\(\int \log (x) \sin ^3(a+b x) \, dx\) [156]

Optimal result

Integrand size = 11, antiderivative size = 89 \[ \int \log (x) \sin ^3(a+b x) \, dx=\frac {3 \cos (a) \operatorname {CosIntegral}(b x)}{4 b}-\frac {\cos (3 a) \operatorname {CosIntegral}(3 b x)}{12 b}-\frac {\cos (a+b x) \log (x)}{b}+\frac {\cos ^3(a+b x) \log (x)}{3 b}-\frac {3 \sin (a) \text {Si}(b x)}{4 b}+\frac {\sin (3 a) \text {Si}(3 b x)}{12 b} \]

[Out]

3/4*Ci(b*x)*cos(a)/b-1/12*Ci(3*b*x)*cos(3*a)/b-cos(b*x+a)*ln(x)/b+1/3*cos(b*x+a)^3*ln(x)/b-3/4*Si(b*x)*sin(a)/
b+1/12*Si(3*b*x)*sin(3*a)/b

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.727, Rules used = {2713, 2634, 12, 6874, 3384, 3380, 3383, 3393} \[ \int \log (x) \sin ^3(a+b x) \, dx=\frac {3 \cos (a) \operatorname {CosIntegral}(b x)}{4 b}-\frac {\cos (3 a) \operatorname {CosIntegral}(3 b x)}{12 b}-\frac {3 \sin (a) \text {Si}(b x)}{4 b}+\frac {\sin (3 a) \text {Si}(3 b x)}{12 b}+\frac {\log (x) \cos ^3(a+b x)}{3 b}-\frac {\log (x) \cos (a+b x)}{b} \]

[In]

Int[Log[x]*Sin[a + b*x]^3,x]

[Out]

(3*Cos[a]*CosIntegral[b*x])/(4*b) - (Cos[3*a]*CosIntegral[3*b*x])/(12*b) - (Cos[a + b*x]*Log[x])/b + (Cos[a +
b*x]^3*Log[x])/(3*b) - (3*Sin[a]*SinIntegral[b*x])/(4*b) + (Sin[3*a]*SinIntegral[3*b*x])/(12*b)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2634

Int[Log[u_]*(v_), x_Symbol] :> With[{w = IntHide[v, x]}, Dist[Log[u], w, x] - Int[SimplifyIntegrand[w*(D[u, x]
/u), x], x] /; InverseFunctionFreeQ[w, x]] /; InverseFunctionFreeQ[u, x]

Rule 2713

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]

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 3393

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1])
)

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps \begin{align*} \text {integral}& = -\frac {\cos (a+b x) \log (x)}{b}+\frac {\cos ^3(a+b x) \log (x)}{3 b}-\int \frac {\cos (a+b x) \left (-3+\cos ^2(a+b x)\right )}{3 b x} \, dx \\ & = -\frac {\cos (a+b x) \log (x)}{b}+\frac {\cos ^3(a+b x) \log (x)}{3 b}-\frac {\int \frac {\cos (a+b x) \left (-3+\cos ^2(a+b x)\right )}{x} \, dx}{3 b} \\ & = -\frac {\cos (a+b x) \log (x)}{b}+\frac {\cos ^3(a+b x) \log (x)}{3 b}-\frac {\int \left (-\frac {3 \cos (a+b x)}{x}+\frac {\cos ^3(a+b x)}{x}\right ) \, dx}{3 b} \\ & = -\frac {\cos (a+b x) \log (x)}{b}+\frac {\cos ^3(a+b x) \log (x)}{3 b}-\frac {\int \frac {\cos ^3(a+b x)}{x} \, dx}{3 b}+\frac {\int \frac {\cos (a+b x)}{x} \, dx}{b} \\ & = -\frac {\cos (a+b x) \log (x)}{b}+\frac {\cos ^3(a+b x) \log (x)}{3 b}-\frac {\int \left (\frac {3 \cos (a+b x)}{4 x}+\frac {\cos (3 a+3 b x)}{4 x}\right ) \, dx}{3 b}+\frac {\cos (a) \int \frac {\cos (b x)}{x} \, dx}{b}-\frac {\sin (a) \int \frac {\sin (b x)}{x} \, dx}{b} \\ & = \frac {\cos (a) \text {Ci}(b x)}{b}-\frac {\cos (a+b x) \log (x)}{b}+\frac {\cos ^3(a+b x) \log (x)}{3 b}-\frac {\sin (a) \text {Si}(b x)}{b}-\frac {\int \frac {\cos (3 a+3 b x)}{x} \, dx}{12 b}-\frac {\int \frac {\cos (a+b x)}{x} \, dx}{4 b} \\ & = \frac {\cos (a) \text {Ci}(b x)}{b}-\frac {\cos (a+b x) \log (x)}{b}+\frac {\cos ^3(a+b x) \log (x)}{3 b}-\frac {\sin (a) \text {Si}(b x)}{b}-\frac {\cos (a) \int \frac {\cos (b x)}{x} \, dx}{4 b}-\frac {\cos (3 a) \int \frac {\cos (3 b x)}{x} \, dx}{12 b}+\frac {\sin (a) \int \frac {\sin (b x)}{x} \, dx}{4 b}+\frac {\sin (3 a) \int \frac {\sin (3 b x)}{x} \, dx}{12 b} \\ & = \frac {3 \cos (a) \text {Ci}(b x)}{4 b}-\frac {\cos (3 a) \text {Ci}(3 b x)}{12 b}-\frac {\cos (a+b x) \log (x)}{b}+\frac {\cos ^3(a+b x) \log (x)}{3 b}-\frac {3 \sin (a) \text {Si}(b x)}{4 b}+\frac {\sin (3 a) \text {Si}(3 b x)}{12 b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.74 \[ \int \log (x) \sin ^3(a+b x) \, dx=\frac {9 \cos (a) \operatorname {CosIntegral}(b x)-\cos (3 a) \operatorname {CosIntegral}(3 b x)-9 \cos (a+b x) \log (x)+\cos (3 (a+b x)) \log (x)-9 \sin (a) \text {Si}(b x)+\sin (3 a) \text {Si}(3 b x)}{12 b} \]

