3.80 \(\int x^m \sin (a+b x) \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.07, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3308, 2181} \[ -\frac {e^{i a} x^m (-i b x)^{-m} \text {Gamma}(m+1,-i b x)}{2 b}-\frac {e^{-i a} x^m (i b x)^{-m} \text {Gamma}(m+1,i b x)}{2 b} \]

Antiderivative was successfully verified.

[In]

Int[x^m*Sin[a + b*x],x]

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 75, normalized size = 1.00 \[ -\frac {e^{i a} x^m (-i b x)^{-m} \Gamma (m+1,-i b x)}{2 b}-\frac {e^{-i a} x^m (i b x)^{-m} \Gamma (m+1,i b x)}{2 b} \]

Antiderivative was successfully verified.

[In]

Integrate[x^m*Sin[a + b*x],x]

[Out]

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

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fricas [A]  time = 0.59, size = 48, normalized size = 0.64 \[ -\frac {e^{\left (-m \log \left (i \, b\right ) - i \, a\right )} \Gamma \left (m + 1, i \, b x\right ) + e^{\left (-m \log \left (-i \, b\right ) + i \, a\right )} \Gamma \left (m + 1, -i \, b x\right )}{2 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^m*sin(b*x + a), x)

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maple [C]  time = 0.08, size = 378, normalized size = 5.04 \[ 2^{m} \left (b^{2}\right )^{-\frac {1}{2}-\frac {m}{2}} \sqrt {\pi }\, \left (\frac {3 \,2^{-1-m} \left (b^{2}\right )^{\frac {1}{2}+\frac {m}{2}} x^{m} \left (6+2 m \right ) \sin \left (b x \right )}{\sqrt {\pi }\, \left (1+m \right ) \left (9+3 m \right ) b}+\frac {\left (b^{2}\right )^{\frac {1}{2}+\frac {m}{2}} x^{m} 2^{-m} \left (\cos \left (b x \right ) x b -\sin \left (b x \right )\right )}{\sqrt {\pi }\, \left (1+m \right ) b}+\frac {2^{-m} x^{2+m} \left (b^{2}\right )^{\frac {1}{2}+\frac {m}{2}} b m \left (b x \right )^{-\frac {3}{2}-m} \LommelS 1 \left (m +\frac {1}{2}, \frac {3}{2}, b x \right ) \sin \left (b x \right )}{\sqrt {\pi }\, \left (1+m \right )}-\frac {2^{-m} x^{2+m} \left (b^{2}\right )^{\frac {1}{2}+\frac {m}{2}} b \left (b x \right )^{-\frac {5}{2}-m} \left (\cos \left (b x \right ) x b -\sin \left (b x \right )\right ) \LommelS 1 \left (m +\frac {3}{2}, \frac {1}{2}, b x \right )}{\sqrt {\pi }\, \left (1+m \right )}\right ) \sin \relax (a )+2^{m} b^{-1-m} \sqrt {\pi }\, \left (\frac {x^{1+m} b^{1+m} 2^{-m} \sin \left (b x \right )}{\sqrt {\pi }\, \left (2+m \right )}-\frac {2^{-m} x^{2+m} b^{2+m} \left (b x \right )^{-\frac {3}{2}-m} \LommelS 1 \left (m +\frac {3}{2}, \frac {3}{2}, b x \right ) \sin \left (b x \right )}{\sqrt {\pi }\, \left (2+m \right )}-\frac {3 \,2^{-1-m} x^{2+m} b^{2+m} \left (\frac {4}{3}+\frac {2 m}{3}\right ) \left (b x \right )^{-\frac {5}{2}-m} \left (\cos \left (b x \right ) x b -\sin \left (b x \right )\right ) \LommelS 1 \left (m +\frac {1}{2}, \frac {1}{2}, b x \right )}{\sqrt {\pi }\, \left (2+m \right )}\right ) \cos \relax (a ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2^m*(b^2)^(-1/2-1/2*m)*Pi^(1/2)*(3*2^(-1-m)/Pi^(1/2)/(1+m)*(b^2)^(1/2+1/2*m)*x^m*(6+2*m)/(9+3*m)/b*sin(b*x)+1/
Pi^(1/2)/(1+m)*(b^2)^(1/2+1/2*m)*x^m*2^(-m)/b*(cos(b*x)*x*b-sin(b*x))+2^(-m)/Pi^(1/2)/(1+m)*x^(2+m)*(b^2)^(1/2
+1/2*m)*b*m*(b*x)^(-3/2-m)*LommelS1(m+1/2,3/2,b*x)*sin(b*x)-2^(-m)/Pi^(1/2)/(1+m)*x^(2+m)*(b^2)^(1/2+1/2*m)*b*
(b*x)^(-5/2-m)*(cos(b*x)*x*b-sin(b*x))*LommelS1(m+3/2,1/2,b*x))*sin(a)+2^m*b^(-1-m)*Pi^(1/2)*(1/Pi^(1/2)/(2+m)
*x^(1+m)*b^(1+m)*2^(-m)*sin(b*x)-2^(-m)/Pi^(1/2)/(2+m)*x^(2+m)*b^(2+m)*(b*x)^(-3/2-m)*LommelS1(m+3/2,3/2,b*x)*
sin(b*x)-3*2^(-1-m)/Pi^(1/2)/(2+m)*x^(2+m)*b^(2+m)*(4/3+2/3*m)*(b*x)^(-5/2-m)*(cos(b*x)*x*b-sin(b*x))*LommelS1
(m+1/2,1/2,b*x))*cos(a)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^m*sin(b*x + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*sin(a + b*x),x)

[Out]

int(x^m*sin(a + b*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**m*sin(a + b*x), x)

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