\(\int (c+d x)^4 \cos (a+b x) \sin ^3(a+b x) \, dx\) [23]

Optimal result

Integrand size = 22, antiderivative size = 252 \[ \int (c+d x)^4 \cos (a+b x) \sin ^3(a+b x) \, dx=\frac {45 d^2 (c+d x)^2}{128 b^3}-\frac {3 (c+d x)^4}{32 b}-\frac {45 d^3 (c+d x) \cos (a+b x) \sin (a+b x)}{64 b^4}+\frac {3 d (c+d x)^3 \cos (a+b x) \sin (a+b x)}{8 b^2}+\frac {45 d^4 \sin ^2(a+b x)}{128 b^5}-\frac {9 d^2 (c+d x)^2 \sin ^2(a+b x)}{16 b^3}-\frac {3 d^3 (c+d x) \cos (a+b x) \sin ^3(a+b x)}{32 b^4}+\frac {d (c+d x)^3 \cos (a+b x) \sin ^3(a+b x)}{4 b^2}+\frac {3 d^4 \sin ^4(a+b x)}{128 b^5}-\frac {3 d^2 (c+d x)^2 \sin ^4(a+b x)}{16 b^3}+\frac {(c+d x)^4 \sin ^4(a+b x)}{4 b} \] Output:

45/128*d^2*(d*x+c)^2/b^3-3/32*(d*x+c)^4/b-45/64*d^3*(d*x+c)*cos(b*x+a)*sin 
(b*x+a)/b^4+3/8*d*(d*x+c)^3*cos(b*x+a)*sin(b*x+a)/b^2+45/128*d^4*sin(b*x+a 
)^2/b^5-9/16*d^2*(d*x+c)^2*sin(b*x+a)^2/b^3-3/32*d^3*(d*x+c)*cos(b*x+a)*si 
n(b*x+a)^3/b^4+1/4*d*(d*x+c)^3*cos(b*x+a)*sin(b*x+a)^3/b^2+3/128*d^4*sin(b 
*x+a)^4/b^5-3/16*d^2*(d*x+c)^2*sin(b*x+a)^4/b^3+1/4*(d*x+c)^4*sin(b*x+a)^4 
/b
 

Mathematica [A] (verified)

Time = 1.15 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.63 \[ \int (c+d x)^4 \cos (a+b x) \sin ^3(a+b x) \, dx=\frac {-64 \left (3 d^4-6 b^2 d^2 (c+d x)^2+2 b^4 (c+d x)^4\right ) \cos (2 (a+b x))+\left (3 d^4-24 b^2 d^2 (c+d x)^2+32 b^4 (c+d x)^4\right ) \cos (4 (a+b x))-8 b d (c+d x) \left (-16 \left (-3 d^2+2 b^2 (c+d x)^2\right )+\left (-3 d^2+8 b^2 (c+d x)^2\right ) \cos (2 (a+b x))\right ) \sin (2 (a+b x))}{1024 b^5} \] Input:

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

Output:

(-64*(3*d^4 - 6*b^2*d^2*(c + d*x)^2 + 2*b^4*(c + d*x)^4)*Cos[2*(a + b*x)] 
+ (3*d^4 - 24*b^2*d^2*(c + d*x)^2 + 32*b^4*(c + d*x)^4)*Cos[4*(a + b*x)] - 
 8*b*d*(c + d*x)*(-16*(-3*d^2 + 2*b^2*(c + d*x)^2) + (-3*d^2 + 8*b^2*(c + 
d*x)^2)*Cos[2*(a + b*x)])*Sin[2*(a + b*x)])/(1024*b^5)
 

Rubi [A] (verified)

Time = 0.98 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.29, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.591, Rules used = {4904, 3042, 3792, 3042, 3791, 3042, 3791, 17, 3792, 17, 3042, 3791, 17}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 4904

\(\displaystyle \frac {(c+d x)^4 \sin ^4(a+b x)}{4 b}-\frac {d \int (c+d x)^3 \sin ^4(a+b x)dx}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {(c+d x)^4 \sin ^4(a+b x)}{4 b}-\frac {d \int (c+d x)^3 \sin (a+b x)^4dx}{b}\)

\(\Big \downarrow \) 3792

\(\displaystyle \frac {(c+d x)^4 \sin ^4(a+b x)}{4 b}-\frac {d \left (-\frac {3 d^2 \int (c+d x) \sin ^4(a+b x)dx}{8 b^2}+\frac {3}{4} \int (c+d x)^3 \sin ^2(a+b x)dx+\frac {3 d (c+d x)^2 \sin ^4(a+b x)}{16 b^2}-\frac {(c+d x)^3 \sin ^3(a+b x) \cos (a+b x)}{4 b}\right )}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {(c+d x)^4 \sin ^4(a+b x)}{4 b}-\frac {d \left (-\frac {3 d^2 \int (c+d x) \sin (a+b x)^4dx}{8 b^2}+\frac {3}{4} \int (c+d x)^3 \sin (a+b x)^2dx+\frac {3 d (c+d x)^2 \sin ^4(a+b x)}{16 b^2}-\frac {(c+d x)^3 \sin ^3(a+b x) \cos (a+b x)}{4 b}\right )}{b}\)