[In]

Integrate[Log[x]*Sin[a + b*x]^3,x]

[Out]

(9*Cos[a]*CosIntegral[b*x] - Cos[3*a]*CosIntegral[3*b*x] - 9*Cos[a + b*x]*Log[x] + Cos[3*(a + b*x)]*Log[x] - 9
*Sin[a]*SinIntegral[b*x] + Sin[3*a]*SinIntegral[3*b*x])/(12*b)

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 1.90 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.82

method	result	size
risch	\(-\frac {3 \cos \left (b x +a \right ) \ln \left (x \right )}{4 b}+\frac {\ln \left (x \right ) \cos \left (3 b x +3 a \right )}{12 b}-\frac {i {\mathrm e}^{-3 i a} \pi \,\operatorname {csgn}\left (b x \right )}{24 b}+\frac {i {\mathrm e}^{-3 i a} \operatorname {Si}\left (3 b x \right )}{12 b}+\frac {{\mathrm e}^{-3 i a} \operatorname {Ei}_{1}\left (-3 i b x \right )}{24 b}+\frac {3 i {\mathrm e}^{-i a} \pi \,\operatorname {csgn}\left (b x \right )}{8 b}-\frac {3 i {\mathrm e}^{-i a} \operatorname {Si}\left (b x \right )}{4 b}-\frac {3 \,{\mathrm e}^{-i a} \operatorname {Ei}_{1}\left (-i b x \right )}{8 b}-\frac {3 \,{\mathrm e}^{i a} \operatorname {Ei}_{1}\left (-i b x \right )}{8 b}+\frac {{\mathrm e}^{3 i a} \operatorname {Ei}_{1}\left (-3 i b x \right )}{24 b}\)	\(162\)

[In]

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

[Out]

-3/4*cos(b*x+a)*ln(x)/b+1/12/b*ln(x)*cos(3*b*x+3*a)-1/24*I/b*exp(-3*I*a)*Pi*csgn(b*x)+1/12*I/b*exp(-3*I*a)*Si(
3*b*x)+1/24/b*exp(-3*I*a)*Ei(1,-3*I*b*x)+3/8*I/b*exp(-I*a)*Pi*csgn(b*x)-3/4*I/b*exp(-I*a)*Si(b*x)-3/8/b*exp(-I
*a)*Ei(1,-I*b*x)-3/8/b*exp(I*a)*Ei(1,-I*b*x)+1/24/b*exp(3*I*a)*Ei(1,-3*I*b*x)

Fricas [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.72 \[ \int \log (x) \sin ^3(a+b x) \, dx=-\frac {\cos \left (3 \, a\right ) \operatorname {Ci}\left (3 \, b x\right ) - 9 \, \cos \left (a\right ) \operatorname {Ci}\left (b x\right ) - 4 \, {\left (\cos \left (b x + a\right )^{3} - 3 \, \cos \left (b x + a\right )\right )} \log \left (x\right ) - \sin \left (3 \, a\right ) \operatorname {Si}\left (3 \, b x\right ) + 9 \, \sin \left (a\right ) \operatorname {Si}\left (b x\right )}{12 \, b} \]

[In]

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

[Out]

-1/12*(cos(3*a)*cos_integral(3*b*x) - 9*cos(a)*cos_integral(b*x) - 4*(cos(b*x + a)^3 - 3*cos(b*x + a))*log(x)
- sin(3*a)*sin_integral(3*b*x) + 9*sin(a)*sin_integral(b*x))/b

Sympy [F]

\[ \int \log (x) \sin ^3(a+b x) \, dx=\int \log {\left (x \right )} \sin ^{3}{\left (a + b x \right )}\, dx \]

