3.2.35 \(\int \frac {\sin (a+b x^n)}{x} \, dx\) [135]

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

[Out]

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

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Rubi [A]
time = 0.02, 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 = {3458, 3457, 3456} \begin {gather*} \frac {\sin (a) \text {CosIntegral}\left (b x^n\right )}{n}+\frac {\cos (a) \text {Si}\left (b x^n\right )}{n} \end {gather*}

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 3456

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

Rule 3457

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

Rule 3458

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 = 0.04, size = 23, normalized size = 0.92 \begin {gather*} \frac {\text {Ci}\left (b x^n\right ) \sin (a)+\cos (a) \text {Si}\left (b x^n\right )}{n} \end {gather*}

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 = 0.07, size = 24, normalized size = 0.96

method	result	size
derivativedivides	\(\frac {\sinIntegral \left (b \,x^{n}\right ) \cos \left (a \right )+\cosineIntegral \left (b \,x^{n}\right ) \sin \left (a \right )}{n}\)	\(24\)
default	\(\frac {\sinIntegral \left (b \,x^{n}\right ) \cos \left (a \right )+\cosineIntegral \left (b \,x^{n}\right ) \sin \left (a \right )}{n}\)	\(24\)
risch	\(\frac {i {\mathrm e}^{i a} \expIntegral \left (1, -i b \,x^{n}\right )}{2 n}-\frac {{\mathrm e}^{-i a} \pi \,\mathrm {csgn}\left (b \,x^{n}\right )}{2 n}+\frac {{\mathrm e}^{-i a} \sinIntegral \left (b \,x^{n}\right )}{n}-\frac {i {\mathrm e}^{-i a} \expIntegral \left (1, -i b \,x^{n}\right )}{2 n}\)	\(74\)
meijerg	\(\frac {\sqrt {\pi }\, \left (\frac {2 \gamma +2 n \ln \left (x \right )+\ln \left (b^{2}\right )}{\sqrt {\pi }}-\frac {2 \gamma }{\sqrt {\pi }}-\frac {2 \ln \left (2\right )}{\sqrt {\pi }}-\frac {2 \ln \left (\frac {b \,x^{n}}{2}\right )}{\sqrt {\pi }}+\frac {2 \cosineIntegral \left (b \,x^{n}\right )}{\sqrt {\pi }}\right ) \sin \left (a \right )}{2 n}+\frac {\cos \left (a \right ) \sinIntegral \left (b \,x^{n}\right )}{n}\)	\(78\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.42, size = 91, normalized size = 3.64 \begin {gather*} -\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 \left (x\right )}\right )}\right ) - i \, {\rm Ei}\left (-i \, b e^{\left (n \overline {\log \left (x\right )}\right )}\right )\right )} \cos \left (a\right ) - {\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 \left (x\right )}\right )}\right ) + {\rm Ei}\left (-i \, b e^{\left (n \overline {\log \left (x\right )}\right )}\right )\right )} \sin \left (a\right )}{4 \, n} \end {gather*}

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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Fricas [A]
time = 0.38, size = 35, normalized size = 1.40 \begin {gather*} \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 {gather*}

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

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 = 0.00, 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,x, algorithm="giac")

[Out]

integrate(sin(b*x^n + a)/x, x)

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

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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