\(\int (c+d x)^4 \cos ^3(a+b x) \sin (a+b x) \, dx\) [137]

Optimal result

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

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

Mathematica [A] (verified)

Time = 1.19 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.63 \[ \int (c+d x)^4 \cos ^3(a+b x) \sin (a+b x) \, dx=-\frac {64 \left (3 d^4-6 b^2 d^2 (c+d x)^2+2 b^4 (c+d x)^4\right ) \cos (2 (a+b x))+\left (3 d^4-24 b^2 d^2 (c+d x)^2+32 b^4 (c+d x)^4\right ) \cos (4 (a+b x))-8 b d (c+d x) \left (16 \left (-3 d^2+2 b^2 (c+d x)^2\right )+\left (-3 d^2+8 b^2 (c+d x)^2\right ) \cos (2 (a+b x))\right ) \sin (2 (a+b x))}{1024 b^5} \] Input:

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

Output:

-1/1024*(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)])/b^5
 

Rubi [A] (verified)

Time = 1.01 (sec) , antiderivative size = 325, 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 = {4905, 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.

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

\(\Big \downarrow \) 4905

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3792

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3791

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3791

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

\(\Big \downarrow \) 17

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

\(\Big \downarrow \) 3792

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

\(\Big \downarrow \) 17

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3791

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

\(\Big \downarrow \) 17

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

Input:

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

Output:

-1/4*((c + d*x)^4*Cos[a + b*x]^4)/b + (d*((3*d*(c + d*x)^2*Cos[a + b*x]^4) 
/(16*b^2) + ((c + d*x)^3*Cos[a + b*x]^3*Sin[a + b*x])/(4*b) - (3*d^2*((d*C 
os[a + b*x]^4)/(16*b^2) + ((c + d*x)*Cos[a + b*x]^3*Sin[a + b*x])/(4*b) + 
(3*((c + d*x)^2/(4*d) + (d*Cos[a + b*x]^2)/(4*b^2) + ((c + d*x)*Cos[a + b* 
x]*Sin[a + b*x])/(2*b)))/4))/(8*b^2) + (3*((c + d*x)^4/(8*d) + (3*d*(c + d 
*x)^2*Cos[a + b*x]^2)/(4*b^2) + ((c + d*x)^3*Cos[a + b*x]*Sin[a + b*x])/(2 
*b) - (3*d^2*((c + d*x)^2/(4*d) + (d*Cos[a + b*x]^2)/(4*b^2) + ((c + d*x)* 
Cos[a + b*x]*Sin[a + b*x])/(2*b)))/(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 4905

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

Maple [A] (verified)

Time = 2.28 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.74

method	result	size
parallelrisch	\(\frac {\left (-128 \left (d x +c \right )^{4} b^{4}+384 d^{2} \left (d x +c \right )^{2} b^{2}-192 d^{4}\right ) \cos \left (2 b x +2 a \right )+\left (-32 \left (d x +c \right )^{4} b^{4}+24 d^{2} \left (d x +c \right )^{2} b^{2}-3 d^{4}\right ) \cos \left (4 b x +4 a \right )+256 d \left (\left (d x +c \right )^{2} b^{2}-\frac {3 d^{2}}{2}\right ) \left (d x +c \right ) b \sin \left (2 b x +2 a \right )+32 d \left (d x +c \right ) \left (\left (d x +c \right )^{2} b^{2}-\frac {3 d^{2}}{8}\right ) b \sin \left (4 b x +4 a \right )+160 b^{4} c^{4}-408 b^{2} c^{2} d^{2}+195 d^{4}}{1024 b^{5}}\)	\(187\)
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	\(\text {Expression too large to display}\)	\(1004\)
derivativedivides	\(\text {Expression too large to display}\)	\(1134\)
default	\(\text {Expression too large to display}\)	\(1134\)

