3.152 \(\int x^{-1-2 n} \sin ^3(a+b x^n) \, dx\)

Optimal. Leaf size=165 \[ -\frac {3 b^2 \sin (a) \text {Ci}\left (b x^n\right )}{8 n}+\frac {9 b^2 \sin (3 a) \text {Ci}\left (3 b x^n\right )}{8 n}-\frac {3 b^2 \cos (a) \text {Si}\left (b x^n\right )}{8 n}+\frac {9 b^2 \cos (3 a) \text {Si}\left (3 b x^n\right )}{8 n}-\frac {3 x^{-2 n} \sin \left (a+b x^n\right )}{8 n}+\frac {x^{-2 n} \sin \left (3 \left (a+b x^n\right )\right )}{8 n}-\frac {3 b x^{-n} \cos \left (a+b x^n\right )}{8 n}+\frac {3 b x^{-n} \cos \left (3 \left (a+b x^n\right )\right )}{8 n} \]

[Out]

-3/8*b*cos(a+b*x^n)/n/(x^n)+3/8*b*cos(3*a+3*b*x^n)/n/(x^n)-3/8*b^2*cos(a)*Si(b*x^n)/n+9/8*b^2*cos(3*a)*Si(3*b*
x^n)/n-3/8*b^2*Ci(b*x^n)*sin(a)/n+9/8*b^2*Ci(3*b*x^n)*sin(3*a)/n-3/8*sin(a+b*x^n)/n/(x^(2*n))+1/8*sin(3*a+3*b*
x^n)/n/(x^(2*n))

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Rubi [A]  time = 0.26, antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 6, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3425, 3379, 3297, 3303, 3299, 3302} \[ -\frac {3 b^2 \sin (a) \text {CosIntegral}\left (b x^n\right )}{8 n}+\frac {9 b^2 \sin (3 a) \text {CosIntegral}\left (3 b x^n\right )}{8 n}-\frac {3 b^2 \cos (a) \text {Si}\left (b x^n\right )}{8 n}+\frac {9 b^2 \cos (3 a) \text {Si}\left (3 b x^n\right )}{8 n}-\frac {3 x^{-2 n} \sin \left (a+b x^n\right )}{8 n}+\frac {x^{-2 n} \sin \left (3 \left (a+b x^n\right )\right )}{8 n}-\frac {3 b x^{-n} \cos \left (a+b x^n\right )}{8 n}+\frac {3 b x^{-n} \cos \left (3 \left (a+b x^n\right )\right )}{8 n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 - 2*n)*Sin[a + b*x^n]^3,x]

[Out]

(-3*b*Cos[a + b*x^n])/(8*n*x^n) + (3*b*Cos[3*(a + b*x^n)])/(8*n*x^n) - (3*b^2*CosIntegral[b*x^n]*Sin[a])/(8*n)
+ (9*b^2*CosIntegral[3*b*x^n]*Sin[3*a])/(8*n) - (3*Sin[a + b*x^n])/(8*n*x^(2*n)) + Sin[3*(a + b*x^n)]/(8*n*x^
(2*n)) - (3*b^2*Cos[a]*SinIntegral[b*x^n])/(8*n) + (9*b^2*Cos[3*a]*SinIntegral[3*b*x^n])/(8*n)

Rule 3297

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[((c + d*x)^(m + 1)*Sin[e + f*x])/(d*(
m + 1)), x] - Dist[f/(d*(m + 1)), Int[(c + d*x)^(m + 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && LtQ[
m, -1]

Rule 3299

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinIntegral[e + f*x]/d, x] /; FreeQ[{c, d,
e, f}, x] && EqQ[d*e - c*f, 0]

Rule 3302

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosIntegral[e - Pi/2 + f*x]/d, x] /; FreeQ
[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]

Rule 3303

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[(c*f)/d + f*x]
/(c + d*x), x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[(c*f)/d + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f},
x] && NeQ[d*e - c*f, 0]

Rule 3379

Int[(x_)^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*Sin[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IntegerQ[Simpl
ify[(m + 1)/n]] && (EqQ[p, 1] || EqQ[m, n - 1] || (IntegerQ[p] && GtQ[Simplify[(m + 1)/n], 0]))

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]

