Integrand size = 9, antiderivative size = 261 \[ \int d^x x^3 \sin (x) \, dx=-\frac {24 d^x \cos (x) \log (d)}{\left (1+\log ^2(d)\right )^4}+\frac {24 d^x \cos (x) \log ^3(d)}{\left (1+\log ^2(d)\right )^4}+\frac {6 d^x x \cos (x)}{\left (1+\log ^2(d)\right )^3}-\frac {18 d^x x \cos (x) \log ^2(d)}{\left (1+\log ^2(d)\right )^3}+\frac {6 d^x x^2 \cos (x) \log (d)}{\left (1+\log ^2(d)\right )^2}-\frac {d^x x^3 \cos (x)}{1+\log ^2(d)}-\frac {6 d^x \sin (x)}{\left (1+\log ^2(d)\right )^4}+\frac {36 d^x \log ^2(d) \sin (x)}{\left (1+\log ^2(d)\right )^4}-\frac {6 d^x \log ^4(d) \sin (x)}{\left (1+\log ^2(d)\right )^4}-\frac {18 d^x x \log (d) \sin (x)}{\left (1+\log ^2(d)\right )^3}+\frac {6 d^x x \log ^3(d) \sin (x)}{\left (1+\log ^2(d)\right )^3}+\frac {3 d^x x^2 \sin (x)}{\left (1+\log ^2(d)\right )^2}-\frac {3 d^x x^2 \log ^2(d) \sin (x)}{\left (1+\log ^2(d)\right )^2}+\frac {d^x x^3 \log (d) \sin (x)}{1+\log ^2(d)} \] Output:
-24*d^x*cos(x)*ln(d)/(1+ln(d)^2)^4+24*d^x*cos(x)*ln(d)^3/(1+ln(d)^2)^4+6*d ^x*x*cos(x)/(1+ln(d)^2)^3-18*d^x*x*cos(x)*ln(d)^2/(1+ln(d)^2)^3+6*d^x*x^2* cos(x)*ln(d)/(1+ln(d)^2)^2-d^x*x^3*cos(x)/(1+ln(d)^2)-6*d^x*sin(x)/(1+ln(d )^2)^4+36*d^x*ln(d)^2*sin(x)/(1+ln(d)^2)^4-6*d^x*ln(d)^4*sin(x)/(1+ln(d)^2 )^4-18*d^x*x*ln(d)*sin(x)/(1+ln(d)^2)^3+6*d^x*x*ln(d)^3*sin(x)/(1+ln(d)^2) ^3+3*d^x*x^2*sin(x)/(1+ln(d)^2)^2-3*d^x*x^2*ln(d)^2*sin(x)/(1+ln(d)^2)^2+d ^x*x^3*ln(d)*sin(x)/(1+ln(d)^2)
Time = 0.09 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.65 \[ \int d^x x^3 \sin (x) \, dx=\frac {d^x \left (-\cos (x) \left (x \left (-6+x^2\right )-6 \left (-4+x^2\right ) \log (d)+3 x \left (4+x^2\right ) \log ^2(d)-12 \left (2+x^2\right ) \log ^3(d)+3 x \left (6+x^2\right ) \log ^4(d)-6 x^2 \log ^5(d)+x^3 \log ^6(d)\right )+\left (3 \left (-2+x^2\right )+x \left (-18+x^2\right ) \log (d)+3 \left (12+x^2\right ) \log ^2(d)+3 x \left (-4+x^2\right ) \log ^3(d)-3 \left (2+x^2\right ) \log ^4(d)+3 x \left (2+x^2\right ) \log ^5(d)-3 x^2 \log ^6(d)+x^3 \log ^7(d)\right ) \sin (x)\right )}{\left (1+\log ^2(d)\right )^4} \] Input:
Integrate[d^x*x^3*Sin[x],x]
Output:
(d^x*(-(Cos[x]*(x*(-6 + x^2) - 6*(-4 + x^2)*Log[d] + 3*x*(4 + x^2)*Log[d]^ 2 - 12*(2 + x^2)*Log[d]^3 + 3*x*(6 + x^2)*Log[d]^4 - 6*x^2*Log[d]^5 + x^3* Log[d]^6)) + (3*(-2 + x^2) + x*(-18 + x^2)*Log[d] + 3*(12 + x^2)*Log[d]^2 + 3*x*(-4 + x^2)*Log[d]^3 - 3*(2 + x^2)*Log[d]^4 + 3*x*(2 + x^2)*Log[d]^5 - 3*x^2*Log[d]^6 + x^3*Log[d]^7)*Sin[x]))/(1 + Log[d]^2)^4
Time = 0.58 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.01, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {4968, 25, 2010, 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.
