3.81 \(\int x^{-1-n} \cos ^3(a+b x^n) \, dx\)

Optimal. Leaf size=113 \[ -\frac {3 b \sin (a) \text {Ci}\left (b x^n\right )}{4 n}-\frac {3 b \sin (3 a) \text {Ci}\left (3 b x^n\right )}{4 n}-\frac {3 b \cos (a) \text {Si}\left (b x^n\right )}{4 n}-\frac {3 b \cos (3 a) \text {Si}\left (3 b x^n\right )}{4 n}-\frac {3 x^{-n} \cos \left (a+b x^n\right )}{4 n}-\frac {x^{-n} \cos \left (3 \left (a+b x^n\right )\right )}{4 n} \]

[Out]

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

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Rubi [A]  time = 0.20, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 6, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3426, 3380, 3297, 3303, 3299, 3302} \[ -\frac {3 b \sin (a) \text {CosIntegral}\left (b x^n\right )}{4 n}-\frac {3 b \sin (3 a) \text {CosIntegral}\left (3 b x^n\right )}{4 n}-\frac {3 b \cos (a) \text {Si}\left (b x^n\right )}{4 n}-\frac {3 b \cos (3 a) \text {Si}\left (3 b x^n\right )}{4 n}-\frac {3 x^{-n} \cos \left (a+b x^n\right )}{4 n}-\frac {x^{-n} \cos \left (3 \left (a+b x^n\right )\right )}{4 n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 - n)*Cos[a + b*x^n]^3,x]

[Out]

(-3*Cos[a + b*x^n])/(4*n*x^n) - Cos[3*(a + b*x^n)]/(4*n*x^n) - (3*b*CosIntegral[b*x^n]*Sin[a])/(4*n) - (3*b*Co
sIntegral[3*b*x^n]*Sin[3*a])/(4*n) - (3*b*Cos[a]*SinIntegral[b*x^n])/(4*n) - (3*b*Cos[3*a]*SinIntegral[3*b*x^n
])/(4*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 3380

Int[((a_.) + Cos[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*Cos[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 3426

Int[((a_.) + Cos[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_)*((e_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandTrigReduce[(e
*x)^m, (a + b*Cos[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-n} \cos ^3\left (a+b x^n\right ) \, dx &=\int \left (\frac {3}{4} x^{-1-n} \cos \left (a+b x^n\right )+\frac {1}{4} x^{-1-n} \cos \left (3 a+3 b x^n\right )\right ) \, dx\\ &=\frac {1}{4} \int x^{-1-n} \cos \left (3 a+3 b x^n\right ) \, dx+\frac {3}{4} \int x^{-1-n} \cos \left (a+b x^n\right ) \, dx\\ &=\frac {\operatorname {Subst}\left (\int \frac {\cos (3 a+3 b x)}{x^2} \, dx,x,x^n\right )}{4 n}+\frac {3 \operatorname {Subst}\left (\int \frac {\cos (a+b x)}{x^2} \, dx,x,x^n\right )}{4 n}\\ &=-\frac {3 x^{-n} \cos \left (a+b x^n\right )}{4 n}-\frac {x^{-n} \cos \left (3 \left (a+b x^n\right )\right )}{4 n}-\frac {(3 b) \operatorname {Subst}\left (\int \frac {\sin (a+b x)}{x} \, dx,x,x^n\right )}{4 n}-\frac {(3 b) \operatorname {Subst}\left (\int \frac {\sin (3 a+3 b x)}{x} \, dx,x,x^n\right )}{4 n}\\ &=-\frac {3 x^{-n} \cos \left (a+b x^n\right )}{4 n}-\frac {x^{-n} \cos \left (3 \left (a+b x^n\right )\right )}{4 n}-\frac {(3 b \cos (a)) \operatorname {Subst}\left (\int \frac {\sin (b x)}{x} \, dx,x,x^n\right )}{4 n}-\frac {(3 b \cos (3 a)) \operatorname {Subst}\left (\int \frac {\sin (3 b x)}{x} \, dx,x,x^n\right )}{4 n}-\frac {(3 b \sin (a)) \operatorname {Subst}\left (\int \frac {\cos (b x)}{x} \, dx,x,x^n\right )}{4 n}-\frac {(3 b \sin (3 a)) \operatorname {Subst}\left (\int \frac {\cos (3 b x)}{x} \, dx,x,x^n\right )}{4 n}\\ &=-\frac {3 x^{-n} \cos \left (a+b x^n\right )}{4 n}-\frac {x^{-n} \cos \left (3 \left (a+b x^n\right )\right )}{4 n}-\frac {3 b \text {Ci}\left (b x^n\right ) \sin (a)}{4 n}-\frac {3 b \text {Ci}\left (3 b x^n\right ) \sin (3 a)}{4 n}-\frac {3 b \cos (a) \text {Si}\left (b x^n\right )}{4 n}-\frac {3 b \cos (3 a) \text {Si}\left (3 b x^n\right )}{4 n}\\ \end {align*}

