3.26 \(\int \frac{\sin ^3(a+b x^2)}{x} \, dx\)

Optimal. Leaf size=55 \[ \frac{3}{8} \sin (a) \text{CosIntegral}\left (b x^2\right )-\frac{1}{8} \sin (3 a) \text{CosIntegral}\left (3 b x^2\right )+\frac{3}{8} \cos (a) \text{Si}\left (b x^2\right )-\frac{1}{8} \cos (3 a) \text{Si}\left (3 b x^2\right ) \]

[Out]

(3*CosIntegral[b*x^2]*Sin[a])/8 - (CosIntegral[3*b*x^2]*Sin[3*a])/8 + (3*Cos[a]*SinIntegral[b*x^2])/8 - (Cos[3
*a]*SinIntegral[3*b*x^2])/8

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Rubi [A]  time = 0.0953006, antiderivative size = 55, 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 = {3403, 3377, 3376, 3375} \[ \frac{3}{8} \sin (a) \text{CosIntegral}\left (b x^2\right )-\frac{1}{8} \sin (3 a) \text{CosIntegral}\left (3 b x^2\right )+\frac{3}{8} \cos (a) \text{Si}\left (b x^2\right )-\frac{1}{8} \cos (3 a) \text{Si}\left (3 b x^2\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sin[a + b*x^2]^3/x,x]

[Out]

(3*CosIntegral[b*x^2]*Sin[a])/8 - (CosIntegral[3*b*x^2]*Sin[3*a])/8 + (3*Cos[a]*SinIntegral[b*x^2])/8 - (Cos[3
*a]*SinIntegral[3*b*x^2])/8

Rule 3403

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}, x] && IGtQ[p, 1] && IGtQ[n, 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^2\right )}{x} \, dx &=\int \left (\frac{3 \sin \left (a+b x^2\right )}{4 x}-\frac{\sin \left (3 a+3 b x^2\right )}{4 x}\right ) \, dx\\ &=-\left (\frac{1}{4} \int \frac{\sin \left (3 a+3 b x^2\right )}{x} \, dx\right )+\frac{3}{4} \int \frac{\sin \left (a+b x^2\right )}{x} \, dx\\ &=\frac{1}{4} (3 \cos (a)) \int \frac{\sin \left (b x^2\right )}{x} \, dx-\frac{1}{4} \cos (3 a) \int \frac{\sin \left (3 b x^2\right )}{x} \, dx+\frac{1}{4} (3 \sin (a)) \int \frac{\cos \left (b x^2\right )}{x} \, dx-\frac{1}{4} \sin (3 a) \int \frac{\cos \left (3 b x^2\right )}{x} \, dx\\ &=\frac{3}{8} \text{Ci}\left (b x^2\right ) \sin (a)-\frac{1}{8} \text{Ci}\left (3 b x^2\right ) \sin (3 a)+\frac{3}{8} \cos (a) \text{Si}\left (b x^2\right )-\frac{1}{8} \cos (3 a) \text{Si}\left (3 b x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0738422, size = 51, normalized size = 0.93 \[ \frac{1}{8} \left (3 \sin (a) \text{CosIntegral}\left (b x^2\right )-\sin (3 a) \text{CosIntegral}\left (3 b x^2\right )+3 \cos (a) \text{Si}\left (b x^2\right )-\cos (3 a) \text{Si}\left (3 b x^2\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[a + b*x^2]^3/x,x]

[Out]

(3*CosIntegral[b*x^2]*Sin[a] - CosIntegral[3*b*x^2]*Sin[3*a] + 3*Cos[a]*SinIntegral[b*x^2] - Cos[3*a]*SinInteg
ral[3*b*x^2])/8

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Maple [A]  time = 0.027, size = 48, normalized size = 0.9 \begin{align*}{\frac{3\,\cos \left ( a \right ){\it Si} \left ( b{x}^{2} \right ) }{8}}-{\frac{\cos \left ( 3\,a \right ){\it Si} \left ( 3\,b{x}^{2} \right ) }{8}}+{\frac{3\,{\it Ci} \left ( b{x}^{2} \right ) \sin \left ( a \right ) }{8}}-{\frac{{\it Ci} \left ( 3\,b{x}^{2} \right ) \sin \left ( 3\,a \right ) }{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(b*x^2+a)^3/x,x)

