{3}+75 b^{2} c \,d^{2} x^{2}+75 b^{2} c^{2} d x +25 b^{2} c^{3}-6 d^{3} x -6 c \,d^{2}\right ) \cos \left (5 x b +5 a \right )}{2500 b^{4}}-\frac {\left (625 d^{4} x^{4} b^{4}+2500 b^{4} c \,d^{3} x^{3}+3750 b^{4} c^{2} d^{2} x^{2}+2500 b^{4} c^{3} d x +625 b^{4} c^{4}-300 b^{2} d^{4} x^{2}-600 b^{2} c \,d^{3} x -300 b^{2} c^{2} d^{2}+24 d^{4}\right ) \sin \left (5 x b +5 a \right )}{50000 b^{5}}-\frac {d \left (3 b^{2} d^{3} x^{3}+9 b^{2} c \,d^{2} x^{2}+9 b^{2} c^{2} d x +3 b^{2} c^{3}-2 d^{3} x -2 c \,d^{2}\right ) \cos \left (3 x b +3 a \right )}{108 b^{4}}-\frac {\left (27 d^{4} x^{4} b^{4}+108 b^{4} c \,d^{3} x^{3}+162 b^{4} c^{2} d^{2} x^{2}+108 b^{4} c^{3} d x +27 b^{4} c^{4}-36 b^{2} d^{4} x^{2}-72 b^{2} c \,d^{3} x -36 b^{2} c^{2} d^{2}+8 d^{4}\right ) \sin \left (3 x b +3 a \right )}{1296 b^{5}}\)	\(520\)
derivativedivides	\(\text {Expression too large to display}\)	\(1842\)
default	\(\text {Expression too large to display}\)	\(1842\)

[In]

int((d*x+c)^4*cos(b*x+a)^3*sin(b*x+a)^2,x,method=_RETURNVERBOSE)

[Out]

1/4050000*((-84375*b^4*(d*x+c)^4+112500*d^2*(d*x+c)^2*b^2-25000*d^4)*sin(3*b*x+3*a)+(-50625*b^4*(d*x+c)^4+2430
0*d^2*(d*x+c)^2*b^2-1944*d^4)*sin(5*b*x+5*a)-112500*b*((d*x+c)^2*b^2-2/3*d^2)*d*(d*x+c)*cos(3*b*x+3*a)-40500*b
*((d*x+c)^2*b^2-6/25*d^2)*d*(d*x+c)*cos(5*b*x+5*a)+(506250*b^4*(d*x+c)^4-6075000*d^2*(d*x+c)^2*b^2+12150000*d^
4)*sin(b*x+a)+2025000*b*d*(((d*x+c)^2*b^2-6*d^2)*(d*x+c)*cos(b*x+a)-208/225*c*(b^2*c^2-6284/975*d^2)))/b^5

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 527, normalized size of antiderivative = 1.60 \[ \int (c+d x)^4 \cos ^3(a+b x) \sin ^2(a+b x) \, dx=-\frac {1620 \, {\left (25 \, b^{3} d^{4} x^{3} + 75 \, b^{3} c d^{3} x^{2} + 25 \, b^{3} c^{3} d - 6 \, b c d^{3} + 3 \, {\left (25 \, b^{3} c^{2} d^{2} - 2 \, b d^{4}\right )} x\right )} \cos \left (b x + a\right )^{5} - 300 \, {\left (75 \, b^{3} d^{4} x^{3} + 225 \, b^{3} c d^{3} x^{2} + 75 \, b^{3} c^{3} d + 22 \, b c d^{3} + {\left (225 \, b^{3} c^{2} d^{2} + 22 \, b d^{4}\right )} x\right )} \cos \left (b x + a\right )^{3} - 1800 \, {\left (75 \, b^{3} d^{4} x^{3} + 225 \, b^{3} c d^{3} x^{2} + 75 \, b^{3} c^{3} d - 428 \, b c d^{3} + {\left (225 \, b^{3} c^{2} d^{2} - 428 \, b d^{4}\right )} x\right )} \cos \left (b x + a\right ) - {\left (33750 \, b^{4} d^{4} x^{4} + 135000 \, b^{4} c d^{3} x^{3} + 33750 \, b^{4} c^{4} - 385200 \, b^{2} c^{2} d^{2} - 81 \, {\left (625 \, b^{4} d^{4} x^{4} + 2500 \, b^{4} c d^{3} x^{3} + 625 \, b^{4} c^{4} - 300 \, b^{2} c^{2} d^{2} + 24 \, d^{4} + 150 \, {\left (25 \, b^{4} c^{2} d^{2} - 2 \, b^{2} d^{4}\right )} x^{2} + 100 \, {\left (25 \, b^{4} c^{3} d - 6 \, b^{2} c d^{3}\right )} x\right )} \cos \left (b x + a\right )^{4} + 760816 \, d^{4} + 900 \, {\left (225 \, b^{4} c^{2} d^{2} - 428 \, b^{2} d^{4}\right )} x^{2} + {\left (16875 \, b^{4} d^{4} x^{4} + 67500 \, b^{4} c d^{3} x^{3} + 16875 \, b^{4} c^{4} + 9900 \, b^{2} c^{2} d^{2} - 4792 \, d^{4} + 450 \, {\left (225 \, b^{4} c^{2} d^{2} + 22 \, b^{2} d^{4}\right )} x^{2} + 900 \, {\left (75 \, b^{4} c^{3} d + 22 \, b^{2} c d^{3}\right )} x\right )} \cos \left (b x + a\right )^{2} + 1800 \, {\left (75 \, b^{4} c^{3} d - 428 \, b^{2} c d^{3}\right )} x\right )} \sin \left (b x + a\right )}{253125 \, b^{5}} \]

