3.79 \(\int x^{1+m} \sin (a+b x) \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0708763, 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} \[ \frac{i e^{-i a} x^m (i b x)^{-m} \text{Gamma}(m+2,i b x)}{2 b^2}-\frac{i e^{i a} x^m (-i b x)^{-m} \text{Gamma}(m+2,-i b x)}{2 b^2} \]

Antiderivative was successfully verified.

[In]

Int[x^(1 + m)*Sin[a + b*x],x]

[Out]

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

Mathematica [A]  time = 0.0161029, size = 79, normalized size = 1. \[ \frac{i e^{-i a} x^m (i b x)^{-m} \text{Gamma}(m+2,i b x)}{2 b^2}-\frac{i e^{i a} x^m (-i b x)^{-m} \text{Gamma}(m+2,-i b x)}{2 b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(1 + m)*Sin[a + b*x],x]

[Out]

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

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Maple [C]  time = 0.063, size = 290, normalized size = 3.7 \begin{align*}{\frac{{2}^{1+m}\sqrt{\pi }\sin \left ( a \right ) }{{b}^{2}} \left ({b}^{2} \right ) ^{-{\frac{m}{2}}} \left ({\frac{{2}^{-1-m}{x}^{1+m}b\sin \left ( bx \right ) }{\sqrt{\pi } \left ( 2+m \right ) } \left ({b}^{2} \right ) ^{{\frac{m}{2}}}}+3\,{\frac{{2}^{-2-m}{x}^{2+m}{b}^{2} \left ({b}^{2} \right ) ^{m/2} \left ( 2/3+2/3\,m \right ) \left ( bx \right ) ^{-3/2-m}{\it LommelS1} \left ( m+3/2,3/2,bx \right ) \sin \left ( bx \right ) }{\sqrt{\pi } \left ( 2+m \right ) }}+{\frac{{2}^{-1-m}{x}^{2+m}{b}^{2} \left ( 1+m \right ) \left ( \cos \left ( bx \right ) xb-\sin \left ( bx \right ) \right ) }{\sqrt{\pi }} \left ({b}^{2} \right ) ^{{\frac{m}{2}}} \left ( bx \right ) ^{-{\frac{5}{2}}-m}{\it LommelS1} \left ( m+{\frac{1}{2}},{\frac{1}{2}},bx \right ) } \right ) }+{2}^{1+m}{b}^{-2-m}\sqrt{\pi } \left ({\frac{{2}^{-1-m}{x}^{2+m}{b}^{2+m}m\sin \left ( bx \right ) }{\sqrt{\pi }} \left ( bx \right ) ^{-{\frac{3}{2}}-m}{\it LommelS1} \left ( m+{\frac{1}{2}},{\frac{3}{2}},bx \right ) }-{\frac{{2}^{-1-m}{x}^{2+m}{b}^{2+m} \left ( \cos \left ( bx \right ) xb-\sin \left ( bx \right ) \right ) }{\sqrt{\pi }} \left ( bx \right ) ^{-{\frac{5}{2}}-m}{\it LommelS1} \left ( m+{\frac{3}{2}},{\frac{1}{2}},bx \right ) } \right ) \cos \left ( a \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(1+m)*sin(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))*sin(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))*cos(a)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{m + 1} \sin \left (b x + a\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^(m + 1)*sin(b*x + a), x)

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Fricas [A]  time = 1.71925, size = 149, normalized size = 1.89 \begin{align*} -\frac{e^{\left (-{\left (m + 1\right )} \log \left (i \, b\right ) - i \, a\right )} \Gamma \left (m + 2, i \, b x\right ) + e^{\left (-{\left (m + 1\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^(1+m)*sin(b*x+a),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{m + 1} \sin{\left (a + b x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**(m + 1)*sin(a + b*x), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{m + 1} \sin \left (b x + a\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^(m + 1)*sin(b*x + a), x)