\(\int x^{-2+m} \cos (a+b x) \, dx\) [109]

Optimal result

Integrand size = 12, antiderivative size = 75 \[ \int x^{-2+m} \cos (a+b x) \, dx=\frac {1}{2} i b e^{i a} x^m (-i b x)^{-m} \Gamma (-1+m,-i b x)-\frac {1}{2} i b e^{-i a} x^m (i b x)^{-m} \Gamma (-1+m,i b x) \] Output:

1/2*I*b*exp(I*a)*x^m*GAMMA(-1+m,-I*b*x)/((-I*b*x)^m)-1/2*I*b*x^m*GAMMA(-1+ 
m,I*b*x)/exp(I*a)/((I*b*x)^m)
 

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00 \[ \int x^{-2+m} \cos (a+b x) \, dx=\frac {1}{2} i b e^{i a} x^m (-i b x)^{-m} \Gamma (-1+m,-i b x)-\frac {1}{2} i b e^{-i a} x^m (i b x)^{-m} \Gamma (-1+m,i b x) \] Input:

Integrate[x^(-2 + m)*Cos[a + b*x],x]
 

Output:

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

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, 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.

\(\displaystyle \int x^{m-2} \cos (a+b x) \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int x^{m-2} \sin \left (a+b x+\frac {\pi }{2}\right )dx\)

\(\Big \downarrow \) 3788

\(\displaystyle \frac {1}{2} i \int -i e^{-i (a+b x)} x^{m-2}dx-\frac {1}{2} i \int i e^{i (a+b x)} x^{m-2}dx\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {1}{2} \int e^{-i (a+b x)} x^{m-2}dx+\frac {1}{2} \int e^{i (a+b x)} x^{m-2}dx\)

\(\Big \downarrow \) 2612

\(\displaystyle \frac {1}{2} i e^{i a} b x^m (-i b x)^{-m} \Gamma (m-1,-i b x)-\frac {1}{2} i e^{-i a} b x^m (i b x)^{-m} \Gamma (m-1,i b x)\)

Input:

Int[x^(-2 + m)*Cos[a + b*x],x]
 

Output:

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

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.76 (sec) , antiderivative size = 530, normalized size of antiderivative = 7.07

method	result	size
meijerg	\(2^{-2+m} b^{2} \left (b^{2}\right )^{-\frac {1}{2}-\frac {m}{2}} \sqrt {\pi }\, \left (\frac {3 \,2^{1-m} x^{-2+m} \left (b^{2}\right )^{-\frac {1}{2}+\frac {m}{2}} \left (2 x^{2} b^{2}+2 m +2\right ) \sin \left (b x \right )}{\sqrt {\pi }\, \left (-1+m \right ) \left (3+3 m \right ) b}-\frac {2^{2-m} x^{-2+m} \left (b^{2}\right )^{-\frac {1}{2}+\frac {m}{2}} \left (x^{2} b^{2}-m^{2}-m \right ) \left (\cos \left (b x \right ) x b -\sin \left (b x \right )\right )}{\sqrt {\pi }\, \left (-1+m \right ) b \left (1+m \right ) m}-\frac {3 \,2^{2-m} x^{2+m} \left (b^{2}\right )^{-\frac {1}{2}+\frac {m}{2}} b^{3} \left (b x \right )^{-\frac {3}{2}-m} \operatorname {LommelS1}\left (m +\frac {1}{2}, \frac {3}{2}, b x \right ) \sin \left (b x \right )}{\sqrt {\pi }\, \left (-1+m \right ) \left (3+3 m \right )}+\frac {2^{2-m} x^{2+m} \left (b^{2}\right )^{-\frac {1}{2}+\frac {m}{2}} b^{3} \left (b x \right )^{-\frac {5}{2}-m} \left (\cos \left (b x \right ) x b -\sin \left (b x \right )\right ) \operatorname {LommelS1}\left (m +\frac {3}{2}, \frac {1}{2}, b x \right )}{\sqrt {\pi }\, \left (-1+m \right ) \left (1+m \right ) m}\right ) \cos \left (a \right )-2^{-2+m} b^{1-m} \sqrt {\pi }\, \left (\frac {2^{1-m} x^{-1+m} b^{-1+m} \left (-2 x^{2} b^{2}+2 m^{2}+2 m -4\right ) \sin \left (b x \right )}{\sqrt {\pi }\, m \left (2+m \right ) \left (-1+m \right )}-\frac {3 \,2^{2-m} x^{-1+m} b^{-1+m} \left (\cos \left (b x \right ) x b -\sin \left (b x \right )\right )}{\sqrt {\pi }\, m \left (-3+3 m \right )}+\frac {2^{2-m} x^{2+m} b^{2+m} \left (b x \right )^{-\frac {3}{2}-m} \operatorname {LommelS1}\left (m +\frac {3}{2}, \frac {3}{2}, b x \right ) \sin \left (b x \right )}{\sqrt {\pi }\, m \left (2+m \right ) \left (-1+m \right )}+\frac {3 \,2^{2-m} x^{2+m} b^{2+m} \left (b x \right )^{-\frac {5}{2}-m} \left (\cos \left (b x \right ) x b -\sin \left (b x \right )\right ) \operatorname {LommelS1}\left (m +\frac {1}{2}, \frac {1}{2}, b x \right )}{\sqrt {\pi }\, m \left (-3+3 m \right )}\right ) \sin \left (a \right )\)	\(530\)

