3.137 \(\int \frac{\sin ^3(a+b x^n)}{x} \, dx\)

Optimal. Leaf size=67 \[ \frac{3 \sin (a) \text{CosIntegral}\left (b x^n\right )}{4 n}-\frac{\sin (3 a) \text{CosIntegral}\left (3 b x^n\right )}{4 n}+\frac{3 \cos (a) \text{Si}\left (b x^n\right )}{4 n}-\frac{\cos (3 a) \text{Si}\left (3 b x^n\right )}{4 n} \]

[Out]

(3*CosIntegral[b*x^n]*Sin[a])/(4*n) - (CosIntegral[3*b*x^n]*Sin[3*a])/(4*n) + (3*Cos[a]*SinIntegral[b*x^n])/(4
*n) - (Cos[3*a]*SinIntegral[3*b*x^n])/(4*n)

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Rubi [A]  time = 0.0922735, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {3425, 3377, 3376, 3375} \[ \frac{3 \sin (a) \text{CosIntegral}\left (b x^n\right )}{4 n}-\frac{\sin (3 a) \text{CosIntegral}\left (3 b x^n\right )}{4 n}+\frac{3 \cos (a) \text{Si}\left (b x^n\right )}{4 n}-\frac{\cos (3 a) \text{Si}\left (3 b x^n\right )}{4 n} \]

Antiderivative was successfully verified.

[In]

Int[Sin[a + b*x^n]^3/x,x]

[Out]

(3*CosIntegral[b*x^n]*Sin[a])/(4*n) - (CosIntegral[3*b*x^n]*Sin[3*a])/(4*n) + (3*Cos[a]*SinIntegral[b*x^n])/(4
*n) - (Cos[3*a]*SinIntegral[3*b*x^n])/(4*n)

Rule 3425

Int[((e_.)*(x_))^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*(x_)^(n_)])^(p_), x_Symbol] :> Int[ExpandTrigReduce[(e
*x)^m, (a + b*Sin[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]

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 ^3\left (a+b x^n\right )}{x} \, dx &=\int \left (\frac{3 \sin \left (a+b x^n\right )}{4 x}-\frac{\sin \left (3 a+3 b x^n\right )}{4 x}\right ) \, dx\\ &=-\left (\frac{1}{4} \int \frac{\sin \left (3 a+3 b x^n\right )}{x} \, dx\right )+\frac{3}{4} \int \frac{\sin \left (a+b x^n\right )}{x} \, dx\\ &=\frac{1}{4} (3 \cos (a)) \int \frac{\sin \left (b x^n\right )}{x} \, dx-\frac{1}{4} \cos (3 a) \int \frac{\sin \left (3 b x^n\right )}{x} \, dx+\frac{1}{4} (3 \sin (a)) \int \frac{\cos \left (b x^n\right )}{x} \, dx-\frac{1}{4} \sin (3 a) \int \frac{\cos \left (3 b x^n\right )}{x} \, dx\\ &=\frac{3 \text{Ci}\left (b x^n\right ) \sin (a)}{4 n}-\frac{\text{Ci}\left (3 b x^n\right ) \sin (3 a)}{4 n}+\frac{3 \cos (a) \text{Si}\left (b x^n\right )}{4 n}-\frac{\cos (3 a) \text{Si}\left (3 b x^n\right )}{4 n}\\ \end{align*}

Mathematica [A]  time = 0.105854, size = 54, normalized size = 0.81 \[ \frac{3 \sin (a) \text{CosIntegral}\left (b x^n\right )-\sin (3 a) \text{CosIntegral}\left (3 b x^n\right )+3 \cos (a) \text{Si}\left (b x^n\right )-\cos (3 a) \text{Si}\left (3 b x^n\right )}{4 n} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[a + b*x^n]^3/x,x]

[Out]

(3*CosIntegral[b*x^n]*Sin[a] - CosIntegral[3*b*x^n]*Sin[3*a] + 3*Cos[a]*SinIntegral[b*x^n] - Cos[3*a]*SinInteg
ral[3*b*x^n])/(4*n)

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Maple [A]  time = 0.012, size = 52, normalized size = 0.8 \begin{align*}{\frac{1}{n} \left ( -{\frac{{\it Si} \left ( 3\,b{x}^{n} \right ) \cos \left ( 3\,a \right ) }{4}}-{\frac{{\it Ci} \left ( 3\,b{x}^{n} \right ) \sin \left ( 3\,a \right ) }{4}}+{\frac{3\,{\it Si} \left ( b{x}^{n} \right ) \cos \left ( a \right ) }{4}}+{\frac{3\,{\it Ci} \left ( b{x}^{n} \right ) \sin \left ( a \right ) }{4}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(a+b*x^n)^3/x,x)

[Out]

1/n*(-1/4*Si(3*b*x^n)*cos(3*a)-1/4*Ci(3*b*x^n)*sin(3*a)+3/4*Si(b*x^n)*cos(a)+3/4*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)^3/x,x, algorithm="maxima")

[Out]

Exception raised: IndexError

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Fricas [A]  time = 1.72273, size = 275, normalized size = 4.1 \begin{align*} -\frac{\operatorname{Ci}\left (3 \, b x^{n}\right ) \sin \left (3 \, a\right ) + \operatorname{Ci}\left (-3 \, b x^{n}\right ) \sin \left (3 \, a\right ) - 3 \, \operatorname{Ci}\left (b x^{n}\right ) \sin \left (a\right ) - 3 \, \operatorname{Ci}\left (-b x^{n}\right ) \sin \left (a\right ) + 2 \, \cos \left (3 \, a\right ) \operatorname{Si}\left (3 \, b x^{n}\right ) - 6 \, \cos \left (a\right ) \operatorname{Si}\left (b x^{n}\right )}{8 \, n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(a+b*x^n)^3/x,x, algorithm="fricas")

[Out]

-1/8*(cos_integral(3*b*x^n)*sin(3*a) + cos_integral(-3*b*x^n)*sin(3*a) - 3*cos_integral(b*x^n)*sin(a) - 3*cos_
integral(-b*x^n)*sin(a) + 2*cos(3*a)*sin_integral(3*b*x^n) - 6*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 ^{3}{\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)**3/x,x)

[Out]

Integral(sin(a + b*x**n)**3/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 )^{3}}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(a+b*x^n)^3/x,x, algorithm="giac")

[Out]

integrate(sin(b*x^n + a)^3/x, x)