\(\int (c+d x)^m \cos (a+b x) \sin ^2(a+b x) \, dx\) [13]

Optimal result

Integrand size = 22, antiderivative size = 275 \[ \int (c+d x)^m \cos (a+b x) \sin ^2(a+b x) \, dx=-\frac {i e^{i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (-\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {i b (c+d x)}{d}\right )}{8 b}+\frac {i e^{-i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {i b (c+d x)}{d}\right )}{8 b}+\frac {i 3^{-1-m} e^{3 i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (-\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {3 i b (c+d x)}{d}\right )}{8 b}-\frac {i 3^{-1-m} e^{-3 i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {3 i b (c+d x)}{d}\right )}{8 b} \] Output:

-1/8*I*exp(I*(a-b*c/d))*(d*x+c)^m*GAMMA(1+m,-I*b*(d*x+c)/d)/b/((-I*b*(d*x+ 
c)/d)^m)+1/8*I*(d*x+c)^m*GAMMA(1+m,I*b*(d*x+c)/d)/b/exp(I*(a-b*c/d))/((I*b 
*(d*x+c)/d)^m)+1/8*I*3^(-1-m)*exp(3*I*(a-b*c/d))*(d*x+c)^m*GAMMA(1+m,-3*I* 
b*(d*x+c)/d)/b/((-I*b*(d*x+c)/d)^m)-1/8*I*3^(-1-m)*(d*x+c)^m*GAMMA(1+m,3*I 
*b*(d*x+c)/d)/b/exp(3*I*(a-b*c/d))/((I*b*(d*x+c)/d)^m)
 

Mathematica [A] (verified)

Time = 0.67 (sec) , antiderivative size = 237, normalized size of antiderivative = 0.86 \[ \int (c+d x)^m \cos (a+b x) \sin ^2(a+b x) \, dx=-\frac {i e^{-\frac {3 i (b c+a d)}{d}} (c+d x)^m \left (3 e^{2 i \left (2 a+\frac {b c}{d}\right )} \left (-\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {i b (c+d x)}{d}\right )+\left (\frac {i b (c+d x)}{d}\right )^{-m} \left (-3 e^{2 i a+\frac {4 i b c}{d}} \Gamma \left (1+m,\frac {i b (c+d x)}{d}\right )+3^{-m} \left (-e^{6 i a} \left (\frac {i b (c+d x)}{d}\right )^{2 m} \left (\frac {b^2 (c+d x)^2}{d^2}\right )^{-m} \Gamma \left (1+m,-\frac {3 i b (c+d x)}{d}\right )+e^{\frac {6 i b c}{d}} \Gamma \left (1+m,\frac {3 i b (c+d x)}{d}\right )\right )\right )\right )}{24 b} \] Input:

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

Output:

((-1/24*I)*(c + d*x)^m*((3*E^((2*I)*(2*a + (b*c)/d))*Gamma[1 + m, ((-I)*b* 
(c + d*x))/d])/(((-I)*b*(c + d*x))/d)^m + (-3*E^((2*I)*a + ((4*I)*b*c)/d)* 
Gamma[1 + m, (I*b*(c + d*x))/d] + (-((E^((6*I)*a)*((I*b*(c + d*x))/d)^(2*m 
)*Gamma[1 + m, ((-3*I)*b*(c + d*x))/d])/((b^2*(c + d*x)^2)/d^2)^m) + E^((( 
6*I)*b*c)/d)*Gamma[1 + m, ((3*I)*b*(c + d*x))/d])/3^m)/((I*b*(c + d*x))/d) 
^m))/(b*E^(((3*I)*(b*c + a*d))/d))
 

Rubi [A] (verified)

Time = 0.55 (sec) , antiderivative size = 275, 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.