\(\Big \downarrow \) 3791

\(\displaystyle \frac {(c+d x)^4 \sin ^4(a+b x)}{4 b}-\frac {d \left (-\frac {3 d^2 \left (\frac {3}{4} \int (c+d x) \sin ^2(a+b x)dx+\frac {d \sin ^4(a+b x)}{16 b^2}-\frac {(c+d x) \sin ^3(a+b x) \cos (a+b x)}{4 b}\right )}{8 b^2}+\frac {3}{4} \int (c+d x)^3 \sin (a+b x)^2dx+\frac {3 d (c+d x)^2 \sin ^4(a+b x)}{16 b^2}-\frac {(c+d x)^3 \sin ^3(a+b x) \cos (a+b x)}{4 b}\right )}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {(c+d x)^4 \sin ^4(a+b x)}{4 b}-\frac {d \left (-\frac {3 d^2 \left (\frac {3}{4} \int (c+d x) \sin (a+b x)^2dx+\frac {d \sin ^4(a+b x)}{16 b^2}-\frac {(c+d x) \sin ^3(a+b x) \cos (a+b x)}{4 b}\right )}{8 b^2}+\frac {3}{4} \int (c+d x)^3 \sin (a+b x)^2dx+\frac {3 d (c+d x)^2 \sin ^4(a+b x)}{16 b^2}-\frac {(c+d x)^3 \sin ^3(a+b x) \cos (a+b x)}{4 b}\right )}{b}\)

\(\Big \downarrow \) 3791

\(\displaystyle \frac {(c+d x)^4 \sin ^4(a+b x)}{4 b}-\frac {d \left (-\frac {3 d^2 \left (\frac {3}{4} \left (\frac {1}{2} \int (c+d x)dx+\frac {d \sin ^2(a+b x)}{4 b^2}-\frac {(c+d x) \sin (a+b x) \cos (a+b x)}{2 b}\right )+\frac {d \sin ^4(a+b x)}{16 b^2}-\frac {(c+d x) \sin ^3(a+b x) \cos (a+b x)}{4 b}\right )}{8 b^2}+\frac {3}{4} \int (c+d x)^3 \sin (a+b x)^2dx+\frac {3 d (c+d x)^2 \sin ^4(a+b x)}{16 b^2}-\frac {(c+d x)^3 \sin ^3(a+b x) \cos (a+b x)}{4 b}\right )}{b}\)

\(\Big \downarrow \) 17

\(\displaystyle \frac {(c+d x)^4 \sin ^4(a+b x)}{4 b}-\frac {d \left (\frac {3}{4} \int (c+d x)^3 \sin (a+b x)^2dx-\frac {3 d^2 \left (\frac {3}{4} \left (\frac {d \sin ^2(a+b x)}{4 b^2}-\frac {(c+d x) \sin (a+b x) \cos (a+b x)}{2 b}+\frac {(c+d x)^2}{4 d}\right )+\frac {d \sin ^4(a+b x)}{16 b^2}-\frac {(c+d x) \sin ^3(a+b x) \cos (a+b x)}{4 b}\right )}{8 b^2}+\frac {3 d (c+d x)^2 \sin ^4(a+b x)}{16 b^2}-\frac {(c+d x)^3 \sin ^3(a+b x) \cos (a+b x)}{4 b}\right )}{b}\)

\(\Big \downarrow \) 3792

\(\displaystyle \frac {(c+d x)^4 \sin ^4(a+b x)}{4 b}-\frac {d \left (\frac {3}{4} \left (-\frac {3 d^2 \int (c+d x) \sin ^2(a+b x)dx}{2 b^2}+\frac {1}{2} \int (c+d x)^3dx+\frac {3 d (c+d x)^2 \sin ^2(a+b x)}{4 b^2}-\frac {(c+d x)^3 \sin (a+b x) \cos (a+b x)}{2 b}\right )-\frac {3 d^2 \left (\frac {3}{4} \left (\frac {d \sin ^2(a+b x)}{4 b^2}-\frac {(c+d x) \sin (a+b x) \cos (a+b x)}{2 b}+\frac {(c+d x)^2}{4 d}\right )+\frac {d \sin ^4(a+b x)}{16 b^2}-\frac {(c+d x) \sin ^3(a+b x) \cos (a+b x)}{4 b}\right )}{8 b^2}+\frac {3 d (c+d x)^2 \sin ^4(a+b x)}{16 b^2}-\frac {(c+d x)^3 \sin ^3(a+b x) \cos (a+b x)}{4 b}\right )}{b}\)

\(\Big \downarrow \) 17

\(\displaystyle \frac {(c+d x)^4 \sin ^4(a+b x)}{4 b}-\frac {d \left (\frac {3}{4} \left (-\frac {3 d^2 \int (c+d x) \sin ^2(a+b x)dx}{2 b^2}+\frac {3 d (c+d x)^2 \sin ^2(a+b x)}{4 b^2}-\frac {(c+d x)^3 \sin (a+b x) \cos (a+b x)}{2 b}+\frac {(c+d x)^4}{8 d}\right )-\frac {3 d^2 \left (\frac {3}{4} \left (\frac {d \sin ^2(a+b x)}{4 b^2}-\frac {(c+d x) \sin (a+b x) \cos (a+b x)}{2 b}+\frac {(c+d x)^2}{4 d}\right )+\frac {d \sin ^4(a+b x)}{16 b^2}-\frac {(c+d x) \sin ^3(a+b x) \cos (a+b x)}{4 b}\right )}{8 b^2}+\frac {3 d (c+d x)^2 \sin ^4(a+b x)}{16 b^2}-\frac {(c+d x)^3 \sin ^3(a+b x) \cos (a+b x)}{4 b}\right )}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {(c+d x)^4 \sin ^4(a+b x)}{4 b}-\frac {d \left (\frac {3}{4} \left (-\frac {3 d^2 \int (c+d x) \sin (a+b x)^2dx}{2 b^2}+\frac {3 d (c+d x)^2 \sin ^2(a+b x)}{4 b^2}-\frac {(c+d x)^3 \sin (a+b x) \cos (a+b x)}{2 b}+\frac {(c+d x)^4}{8 d}\right )-\frac {3 d^2 \left (\frac {3}{4} \left (\frac {d \sin ^2(a+b x)}{4 b^2}-\frac {(c+d x) \sin (a+b x) \cos (a+b x)}{2 b}+\frac {(c+d x)^2}{4 d}\right )+\frac {d \sin ^4(a+b x)}{16 b^2}-\frac {(c+d x) \sin ^3(a+b x) \cos (a+b x)}{4 b}\right )}{8 b^2}+\frac {3 d (c+d x)^2 \sin ^4(a+b x)}{16 b^2}-\frac {(c+d x)^3 \sin ^3(a+b x) \cos (a+b x)}{4 b}\right )}{b}\)