Input:

int(x^(-2+m)*cos(b*x+a),x,method=_RETURNVERBOSE)
 

Output:

2^(-2+m)*b^2*(b^2)^(-1/2-1/2*m)*Pi^(1/2)*(3*2^(1-m)/Pi^(1/2)/(-1+m)*x^(-2+ 
m)*(b^2)^(-1/2+1/2*m)*(2*b^2*x^2+2*m+2)/(3+3*m)/b*sin(b*x)-2^(2-m)/Pi^(1/2 
)/(-1+m)*x^(-2+m)*(b^2)^(-1/2+1/2*m)/b*(b^2*x^2-m^2-m)/(1+m)/m*(cos(b*x)*x 
*b-sin(b*x))-3*2^(2-m)/Pi^(1/2)/(-1+m)*x^(2+m)*(b^2)^(-1/2+1/2*m)*b^3/(3+3 
*m)*(b*x)^(-3/2-m)*LommelS1(m+1/2,3/2,b*x)*sin(b*x)+2^(2-m)/Pi^(1/2)/(-1+m 
)*x^(2+m)*(b^2)^(-1/2+1/2*m)*b^3/(1+m)/m*(b*x)^(-5/2-m)*(cos(b*x)*x*b-sin( 
b*x))*LommelS1(m+3/2,1/2,b*x))*cos(a)-2^(-2+m)*b^(1-m)*Pi^(1/2)*(2^(1-m)/P 
i^(1/2)/m*x^(-1+m)*b^(-1+m)*(-2*b^2*x^2+2*m^2+2*m-4)/(2+m)/(-1+m)*sin(b*x) 
-3*2^(2-m)/Pi^(1/2)/m*x^(-1+m)*b^(-1+m)/(-3+3*m)*(cos(b*x)*x*b-sin(b*x))+2 
^(2-m)/Pi^(1/2)/m*x^(2+m)*b^(2+m)/(2+m)/(-1+m)*(b*x)^(-3/2-m)*LommelS1(m+3 
/2,3/2,b*x)*sin(b*x)+3*2^(2-m)/Pi^(1/2)/m*x^(2+m)*b^(2+m)/(-3+3*m)*(b*x)^( 
-5/2-m)*(cos(b*x)*x*b-sin(b*x))*LommelS1(m+1/2,1/2,b*x))*sin(a)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.72 \[ \int x^{-2+m} \cos (a+b x) \, dx=\frac {i \, e^{\left (-{\left (m - 2\right )} \log \left (i \, b\right ) - i \, a\right )} \Gamma \left (m - 1, i \, b x\right ) - i \, e^{\left (-{\left (m - 2\right )} \log \left (-i \, b\right ) + i \, a\right )} \Gamma \left (m - 1, -i \, b x\right )}{2 \, b} \] Input:

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

Output:

1/2*(I*e^(-(m - 2)*log(I*b) - I*a)*gamma(m - 1, I*b*x) - I*e^(-(m - 2)*log 
(-I*b) + I*a)*gamma(m - 1, -I*b*x))/b
 

Sympy [F]

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

integrate(x**(-2+m)*cos(b*x+a),x)
 

Output:

Integral(x**(m - 2)*cos(a + b*x), x)
 

Maxima [F]

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

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

Output:

integrate(x^(m - 2)*cos(b*x + a), x)
 

Giac [F]

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

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

Output:

integrate(x^(m - 2)*cos(b*x + a), x)
 

Mupad [F(-1)]

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

int(x^(m - 2)*cos(a + b*x),x)
 

Output:

int(x^(m - 2)*cos(a + b*x), x)
 

Reduce [F]

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

int(x^(-2+m)*cos(b*x+a),x)
                                                                                    
                                                                                    
 

Output:

( - x**m + int(x**m/x**2,x)*m*x - int(x**m/x**2,x)*x + int((x**m*cos(a + b 
*x))/x**2,x)*m*x - int((x**m*cos(a + b*x))/x**2,x)*x)/(x*(m - 1))