\(\int (c+d x)^3 \cos ^2(a+b x) \sin ^3(a+b x) \, dx\) [90]

Optimal result

Integrand size = 24, antiderivative size = 259 \[ \int (c+d x)^3 \cos ^2(a+b x) \sin ^3(a+b x) \, dx=\frac {3 d^2 (c+d x) \cos (a+b x)}{4 b^3}-\frac {(c+d x)^3 \cos (a+b x)}{8 b}+\frac {d^2 (c+d x) \cos (3 a+3 b x)}{72 b^3}-\frac {(c+d x)^3 \cos (3 a+3 b x)}{48 b}-\frac {3 d^2 (c+d x) \cos (5 a+5 b x)}{1000 b^3}+\frac {(c+d x)^3 \cos (5 a+5 b x)}{80 b}-\frac {3 d^3 \sin (a+b x)}{4 b^4}+\frac {3 d (c+d x)^2 \sin (a+b x)}{8 b^2}-\frac {d^3 \sin (3 a+3 b x)}{216 b^4}+\frac {d (c+d x)^2 \sin (3 a+3 b x)}{48 b^2}+\frac {3 d^3 \sin (5 a+5 b x)}{5000 b^4}-\frac {3 d (c+d x)^2 \sin (5 a+5 b x)}{400 b^2} \]

[Out]

3/4*d^2*(d*x+c)*cos(b*x+a)/b^3-1/8*(d*x+c)^3*cos(b*x+a)/b+1/72*d^2*(d*x+c)*cos(3*b*x+3*a)/b^3-1/48*(d*x+c)^3*c
os(3*b*x+3*a)/b-3/1000*d^2*(d*x+c)*cos(5*b*x+5*a)/b^3+1/80*(d*x+c)^3*cos(5*b*x+5*a)/b-3/4*d^3*sin(b*x+a)/b^4+3
/8*d*(d*x+c)^2*sin(b*x+a)/b^2-1/216*d^3*sin(3*b*x+3*a)/b^4+1/48*d*(d*x+c)^2*sin(3*b*x+3*a)/b^2+3/5000*d^3*sin(
5*b*x+5*a)/b^4-3/400*d*(d*x+c)^2*sin(5*b*x+5*a)/b^2

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {4491, 3377, 2717} \[ \int (c+d x)^3 \cos ^2(a+b x) \sin ^3(a+b x) \, dx=-\frac {3 d^3 \sin (a+b x)}{4 b^4}-\frac {d^3 \sin (3 a+3 b x)}{216 b^4}+\frac {3 d^3 \sin (5 a+5 b x)}{5000 b^4}+\frac {3 d^2 (c+d x) \cos (a+b x)}{4 b^3}+\frac {d^2 (c+d x) \cos (3 a+3 b x)}{72 b^3}-\frac {3 d^2 (c+d x) \cos (5 a+5 b x)}{1000 b^3}+\frac {3 d (c+d x)^2 \sin (a+b x)}{8 b^2}+\frac {d (c+d x)^2 \sin (3 a+3 b x)}{48 b^2}-\frac {3 d (c+d x)^2 \sin (5 a+5 b x)}{400 b^2}-\frac {(c+d x)^3 \cos (a+b x)}{8 b}-\frac {(c+d x)^3 \cos (3 a+3 b x)}{48 b}+\frac {(c+d x)^3 \cos (5 a+5 b x)}{80 b} \]

[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 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)^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] (verified)

Time = 1.51 (sec) , antiderivative size = 369, normalized size of antiderivative = 1.42 \[ \int (c+d x)^3 \cos ^2(a+b x) \sin ^3(a+b x) \, dx=\frac {-33750 b (c+d x) \left (-6 d^2+b^2 (c+d x)^2\right ) \cos (a+b x)-1875 b (c+d x) \left (-2 d^2+3 b^2 (c+d x)^2\right ) \cos (3 (a+b x))+3375 b^3 c^3 \cos (5 (a+b x))-810 b c d^2 \cos (5 (a+b x))+10125 b^3 c^2 d x \cos (5 (a+b x))-810 b d^3 x \cos (5 (a+b x))+10125 b^3 c d^2 x^2 \cos (5 (a+b x))+3375 b^3 d^3 x^3 \cos (5 (a+b x))+101250 b^2 c^2 d \sin (a+b x)-202500 d^3 \sin (a+b x)+202500 b^2 c d^2 x \sin (a+b x)+101250 b^2 d^3 x^2 \sin (a+b x)+5625 b^2 c^2 d \sin (3 (a+b x))-1250 d^3 \sin (3 (a+b x))+11250 b^2 c d^2 x \sin (3 (a+b x))+5625 b^2 d^3 x^2 \sin (3 (a+b x))-2025 b^2 c^2 d \sin (5 (a+b x))+162 d^3 \sin (5 (a+b x))-4050 b^2 c d^2 x \sin (5 (a+b x))-2025 b^2 d^3 x^2 \sin (5 (a+b x))}{270000 b^4} \]

