3.2.37 \(\int \frac {\sin ^3(a+b x^n)}{x} \, dx\) [137]

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

[Out]

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

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Rubi [A]
time = 0.07, antiderivative size = 67, normalized size of antiderivative = 1.00, 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 = {3506, 3458, 3457, 3456} \begin {gather*} \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} \end {gather*}

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 3456

Int[Sin[(d_.)*(x_)^(n_)]/(x_), x_Symbol] :> Simp[SinIntegral[d*x^n]/n, x] /; FreeQ[{d, n}, x]

Rule 3457

Int[Cos[(d_.)*(x_)^(n_)]/(x_), x_Symbol] :> Simp[CosIntegral[d*x^n]/n, x] /; FreeQ[{d, n}, x]

Rule 3458

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 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 \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*}

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Mathematica [A]
time = 0.08, size = 54, normalized size = 0.81 \begin {gather*} \frac {3 \text {Ci}\left (b x^n\right ) \sin (a)-\text {Ci}\left (3 b x^n\right ) \sin (3 a)+3 \cos (a) \text {Si}\left (b x^n\right )-\cos (3 a) \text {Si}\left (3 b x^n\right )}{4 n} \end {gather*}

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 = 0.06, size = 52, normalized size = 0.78

method	result	size
derivativedivides	\(\frac {-\frac {\sinIntegral \left (3 b \,x^{n}\right ) \cos \left (3 a \right )}{4}-\frac {\cosineIntegral \left (3 b \,x^{n}\right ) \sin \left (3 a \right )}{4}+\frac {3 \sinIntegral \left (b \,x^{n}\right ) \cos \left (a \right )}{4}+\frac {3 \cosineIntegral \left (b \,x^{n}\right ) \sin \left (a \right )}{4}}{n}\)	\(52\)
default	\(\frac {-\frac {\sinIntegral \left (3 b \,x^{n}\right ) \cos \left (3 a \right )}{4}-\frac {\cosineIntegral \left (3 b \,x^{n}\right ) \sin \left (3 a \right )}{4}+\frac {3 \sinIntegral \left (b \,x^{n}\right ) \cos \left (a \right )}{4}+\frac {3 \cosineIntegral \left (b \,x^{n}\right ) \sin \left (a \right )}{4}}{n}\)	\(52\)
risch	\(-\frac {i {\mathrm e}^{3 i a} \expIntegral \left (1, -3 i b \,x^{n}\right )}{8 n}+\frac {{\mathrm e}^{-3 i a} \pi \,\mathrm {csgn}\left (b \,x^{n}\right )}{8 n}-\frac {{\mathrm e}^{-3 i a} \sinIntegral \left (3 b \,x^{n}\right )}{4 n}+\frac {i {\mathrm e}^{-3 i a} \expIntegral \left (1, -3 i b \,x^{n}\right )}{8 n}-\frac {3 \,{\mathrm e}^{-i a} \pi \,\mathrm {csgn}\left (b \,x^{n}\right )}{8 n}+\frac {3 \,{\mathrm e}^{-i a} \sinIntegral \left (b \,x^{n}\right )}{4 n}-\frac {3 i {\mathrm e}^{-i a} \expIntegral \left (1, -i b \,x^{n}\right )}{8 n}+\frac {3 i {\mathrm e}^{i a} \expIntegral \left (1, -i b \,x^{n}\right )}{8 n}\)	\(149\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.50, size = 181, normalized size = 2.70 \begin {gather*} \frac {{\left (i \, {\rm Ei}\left (3 i \, b x^{n}\right ) - i \, {\rm Ei}\left (-3 i \, b x^{n}\right ) + i \, {\rm Ei}\left (3 i \, b e^{\left (n \overline {\log \left (x\right )}\right )}\right ) - i \, {\rm Ei}\left (-3 i \, b e^{\left (n \overline {\log \left (x\right )}\right )}\right )\right )} \cos \left (3 \, a\right ) - 3 \, {\left (i \, {\rm Ei}\left (i \, b x^{n}\right ) - i \, {\rm Ei}\left (-i \, b x^{n}\right ) + i \, {\rm Ei}\left (i \, b e^{\left (n \overline {\log \left (x\right )}\right )}\right ) - i \, {\rm Ei}\left (-i \, b e^{\left (n \overline {\log \left (x\right )}\right )}\right )\right )} \cos \left (a\right ) - {\left ({\rm Ei}\left (3 i \, b x^{n}\right ) + {\rm Ei}\left (-3 i \, b x^{n}\right ) + {\rm Ei}\left (3 i \, b e^{\left (n \overline {\log \left (x\right )}\right )}\right ) + {\rm Ei}\left (-3 i \, b e^{\left (n \overline {\log \left (x\right )}\right )}\right )\right )} \sin \left (3 \, a\right ) + 3 \, {\left ({\rm Ei}\left (i \, b x^{n}\right ) + {\rm Ei}\left (-i \, b x^{n}\right ) + {\rm Ei}\left (i \, b e^{\left (n \overline {\log \left (x\right )}\right )}\right ) + {\rm Ei}\left (-i \, b e^{\left (n \overline {\log \left (x\right )}\right )}\right )\right )} \sin \left (a\right )}{16 \, n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*((I*Ei(3*I*b*x^n) - I*Ei(-3*I*b*x^n) + I*Ei(3*I*b*e^(n*conjugate(log(x)))) - I*Ei(-3*I*b*e^(n*conjugate(l
og(x)))))*cos(3*a) - 3*(I*Ei(I*b*x^n) - I*Ei(-I*b*x^n) + I*Ei(I*b*e^(n*conjugate(log(x)))) - I*Ei(-I*b*e^(n*co
njugate(log(x)))))*cos(a) - (Ei(3*I*b*x^n) + Ei(-3*I*b*x^n) + Ei(3*I*b*e^(n*conjugate(log(x)))) + Ei(-3*I*b*e^
(n*conjugate(log(x)))))*sin(3*a) + 3*(Ei(I*b*x^n) + Ei(-I*b*x^n) + Ei(I*b*e^(n*conjugate(log(x)))) + Ei(-I*b*e
^(n*conjugate(log(x)))))*sin(a))/n

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Fricas [A]
time = 0.35, size = 74, normalized size = 1.10 \begin {gather*} -\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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sin ^{3}{\left (a + b x^{n} \right )}}{x}\, dx \end {gather*}

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 = 0.00, 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(sin(a+b*x^n)^3/x,x, algorithm="giac")

[Out]

integrate(sin(b*x^n + a)^3/x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(a + b*x^n)^3/x,x)

[Out]

int(sin(a + b*x^n)^3/x, x)

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