3.263 \(\int \sin (a+b (c+d x)^n) \, dx\)

Optimal. Leaf size=117 \[ \frac{i e^{i a} (c+d x) \left (-i b (c+d x)^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},-i b (c+d x)^n\right )}{2 d n}-\frac{i e^{-i a} (c+d x) \left (i b (c+d x)^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},i b (c+d x)^n\right )}{2 d n} \]

[Out]

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

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Rubi [A]  time = 0.0293592, antiderivative size = 117, 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 = {3365, 2208} \[ \frac{i e^{i a} (c+d x) \left (-i b (c+d x)^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},-i b (c+d x)^n\right )}{2 d n}-\frac{i e^{-i a} (c+d x) \left (i b (c+d x)^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},i b (c+d x)^n\right )}{2 d n} \]

Antiderivative was successfully verified.

[In]

Int[Sin[a + b*(c + d*x)^n],x]

[Out]

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

Mathematica [A]  time = 0.095625, size = 121, normalized size = 1.03 \[ \frac{i (\cos (a)+i \sin (a)) (c+d x) \left (-i b (c+d x)^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},-i b (c+d x)^n\right )}{2 d n}-\frac{i (\cos (a)-i \sin (a)) (c+d x) \left (i b (c+d x)^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},i b (c+d x)^n\right )}{2 d n} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[a + b*(c + d*x)^n],x]

[Out]

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

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Maple [F]  time = 0.06, size = 0, normalized size = 0. \begin{align*} \int \sin \left ( a+b \left ( dx+c \right ) ^{n} \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sin((d*x + c)^n*b + a), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sin \left ({\left (d x + c\right )}^{n} b + a\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sin((d*x + c)^n*b + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sin(a + b*(c + d*x)**n), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sin((d*x + c)^n*b + a), x)