\(\Big \downarrow \) 3791

\(\displaystyle \frac {(c+d x)^4 \sin ^4(a+b x)}{4 b}-\frac {d \left (\frac {3}{4} \left (-\frac {3 d^2 \left (\frac {1}{2} \int (c+d x)dx+\frac {d \sin ^2(a+b x)}{4 b^2}-\frac {(c+d x) \sin (a+b x) \cos (a+b x)}{2 b}\right )}{2 b^2}+\frac {3 d (c+d x)^2 \sin ^2(a+b x)}{4 b^2}-\frac {(c+d x)^3 \sin (a+b x) \cos (a+b x)}{2 b}+\frac {(c+d x)^4}{8 d}\right )-\frac {3 d^2 \left (\frac {3}{4} \left (\frac {d \sin ^2(a+b x)}{4 b^2}-\frac {(c+d x) \sin (a+b x) \cos (a+b x)}{2 b}+\frac {(c+d x)^2}{4 d}\right )+\frac {d \sin ^4(a+b x)}{16 b^2}-\frac {(c+d x) \sin ^3(a+b x) \cos (a+b x)}{4 b}\right )}{8 b^2}+\frac {3 d (c+d x)^2 \sin ^4(a+b x)}{16 b^2}-\frac {(c+d x)^3 \sin ^3(a+b x) \cos (a+b x)}{4 b}\right )}{b}\)

\(\Big \downarrow \) 17

\(\displaystyle \frac {(c+d x)^4 \sin ^4(a+b x)}{4 b}-\frac {d \left (\frac {3}{4} \left (-\frac {3 d^2 \left (\frac {d \sin ^2(a+b x)}{4 b^2}-\frac {(c+d x) \sin (a+b x) \cos (a+b x)}{2 b}+\frac {(c+d x)^2}{4 d}\right )}{2 b^2}+\frac {3 d (c+d x)^2 \sin ^2(a+b x)}{4 b^2}-\frac {(c+d x)^3 \sin (a+b x) \cos (a+b x)}{2 b}+\frac {(c+d x)^4}{8 d}\right )-\frac {3 d^2 \left (\frac {3}{4} \left (\frac {d \sin ^2(a+b x)}{4 b^2}-\frac {(c+d x) \sin (a+b x) \cos (a+b x)}{2 b}+\frac {(c+d x)^2}{4 d}\right )+\frac {d \sin ^4(a+b x)}{16 b^2}-\frac {(c+d x) \sin ^3(a+b x) \cos (a+b x)}{4 b}\right )}{8 b^2}+\frac {3 d (c+d x)^2 \sin ^4(a+b x)}{16 b^2}-\frac {(c+d x)^3 \sin ^3(a+b x) \cos (a+b x)}{4 b}\right )}{b}\)

Input:

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

Output:

((c + d*x)^4*Sin[a + b*x]^4)/(4*b) - (d*(-1/4*((c + d*x)^3*Cos[a + b*x]*Si 
n[a + b*x]^3)/b + (3*d*(c + d*x)^2*Sin[a + b*x]^4)/(16*b^2) - (3*d^2*(-1/4 
*((c + d*x)*Cos[a + b*x]*Sin[a + b*x]^3)/b + (d*Sin[a + b*x]^4)/(16*b^2) + 
 (3*((c + d*x)^2/(4*d) - ((c + d*x)*Cos[a + b*x]*Sin[a + b*x])/(2*b) + (d* 
Sin[a + b*x]^2)/(4*b^2)))/4))/(8*b^2) + (3*((c + d*x)^4/(8*d) - ((c + d*x) 
^3*Cos[a + b*x]*Sin[a + b*x])/(2*b) + (3*d*(c + d*x)^2*Sin[a + b*x]^2)/(4* 
b^2) - (3*d^2*((c + d*x)^2/(4*d) - ((c + d*x)*Cos[a + b*x]*Sin[a + b*x])/( 
2*b) + (d*Sin[a + b*x]^2)/(4*b^2)))/(2*b^2)))/4))/b
 

Defintions of rubi rules used

rule 17

Int[(c_.)*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[c*((a + b*x)^(m + 1 
)/(b*(m + 1))), x] /; FreeQ[{a, b, c, m}, x] && NeQ[m, -1]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3791

Int[((c_.) + (d_.)*(x_))*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> 
 Simp[d*((b*Sin[e + f*x])^n/(f^2*n^2)), x] + (-Simp[b*(c + d*x)*Cos[e + f*x 
]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x] + Simp[b^2*((n - 1)/n)   Int[(c + d* 
x)*(b*Sin[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 
 1]
 

rule 3792

Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbo 
l] :> Simp[d*m*(c + d*x)^(m - 1)*((b*Sin[e + f*x])^n/(f^2*n^2)), x] + (-Sim 
p[b*(c + d*x)^m*Cos[e + f*x]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x] + Simp[b^ 
2*((n - 1)/n)   Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[d^2 
*m*((m - 1)/(f^2*n^2))   Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x]) 
/; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 1]
 

rule 4904

Int[Cos[(a_.) + (b_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x 
_)]^(n_.), x_Symbol] :> Simp[(c + d*x)^m*(Sin[a + b*x]^(n + 1)/(b*(n + 1))) 
, x] - Simp[d*(m/(b*(n + 1)))   Int[(c + d*x)^(m - 1)*Sin[a + b*x]^(n + 1), 
 x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && NeQ[n, -1]
 

Maple [A] (verified)

