\(\int (c+d x)^m \cos (a+b x) \, dx\) [101]

Optimal result

Integrand size = 14, antiderivative size = 131 \[ \int (c+d x)^m \cos (a+b x) \, dx=-\frac {i e^{i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (-\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {i b (c+d x)}{d}\right )}{2 b}+\frac {i e^{-i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {i b (c+d x)}{d}\right )}{2 b} \] Output:

-1/2*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)+1/2*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)
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.93 \[ \int (c+d x)^m \cos (a+b x) \, dx=-\frac {i e^{-\frac {i (b c+a d)}{d}} (c+d x)^m \left (e^{2 i a} \left (-\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {i b (c+d x)}{d}\right )-e^{\frac {2 i b c}{d}} \left (\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {i b (c+d x)}{d}\right )\right )}{2 b} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3042, 3788, 26, 2612}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3788

\(\displaystyle \frac {1}{2} i \int -i e^{-i (a+b x)} (c+d x)^mdx-\frac {1}{2} i \int i e^{i (a+b x)} (c+d x)^mdx\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {1}{2} \int e^{-i (a+b x)} (c+d x)^mdx+\frac {1}{2} \int e^{i (a+b x)} (c+d x)^mdx\)

\(\Big \downarrow \) 2612

\(\displaystyle \frac {i e^{-i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,\frac {i b (c+d x)}{d}\right )}{2 b}-\frac {i e^{i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (-\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,-\frac {i b (c+d x)}{d}\right )}{2 b}\)

Input:

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

Output:

((-1/2*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/2)*(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)
 

Defintions of rubi rules used

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 2612

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))*((c_.) + (d_.)*(x_))^(m_), x_Symbol] 
:> Simp[(-F^(g*(e - c*(f/d))))*((c + d*x)^FracPart[m]/(d*((-f)*g*(Log[F]/d) 
)^(IntPart[m] + 1)*((-f)*g*Log[F]*((c + d*x)/d))^FracPart[m]))*Gamma[m + 1, 
 ((-f)*g*(Log[F]/d))*(c + d*x)], x] /; FreeQ[{F, c, d, e, f, g, m}, x] && 
!IntegerQ[m]
 

rule 3042

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

rule 3788

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol 
] :> Simp[I/2   Int[(c + d*x)^m/(E^(I*k*Pi)*E^(I*(e + f*x))), x], x] - Simp 
[I/2   Int[(c + d*x)^m*E^(I*k*Pi)*E^(I*(e + f*x)), x], x] /; FreeQ[{c, d, e 
, f, m}, x] && IntegerQ[2*k]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.73 \[ \int (c+d x)^m \cos (a+b x) \, dx=\frac {i \, e^{\left (-\frac {d m \log \left (\frac {i \, b}{d}\right ) - i \, b c + i \, a d}{d}\right )} \Gamma \left (m + 1, \frac {i \, b d x + i \, b c}{d}\right ) - i \, e^{\left (-\frac {d m \log \left (-\frac {i \, b}{d}\right ) + i \, b c - i \, a d}{d}\right )} \Gamma \left (m + 1, \frac {-i \, b d x - i \, b c}{d}\right )}{2 \, b} \] Input:

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

Output:

1/2*(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(-I*b/d) + I*b*c - I*a*d)/d)*gamma(m + 1, (-I*b*d*x 
 - I*b*c)/d))/b
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (c+d x)^m \cos (a+b x) \, dx=\frac {2 \left (d x +c \right )^{m} \tan \left (\frac {b x}{2}+\frac {a}{2}\right )-2 \left (\int \frac {\left (d x +c \right )^{m} \tan \left (\frac {b x}{2}+\frac {a}{2}\right )}{\tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{2} c +\tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{2} d x +c +d x}d x \right ) \tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{2} d m -2 \left (\int \frac {\left (d x +c \right )^{m} \tan \left (\frac {b x}{2}+\frac {a}{2}\right )}{\tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{2} c +\tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{2} d x +c +d x}d x \right ) d m}{b \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}+1\right )} \] Input:

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

Output:

(2*((c + d*x)**m*tan((a + b*x)/2) - int(((c + d*x)**m*tan((a + b*x)/2))/(t 
an((a + b*x)/2)**2*c + tan((a + b*x)/2)**2*d*x + c + d*x),x)*tan((a + b*x) 
/2)**2*d*m - int(((c + d*x)**m*tan((a + b*x)/2))/(tan((a + b*x)/2)**2*c + 
tan((a + b*x)/2)**2*d*x + c + d*x),x)*d*m))/(b*(tan((a + b*x)/2)**2 + 1))