Integrand size = 11, antiderivative size = 89 \[ \int \log (x) \sin ^3(a+b x) \, dx=\frac {3 \cos (a) \operatorname {CosIntegral}(b x)}{4 b}-\frac {\cos (3 a) \operatorname {CosIntegral}(3 b x)}{12 b}-\frac {\cos (a+b x) \log (x)}{b}+\frac {\cos ^3(a+b x) \log (x)}{3 b}-\frac {3 \sin (a) \text {Si}(b x)}{4 b}+\frac {\sin (3 a) \text {Si}(3 b x)}{12 b} \] Output:
3/4*cos(a)*Ci(b*x)/b-1/12*cos(3*a)*Ci(3*b*x)/b-cos(b*x+a)*ln(x)/b+1/3*cos( b*x+a)^3*ln(x)/b-3/4*sin(a)*Si(b*x)/b+1/12*sin(3*a)*Si(3*b*x)/b
Time = 0.12 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.74 \[ \int \log (x) \sin ^3(a+b x) \, dx=\frac {9 \cos (a) \operatorname {CosIntegral}(b x)-\cos (3 a) \operatorname {CosIntegral}(3 b x)-9 \cos (a+b x) \log (x)+\cos (3 (a+b x)) \log (x)-9 \sin (a) \text {Si}(b x)+\sin (3 a) \text {Si}(3 b x)}{12 b} \] Input:
Integrate[Log[x]*Sin[a + b*x]^3,x]
Output:
(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]*Si nIntegral[3*b*x])/(12*b)
Time = 0.78 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.96, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {3034, 27, 25, 7293, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \log (x) \sin ^3(a+b x) \, dx\) |
\(\Big \downarrow \) 3034 |
\(\displaystyle -\int \frac {\cos (a+b x) \left (\cos ^2(a+b x)-3\right )}{3 b x}dx+\frac {\log (x) \cos ^3(a+b x)}{3 b}-\frac {\log (x) \cos (a+b x)}{b}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\int -\frac {\cos (a+b x) \left (3-\cos ^2(a+b x)\right )}{x}dx}{3 b}+\frac {\log (x) \cos ^3(a+b x)}{3 b}-\frac {\log (x) \cos (a+b x)}{b}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int \frac {\cos (a+b x) \left (3-\cos ^2(a+b x)\right )}{x}dx}{3 b}+\frac {\log (x) \cos ^3(a+b x)}{3 b}-\frac {\log (x) \cos (a+b x)}{b}\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {\int \left (\frac {3 \cos (a+b x)}{x}-\frac {\cos ^3(a+b x)}{x}\right )dx}{3 b}+\frac {\log (x) \cos ^3(a+b x)}{3 b}-\frac {\log (x) \cos (a+b x)}{b}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\frac {9}{4} \cos (a) \operatorname {CosIntegral}(b x)-\frac {1}{4} \cos (3 a) \operatorname {CosIntegral}(3 b x)-\frac {9}{4} \sin (a) \text {Si}(b x)+\frac {1}{4} \sin (3 a) \text {Si}(3 b x)}{3 b}+\frac {\log (x) \cos ^3(a+b x)}{3 b}-\frac {\log (x) \cos (a+b x)}{b}\) |
Input:
Int[Log[x]*Sin[a + b*x]^3,x]
Output:
-((Cos[a + b*x]*Log[x])/b) + (Cos[a + b*x]^3*Log[x])/(3*b) + ((9*Cos[a]*Co sIntegral[b*x])/4 - (Cos[3*a]*CosIntegral[3*b*x])/4 - (9*Sin[a]*SinIntegra l[b*x])/4 + (Sin[3*a]*SinIntegral[3*b*x])/4)/(3*b)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Log[u_]*(v_), x_Symbol] :> With[{w = IntHide[v, x]}, Simp[Log[u] w, x ] - Int[SimplifyIntegrand[w*(D[u, x]/u), x], x] /; InverseFunctionFreeQ[w, x]] /; InverseFunctionFreeQ[u, x]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 267.45 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.82
method | result | size |
risch | \(-\frac {3 \cos \left (b x +a \right ) \ln \left (x \right )}{4 b}+\frac {\ln \left (x \right ) \cos \left (3 b x +3 a \right )}{12 b}-\frac {i {\mathrm e}^{-3 i a} \pi \,\operatorname {csgn}\left (b x \right )}{24 b}+\frac {i {\mathrm e}^{-3 i a} \operatorname {Si}\left (3 b x \right )}{12 b}+\frac {{\mathrm e}^{-3 i a} \operatorname {expIntegral}_{1}\left (-3 i b x \right )}{24 b}+\frac {3 i {\mathrm e}^{-i a} \pi \,\operatorname {csgn}\left (b x \right )}{8 b}-\frac {3 i {\mathrm e}^{-i a} \operatorname {Si}\left (b x \right )}{4 b}-\frac {3 \,{\mathrm e}^{-i a} \operatorname {expIntegral}_{1}\left (-i b x \right )}{8 b}-\frac {3 \,{\mathrm e}^{i a} \operatorname {expIntegral}_{1}\left (-i b x \right )}{8 b}+\frac {{\mathrm e}^{3 i a} \operatorname {expIntegral}_{1}\left (-3 i b x \right )}{24 b}\) | \(162\) |