[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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1098 vs. \(2 (325) = 650\).

Time = 1.15 (sec) , antiderivative size = 1098, normalized size of antiderivative = 3.33 \[ \int (c+d x)^4 \cos ^3(a+b x) \sin ^2(a+b x) \, dx=\text {Too large to display} \]

[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))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1339 vs. \(2 (304) = 608\).

Time = 0.32 (sec) , antiderivative size = 1339, normalized size of antiderivative = 4.06 \[ \int (c+d x)^4 \cos ^3(a+b x) \sin ^2(a+b x) \, dx=\text {Too large to display} \]

[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

Giac [A] (verification not implemented)

none

Time = 0.39 (sec) , antiderivative size = 531, normalized size of antiderivative = 1.61 \[ \int (c+d x)^4 \cos ^3(a+b x) \sin ^2(a+b x) \, dx=-\frac {{\left (25 \, b^{3} d^{4} x^{3} + 75 \, b^{3} c d^{3} x^{2} + 75 \, b^{3} c^{2} d^{2} x + 25 \, b^{3} c^{3} d - 6 \, b d^{4} x - 6 \, b c d^{3}\right )} \cos \left (5 \, b x + 5 \, a\right )}{2500 \, b^{5}} - \frac {{\left (3 \, b^{3} d^{4} x^{3} + 9 \, b^{3} c d^{3} x^{2} + 9 \, b^{3} c^{2} d^{2} x + 3 \, b^{3} c^{3} d - 2 \, b d^{4} x - 2 \, b c d^{3}\right )} \cos \left (3 \, b x + 3 \, a\right )}{108 \, b^{5}} + \frac {{\left (b^{3} d^{4} x^{3} + 3 \, b^{3} c d^{3} x^{2} + 3 \, b^{3} c^{2} d^{2} x + b^{3} c^{3} d - 6 \, b d^{4} x - 6 \, b c d^{3}\right )} \cos \left (b x + a\right )}{2 \, b^{5}} - \frac {{\left (625 \, b^{4} d^{4} x^{4} + 2500 \, b^{4} c d^{3} x^{3} + 3750 \, b^{4} c^{2} d^{2} x^{2} + 2500 \, b^{4} c^{3} d x + 625 \, b^{4} c^{4} - 300 \, b^{2} d^{4} x^{2} - 600 \, b^{2} c d^{3} x - 300 \, b^{2} c^{2} d^{2} + 24 \, d^{4}\right )} \sin \left (5 \, b x + 5 \, a\right )}{50000 \, b^{5}} - \frac {{\left (27 \, b^{4} d^{4} x^{4} + 108 \, b^{4} c d^{3} x^{3} + 162 \, b^{4} c^{2} d^{2} x^{2} + 108 \, b^{4} c^{3} d x + 27 \, b^{4} c^{4} - 36 \, b^{2} d^{4} x^{2} - 72 \, b^{2} c d^{3} x - 36 \, b^{2} c^{2} d^{2} + 8 \, d^{4}\right )} \sin \left (3 \, b x + 3 \, a\right )}{1296 \, b^{5}} + \frac {{\left (b^{4} d^{4} x^{4} + 4 \, b^{4} c d^{3} x^{3} + 6 \, b^{4} c^{2} d^{2} x^{2} + 4 \, b^{4} c^{3} d x + b^{4} c^{4} - 12 \, b^{2} d^{4} x^{2} - 24 \, b^{2} c d^{3} x - 12 \, b^{2} c^{2} d^{2} + 24 \, d^{4}\right )} \sin \left (b x + a\right )}{8 \, b^{5}} \]

