\(\int (c+d x)^4 \cos ^3(a+b x) \, dx\) [16]

Optimal result

Integrand size = 16, antiderivative size = 225 \[ \int (c+d x)^4 \cos ^3(a+b x) \, dx=-\frac {160 d^3 (c+d x) \cos (a+b x)}{9 b^4}+\frac {8 d (c+d x)^3 \cos (a+b x)}{3 b^2}-\frac {8 d^3 (c+d x) \cos ^3(a+b x)}{27 b^4}+\frac {4 d (c+d x)^3 \cos ^3(a+b x)}{9 b^2}+\frac {488 d^4 \sin (a+b x)}{27 b^5}-\frac {80 d^2 (c+d x)^2 \sin (a+b x)}{9 b^3}+\frac {2 (c+d x)^4 \sin (a+b x)}{3 b}-\frac {4 d^2 (c+d x)^2 \cos ^2(a+b x) \sin (a+b x)}{9 b^3}+\frac {(c+d x)^4 \cos ^2(a+b x) \sin (a+b x)}{3 b}-\frac {8 d^4 \sin ^3(a+b x)}{81 b^5} \] Output:

-160/9*d^3*(d*x+c)*cos(b*x+a)/b^4+8/3*d*(d*x+c)^3*cos(b*x+a)/b^2-8/27*d^3* 
(d*x+c)*cos(b*x+a)^3/b^4+4/9*d*(d*x+c)^3*cos(b*x+a)^3/b^2+488/27*d^4*sin(b 
*x+a)/b^5-80/9*d^2*(d*x+c)^2*sin(b*x+a)/b^3+2/3*(d*x+c)^4*sin(b*x+a)/b-4/9 
*d^2*(d*x+c)^2*cos(b*x+a)^2*sin(b*x+a)/b^3+1/3*(d*x+c)^4*cos(b*x+a)^2*sin( 
b*x+a)/b-8/81*d^4*sin(b*x+a)^3/b^5
 

Mathematica [A] (verified)

Time = 1.09 (sec) , antiderivative size = 385, normalized size of antiderivative = 1.71 \[ \int (c+d x)^4 \cos ^3(a+b x) \, dx=\frac {972 b d (c+d x) \left (-6 d^2+b^2 (c+d x)^2\right ) \cos (a+b x)+12 b d (c+d x) \left (-2 d^2+3 b^2 (c+d x)^2\right ) \cos (3 (a+b x))+243 b^4 c^4 \sin (a+b x)-2916 b^2 c^2 d^2 \sin (a+b x)+5832 d^4 \sin (a+b x)+972 b^4 c^3 d x \sin (a+b x)-5832 b^2 c d^3 x \sin (a+b x)+1458 b^4 c^2 d^2 x^2 \sin (a+b x)-2916 b^2 d^4 x^2 \sin (a+b x)+972 b^4 c d^3 x^3 \sin (a+b x)+243 b^4 d^4 x^4 \sin (a+b x)+27 b^4 c^4 \sin (3 (a+b x))-36 b^2 c^2 d^2 \sin (3 (a+b x))+8 d^4 \sin (3 (a+b x))+108 b^4 c^3 d x \sin (3 (a+b x))-72 b^2 c d^3 x \sin (3 (a+b x))+162 b^4 c^2 d^2 x^2 \sin (3 (a+b x))-36 b^2 d^4 x^2 \sin (3 (a+b x))+108 b^4 c d^3 x^3 \sin (3 (a+b x))+27 b^4 d^4 x^4 \sin (3 (a+b x))}{324 b^5} \] Input:

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

Output:

(972*b*d*(c + d*x)*(-6*d^2 + b^2*(c + d*x)^2)*Cos[a + b*x] + 12*b*d*(c + d 
*x)*(-2*d^2 + 3*b^2*(c + d*x)^2)*Cos[3*(a + b*x)] + 243*b^4*c^4*Sin[a + b* 
x] - 2916*b^2*c^2*d^2*Sin[a + b*x] + 5832*d^4*Sin[a + b*x] + 972*b^4*c^3*d 
*x*Sin[a + b*x] - 5832*b^2*c*d^3*x*Sin[a + b*x] + 1458*b^4*c^2*d^2*x^2*Sin 
[a + b*x] - 2916*b^2*d^4*x^2*Sin[a + b*x] + 972*b^4*c*d^3*x^3*Sin[a + b*x] 
 + 243*b^4*d^4*x^4*Sin[a + b*x] + 27*b^4*c^4*Sin[3*(a + b*x)] - 36*b^2*c^2 
*d^2*Sin[3*(a + b*x)] + 8*d^4*Sin[3*(a + b*x)] + 108*b^4*c^3*d*x*Sin[3*(a 
+ b*x)] - 72*b^2*c*d^3*x*Sin[3*(a + b*x)] + 162*b^4*c^2*d^2*x^2*Sin[3*(a + 
 b*x)] - 36*b^2*d^4*x^2*Sin[3*(a + b*x)] + 108*b^4*c*d^3*x^3*Sin[3*(a + b* 
x)] + 27*b^4*d^4*x^4*Sin[3*(a + b*x)])/(324*b^5)
 

Rubi [A] (verified)

Time = 1.80 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.35, number of steps used = 25, number of rules used = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.500, Rules used = {3042, 3792, 3042, 3777, 25, 3042, 3777, 3042, 3777, 25, 3042, 3777, 3042, 3117, 3792, 3042, 3113, 2009, 3777, 25, 3042, 3777, 3042, 3117}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3792

\(\displaystyle -\frac {4 d^2 \int (c+d x)^2 \cos ^3(a+b x)dx}{3 b^2}+\frac {2}{3} \int (c+d x)^4 \cos (a+b x)dx+\frac {4 d (c+d x)^3 \cos ^3(a+b x)}{9 b^2}+\frac {(c+d x)^4 \sin (a+b x) \cos ^2(a+b x)}{3 b}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {4 d^2 \int (c+d x)^2 \sin \left (a+b x+\frac {\pi }{2}\right )^3dx}{3 b^2}+\frac {2}{3} \int (c+d x)^4 \sin \left (a+b x+\frac {\pi }{2}\right )dx+\frac {4 d (c+d x)^3 \cos ^3(a+b x)}{9 b^2}+\frac {(c+d x)^4 \sin (a+b x) \cos ^2(a+b x)}{3 b}\)

\(\Big \downarrow \) 3777

\(\displaystyle -\frac {4 d^2 \int (c+d x)^2 \sin \left (a+b x+\frac {\pi }{2}\right )^3dx}{3 b^2}+\frac {2}{3} \left (\frac {4 d \int -(c+d x)^3 \sin (a+b x)dx}{b}+\frac {(c+d x)^4 \sin (a+b x)}{b}\right )+\frac {4 d (c+d x)^3 \cos ^3(a+b x)}{9 b^2}+\frac {(c+d x)^4 \sin (a+b x) \cos ^2(a+b x)}{3 b}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {4 d^2 \int (c+d x)^2 \sin \left (a+b x+\frac {\pi }{2}\right )^3dx}{3 b^2}+\frac {2}{3} \left (\frac {(c+d x)^4 \sin (a+b x)}{b}-\frac {4 d \int (c+d x)^3 \sin (a+b x)dx}{b}\right )+\frac {4 d (c+d x)^3 \cos ^3(a+b x)}{9 b^2}+\frac {(c+d x)^4 \sin (a+b x) \cos ^2(a+b x)}{3 b}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {4 d^2 \int (c+d x)^2 \sin \left (a+b x+\frac {\pi }{2}\right )^3dx}{3 b^2}+\frac {2}{3} \left (\frac {(c+d x)^4 \sin (a+b x)}{b}-\frac {4 d \int (c+d x)^3 \sin (a+b x)dx}{b}\right )+\frac {4 d (c+d x)^3 \cos ^3(a+b x)}{9 b^2}+\frac {(c+d x)^4 \sin (a+b x) \cos ^2(a+b x)}{3 b}\)

