\(\int (c+d x) \cos ^3(a+b x) \sin ^2(a+b x) \, dx\) [149]

Optimal result

Integrand size = 22, antiderivative size = 109 \[ \int (c+d x) \cos ^3(a+b x) \sin ^2(a+b x) \, dx=\frac {d \cos (a+b x)}{8 b^2}-\frac {d \cos (3 a+3 b x)}{144 b^2}-\frac {d \cos (5 a+5 b x)}{400 b^2}+\frac {(c+d x) \sin (a+b x)}{8 b}-\frac {(c+d x) \sin (3 a+3 b x)}{48 b}-\frac {(c+d x) \sin (5 a+5 b x)}{80 b} \] Output:

1/8*d*cos(b*x+a)/b^2-1/144*d*cos(3*b*x+3*a)/b^2-1/400*d*cos(5*b*x+5*a)/b^2 
+1/8*(d*x+c)*sin(b*x+a)/b-1/48*(d*x+c)*sin(3*b*x+3*a)/b-1/80*(d*x+c)*sin(5 
*b*x+5*a)/b
 

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.01 \[ \int (c+d x) \cos ^3(a+b x) \sin ^2(a+b x) \, dx=-\frac {-450 d \cos (a+b x)+25 d \cos (3 (a+b x))+9 d \cos (5 (a+b x))-450 b c \sin (a+b x)-450 b d x \sin (a+b x)+75 b c \sin (3 (a+b x))+75 b d x \sin (3 (a+b x))+45 b c \sin (5 (a+b x))+45 b d x \sin (5 (a+b x))}{3600 b^2} \] Input:

Integrate[(c + d*x)*Cos[a + b*x]^3*Sin[a + b*x]^2,x]
 

Output:

-1/3600*(-450*d*Cos[a + b*x] + 25*d*Cos[3*(a + b*x)] + 9*d*Cos[5*(a + b*x) 
] - 450*b*c*Sin[a + b*x] - 450*b*d*x*Sin[a + b*x] + 75*b*c*Sin[3*(a + b*x) 
] + 75*b*d*x*Sin[3*(a + b*x)] + 45*b*c*Sin[5*(a + b*x)] + 45*b*d*x*Sin[5*( 
a + b*x)])/b^2
 

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {4906, 2009}

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.

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

\(\Big \downarrow \) 4906

\(\displaystyle \int \left (\frac {1}{8} (c+d x) \cos (a+b x)-\frac {1}{16} (c+d x) \cos (3 a+3 b x)-\frac {1}{16} (c+d x) \cos (5 a+5 b x)\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {d \cos (a+b x)}{8 b^2}-\frac {d \cos (3 a+3 b x)}{144 b^2}-\frac {d \cos (5 a+5 b x)}{400 b^2}+\frac {(c+d x) \sin (a+b x)}{8 b}-\frac {(c+d x) \sin (3 a+3 b x)}{48 b}-\frac {(c+d x) \sin (5 a+5 b x)}{80 b}\)

Input:

Int[(c + d*x)*Cos[a + b*x]^3*Sin[a + b*x]^2,x]
 

Output:

(d*Cos[a + b*x])/(8*b^2) - (d*Cos[3*a + 3*b*x])/(144*b^2) - (d*Cos[5*a + 5 
*b*x])/(400*b^2) + ((c + d*x)*Sin[a + b*x])/(8*b) - ((c + d*x)*Sin[3*a + 3 
*b*x])/(48*b) - ((c + d*x)*Sin[5*a + 5*b*x])/(80*b)
 

Defintions of rubi rules used

rule 2009

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

rule 4906

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

Maple [A] (verified)