[Out]

3/8*cos(a)*Si(b*x^2)-1/8*cos(3*a)*Si(3*b*x^2)+3/8*Ci(b*x^2)*sin(a)-1/8*Ci(3*b*x^2)*sin(3*a)

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Maxima [C]  time = 1.22061, size = 120, normalized size = 2.18 \begin{align*} \frac{1}{16} \,{\left (i \,{\rm Ei}\left (3 i \, b x^{2}\right ) - i \,{\rm Ei}\left (-3 i \, b x^{2}\right )\right )} \cos \left (3 \, a\right ) + \frac{1}{16} \,{\left (-3 i \,{\rm Ei}\left (i \, b x^{2}\right ) + 3 i \,{\rm Ei}\left (-i \, b x^{2}\right )\right )} \cos \left (a\right ) - \frac{1}{16} \,{\left ({\rm Ei}\left (3 i \, b x^{2}\right ) +{\rm Ei}\left (-3 i \, b x^{2}\right )\right )} \sin \left (3 \, a\right ) + \frac{3}{16} \,{\left ({\rm Ei}\left (i \, b x^{2}\right ) +{\rm Ei}\left (-i \, b x^{2}\right )\right )} \sin \left (a\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(b*x^2+a)^3/x,x, algorithm="maxima")

[Out]

1/16*(I*Ei(3*I*b*x^2) - I*Ei(-3*I*b*x^2))*cos(3*a) + 1/16*(-3*I*Ei(I*b*x^2) + 3*I*Ei(-I*b*x^2))*cos(a) - 1/16*
(Ei(3*I*b*x^2) + Ei(-3*I*b*x^2))*sin(3*a) + 3/16*(Ei(I*b*x^2) + Ei(-I*b*x^2))*sin(a)

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Fricas [A]  time = 2.30419, size = 262, normalized size = 4.76 \begin{align*} -\frac{1}{16} \,{\left (\operatorname{Ci}\left (3 \, b x^{2}\right ) + \operatorname{Ci}\left (-3 \, b x^{2}\right )\right )} \sin \left (3 \, a\right ) + \frac{3}{16} \,{\left (\operatorname{Ci}\left (b x^{2}\right ) + \operatorname{Ci}\left (-b x^{2}\right )\right )} \sin \left (a\right ) - \frac{1}{8} \, \cos \left (3 \, a\right ) \operatorname{Si}\left (3 \, b x^{2}\right ) + \frac{3}{8} \, \cos \left (a\right ) \operatorname{Si}\left (b x^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(b*x^2+a)^3/x,x, algorithm="fricas")

[Out]

-1/16*(cos_integral(3*b*x^2) + cos_integral(-3*b*x^2))*sin(3*a) + 3/16*(cos_integral(b*x^2) + cos_integral(-b*
x^2))*sin(a) - 1/8*cos(3*a)*sin_integral(3*b*x^2) + 3/8*cos(a)*sin_integral(b*x^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin ^{3}{\left (a + b x^{2} \right )}}{x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(b*x**2+a)**3/x,x)

[Out]

Integral(sin(a + b*x**2)**3/x, x)

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Giac [A]  time = 1.12242, size = 63, normalized size = 1.15 \begin{align*} -\frac{1}{8} \, \operatorname{Ci}\left (3 \, b x^{2}\right ) \sin \left (3 \, a\right ) + \frac{3}{8} \, \operatorname{Ci}\left (b x^{2}\right ) \sin \left (a\right ) + \frac{3}{8} \, \cos \left (a\right ) \operatorname{Si}\left (b x^{2}\right ) + \frac{1}{8} \, \cos \left (3 \, a\right ) \operatorname{Si}\left (-3 \, b x^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(b*x^2+a)^3/x,x, algorithm="giac")

[Out]

-1/8*cos_integral(3*b*x^2)*sin(3*a) + 3/8*cos_integral(b*x^2)*sin(a) + 3/8*cos(a)*sin_integral(b*x^2) + 1/8*co
s(3*a)*sin_integral(-3*b*x^2)