[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)

Maple [A] (verified)

Time = 2.25 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.79

method	result	size
parallelrisch	\(\frac {-375 b \left (\left (d x +c \right )^{2} b^{2}-\frac {2 d^{2}}{3}\right ) \left (d x +c \right ) \cos \left (3 x b +3 a \right )+225 b \left (\left (d x +c \right )^{2} b^{2}-\frac {6 d^{2}}{25}\right ) \left (d x +c \right ) \cos \left (5 x b +5 a \right )+375 d \left (\left (d x +c \right )^{2} b^{2}-\frac {2 d^{2}}{9}\right ) \sin \left (3 x b +3 a \right )-135 d \left (\left (d x +c \right )^{2} b^{2}-\frac {2 d^{2}}{25}\right ) \sin \left (5 x b +5 a \right )-2250 b \left (\left (d x +c \right )^{2} b^{2}-6 d^{2}\right ) \left (d x +c \right ) \cos \left (x b +a \right )+6750 d \left (\left (d x +c \right )^{2} b^{2}-2 d^{2}\right ) \sin \left (x b +a \right )-2400 b^{3} c^{3}+13696 c \,d^{2} b}{18000 b^{4}}\)	\(205\)
risch	\(-\frac {\left (b^{2} d^{3} x^{3}+3 b^{2} c \,d^{2} x^{2}+3 b^{2} c^{2} d x +b^{2} c^{3}-6 d^{3} x -6 c \,d^{2}\right ) \cos \left (x b +a \right )}{8 b^{3}}+\frac {3 d \left (x^{2} d^{2} b^{2}+2 b^{2} c d x +b^{2} c^{2}-2 d^{2}\right ) \sin \left (x b +a \right )}{8 b^{4}}+\frac {\left (25 b^{2} d^{3} x^{3}+75 b^{2} c \,d^{2} x^{2}+75 b^{2} c^{2} d x +25 b^{2} c^{3}-6 d^{3} x -6 c \,d^{2}\right ) \cos \left (5 x b +5 a \right )}{2000 b^{3}}-\frac {3 d \left (25 x^{2} d^{2} b^{2}+50 b^{2} c d x +25 b^{2} c^{2}-2 d^{2}\right ) \sin \left (5 x b +5 a \right )}{10000 b^{4}}-\frac {\left (3 b^{2} d^{3} x^{3}+9 b^{2} c \,d^{2} x^{2}+9 b^{2} c^{2} d x +3 b^{2} c^{3}-2 d^{3} x -2 c \,d^{2}\right ) \cos \left (3 x b +3 a \right )}{144 b^{3}}+\frac {d \left (9 x^{2} d^{2} b^{2}+18 b^{2} c d x +9 b^{2} c^{2}-2 d^{2}\right ) \sin \left (3 x b +3 a \right )}{432 b^{4}}\)	\(340\)
derivativedivides	\(\text {Expression too large to display}\)	\(992\)
default	\(\text {Expression too large to display}\)	\(992\)

[In]

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

[Out]

1/18000*(-375*b*((d*x+c)^2*b^2-2/3*d^2)*(d*x+c)*cos(3*b*x+3*a)+225*b*((d*x+c)^2*b^2-6/25*d^2)*(d*x+c)*cos(5*b*
x+5*a)+375*d*((d*x+c)^2*b^2-2/9*d^2)*sin(3*b*x+3*a)-135*d*((d*x+c)^2*b^2-2/25*d^2)*sin(5*b*x+5*a)-2250*b*((d*x
+c)^2*b^2-6*d^2)*(d*x+c)*cos(b*x+a)+6750*d*((d*x+c)^2*b^2-2*d^2)*sin(b*x+a)-2400*b^3*c^3+13696*c*d^2*b)/b^4