Time = 2.43 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.83

method	result	size
parallelrisch	\(\frac {-75 b \left (d x +c \right ) \sin \left (3 b x +3 a \right )-45 b \left (d x +c \right ) \sin \left (5 b x +5 a \right )-25 d \cos \left (3 b x +3 a \right )-9 d \cos \left (5 b x +5 a \right )+450 \left (d x +c \right ) b \sin \left (b x +a \right )+450 d \cos \left (b x +a \right )+416 d}{3600 b^{2}}\)	\(91\)
risch	\(\frac {d \cos \left (b x +a \right )}{8 b^{2}}-\frac {d \cos \left (3 b x +3 a \right )}{144 b^{2}}-\frac {d \cos \left (5 b x +5 a \right )}{400 b^{2}}+\frac {\left (d x +c \right ) \sin \left (b x +a \right )}{8 b}-\frac {\left (d x +c \right ) \sin \left (3 b x +3 a \right )}{48 b}-\frac {\left (d x +c \right ) \sin \left (5 b x +5 a \right )}{80 b}\)	\(98\)
derivativedivides	\(\frac {-\frac {d a \left (-\frac {\sin \left (b x +a \right ) \cos \left (b x +a \right )^{4}}{5}+\frac {\left (2+\cos \left (b x +a \right )^{2}\right ) \sin \left (b x +a \right )}{15}\right )}{b}+c \left (-\frac {\sin \left (b x +a \right ) \cos \left (b x +a \right )^{4}}{5}+\frac {\left (2+\cos \left (b x +a \right )^{2}\right ) \sin \left (b x +a \right )}{15}\right )+\frac {d \left (\frac {\left (b x +a \right ) \left (2+\cos \left (b x +a \right )^{2}\right ) \sin \left (b x +a \right )}{3}+\frac {\cos \left (b x +a \right )^{3}}{45}+\frac {2 \cos \left (b x +a \right )}{15}-\frac {\left (b x +a \right ) \left (\frac {8}{3}+\cos \left (b x +a \right )^{4}+\frac {4 \cos \left (b x +a \right )^{2}}{3}\right ) \sin \left (b x +a \right )}{5}-\frac {\cos \left (b x +a \right )^{5}}{25}\right )}{b}}{b}\)	\(175\)
default	\(\frac {-\frac {d a \left (-\frac {\sin \left (b x +a \right ) \cos \left (b x +a \right )^{4}}{5}+\frac {\left (2+\cos \left (b x +a \right )^{2}\right ) \sin \left (b x +a \right )}{15}\right )}{b}+c \left (-\frac {\sin \left (b x +a \right ) \cos \left (b x +a \right )^{4}}{5}+\frac {\left (2+\cos \left (b x +a \right )^{2}\right ) \sin \left (b x +a \right )}{15}\right )+\frac {d \left (\frac {\left (b x +a \right ) \left (2+\cos \left (b x +a \right )^{2}\right ) \sin \left (b x +a \right )}{3}+\frac {\cos \left (b x +a \right )^{3}}{45}+\frac {2 \cos \left (b x +a \right )}{15}-\frac {\left (b x +a \right ) \left (\frac {8}{3}+\cos \left (b x +a \right )^{4}+\frac {4 \cos \left (b x +a \right )^{2}}{3}\right ) \sin \left (b x +a \right )}{5}-\frac {\cos \left (b x +a \right )^{5}}{25}\right )}{b}}{b}\)	\(175\)
norman	\(\frac {\frac {52 d}{225 b^{2}}+\frac {8 c \tan \left (\frac {a}{2}+\frac {b x}{2}\right )^{3}}{3 b}-\frac {16 c \tan \left (\frac {a}{2}+\frac {b x}{2}\right )^{5}}{15 b}+\frac {8 c \tan \left (\frac {a}{2}+\frac {b x}{2}\right )^{7}}{3 b}+\frac {4 d \tan \left (\frac {a}{2}+\frac {b x}{2}\right )^{6}}{3 b^{2}}+\frac {44 d \tan \left (\frac {a}{2}+\frac {b x}{2}\right )^{4}}{45 b^{2}}+\frac {52 d \tan \left (\frac {a}{2}+\frac {b x}{2}\right )^{2}}{45 b^{2}}+\frac {8 d x \tan \left (\frac {a}{2}+\frac {b x}{2}\right )^{3}}{3 b}-\frac {16 d x \tan \left (\frac {a}{2}+\frac {b x}{2}\right )^{5}}{15 b}+\frac {8 d x \tan \left (\frac {a}{2}+\frac {b x}{2}\right )^{7}}{3 b}}{\left (1+\tan \left (\frac {a}{2}+\frac {b x}{2}\right )^{2}\right )^{5}}\)	\(180\)