| \(\displaystyle \int x^3 d^x \sin (x) \, dx\) |
| \(\Big \downarrow \) 4968 |
| \(\displaystyle -3 \int -x^2 \left (\frac {d^x \cos (x)}{\log ^2(d)+1}-\frac {d^x \log (d) \sin (x)}{\log ^2(d)+1}\right )dx+\frac {x^3 d^x \log (d) \sin (x)}{\log ^2(d)+1}-\frac {x^3 d^x \cos (x)}{\log ^2(d)+1}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 3 \int x^2 \left (\frac {d^x \cos (x)}{\log ^2(d)+1}-\frac {d^x \log (d) \sin (x)}{\log ^2(d)+1}\right )dx+\frac {x^3 d^x \log (d) \sin (x)}{\log ^2(d)+1}-\frac {x^3 d^x \cos (x)}{\log ^2(d)+1}\) |
| \(\Big \downarrow \) 2010 |
| \(\displaystyle 3 \int \left (\frac {d^x x^2 \cos (x)}{\log ^2(d)+1}-\frac {d^x x^2 \log (d) \sin (x)}{\log ^2(d)+1}\right )dx+\frac {x^3 d^x \log (d) \sin (x)}{\log ^2(d)+1}-\frac {x^3 d^x \cos (x)}{\log ^2(d)+1}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {x^3 d^x \log (d) \sin (x)}{\log ^2(d)+1}-\frac {x^3 d^x \cos (x)}{\log ^2(d)+1}+3 \left (\frac {x^2 d^x \sin (x)}{\left (\log ^2(d)+1\right )^2}-\frac {x^2 d^x \log ^2(d) \sin (x)}{\left (\log ^2(d)+1\right )^2}+\frac {2 x^2 d^x \log (d) \cos (x)}{\left (\log ^2(d)+1\right )^2}-\frac {6 x d^x \log (d) \sin (x)}{\left (\log ^2(d)+1\right )^3}+\frac {12 d^x \log ^2(d) \sin (x)}{\left (\log ^2(d)+1\right )^4}-\frac {2 d^x \sin (x)}{\left (\log ^2(d)+1\right )^4}-\frac {2 d^x \log ^4(d) \sin (x)}{\left (\log ^2(d)+1\right )^4}+\frac {2 x d^x \log ^3(d) \sin (x)}{\left (\log ^2(d)+1\right )^3}-\frac {6 x d^x \log ^2(d) \cos (x)}{\left (\log ^2(d)+1\right )^3}+\frac {2 x d^x \cos (x)}{\left (\log ^2(d)+1\right )^3}-\frac {8 d^x \log (d) \cos (x)}{\left (\log ^2(d)+1\right )^4}+\frac {8 d^x \log ^3(d) \cos (x)}{\left (\log ^2(d)+1\right )^4}\right )\) |
Input:
Int[d^x*x^3*Sin[x],x]
Output:
-((d^x*x^3*Cos[x])/(1 + Log[d]^2)) + (d^x*x^3*Log[d]*Sin[x])/(1 + Log[d]^2 ) + 3*((-8*d^x*Cos[x]*Log[d])/(1 + Log[d]^2)^4 + (8*d^x*Cos[x]*Log[d]^3)/( 1 + Log[d]^2)^4 + (2*d^x*x*Cos[x])/(1 + Log[d]^2)^3 - (6*d^x*x*Cos[x]*Log[ d]^2)/(1 + Log[d]^2)^3 + (2*d^x*x^2*Cos[x]*Log[d])/(1 + Log[d]^2)^2 - (2*d ^x*Sin[x])/(1 + Log[d]^2)^4 + (12*d^x*Log[d]^2*Sin[x])/(1 + Log[d]^2)^4 - (2*d^x*Log[d]^4*Sin[x])/(1 + Log[d]^2)^4 - (6*d^x*x*Log[d]*Sin[x])/(1 + Lo g[d]^2)^3 + (2*d^x*x*Log[d]^3*Sin[x])/(1 + Log[d]^2)^3 + (d^x*x^2*Sin[x])/ (1 + Log[d]^2)^2 - (d^x*x^2*Log[d]^2*Sin[x])/(1 + Log[d]^2)^2)
Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x] , x] /; FreeQ[{c, m}, x] && SumQ[u] && !LinearQ[u, x] && !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]
Int[(F_)^((c_.)*((a_.) + (b_.)*(x_)))*((f_.)*(x_))^(m_.)*Sin[(d_.) + (e_.)* (x_)]^(n_.), x_Symbol] :> Module[{u = IntHide[F^(c*(a + b*x))*Sin[d + e*x]^ n, x]}, Simp[(f*x)^m u, x] - Simp[f*m Int[(f*x)^(m - 1)*u, x], x]] /; F reeQ[{F, a, b, c, d, e, f}, x] && IGtQ[n, 0] && GtQ[m, 0]
Result contains complex when optimal does not.