Time = 2.00 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.74

method	result	size
parallelrisch	\(\frac {\left (-128 \left (d x +c \right )^{4} b^{4}+384 d^{2} \left (d x +c \right )^{2} b^{2}-192 d^{4}\right ) \cos \left (2 b x +2 a \right )+\left (32 \left (d x +c \right )^{4} b^{4}-24 d^{2} \left (d x +c \right )^{2} b^{2}+3 d^{4}\right ) \cos \left (4 b x +4 a \right )+256 d \left (\left (d x +c \right )^{2} b^{2}-\frac {3 d^{2}}{2}\right ) \left (d x +c \right ) b \sin \left (2 b x +2 a \right )-32 d \left (\left (d x +c \right )^{2} b^{2}-\frac {3 d^{2}}{8}\right ) \left (d x +c \right ) b \sin \left (4 b x +4 a \right )+96 b^{4} c^{4}-360 b^{2} c^{2} d^{2}+189 d^{4}}{1024 b^{5}}\)	\(187\)
risch	\(\frac {\left (32 d^{4} x^{4} b^{4}+128 b^{4} c \,d^{3} x^{3}+192 b^{4} c^{2} d^{2} x^{2}+128 b^{4} c^{3} d x +32 b^{4} c^{4}-24 b^{2} d^{4} x^{2}-48 b^{2} c \,d^{3} x -24 b^{2} c^{2} d^{2}+3 d^{4}\right ) \cos \left (4 b x +4 a \right )}{1024 b^{5}}-\frac {d \left (8 b^{2} d^{3} x^{3}+24 b^{2} c \,d^{2} x^{2}+24 b^{2} c^{2} d x +8 b^{2} c^{3}-3 d^{3} x -3 c \,d^{2}\right ) \sin \left (4 b x +4 a \right )}{256 b^{4}}-\frac {\left (2 d^{4} x^{4} b^{4}+8 b^{4} c \,d^{3} x^{3}+12 b^{4} c^{2} d^{2} x^{2}+8 b^{4} c^{3} d x +2 b^{4} c^{4}-6 b^{2} d^{4} x^{2}-12 b^{2} c \,d^{3} x -6 b^{2} c^{2} d^{2}+3 d^{4}\right ) \cos \left (2 b x +2 a \right )}{16 b^{5}}+\frac {d \left (2 b^{2} d^{3} x^{3}+6 b^{2} c \,d^{2} x^{2}+6 b^{2} c^{2} d x +2 b^{2} c^{3}-3 d^{3} x -3 c \,d^{2}\right ) \sin \left (2 b x +2 a \right )}{8 b^{4}}\)	\(354\)
orering	\(\text {Expression too large to display}\)	\(1005\)
derivativedivides	\(\text {Expression too large to display}\)	\(1137\)
default	\(\text {Expression too large to display}\)	\(1137\)

Input:

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

Output:

1/1024*((-128*(d*x+c)^4*b^4+384*d^2*(d*x+c)^2*b^2-192*d^4)*cos(2*b*x+2*a)+ 
(32*(d*x+c)^4*b^4-24*d^2*(d*x+c)^2*b^2+3*d^4)*cos(4*b*x+4*a)+256*d*((d*x+c 
)^2*b^2-3/2*d^2)*(d*x+c)*b*sin(2*b*x+2*a)-32*d*((d*x+c)^2*b^2-3/8*d^2)*(d* 
x+c)*b*sin(4*b*x+4*a)+96*b^4*c^4-360*b^2*c^2*d^2+189*d^4)/b^5
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 434, normalized size of antiderivative = 1.72 \[ \int (c+d x)^4 \cos (a+b x) \sin ^3(a+b x) \, dx=\frac {20 \, b^{4} d^{4} x^{4} + 80 \, b^{4} c d^{3} x^{3} + {\left (32 \, b^{4} d^{4} x^{4} + 128 \, b^{4} c d^{3} x^{3} + 32 \, b^{4} c^{4} - 24 \, b^{2} c^{2} d^{2} + 3 \, d^{4} + 24 \, {\left (8 \, b^{4} c^{2} d^{2} - b^{2} d^{4}\right )} x^{2} + 16 \, {\left (8 \, b^{4} c^{3} d - 3 \, b^{2} c d^{3}\right )} x\right )} \cos \left (b x + a\right )^{4} + 3 \, {\left (40 \, b^{4} c^{2} d^{2} - 17 \, b^{2} d^{4}\right )} x^{2} - {\left (64 \, b^{4} d^{4} x^{4} + 256 \, b^{4} c d^{3} x^{3} + 64 \, b^{4} c^{4} - 120 \, b^{2} c^{2} d^{2} + 51 \, d^{4} + 24 \, {\left (16 \, b^{4} c^{2} d^{2} - 5 \, b^{2} d^{4}\right )} x^{2} + 16 \, {\left (16 \, b^{4} c^{3} d - 15 \, b^{2} c d^{3}\right )} x\right )} \cos \left (b x + a\right )^{2} + 2 \, {\left (40 \, b^{4} c^{3} d - 51 \, b^{2} c d^{3}\right )} x - 2 \, {\left (2 \, {\left (8 \, b^{3} d^{4} x^{3} + 24 \, b^{3} c d^{3} x^{2} + 8 \, b^{3} c^{3} d - 3 \, b c d^{3} + 3 \, {\left (8 \, b^{3} c^{2} d^{2} - b d^{4}\right )} x\right )} \cos \left (b x + a\right )^{3} - {\left (40 \, b^{3} d^{4} x^{3} + 120 \, b^{3} c d^{3} x^{2} + 40 \, b^{3} c^{3} d - 51 \, b c d^{3} + 3 \, {\left (40 \, b^{3} c^{2} d^{2} - 17 \, b d^{4}\right )} x\right )} \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )}{128 \, b^{5}} \] Input:

integrate((d*x+c)^4*cos(b*x+a)*sin(b*x+a)^3,x, algorithm="fricas")
 