orering	\(\frac {2 d \left (259 d^{4} x^{4} b^{4}+1036 b^{4} c \,d^{3} x^{3}+1554 b^{4} c^{2} d^{2} x^{2}+1036 b^{4} c^{3} d x +259 b^{4} c^{4}+140 b^{2} d^{4} x^{2}+280 b^{2} c \,d^{3} x +140 b^{2} c^{2} d^{2}+72 d^{4}\right ) \cos \left (b x +a \right )^{3} \sin \left (b x +a \right )^{2}}{225 b^{6} \left (d x +c \right )^{4}}-\frac {\left (259 d^{4} x^{4} b^{4}+1036 b^{4} c \,d^{3} x^{3}+1554 b^{4} c^{2} d^{2} x^{2}+1036 b^{4} c^{3} d x +259 b^{4} c^{4}+280 b^{2} d^{4} x^{2}+560 b^{2} c \,d^{3} x +280 b^{2} c^{2} d^{2}+144 d^{4}\right ) \left (d \cos \left (b x +a \right )^{3} \sin \left (b x +a \right )^{2}-3 \left (d x +c \right ) \cos \left (b x +a \right )^{2} \sin \left (b x +a \right )^{3} b +2 \left (d x +c \right ) \cos \left (b x +a \right )^{4} \sin \left (b x +a \right ) b \right )}{225 b^{6} \left (d x +c \right )^{4}}+\frac {4 d \left (35 x^{2} d^{2} b^{2}+70 b^{2} c d x +35 b^{2} c^{2}+18 d^{2}\right ) \left (-6 d \cos \left (b x +a \right )^{2} \sin \left (b x +a \right )^{3} b +4 d \cos \left (b x +a \right )^{4} \sin \left (b x +a \right ) b +6 \left (d x +c \right ) \cos \left (b x +a \right ) \sin \left (b x +a \right )^{4} b^{2}-17 \left (d x +c \right ) \cos \left (b x +a \right )^{3} \sin \left (b x +a \right )^{2} b^{2}+2 \left (d x +c \right ) \cos \left (b x +a \right )^{5} b^{2}\right )}{225 b^{6} \left (d x +c \right )^{3}}-\frac {\left (35 x^{2} d^{2} b^{2}+70 b^{2} c d x +35 b^{2} c^{2}+24 d^{2}\right ) \left (18 d \cos \left (b x +a \right ) \sin \left (b x +a \right )^{4} b^{2}-51 d \cos \left (b x +a \right )^{3} \sin \left (b x +a \right )^{2} b^{2}+6 d \cos \left (b x +a \right )^{5} b^{2}-6 \left (d x +c \right ) b^{3} \sin \left (b x +a \right )^{5}+75 \left (d x +c \right ) \cos \left (b x +a \right )^{2} \sin \left (b x +a \right )^{3} b^{3}-44 \left (d x +c \right ) \cos \left (b x +a \right )^{4} \sin \left (b x +a \right ) b^{3}\right )}{225 b^{6} \left (d x +c \right )^{2}}+\frac {2 d \left (-24 d \,b^{3} \sin \left (b x +a \right )^{5}+300 d \cos \left (b x +a \right )^{2} \sin \left (b x +a \right )^{3} b^{3}-176 d \cos \left (b x +a \right )^{4} \sin \left (b x +a \right ) b^{3}-180 \left (d x +c \right ) b^{4} \sin \left (b x +a \right )^{4} \cos \left (b x +a \right )+401 \left (d x +c \right ) \cos \left (b x +a \right )^{3} \sin \left (b x +a \right )^{2} b^{4}-44 \left (d x +c \right ) \cos \left (b x +a \right )^{5} b^{4}\right )}{75 b^{6} \left (d x +c \right )}-\frac {-900 d \,b^{4} \sin \left (b x +a \right )^{4} \cos \left (b x +a \right )+2005 d \cos \left (b x +a \right )^{3} \sin \left (b x +a \right )^{2} b^{4}-220 d \cos \left (b x +a \right )^{5} b^{4}-1923 \left (d x +c \right ) b^{5} \sin \left (b x +a \right )^{3} \cos \left (b x +a \right )^{2}+180 \left (d x +c \right ) b^{5} \sin \left (b x +a \right )^{5}+1022 \left (d x +c \right ) \cos \left (b x +a \right )^{4} \sin \left (b x +a \right ) b^{5}}{225 b^{6}}\)	\(885\)

