3.2.39 \(\int \sin (a+b x^n) \, dx\) [139]

Optimal. Leaf size=87 \[ \frac {i e^{i a} x \left (-i b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},-i b x^n\right )}{2 n}-\frac {i e^{-i a} x \left (i b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},i b x^n\right )}{2 n} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3446, 2239} \begin {gather*} \frac {i e^{i a} x \left (-i b x^n\right )^{-1/n} \text {Gamma}\left (\frac {1}{n},-i b x^n\right )}{2 n}-\frac {i e^{-i a} x \left (i b x^n\right )^{-1/n} \text {Gamma}\left (\frac {1}{n},i b x^n\right )}{2 n} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sin[a + b*x^n],x]

[Out]

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

Rule 2239

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_)), x_Symbol] :> Simp[(-F^a)*(c + d*x)*(Gamma[1/n, (-b)*(c + d
*x)^n*Log[F]]/(d*n*((-b)*(c + d*x)^n*Log[F])^(1/n))), x] /; FreeQ[{F, a, b, c, d, n}, x] &&  !IntegerQ[2/n]

Rule 3446

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

Rubi steps

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

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Mathematica [A]
time = 0.06, size = 95, normalized size = 1.09 \begin {gather*} \frac {i x \left (b^2 x^{2 n}\right )^{-1/n} \left (-\left (-i b x^n\right )^{\frac {1}{n}} \Gamma \left (\frac {1}{n},i b x^n\right ) (\cos (a)-i \sin (a))+\left (i b x^n\right )^{\frac {1}{n}} \Gamma \left (\frac {1}{n},-i b x^n\right ) (\cos (a)+i \sin (a))\right )}{2 n} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sin[a + b*x^n],x]

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 5 vs. order 4.
time = 0.07, size = 74, normalized size = 0.85

method	result	size
meijerg	\(x \hypergeom \left (\left [\frac {1}{2 n}\right ], \left [\frac {1}{2}, 1+\frac {1}{2 n}\right ], -\frac {x^{2 n} b^{2}}{4}\right ) \sin \left (a \right )+\frac {b \,x^{1+n} \hypergeom \left (\left [\frac {1}{2}+\frac {1}{2 n}\right ], \left [\frac {3}{2}, \frac {3}{2}+\frac {1}{2 n}\right ], -\frac {x^{2 n} b^{2}}{4}\right ) \cos \left (a \right )}{1+n}\)	\(74\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*hypergeom([1/2/n],[1/2,1+1/2/n],-1/4*x^(2*n)*b^2)*sin(a)+b/(1+n)*x^(1+n)*hypergeom([1/2+1/2/n],[3/2,3/2+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(sin(a+b*x^n),x, algorithm="maxima")

[Out]

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

[Out]

integral(sin(b*x^n + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

integrate(sin(b*x^n + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(a + b*x^n),x)

[Out]

int(sin(a + b*x^n), x)

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