\(\int (c+d x)^4 \sin (a+b x) \, dx\) [1]

Optimal result

Integrand size = 14, antiderivative size = 92 \[ \int (c+d x)^4 \sin (a+b x) \, dx=-\frac {24 d^4 \cos (a+b x)}{b^5}+\frac {12 d^2 (c+d x)^2 \cos (a+b x)}{b^3}-\frac {(c+d x)^4 \cos (a+b x)}{b}-\frac {24 d^3 (c+d x) \sin (a+b x)}{b^4}+\frac {4 d (c+d x)^3 \sin (a+b x)}{b^2} \] Output:

-24*d^4*cos(b*x+a)/b^5+12*d^2*(d*x+c)^2*cos(b*x+a)/b^3-(d*x+c)^4*cos(b*x+a 
)/b-24*d^3*(d*x+c)*sin(b*x+a)/b^4+4*d*(d*x+c)^3*sin(b*x+a)/b^2
 

Mathematica [A] (verified)

Time = 0.34 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.84 \[ \int (c+d x)^4 \sin (a+b x) \, dx=\frac {-\left (\left (24 d^4-12 b^2 d^2 (c+d x)^2+b^4 (c+d x)^4\right ) \cos (a+b x)\right )+4 b d (c+d x) \left (-6 d^2+b^2 (c+d x)^2\right ) \sin (a+b x)}{b^5} \] Input:

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

Output:

(-((24*d^4 - 12*b^2*d^2*(c + d*x)^2 + b^4*(c + d*x)^4)*Cos[a + b*x]) + 4*b 
*d*(c + d*x)*(-6*d^2 + b^2*(c + d*x)^2)*Sin[a + b*x])/b^5
 

Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.10, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {3042, 3777, 3042, 3777, 25, 3042, 3777, 3042, 3777, 25, 3042, 3118}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3118

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 25

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

rule 3042

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

rule 3118

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Cos[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]
 

Maple [A] (verified)