Rubi steps

\begin {align*} \int x^{-1-2 n} \sin ^3\left (a+b x^n\right ) \, dx &=\int \left (\frac {3}{4} x^{-1-2 n} \sin \left (a+b x^n\right )-\frac {1}{4} x^{-1-2 n} \sin \left (3 a+3 b x^n\right )\right ) \, dx\\ &=-\left (\frac {1}{4} \int x^{-1-2 n} \sin \left (3 a+3 b x^n\right ) \, dx\right )+\frac {3}{4} \int x^{-1-2 n} \sin \left (a+b x^n\right ) \, dx\\ &=-\frac {\operatorname {Subst}\left (\int \frac {\sin (3 a+3 b x)}{x^3} \, dx,x,x^n\right )}{4 n}+\frac {3 \operatorname {Subst}\left (\int \frac {\sin (a+b x)}{x^3} \, dx,x,x^n\right )}{4 n}\\ &=-\frac {3 x^{-2 n} \sin \left (a+b x^n\right )}{8 n}+\frac {x^{-2 n} \sin \left (3 \left (a+b x^n\right )\right )}{8 n}+\frac {(3 b) \operatorname {Subst}\left (\int \frac {\cos (a+b x)}{x^2} \, dx,x,x^n\right )}{8 n}-\frac {(3 b) \operatorname {Subst}\left (\int \frac {\cos (3 a+3 b x)}{x^2} \, dx,x,x^n\right )}{8 n}\\ &=-\frac {3 b x^{-n} \cos \left (a+b x^n\right )}{8 n}+\frac {3 b x^{-n} \cos \left (3 \left (a+b x^n\right )\right )}{8 n}-\frac {3 x^{-2 n} \sin \left (a+b x^n\right )}{8 n}+\frac {x^{-2 n} \sin \left (3 \left (a+b x^n\right )\right )}{8 n}-\frac {\left (3 b^2\right ) \operatorname {Subst}\left (\int \frac {\sin (a+b x)}{x} \, dx,x,x^n\right )}{8 n}+\frac {\left (9 b^2\right ) \operatorname {Subst}\left (\int \frac {\sin (3 a+3 b x)}{x} \, dx,x,x^n\right )}{8 n}\\ &=-\frac {3 b x^{-n} \cos \left (a+b x^n\right )}{8 n}+\frac {3 b x^{-n} \cos \left (3 \left (a+b x^n\right )\right )}{8 n}-\frac {3 x^{-2 n} \sin \left (a+b x^n\right )}{8 n}+\frac {x^{-2 n} \sin \left (3 \left (a+b x^n\right )\right )}{8 n}-\frac {\left (3 b^2 \cos (a)\right ) \operatorname {Subst}\left (\int \frac {\sin (b x)}{x} \, dx,x,x^n\right )}{8 n}+\frac {\left (9 b^2 \cos (3 a)\right ) \operatorname {Subst}\left (\int \frac {\sin (3 b x)}{x} \, dx,x,x^n\right )}{8 n}-\frac {\left (3 b^2 \sin (a)\right ) \operatorname {Subst}\left (\int \frac {\cos (b x)}{x} \, dx,x,x^n\right )}{8 n}+\frac {\left (9 b^2 \sin (3 a)\right ) \operatorname {Subst}\left (\int \frac {\cos (3 b x)}{x} \, dx,x,x^n\right )}{8 n}\\ &=-\frac {3 b x^{-n} \cos \left (a+b x^n\right )}{8 n}+\frac {3 b x^{-n} \cos \left (3 \left (a+b x^n\right )\right )}{8 n}-\frac {3 b^2 \text {Ci}\left (b x^n\right ) \sin (a)}{8 n}+\frac {9 b^2 \text {Ci}\left (3 b x^n\right ) \sin (3 a)}{8 n}-\frac {3 x^{-2 n} \sin \left (a+b x^n\right )}{8 n}+\frac {x^{-2 n} \sin \left (3 \left (a+b x^n\right )\right )}{8 n}-\frac {3 b^2 \cos (a) \text {Si}\left (b x^n\right )}{8 n}+\frac {9 b^2 \cos (3 a) \text {Si}\left (3 b x^n\right )}{8 n}\\ \end {align*}

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Mathematica [A]  time = 0.30, size = 141, normalized size = 0.85 \[ \frac {x^{-2 n} \left (-3 b^2 \sin (a) x^{2 n} \text {Ci}\left (b x^n\right )+9 b^2 \sin (3 a) x^{2 n} \text {Ci}\left (3 b x^n\right )-3 b^2 \cos (a) x^{2 n} \text {Si}\left (b x^n\right )+9 b^2 \cos (3 a) x^{2 n} \text {Si}\left (3 b x^n\right )-3 \sin \left (a+b x^n\right )+\sin \left (3 \left (a+b x^n\right )\right )-3 b x^n \cos \left (a+b x^n\right )+3 b x^n \cos \left (3 \left (a+b x^n\right )\right )\right )}{8 n} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 - 2*n)*Sin[a + b*x^n]^3,x]

[Out]

(-3*b*x^n*Cos[a + b*x^n] + 3*b*x^n*Cos[3*(a + b*x^n)] - 3*b^2*x^(2*n)*CosIntegral[b*x^n]*Sin[a] + 9*b^2*x^(2*n
)*CosIntegral[3*b*x^n]*Sin[3*a] - 3*Sin[a + b*x^n] + Sin[3*(a + b*x^n)] - 3*b^2*x^(2*n)*Cos[a]*SinIntegral[b*x
^n] + 9*b^2*x^(2*n)*Cos[3*a]*SinIntegral[3*b*x^n])/(8*n*x^(2*n))