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.14 \[ \int (c+d x)^3 \cos ^2(a+b x) \sin ^3(a+b x) \, dx=\frac {135 \, {\left (25 \, b^{3} d^{3} x^{3} + 75 \, b^{3} c d^{2} x^{2} + 25 \, b^{3} c^{3} - 6 \, b c d^{2} + 3 \, {\left (25 \, b^{3} c^{2} d - 2 \, b d^{3}\right )} x\right )} \cos \left (b x + a\right )^{5} - 75 \, {\left (75 \, b^{3} d^{3} x^{3} + 225 \, b^{3} c d^{2} x^{2} + 75 \, b^{3} c^{3} - 26 \, b c d^{2} + {\left (225 \, b^{3} c^{2} d - 26 \, b d^{3}\right )} x\right )} \cos \left (b x + a\right )^{3} + 11700 \, {\left (b d^{3} x + b c d^{2}\right )} \cos \left (b x + a\right ) + {\left (5850 \, b^{2} d^{3} x^{2} + 11700 \, b^{2} c d^{2} x + 5850 \, b^{2} c^{2} d - 81 \, {\left (25 \, b^{2} d^{3} x^{2} + 50 \, b^{2} c d^{2} x + 25 \, b^{2} c^{2} d - 2 \, d^{3}\right )} \cos \left (b x + a\right )^{4} - 12568 \, d^{3} + {\left (2925 \, b^{2} d^{3} x^{2} + 5850 \, b^{2} c d^{2} x + 2925 \, b^{2} c^{2} d - 434 \, d^{3}\right )} \cos \left (b x + a\right )^{2}\right )} \sin \left (b x + a\right )}{16875 \, b^{4}} \]

[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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 690 vs. \(2 (255) = 510\).

Time = 0.84 (sec) , antiderivative size = 690, normalized size of antiderivative = 2.66 \[ \int (c+d x)^3 \cos ^2(a+b x) \sin ^3(a+b x) \, dx=\begin {cases} - \frac {c^{3} \sin ^{2}{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{3 b} - \frac {2 c^{3} \cos ^{5}{\left (a + b x \right )}}{15 b} - \frac {c^{2} d x \sin ^{2}{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{b} - \frac {2 c^{2} d x \cos ^{5}{\left (a + b x \right )}}{5 b} - \frac {c d^{2} x^{2} \sin ^{2}{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{b} - \frac {2 c d^{2} x^{2} \cos ^{5}{\left (a + b x \right )}}{5 b} - \frac {d^{3} x^{3} \sin ^{2}{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{3 b} - \frac {2 d^{3} x^{3} \cos ^{5}{\left (a + b x \right )}}{15 b} + \frac {26 c^{2} d \sin ^{5}{\left (a + b x \right )}}{75 b^{2}} + \frac {13 c^{2} d \sin ^{3}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{15 b^{2}} + \frac {2 c^{2} d \sin {\left (a + b x \right )} \cos ^{4}{\left (a + b x \right )}}{5 b^{2}} + \frac {52 c d^{2} x \sin ^{5}{\left (a + b x \right )}}{75 b^{2}} + \frac {26 c d^{2} x \sin ^{3}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{15 b^{2}} + \frac {4 c d^{2} x \sin {\left (a + b x \right )} \cos ^{4}{\left (a + b x \right )}}{5 b^{2}} + \frac {26 d^{3} x^{2} \sin ^{5}{\left (a + b x \right )}}{75 b^{2}} + \frac {13 d^{3} x^{2} \sin ^{3}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{15 b^{2}} + \frac {2 d^{3} x^{2} \sin {\left (a + b x \right )} \cos ^{4}{\left (a + b x \right )}}{5 b^{2}} + \frac {52 c d^{2} \sin ^{4}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{75 b^{3}} + \frac {338 c d^{2} \sin ^{2}{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{225 b^{3}} + \frac {856 c d^{2} \cos ^{5}{\left (a + b x \right )}}{1125 b^{3}} + \frac {52 d^{3} x \sin ^{4}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{75 b^{3}} + \frac {338 d^{3} x \sin ^{2}{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{225 b^{3}} + \frac {856 d^{3} x \cos ^{5}{\left (a + b x \right )}}{1125 b^{3}} - \frac {12568 d^{3} \sin ^{5}{\left (a + b x \right )}}{16875 b^{4}} - \frac {5114 d^{3} \sin ^{3}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{3375 b^{4}} - \frac {856 d^{3} \sin {\left (a + b x \right )} \cos ^{4}{\left (a + b x \right )}}{1125 b^{4}} & \text {for}\: b \neq 0 \\\left (c^{3} x + \frac {3 c^{2} d x^{2}}{2} + c d^{2} x^{3} + \frac {d^{3} x^{4}}{4}\right ) \sin ^{3}{\left (a \right )} \cos ^{2}{\left (a \right )} & \text {otherwise} \end {cases} \]