Time = 0.97 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.83

method	result	size
risch	\(-\frac {\left (d^{4} x^{4} b^{4}+4 b^{4} c \,d^{3} x^{3}+6 b^{4} c^{2} d^{2} x^{2}+4 b^{4} c^{3} d x +b^{4} c^{4}-12 b^{2} d^{4} x^{2}-24 b^{2} c \,d^{3} x -12 b^{2} c^{2} d^{2}+24 d^{4}\right ) \cos \left (b x +a \right )}{b^{5}}+\frac {4 d \left (b^{2} d^{3} x^{3}+3 b^{2} c \,d^{2} x^{2}+3 b^{2} c^{2} d x +b^{2} c^{3}-6 d^{3} x -6 c \,d^{2}\right ) \sin \left (b x +a \right )}{b^{4}}\)	\(168\)
parallelrisch	\(\frac {\left (\left (d^{4} x^{4}+4 c \,d^{3} x^{3}+6 c^{2} d^{2} x^{2}+4 c^{3} d x +2 c^{4}\right ) b^{4}+\left (-12 d^{4} x^{2}-24 c \,d^{3} x -24 c^{2} d^{2}\right ) b^{2}+48 d^{4}\right ) \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}+8 \left (d x +c \right ) d b \left (\left (d x +c \right )^{2} b^{2}-6 d^{2}\right ) \tan \left (\frac {b x}{2}+\frac {a}{2}\right )-4 \left (\frac {d x}{2}+c \right ) d \,b^{2} x \left (\left (\frac {1}{2} x^{2} d^{2}+c d x +c^{2}\right ) b^{2}-6 d^{2}\right )}{b^{5} \left (1+\tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}\right )}\)	\(185\)
orering	\(\frac {8 d \left (d^{4} x^{4} b^{4}+4 b^{4} c \,d^{3} x^{3}+6 b^{4} c^{2} d^{2} x^{2}+4 b^{4} c^{3} d x +b^{4} c^{4}-9 b^{2} d^{4} x^{2}-18 b^{2} c \,d^{3} x -9 b^{2} c^{2} d^{2}+12 d^{4}\right ) \sin \left (b x +a \right )}{b^{6} \left (d x +c \right )}-\frac {\left (d^{4} x^{4} b^{4}+4 b^{4} c \,d^{3} x^{3}+6 b^{4} c^{2} d^{2} x^{2}+4 b^{4} c^{3} d x +b^{4} c^{4}-12 b^{2} d^{4} x^{2}-24 b^{2} c \,d^{3} x -12 b^{2} c^{2} d^{2}+24 d^{4}\right ) \left (4 \left (d x +c \right )^{3} \sin \left (b x +a \right ) d +\left (d x +c \right )^{4} b \cos \left (b x +a \right )\right )}{b^{6} \left (d x +c \right )^{4}}\)	\(247\)
norman	\(\frac {\frac {\left (2 b^{4} c^{4}-24 b^{2} c^{2} d^{2}+48 d^{4}\right ) \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}{b^{5}}+\frac {d^{4} x^{4} \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}{b}-\frac {d^{4} x^{4}}{b}-\frac {6 d^{2} \left (b^{2} c^{2}-2 d^{2}\right ) x^{2}}{b^{3}}-\frac {4 d^{3} c \,x^{3}}{b}+\frac {8 d^{4} x^{3} \tan \left (\frac {b x}{2}+\frac {a}{2}\right )}{b^{2}}-\frac {4 c d \left (b^{2} c^{2}-6 d^{2}\right ) x}{b^{3}}+\frac {6 d^{2} \left (b^{2} c^{2}-2 d^{2}\right ) x^{2} \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}{b^{3}}+\frac {4 d^{3} c \,x^{3} \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}{b}+\frac {8 c d \left (b^{2} c^{2}-6 d^{2}\right ) \tan \left (\frac {b x}{2}+\frac {a}{2}\right )}{b^{4}}+\frac {24 d^{2} \left (b^{2} c^{2}-2 d^{2}\right ) x \tan \left (\frac {b x}{2}+\frac {a}{2}\right )}{b^{4}}+\frac {24 d^{3} c \,x^{2} \tan \left (\frac {b x}{2}+\frac {a}{2}\right )}{b^{2}}+\frac {4 c d \left (b^{2} c^{2}-6 d^{2}\right ) x \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}{b^{3}}}{1+\tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}\)	\(338\)
parts	\(-\frac {\cos \left (b x +a \right ) d^{4} x^{4}}{b}-\frac {4 \cos \left (b x +a \right ) c \,d^{3} x^{3}}{b}-\frac {6 \cos \left (b x +a \right ) c^{2} d^{2} x^{2}}{b}-\frac {4 \cos \left (b x +a \right ) c^{3} d x}{b}-\frac {\cos \left (b x +a \right ) c^{4}}{b}+\frac {4 d \left (-\frac {a^{3} d^{3} \sin \left (b x +a \right )}{b^{3}}+\frac {3 a^{2} c \,d^{2} \sin \left (b x +a \right )}{b^{2}}+\frac {3 a^{2} d^{3} \left (\cos \left (b x +a \right )+\left (b x +a \right ) \sin \left (b x +a \right )\right )}{b^{3}}-\frac {3 a \,c^{2} d \sin \left (b x +a \right )}{b}-\frac {6 a c \,d^{2} \left (\cos \left (b x +a \right )+\left (b x +a \right ) \sin \left (b x +a \right )\right )}{b^{2}}-\frac {3 a \,d^{3} \left (\left (b x +a \right )^{2} \sin \left (b x +a \right )-2 \sin \left (b x +a \right )+2 \left (b x +a \right ) \cos \left (b x +a \right )\right )}{b^{3}}+c^{3} \sin \left (b x +a \right )+\frac {3 c^{2} d \left (\cos \left (b x +a \right )+\left (b x +a \right ) \sin \left (b x +a \right )\right )}{b}+\frac {3 c \,d^{2} \left (\left (b x +a \right )^{2} \sin \left (b x +a \right )-2 \sin \left (b x +a \right )+2 \left (b x +a \right ) \cos \left (b x +a \right )\right )}{b^{2}}+\frac {d^{3} \left (\left (b x +a \right )^{3} \sin \left (b x +a \right )+3 \left (b x +a \right )^{2} \cos \left (b x +a \right )-6 \cos \left (b x +a \right )-6 \left (b x +a \right ) \sin \left (b x +a \right )\right )}{b^{3}}\right )}{b^{2}}\)	\(390\)
meijerg	\(\frac {16 d^{4} \sin \left (a \right ) \sqrt {\pi }\, \left (-\frac {x \left (b^{2}\right )^{\frac {5}{2}} \left (-\frac {5 x^{2} b^{2}}{2}+15\right ) \cos \left (b x \right )}{10 \sqrt {\pi }\, b^{4}}+\frac {\left (b^{2}\right )^{\frac {5}{2}} \left (\frac {5}{8} x^{4} b^{4}-\frac {15}{2} x^{2} b^{2}+15\right ) \sin \left (b x \right )}{10 \sqrt {\pi }\, b^{5}}\right )}{b^{4} \sqrt {b^{2}}}+\frac {16 d^{4} \cos \left (a \right ) \sqrt {\pi }\, \left (\frac {3}{2 \sqrt {\pi }}-\frac {\left (\frac {3}{8} x^{4} b^{4}-\frac {9}{2} x^{2} b^{2}+9\right ) \cos \left (b x \right )}{6 \sqrt {\pi }}-\frac {x b \left (-\frac {3 x^{2} b^{2}}{2}+9\right ) \sin \left (b x \right )}{6 \sqrt {\pi }}\right )}{b^{5}}+\frac {32 d^{3} c \sin \left (a \right ) \sqrt {\pi }\, \left (\frac {3}{4 \sqrt {\pi }}-\frac {\left (-\frac {3 x^{2} b^{2}}{2}+3\right ) \cos \left (b x \right )}{4 \sqrt {\pi }}-\frac {x b \left (-\frac {x^{2} b^{2}}{2}+3\right ) \sin \left (b x \right )}{4 \sqrt {\pi }}\right )}{b^{4}}+\frac {32 d^{3} c \cos \left (a \right ) \sqrt {\pi }\, \left (\frac {x b \left (-\frac {5 x^{2} b^{2}}{2}+15\right ) \cos \left (b x \right )}{20 \sqrt {\pi }}-\frac {\left (-\frac {15 x^{2} b^{2}}{2}+15\right ) \sin \left (b x \right )}{20 \sqrt {\pi }}\right )}{b^{4}}+\frac {24 c^{2} d^{2} \sin \left (a \right ) \sqrt {\pi }\, \left (\frac {x \left (b^{2}\right )^{\frac {3}{2}} \cos \left (b x \right )}{2 \sqrt {\pi }\, b^{2}}-\frac {\left (b^{2}\right )^{\frac {3}{2}} \left (-\frac {3 x^{2} b^{2}}{2}+3\right ) \sin \left (b x \right )}{6 \sqrt {\pi }\, b^{3}}\right )}{b^{2} \sqrt {b^{2}}}+\frac {24 c^{2} d^{2} \cos \left (a \right ) \sqrt {\pi }\, \left (-\frac {1}{2 \sqrt {\pi }}+\frac {\left (-\frac {x^{2} b^{2}}{2}+1\right ) \cos \left (b x \right )}{2 \sqrt {\pi }}+\frac {x b \sin \left (b x \right )}{2 \sqrt {\pi }}\right )}{b^{3}}+\frac {8 d \,c^{3} \sin \left (a \right ) \sqrt {\pi }\, \left (-\frac {1}{2 \sqrt {\pi }}+\frac {\cos \left (b x \right )}{2 \sqrt {\pi }}+\frac {x b \sin \left (b x \right )}{2 \sqrt {\pi }}\right )}{b^{2}}+\frac {8 d \,c^{3} \cos \left (a \right ) \sqrt {\pi }\, \left (-\frac {x b \cos \left (b x \right )}{2 \sqrt {\pi }}+\frac {\sin \left (b x \right )}{2 \sqrt {\pi }}\right )}{b^{2}}+\frac {c^{4} \sin \left (a \right ) \sin \left (b x \right )}{b}+\frac {c^{4} \cos \left (a \right ) \sqrt {\pi }\, \left (\frac {1}{\sqrt {\pi }}-\frac {\cos \left (b x \right )}{\sqrt {\pi }}\right )}{b}\)	\(498\)