Output:

1/128*(20*b^4*d^4*x^4 + 80*b^4*c*d^3*x^3 + (32*b^4*d^4*x^4 + 128*b^4*c*d^3 
*x^3 + 32*b^4*c^4 - 24*b^2*c^2*d^2 + 3*d^4 + 24*(8*b^4*c^2*d^2 - b^2*d^4)* 
x^2 + 16*(8*b^4*c^3*d - 3*b^2*c*d^3)*x)*cos(b*x + a)^4 + 3*(40*b^4*c^2*d^2 
 - 17*b^2*d^4)*x^2 - (64*b^4*d^4*x^4 + 256*b^4*c*d^3*x^3 + 64*b^4*c^4 - 12 
0*b^2*c^2*d^2 + 51*d^4 + 24*(16*b^4*c^2*d^2 - 5*b^2*d^4)*x^2 + 16*(16*b^4* 
c^3*d - 15*b^2*c*d^3)*x)*cos(b*x + a)^2 + 2*(40*b^4*c^3*d - 51*b^2*c*d^3)* 
x - 2*(2*(8*b^3*d^4*x^3 + 24*b^3*c*d^3*x^2 + 8*b^3*c^3*d - 3*b*c*d^3 + 3*( 
8*b^3*c^2*d^2 - b*d^4)*x)*cos(b*x + a)^3 - (40*b^3*d^4*x^3 + 120*b^3*c*d^3 
*x^2 + 40*b^3*c^3*d - 51*b*c*d^3 + 3*(40*b^3*c^2*d^2 - 17*b*d^4)*x)*cos(b* 
x + a))*sin(b*x + a))/b^5
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 935 vs. \(2 (252) = 504\).

Time = 0.93 (sec) , antiderivative size = 935, normalized size of antiderivative = 3.71 \[ \int (c+d x)^4 \cos (a+b x) \sin ^3(a+b x) \, dx =\text {Too large to display} \] Input:

integrate((d*x+c)**4*cos(b*x+a)*sin(b*x+a)**3,x)
 

Output:

Piecewise((c**4*sin(a + b*x)**4/(4*b) + 5*c**3*d*x*sin(a + b*x)**4/(8*b) - 
 3*c**3*d*x*sin(a + b*x)**2*cos(a + b*x)**2/(4*b) - 3*c**3*d*x*cos(a + b*x 
)**4/(8*b) + 15*c**2*d**2*x**2*sin(a + b*x)**4/(16*b) - 9*c**2*d**2*x**2*s 
in(a + b*x)**2*cos(a + b*x)**2/(8*b) - 9*c**2*d**2*x**2*cos(a + b*x)**4/(1 
6*b) + 5*c*d**3*x**3*sin(a + b*x)**4/(8*b) - 3*c*d**3*x**3*sin(a + b*x)**2 
*cos(a + b*x)**2/(4*b) - 3*c*d**3*x**3*cos(a + b*x)**4/(8*b) + 5*d**4*x**4 
*sin(a + b*x)**4/(32*b) - 3*d**4*x**4*sin(a + b*x)**2*cos(a + b*x)**2/(16* 
b) - 3*d**4*x**4*cos(a + b*x)**4/(32*b) + 5*c**3*d*sin(a + b*x)**3*cos(a + 
 b*x)/(8*b**2) + 3*c**3*d*sin(a + b*x)*cos(a + b*x)**3/(8*b**2) + 15*c**2* 
d**2*x*sin(a + b*x)**3*cos(a + b*x)/(8*b**2) + 9*c**2*d**2*x*sin(a + b*x)* 
cos(a + b*x)**3/(8*b**2) + 15*c*d**3*x**2*sin(a + b*x)**3*cos(a + b*x)/(8* 
b**2) + 9*c*d**3*x**2*sin(a + b*x)*cos(a + b*x)**3/(8*b**2) + 5*d**4*x**3* 
sin(a + b*x)**3*cos(a + b*x)/(8*b**2) + 3*d**4*x**3*sin(a + b*x)*cos(a + b 
*x)**3/(8*b**2) - 15*c**2*d**2*sin(a + b*x)**4/(32*b**3) + 9*c**2*d**2*cos 
(a + b*x)**4/(32*b**3) - 51*c*d**3*x*sin(a + b*x)**4/(64*b**3) + 9*c*d**3* 
x*sin(a + b*x)**2*cos(a + b*x)**2/(32*b**3) + 45*c*d**3*x*cos(a + b*x)**4/ 
(64*b**3) - 51*d**4*x**2*sin(a + b*x)**4/(128*b**3) + 9*d**4*x**2*sin(a + 
b*x)**2*cos(a + b*x)**2/(64*b**3) + 45*d**4*x**2*cos(a + b*x)**4/(128*b**3 
) - 51*c*d**3*sin(a + b*x)**3*cos(a + b*x)/(64*b**4) - 45*c*d**3*sin(a + b 
*x)*cos(a + b*x)**3/(64*b**4) - 51*d**4*x*sin(a + b*x)**3*cos(a + b*x)/...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 967 vs. \(2 (230) = 460\).