[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))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 766 vs. \(2 (235) = 470\).

Time = 0.27 (sec) , antiderivative size = 766, normalized size of antiderivative = 2.96 \[ \int (c+d x)^3 \cos ^2(a+b x) \sin ^3(a+b x) \, dx=\text {Too large to display} \]

[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

Giac [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.36 \[ \int (c+d x)^3 \cos ^2(a+b x) \sin ^3(a+b x) \, dx=\frac {{\left (25 \, b^{3} d^{3} x^{3} + 75 \, b^{3} c d^{2} x^{2} + 75 \, b^{3} c^{2} d x + 25 \, b^{3} c^{3} - 6 \, b d^{3} x - 6 \, b c d^{2}\right )} \cos \left (5 \, b x + 5 \, a\right )}{2000 \, b^{4}} - \frac {{\left (3 \, b^{3} d^{3} x^{3} + 9 \, b^{3} c d^{2} x^{2} + 9 \, b^{3} c^{2} d x + 3 \, b^{3} c^{3} - 2 \, b d^{3} x - 2 \, b c d^{2}\right )} \cos \left (3 \, b x + 3 \, a\right )}{144 \, b^{4}} - \frac {{\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + b^{3} c^{3} - 6 \, b d^{3} x - 6 \, b c d^{2}\right )} \cos \left (b x + a\right )}{8 \, b^{4}} - \frac {3 \, {\left (25 \, b^{2} d^{3} x^{2} + 50 \, b^{2} c d^{2} x + 25 \, b^{2} c^{2} d - 2 \, d^{3}\right )} \sin \left (5 \, b x + 5 \, a\right )}{10000 \, b^{4}} + \frac {{\left (9 \, b^{2} d^{3} x^{2} + 18 \, b^{2} c d^{2} x + 9 \, b^{2} c^{2} d - 2 \, d^{3}\right )} \sin \left (3 \, b x + 3 \, a\right )}{432 \, b^{4}} + \frac {3 \, {\left (b^{2} d^{3} x^{2} + 2 \, b^{2} c d^{2} x + b^{2} c^{2} d - 2 \, d^{3}\right )} \sin \left (b x + a\right )}{8 \, b^{4}} \]

[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

Mupad [B] (verification not implemented)