\(\Big \downarrow \) 3777

\(\displaystyle -\frac {4 d^2 \int (c+d x)^2 \sin \left (a+b x+\frac {\pi }{2}\right )^3dx}{3 b^2}+\frac {2}{3} \left (\frac {(c+d x)^4 \sin (a+b x)}{b}-\frac {4 d \left (\frac {3 d \int (c+d x)^2 \cos (a+b x)dx}{b}-\frac {(c+d x)^3 \cos (a+b x)}{b}\right )}{b}\right )+\frac {4 d (c+d x)^3 \cos ^3(a+b x)}{9 b^2}+\frac {(c+d x)^4 \sin (a+b x) \cos ^2(a+b x)}{3 b}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {4 d^2 \int (c+d x)^2 \sin \left (a+b x+\frac {\pi }{2}\right )^3dx}{3 b^2}+\frac {2}{3} \left (\frac {(c+d x)^4 \sin (a+b x)}{b}-\frac {4 d \left (\frac {3 d \int (c+d x)^2 \sin \left (a+b x+\frac {\pi }{2}\right )dx}{b}-\frac {(c+d x)^3 \cos (a+b x)}{b}\right )}{b}\right )+\frac {4 d (c+d x)^3 \cos ^3(a+b x)}{9 b^2}+\frac {(c+d x)^4 \sin (a+b x) \cos ^2(a+b x)}{3 b}\)

\(\Big \downarrow \) 3777

\(\displaystyle -\frac {4 d^2 \int (c+d x)^2 \sin \left (a+b x+\frac {\pi }{2}\right )^3dx}{3 b^2}+\frac {2}{3} \left (\frac {(c+d x)^4 \sin (a+b x)}{b}-\frac {4 d \left (\frac {3 d \left (\frac {2 d \int -((c+d x) \sin (a+b x))dx}{b}+\frac {(c+d x)^2 \sin (a+b x)}{b}\right )}{b}-\frac {(c+d x)^3 \cos (a+b x)}{b}\right )}{b}\right )+\frac {4 d (c+d x)^3 \cos ^3(a+b x)}{9 b^2}+\frac {(c+d x)^4 \sin (a+b x) \cos ^2(a+b x)}{3 b}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {4 d^2 \int (c+d x)^2 \sin \left (a+b x+\frac {\pi }{2}\right )^3dx}{3 b^2}+\frac {2}{3} \left (\frac {(c+d x)^4 \sin (a+b x)}{b}-\frac {4 d \left (\frac {3 d \left (\frac {(c+d x)^2 \sin (a+b x)}{b}-\frac {2 d \int (c+d x) \sin (a+b x)dx}{b}\right )}{b}-\frac {(c+d x)^3 \cos (a+b x)}{b}\right )}{b}\right )+\frac {4 d (c+d x)^3 \cos ^3(a+b x)}{9 b^2}+\frac {(c+d x)^4 \sin (a+b x) \cos ^2(a+b x)}{3 b}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {4 d^2 \int (c+d x)^2 \sin \left (a+b x+\frac {\pi }{2}\right )^3dx}{3 b^2}+\frac {2}{3} \left (\frac {(c+d x)^4 \sin (a+b x)}{b}-\frac {4 d \left (\frac {3 d \left (\frac {(c+d x)^2 \sin (a+b x)}{b}-\frac {2 d \int (c+d x) \sin (a+b x)dx}{b}\right )}{b}-\frac {(c+d x)^3 \cos (a+b x)}{b}\right )}{b}\right )+\frac {4 d (c+d x)^3 \cos ^3(a+b x)}{9 b^2}+\frac {(c+d x)^4 \sin (a+b x) \cos ^2(a+b x)}{3 b}\)

\(\Big \downarrow \) 3777

\(\displaystyle -\frac {4 d^2 \int (c+d x)^2 \sin \left (a+b x+\frac {\pi }{2}\right )^3dx}{3 b^2}+\frac {2}{3} \left (\frac {(c+d x)^4 \sin (a+b x)}{b}-\frac {4 d \left (\frac {3 d \left (\frac {(c+d x)^2 \sin (a+b x)}{b}-\frac {2 d \left (\frac {d \int \cos (a+b x)dx}{b}-\frac {(c+d x) \cos (a+b x)}{b}\right )}{b}\right )}{b}-\frac {(c+d x)^3 \cos (a+b x)}{b}\right )}{b}\right )+\frac {4 d (c+d x)^3 \cos ^3(a+b x)}{9 b^2}+\frac {(c+d x)^4 \sin (a+b x) \cos ^2(a+b x)}{3 b}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {4 d^2 \int (c+d x)^2 \sin \left (a+b x+\frac {\pi }{2}\right )^3dx}{3 b^2}+\frac {2}{3} \left (\frac {(c+d x)^4 \sin (a+b x)}{b}-\frac {4 d \left (\frac {3 d \left (\frac {(c+d x)^2 \sin (a+b x)}{b}-\frac {2 d \left (\frac {d \int \sin \left (a+b x+\frac {\pi }{2}\right )dx}{b}-\frac {(c+d x) \cos (a+b x)}{b}\right )}{b}\right )}{b}-\frac {(c+d x)^3 \cos (a+b x)}{b}\right )}{b}\right )+\frac {4 d (c+d x)^3 \cos ^3(a+b x)}{9 b^2}+\frac {(c+d x)^4 \sin (a+b x) \cos ^2(a+b x)}{3 b}\)

\(\Big \downarrow \) 3117

\(\displaystyle -\frac {4 d^2 \int (c+d x)^2 \sin \left (a+b x+\frac {\pi }{2}\right )^3dx}{3 b^2}+\frac {4 d (c+d x)^3 \cos ^3(a+b x)}{9 b^2}+\frac {2}{3} \left (\frac {(c+d x)^4 \sin (a+b x)}{b}-\frac {4 d \left (\frac {3 d \left (\frac {(c+d x)^2 \sin (a+b x)}{b}-\frac {2 d \left (\frac {d \sin (a+b x)}{b^2}-\frac {(c+d x) \cos (a+b x)}{b}\right )}{b}\right )}{b}-\frac {(c+d x)^3 \cos (a+b x)}{b}\right )}{b}\right )+\frac {(c+d x)^4 \sin (a+b x) \cos ^2(a+b x)}{3 b}\)