derivativedivides	\(\frac {-\frac {a^{4} d^{4} \cos \left (b x +a \right )}{b^{4}}+\frac {4 a^{3} c \,d^{3} \cos \left (b x +a \right )}{b^{3}}-\frac {4 a^{3} d^{4} \left (\sin \left (b x +a \right )-\left (b x +a \right ) \cos \left (b x +a \right )\right )}{b^{4}}-\frac {6 a^{2} c^{2} d^{2} \cos \left (b x +a \right )}{b^{2}}+\frac {12 a^{2} c \,d^{3} \left (\sin \left (b x +a \right )-\left (b x +a \right ) \cos \left (b x +a \right )\right )}{b^{3}}+\frac {6 a^{2} d^{4} \left (-\left (b x +a \right )^{2} \cos \left (b x +a \right )+2 \cos \left (b x +a \right )+2 \left (b x +a \right ) \sin \left (b x +a \right )\right )}{b^{4}}+\frac {4 a \,c^{3} d \cos \left (b x +a \right )}{b}-\frac {12 a \,c^{2} d^{2} \left (\sin \left (b x +a \right )-\left (b x +a \right ) \cos \left (b x +a \right )\right )}{b^{2}}-\frac {12 a c \,d^{3} \left (-\left (b x +a \right )^{2} \cos \left (b x +a \right )+2 \cos \left (b x +a \right )+2 \left (b x +a \right ) \sin \left (b x +a \right )\right )}{b^{3}}-\frac {4 a \,d^{4} \left (-\left (b x +a \right )^{3} \cos \left (b x +a \right )+3 \left (b x +a \right )^{2} \sin \left (b x +a \right )-6 \sin \left (b x +a \right )+6 \left (b x +a \right ) \cos \left (b x +a \right )\right )}{b^{4}}-c^{4} \cos \left (b x +a \right )+\frac {4 c^{3} d \left (\sin \left (b x +a \right )-\left (b x +a \right ) \cos \left (b x +a \right )\right )}{b}+\frac {6 c^{2} d^{2} \left (-\left (b x +a \right )^{2} \cos \left (b x +a \right )+2 \cos \left (b x +a \right )+2 \left (b x +a \right ) \sin \left (b x +a \right )\right )}{b^{2}}+\frac {4 c \,d^{3} \left (-\left (b x +a \right )^{3} \cos \left (b x +a \right )+3 \left (b x +a \right )^{2} \sin \left (b x +a \right )-6 \sin \left (b x +a \right )+6 \left (b x +a \right ) \cos \left (b x +a \right )\right )}{b^{3}}+\frac {d^{4} \left (-\left (b x +a \right )^{4} \cos \left (b x +a \right )+4 \left (b x +a \right )^{3} \sin \left (b x +a \right )+12 \left (b x +a \right )^{2} \cos \left (b x +a \right )-24 \cos \left (b x +a \right )-24 \left (b x +a \right ) \sin \left (b x +a \right )\right )}{b^{4}}}{b}\)	\(551\)
default	\(\frac {-\frac {a^{4} d^{4} \cos \left (b x +a \right )}{b^{4}}+\frac {4 a^{3} c \,d^{3} \cos \left (b x +a \right )}{b^{3}}-\frac {4 a^{3} d^{4} \left (\sin \left (b x +a \right )-\left (b x +a \right ) \cos \left (b x +a \right )\right )}{b^{4}}-\frac {6 a^{2} c^{2} d^{2} \cos \left (b x +a \right )}{b^{2}}+\frac {12 a^{2} c \,d^{3} \left (\sin \left (b x +a \right )-\left (b x +a \right ) \cos \left (b x +a \right )\right )}{b^{3}}+\frac {6 a^{2} d^{4} \left (-\left (b x +a \right )^{2} \cos \left (b x +a \right )+2 \cos \left (b x +a \right )+2 \left (b x +a \right ) \sin \left (b x +a \right )\right )}{b^{4}}+\frac {4 a \,c^{3} d \cos \left (b x +a \right )}{b}-\frac {12 a \,c^{2} d^{2} \left (\sin \left (b x +a \right )-\left (b x +a \right ) \cos \left (b x +a \right )\right )}{b^{2}}-\frac {12 a c \,d^{3} \left (-\left (b x +a \right )^{2} \cos \left (b x +a \right )+2 \cos \left (b x +a \right )+2 \left (b x +a \right ) \sin \left (b x +a \right )\right )}{b^{3}}-\frac {4 a \,d^{4} \left (-\left (b x +a \right )^{3} \cos \left (b x +a \right )+3 \left (b x +a \right )^{2} \sin \left (b x +a \right )-6 \sin \left (b x +a \right )+6 \left (b x +a \right ) \cos \left (b x +a \right )\right )}{b^{4}}-c^{4} \cos \left (b x +a \right )+\frac {4 c^{3} d \left (\sin \left (b x +a \right )-\left (b x +a \right ) \cos \left (b x +a \right )\right )}{b}+\frac {6 c^{2} d^{2} \left (-\left (b x +a \right )^{2} \cos \left (b x +a \right )+2 \cos \left (b x +a \right )+2 \left (b x +a \right ) \sin \left (b x +a \right )\right )}{b^{2}}+\frac {4 c \,d^{3} \left (-\left (b x +a \right )^{3} \cos \left (b x +a \right )+3 \left (b x +a \right )^{2} \sin \left (b x +a \right )-6 \sin \left (b x +a \right )+6 \left (b x +a \right ) \cos \left (b x +a \right )\right )}{b^{3}}+\frac {d^{4} \left (-\left (b x +a \right )^{4} \cos \left (b x +a \right )+4 \left (b x +a \right )^{3} \sin \left (b x +a \right )+12 \left (b x +a \right )^{2} \cos \left (b x +a \right )-24 \cos \left (b x +a \right )-24 \left (b x +a \right ) \sin \left (b x +a \right )\right )}{b^{4}}}{b}\)	\(551\)