Time = 0.46 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.64
| method | result | size |
| risch | \(-\frac {i \left (-6+\ln \left (d \right )^{3} x^{3}+3 i \ln \left (d \right )^{2} x^{3}-3 \ln \left (d \right ) x^{3}-i x^{3}+6 x \ln \left (d \right )+6 i x -3 \ln \left (d \right )^{2} x^{2}-6 i \ln \left (d \right ) x^{2}+3 x^{2}\right ) d^{x} {\mathrm e}^{i x}}{2 \left (\ln \left (d \right )+i\right )^{4}}+\frac {i \left (-6+6 x \ln \left (d \right )-6 i x -3 \ln \left (d \right )^{2} x^{2}+6 i \ln \left (d \right ) x^{2}+3 x^{2}+\ln \left (d \right )^{3} x^{3}-3 i \ln \left (d \right )^{2} x^{3}-3 \ln \left (d \right ) x^{3}+i x^{3}\right ) d^{x} {\mathrm e}^{-i x}}{2 \left (\ln \left (d \right )-i\right )^{4}}\) | \(166\) |
| parallelrisch | \(\frac {\left (\ln \left (d \right )^{7} x^{3} \sin \left (x \right )+\left (-x^{3} \cos \left (x \right )-3 x^{2} \sin \left (x \right )\right ) \ln \left (d \right )^{6}+3 \left (\left (x^{3}+2 x \right ) \sin \left (x \right )+2 x^{2} \cos \left (x \right )\right ) \ln \left (d \right )^{5}+3 \left (-x^{3} \cos \left (x \right )-x^{2} \sin \left (x \right )-6 x \cos \left (x \right )-2 \sin \left (x \right )\right ) \ln \left (d \right )^{4}+3 \left (\left (x^{3}-4 x \right ) \sin \left (x \right )+4 \cos \left (x \right ) \left (x^{2}+2\right )\right ) \ln \left (d \right )^{3}+3 \left (-x^{3} \cos \left (x \right )+x^{2} \sin \left (x \right )-4 x \cos \left (x \right )+12 \sin \left (x \right )\right ) \ln \left (d \right )^{2}+\left (\left (x^{3}-18 x \right ) \sin \left (x \right )+6 \cos \left (x \right ) \left (x^{2}-4\right )\right ) \ln \left (d \right )-x^{3} \cos \left (x \right )+3 x^{2} \sin \left (x \right )+6 x \cos \left (x \right )-6 \sin \left (x \right )\right ) d^{x}}{\left (1+\ln \left (d \right )^{2}\right )^{4}}\) | \(199\) |
| orering | \(\frac {2 \left (\ln \left (d \right )^{7} x^{4}-3 \ln \left (d \right )^{6} x^{3}+3 \ln \left (d \right )^{5} x^{4}+3 \ln \left (d \right )^{5} x^{2}-3 \ln \left (d \right )^{4} x^{3}+3 \ln \left (d \right )^{3} x^{4}+12 \ln \left (d \right )^{4} x -18 \ln \left (d \right )^{3} x^{2}+3 \ln \left (d \right )^{2} x^{3}+\ln \left (d \right ) x^{4}-36 \ln \left (d \right )^{3}+48 \ln \left (d \right )^{2} x -21 \ln \left (d \right ) x^{2}+3 x^{3}+36 \ln \left (d \right )-12 x \right ) d^{x} \sin \left (x \right )}{\left (\ln \left (d \right )^{4}+2 \ln \left (d \right )^{2}+1\right ) \left (1+\ln \left (d \right )^{2}\right )^{2} x}-\frac {\left (\ln \left (d \right )^{6} x^{3}+3 \ln \left (d \right )^{4} x^{3}+3 \ln \left (d \right )^{2} x^{3}-6 \ln \left (d \right )^{5} x^{2}-12 \ln \left (d \right )^{3} x^{2}-6 \ln \left (d \right ) x^{2}+x^{3}+18 \ln \left (d \right )^{4} x +12 \ln \left (d \right )^{2} x -6 x -24 \ln \left (d \right )^{3}+24 \ln \left (d \right )\right ) \left (d^{x} \ln \left (d \right ) x^{3} \sin \left (x \right )+3 d^{x} x^{2} \sin \left (x \right )+d^{x} x^{3} \cos \left (x \right )\right )}{\left (\ln \left (d \right )^{4}+2 \ln \left (d \right )^{2}+1\right ) \left (1+\ln \left (d \right )^{2}\right )^{2} x^{3}}\) | \(291\) |
