3.139 \(\int \sin (a+b x^n) \, dx\)

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

[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))

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Rubi [A]  time = 0.0270594, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3365, 2208} \[ \frac{i e^{i a} x \left (-i b x^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},-i b x^n\right )}{2 n}-\frac{i e^{-i a} x \left (i b x^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},i b x^n\right )}{2 n} \]

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 3365

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]

Rule 2208

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]

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*}

Mathematica [A]  time = 0.08404, size = 95, normalized size = 1.09 \[ \frac{i x \left (b^2 x^{2 n}\right )^{-1/n} \left ((\cos (a)+i \sin (a)) \left (i b x^n\right )^{\frac{1}{n}} \text{Gamma}\left (\frac{1}{n},-i b x^n\right )-(\cos (a)-i \sin (a)) \left (-i b x^n\right )^{\frac{1}{n}} \text{Gamma}\left (\frac{1}{n},i b x^n\right )\right )}{2 n} \]

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]  time = 0.077, size = 74, normalized size = 0.9 \begin{align*} x{\mbox{$_1$F$_2$}({\frac{1}{2\,n}};\,{\frac{1}{2}},1+{\frac{1}{2\,n}};\,-{\frac{{x}^{2\,n}{b}^{2}}{4}})}\sin \left ( a \right ) +{\frac{b{x}^{1+n}\cos \left ( a \right ) }{1+n}{\mbox{$_1$F$_2$}({\frac{1}{2}}+{\frac{1}{2\,n}};\,{\frac{3}{2}},{\frac{3}{2}}+{\frac{1}{2\,n}};\,-{\frac{{x}^{2\,n}{b}^{2}}{4}})}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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., size = 0, normalized size = 0. \begin{align*} \int \sin \left (b x^{n} + a\right )\,{d x} \end{align*}

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., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sin \left (b x^{n} + a\right ), x\right ) \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \int \sin{\left (a + b x^{n} \right )}\, dx \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \int \sin \left (b x^{n} + a\right )\,{d x} \end{align*}

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)