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

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

[Out]

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

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Rubi [A]  time = 0.04, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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 3375

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

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*}

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Mathematica [A]  time = 0.07, size = 23, normalized size = 0.92 \[ \frac {\sin (a) \text {Ci}\left (b x^n\right )+\cos (a) \text {Si}\left (b x^n\right )}{n} \]

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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fricas [A]  time = 0.66, size = 35, normalized size = 1.40 \[ \frac {\operatorname {Ci}\left (b x^{n}\right ) \sin \relax (a) + \operatorname {Ci}\left (-b x^{n}\right ) \sin \relax (a) + 2 \, \cos \relax (a) \operatorname {Si}\left (b x^{n}\right )}{2 \, n} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin \left (b x^{n} + a\right )}{x}\,{d x} \]

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)

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maple [A]  time = 0.03, size = 24, normalized size = 0.96 \[ \frac {\Si \left (b \,x^{n}\right ) \cos \relax (a )+\Ci \left (b \,x^{n}\right ) \sin \relax (a )}{n} \]

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 [C]  time = 1.02, size = 91, normalized size = 3.64 \[ -\frac {{\left (i \, {\rm Ei}\left (i \, b x^{n}\right ) - i \, {\rm Ei}\left (-i \, b x^{n}\right ) + i \, {\rm Ei}\left (i \, b e^{\left (n \overline {\log \relax (x)}\right )}\right ) - i \, {\rm Ei}\left (-i \, b e^{\left (n \overline {\log \relax (x)}\right )}\right )\right )} \cos \relax (a) - {\left ({\rm Ei}\left (i \, b x^{n}\right ) + {\rm Ei}\left (-i \, b x^{n}\right ) + {\rm Ei}\left (i \, b e^{\left (n \overline {\log \relax (x)}\right )}\right ) + {\rm Ei}\left (-i \, b e^{\left (n \overline {\log \relax (x)}\right )}\right )\right )} \sin \relax (a)}{4 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*((I*Ei(I*b*x^n) - I*Ei(-I*b*x^n) + I*Ei(I*b*e^(n*conjugate(log(x)))) - I*Ei(-I*b*e^(n*conjugate(log(x))))
)*cos(a) - (Ei(I*b*x^n) + Ei(-I*b*x^n) + Ei(I*b*e^(n*conjugate(log(x)))) + Ei(-I*b*e^(n*conjugate(log(x)))))*s
in(a))/n

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {\sin \left (a+b\,x^n\right )}{x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(a + b*x^n)/x,x)

[Out]

int(sin(a + b*x^n)/x, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin {\left (a + b x^{n} \right )}}{x}\, dx \]

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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