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Mathematica [A]  time = 0.24, size = 95, normalized size = 0.84 \[ -\frac {x^{-n} \left (3 b \sin (a) x^n \text {Ci}\left (b x^n\right )+3 b \sin (3 a) x^n \text {Ci}\left (3 b x^n\right )+3 b \cos (a) x^n \text {Si}\left (b x^n\right )+3 b \cos (3 a) x^n \text {Si}\left (3 b x^n\right )+3 \cos \left (a+b x^n\right )+\cos \left (3 \left (a+b x^n\right )\right )\right )}{4 n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*(3*Cos[a + b*x^n] + Cos[3*(a + b*x^n)] + 3*b*x^n*CosIntegral[b*x^n]*Sin[a] + 3*b*x^n*CosIntegral[3*b*x^n]
*Sin[3*a] + 3*b*x^n*Cos[a]*SinIntegral[b*x^n] + 3*b*x^n*Cos[3*a]*SinIntegral[3*b*x^n])/(n*x^n)

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fricas [A]  time = 1.04, size = 117, normalized size = 1.04 \[ -\frac {3 \, b x^{n} \operatorname {Ci}\left (3 \, b x^{n}\right ) \sin \left (3 \, a\right ) + 3 \, b x^{n} \operatorname {Ci}\left (-3 \, b x^{n}\right ) \sin \left (3 \, a\right ) + 3 \, b x^{n} \operatorname {Ci}\left (b x^{n}\right ) \sin \relax (a) + 3 \, b x^{n} \operatorname {Ci}\left (-b x^{n}\right ) \sin \relax (a) + 6 \, b x^{n} \cos \left (3 \, a\right ) \operatorname {Si}\left (3 \, b x^{n}\right ) + 6 \, b x^{n} \cos \relax (a) \operatorname {Si}\left (b x^{n}\right ) + 8 \, \cos \left (b x^{n} + a\right )^{3}}{8 \, n x^{n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(3*b*x^n*cos_integral(3*b*x^n)*sin(3*a) + 3*b*x^n*cos_integral(-3*b*x^n)*sin(3*a) + 3*b*x^n*cos_integral(
b*x^n)*sin(a) + 3*b*x^n*cos_integral(-b*x^n)*sin(a) + 6*b*x^n*cos(3*a)*sin_integral(3*b*x^n) + 6*b*x^n*cos(a)*
sin_integral(b*x^n) + 8*cos(b*x^n + a)^3)/(n*x^n)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.06, size = 101, normalized size = 0.89 \[ \frac {3 b \left (-\frac {\cos \left (a +b \,x^{n}\right ) x^{-n}}{b}-\Si \left (b \,x^{n}\right ) \cos \relax (a )-\Ci \left (b \,x^{n}\right ) \sin \relax (a )\right )}{4 n}+\frac {3 b \left (-\frac {\cos \left (3 a +3 b \,x^{n}\right ) x^{-n}}{3 b}-\Si \left (3 b \,x^{n}\right ) \cos \left (3 a \right )-\Ci \left (3 b \,x^{n}\right ) \sin \left (3 a \right )\right )}{4 n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(a + b*x^n)^3/x^(n + 1),x)

[Out]

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

[Out]

Timed out

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