[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

Mupad [B] (verification not implemented)

Time = 28.90 (sec) , antiderivative size = 816, normalized size of antiderivative = 2.47 \[ \int (c+d x)^4 \cos ^3(a+b x) \sin ^2(a+b x) \, dx=-\frac {\frac {d^4\,\sin \left (3\,a+3\,b\,x\right )}{162}-3\,d^4\,\sin \left (a+b\,x\right )+\frac {3\,d^4\,\sin \left (5\,a+5\,b\,x\right )}{6250}-\frac {b^4\,c^4\,\sin \left (a+b\,x\right )}{8}+\frac {b^4\,c^4\,\sin \left (3\,a+3\,b\,x\right )}{48}+\frac {b^4\,c^4\,\sin \left (5\,a+5\,b\,x\right )}{80}+\frac {b^3\,c^3\,d\,\cos \left (3\,a+3\,b\,x\right )}{36}+\frac {b^3\,c^3\,d\,\cos \left (5\,a+5\,b\,x\right )}{100}+\frac {3\,b^2\,c^2\,d^2\,\sin \left (a+b\,x\right )}{2}-\frac {b^3\,d^4\,x^3\,\cos \left (a+b\,x\right )}{2}+\frac {3\,b^2\,d^4\,x^2\,\sin \left (a+b\,x\right )}{2}-\frac {b^4\,d^4\,x^4\,\sin \left (a+b\,x\right )}{8}+3\,b\,c\,d^3\,\cos \left (a+b\,x\right )+3\,b\,d^4\,x\,\cos \left (a+b\,x\right )-\frac {b^2\,c^2\,d^2\,\sin \left (3\,a+3\,b\,x\right )}{36}-\frac {3\,b^2\,c^2\,d^2\,\sin \left (5\,a+5\,b\,x\right )}{500}+\frac {b^3\,d^4\,x^3\,\cos \left (3\,a+3\,b\,x\right )}{36}+\frac {b^3\,d^4\,x^3\,\cos \left (5\,a+5\,b\,x\right )}{100}-\frac {b^2\,d^4\,x^2\,\sin \left (3\,a+3\,b\,x\right )}{36}-\frac {3\,b^2\,d^4\,x^2\,\sin \left (5\,a+5\,b\,x\right )}{500}+\frac {b^4\,d^4\,x^4\,\sin \left (3\,a+3\,b\,x\right )}{48}+\frac {b^4\,d^4\,x^4\,\sin \left (5\,a+5\,b\,x\right )}{80}-\frac {b\,c\,d^3\,\cos \left (3\,a+3\,b\,x\right )}{54}-\frac {3\,b\,c\,d^3\,\cos \left (5\,a+5\,b\,x\right )}{1250}-\frac {b^3\,c^3\,d\,\cos \left (a+b\,x\right )}{2}-\frac {b\,d^4\,x\,\cos \left (3\,a+3\,b\,x\right )}{54}-\frac {3\,b\,d^4\,x\,\cos \left (5\,a+5\,b\,x\right )}{1250}+3\,b^2\,c\,d^3\,x\,\sin \left (a+b\,x\right )-\frac {b^4\,c^3\,d\,x\,\sin \left (a+b\,x\right )}{2}+\frac {b^4\,c^2\,d^2\,x^2\,\sin \left (3\,a+3\,b\,x\right )}{8}+\frac {3\,b^4\,c^2\,d^2\,x^2\,\sin \left (5\,a+5\,b\,x\right )}{40}-\frac {3\,b^3\,c^2\,d^2\,x\,\cos \left (a+b\,x\right )}{2}-\frac {3\,b^3\,c\,d^3\,x^2\,\cos \left (a+b\,x\right )}{2}-\frac {b^2\,c\,d^3\,x\,\sin \left (3\,a+3\,b\,x\right )}{18}+\frac {b^4\,c^3\,d\,x\,\sin \left (3\,a+3\,b\,x\right )}{12}-\frac {3\,b^2\,c\,d^3\,x\,\sin \left (5\,a+5\,b\,x\right )}{250}+\frac {b^4\,c^3\,d\,x\,\sin \left (5\,a+5\,b\,x\right )}{20}-\frac {b^4\,c\,d^3\,x^3\,\sin \left (a+b\,x\right )}{2}+\frac {b^3\,c^2\,d^2\,x\,\cos \left (3\,a+3\,b\,x\right )}{12}+\frac {b^3\,c\,d^3\,x^2\,\cos \left (3\,a+3\,b\,x\right )}{12}+\frac {3\,b^3\,c^2\,d^2\,x\,\cos \left (5\,a+5\,b\,x\right )}{100}+\frac {3\,b^3\,c\,d^3\,x^2\,\cos \left (5\,a+5\,b\,x\right )}{100}+\frac {b^4\,c\,d^3\,x^3\,\sin \left (3\,a+3\,b\,x\right )}{12}+\frac {b^4\,c\,d^3\,x^3\,\sin \left (5\,a+5\,b\,x\right )}{20}-\frac {3\,b^4\,c^2\,d^2\,x^2\,\sin \left (a+b\,x\right )}{4}}{b^5} \]