Time = 0.08 (sec) , antiderivative size = 967, normalized size of antiderivative = 3.84 \[ \int (c+d x)^4 \cos (a+b x) \sin ^3(a+b x) \, dx=\text {Too large to display} \] Input:

integrate((d*x+c)^4*cos(b*x+a)*sin(b*x+a)^3,x, algorithm="maxima")
 

Output:

1/1024*(256*c^4*sin(b*x + a)^4 - 1024*a*c^3*d*sin(b*x + a)^4/b + 1536*a^2* 
c^2*d^2*sin(b*x + a)^4/b^2 - 1024*a^3*c*d^3*sin(b*x + a)^4/b^3 + 256*a^4*d 
^4*sin(b*x + a)^4/b^4 + 32*(4*(b*x + a)*cos(4*b*x + 4*a) - 16*(b*x + a)*co 
s(2*b*x + 2*a) - sin(4*b*x + 4*a) + 8*sin(2*b*x + 2*a))*c^3*d/b - 96*(4*(b 
*x + a)*cos(4*b*x + 4*a) - 16*(b*x + a)*cos(2*b*x + 2*a) - sin(4*b*x + 4*a 
) + 8*sin(2*b*x + 2*a))*a*c^2*d^2/b^2 + 96*(4*(b*x + a)*cos(4*b*x + 4*a) - 
 16*(b*x + a)*cos(2*b*x + 2*a) - sin(4*b*x + 4*a) + 8*sin(2*b*x + 2*a))*a^ 
2*c*d^3/b^3 - 32*(4*(b*x + a)*cos(4*b*x + 4*a) - 16*(b*x + a)*cos(2*b*x + 
2*a) - sin(4*b*x + 4*a) + 8*sin(2*b*x + 2*a))*a^3*d^4/b^4 + 24*((8*(b*x + 
a)^2 - 1)*cos(4*b*x + 4*a) - 16*(2*(b*x + a)^2 - 1)*cos(2*b*x + 2*a) - 4*( 
b*x + a)*sin(4*b*x + 4*a) + 32*(b*x + a)*sin(2*b*x + 2*a))*c^2*d^2/b^2 - 4 
8*((8*(b*x + a)^2 - 1)*cos(4*b*x + 4*a) - 16*(2*(b*x + a)^2 - 1)*cos(2*b*x 
 + 2*a) - 4*(b*x + a)*sin(4*b*x + 4*a) + 32*(b*x + a)*sin(2*b*x + 2*a))*a* 
c*d^3/b^3 + 24*((8*(b*x + a)^2 - 1)*cos(4*b*x + 4*a) - 16*(2*(b*x + a)^2 - 
 1)*cos(2*b*x + 2*a) - 4*(b*x + a)*sin(4*b*x + 4*a) + 32*(b*x + a)*sin(2*b 
*x + 2*a))*a^2*d^4/b^4 + 4*(4*(8*(b*x + a)^3 - 3*b*x - 3*a)*cos(4*b*x + 4* 
a) - 64*(2*(b*x + a)^3 - 3*b*x - 3*a)*cos(2*b*x + 2*a) - 3*(8*(b*x + a)^2 
- 1)*sin(4*b*x + 4*a) + 96*(2*(b*x + a)^2 - 1)*sin(2*b*x + 2*a))*c*d^3/b^3 
 - 4*(4*(8*(b*x + a)^3 - 3*b*x - 3*a)*cos(4*b*x + 4*a) - 64*(2*(b*x + a)^3 
 - 3*b*x - 3*a)*cos(2*b*x + 2*a) - 3*(8*(b*x + a)^2 - 1)*sin(4*b*x + 4*...
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 361, normalized size of antiderivative = 1.43 \[ \int (c+d x)^4 \cos (a+b x) \sin ^3(a+b x) \, dx=\frac {{\left (32 \, b^{4} d^{4} x^{4} + 128 \, b^{4} c d^{3} x^{3} + 192 \, b^{4} c^{2} d^{2} x^{2} + 128 \, b^{4} c^{3} d x + 32 \, b^{4} c^{4} - 24 \, b^{2} d^{4} x^{2} - 48 \, b^{2} c d^{3} x - 24 \, b^{2} c^{2} d^{2} + 3 \, d^{4}\right )} \cos \left (4 \, b x + 4 \, a\right )}{1024 \, b^{5}} - \frac {{\left (2 \, b^{4} d^{4} x^{4} + 8 \, b^{4} c d^{3} x^{3} + 12 \, b^{4} c^{2} d^{2} x^{2} + 8 \, b^{4} c^{3} d x + 2 \, b^{4} c^{4} - 6 \, b^{2} d^{4} x^{2} - 12 \, b^{2} c d^{3} x - 6 \, b^{2} c^{2} d^{2} + 3 \, d^{4}\right )} \cos \left (2 \, b x + 2 \, a\right )}{16 \, b^{5}} - \frac {{\left (8 \, b^{3} d^{4} x^{3} + 24 \, b^{3} c d^{3} x^{2} + 24 \, b^{3} c^{2} d^{2} x + 8 \, b^{3} c^{3} d - 3 \, b d^{4} x - 3 \, b c d^{3}\right )} \sin \left (4 \, b x + 4 \, a\right )}{256 \, b^{5}} + \frac {{\left (2 \, b^{3} d^{4} x^{3} + 6 \, b^{3} c d^{3} x^{2} + 6 \, b^{3} c^{2} d^{2} x + 2 \, b^{3} c^{3} d - 3 \, b d^{4} x - 3 \, b c d^{3}\right )} \sin \left (2 \, b x + 2 \, a\right )}{8 \, b^{5}} \] Input:

integrate((d*x+c)^4*cos(b*x+a)*sin(b*x+a)^3,x, algorithm="giac")
 

Output:

1/1024*(32*b^4*d^4*x^4 + 128*b^4*c*d^3*x^3 + 192*b^4*c^2*d^2*x^2 + 128*b^4 
*c^3*d*x + 32*b^4*c^4 - 24*b^2*d^4*x^2 - 48*b^2*c*d^3*x - 24*b^2*c^2*d^2 + 
 3*d^4)*cos(4*b*x + 4*a)/b^5 - 1/16*(2*b^4*d^4*x^4 + 8*b^4*c*d^3*x^3 + 12* 
b^4*c^2*d^2*x^2 + 8*b^4*c^3*d*x + 2*b^4*c^4 - 6*b^2*d^4*x^2 - 12*b^2*c*d^3 
*x - 6*b^2*c^2*d^2 + 3*d^4)*cos(2*b*x + 2*a)/b^5 - 1/256*(8*b^3*d^4*x^3 + 
24*b^3*c*d^3*x^2 + 24*b^3*c^2*d^2*x + 8*b^3*c^3*d - 3*b*d^4*x - 3*b*c*d^3) 
*sin(4*b*x + 4*a)/b^5 + 1/8*(2*b^3*d^4*x^3 + 6*b^3*c*d^3*x^2 + 6*b^3*c^2*d 
^2*x + 2*b^3*c^3*d - 3*b*d^4*x - 3*b*c*d^3)*sin(2*b*x + 2*a)/b^5
 

