3.2.49 \(\int x^{-1-n} \sin ^3(a+b x^n) \, dx\) [149]

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.14, 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 = {3506, 3460, 3378, 3384, 3380, 3383} \begin {gather*} \frac {3 b \cos (a) \text {CosIntegral}\left (b x^n\right )}{4 n}-\frac {3 b \cos (3 a) \text {CosIntegral}\left (3 b x^n\right )}{4 n}-\frac {3 b \sin (a) \text {Si}\left (b x^n\right )}{4 n}+\frac {3 b \sin (3 a) \text {Si}\left (3 b x^n\right )}{4 n}-\frac {3 x^{-n} \sin \left (a+b x^n\right )}{4 n}+\frac {x^{-n} \sin \left (3 \left (a+b x^n\right )\right )}{4 n} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^(-1 - n)*Sin[a + b*x^n]^3,x]

[Out]

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

Rule 3378

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 3380

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 3383

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 3384

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 3460

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 3506

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

________________________________________________________________________________________

Mathematica [A]
time = 0.13, size = 95, normalized size = 0.84 \begin {gather*} \frac {x^{-n} \left (3 b x^n \cos (a) \text {Ci}\left (b x^n\right )-3 b x^n \cos (3 a) \text {Ci}\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 \sin (a) \text {Si}\left (b x^n\right )+3 b x^n \sin (3 a) \text {Si}\left (3 b x^n\right )\right )}{4 n} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 0.08, size = 99, normalized size = 0.88

method	result	size
default	\(\frac {3 b \left (-\frac {\sin \left (a +b \,x^{n}\right ) x^{-n}}{b}-\sinIntegral \left (b \,x^{n}\right ) \sin \left (a \right )+\cosineIntegral \left (b \,x^{n}\right ) \cos \left (a \right )\right )}{4 n}-\frac {3 b \left (-\frac {\sin \left (3 a +3 b \,x^{n}\right ) x^{-n}}{3 b}-\sinIntegral \left (3 b \,x^{n}\right ) \sin \left (3 a \right )+\cosineIntegral \left (3 b \,x^{n}\right ) \cos \left (3 a \right )\right )}{4 n}\)	\(99\)
risch	\(\frac {3 b \,{\mathrm e}^{3 i a} \expIntegral \left (1, -3 i b \,x^{n}\right )}{8 n}-\frac {3 i b \,{\mathrm e}^{-3 i a} \pi \,\mathrm {csgn}\left (b \,x^{n}\right )}{8 n}+\frac {3 i b \,{\mathrm e}^{-3 i a} \sinIntegral \left (3 b \,x^{n}\right )}{4 n}+\frac {3 b \,{\mathrm e}^{-3 i a} \expIntegral \left (1, -3 i b \,x^{n}\right )}{8 n}+\frac {3 i b \,{\mathrm e}^{-i a} \pi \,\mathrm {csgn}\left (b \,x^{n}\right )}{8 n}-\frac {3 i b \,{\mathrm e}^{-i a} \sinIntegral \left (b \,x^{n}\right )}{4 n}-\frac {3 b \,{\mathrm e}^{-i a} \expIntegral \left (1, -i b \,x^{n}\right )}{8 n}-\frac {3 b \,{\mathrm e}^{i a} \expIntegral \left (1, -i b \,x^{n}\right )}{8 n}-\frac {3 \sin \left (a +b \,x^{n}\right ) x^{-n}}{4 n}+\frac {\sin \left (3 a +3 b \,x^{n}\right ) x^{-n}}{4 n}\)	\(196\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-1-n)*sin(a+b*x^n)^3,x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 0.59, size = 127, normalized size = 1.12 \begin {gather*} -\frac {3 \, b x^{n} \cos \left (3 \, a\right ) \operatorname {Ci}\left (3 \, b x^{n}\right ) - 3 \, b x^{n} \cos \left (a\right ) \operatorname {Ci}\left (b x^{n}\right ) - 3 \, b x^{n} \cos \left (a\right ) \operatorname {Ci}\left (-b x^{n}\right ) + 3 \, b x^{n} \cos \left (3 \, a\right ) \operatorname {Ci}\left (-3 \, b x^{n}\right ) - 6 \, b x^{n} \sin \left (3 \, a\right ) \operatorname {Si}\left (3 \, b x^{n}\right ) + 6 \, b x^{n} \sin \left (a\right ) \operatorname {Si}\left (b x^{n}\right ) - 8 \, {\left (\cos \left (b x^{n} + a\right )^{2} - 1\right )} \sin \left (b x^{n} + a\right )}{8 \, n x^{n}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(3*b*x^n*cos(3*a)*cos_integral(3*b*x^n) - 3*b*x^n*cos(a)*cos_integral(b*x^n) - 3*b*x^n*cos(a)*cos_integra
l(-b*x^n) + 3*b*x^n*cos(3*a)*cos_integral(-3*b*x^n) - 6*b*x^n*sin(3*a)*sin_integral(3*b*x^n) + 6*b*x^n*sin(a)*
sin_integral(b*x^n) - 8*(cos(b*x^n + a)^2 - 1)*sin(b*x^n + a))/(n*x^n)

________________________________________________________________________________________

Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3005 deep

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(a + b*x^n)^3/x^(n + 1),x)

[Out]

int(sin(a + b*x^n)^3/x^(n + 1), x)

________________________________________________________________________________________