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fricas [A]  time = 0.80, size = 183, normalized size = 1.11 \[ \frac {24 \, b x^{n} \cos \left (b x^{n} + a\right )^{3} + 9 \, b^{2} x^{2 \, n} \operatorname {Ci}\left (3 \, b x^{n}\right ) \sin \left (3 \, a\right ) + 9 \, b^{2} x^{2 \, n} \operatorname {Ci}\left (-3 \, b x^{n}\right ) \sin \left (3 \, a\right ) - 3 \, b^{2} x^{2 \, n} \operatorname {Ci}\left (b x^{n}\right ) \sin \relax (a) - 3 \, b^{2} x^{2 \, n} \operatorname {Ci}\left (-b x^{n}\right ) \sin \relax (a) + 18 \, b^{2} x^{2 \, n} \cos \left (3 \, a\right ) \operatorname {Si}\left (3 \, b x^{n}\right ) - 6 \, b^{2} x^{2 \, n} \cos \relax (a) \operatorname {Si}\left (b x^{n}\right ) - 24 \, b x^{n} \cos \left (b x^{n} + a\right ) + 8 \, {\left (\cos \left (b x^{n} + a\right )^{2} - 1\right )} \sin \left (b x^{n} + a\right )}{16 \, n x^{2 \, n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-2*n)*sin(a+b*x^n)^3,x, algorithm="fricas")

[Out]

1/16*(24*b*x^n*cos(b*x^n + a)^3 + 9*b^2*x^(2*n)*cos_integral(3*b*x^n)*sin(3*a) + 9*b^2*x^(2*n)*cos_integral(-3
*b*x^n)*sin(3*a) - 3*b^2*x^(2*n)*cos_integral(b*x^n)*sin(a) - 3*b^2*x^(2*n)*cos_integral(-b*x^n)*sin(a) + 18*b
^2*x^(2*n)*cos(3*a)*sin_integral(3*b*x^n) - 6*b^2*x^(2*n)*cos(a)*sin_integral(b*x^n) - 24*b*x^n*cos(b*x^n + a)
+ 8*(cos(b*x^n + a)^2 - 1)*sin(b*x^n + a))/(n*x^(2*n))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{-2 \, n - 1} \sin \left (b x^{n} + a\right )^{3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-2*n)*sin(a+b*x^n)^3,x, algorithm="giac")

[Out]

integrate(x^(-2*n - 1)*sin(b*x^n + a)^3, x)

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maple [A]  time = 0.05, size = 144, normalized size = 0.87 \[ \frac {3 b^{2} \left (-\frac {\sin \left (a +b \,x^{n}\right ) x^{-2 n}}{2 b^{2}}-\frac {\cos \left (a +b \,x^{n}\right ) x^{-n}}{2 b}-\frac {\Si \left (b \,x^{n}\right ) \cos \relax (a )}{2}-\frac {\Ci \left (b \,x^{n}\right ) \sin \relax (a )}{2}\right )}{4 n}-\frac {9 b^{2} \left (-\frac {\sin \left (3 a +3 b \,x^{n}\right ) x^{-2 n}}{18 b^{2}}-\frac {\cos \left (3 a +3 b \,x^{n}\right ) x^{-n}}{6 b}-\frac {\Si \left (3 b \,x^{n}\right ) \cos \left (3 a \right )}{2}-\frac {\Ci \left (3 b \,x^{n}\right ) \sin \left (3 a \right )}{2}\right )}{4 n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-1-2*n)*sin(a+b*x^n)^3,x)

[Out]

3/4/n*b^2*(-1/2*sin(a+b*x^n)/(x^n)^2/b^2-1/2*cos(a+b*x^n)/(x^n)/b-1/2*Si(b*x^n)*cos(a)-1/2*Ci(b*x^n)*sin(a))-9
/4/n*b^2*(-1/18*sin(3*a+3*b*x^n)/(x^n)^2/b^2-1/6*cos(3*a+3*b*x^n)/(x^n)/b-1/2*Si(3*b*x^n)*cos(3*a)-1/2*Ci(3*b*
x^n)*sin(3*a))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{-2 \, n - 1} \sin \left (b x^{n} + a\right )^{3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-2*n)*sin(a+b*x^n)^3,x, algorithm="maxima")

[Out]

integrate(x^(-2*n - 1)*sin(b*x^n + a)^3, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\sin \left (a+b\,x^n\right )}^3}{x^{2\,n+1}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(a + b*x^n)^3/x^(2*n + 1),x)

[Out]

int(sin(a + b*x^n)^3/x^(2*n + 1), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(-1-2*n)*sin(a+b*x**n)**3,x)

[Out]

Timed out

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