Mupad [B] (verification not implemented)

Time = 21.00 (sec) , antiderivative size = 576, normalized size of antiderivative = 2.29 \[ \int (c+d x)^4 \cos (a+b x) \sin ^3(a+b x) \, dx=-\frac {192\,d^4\,\cos \left (2\,a+2\,b\,x\right )-3\,d^4\,\cos \left (4\,a+4\,b\,x\right )+128\,b^4\,c^4\,\cos \left (2\,a+2\,b\,x\right )-32\,b^4\,c^4\,\cos \left (4\,a+4\,b\,x\right )-256\,b^3\,c^3\,d\,\sin \left (2\,a+2\,b\,x\right )+32\,b^3\,c^3\,d\,\sin \left (4\,a+4\,b\,x\right )-384\,b^2\,c^2\,d^2\,\cos \left (2\,a+2\,b\,x\right )+24\,b^2\,c^2\,d^2\,\cos \left (4\,a+4\,b\,x\right )-384\,b^2\,d^4\,x^2\,\cos \left (2\,a+2\,b\,x\right )+24\,b^2\,d^4\,x^2\,\cos \left (4\,a+4\,b\,x\right )+128\,b^4\,d^4\,x^4\,\cos \left (2\,a+2\,b\,x\right )-32\,b^4\,d^4\,x^4\,\cos \left (4\,a+4\,b\,x\right )-256\,b^3\,d^4\,x^3\,\sin \left (2\,a+2\,b\,x\right )+32\,b^3\,d^4\,x^3\,\sin \left (4\,a+4\,b\,x\right )+384\,b\,c\,d^3\,\sin \left (2\,a+2\,b\,x\right )-12\,b\,c\,d^3\,\sin \left (4\,a+4\,b\,x\right )+384\,b\,d^4\,x\,\sin \left (2\,a+2\,b\,x\right )-12\,b\,d^4\,x\,\sin \left (4\,a+4\,b\,x\right )+768\,b^4\,c^2\,d^2\,x^2\,\cos \left (2\,a+2\,b\,x\right )-192\,b^4\,c^2\,d^2\,x^2\,\cos \left (4\,a+4\,b\,x\right )-768\,b^2\,c\,d^3\,x\,\cos \left (2\,a+2\,b\,x\right )+512\,b^4\,c^3\,d\,x\,\cos \left (2\,a+2\,b\,x\right )+48\,b^2\,c\,d^3\,x\,\cos \left (4\,a+4\,b\,x\right )-128\,b^4\,c^3\,d\,x\,\cos \left (4\,a+4\,b\,x\right )+512\,b^4\,c\,d^3\,x^3\,\cos \left (2\,a+2\,b\,x\right )-128\,b^4\,c\,d^3\,x^3\,\cos \left (4\,a+4\,b\,x\right )-768\,b^3\,c^2\,d^2\,x\,\sin \left (2\,a+2\,b\,x\right )-768\,b^3\,c\,d^3\,x^2\,\sin \left (2\,a+2\,b\,x\right )+96\,b^3\,c^2\,d^2\,x\,\sin \left (4\,a+4\,b\,x\right )+96\,b^3\,c\,d^3\,x^2\,\sin \left (4\,a+4\,b\,x\right )}{1024\,b^5} \] Input:

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

Output:

-(192*d^4*cos(2*a + 2*b*x) - 3*d^4*cos(4*a + 4*b*x) + 128*b^4*c^4*cos(2*a 
+ 2*b*x) - 32*b^4*c^4*cos(4*a + 4*b*x) - 256*b^3*c^3*d*sin(2*a + 2*b*x) + 
32*b^3*c^3*d*sin(4*a + 4*b*x) - 384*b^2*c^2*d^2*cos(2*a + 2*b*x) + 24*b^2* 
c^2*d^2*cos(4*a + 4*b*x) - 384*b^2*d^4*x^2*cos(2*a + 2*b*x) + 24*b^2*d^4*x 
^2*cos(4*a + 4*b*x) + 128*b^4*d^4*x^4*cos(2*a + 2*b*x) - 32*b^4*d^4*x^4*co 
s(4*a + 4*b*x) - 256*b^3*d^4*x^3*sin(2*a + 2*b*x) + 32*b^3*d^4*x^3*sin(4*a 
 + 4*b*x) + 384*b*c*d^3*sin(2*a + 2*b*x) - 12*b*c*d^3*sin(4*a + 4*b*x) + 3 
84*b*d^4*x*sin(2*a + 2*b*x) - 12*b*d^4*x*sin(4*a + 4*b*x) + 768*b^4*c^2*d^ 
2*x^2*cos(2*a + 2*b*x) - 192*b^4*c^2*d^2*x^2*cos(4*a + 4*b*x) - 768*b^2*c* 
d^3*x*cos(2*a + 2*b*x) + 512*b^4*c^3*d*x*cos(2*a + 2*b*x) + 48*b^2*c*d^3*x 
*cos(4*a + 4*b*x) - 128*b^4*c^3*d*x*cos(4*a + 4*b*x) + 512*b^4*c*d^3*x^3*c 
os(2*a + 2*b*x) - 128*b^4*c*d^3*x^3*cos(4*a + 4*b*x) - 768*b^3*c^2*d^2*x*s 
in(2*a + 2*b*x) - 768*b^3*c*d^3*x^2*sin(2*a + 2*b*x) + 96*b^3*c^2*d^2*x*si 
n(4*a + 4*b*x) + 96*b^3*c*d^3*x^2*sin(4*a + 4*b*x))/(1024*b^5)
                                                                                    