\(\Big \downarrow \) 3792

\(\displaystyle -\frac {4 d^2 \left (-\frac {2 d^2 \int \cos ^3(a+b x)dx}{9 b^2}+\frac {2}{3} \int (c+d x)^2 \cos (a+b x)dx+\frac {2 d (c+d x) \cos ^3(a+b x)}{9 b^2}+\frac {(c+d x)^2 \sin (a+b x) \cos ^2(a+b x)}{3 b}\right )}{3 b^2}+\frac {4 d (c+d x)^3 \cos ^3(a+b x)}{9 b^2}+\frac {2}{3} \left (\frac {(c+d x)^4 \sin (a+b x)}{b}-\frac {4 d \left (\frac {3 d \left (\frac {(c+d x)^2 \sin (a+b x)}{b}-\frac {2 d \left (\frac {d \sin (a+b x)}{b^2}-\frac {(c+d x) \cos (a+b x)}{b}\right )}{b}\right )}{b}-\frac {(c+d x)^3 \cos (a+b x)}{b}\right )}{b}\right )+\frac {(c+d x)^4 \sin (a+b x) \cos ^2(a+b x)}{3 b}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {4 d^2 \left (-\frac {2 d^2 \int \sin \left (a+b x+\frac {\pi }{2}\right )^3dx}{9 b^2}+\frac {2}{3} \int (c+d x)^2 \sin \left (a+b x+\frac {\pi }{2}\right )dx+\frac {2 d (c+d x) \cos ^3(a+b x)}{9 b^2}+\frac {(c+d x)^2 \sin (a+b x) \cos ^2(a+b x)}{3 b}\right )}{3 b^2}+\frac {4 d (c+d x)^3 \cos ^3(a+b x)}{9 b^2}+\frac {2}{3} \left (\frac {(c+d x)^4 \sin (a+b x)}{b}-\frac {4 d \left (\frac {3 d \left (\frac {(c+d x)^2 \sin (a+b x)}{b}-\frac {2 d \left (\frac {d \sin (a+b x)}{b^2}-\frac {(c+d x) \cos (a+b x)}{b}\right )}{b}\right )}{b}-\frac {(c+d x)^3 \cos (a+b x)}{b}\right )}{b}\right )+\frac {(c+d x)^4 \sin (a+b x) \cos ^2(a+b x)}{3 b}\)

\(\Big \downarrow \) 3113

\(\displaystyle -\frac {4 d^2 \left (\frac {2 d^2 \int \left (1-\sin ^2(a+b x)\right )d(-\sin (a+b x))}{9 b^3}+\frac {2}{3} \int (c+d x)^2 \sin \left (a+b x+\frac {\pi }{2}\right )dx+\frac {2 d (c+d x) \cos ^3(a+b x)}{9 b^2}+\frac {(c+d x)^2 \sin (a+b x) \cos ^2(a+b x)}{3 b}\right )}{3 b^2}+\frac {4 d (c+d x)^3 \cos ^3(a+b x)}{9 b^2}+\frac {2}{3} \left (\frac {(c+d x)^4 \sin (a+b x)}{b}-\frac {4 d \left (\frac {3 d \left (\frac {(c+d x)^2 \sin (a+b x)}{b}-\frac {2 d \left (\frac {d \sin (a+b x)}{b^2}-\frac {(c+d x) \cos (a+b x)}{b}\right )}{b}\right )}{b}-\frac {(c+d x)^3 \cos (a+b x)}{b}\right )}{b}\right )+\frac {(c+d x)^4 \sin (a+b x) \cos ^2(a+b x)}{3 b}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {4 d^2 \left (\frac {2}{3} \int (c+d x)^2 \sin \left (a+b x+\frac {\pi }{2}\right )dx+\frac {2 d^2 \left (\frac {1}{3} \sin ^3(a+b x)-\sin (a+b x)\right )}{9 b^3}+\frac {2 d (c+d x) \cos ^3(a+b x)}{9 b^2}+\frac {(c+d x)^2 \sin (a+b x) \cos ^2(a+b x)}{3 b}\right )}{3 b^2}+\frac {4 d (c+d x)^3 \cos ^3(a+b x)}{9 b^2}+\frac {2}{3} \left (\frac {(c+d x)^4 \sin (a+b x)}{b}-\frac {4 d \left (\frac {3 d \left (\frac {(c+d x)^2 \sin (a+b x)}{b}-\frac {2 d \left (\frac {d \sin (a+b x)}{b^2}-\frac {(c+d x) \cos (a+b x)}{b}\right )}{b}\right )}{b}-\frac {(c+d x)^3 \cos (a+b x)}{b}\right )}{b}\right )+\frac {(c+d x)^4 \sin (a+b x) \cos ^2(a+b x)}{3 b}\)

\(\Big \downarrow \) 3777

\(\displaystyle -\frac {4 d^2 \left (\frac {2}{3} \left (\frac {2 d \int -((c+d x) \sin (a+b x))dx}{b}+\frac {(c+d x)^2 \sin (a+b x)}{b}\right )+\frac {2 d^2 \left (\frac {1}{3} \sin ^3(a+b x)-\sin (a+b x)\right )}{9 b^3}+\frac {2 d (c+d x) \cos ^3(a+b x)}{9 b^2}+\frac {(c+d x)^2 \sin (a+b x) \cos ^2(a+b x)}{3 b}\right )}{3 b^2}+\frac {4 d (c+d x)^3 \cos ^3(a+b x)}{9 b^2}+\frac {2}{3} \left (\frac {(c+d x)^4 \sin (a+b x)}{b}-\frac {4 d \left (\frac {3 d \left (\frac {(c+d x)^2 \sin (a+b x)}{b}-\frac {2 d \left (\frac {d \sin (a+b x)}{b^2}-\frac {(c+d x) \cos (a+b x)}{b}\right )}{b}\right )}{b}-\frac {(c+d x)^3 \cos (a+b x)}{b}\right )}{b}\right )+\frac {(c+d x)^4 \sin (a+b x) \cos ^2(a+b x)}{3 b}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {4 d^2 \left (\frac {2}{3} \left (\frac {(c+d x)^2 \sin (a+b x)}{b}-\frac {2 d \int (c+d x) \sin (a+b x)dx}{b}\right )+\frac {2 d^2 \left (\frac {1}{3} \sin ^3(a+b x)-\sin (a+b x)\right )}{9 b^3}+\frac {2 d (c+d x) \cos ^3(a+b x)}{9 b^2}+\frac {(c+d x)^2 \sin (a+b x) \cos ^2(a+b x)}{3 b}\right )}{3 b^2}+\frac {4 d (c+d x)^3 \cos ^3(a+b x)}{9 b^2}+\frac {2}{3} \left (\frac {(c+d x)^4 \sin (a+b x)}{b}-\frac {4 d \left (\frac {3 d \left (\frac {(c+d x)^2 \sin (a+b x)}{b}-\frac {2 d \left (\frac {d \sin (a+b x)}{b^2}-\frac {(c+d x) \cos (a+b x)}{b}\right )}{b}\right )}{b}-\frac {(c+d x)^3 \cos (a+b x)}{b}\right )}{b}\right )+\frac {(c+d x)^4 \sin (a+b x) \cos ^2(a+b x)}{3 b}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {4 d^2 \left (\frac {2}{3} \left (\frac {(c+d x)^2 \sin (a+b x)}{b}-\frac {2 d \int (c+d x) \sin (a+b x)dx}{b}\right )+\frac {2 d^2 \left (\frac {1}{3} \sin ^3(a+b x)-\sin (a+b x)\right )}{9 b^3}+\frac {2 d (c+d x) \cos ^3(a+b x)}{9 b^2}+\frac {(c+d x)^2 \sin (a+b x) \cos ^2(a+b x)}{3 b}\right )}{3 b^2}+\frac {4 d (c+d x)^3 \cos ^3(a+b x)}{9 b^2}+\frac {2}{3} \left (\frac {(c+d x)^4 \sin (a+b x)}{b}-\frac {4 d \left (\frac {3 d \left (\frac {(c+d x)^2 \sin (a+b x)}{b}-\frac {2 d \left (\frac {d \sin (a+b x)}{b^2}-\frac {(c+d x) \cos (a+b x)}{b}\right )}{b}\right )}{b}-\frac {(c+d x)^3 \cos (a+b x)}{b}\right )}{b}\right )+\frac {(c+d x)^4 \sin (a+b x) \cos ^2(a+b x)}{3 b}\)