Time = 3.00 (sec) , antiderivative size = 516, normalized size of antiderivative = 1.99 \[ \int (c+d x)^3 \cos ^2(a+b x) \sin ^3(a+b x) \, dx=-\frac {\frac {3\,d^3\,\sin \left (a+b\,x\right )}{4}+\frac {d^3\,\sin \left (3\,a+3\,b\,x\right )}{216}-\frac {3\,d^3\,\sin \left (5\,a+5\,b\,x\right )}{5000}+\frac {b^3\,c^3\,\cos \left (a+b\,x\right )}{8}+\frac {b^3\,c^3\,\cos \left (3\,a+3\,b\,x\right )}{48}-\frac {b^3\,c^3\,\cos \left (5\,a+5\,b\,x\right )}{80}-\frac {b^2\,c^2\,d\,\sin \left (3\,a+3\,b\,x\right )}{48}+\frac {3\,b^2\,c^2\,d\,\sin \left (5\,a+5\,b\,x\right )}{400}+\frac {b^3\,d^3\,x^3\,\cos \left (a+b\,x\right )}{8}-\frac {3\,b^2\,d^3\,x^2\,\sin \left (a+b\,x\right )}{8}-\frac {3\,b\,c\,d^2\,\cos \left (a+b\,x\right )}{4}-\frac {3\,b\,d^3\,x\,\cos \left (a+b\,x\right )}{4}+\frac {b^3\,d^3\,x^3\,\cos \left (3\,a+3\,b\,x\right )}{48}-\frac {b^3\,d^3\,x^3\,\cos \left (5\,a+5\,b\,x\right )}{80}-\frac {b^2\,d^3\,x^2\,\sin \left (3\,a+3\,b\,x\right )}{48}+\frac {3\,b^2\,d^3\,x^2\,\sin \left (5\,a+5\,b\,x\right )}{400}-\frac {b\,c\,d^2\,\cos \left (3\,a+3\,b\,x\right )}{72}+\frac {3\,b\,c\,d^2\,\cos \left (5\,a+5\,b\,x\right )}{1000}-\frac {3\,b^2\,c^2\,d\,\sin \left (a+b\,x\right )}{8}-\frac {b\,d^3\,x\,\cos \left (3\,a+3\,b\,x\right )}{72}+\frac {3\,b\,d^3\,x\,\cos \left (5\,a+5\,b\,x\right )}{1000}+\frac {3\,b^3\,c^2\,d\,x\,\cos \left (a+b\,x\right )}{8}-\frac {3\,b^2\,c\,d^2\,x\,\sin \left (a+b\,x\right )}{4}+\frac {b^3\,c^2\,d\,x\,\cos \left (3\,a+3\,b\,x\right )}{16}-\frac {3\,b^3\,c^2\,d\,x\,\cos \left (5\,a+5\,b\,x\right )}{80}+\frac {3\,b^3\,c\,d^2\,x^2\,\cos \left (a+b\,x\right )}{8}-\frac {b^2\,c\,d^2\,x\,\sin \left (3\,a+3\,b\,x\right )}{24}+\frac {3\,b^2\,c\,d^2\,x\,\sin \left (5\,a+5\,b\,x\right )}{200}+\frac {b^3\,c\,d^2\,x^2\,\cos \left (3\,a+3\,b\,x\right )}{16}-\frac {3\,b^3\,c\,d^2\,x^2\,\cos \left (5\,a+5\,b\,x\right )}{80}}{b^4} \]

[In]

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

[Out]

-((3*d^3*sin(a + b*x))/4 + (d^3*sin(3*a + 3*b*x))/216 - (3*d^3*sin(5*a + 5*b*x))/5000 + (b^3*c^3*cos(a + b*x))
/8 + (b^3*c^3*cos(3*a + 3*b*x))/48 - (b^3*c^3*cos(5*a + 5*b*x))/80 - (b^2*c^2*d*sin(3*a + 3*b*x))/48 + (3*b^2*
c^2*d*sin(5*a + 5*b*x))/400 + (b^3*d^3*x^3*cos(a + b*x))/8 - (3*b^2*d^3*x^2*sin(a + b*x))/8 - (3*b*c*d^2*cos(a
+ b*x))/4 - (3*b*d^3*x*cos(a + b*x))/4 + (b^3*d^3*x^3*cos(3*a + 3*b*x))/48 - (b^3*d^3*x^3*cos(5*a + 5*b*x))/8
0 - (b^2*d^3*x^2*sin(3*a + 3*b*x))/48 + (3*b^2*d^3*x^2*sin(5*a + 5*b*x))/400 - (b*c*d^2*cos(3*a + 3*b*x))/72 +
(3*b*c*d^2*cos(5*a + 5*b*x))/1000 - (3*b^2*c^2*d*sin(a + b*x))/8 - (b*d^3*x*cos(3*a + 3*b*x))/72 + (3*b*d^3*x
*cos(5*a + 5*b*x))/1000 + (3*b^3*c^2*d*x*cos(a + b*x))/8 - (3*b^2*c*d^2*x*sin(a + b*x))/4 + (b^3*c^2*d*x*cos(3
*a + 3*b*x))/16 - (3*b^3*c^2*d*x*cos(5*a + 5*b*x))/80 + (3*b^3*c*d^2*x^2*cos(a + b*x))/8 - (b^2*c*d^2*x*sin(3*
a + 3*b*x))/24 + (3*b^2*c*d^2*x*sin(5*a + 5*b*x))/200 + (b^3*c*d^2*x^2*cos(3*a + 3*b*x))/16 - (3*b^3*c*d^2*x^2
*cos(5*a + 5*b*x))/80)/b^4