| norman | \(\frac {\frac {x^{3} {\mathrm e}^{x \ln \left (d \right )} \tan \left (\frac {x}{2}\right )^{2}}{1+\ln \left (d \right )^{2}}-\frac {x^{3} {\mathrm e}^{x \ln \left (d \right )}}{1+\ln \left (d \right )^{2}}+\frac {6 \ln \left (d \right ) x^{2} {\mathrm e}^{x \ln \left (d \right )}}{\ln \left (d \right )^{4}+2 \ln \left (d \right )^{2}+1}-\frac {6 \left (\ln \left (d \right )^{2}-1\right ) x^{2} {\mathrm e}^{x \ln \left (d \right )} \tan \left (\frac {x}{2}\right )}{\ln \left (d \right )^{4}+2 \ln \left (d \right )^{2}+1}-\frac {6 \left (3 \ln \left (d \right )^{2}-1\right ) x \,{\mathrm e}^{x \ln \left (d \right )}}{\left (1+\ln \left (d \right )^{2}\right ) \left (\ln \left (d \right )^{4}+2 \ln \left (d \right )^{2}+1\right )}-\frac {12 \left (\ln \left (d \right )^{4}-6 \ln \left (d \right )^{2}+1\right ) {\mathrm e}^{x \ln \left (d \right )} \tan \left (\frac {x}{2}\right )}{\left (\ln \left (d \right )^{4}+2 \ln \left (d \right )^{2}+1\right ) \left (1+\ln \left (d \right )^{2}\right )^{2}}+\frac {24 \ln \left (d \right ) \left (\ln \left (d \right )^{2}-1\right ) {\mathrm e}^{x \ln \left (d \right )}}{\left (\ln \left (d \right )^{6}+3 \ln \left (d \right )^{4}+3 \ln \left (d \right )^{2}+1\right ) \left (1+\ln \left (d \right )^{2}\right )}+\frac {2 \ln \left (d \right ) x^{3} {\mathrm e}^{x \ln \left (d \right )} \tan \left (\frac {x}{2}\right )}{1+\ln \left (d \right )^{2}}-\frac {6 \ln \left (d \right ) x^{2} {\mathrm e}^{x \ln \left (d \right )} \tan \left (\frac {x}{2}\right )^{2}}{\ln \left (d \right )^{4}+2 \ln \left (d \right )^{2}+1}+\frac {6 \left (3 \ln \left (d \right )^{2}-1\right ) x \,{\mathrm e}^{x \ln \left (d \right )} \tan \left (\frac {x}{2}\right )^{2}}{\left (1+\ln \left (d \right )^{2}\right ) \left (\ln \left (d \right )^{4}+2 \ln \left (d \right )^{2}+1\right )}-\frac {24 \ln \left (d \right ) \left (\ln \left (d \right )^{2}-1\right ) {\mathrm e}^{x \ln \left (d \right )} \tan \left (\frac {x}{2}\right )^{2}}{\left (\ln \left (d \right )^{6}+3 \ln \left (d \right )^{4}+3 \ln \left (d \right )^{2}+1\right ) \left (1+\ln \left (d \right )^{2}\right )}+\frac {12 \ln \left (d \right ) \left (\ln \left (d \right )^{2}-3\right ) x \,{\mathrm e}^{x \ln \left (d \right )} \tan \left (\frac {x}{2}\right )}{\left (1+\ln \left (d \right )^{2}\right ) \left (\ln \left (d \right )^{4}+2 \ln \left (d \right )^{2}+1\right )}}{1+\tan \left (\frac {x}{2}\right )^{2}}\) | \(431\) |
Input:
int(d^x*x^3*sin(x),x,method=_RETURNVERBOSE)
Output:
-1/2*I*(-6+ln(d)^3*x^3+3*I*ln(d)^2*x^3-3*ln(d)*x^3-I*x^3+6*x*ln(d)+6*I*x-3 *ln(d)^2*x^2-6*I*ln(d)*x^2+3*x^2)*d^x/(ln(d)+I)^4*exp(I*x)+1/2*I*(-6+6*x*l n(d)-6*I*x-3*ln(d)^2*x^2+6*I*ln(d)*x^2+3*x^2+ln(d)^3*x^3-3*I*ln(d)^2*x^3-3 *ln(d)*x^3+I*x^3)*d^x/(ln(d)-I)^4*exp(-I*x)