\(\Big \downarrow \) 3777

\(\displaystyle -\frac {4 d^2 \left (\frac {2}{3} \left (\frac {(c+d x)^2 \sin (a+b x)}{b}-\frac {2 d \left (\frac {d \int \cos (a+b x)dx}{b}-\frac {(c+d x) \cos (a+b x)}{b}\right )}{b}\right )+\frac {2 d^2 \left (\frac {1}{3} \sin ^3(a+b x)-\sin (a+b x)\right )}{9 b^3}+\frac {2 d (c+d x) \cos ^3(a+b x)}{9 b^2}+\frac {(c+d x)^2 \sin (a+b x) \cos ^2(a+b x)}{3 b}\right )}{3 b^2}+\frac {4 d (c+d x)^3 \cos ^3(a+b x)}{9 b^2}+\frac {2}{3} \left (\frac {(c+d x)^4 \sin (a+b x)}{b}-\frac {4 d \left (\frac {3 d \left (\frac {(c+d x)^2 \sin (a+b x)}{b}-\frac {2 d \left (\frac {d \sin (a+b x)}{b^2}-\frac {(c+d x) \cos (a+b x)}{b}\right )}{b}\right )}{b}-\frac {(c+d x)^3 \cos (a+b x)}{b}\right )}{b}\right )+\frac {(c+d x)^4 \sin (a+b x) \cos ^2(a+b x)}{3 b}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {4 d^2 \left (\frac {2}{3} \left (\frac {(c+d x)^2 \sin (a+b x)}{b}-\frac {2 d \left (\frac {d \int \sin \left (a+b x+\frac {\pi }{2}\right )dx}{b}-\frac {(c+d x) \cos (a+b x)}{b}\right )}{b}\right )+\frac {2 d^2 \left (\frac {1}{3} \sin ^3(a+b x)-\sin (a+b x)\right )}{9 b^3}+\frac {2 d (c+d x) \cos ^3(a+b x)}{9 b^2}+\frac {(c+d x)^2 \sin (a+b x) \cos ^2(a+b x)}{3 b}\right )}{3 b^2}+\frac {4 d (c+d x)^3 \cos ^3(a+b x)}{9 b^2}+\frac {2}{3} \left (\frac {(c+d x)^4 \sin (a+b x)}{b}-\frac {4 d \left (\frac {3 d \left (\frac {(c+d x)^2 \sin (a+b x)}{b}-\frac {2 d \left (\frac {d \sin (a+b x)}{b^2}-\frac {(c+d x) \cos (a+b x)}{b}\right )}{b}\right )}{b}-\frac {(c+d x)^3 \cos (a+b x)}{b}\right )}{b}\right )+\frac {(c+d x)^4 \sin (a+b x) \cos ^2(a+b x)}{3 b}\)

\(\Big \downarrow \) 3117

\(\displaystyle \frac {4 d (c+d x)^3 \cos ^3(a+b x)}{9 b^2}+\frac {2}{3} \left (\frac {(c+d x)^4 \sin (a+b x)}{b}-\frac {4 d \left (\frac {3 d \left (\frac {(c+d x)^2 \sin (a+b x)}{b}-\frac {2 d \left (\frac {d \sin (a+b x)}{b^2}-\frac {(c+d x) \cos (a+b x)}{b}\right )}{b}\right )}{b}-\frac {(c+d x)^3 \cos (a+b x)}{b}\right )}{b}\right )-\frac {4 d^2 \left (\frac {2 d^2 \left (\frac {1}{3} \sin ^3(a+b x)-\sin (a+b x)\right )}{9 b^3}+\frac {2 d (c+d x) \cos ^3(a+b x)}{9 b^2}+\frac {2}{3} \left (\frac {(c+d x)^2 \sin (a+b x)}{b}-\frac {2 d \left (\frac {d \sin (a+b x)}{b^2}-\frac {(c+d x) \cos (a+b x)}{b}\right )}{b}\right )+\frac {(c+d x)^2 \sin (a+b x) \cos ^2(a+b x)}{3 b}\right )}{3 b^2}+\frac {(c+d x)^4 \sin (a+b x) \cos ^2(a+b x)}{3 b}\)

Input:

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

Output:

(4*d*(c + d*x)^3*Cos[a + b*x]^3)/(9*b^2) + ((c + d*x)^4*Cos[a + b*x]^2*Sin 
[a + b*x])/(3*b) - (4*d^2*((2*d*(c + d*x)*Cos[a + b*x]^3)/(9*b^2) + ((c + 
d*x)^2*Cos[a + b*x]^2*Sin[a + b*x])/(3*b) + (2*d^2*(-Sin[a + b*x] + Sin[a 
+ b*x]^3/3))/(9*b^3) + (2*(((c + d*x)^2*Sin[a + b*x])/b - (2*d*(-(((c + d* 
x)*Cos[a + b*x])/b) + (d*Sin[a + b*x])/b^2))/b))/3))/(3*b^2) + (2*(((c + d 
*x)^4*Sin[a + b*x])/b - (4*d*(-(((c + d*x)^3*Cos[a + b*x])/b) + (3*d*(((c 
+ d*x)^2*Sin[a + b*x])/b - (2*d*(-(((c + d*x)*Cos[a + b*x])/b) + (d*Sin[a 
+ b*x])/b^2))/b))/b))/b))/3
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

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

rule 3113

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1)   Subst[Int[Exp 
and[(1 - x^2)^((n - 1)/2), x], x], x, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] 
 && IGtQ[(n - 1)/2, 0]
 

rule 3117

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

rule 3777

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[( 
-(c + d*x)^m)*(Cos[e + f*x]/f), x] + Simp[d*(m/f)   Int[(c + d*x)^(m - 1)*C 
os[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
 

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]
 

Maple [A] (verified)