                                                                                    
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 613, normalized size of antiderivative = 2.43 \[ \int (c+d x)^4 \cos (a+b x) \sin ^3(a+b x) \, dx=\frac {-72 \sin \left (b x +a \right )^{2} b^{2} d^{4} x^{2}+48 \cos \left (b x +a \right ) \sin \left (b x +a \right ) b^{3} c^{3} d +48 \cos \left (b x +a \right ) \sin \left (b x +a \right ) b^{3} d^{4} x^{3}-90 \cos \left (b x +a \right ) \sin \left (b x +a \right ) b c \,d^{3}-90 \cos \left (b x +a \right ) \sin \left (b x +a \right ) b \,d^{4} x -144 \sin \left (b x +a \right )^{2} b^{2} c \,d^{3} x +96 \cos \left (b x +a \right ) \sin \left (b x +a \right )^{3} b^{3} c^{2} d^{2} x +96 \cos \left (b x +a \right ) \sin \left (b x +a \right )^{3} b^{3} c \,d^{3} x^{2}+32 \sin \left (b x +a \right )^{4} b^{4} d^{4} x^{4}-24 \sin \left (b x +a \right )^{4} b^{2} c^{2} d^{2}-24 \sin \left (b x +a \right )^{4} b^{2} d^{4} x^{2}-72 \sin \left (b x +a \right )^{2} b^{2} c^{2} d^{2}-48 a \,b^{3} c^{3} d -54 a b c \,d^{3}-48 b^{4} c^{3} d x -72 b^{4} c^{2} d^{2} x^{2}-48 b^{4} c \,d^{3} x^{3}+90 b^{2} c \,d^{3} x +3 \sin \left (b x +a \right )^{4} d^{4}+45 \sin \left (b x +a \right )^{2} d^{4}+144 \cos \left (b x +a \right ) \sin \left (b x +a \right ) b^{3} c^{2} d^{2} x +144 \cos \left (b x +a \right ) \sin \left (b x +a \right ) b^{3} c \,d^{3} x^{2}-45 d^{4}+32 \sin \left (b x +a \right )^{4} b^{4} c^{4}-12 b^{4} d^{4} x^{4}+72 b^{2} c^{2} d^{2}+45 b^{2} d^{4} x^{2}+32 \cos \left (b x +a \right ) \sin \left (b x +a \right )^{3} b^{3} c^{3} d +32 \cos \left (b x +a \right ) \sin \left (b x +a \right )^{3} b^{3} d^{4} x^{3}-12 \cos \left (b x +a \right ) \sin \left (b x +a \right )^{3} b c \,d^{3}-12 \cos \left (b x +a \right ) \sin \left (b x +a \right )^{3} b \,d^{4} x +128 \sin \left (b x +a \right )^{4} b^{4} c^{3} d x +192 \sin \left (b x +a \right )^{4} b^{4} c^{2} d^{2} x^{2}+128 \sin \left (b x +a \right )^{4} b^{4} c \,d^{3} x^{3}-48 \sin \left (b x +a \right )^{4} b^{2} c \,d^{3} x}{128 b^{5}} \] Input:

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

Output:

(32*cos(a + b*x)*sin(a + b*x)**3*b**3*c**3*d + 96*cos(a + b*x)*sin(a + b*x 
)**3*b**3*c**2*d**2*x + 96*cos(a + b*x)*sin(a + b*x)**3*b**3*c*d**3*x**2 + 
 32*cos(a + b*x)*sin(a + b*x)**3*b**3*d**4*x**3 - 12*cos(a + b*x)*sin(a + 
b*x)**3*b*c*d**3 - 12*cos(a + b*x)*sin(a + b*x)**3*b*d**4*x + 48*cos(a + b 
*x)*sin(a + b*x)*b**3*c**3*d + 144*cos(a + b*x)*sin(a + b*x)*b**3*c**2*d** 
2*x + 144*cos(a + b*x)*sin(a + b*x)*b**3*c*d**3*x**2 + 48*cos(a + b*x)*sin 
(a + b*x)*b**3*d**4*x**3 - 90*cos(a + b*x)*sin(a + b*x)*b*c*d**3 - 90*cos( 
a + b*x)*sin(a + b*x)*b*d**4*x + 32*sin(a + b*x)**4*b**4*c**4 + 128*sin(a 
+ b*x)**4*b**4*c**3*d*x + 192*sin(a + b*x)**4*b**4*c**2*d**2*x**2 + 128*si 
n(a + b*x)**4*b**4*c*d**3*x**3 + 32*sin(a + b*x)**4*b**4*d**4*x**4 - 24*si 
n(a + b*x)**4*b**2*c**2*d**2 - 48*sin(a + b*x)**4*b**2*c*d**3*x - 24*sin(a 
 + b*x)**4*b**2*d**4*x**2 + 3*sin(a + b*x)**4*d**4 - 72*sin(a + b*x)**2*b* 
*2*c**2*d**2 - 144*sin(a + b*x)**2*b**2*c*d**3*x - 72*sin(a + b*x)**2*b**2 
*d**4*x**2 + 45*sin(a + b*x)**2*d**4 - 48*a*b**3*c**3*d - 54*a*b*c*d**3 - 
48*b**4*c**3*d*x - 72*b**4*c**2*d**2*x**2 - 48*b**4*c*d**3*x**3 - 12*b**4* 
d**4*x**4 + 72*b**2*c**2*d**2 + 90*b**2*c*d**3*x + 45*b**2*d**4*x**2 - 45* 
d**4)/(128*b**5)