Time = 0.08 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.78 \[ \int d^x x^3 \sin (x) \, dx=-\frac {{\left (x^{3} \cos \left (x\right ) \log \left (d\right )^{6} - 6 \, x^{2} \cos \left (x\right ) \log \left (d\right )^{5} + 3 \, {\left (x^{3} + 6 \, x\right )} \cos \left (x\right ) \log \left (d\right )^{4} - 12 \, {\left (x^{2} + 2\right )} \cos \left (x\right ) \log \left (d\right )^{3} + 3 \, {\left (x^{3} + 4 \, x\right )} \cos \left (x\right ) \log \left (d\right )^{2} - 6 \, {\left (x^{2} - 4\right )} \cos \left (x\right ) \log \left (d\right ) + {\left (x^{3} - 6 \, x\right )} \cos \left (x\right ) - {\left (x^{3} \log \left (d\right )^{7} - 3 \, x^{2} \log \left (d\right )^{6} + 3 \, {\left (x^{3} + 2 \, x\right )} \log \left (d\right )^{5} - 3 \, {\left (x^{2} + 2\right )} \log \left (d\right )^{4} + 3 \, {\left (x^{3} - 4 \, x\right )} \log \left (d\right )^{3} + 3 \, {\left (x^{2} + 12\right )} \log \left (d\right )^{2} + 3 \, x^{2} + {\left (x^{3} - 18 \, x\right )} \log \left (d\right ) - 6\right )} \sin \left (x\right )\right )} d^{x}}{\log \left (d\right )^{8} + 4 \, \log \left (d\right )^{6} + 6 \, \log \left (d\right )^{4} + 4 \, \log \left (d\right )^{2} + 1} \] Input:
integrate(d^x*x^3*sin(x),x, algorithm="fricas")
Output:
-(x^3*cos(x)*log(d)^6 - 6*x^2*cos(x)*log(d)^5 + 3*(x^3 + 6*x)*cos(x)*log(d )^4 - 12*(x^2 + 2)*cos(x)*log(d)^3 + 3*(x^3 + 4*x)*cos(x)*log(d)^2 - 6*(x^ 2 - 4)*cos(x)*log(d) + (x^3 - 6*x)*cos(x) - (x^3*log(d)^7 - 3*x^2*log(d)^6 + 3*(x^3 + 2*x)*log(d)^5 - 3*(x^2 + 2)*log(d)^4 + 3*(x^3 - 4*x)*log(d)^3 + 3*(x^2 + 12)*log(d)^2 + 3*x^2 + (x^3 - 18*x)*log(d) - 6)*sin(x))*d^x/(lo g(d)^8 + 4*log(d)^6 + 6*log(d)^4 + 4*log(d)^2 + 1)
Result contains complex when optimal does not.
Time = 1.35 (sec) , antiderivative size = 1355, normalized size of antiderivative = 5.19 \[ \int d^x x^3 \sin (x) \, dx=\text {Too large to display} \] Input:
integrate(d**x*x**3*sin(x),x)
Output:
Piecewise((x**4*exp(-I*x)*sin(x)/8 - I*x**4*exp(-I*x)*cos(x)/8 + I*x**3*ex p(-I*x)*sin(x)/4 - x**3*exp(-I*x)*cos(x)/4 + 3*x**2*exp(-I*x)*sin(x)/8 + 3 *I*x**2*exp(-I*x)*cos(x)/8 - 3*I*x*exp(-I*x)*sin(x)/8 + 3*x*exp(-I*x)*cos( x)/8 - 3*I*exp(-I*x)*cos(x)/8, Eq(d, exp(-I))), (x**4*exp(I*x)*sin(x)/8 + I*x**4*exp(I*x)*cos(x)/8 - I*x**3*exp(I*x)*sin(x)/4 - x**3*exp(I*x)*cos(x) /4 + 3*x**2*exp(I*x)*sin(x)/8 - 3*I*x**2*exp(I*x)*cos(x)/8 + 3*I*x*exp(I*x )*sin(x)/8 + 3*x*exp(I*x)*cos(x)/8 + 3*I*exp(I*x)*cos(x)/8, Eq(d, exp(I))) , (d**x*x**3*log(d)**7*sin(x)/(log(d)**8 + 4*log(d)**6 + 6*log(d)**4 + 4*l og(d)**2 + 1) - d**x*x**3*log(d)**6*cos(x)/(log(d)**8 + 4*log(d)**6 + 6*lo g(d)**4 + 4*log(d)**2 + 1) + 3*d**x*x**3*log(d)**5*sin(x)/(log(d)**8 + 4*l og(d)**6 + 6*log(d)**4 + 4*log(d)**2 + 1) - 3*d**x*x**3*log(d)**4*cos(x)/( log(d)**8 + 4*log(d)**6 + 6*log(d)**4 + 4*log(d)**2 + 1) + 3*d**x*x**3*log (d)**3*sin(x)/(log(d)**8 + 4*log(d)**6 + 6*log(d)**4 + 4*log(d)**2 + 1) - 3*d**x*x**3*log(d)**2*cos(x)/(log(d)**8 + 4*log(d)**6 + 6*log(d)**4 + 4*lo g(d)**2 + 1) + d**x*x**3*log(d)*sin(x)/(log(d)**8 + 4*log(d)**6 + 6*log(d) **4 + 4*log(d)**2 + 1) - d**x*x**3*cos(x)/(log(d)**8 + 4*log(d)**6 + 6*log (d)**4 + 4*log(d)**2 + 1) - 3*d**x*x**2*log(d)**6*sin(x)/(log(d)**8 + 4*lo g(d)**6 + 6*log(d)**4 + 4*log(d)**2 + 1) + 6*d**x*x**2*log(d)**5*cos(x)/(l og(d)**8 + 4*log(d)**6 + 6*log(d)**4 + 4*log(d)**2 + 1) - 3*d**x*x**2*log( d)**4*sin(x)/(log(d)**8 + 4*log(d)**6 + 6*log(d)**4 + 4*log(d)**2 + 1) ...