Input:

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

Output:

1/3600*(-75*b*(d*x+c)*sin(3*b*x+3*a)-45*b*(d*x+c)*sin(5*b*x+5*a)-25*d*cos( 
3*b*x+3*a)-9*d*cos(5*b*x+5*a)+450*(d*x+c)*b*sin(b*x+a)+450*d*cos(b*x+a)+41 
6*d)/b^2
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.83 \[ \int (c+d x) \cos ^3(a+b x) \sin ^2(a+b x) \, dx=-\frac {9 \, d \cos \left (b x + a\right )^{5} - 5 \, d \cos \left (b x + a\right )^{3} - 30 \, d \cos \left (b x + a\right ) + 15 \, {\left (3 \, {\left (b d x + b c\right )} \cos \left (b x + a\right )^{4} - 2 \, b d x - {\left (b d x + b c\right )} \cos \left (b x + a\right )^{2} - 2 \, b c\right )} \sin \left (b x + a\right )}{225 \, b^{2}} \] Input:

integrate((d*x+c)*cos(b*x+a)^3*sin(b*x+a)^2,x, algorithm="fricas")
 

Output:

-1/225*(9*d*cos(b*x + a)^5 - 5*d*cos(b*x + a)^3 - 30*d*cos(b*x + a) + 15*( 
3*(b*d*x + b*c)*cos(b*x + a)^4 - 2*b*d*x - (b*d*x + b*c)*cos(b*x + a)^2 - 
2*b*c)*sin(b*x + a))/b^2
 

Sympy [A] (verification not implemented)

Time = 0.42 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.50 \[ \int (c+d x) \cos ^3(a+b x) \sin ^2(a+b x) \, dx=\begin {cases} \frac {2 c \sin ^{5}{\left (a + b x \right )}}{15 b} + \frac {c \sin ^{3}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{3 b} + \frac {2 d x \sin ^{5}{\left (a + b x \right )}}{15 b} + \frac {d x \sin ^{3}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{3 b} + \frac {2 d \sin ^{4}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{15 b^{2}} + \frac {13 d \sin ^{2}{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{45 b^{2}} + \frac {26 d \cos ^{5}{\left (a + b x \right )}}{225 b^{2}} & \text {for}\: b \neq 0 \\\left (c x + \frac {d x^{2}}{2}\right ) \sin ^{2}{\left (a \right )} \cos ^{3}{\left (a \right )} & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((2*c*sin(a + b*x)**5/(15*b) + c*sin(a + b*x)**3*cos(a + b*x)**2/ 
(3*b) + 2*d*x*sin(a + b*x)**5/(15*b) + d*x*sin(a + b*x)**3*cos(a + b*x)**2 
/(3*b) + 2*d*sin(a + b*x)**4*cos(a + b*x)/(15*b**2) + 13*d*sin(a + b*x)**2 
*cos(a + b*x)**3/(45*b**2) + 26*d*cos(a + b*x)**5/(225*b**2), Ne(b, 0)), ( 
(c*x + d*x**2/2)*sin(a)**2*cos(a)**3, True))
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.28 \[ \int (c+d x) \cos ^3(a+b x) \sin ^2(a+b x) \, dx=-\frac {240 \, {\left (3 \, \sin \left (b x + a\right )^{5} - 5 \, \sin \left (b x + a\right )^{3}\right )} c - \frac {240 \, {\left (3 \, \sin \left (b x + a\right )^{5} - 5 \, \sin \left (b x + a\right )^{3}\right )} a d}{b} + \frac {{\left (45 \, {\left (b x + a\right )} \sin \left (5 \, b x + 5 \, a\right ) + 75 \, {\left (b x + a\right )} \sin \left (3 \, b x + 3 \, a\right ) - 450 \, {\left (b x + a\right )} \sin \left (b x + a\right ) + 9 \, \cos \left (5 \, b x + 5 \, a\right ) + 25 \, \cos \left (3 \, b x + 3 \, a\right ) - 450 \, \cos \left (b x + a\right )\right )} d}{b}}{3600 \, b} \] Input:

integrate((d*x+c)*cos(b*x+a)^3*sin(b*x+a)^2,x, algorithm="maxima")
 

Output:

-1/3600*(240*(3*sin(b*x + a)^5 - 5*sin(b*x + a)^3)*c - 240*(3*sin(b*x + a) 
^5 - 5*sin(b*x + a)^3)*a*d/b + (45*(b*x + a)*sin(5*b*x + 5*a) + 75*(b*x + 
a)*sin(3*b*x + 3*a) - 450*(b*x + a)*sin(b*x + a) + 9*cos(5*b*x + 5*a) + 25 
*cos(3*b*x + 3*a) - 450*cos(b*x + a))*d/b)/b
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.97 \[ \int (c+d x) \cos ^3(a+b x) \sin ^2(a+b x) \, dx=-\frac {d \cos \left (5 \, b x + 5 \, a\right )}{400 \, b^{2}} - \frac {d \cos \left (3 \, b x + 3 \, a\right )}{144 \, b^{2}} + \frac {d \cos \left (b x + a\right )}{8 \, b^{2}} - \frac {{\left (b d x + b c\right )} \sin \left (5 \, b x + 5 \, a\right )}{80 \, b^{2}} - \frac {{\left (b d x + b c\right )} \sin \left (3 \, b x + 3 \, a\right )}{48 \, b^{2}} + \frac {{\left (b d x + b c\right )} \sin \left (b x + a\right )}{8 \, b^{2}} \] Input:

integrate((d*x+c)*cos(b*x+a)^3*sin(b*x+a)^2,x, algorithm="giac")
                                                                                    
                                                                                    
 

Output:

-1/400*d*cos(5*b*x + 5*a)/b^2 - 1/144*d*cos(3*b*x + 3*a)/b^2 + 1/8*d*cos(b 
*x + a)/b^2 - 1/80*(b*d*x + b*c)*sin(5*b*x + 5*a)/b^2 - 1/48*(b*d*x + b*c) 
*sin(3*b*x + 3*a)/b^2 + 1/8*(b*d*x + b*c)*sin(b*x + a)/b^2
 

Mupad [B] (verification not implemented)

Time = 0.46 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.09 \[ \int (c+d x) \cos ^3(a+b x) \sin ^2(a+b x) \, dx=\frac {26\,d\,{\cos \left (a+b\,x\right )}^5+65\,d\,{\cos \left (a+b\,x\right )}^3\,{\sin \left (a+b\,x\right )}^2+30\,d\,\cos \left (a+b\,x\right )\,{\sin \left (a+b\,x\right )}^4+30\,b\,c\,{\sin \left (a+b\,x\right )}^5+30\,b\,d\,x\,{\sin \left (a+b\,x\right )}^5+75\,b\,c\,{\cos \left (a+b\,x\right )}^2\,{\sin \left (a+b\,x\right )}^3+75\,b\,d\,x\,{\cos \left (a+b\,x\right )}^2\,{\sin \left (a+b\,x\right )}^3}{225\,b^2} \] Input:

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

Output:

(26*d*cos(a + b*x)^5 + 65*d*cos(a + b*x)^3*sin(a + b*x)^2 + 30*d*cos(a + b 
*x)*sin(a + b*x)^4 + 30*b*c*sin(a + b*x)^5 + 30*b*d*x*sin(a + b*x)^5 + 75* 
b*c*cos(a + b*x)^2*sin(a + b*x)^3 + 75*b*d*x*cos(a + b*x)^2*sin(a + b*x)^3 
)/(225*b^2)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.94 \[ \int (c+d x) \cos ^3(a+b x) \sin ^2(a+b x) \, dx=\frac {-9 \cos \left (b x +a \right ) \sin \left (b x +a \right )^{4} d +13 \cos \left (b x +a \right ) \sin \left (b x +a \right )^{2} d +26 \cos \left (b x +a \right ) d -45 \sin \left (b x +a \right )^{5} b c -45 \sin \left (b x +a \right )^{5} b d x +75 \sin \left (b x +a \right )^{3} b c +75 \sin \left (b x +a \right )^{3} b d x +26 d}{225 b^{2}} \] Input:

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

Output:

( - 9*cos(a + b*x)*sin(a + b*x)**4*d + 13*cos(a + b*x)*sin(a + b*x)**2*d + 
 26*cos(a + b*x)*d - 45*sin(a + b*x)**5*b*c - 45*sin(a + b*x)**5*b*d*x + 7 
5*sin(a + b*x)**3*b*c + 75*sin(a + b*x)**3*b*d*x + 26*d)/(225*b**2)