Input:
int(ln(x)*sin(b*x+a)^3,x,method=_RETURNVERBOSE)
Output:
-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)*P i*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 )
Time = 0.11 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.72 \[ \int \log (x) \sin ^3(a+b x) \, dx=-\frac {\cos \left (3 \, a\right ) \operatorname {Ci}\left (3 \, b x\right ) - 9 \, \cos \left (a\right ) \operatorname {Ci}\left (b x\right ) - 4 \, {\left (\cos \left (b x + a\right )^{3} - 3 \, \cos \left (b x + a\right )\right )} \log \left (x\right ) - \sin \left (3 \, a\right ) \operatorname {Si}\left (3 \, b x\right ) + 9 \, \sin \left (a\right ) \operatorname {Si}\left (b x\right )}{12 \, b} \] Input:
integrate(log(x)*sin(b*x+a)^3,x, algorithm="fricas")
Output:
-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
\[ \int \log (x) \sin ^3(a+b x) \, dx=\int \log {\left (x \right )} \sin ^{3}{\left (a + b x \right )}\, dx \] Input:
integrate(ln(x)*sin(b*x+a)**3,x)
Output:
Integral(log(x)*sin(a + b*x)**3, x)
Result contains complex when optimal does not.
Time = 0.08 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.24 \[ \int \log (x) \sin ^3(a+b x) \, dx=\frac {{\left (\cos \left (b x + a\right )^{3} - 3 \, \cos \left (b x + a\right )\right )} \log \left (x\right )}{3 \, b} + \frac {{\left (E_{1}\left (3 i \, b x\right ) + E_{1}\left (-3 i \, b x\right )\right )} \cos \left (3 \, a\right ) - 9 \, {\left (E_{1}\left (i \, b x\right ) + E_{1}\left (-i \, b x\right )\right )} \cos \left (a\right ) - {\left (i \, E_{1}\left (3 i \, b x\right ) - i \, E_{1}\left (-3 i \, b x\right )\right )} \sin \left (3 \, a\right ) + 9 \, {\left (i \, E_{1}\left (i \, b x\right ) - i \, E_{1}\left (-i \, b x\right )\right )} \sin \left (a\right )}{24 \, b} \] Input:
integrate(log(x)*sin(b*x+a)^3,x, algorithm="maxima")
Output:
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
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.12 (sec) , antiderivative size = 454, normalized size of antiderivative = 5.10 \[ \int \log (x) \sin ^3(a+b x) \, dx=\text {Too large to display} \] Input:
integrate(log(x)*sin(b*x+a)^3,x, algorithm="giac")
Output:
1/3*(cos(b*x + a)^3/b - 3*cos(b*x + a)/b)*log(x) + 1/24*(real_part(cos_int egral(3*b*x))*tan(3/2*a)^2*tan(1/2*a)^2 - 9*real_part(cos_integral(b*x))*t an(3/2*a)^2*tan(1/2*a)^2 - 9*real_part(cos_integral(-b*x))*tan(3/2*a)^2*ta n(1/2*a)^2 + real_part(cos_integral(-3*b*x))*tan(3/2*a)^2*tan(1/2*a)^2 - 1 8*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_integr al(-b*x))*tan(3/2*a)^2 + real_part(cos_integral(-3*b*x))*tan(3/2*a)^2 - re al_part(cos_integral(3*b*x))*tan(1/2*a)^2 - 9*real_part(cos_integral(b*x)) *tan(1/2*a)^2 - 9*real_part(cos_integral(-b*x))*tan(1/2*a)^2 - real_part(c os_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_integral(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_pa rt(cos_integral(3*b*x)) + 9*real_part(cos_integral(b*x)) + 9*real_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)
Timed out. \[ \int \log (x) \sin ^3(a+b x) \, dx=\int {\sin \left (a+b\,x\right )}^3\,\ln \left (x\right ) \,d x \] Input:
int(sin(a + b*x)^3*log(x),x)
Output:
int(sin(a + b*x)^3*log(x), x)
\[ \int \log (x) \sin ^3(a+b x) \, dx=\int \mathrm {log}\left (x \right ) \sin \left (b x +a \right )^{3}d x \] Input:
int(log(x)*sin(b*x+a)^3,x)
Output:
int(log(x)*sin(a + b*x)**3,x)