3.106 \(\int x^{1+m} \cos (a+b x) \, dx\)

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.08, antiderivative size = 75, normalized size of antiderivative = 1.00, 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} \[ \frac {e^{i a} x^m (-i b x)^{-m} \text {Gamma}(m+2,-i b x)}{2 b^2}+\frac {e^{-i a} x^m (i b x)^{-m} \text {Gamma}(m+2,i b x)}{2 b^2} \]

Antiderivative was successfully verified.

[In]

Int[x^(1 + m)*Cos[a + b*x],x]

[Out]

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

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]

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]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 75, normalized size = 1.00 \[ \frac {e^{i a} x^m (-i b x)^{-m} \Gamma (m+2,-i b x)}{2 b^2}+\frac {e^{-i a} x^m (i b x)^{-m} \Gamma (m+2,i b x)}{2 b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(1 + m)*Cos[a + b*x],x]

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.44, size = 54, normalized size = 0.72 \[ \frac {i \, e^{\left (-{\left (m + 1\right )} \log \left (i \, b\right ) - i \, a\right )} \Gamma \left (m + 2, i \, b x\right ) - i \, e^{\left (-{\left (m + 1\right )} \log \left (-i \, b\right ) + i \, a\right )} \Gamma \left (m + 2, -i \, b x\right )}{2 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(1+m)*cos(b*x+a),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{m + 1} \cos \left (b x + a\right )\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(1+m)*cos(b*x+a),x, algorithm="giac")

[Out]

integrate(x^(m + 1)*cos(b*x + a), x)

________________________________________________________________________________________

maple [C]  time = 0.11, size = 291, normalized size = 3.88 \[ \frac {2^{1+m} \left (b^{2}\right )^{-\frac {m}{2}} \sqrt {\pi }\, \left (\frac {2^{-1-m} x^{1+m} b \left (b^{2}\right )^{\frac {m}{2}} \sin \left (b x \right )}{\sqrt {\pi }\, \left (2+m \right )}+\frac {3 \,2^{-2-m} x^{2+m} b^{2} \left (b^{2}\right )^{\frac {m}{2}} \left (\frac {2}{3}+\frac {2 m}{3}\right ) \left (b x \right )^{-\frac {3}{2}-m} \LommelS 1 \left (m +\frac {3}{2}, \frac {3}{2}, b x \right ) \sin \left (b x \right )}{\sqrt {\pi }\, \left (2+m \right )}+\frac {2^{-1-m} x^{2+m} b^{2} \left (b^{2}\right )^{\frac {m}{2}} \left (1+m \right ) \left (b x \right )^{-\frac {5}{2}-m} \left (\cos \left (b x \right ) x b -\sin \left (b x \right )\right ) \LommelS 1 \left (m +\frac {1}{2}, \frac {1}{2}, b x \right )}{\sqrt {\pi }}\right ) \cos \relax (a )}{b^{2}}-2^{1+m} b^{-2-m} \sqrt {\pi }\, \left (\frac {2^{-1-m} x^{2+m} b^{2+m} m \left (b x \right )^{-\frac {3}{2}-m} \LommelS 1 \left (m +\frac {1}{2}, \frac {3}{2}, b x \right ) \sin \left (b x \right )}{\sqrt {\pi }}-\frac {2^{-1-m} x^{2+m} b^{2+m} \left (b x \right )^{-\frac {5}{2}-m} \left (\cos \left (b x \right ) x b -\sin \left (b x \right )\right ) \LommelS 1 \left (m +\frac {3}{2}, \frac {1}{2}, b x \right )}{\sqrt {\pi }}\right ) \sin \relax (a ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(1+m)*cos(b*x+a),x)

[Out]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{m + 1} \cos \left (b x + a\right )\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(1+m)*cos(b*x+a),x, algorithm="maxima")

[Out]

integrate(x^(m + 1)*cos(b*x + a), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int x^{m+1}\,\cos \left (a+b\,x\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(m + 1)*cos(a + b*x),x)

[Out]

int(x^(m + 1)*cos(a + b*x), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{m + 1} \cos {\left (a + b x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(1+m)*cos(b*x+a),x)

[Out]

Integral(x**(m + 1)*cos(a + b*x), x)

________________________________________________________________________________________