Input:

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

Output:

-(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)/b^5*cos(b*x+a)+4/b^4*d*(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)*sin(b*x+ 
a)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.85 \[ \int (c+d x)^4 \sin (a+b x) \, dx=-\frac {{\left (b^{4} d^{4} x^{4} + 4 \, b^{4} c d^{3} x^{3} + b^{4} c^{4} - 12 \, b^{2} c^{2} d^{2} + 24 \, d^{4} + 6 \, {\left (b^{4} c^{2} d^{2} - 2 \, b^{2} d^{4}\right )} x^{2} + 4 \, {\left (b^{4} c^{3} d - 6 \, b^{2} c d^{3}\right )} x\right )} \cos \left (b x + a\right ) - 4 \, {\left (b^{3} d^{4} x^{3} + 3 \, b^{3} c d^{3} x^{2} + b^{3} c^{3} d - 6 \, b c d^{3} + 3 \, {\left (b^{3} c^{2} d^{2} - 2 \, b d^{4}\right )} x\right )} \sin \left (b x + a\right )}{b^{5}} \] Input:

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

Output:

-((b^4*d^4*x^4 + 4*b^4*c*d^3*x^3 + b^4*c^4 - 12*b^2*c^2*d^2 + 24*d^4 + 6*( 
b^4*c^2*d^2 - 2*b^2*d^4)*x^2 + 4*(b^4*c^3*d - 6*b^2*c*d^3)*x)*cos(b*x + a) 
 - 4*(b^3*d^4*x^3 + 3*b^3*c*d^3*x^2 + b^3*c^3*d - 6*b*c*d^3 + 3*(b^3*c^2*d 
^2 - 2*b*d^4)*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. 311 vs. \(2 (92) = 184\).