[In]

integrate(ln(x)*sin(b*x+a)**3,x)

[Out]

Integral(log(x)*sin(a + b*x)**3, x)

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.26 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.24 \[ \int \log (x) \sin ^3(a+b x) \, dx=\frac {{\left (\cos \left (b x + a\right )^{3} - 3 \, \cos \left (b x + a\right )\right )} \log \left (x\right )}{3 \, b} + \frac {{\left (E_{1}\left (3 i \, b x\right ) + E_{1}\left (-3 i \, b x\right )\right )} \cos \left (3 \, a\right ) - 9 \, {\left (E_{1}\left (i \, b x\right ) + E_{1}\left (-i \, b x\right )\right )} \cos \left (a\right ) - {\left (i \, E_{1}\left (3 i \, b x\right ) - i \, E_{1}\left (-3 i \, b x\right )\right )} \sin \left (3 \, a\right ) + 9 \, {\left (i \, E_{1}\left (i \, b x\right ) - i \, E_{1}\left (-i \, b x\right )\right )} \sin \left (a\right )}{24 \, b} \]

[In]

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

[Out]

1/3*(cos(b*x + a)^3 - 3*cos(b*x + a))*log(x)/b + 1/24*((exp_integral_e(1, 3*I*b*x) + exp_integral_e(1, -3*I*b*
x))*cos(3*a) - 9*(exp_integral_e(1, I*b*x) + exp_integral_e(1, -I*b*x))*cos(a) - (I*exp_integral_e(1, 3*I*b*x)
- I*exp_integral_e(1, -3*I*b*x))*sin(3*a) + 9*(I*exp_integral_e(1, I*b*x) - I*exp_integral_e(1, -I*b*x))*sin(
a))/b

Giac [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.33 (sec) , antiderivative size = 454, normalized size of antiderivative = 5.10 \[ \int \log (x) \sin ^3(a+b x) \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/3*(cos(b*x + a)^3/b - 3*cos(b*x + a)/b)*log(x) + 1/24*(real_part(cos_integral(3*b*x))*tan(3/2*a)^2*tan(1/2*a
)^2 - 9*real_part(cos_integral(b*x))*tan(3/2*a)^2*tan(1/2*a)^2 - 9*real_part(cos_integral(-b*x))*tan(3/2*a)^2*
tan(1/2*a)^2 + real_part(cos_integral(-3*b*x))*tan(3/2*a)^2*tan(1/2*a)^2 - 18*imag_part(cos_integral(b*x))*tan
(3/2*a)^2*tan(1/2*a) + 18*imag_part(cos_integral(-b*x))*tan(3/2*a)^2*tan(1/2*a) - 36*sin_integral(b*x)*tan(3/2
*a)^2*tan(1/2*a) + 2*imag_part(cos_integral(3*b*x))*tan(3/2*a)*tan(1/2*a)^2 - 2*imag_part(cos_integral(-3*b*x)
)*tan(3/2*a)*tan(1/2*a)^2 + 4*sin_integral(3*b*x)*tan(3/2*a)*tan(1/2*a)^2 + real_part(cos_integral(3*b*x))*tan
(3/2*a)^2 + 9*real_part(cos_integral(b*x))*tan(3/2*a)^2 + 9*real_part(cos_integral(-b*x))*tan(3/2*a)^2 + real_
part(cos_integral(-3*b*x))*tan(3/2*a)^2 - real_part(cos_integral(3*b*x))*tan(1/2*a)^2 - 9*real_part(cos_integr
al(b*x))*tan(1/2*a)^2 - 9*real_part(cos_integral(-b*x))*tan(1/2*a)^2 - real_part(cos_integral(-3*b*x))*tan(1/2
*a)^2 + 2*imag_part(cos_integral(3*b*x))*tan(3/2*a) - 2*imag_part(cos_integral(-3*b*x))*tan(3/2*a) + 4*sin_int
egral(3*b*x)*tan(3/2*a) - 18*imag_part(cos_integral(b*x))*tan(1/2*a) + 18*imag_part(cos_integral(-b*x))*tan(1/
2*a) - 36*sin_integral(b*x)*tan(1/2*a) - real_part(cos_integral(3*b*x)) + 9*real_part(cos_integral(b*x)) + 9*r
eal_part(cos_integral(-b*x)) - real_part(cos_integral(-3*b*x)))/(b*tan(3/2*a)^2*tan(1/2*a)^2 + b*tan(3/2*a)^2
+ b*tan(1/2*a)^2 + b)

Mupad [F(-1)]

Timed out. \[ \int \log (x) \sin ^3(a+b x) \, dx=\int {\sin \left (a+b\,x\right )}^3\,\ln \left (x\right ) \,d x \]

[In]

int(sin(a + b*x)^3*log(x),x)

[Out]

int(sin(a + b*x)^3*log(x), x)