3.83 \(\int x^{-3+m} \sin (a+b x) \, dx\)

Optimal. Leaf size=79 \[ \frac{1}{2} i e^{-i a} b^2 x^m (i b x)^{-m} \text{Gamma}(m-2,i b x)-\frac{1}{2} i e^{i a} b^2 x^m (-i b x)^{-m} \text{Gamma}(m-2,-i b x) \]

[Out]

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

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Rubi [A]  time = 0.0717161, antiderivative size = 79, 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 = {3308, 2181} \[ \frac{1}{2} i e^{-i a} b^2 x^m (i b x)^{-m} \text{Gamma}(m-2,i b x)-\frac{1}{2} i e^{i a} b^2 x^m (-i b x)^{-m} \text{Gamma}(m-2,-i b x) \]

Antiderivative was successfully verified.

[In]

Int[x^(-3 + m)*Sin[a + b*x],x]

[Out]

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

Rule 3308

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

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^{-3+m} \sin (a+b x) \, dx &=\frac{1}{2} i \int e^{-i (a+b x)} x^{-3+m} \, dx-\frac{1}{2} i \int e^{i (a+b x)} x^{-3+m} \, dx\\ &=-\frac{1}{2} i b^2 e^{i a} x^m (-i b x)^{-m} \Gamma (-2+m,-i b x)+\frac{1}{2} i b^2 e^{-i a} x^m (i b x)^{-m} \Gamma (-2+m,i b x)\\ \end{align*}

Mathematica [A]  time = 0.0156524, size = 79, normalized size = 1. \[ \frac{1}{2} i e^{-i a} b^2 x^m (i b x)^{-m} \text{Gamma}(m-2,i b x)-\frac{1}{2} i e^{i a} b^2 x^m (-i b x)^{-m} \text{Gamma}(m-2,-i b x) \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-3 + m)*Sin[a + b*x],x]

[Out]

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

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Maple [C]  time = 0.079, size = 599, normalized size = 7.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(m-3)*sin(b*x+a),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{m - 3} \sin \left (b x + a\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-3+m)*sin(b*x+a),x, algorithm="maxima")

[Out]

integrate(x^(m - 3)*sin(b*x + a), x)

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Fricas [A]  time = 1.7661, size = 149, normalized size = 1.89 \begin{align*} -\frac{e^{\left (-{\left (m - 3\right )} \log \left (i \, b\right ) - i \, a\right )} \Gamma \left (m - 2, i \, b x\right ) + e^{\left (-{\left (m - 3\right )} \log \left (-i \, b\right ) + i \, a\right )} \Gamma \left (m - 2, -i \, b x\right )}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-3+m)*sin(b*x+a),x, algorithm="fricas")

[Out]

-1/2*(e^(-(m - 3)*log(I*b) - I*a)*gamma(m - 2, I*b*x) + e^(-(m - 3)*log(-I*b) + I*a)*gamma(m - 2, -I*b*x))/b

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(-3+m)*sin(b*x+a),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{m - 3} \sin \left (b x + a\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-3+m)*sin(b*x+a),x, algorithm="giac")

[Out]

integrate(x^(m - 3)*sin(b*x + a), x)