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

\(\Big \downarrow \) 4906

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {i e^{i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (-\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,-\frac {i b (c+d x)}{d}\right )}{8 b}+\frac {i 3^{-m-1} e^{3 i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (-\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,-\frac {3 i b (c+d x)}{d}\right )}{8 b}+\frac {i e^{-i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,\frac {i b (c+d x)}{d}\right )}{8 b}-\frac {i 3^{-m-1} e^{-3 i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,\frac {3 i b (c+d x)}{d}\right )}{8 b}\)

Input:

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

Output:

((-1/8*I)*E^(I*(a - (b*c)/d))*(c + d*x)^m*Gamma[1 + m, ((-I)*b*(c + d*x))/ 
d])/(b*(((-I)*b*(c + d*x))/d)^m) + ((I/8)*(c + d*x)^m*Gamma[1 + m, (I*b*(c 
 + d*x))/d])/(b*E^(I*(a - (b*c)/d))*((I*b*(c + d*x))/d)^m) + ((I/8)*3^(-1 
- m)*E^((3*I)*(a - (b*c)/d))*(c + d*x)^m*Gamma[1 + m, ((-3*I)*b*(c + d*x)) 
/d])/(b*(((-I)*b*(c + d*x))/d)^m) - ((I/8)*3^(-1 - m)*(c + d*x)^m*Gamma[1 
+ m, ((3*I)*b*(c + d*x))/d])/(b*E^((3*I)*(a - (b*c)/d))*((I*b*(c + d*x))/d 
)^m)
 

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 [F]

\[\int \left (d x +c \right )^{m} \cos \left (b x +a \right ) \sin \left (b x +a \right )^{2}d x\]

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.68 \[ \int (c+d x)^m \cos (a+b x) \sin ^2(a+b x) \, dx=\frac {3 i \, e^{\left (-\frac {d m \log \left (\frac {i \, b}{d}\right ) - i \, b c + i \, a d}{d}\right )} \Gamma \left (m + 1, \frac {i \, b d x + i \, b c}{d}\right ) + i \, e^{\left (-\frac {d m \log \left (-\frac {3 i \, b}{d}\right ) + 3 i \, b c - 3 i \, a d}{d}\right )} \Gamma \left (m + 1, -\frac {3 \, {\left (i \, b d x + i \, b c\right )}}{d}\right ) - 3 i \, e^{\left (-\frac {d m \log \left (-\frac {i \, b}{d}\right ) + i \, b c - i \, a d}{d}\right )} \Gamma \left (m + 1, \frac {-i \, b d x - i \, b c}{d}\right ) - i \, e^{\left (-\frac {d m \log \left (\frac {3 i \, b}{d}\right ) - 3 i \, b c + 3 i \, a d}{d}\right )} \Gamma \left (m + 1, -\frac {3 \, {\left (-i \, b d x - i \, b c\right )}}{d}\right )}{24 \, b} \] Input:

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

Output:

1/24*(3*I*e^(-(d*m*log(I*b/d) - I*b*c + I*a*d)/d)*gamma(m + 1, (I*b*d*x + 
I*b*c)/d) + I*e^(-(d*m*log(-3*I*b/d) + 3*I*b*c - 3*I*a*d)/d)*gamma(m + 1, 
-3*(I*b*d*x + I*b*c)/d) - 3*I*e^(-(d*m*log(-I*b/d) + I*b*c - I*a*d)/d)*gam 
ma(m + 1, (-I*b*d*x - I*b*c)/d) - I*e^(-(d*m*log(3*I*b/d) - 3*I*b*c + 3*I* 
a*d)/d)*gamma(m + 1, -3*(-I*b*d*x - I*b*c)/d))/b
 

Sympy [F]

\[ \int (c+d x)^m \cos (a+b x) \sin ^2(a+b x) \, dx=\int \left (c + d x\right )^{m} \sin ^{2}{\left (a + b x \right )} \cos {\left (a + b x \right )}\, dx \] Input:

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

Output:

Integral((c + d*x)**m*sin(a + b*x)**2*cos(a + b*x), x)
 

Maxima [F]

\[ \int (c+d x)^m \cos (a+b x) \sin ^2(a+b x) \, dx=\int { {\left (d x + c\right )}^{m} \cos \left (b x + a\right ) \sin \left (b x + a\right )^{2} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int (c+d x)^m \cos (a+b x) \sin ^2(a+b x) \, dx=\int { {\left (d x + c\right )}^{m} \cos \left (b x + a\right ) \sin \left (b x + a\right )^{2} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int (c+d x)^m \cos (a+b x) \sin ^2(a+b x) \, dx=\int \cos \left (a+b\,x\right )\,{\sin \left (a+b\,x\right )}^2\,{\left (c+d\,x\right )}^m \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int (c+d x)^m \cos (a+b x) \sin ^2(a+b x) \, dx=\int \left (d x +c \right )^{m} \cos \left (b x +a \right ) \sin \left (b x +a \right )^{2}d x \] Input:

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

Output:

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