Time = 0.36 (sec) , antiderivative size = 311, normalized size of antiderivative = 3.38 \[ \int (c+d x)^4 \sin (a+b x) \, dx=\begin {cases} - \frac {c^{4} \cos {\left (a + b x \right )}}{b} - \frac {4 c^{3} d x \cos {\left (a + b x \right )}}{b} - \frac {6 c^{2} d^{2} x^{2} \cos {\left (a + b x \right )}}{b} - \frac {4 c d^{3} x^{3} \cos {\left (a + b x \right )}}{b} - \frac {d^{4} x^{4} \cos {\left (a + b x \right )}}{b} + \frac {4 c^{3} d \sin {\left (a + b x \right )}}{b^{2}} + \frac {12 c^{2} d^{2} x \sin {\left (a + b x \right )}}{b^{2}} + \frac {12 c d^{3} x^{2} \sin {\left (a + b x \right )}}{b^{2}} + \frac {4 d^{4} x^{3} \sin {\left (a + b x \right )}}{b^{2}} + \frac {12 c^{2} d^{2} \cos {\left (a + b x \right )}}{b^{3}} + \frac {24 c d^{3} x \cos {\left (a + b x \right )}}{b^{3}} + \frac {12 d^{4} x^{2} \cos {\left (a + b x \right )}}{b^{3}} - \frac {24 c d^{3} \sin {\left (a + b x \right )}}{b^{4}} - \frac {24 d^{4} x \sin {\left (a + b x \right )}}{b^{4}} - \frac {24 d^{4} \cos {\left (a + b x \right )}}{b^{5}} & \text {for}\: b \neq 0 \\\left (c^{4} x + 2 c^{3} d x^{2} + 2 c^{2} d^{2} x^{3} + c d^{3} x^{4} + \frac {d^{4} x^{5}}{5}\right ) \sin {\left (a \right )} & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((-c**4*cos(a + b*x)/b - 4*c**3*d*x*cos(a + b*x)/b - 6*c**2*d**2* 
x**2*cos(a + b*x)/b - 4*c*d**3*x**3*cos(a + b*x)/b - d**4*x**4*cos(a + b*x 
)/b + 4*c**3*d*sin(a + b*x)/b**2 + 12*c**2*d**2*x*sin(a + b*x)/b**2 + 12*c 
*d**3*x**2*sin(a + b*x)/b**2 + 4*d**4*x**3*sin(a + b*x)/b**2 + 12*c**2*d** 
2*cos(a + b*x)/b**3 + 24*c*d**3*x*cos(a + b*x)/b**3 + 12*d**4*x**2*cos(a + 
 b*x)/b**3 - 24*c*d**3*sin(a + b*x)/b**4 - 24*d**4*x*sin(a + b*x)/b**4 - 2 
4*d**4*cos(a + b*x)/b**5, Ne(b, 0)), ((c**4*x + 2*c**3*d*x**2 + 2*c**2*d** 
2*x**3 + c*d**3*x**4 + d**4*x**5/5)*sin(a), True))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 490 vs. \(2 (92) = 184\).

Time = 0.07 (sec) , antiderivative size = 490, normalized size of antiderivative = 5.33 \[ \int (c+d x)^4 \sin (a+b x) \, dx=-\frac {c^{4} \cos \left (b x + a\right ) - \frac {4 \, a c^{3} d \cos \left (b x + a\right )}{b} + \frac {6 \, a^{2} c^{2} d^{2} \cos \left (b x + a\right )}{b^{2}} - \frac {4 \, a^{3} c d^{3} \cos \left (b x + a\right )}{b^{3}} + \frac {a^{4} d^{4} \cos \left (b x + a\right )}{b^{4}} + \frac {4 \, {\left ({\left (b x + a\right )} \cos \left (b x + a\right ) - \sin \left (b x + a\right )\right )} c^{3} d}{b} - \frac {12 \, {\left ({\left (b x + a\right )} \cos \left (b x + a\right ) - \sin \left (b x + a\right )\right )} a c^{2} d^{2}}{b^{2}} + \frac {12 \, {\left ({\left (b x + a\right )} \cos \left (b x + a\right ) - \sin \left (b x + a\right )\right )} a^{2} c d^{3}}{b^{3}} - \frac {4 \, {\left ({\left (b x + a\right )} \cos \left (b x + a\right ) - \sin \left (b x + a\right )\right )} a^{3} d^{4}}{b^{4}} + \frac {6 \, {\left ({\left ({\left (b x + a\right )}^{2} - 2\right )} \cos \left (b x + a\right ) - 2 \, {\left (b x + a\right )} \sin \left (b x + a\right )\right )} c^{2} d^{2}}{b^{2}} - \frac {12 \, {\left ({\left ({\left (b x + a\right )}^{2} - 2\right )} \cos \left (b x + a\right ) - 2 \, {\left (b x + a\right )} \sin \left (b x + a\right )\right )} a c d^{3}}{b^{3}} + \frac {6 \, {\left ({\left ({\left (b x + a\right )}^{2} - 2\right )} \cos \left (b x + a\right ) - 2 \, {\left (b x + a\right )} \sin \left (b x + a\right )\right )} a^{2} d^{4}}{b^{4}} + \frac {4 \, {\left ({\left ({\left (b x + a\right )}^{3} - 6 \, b x - 6 \, a\right )} \cos \left (b x + a\right ) - 3 \, {\left ({\left (b x + a\right )}^{2} - 2\right )} \sin \left (b x + a\right )\right )} c d^{3}}{b^{3}} - \frac {4 \, {\left ({\left ({\left (b x + a\right )}^{3} - 6 \, b x - 6 \, a\right )} \cos \left (b x + a\right ) - 3 \, {\left ({\left (b x + a\right )}^{2} - 2\right )} \sin \left (b x + a\right )\right )} a d^{4}}{b^{4}} + \frac {{\left ({\left ({\left (b x + a\right )}^{4} - 12 \, {\left (b x + a\right )}^{2} + 24\right )} \cos \left (b x + a\right ) - 4 \, {\left ({\left (b x + a\right )}^{3} - 6 \, b x - 6 \, a\right )} \sin \left (b x + a\right )\right )} d^{4}}{b^{4}}}{b} \] Input:

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

Output:

-(c^4*cos(b*x + a) - 4*a*c^3*d*cos(b*x + a)/b + 6*a^2*c^2*d^2*cos(b*x + a) 
/b^2 - 4*a^3*c*d^3*cos(b*x + a)/b^3 + a^4*d^4*cos(b*x + a)/b^4 + 4*((b*x + 
 a)*cos(b*x + a) - sin(b*x + a))*c^3*d/b - 12*((b*x + a)*cos(b*x + a) - si 
n(b*x + a))*a*c^2*d^2/b^2 + 12*((b*x + a)*cos(b*x + a) - sin(b*x + a))*a^2 
*c*d^3/b^3 - 4*((b*x + a)*cos(b*x + a) - sin(b*x + a))*a^3*d^4/b^4 + 6*((( 
b*x + a)^2 - 2)*cos(b*x + a) - 2*(b*x + a)*sin(b*x + a))*c^2*d^2/b^2 - 12* 
(((b*x + a)^2 - 2)*cos(b*x + a) - 2*(b*x + a)*sin(b*x + a))*a*c*d^3/b^3 + 
6*(((b*x + a)^2 - 2)*cos(b*x + a) - 2*(b*x + a)*sin(b*x + a))*a^2*d^4/b^4 
+ 4*(((b*x + a)^3 - 6*b*x - 6*a)*cos(b*x + a) - 3*((b*x + a)^2 - 2)*sin(b* 
x + a))*c*d^3/b^3 - 4*(((b*x + a)^3 - 6*b*x - 6*a)*cos(b*x + a) - 3*((b*x 
+ a)^2 - 2)*sin(b*x + a))*a*d^4/b^4 + (((b*x + a)^4 - 12*(b*x + a)^2 + 24) 
*cos(b*x + a) - 4*((b*x + a)^3 - 6*b*x - 6*a)*sin(b*x + a))*d^4/b^4)/b
 

Giac [A] (verification not implemented)

Time = 0.34 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.86 \[ \int (c+d x)^4 \sin (a+b x) \, dx=-\frac {{\left (b^{4} d^{4} x^{4} + 4 \, b^{4} c d^{3} x^{3} + 6 \, b^{4} c^{2} d^{2} x^{2} + 4 \, b^{4} c^{3} d x + b^{4} c^{4} - 12 \, b^{2} d^{4} x^{2} - 24 \, b^{2} c d^{3} x - 12 \, b^{2} c^{2} d^{2} + 24 \, d^{4}\right )} \cos \left (b x + a\right )}{b^{5}} + \frac {4 \, {\left (b^{3} d^{4} x^{3} + 3 \, b^{3} c d^{3} x^{2} + 3 \, b^{3} c^{2} d^{2} x + b^{3} c^{3} d - 6 \, b d^{4} x - 6 \, b c d^{3}\right )} \sin \left (b x + a\right )}{b^{5}} \] Input:

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

Output:

-(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)*cos(b*x + 
 a)/b^5 + 4*(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)*sin(b*x + a)/b^5
 

Mupad [B] (verification not implemented)