[In]

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

[Out]

-((d^4*sin(3*a + 3*b*x))/162 - 3*d^4*sin(a + b*x) + (3*d^4*sin(5*a + 5*b*x))/6250 - (b^4*c^4*sin(a + b*x))/8 +
(b^4*c^4*sin(3*a + 3*b*x))/48 + (b^4*c^4*sin(5*a + 5*b*x))/80 + (b^3*c^3*d*cos(3*a + 3*b*x))/36 + (b^3*c^3*d*
cos(5*a + 5*b*x))/100 + (3*b^2*c^2*d^2*sin(a + b*x))/2 - (b^3*d^4*x^3*cos(a + b*x))/2 + (3*b^2*d^4*x^2*sin(a +
b*x))/2 - (b^4*d^4*x^4*sin(a + b*x))/8 + 3*b*c*d^3*cos(a + b*x) + 3*b*d^4*x*cos(a + b*x) - (b^2*c^2*d^2*sin(3
*a + 3*b*x))/36 - (3*b^2*c^2*d^2*sin(5*a + 5*b*x))/500 + (b^3*d^4*x^3*cos(3*a + 3*b*x))/36 + (b^3*d^4*x^3*cos(
5*a + 5*b*x))/100 - (b^2*d^4*x^2*sin(3*a + 3*b*x))/36 - (3*b^2*d^4*x^2*sin(5*a + 5*b*x))/500 + (b^4*d^4*x^4*si
n(3*a + 3*b*x))/48 + (b^4*d^4*x^4*sin(5*a + 5*b*x))/80 - (b*c*d^3*cos(3*a + 3*b*x))/54 - (3*b*c*d^3*cos(5*a +
5*b*x))/1250 - (b^3*c^3*d*cos(a + b*x))/2 - (b*d^4*x*cos(3*a + 3*b*x))/54 - (3*b*d^4*x*cos(5*a + 5*b*x))/1250
+ 3*b^2*c*d^3*x*sin(a + b*x) - (b^4*c^3*d*x*sin(a + b*x))/2 + (b^4*c^2*d^2*x^2*sin(3*a + 3*b*x))/8 + (3*b^4*c^
2*d^2*x^2*sin(5*a + 5*b*x))/40 - (3*b^3*c^2*d^2*x*cos(a + b*x))/2 - (3*b^3*c*d^3*x^2*cos(a + b*x))/2 - (b^2*c*
d^3*x*sin(3*a + 3*b*x))/18 + (b^4*c^3*d*x*sin(3*a + 3*b*x))/12 - (3*b^2*c*d^3*x*sin(5*a + 5*b*x))/250 + (b^4*c
^3*d*x*sin(5*a + 5*b*x))/20 - (b^4*c*d^3*x^3*sin(a + b*x))/2 + (b^3*c^2*d^2*x*cos(3*a + 3*b*x))/12 + (b^3*c*d^
3*x^2*cos(3*a + 3*b*x))/12 + (3*b^3*c^2*d^2*x*cos(5*a + 5*b*x))/100 + (3*b^3*c*d^3*x^2*cos(5*a + 5*b*x))/100 +
(b^4*c*d^3*x^3*sin(3*a + 3*b*x))/12 + (b^4*c*d^3*x^3*sin(5*a + 5*b*x))/20 - (3*b^4*c^2*d^2*x^2*sin(a + b*x))/
4)/b^5