3.135 \(\int \frac{\sin (a+b x^n)}{x} \, dx\)

Optimal. Leaf size=25 \[ \frac{\sin (a) \text{CosIntegral}\left (b x^n\right )}{n}+\frac{\cos (a) \text{Si}\left (b x^n\right )}{n} \]

[Out]

(CosIntegral[b*x^n]*Sin[a])/n + (Cos[a]*SinIntegral[b*x^n])/n

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Rubi [A]  time = 0.038134, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3377, 3376, 3375} \[ \frac{\sin (a) \text{CosIntegral}\left (b x^n\right )}{n}+\frac{\cos (a) \text{Si}\left (b x^n\right )}{n} \]

Antiderivative was successfully verified.

[In]

Int[Sin[a + b*x^n]/x,x]

[Out]

(CosIntegral[b*x^n]*Sin[a])/n + (Cos[a]*SinIntegral[b*x^n])/n

Rule 3377

Int[Sin[(c_) + (d_.)*(x_)^(n_)]/(x_), x_Symbol] :> Dist[Sin[c], Int[Cos[d*x^n]/x, x], x] + Dist[Cos[c], Int[Si
n[d*x^n]/x, x], x] /; FreeQ[{c, d, n}, x]

Rule 3376

Int[Cos[(d_.)*(x_)^(n_)]/(x_), x_Symbol] :> Simp[CosIntegral[d*x^n]/n, x] /; FreeQ[{d, n}, x]

Rule 3375

Int[Sin[(d_.)*(x_)^(n_)]/(x_), x_Symbol] :> Simp[SinIntegral[d*x^n]/n, x] /; FreeQ[{d, n}, x]

Rubi steps

\begin{align*} \int \frac{\sin \left (a+b x^n\right )}{x} \, dx &=\cos (a) \int \frac{\sin \left (b x^n\right )}{x} \, dx+\sin (a) \int \frac{\cos \left (b x^n\right )}{x} \, dx\\ &=\frac{\text{Ci}\left (b x^n\right ) \sin (a)}{n}+\frac{\cos (a) \text{Si}\left (b x^n\right )}{n}\\ \end{align*}

Mathematica [A]  time = 0.0612245, size = 23, normalized size = 0.92 \[ \frac{\sin (a) \text{CosIntegral}\left (b x^n\right )+\cos (a) \text{Si}\left (b x^n\right )}{n} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[a + b*x^n]/x,x]

[Out]

(CosIntegral[b*x^n]*Sin[a] + Cos[a]*SinIntegral[b*x^n])/n

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Maple [A]  time = 0.006, size = 24, normalized size = 1. \begin{align*}{\frac{{\it Si} \left ( b{x}^{n} \right ) \cos \left ( a \right ) +{\it Ci} \left ( b{x}^{n} \right ) \sin \left ( a \right ) }{n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/n*(Si(b*x^n)*cos(a)+Ci(b*x^n)*sin(a))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: IndexError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: IndexError

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Fricas [A]  time = 1.7626, size = 131, normalized size = 5.24 \begin{align*} \frac{\operatorname{Ci}\left (b x^{n}\right ) \sin \left (a\right ) + \operatorname{Ci}\left (-b x^{n}\right ) \sin \left (a\right ) + 2 \, \cos \left (a\right ) \operatorname{Si}\left (b x^{n}\right )}{2 \, n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(cos_integral(b*x^n)*sin(a) + cos_integral(-b*x^n)*sin(a) + 2*cos(a)*sin_integral(b*x^n))/n

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sin(a + b*x**n)/x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sin(b*x^n + a)/x, x)