Time = 0.06 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.71 \[ \int d^x x^3 \sin (x) \, dx=-\frac {{\left ({\left (\log \left (d\right )^{6} + 3 \, \log \left (d\right )^{4} + 3 \, \log \left (d\right )^{2} + 1\right )} x^{3} - 6 \, {\left (\log \left (d\right )^{5} + 2 \, \log \left (d\right )^{3} + \log \left (d\right )\right )} x^{2} - 24 \, \log \left (d\right )^{3} + 6 \, {\left (3 \, \log \left (d\right )^{4} + 2 \, \log \left (d\right )^{2} - 1\right )} x + 24 \, \log \left (d\right )\right )} d^{x} \cos \left (x\right ) - {\left ({\left (\log \left (d\right )^{7} + 3 \, \log \left (d\right )^{5} + 3 \, \log \left (d\right )^{3} + \log \left (d\right )\right )} x^{3} - 6 \, \log \left (d\right )^{4} - 3 \, {\left (\log \left (d\right )^{6} + \log \left (d\right )^{4} - \log \left (d\right )^{2} - 1\right )} x^{2} + 6 \, {\left (\log \left (d\right )^{5} - 2 \, \log \left (d\right )^{3} - 3 \, \log \left (d\right )\right )} x + 36 \, \log \left (d\right )^{2} - 6\right )} d^{x} \sin \left (x\right )}{\log \left (d\right )^{8} + 4 \, \log \left (d\right )^{6} + 6 \, \log \left (d\right )^{4} + 4 \, \log \left (d\right )^{2} + 1} \] Input:
integrate(d^x*x^3*sin(x),x, algorithm="maxima")
Output:
-(((log(d)^6 + 3*log(d)^4 + 3*log(d)^2 + 1)*x^3 - 6*(log(d)^5 + 2*log(d)^3 + log(d))*x^2 - 24*log(d)^3 + 6*(3*log(d)^4 + 2*log(d)^2 - 1)*x + 24*log( d))*d^x*cos(x) - ((log(d)^7 + 3*log(d)^5 + 3*log(d)^3 + log(d))*x^3 - 6*lo g(d)^4 - 3*(log(d)^6 + log(d)^4 - log(d)^2 - 1)*x^2 + 6*(log(d)^5 - 2*log( d)^3 - 3*log(d))*x + 36*log(d)^2 - 6)*d^x*sin(x))/(log(d)^8 + 4*log(d)^6 + 6*log(d)^4 + 4*log(d)^2 + 1)
Result contains complex when optimal does not.