Time = 3.34 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.77

method	result	size
parallelrisch	\(\frac {\left (27 \left (d x +c \right )^{4} b^{4}-36 d^{2} \left (d x +c \right )^{2} b^{2}+8 d^{4}\right ) \sin \left (3 b x +3 a \right )+36 d \left (d x +c \right ) b \left (\left (d x +c \right )^{2} b^{2}-\frac {2 d^{2}}{3}\right ) \cos \left (3 b x +3 a \right )+243 \left (\left (d x +c \right )^{4} b^{4}-12 d^{2} \left (d x +c \right )^{2} b^{2}+24 d^{4}\right ) \sin \left (b x +a \right )+972 \left (\left (d x +c \right ) \left (\left (d x +c \right )^{2} b^{2}-6 d^{2}\right ) \cos \left (b x +a \right )+\frac {28 b^{2} c^{3}}{27}-\frac {488 c \,d^{2}}{81}\right ) d b}{324 b^{5}}\)	\(173\)
risch	\(\frac {3 d \left (b^{2} d^{3} x^{3}+3 b^{2} c \,d^{2} x^{2}+3 b^{2} c^{2} d x +b^{2} c^{3}-6 d^{3} x -6 c \,d^{2}\right ) \cos \left (b x +a \right )}{b^{4}}+\frac {3 \left (d^{4} x^{4} b^{4}+4 b^{4} c \,d^{3} x^{3}+6 b^{4} c^{2} d^{2} x^{2}+4 b^{4} c^{3} d x +b^{4} c^{4}-12 b^{2} d^{4} x^{2}-24 b^{2} c \,d^{3} x -12 b^{2} c^{2} d^{2}+24 d^{4}\right ) \sin \left (b x +a \right )}{4 b^{5}}+\frac {d \left (3 b^{2} d^{3} x^{3}+9 b^{2} c \,d^{2} x^{2}+9 b^{2} c^{2} d x +3 b^{2} c^{3}-2 d^{3} x -2 c \,d^{2}\right ) \cos \left (3 b x +3 a \right )}{27 b^{4}}+\frac {\left (27 d^{4} x^{4} b^{4}+108 b^{4} c \,d^{3} x^{3}+162 b^{4} c^{2} d^{2} x^{2}+108 b^{4} c^{3} d x +27 b^{4} c^{4}-36 b^{2} d^{4} x^{2}-72 b^{2} c \,d^{3} x -36 b^{2} c^{2} d^{2}+8 d^{4}\right ) \sin \left (3 b x +3 a \right )}{324 b^{5}}\)	\(344\)
orering	\(\frac {16 d \left (135 b^{6} d^{6} x^{6}+810 b^{6} c \,d^{5} x^{5}+2025 b^{6} c^{2} d^{4} x^{4}+2700 b^{6} c^{3} d^{3} x^{3}+2025 b^{6} c^{4} d^{2} x^{2}-891 b^{4} d^{6} x^{4}+810 b^{6} c^{5} d x -3564 b^{4} c \,d^{5} x^{3}+135 b^{6} c^{6}-5346 b^{4} c^{2} d^{4} x^{2}-3564 b^{4} c^{3} d^{3} x -891 b^{4} c^{4} d^{2}-1960 b^{2} d^{6} x^{2}-3920 b^{2} c \,d^{5} x -1960 b^{2} c^{2} d^{4}+5460 d^{6}\right ) \cos \left (b x +a \right )^{3}}{243 b^{8} \left (d x +c \right )^{3}}-\frac {2 \left (135 b^{6} d^{6} x^{6}+810 b^{6} c \,d^{5} x^{5}+2025 b^{6} c^{2} d^{4} x^{4}+2700 b^{6} c^{3} d^{3} x^{3}+2025 b^{6} c^{4} d^{2} x^{2}-396 b^{4} d^{6} x^{4}+810 b^{6} c^{5} d x -1584 b^{4} c \,d^{5} x^{3}+135 b^{6} c^{6}-2376 b^{4} c^{2} d^{4} x^{2}-1584 b^{4} c^{3} d^{3} x -396 b^{4} c^{4} d^{2}-10400 b^{2} d^{6} x^{2}-20800 b^{2} c \,d^{5} x -10400 b^{2} c^{2} d^{4}+21840 d^{6}\right ) \left (4 \left (d x +c \right )^{3} \cos \left (b x +a \right )^{3} d -3 \left (d x +c \right )^{4} \cos \left (b x +a \right )^{2} b \sin \left (b x +a \right )\right )}{243 b^{8} \left (d x +c \right )^{6}}+\frac {16 d \left (9 d^{4} x^{4} b^{4}+36 b^{4} c \,d^{3} x^{3}+54 b^{4} c^{2} d^{2} x^{2}+36 b^{4} c^{3} d x +9 b^{4} c^{4}-105 b^{2} d^{4} x^{2}-210 b^{2} c \,d^{3} x -105 b^{2} c^{2} d^{2}+182 d^{4}\right ) \left (12 \left (d x +c \right )^{2} \cos \left (b x +a \right )^{3} d^{2}-24 \left (d x +c \right )^{3} \cos \left (b x +a \right )^{2} d b \sin \left (b x +a \right )+6 \left (d x +c \right )^{4} \cos \left (b x +a \right ) b^{2} \sin \left (b x +a \right )^{2}-3 \left (d x +c \right )^{4} \cos \left (b x +a \right )^{3} b^{2}\right )}{81 b^{8} \left (d x +c \right )^{5}}-\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}-360 b^{2} d^{4} x^{2}-720 b^{2} c \,d^{3} x -360 b^{2} c^{2} d^{2}+728 d^{4}\right ) \left (24 \left (d x +c \right ) \cos \left (b x +a \right )^{3} d^{3}-108 \left (d x +c \right )^{2} \cos \left (b x +a \right )^{2} d^{2} b \sin \left (b x +a \right )+72 \left (d x +c \right )^{3} \cos \left (b x +a \right ) d \,b^{2} \sin \left (b x +a \right )^{2}-36 \left (d x +c \right )^{3} \cos \left (b x +a \right )^{3} d \,b^{2}-6 \left (d x +c \right )^{4} b^{3} \sin \left (b x +a \right )^{3}+21 \left (d x +c \right )^{4} \cos \left (b x +a \right )^{2} b^{3} \sin \left (b x +a \right )\right )}{243 b^{8} \left (d x +c \right )^{4}}\)	\(883\)
derivativedivides	\(\text {Expression too large to display}\)	\(1023\)
default	\(\text {Expression too large to display}\)	\(1023\)

Input:

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

Output:

1/324*((27*(d*x+c)^4*b^4-36*d^2*(d*x+c)^2*b^2+8*d^4)*sin(3*b*x+3*a)+36*d*( 
d*x+c)*b*((d*x+c)^2*b^2-2/3*d^2)*cos(3*b*x+3*a)+243*((d*x+c)^4*b^4-12*d^2* 
(d*x+c)^2*b^2+24*d^4)*sin(b*x+a)+972*((d*x+c)*((d*x+c)^2*b^2-6*d^2)*cos(b* 
x+a)+28/27*b^2*c^3-488/81*c*d^2)*d*b)/b^5
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 350, normalized size of antiderivative = 1.56 \[ \int (c+d x)^4 \cos ^3(a+b x) \, dx=\frac {12 \, {\left (3 \, b^{3} d^{4} x^{3} + 9 \, b^{3} c d^{3} x^{2} + 3 \, b^{3} c^{3} d - 2 \, b c d^{3} + {\left (9 \, b^{3} c^{2} d^{2} - 2 \, b d^{4}\right )} x\right )} \cos \left (b x + a\right )^{3} + 72 \, {\left (3 \, b^{3} d^{4} x^{3} + 9 \, b^{3} c d^{3} x^{2} + 3 \, b^{3} c^{3} d - 20 \, b c d^{3} + {\left (9 \, b^{3} c^{2} d^{2} - 20 \, b d^{4}\right )} x\right )} \cos \left (b x + a\right ) + {\left (54 \, b^{4} d^{4} x^{4} + 216 \, b^{4} c d^{3} x^{3} + 54 \, b^{4} c^{4} - 720 \, b^{2} c^{2} d^{2} + 1456 \, d^{4} + 36 \, {\left (9 \, b^{4} c^{2} d^{2} - 20 \, b^{2} d^{4}\right )} x^{2} + {\left (27 \, b^{4} d^{4} x^{4} + 108 \, b^{4} c d^{3} x^{3} + 27 \, b^{4} c^{4} - 36 \, b^{2} c^{2} d^{2} + 8 \, d^{4} + 18 \, {\left (9 \, b^{4} c^{2} d^{2} - 2 \, b^{2} d^{4}\right )} x^{2} + 36 \, {\left (3 \, b^{4} c^{3} d - 2 \, b^{2} c d^{3}\right )} x\right )} \cos \left (b x + a\right )^{2} + 72 \, {\left (3 \, b^{4} c^{3} d - 20 \, b^{2} c d^{3}\right )} x\right )} \sin \left (b x + a\right )}{81 \, b^{5}} \] Input:

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

Output:

1/81*(12*(3*b^3*d^4*x^3 + 9*b^3*c*d^3*x^2 + 3*b^3*c^3*d - 2*b*c*d^3 + (9*b 
^3*c^2*d^2 - 2*b*d^4)*x)*cos(b*x + a)^3 + 72*(3*b^3*d^4*x^3 + 9*b^3*c*d^3* 
x^2 + 3*b^3*c^3*d - 20*b*c*d^3 + (9*b^3*c^2*d^2 - 20*b*d^4)*x)*cos(b*x + a 
) + (54*b^4*d^4*x^4 + 216*b^4*c*d^3*x^3 + 54*b^4*c^4 - 720*b^2*c^2*d^2 + 1 
456*d^4 + 36*(9*b^4*c^2*d^2 - 20*b^2*d^4)*x^2 + (27*b^4*d^4*x^4 + 108*b^4* 
c*d^3*x^3 + 27*b^4*c^4 - 36*b^2*c^2*d^2 + 8*d^4 + 18*(9*b^4*c^2*d^2 - 2*b^ 
2*d^4)*x^2 + 36*(3*b^4*c^3*d - 2*b^2*c*d^3)*x)*cos(b*x + a)^2 + 72*(3*b^4* 
c^3*d - 20*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. 772 vs. \(2 (226) = 452\).

Time = 0.62 (sec) , antiderivative size = 772, normalized size of antiderivative = 3.43 \[ \int (c+d x)^4 \cos ^3(a+b x) \, dx =\text {Too large to display} \] Input:

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

Output:

Piecewise((2*c**4*sin(a + b*x)**3/(3*b) + c**4*sin(a + b*x)*cos(a + b*x)** 
2/b + 8*c**3*d*x*sin(a + b*x)**3/(3*b) + 4*c**3*d*x*sin(a + b*x)*cos(a + b 
*x)**2/b + 4*c**2*d**2*x**2*sin(a + b*x)**3/b + 6*c**2*d**2*x**2*sin(a + b 
*x)*cos(a + b*x)**2/b + 8*c*d**3*x**3*sin(a + b*x)**3/(3*b) + 4*c*d**3*x** 
3*sin(a + b*x)*cos(a + b*x)**2/b + 2*d**4*x**4*sin(a + b*x)**3/(3*b) + d** 
4*x**4*sin(a + b*x)*cos(a + b*x)**2/b + 8*c**3*d*sin(a + b*x)**2*cos(a + b 
*x)/(3*b**2) + 28*c**3*d*cos(a + b*x)**3/(9*b**2) + 8*c**2*d**2*x*sin(a + 
b*x)**2*cos(a + b*x)/b**2 + 28*c**2*d**2*x*cos(a + b*x)**3/(3*b**2) + 8*c* 
d**3*x**2*sin(a + b*x)**2*cos(a + b*x)/b**2 + 28*c*d**3*x**2*cos(a + b*x)* 
*3/(3*b**2) + 8*d**4*x**3*sin(a + b*x)**2*cos(a + b*x)/(3*b**2) + 28*d**4* 
x**3*cos(a + b*x)**3/(9*b**2) - 80*c**2*d**2*sin(a + b*x)**3/(9*b**3) - 28 
*c**2*d**2*sin(a + b*x)*cos(a + b*x)**2/(3*b**3) - 160*c*d**3*x*sin(a + b* 
x)**3/(9*b**3) - 56*c*d**3*x*sin(a + b*x)*cos(a + b*x)**2/(3*b**3) - 80*d* 
*4*x**2*sin(a + b*x)**3/(9*b**3) - 28*d**4*x**2*sin(a + b*x)*cos(a + b*x)* 
*2/(3*b**3) - 160*c*d**3*sin(a + b*x)**2*cos(a + b*x)/(9*b**4) - 488*c*d** 
3*cos(a + b*x)**3/(27*b**4) - 160*d**4*x*sin(a + b*x)**2*cos(a + b*x)/(9*b 
**4) - 488*d**4*x*cos(a + b*x)**3/(27*b**4) + 1456*d**4*sin(a + b*x)**3/(8 
1*b**5) + 488*d**4*sin(a + b*x)*cos(a + b*x)**2/(27*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)*cos(a 
)**3, True))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 925 vs. \(2 (205) = 410\).

Time = 0.10 (sec) , antiderivative size = 925, normalized size of antiderivative = 4.11 \[ \int (c+d x)^4 \cos ^3(a+b x) \, dx=\text {Too large to display} \] Input:

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

Output:

-1/324*(108*(sin(b*x + a)^3 - 3*sin(b*x + a))*c^4 - 432*(sin(b*x + a)^3 - 
3*sin(b*x + a))*a*c^3*d/b + 648*(sin(b*x + a)^3 - 3*sin(b*x + a))*a^2*c^2* 
d^2/b^2 - 432*(sin(b*x + a)^3 - 3*sin(b*x + a))*a^3*c*d^3/b^3 + 108*(sin(b 
*x + a)^3 - 3*sin(b*x + a))*a^4*d^4/b^4 - 36*(3*(b*x + a)*sin(3*b*x + 3*a) 
 + 27*(b*x + a)*sin(b*x + a) + cos(3*b*x + 3*a) + 27*cos(b*x + a))*c^3*d/b 
 + 108*(3*(b*x + a)*sin(3*b*x + 3*a) + 27*(b*x + a)*sin(b*x + a) + cos(3*b 
*x + 3*a) + 27*cos(b*x + a))*a*c^2*d^2/b^2 - 108*(3*(b*x + a)*sin(3*b*x + 
3*a) + 27*(b*x + a)*sin(b*x + a) + cos(3*b*x + 3*a) + 27*cos(b*x + a))*a^2 
*c*d^3/b^3 + 36*(3*(b*x + a)*sin(3*b*x + 3*a) + 27*(b*x + a)*sin(b*x + a) 
+ cos(3*b*x + 3*a) + 27*cos(b*x + a))*a^3*d^4/b^4 - 18*(6*(b*x + a)*cos(3* 
b*x + 3*a) + 162*(b*x + a)*cos(b*x + a) + (9*(b*x + a)^2 - 2)*sin(3*b*x + 
3*a) + 81*((b*x + a)^2 - 2)*sin(b*x + a))*c^2*d^2/b^2 + 36*(6*(b*x + a)*co 
s(3*b*x + 3*a) + 162*(b*x + a)*cos(b*x + a) + (9*(b*x + a)^2 - 2)*sin(3*b* 
x + 3*a) + 81*((b*x + a)^2 - 2)*sin(b*x + a))*a*c*d^3/b^3 - 18*(6*(b*x + a 
)*cos(3*b*x + 3*a) + 162*(b*x + a)*cos(b*x + a) + (9*(b*x + a)^2 - 2)*sin( 
3*b*x + 3*a) + 81*((b*x + a)^2 - 2)*sin(b*x + a))*a^2*d^4/b^4 - 12*((9*(b* 
x + a)^2 - 2)*cos(3*b*x + 3*a) + 243*((b*x + a)^2 - 2)*cos(b*x + a) + 3*(3 
*(b*x + a)^3 - 2*b*x - 2*a)*sin(3*b*x + 3*a) + 81*((b*x + a)^3 - 6*b*x - 6 
*a)*sin(b*x + a))*c*d^3/b^3 + 12*((9*(b*x + a)^2 - 2)*cos(3*b*x + 3*a) + 2 
43*((b*x + a)^2 - 2)*cos(b*x + a) + 3*(3*(b*x + a)^3 - 2*b*x - 2*a)*sin...
 

Giac [A] (verification not implemented)

Time = 0.39 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.56 \[ \int (c+d x)^4 \cos ^3(a+b x) \, dx=\frac {{\left (3 \, b^{3} d^{4} x^{3} + 9 \, b^{3} c d^{3} x^{2} + 9 \, b^{3} c^{2} d^{2} x + 3 \, b^{3} c^{3} d - 2 \, b d^{4} x - 2 \, b c d^{3}\right )} \cos \left (3 \, b x + 3 \, a\right )}{27 \, b^{5}} + \frac {3 \, {\left (b^{3} d^{4} x^{3} + 3 \, b^{3} c d^{3} x^{2} + 3 \, b^{3} c^{2} d^{2} x + b^{3} c^{3} d - 6 \, b d^{4} x - 6 \, b c d^{3}\right )} \cos \left (b x + a\right )}{b^{5}} + \frac {{\left (27 \, b^{4} d^{4} x^{4} + 108 \, b^{4} c d^{3} x^{3} + 162 \, b^{4} c^{2} d^{2} x^{2} + 108 \, b^{4} c^{3} d x + 27 \, b^{4} c^{4} - 36 \, b^{2} d^{4} x^{2} - 72 \, b^{2} c d^{3} x - 36 \, b^{2} c^{2} d^{2} + 8 \, d^{4}\right )} \sin \left (3 \, b x + 3 \, a\right )}{324 \, b^{5}} + \frac {3 \, {\left (b^{4} d^{4} x^{4} + 4 \, b^{4} c d^{3} x^{3} + 6 \, b^{4} c^{2} d^{2} x^{2} + 4 \, b^{4} c^{3} d x + b^{4} c^{4} - 12 \, b^{2} d^{4} x^{2} - 24 \, b^{2} c d^{3} x - 12 \, b^{2} c^{2} d^{2} + 24 \, d^{4}\right )} \sin \left (b x + a\right )}{4 \, b^{5}} \] Input:

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

Output:

1/27*(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 + 3*(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 
 + 1/324*(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 + 3/4*(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 = 46.21 (sec) , antiderivative size = 532, normalized size of antiderivative = 2.36 \[ \int (c+d x)^4 \cos ^3(a+b x) \, dx=\frac {2\,{\sin \left (a+b\,x\right )}^3\,\left (27\,b^4\,c^4-360\,b^2\,c^2\,d^2+728\,d^4\right )}{81\,b^5}-\frac {4\,{\cos \left (a+b\,x\right )}^3\,\left (122\,c\,d^3-21\,b^2\,c^3\,d\right )}{27\,b^4}+\frac {{\cos \left (a+b\,x\right )}^2\,\sin \left (a+b\,x\right )\,\left (27\,b^4\,c^4-252\,b^2\,c^2\,d^2+488\,d^4\right )}{27\,b^5}-\frac {8\,\cos \left (a+b\,x\right )\,{\sin \left (a+b\,x\right )}^2\,\left (20\,c\,d^3-3\,b^2\,c^3\,d\right )}{9\,b^4}+\frac {28\,d^4\,x^3\,{\cos \left (a+b\,x\right )}^3}{9\,b^2}-\frac {4\,x\,{\cos \left (a+b\,x\right )}^3\,\left (122\,d^4-63\,b^2\,c^2\,d^2\right )}{27\,b^4}+\frac {2\,d^4\,x^4\,{\sin \left (a+b\,x\right )}^3}{3\,b}-\frac {8\,x\,{\sin \left (a+b\,x\right )}^3\,\left (20\,c\,d^3-3\,b^2\,c^3\,d\right )}{9\,b^3}-\frac {4\,x^2\,{\sin \left (a+b\,x\right )}^3\,\left (20\,d^4-9\,b^2\,c^2\,d^2\right )}{9\,b^3}-\frac {2\,x^2\,{\cos \left (a+b\,x\right )}^2\,\sin \left (a+b\,x\right )\,\left (14\,d^4-9\,b^2\,c^2\,d^2\right )}{3\,b^3}+\frac {28\,c\,d^3\,x^2\,{\cos \left (a+b\,x\right )}^3}{3\,b^2}+\frac {d^4\,x^4\,{\cos \left (a+b\,x\right )}^2\,\sin \left (a+b\,x\right )}{b}+\frac {8\,d^4\,x^3\,\cos \left (a+b\,x\right )\,{\sin \left (a+b\,x\right )}^2}{3\,b^2}+\frac {8\,c\,d^3\,x^3\,{\sin \left (a+b\,x\right )}^3}{3\,b}-\frac {8\,x\,\cos \left (a+b\,x\right )\,{\sin \left (a+b\,x\right )}^2\,\left (20\,d^4-9\,b^2\,c^2\,d^2\right )}{9\,b^4}-\frac {4\,x\,{\cos \left (a+b\,x\right )}^2\,\sin \left (a+b\,x\right )\,\left (14\,c\,d^3-3\,b^2\,c^3\,d\right )}{3\,b^3}+\frac {4\,c\,d^3\,x^3\,{\cos \left (a+b\,x\right )}^2\,\sin \left (a+b\,x\right )}{b}+\frac {8\,c\,d^3\,x^2\,\cos \left (a+b\,x\right )\,{\sin \left (a+b\,x\right )}^2}{b^2} \] Input:

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

Output:

(2*sin(a + b*x)^3*(728*d^4 + 27*b^4*c^4 - 360*b^2*c^2*d^2))/(81*b^5) - (4* 
cos(a + b*x)^3*(122*c*d^3 - 21*b^2*c^3*d))/(27*b^4) + (cos(a + b*x)^2*sin( 
a + b*x)*(488*d^4 + 27*b^4*c^4 - 252*b^2*c^2*d^2))/(27*b^5) - (8*cos(a + b 
*x)*sin(a + b*x)^2*(20*c*d^3 - 3*b^2*c^3*d))/(9*b^4) + (28*d^4*x^3*cos(a + 
 b*x)^3)/(9*b^2) - (4*x*cos(a + b*x)^3*(122*d^4 - 63*b^2*c^2*d^2))/(27*b^4 
) + (2*d^4*x^4*sin(a + b*x)^3)/(3*b) - (8*x*sin(a + b*x)^3*(20*c*d^3 - 3*b 
^2*c^3*d))/(9*b^3) - (4*x^2*sin(a + b*x)^3*(20*d^4 - 9*b^2*c^2*d^2))/(9*b^ 
3) - (2*x^2*cos(a + b*x)^2*sin(a + b*x)*(14*d^4 - 9*b^2*c^2*d^2))/(3*b^3) 
+ (28*c*d^3*x^2*cos(a + b*x)^3)/(3*b^2) + (d^4*x^4*cos(a + b*x)^2*sin(a + 
b*x))/b + (8*d^4*x^3*cos(a + b*x)*sin(a + b*x)^2)/(3*b^2) + (8*c*d^3*x^3*s 
in(a + b*x)^3)/(3*b) - (8*x*cos(a + b*x)*sin(a + b*x)^2*(20*d^4 - 9*b^2*c^ 
2*d^2))/(9*b^4) - (4*x*cos(a + b*x)^2*sin(a + b*x)*(14*c*d^3 - 3*b^2*c^3*d 
))/(3*b^3) + (4*c*d^3*x^3*cos(a + b*x)^2*sin(a + b*x))/b + (8*c*d^3*x^2*co 
s(a + b*x)*sin(a + b*x)^2)/b^2
 

Reduce [B] (verification not implemented)

Time = 0.68 (sec) , antiderivative size = 568, normalized size of antiderivative = 2.52 \[ \int (c+d x)^4 \cos ^3(a+b x) \, dx=\frac {-108 \cos \left (b x +a \right ) \sin \left (b x +a \right )^{2} b^{3} c^{2} d^{2} x -108 \cos \left (b x +a \right ) \sin \left (b x +a \right )^{2} b^{3} c \,d^{3} x^{2}-36 \cos \left (b x +a \right ) \sin \left (b x +a \right )^{2} b^{3} c^{3} d -36 \cos \left (b x +a \right ) \sin \left (b x +a \right )^{2} b^{3} d^{4} x^{3}+24 \cos \left (b x +a \right ) \sin \left (b x +a \right )^{2} b c \,d^{3}+24 \cos \left (b x +a \right ) \sin \left (b x +a \right )^{2} b \,d^{4} x -108 \sin \left (b x +a \right )^{3} b^{4} c^{3} d x -162 \sin \left (b x +a \right )^{3} b^{4} c^{2} d^{2} x^{2}-108 \sin \left (b x +a \right )^{3} b^{4} c \,d^{3} x^{3}+72 \sin \left (b x +a \right )^{3} b^{2} c \,d^{3} x +36 b^{3} c^{3} d -456 b c \,d^{3}-27 \sin \left (b x +a \right )^{3} b^{4} c^{4}+756 \cos \left (b x +a \right ) b^{3} c^{2} d^{2} x +756 \cos \left (b x +a \right ) b^{3} c \,d^{3} x^{2}+324 \sin \left (b x +a \right ) b^{4} c^{3} d x +486 \sin \left (b x +a \right ) b^{4} c^{2} d^{2} x^{2}+324 \sin \left (b x +a \right ) b^{4} c \,d^{3} x^{3}-1512 \sin \left (b x +a \right ) b^{2} c \,d^{3} x +81 \sin \left (b x +a \right ) b^{4} c^{4}+81 \sin \left (b x +a \right ) b^{4} d^{4} x^{4}+252 \cos \left (b x +a \right ) b^{3} c^{3} d +252 \cos \left (b x +a \right ) b^{3} d^{4} x^{3}-1464 \cos \left (b x +a \right ) b c \,d^{3}-1464 \cos \left (b x +a \right ) b \,d^{4} x -756 \sin \left (b x +a \right ) b^{2} c^{2} d^{2}-756 \sin \left (b x +a \right ) b^{2} d^{4} x^{2}-8 \sin \left (b x +a \right )^{3} d^{4}+1464 \sin \left (b x +a \right ) d^{4}-27 \sin \left (b x +a \right )^{3} b^{4} d^{4} x^{4}+36 \sin \left (b x +a \right )^{3} b^{2} c^{2} d^{2}+36 \sin \left (b x +a \right )^{3} b^{2} d^{4} x^{2}}{81 b^{5}} \] Input:

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

Output:

( - 36*cos(a + b*x)*sin(a + b*x)**2*b**3*c**3*d - 108*cos(a + b*x)*sin(a + 
 b*x)**2*b**3*c**2*d**2*x - 108*cos(a + b*x)*sin(a + b*x)**2*b**3*c*d**3*x 
**2 - 36*cos(a + b*x)*sin(a + b*x)**2*b**3*d**4*x**3 + 24*cos(a + b*x)*sin 
(a + b*x)**2*b*c*d**3 + 24*cos(a + b*x)*sin(a + b*x)**2*b*d**4*x + 252*cos 
(a + b*x)*b**3*c**3*d + 756*cos(a + b*x)*b**3*c**2*d**2*x + 756*cos(a + b* 
x)*b**3*c*d**3*x**2 + 252*cos(a + b*x)*b**3*d**4*x**3 - 1464*cos(a + b*x)* 
b*c*d**3 - 1464*cos(a + b*x)*b*d**4*x - 27*sin(a + b*x)**3*b**4*c**4 - 108 
*sin(a + b*x)**3*b**4*c**3*d*x - 162*sin(a + b*x)**3*b**4*c**2*d**2*x**2 - 
 108*sin(a + b*x)**3*b**4*c*d**3*x**3 - 27*sin(a + b*x)**3*b**4*d**4*x**4 
+ 36*sin(a + b*x)**3*b**2*c**2*d**2 + 72*sin(a + b*x)**3*b**2*c*d**3*x + 3 
6*sin(a + b*x)**3*b**2*d**4*x**2 - 8*sin(a + b*x)**3*d**4 + 81*sin(a + b*x 
)*b**4*c**4 + 324*sin(a + b*x)*b**4*c**3*d*x + 486*sin(a + b*x)*b**4*c**2* 
d**2*x**2 + 324*sin(a + b*x)*b**4*c*d**3*x**3 + 81*sin(a + b*x)*b**4*d**4* 
x**4 - 756*sin(a + b*x)*b**2*c**2*d**2 - 1512*sin(a + b*x)*b**2*c*d**3*x - 
 756*sin(a + b*x)*b**2*d**4*x**2 + 1464*sin(a + b*x)*d**4 + 36*b**3*c**3*d 
 - 456*b*c*d**3)/(81*b**5)