Input:

int((d*x+c)^4*cos(b*x+a)^3*sin(b*x+a),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)*((d*x+c)^2*b^2-3/8 
*d^2)*b*sin(4*b*x+4*a)+160*b^4*c^4-408*b^2*c^2*d^2+195*d^4)/b^5
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 378, normalized size of antiderivative = 1.50 \[ \int (c+d x)^4 \cos ^3(a+b x) \sin (a+b x) \, dx=\frac {12 \, b^{4} d^{4} x^{4} + 48 \, b^{4} c d^{3} x^{3} - {\left (32 \, b^{4} d^{4} x^{4} + 128 \, b^{4} c d^{3} x^{3} + 32 \, b^{4} c^{4} - 24 \, b^{2} c^{2} d^{2} + 3 \, d^{4} + 24 \, {\left (8 \, b^{4} c^{2} d^{2} - b^{2} d^{4}\right )} x^{2} + 16 \, {\left (8 \, b^{4} c^{3} d - 3 \, b^{2} c d^{3}\right )} x\right )} \cos \left (b x + a\right )^{4} + 9 \, {\left (8 \, b^{4} c^{2} d^{2} - 5 \, b^{2} d^{4}\right )} x^{2} + 9 \, {\left (8 \, b^{2} d^{4} x^{2} + 16 \, b^{2} c d^{3} x + 8 \, b^{2} c^{2} d^{2} - 5 \, d^{4}\right )} \cos \left (b x + a\right )^{2} + 6 \, {\left (8 \, b^{4} c^{3} d - 15 \, b^{2} c d^{3}\right )} x + 2 \, {\left (2 \, {\left (8 \, b^{3} d^{4} x^{3} + 24 \, b^{3} c d^{3} x^{2} + 8 \, b^{3} c^{3} d - 3 \, b c d^{3} + 3 \, {\left (8 \, b^{3} c^{2} d^{2} - b d^{4}\right )} x\right )} \cos \left (b x + a\right )^{3} + 3 \, {\left (8 \, b^{3} d^{4} x^{3} + 24 \, b^{3} c d^{3} x^{2} + 8 \, b^{3} c^{3} d - 15 \, b c d^{3} + 3 \, {\left (8 \, b^{3} c^{2} d^{2} - 5 \, b d^{4}\right )} x\right )} \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )}{128 \, b^{5}} \] Input:

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

Output:

1/128*(12*b^4*d^4*x^4 + 48*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 + 9*(8*b^4*c^2*d^2 
- 5*b^2*d^4)*x^2 + 9*(8*b^2*d^4*x^2 + 16*b^2*c*d^3*x + 8*b^2*c^2*d^2 - 5*d 
^4)*cos(b*x + a)^2 + 6*(8*b^4*c^3*d - 15*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 + 3*(8*b^3*d^4*x^3 + 24*b^3*c*d^3*x^2 + 8*b^3*c^3*d - 1 
5*b*c*d^3 + 3*(8*b^3*c^2*d^2 - 5*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 \[ \int (c+d x)^4 \cos ^3(a+b x) \sin (a+b x) \, dx =\text {Too large to display} \] Input:

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

Output:

Piecewise((-c**4*cos(a + b*x)**4/(4*b) + 3*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) - 5*c**3*d*x*cos(a + b* 
x)**4/(8*b) + 9*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) - 15*c**2*d**2*x**2*cos(a + b*x)**4/( 
16*b) + 3*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) - 5*c*d**3*x**3*cos(a + b*x)**4/(8*b) + 3*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) - 5*d**4*x**4*cos(a + b*x)**4/(32*b) + 3*c**3*d*sin(a + b*x)**3*cos(a 
+ b*x)/(8*b**2) + 5*c**3*d*sin(a + b*x)*cos(a + b*x)**3/(8*b**2) + 9*c**2* 
d**2*x*sin(a + b*x)**3*cos(a + b*x)/(8*b**2) + 15*c**2*d**2*x*sin(a + b*x) 
*cos(a + b*x)**3/(8*b**2) + 9*c*d**3*x**2*sin(a + b*x)**3*cos(a + b*x)/(8* 
b**2) + 15*c*d**3*x**2*sin(a + b*x)*cos(a + b*x)**3/(8*b**2) + 3*d**4*x**3 
*sin(a + b*x)**3*cos(a + b*x)/(8*b**2) + 5*d**4*x**3*sin(a + b*x)*cos(a + 
b*x)**3/(8*b**2) - 9*c**2*d**2*sin(a + b*x)**4/(32*b**3) + 15*c**2*d**2*co 
s(a + b*x)**4/(32*b**3) - 45*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) + 51*c*d**3*x*cos(a + b*x)**4 
/(64*b**3) - 45*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) + 51*d**4*x**2*cos(a + b*x)**4/(128*b** 
3) - 45*c*d**3*sin(a + b*x)**3*cos(a + b*x)/(64*b**4) - 51*c*d**3*sin(a + 
b*x)*cos(a + b*x)**3/(64*b**4) - 45*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 \[ \int (c+d x)^4 \cos ^3(a+b x) \sin (a+b x) \, dx=\text {Too large to display} \] Input:

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

Output:

-1/1024*(256*c^4*cos(b*x + a)^4 - 1024*a*c^3*d*cos(b*x + a)^4/b + 1536*a^2 
*c^2*d^2*cos(b*x + a)^4/b^2 - 1024*a^3*c*d^3*cos(b*x + a)^4/b^3 + 256*a^4* 
d^4*cos(b*x + a)^4/b^4 + 32*(4*(b*x + a)*cos(4*b*x + 4*a) + 16*(b*x + a)*c 
os(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 - 
48*((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 = 0.14 (sec) , antiderivative size = 361, normalized size of antiderivative = 1.43 \[ \int (c+d x)^4 \cos ^3(a+b x) \sin (a+b x) \, dx=-\frac {{\left (32 \, b^{4} d^{4} x^{4} + 128 \, b^{4} c d^{3} x^{3} + 192 \, b^{4} c^{2} d^{2} x^{2} + 128 \, b^{4} c^{3} d x + 32 \, b^{4} c^{4} - 24 \, b^{2} d^{4} x^{2} - 48 \, b^{2} c d^{3} x - 24 \, b^{2} c^{2} d^{2} + 3 \, d^{4}\right )} \cos \left (4 \, b x + 4 \, a\right )}{1024 \, b^{5}} - \frac {{\left (2 \, b^{4} d^{4} x^{4} + 8 \, b^{4} c d^{3} x^{3} + 12 \, b^{4} c^{2} d^{2} x^{2} + 8 \, b^{4} c^{3} d x + 2 \, b^{4} c^{4} - 6 \, b^{2} d^{4} x^{2} - 12 \, b^{2} c d^{3} x - 6 \, b^{2} c^{2} d^{2} + 3 \, d^{4}\right )} \cos \left (2 \, b x + 2 \, a\right )}{16 \, b^{5}} + \frac {{\left (8 \, b^{3} d^{4} x^{3} + 24 \, b^{3} c d^{3} x^{2} + 24 \, b^{3} c^{2} d^{2} x + 8 \, b^{3} c^{3} d - 3 \, b d^{4} x - 3 \, b c d^{3}\right )} \sin \left (4 \, b x + 4 \, a\right )}{256 \, b^{5}} + \frac {{\left (2 \, b^{3} d^{4} x^{3} + 6 \, b^{3} c d^{3} x^{2} + 6 \, b^{3} c^{2} d^{2} x + 2 \, b^{3} c^{3} d - 3 \, b d^{4} x - 3 \, b c d^{3}\right )} \sin \left (2 \, b x + 2 \, a\right )}{8 \, b^{5}} \] Input:

integrate((d*x+c)^4*cos(b*x+a)^3*sin(b*x+a),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 = 19.17 (sec) , antiderivative size = 576, normalized size of antiderivative = 2.29 \[ \int (c+d x)^4 \cos ^3(a+b x) \sin (a+b x) \, dx=-\frac {192\,d^4\,\cos \left (2\,a+2\,b\,x\right )+3\,d^4\,\cos \left (4\,a+4\,b\,x\right )+128\,b^4\,c^4\,\cos \left (2\,a+2\,b\,x\right )+32\,b^4\,c^4\,\cos \left (4\,a+4\,b\,x\right )-256\,b^3\,c^3\,d\,\sin \left (2\,a+2\,b\,x\right )-32\,b^3\,c^3\,d\,\sin \left (4\,a+4\,b\,x\right )-384\,b^2\,c^2\,d^2\,\cos \left (2\,a+2\,b\,x\right )-24\,b^2\,c^2\,d^2\,\cos \left (4\,a+4\,b\,x\right )-384\,b^2\,d^4\,x^2\,\cos \left (2\,a+2\,b\,x\right )-24\,b^2\,d^4\,x^2\,\cos \left (4\,a+4\,b\,x\right )+128\,b^4\,d^4\,x^4\,\cos \left (2\,a+2\,b\,x\right )+32\,b^4\,d^4\,x^4\,\cos \left (4\,a+4\,b\,x\right )-256\,b^3\,d^4\,x^3\,\sin \left (2\,a+2\,b\,x\right )-32\,b^3\,d^4\,x^3\,\sin \left (4\,a+4\,b\,x\right )+384\,b\,c\,d^3\,\sin \left (2\,a+2\,b\,x\right )+12\,b\,c\,d^3\,\sin \left (4\,a+4\,b\,x\right )+384\,b\,d^4\,x\,\sin \left (2\,a+2\,b\,x\right )+12\,b\,d^4\,x\,\sin \left (4\,a+4\,b\,x\right )+768\,b^4\,c^2\,d^2\,x^2\,\cos \left (2\,a+2\,b\,x\right )+192\,b^4\,c^2\,d^2\,x^2\,\cos \left (4\,a+4\,b\,x\right )-768\,b^2\,c\,d^3\,x\,\cos \left (2\,a+2\,b\,x\right )+512\,b^4\,c^3\,d\,x\,\cos \left (2\,a+2\,b\,x\right )-48\,b^2\,c\,d^3\,x\,\cos \left (4\,a+4\,b\,x\right )+128\,b^4\,c^3\,d\,x\,\cos \left (4\,a+4\,b\,x\right )+512\,b^4\,c\,d^3\,x^3\,\cos \left (2\,a+2\,b\,x\right )+128\,b^4\,c\,d^3\,x^3\,\cos \left (4\,a+4\,b\,x\right )-768\,b^3\,c^2\,d^2\,x\,\sin \left (2\,a+2\,b\,x\right )-768\,b^3\,c\,d^3\,x^2\,\sin \left (2\,a+2\,b\,x\right )-96\,b^3\,c^2\,d^2\,x\,\sin \left (4\,a+4\,b\,x\right )-96\,b^3\,c\,d^3\,x^2\,\sin \left (4\,a+4\,b\,x\right )}{1024\,b^5} \] Input:

int(cos(a + b*x)^3*sin(a + b*x)*(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 = 0.17 (sec) , antiderivative size = 698, normalized size of antiderivative = 2.77 \[ \int (c+d x)^4 \cos ^3(a+b x) \sin (a+b x) \, dx =\text {Too large to display} \] Input:

int((d*x+c)^4*cos(b*x+a)^3*sin(b*x+a),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 + 80*cos(a 
+ b*x)*sin(a + b*x)*b**3*c**3*d + 240*cos(a + b*x)*sin(a + b*x)*b**3*c**2* 
d**2*x + 240*cos(a + b*x)*sin(a + b*x)*b**3*c*d**3*x**2 + 80*cos(a + b*x)* 
sin(a + b*x)*b**3*d**4*x**3 - 102*cos(a + b*x)*sin(a + b*x)*b*c*d**3 - 102 
*cos(a + b*x)*sin(a + b*x)*b*d**4*x - 32*sin(a + b*x)**4*b**4*c**4 - 128*s 
in(a + b*x)**4*b**4*c**3*d*x - 192*sin(a + b*x)**4*b**4*c**2*d**2*x**2 - 1 
28*sin(a + b*x)**4*b**4*c*d**3*x**3 - 32*sin(a + b*x)**4*b**4*d**4*x**4 + 
24*sin(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 + 64*sin(a + b*x)* 
*2*b**4*c**4 + 256*sin(a + b*x)**2*b**4*c**3*d*x + 384*sin(a + b*x)**2*b** 
4*c**2*d**2*x**2 + 256*sin(a + b*x)**2*b**4*c*d**3*x**3 + 64*sin(a + b*x)* 
*2*b**4*d**4*x**4 - 120*sin(a + b*x)**2*b**2*c**2*d**2 - 240*sin(a + b*x)* 
*2*b**2*c*d**3*x - 120*sin(a + b*x)**2*b**2*d**4*x**2 + 51*sin(a + b*x)**2 
*d**4 - 64*b**4*c**4 - 80*b**4*c**3*d*x - 120*b**4*c**2*d**2*x**2 - 80*b** 
4*c*d**3*x**3 - 20*b**4*d**4*x**4 + 120*b**2*c**2*d**2 + 102*b**2*c*d**3*x 
 + 51*b**2*d**4*x**2 - 51*d**4)/(128*b**5)