Time = 0.40 (sec) , antiderivative size = 221, normalized size of antiderivative = 2.40 \[ \int (c+d x)^4 \sin (a+b x) \, dx=\frac {4\,x\,\cos \left (a+b\,x\right )\,\left (6\,c\,d^3-b^2\,c^3\,d\right )}{b^3}-\frac {4\,\sin \left (a+b\,x\right )\,\left (6\,c\,d^3-b^2\,c^3\,d\right )}{b^4}-\frac {d^4\,x^4\,\cos \left (a+b\,x\right )}{b}-\frac {\cos \left (a+b\,x\right )\,\left (b^4\,c^4-12\,b^2\,c^2\,d^2+24\,d^4\right )}{b^5}+\frac {4\,d^4\,x^3\,\sin \left (a+b\,x\right )}{b^2}-\frac {12\,x\,\sin \left (a+b\,x\right )\,\left (2\,d^4-b^2\,c^2\,d^2\right )}{b^4}+\frac {6\,x^2\,\cos \left (a+b\,x\right )\,\left (2\,d^4-b^2\,c^2\,d^2\right )}{b^3}-\frac {4\,c\,d^3\,x^3\,\cos \left (a+b\,x\right )}{b}+\frac {12\,c\,d^3\,x^2\,\sin \left (a+b\,x\right )}{b^2} \] Input:

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

Output:

(4*x*cos(a + b*x)*(6*c*d^3 - b^2*c^3*d))/b^3 - (4*sin(a + b*x)*(6*c*d^3 - 
b^2*c^3*d))/b^4 - (d^4*x^4*cos(a + b*x))/b - (cos(a + b*x)*(24*d^4 + b^4*c 
^4 - 12*b^2*c^2*d^2))/b^5 + (4*d^4*x^3*sin(a + b*x))/b^2 - (12*x*sin(a + b 
*x)*(2*d^4 - b^2*c^2*d^2))/b^4 + (6*x^2*cos(a + b*x)*(2*d^4 - b^2*c^2*d^2) 
)/b^3 - (4*c*d^3*x^3*cos(a + b*x))/b + (12*c*d^3*x^2*sin(a + b*x))/b^2
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 245, normalized size of antiderivative = 2.66 \[ \int (c+d x)^4 \sin (a+b x) \, dx=\frac {-\cos \left (b x +a \right ) b^{4} c^{4}-4 \cos \left (b x +a \right ) b^{4} c^{3} d x -6 \cos \left (b x +a \right ) b^{4} c^{2} d^{2} x^{2}-4 \cos \left (b x +a \right ) b^{4} c \,d^{3} x^{3}-\cos \left (b x +a \right ) b^{4} d^{4} x^{4}+12 \cos \left (b x +a \right ) b^{2} c^{2} d^{2}+24 \cos \left (b x +a \right ) b^{2} c \,d^{3} x +12 \cos \left (b x +a \right ) b^{2} d^{4} x^{2}-24 \cos \left (b x +a \right ) d^{4}+4 \sin \left (b x +a \right ) b^{3} c^{3} d +12 \sin \left (b x +a \right ) b^{3} c^{2} d^{2} x +12 \sin \left (b x +a \right ) b^{3} c \,d^{3} x^{2}+4 \sin \left (b x +a \right ) b^{3} d^{4} x^{3}-24 \sin \left (b x +a \right ) b c \,d^{3}-24 \sin \left (b x +a \right ) b \,d^{4} x}{b^{5}} \] Input:

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

Output:

( - cos(a + b*x)*b**4*c**4 - 4*cos(a + b*x)*b**4*c**3*d*x - 6*cos(a + b*x) 
*b**4*c**2*d**2*x**2 - 4*cos(a + b*x)*b**4*c*d**3*x**3 - cos(a + b*x)*b**4 
*d**4*x**4 + 12*cos(a + b*x)*b**2*c**2*d**2 + 24*cos(a + b*x)*b**2*c*d**3* 
x + 12*cos(a + b*x)*b**2*d**4*x**2 - 24*cos(a + b*x)*d**4 + 4*sin(a + b*x) 
*b**3*c**3*d + 12*sin(a + b*x)*b**3*c**2*d**2*x + 12*sin(a + b*x)*b**3*c*d 
**3*x**2 + 4*sin(a + b*x)*b**3*d**4*x**3 - 24*sin(a + b*x)*b*c*d**3 - 24*s 
in(a + b*x)*b*d**4*x)/b**5