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

Optimal. Leaf size=75 \[ \frac{1}{2} i e^{i a} b x^m (-i b x)^{-m} \text{Gamma}(m-1,-i b x)-\frac{1}{2} i e^{-i a} b x^m (i b x)^{-m} \text{Gamma}(m-1,i b x) \]

[Out]

((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)

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Rubi [A]  time = 0.0742731, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3307, 2181} \[ \frac{1}{2} i e^{i a} b x^m (-i b x)^{-m} \text{Gamma}(m-1,-i b x)-\frac{1}{2} i e^{-i a} b x^m (i b x)^{-m} \text{Gamma}(m-1,i b x) \]

Antiderivative was successfully verified.

[In]

Int[x^(-2 + m)*Cos[a + b*x],x]

[Out]

((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)

Rule 3307

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

Rule 2181

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

Rubi steps

\begin{align*} \int x^{-2+m} \cos (a+b x) \, dx &=\frac{1}{2} \int e^{-i (a+b x)} x^{-2+m} \, dx+\frac{1}{2} \int e^{i (a+b x)} x^{-2+m} \, 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)\\ \end{align*}

Mathematica [A]  time = 0.0154093, size = 75, normalized size = 1. \[ \frac{1}{2} i e^{i a} b x^m (-i b x)^{-m} \text{Gamma}(m-1,-i b x)-\frac{1}{2} i e^{-i a} b x^m (i b x)^{-m} \text{Gamma}(m-1,i b x) \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-2 + m)*Cos[a + b*x],x]

[Out]

((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)

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Maple [C]  time = 0.082, size = 530, normalized size = 7.1 \begin{align*}{2}^{m-2} \left ({b}^{2} \right ) ^{-{\frac{1}{2}}-{\frac{m}{2}}}{b}^{2}\sqrt{\pi } \left ( 3\,{\frac{{2}^{1-m}{x}^{m-2} \left ({b}^{2} \right ) ^{-1/2+m/2} \left ( 2\,{x}^{2}{b}^{2}+2\,m+2 \right ) \sin \left ( bx \right ) }{\sqrt{\pi } \left ( -1+m \right ) \left ( 3+3\,m \right ) b}}-{\frac{{2}^{2-m}{x}^{m-2} \left ({x}^{2}{b}^{2}-{m}^{2}-m \right ) \left ( \cos \left ( bx \right ) xb-\sin \left ( bx \right ) \right ) }{\sqrt{\pi } \left ( -1+m \right ) b \left ( 1+m \right ) m} \left ({b}^{2} \right ) ^{-{\frac{1}{2}}+{\frac{m}{2}}}}-3\,{\frac{{2}^{2-m}{x}^{2+m} \left ({b}^{2} \right ) ^{-1/2+m/2}{b}^{3} \left ( bx \right ) ^{-3/2-m}{\it LommelS1} \left ( m+1/2,3/2,bx \right ) \sin \left ( bx \right ) }{\sqrt{\pi } \left ( -1+m \right ) \left ( 3+3\,m \right ) }}+{\frac{{2}^{2-m}{x}^{2+m}{b}^{3} \left ( \cos \left ( bx \right ) xb-\sin \left ( bx \right ) \right ) }{\sqrt{\pi } \left ( -1+m \right ) \left ( 1+m \right ) m} \left ({b}^{2} \right ) ^{-{\frac{1}{2}}+{\frac{m}{2}}} \left ( bx \right ) ^{-{\frac{5}{2}}-m}{\it LommelS1} \left ( m+{\frac{3}{2}},{\frac{1}{2}},bx \right ) } \right ) \cos \left ( a \right ) -{2}^{m-2}{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 ( bx \right ) }{\sqrt{\pi }m \left ( 2+m \right ) \left ( -1+m \right ) }}-3\,{\frac{{2}^{2-m}{x}^{-1+m}{b}^{-1+m} \left ( \cos \left ( bx \right ) xb-\sin \left ( bx \right ) \right ) }{\sqrt{\pi }m \left ( -3+3\,m \right ) }}+{\frac{{2}^{2-m}{x}^{2+m}{b}^{2+m}\sin \left ( bx \right ) }{\sqrt{\pi }m \left ( 2+m \right ) \left ( -1+m \right ) } \left ( bx \right ) ^{-{\frac{3}{2}}-m}{\it LommelS1} \left ( m+{\frac{3}{2}},{\frac{3}{2}},bx \right ) }+3\,{\frac{{2}^{2-m}{x}^{2+m}{b}^{2+m} \left ( bx \right ) ^{-5/2-m} \left ( \cos \left ( bx \right ) xb-\sin \left ( bx \right ) \right ){\it LommelS1} \left ( m+1/2,1/2,bx \right ) }{\sqrt{\pi }m \left ( -3+3\,m \right ) }} \right ) \sin \left ( a \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2^(m-2)*(b^2)^(-1/2-1/2*m)*b^2*Pi^(1/2)*(3*2^(1-m)/Pi^(1/2)/(-1+m)*x^(m-2)*(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^(m-2)*(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^(m-2)*b^(1-m)*Pi^(1/2)*(2^(1-m)/Pi^(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)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{m - 2} \cos \left (b x + a\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^(m - 2)*cos(b*x + a), x)

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Fricas [A]  time = 1.71395, size = 153, normalized size = 2.04 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{m - 2} \cos{\left (a + b x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**(m - 2)*cos(a + b*x), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{m - 2} \cos \left (b x + a\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^(m - 2)*cos(b*x + a), x)