3.2.42 \(\int x^m \sin (a+b x^n) \, dx\) [142]

Optimal. Leaf size=109 \[ \frac {i e^{i a} x^{1+m} \left (-i b x^n\right )^{-\frac {1+m}{n}} \Gamma \left (\frac {1+m}{n},-i b x^n\right )}{2 n}-\frac {i e^{-i a} x^{1+m} \left (i b x^n\right )^{-\frac {1+m}{n}} \Gamma \left (\frac {1+m}{n},i b x^n\right )}{2 n} \]

[Out]

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

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Rubi [A]
time = 0.05, antiderivative size = 109, 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 = {3504, 2250} \begin {gather*} \frac {i e^{i a} x^{m+1} \left (-i b x^n\right )^{-\frac {m+1}{n}} \text {Gamma}\left (\frac {m+1}{n},-i b x^n\right )}{2 n}-\frac {i e^{-i a} x^{m+1} \left (i b x^n\right )^{-\frac {m+1}{n}} \text {Gamma}\left (\frac {m+1}{n},i b x^n\right )}{2 n} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^m*Sin[a + b*x^n],x]

[Out]

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

Rule 2250

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[(-F^a)*((e +
f*x)^(m + 1)/(f*n*((-b)*(c + d*x)^n*Log[F])^((m + 1)/n)))*Gamma[(m + 1)/n, (-b)*(c + d*x)^n*Log[F]], x] /; Fre
eQ[{F, a, b, c, d, e, f, m, n}, x] && EqQ[d*e - c*f, 0]

Rule 3504

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

Rubi steps

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

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Mathematica [A]
time = 0.14, size = 118, normalized size = 1.08 \begin {gather*} \frac {i x^{1+m} \left (b^2 x^{2 n}\right )^{-\frac {1+m}{n}} \left (-\left (-i b x^n\right )^{\frac {1+m}{n}} \Gamma \left (\frac {1+m}{n},i b x^n\right ) (\cos (a)-i \sin (a))+\left (i b x^n\right )^{\frac {1+m}{n}} \Gamma \left (\frac {1+m}{n},-i b x^n\right ) (\cos (a)+i \sin (a))\right )}{2 n} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^m*Sin[a + b*x^n],x]

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 5 vs. order 4.
time = 0.08, size = 110, normalized size = 1.01

method	result	size
meijerg	\(\frac {x^{1+m} \hypergeom \left (\left [\frac {m}{2 n}+\frac {1}{2 n}\right ], \left [\frac {1}{2}, 1+\frac {m}{2 n}+\frac {1}{2 n}\right ], -\frac {x^{2 n} b^{2}}{4}\right ) \sin \left (a \right )}{1+m}+\frac {b \,x^{n +m +1} \hypergeom \left (\left [\frac {1}{2}+\frac {m}{2 n}+\frac {1}{2 n}\right ], \left [\frac {3}{2}, \frac {3}{2}+\frac {m}{2 n}+\frac {1}{2 n}\right ], -\frac {x^{2 n} b^{2}}{4}\right ) \cos \left (a \right )}{n +m +1}\)	\(110\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*sin(a+b*x^n),x,method=_RETURNVERBOSE)

[Out]

1/(1+m)*x^(1+m)*hypergeom([1/2/n*m+1/2/n],[1/2,1+1/2/n*m+1/2/n],-1/4*x^(2*n)*b^2)*sin(a)+b/(n+m+1)*x^(n+m+1)*h
ypergeom([1/2+1/2/n*m+1/2/n],[3/2,3/2+1/2/n*m+1/2/n],-1/4*x^(2*n)*b^2)*cos(a)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^m*sin(b*x^n + a), x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x^m*sin(b*x^n + a), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{m} \sin {\left (a + b x^{n} \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**m*sin(a + b*x**n), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^m*sin(b*x^n + a), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^m\,\sin \left (a+b\,x^n\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*sin(a + b*x^n),x)

[Out]

int(x^m*sin(a + b*x^n), x)

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