Time = 0.18 (sec) , antiderivative size = 5069, normalized size of antiderivative = 19.42 \[ \int d^x x^3 \sin (x) \, dx=\text {Too large to display} \] Input:
integrate(d^x*x^3*sin(x),x, algorithm="giac")
Output:
1/2*(((4*pi + pi^4*sgn(d) - 6*pi^2*log(abs(d))^2*sgn(d) - pi^4 + 6*pi^2*lo g(abs(d))^2 - 2*log(abs(d))^4 - 4*pi^3*sgn(d) + 12*pi*log(abs(d))^2*sgn(d) + 4*pi^3 - 12*pi*log(abs(d))^2 + 6*pi^2*sgn(d) - 6*pi^2 + 12*log(abs(d))^ 2 - 4*pi*sgn(d) - 2)*(pi^3*x^3*sgn(d) - 3*pi*x^3*log(abs(d))^2*sgn(d) - pi ^3*x^3 + 3*pi*x^3*log(abs(d))^2 - 3*pi^2*x^3*sgn(d) + 3*pi^2*x^3 - 6*x^3*l og(abs(d))^2 + 3*pi*x^3*sgn(d) + 6*pi*x^2*log(abs(d))*sgn(d) - 3*pi*x^3 - 6*pi*x^2*log(abs(d)) + 2*x^3 + 12*x^2*log(abs(d)) - 6*pi*x*sgn(d) + 6*pi*x - 12*x)/((4*pi + pi^4*sgn(d) - 6*pi^2*log(abs(d))^2*sgn(d) - pi^4 + 6*pi^ 2*log(abs(d))^2 - 2*log(abs(d))^4 - 4*pi^3*sgn(d) + 12*pi*log(abs(d))^2*sg n(d) + 4*pi^3 - 12*pi*log(abs(d))^2 + 6*pi^2*sgn(d) - 6*pi^2 + 12*log(abs( d))^2 - 4*pi*sgn(d) - 2)^2 + 16*(pi^3*log(abs(d))*sgn(d) - pi*log(abs(d))^ 3*sgn(d) - pi^3*log(abs(d)) + pi*log(abs(d))^3 - 3*pi^2*log(abs(d))*sgn(d) + 3*pi^2*log(abs(d)) - 2*log(abs(d))^3 + 3*pi*log(abs(d))*sgn(d) - 3*pi*l og(abs(d)) + 2*log(abs(d)))^2) + 4*(3*pi^2*x^3*log(abs(d))*sgn(d) - 3*pi^2 *x^3*log(abs(d)) + 2*x^3*log(abs(d))^3 - 6*pi*x^3*log(abs(d))*sgn(d) + 6*p i*x^3*log(abs(d)) - 3*pi^2*x^2*sgn(d) + 3*pi^2*x^2 - 6*x^3*log(abs(d)) - 6 *x^2*log(abs(d))^2 + 6*pi*x^2*sgn(d) - 6*pi*x^2 + 6*x^2 + 12*x*log(abs(d)) - 12)*(pi^3*log(abs(d))*sgn(d) - pi*log(abs(d))^3*sgn(d) - pi^3*log(abs(d )) + pi*log(abs(d))^3 - 3*pi^2*log(abs(d))*sgn(d) + 3*pi^2*log(abs(d)) - 2 *log(abs(d))^3 + 3*pi*log(abs(d))*sgn(d) - 3*pi*log(abs(d)) + 2*log(abs...
Time = 0.36 (sec) , antiderivative size = 231, normalized size of antiderivative = 0.89 \[ \int d^x x^3 \sin (x) \, dx=-\frac {d^x\,\left (6\,\sin \left (x\right )+x^3\,\cos \left (x\right )-3\,x^2\,\sin \left (x\right )-6\,x\,\cos \left (x\right )\right )-d^x\,{\ln \left (d\right )}^5\,\left (6\,x^2\,\cos \left (x\right )+3\,x^3\,\sin \left (x\right )+6\,x\,\sin \left (x\right )\right )+d^x\,{\ln \left (d\right )}^4\,\left (6\,\sin \left (x\right )+3\,x^3\,\cos \left (x\right )+3\,x^2\,\sin \left (x\right )+18\,x\,\cos \left (x\right )\right )-d^x\,{\ln \left (d\right )}^3\,\left (24\,\cos \left (x\right )+12\,x^2\,\cos \left (x\right )+3\,x^3\,\sin \left (x\right )-12\,x\,\sin \left (x\right )\right )-d^x\,{\ln \left (d\right )}^2\,\left (36\,\sin \left (x\right )-3\,x^3\,\cos \left (x\right )+3\,x^2\,\sin \left (x\right )-12\,x\,\cos \left (x\right )\right )+d^x\,{\ln \left (d\right )}^6\,\left (x^3\,\cos \left (x\right )+3\,x^2\,\sin \left (x\right )\right )+d^x\,\ln \left (d\right )\,\left (24\,\cos \left (x\right )-6\,x^2\,\cos \left (x\right )-x^3\,\sin \left (x\right )+18\,x\,\sin \left (x\right )\right )-d^x\,x^3\,{\ln \left (d\right )}^7\,\sin \left (x\right )}{{\left ({\ln \left (d\right )}^2+1\right )}^4} \] Input:
int(d^x*x^3*sin(x),x)
Output:
-(d^x*(6*sin(x) + x^3*cos(x) - 3*x^2*sin(x) - 6*x*cos(x)) - d^x*log(d)^5*( 6*x^2*cos(x) + 3*x^3*sin(x) + 6*x*sin(x)) + d^x*log(d)^4*(6*sin(x) + 3*x^3 *cos(x) + 3*x^2*sin(x) + 18*x*cos(x)) - d^x*log(d)^3*(24*cos(x) + 12*x^2*c os(x) + 3*x^3*sin(x) - 12*x*sin(x)) - d^x*log(d)^2*(36*sin(x) - 3*x^3*cos( x) + 3*x^2*sin(x) - 12*x*cos(x)) + d^x*log(d)^6*(x^3*cos(x) + 3*x^2*sin(x) ) + d^x*log(d)*(24*cos(x) - 6*x^2*cos(x) - x^3*sin(x) + 18*x*sin(x)) - d^x *x^3*log(d)^7*sin(x))/(log(d)^2 + 1)^4
Time = 0.16 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.01 \[ \int d^x x^3 \sin (x) \, dx=\frac {d^{x} \left (-\cos \left (x \right ) \mathrm {log}\left (d \right )^{6} x^{3}+\mathrm {log}\left (d \right )^{7} \sin \left (x \right ) x^{3}+\mathrm {log}\left (d \right ) \sin \left (x \right ) x^{3}+36 \mathrm {log}\left (d \right )^{2} \sin \left (x \right )+24 \cos \left (x \right ) \mathrm {log}\left (d \right )^{3}-3 \cos \left (x \right ) \mathrm {log}\left (d \right )^{4} x^{3}-3 \cos \left (x \right ) \mathrm {log}\left (d \right )^{2} x^{3}-3 \mathrm {log}\left (d \right )^{6} \sin \left (x \right ) x^{2}+3 \mathrm {log}\left (d \right )^{5} \sin \left (x \right ) x^{3}+6 \mathrm {log}\left (d \right )^{5} \sin \left (x \right ) x +3 \mathrm {log}\left (d \right )^{3} \sin \left (x \right ) x^{3}+6 \cos \left (x \right ) \mathrm {log}\left (d \right )^{5} x^{2}+6 \cos \left (x \right ) \mathrm {log}\left (d \right ) x^{2}-3 \mathrm {log}\left (d \right )^{4} \sin \left (x \right ) x^{2}-12 \cos \left (x \right ) \mathrm {log}\left (d \right )^{2} x -18 \,\mathrm {log}\left (d \right ) \sin \left (x \right ) x -12 \mathrm {log}\left (d \right )^{3} \sin \left (x \right ) x +6 \cos \left (x \right ) x -6 \sin \left (x \right )+3 \mathrm {log}\left (d \right )^{2} \sin \left (x \right ) x^{2}-18 \cos \left (x \right ) \mathrm {log}\left (d \right )^{4} x +12 \cos \left (x \right ) \mathrm {log}\left (d \right )^{3} x^{2}+3 \sin \left (x \right ) x^{2}-\cos \left (x \right ) x^{3}-6 \mathrm {log}\left (d \right )^{4} \sin \left (x \right )-24 \cos \left (x \right ) \mathrm {log}\left (d \right )\right )}{\mathrm {log}\left (d \right )^{8}+4 \mathrm {log}\left (d \right )^{6}+6 \mathrm {log}\left (d \right )^{4}+4 \mathrm {log}\left (d \right )^{2}+1} \] Input:
int(d^x*x^3*sin(x),x)
Output:
(d**x*( - cos(x)*log(d)**6*x**3 + 6*cos(x)*log(d)**5*x**2 - 3*cos(x)*log(d )**4*x**3 - 18*cos(x)*log(d)**4*x + 12*cos(x)*log(d)**3*x**2 + 24*cos(x)*l og(d)**3 - 3*cos(x)*log(d)**2*x**3 - 12*cos(x)*log(d)**2*x + 6*cos(x)*log( d)*x**2 - 24*cos(x)*log(d) - cos(x)*x**3 + 6*cos(x)*x + log(d)**7*sin(x)*x **3 - 3*log(d)**6*sin(x)*x**2 + 3*log(d)**5*sin(x)*x**3 + 6*log(d)**5*sin( x)*x - 3*log(d)**4*sin(x)*x**2 - 6*log(d)**4*sin(x) + 3*log(d)**3*sin(x)*x **3 - 12*log(d)**3*sin(x)*x + 3*log(d)**2*sin(x)*x**2 + 36*log(d)**2*sin(x ) + log(d)*sin(x)*x**3 - 18*log(d)*sin(x)*x + 3*sin(x)*x**2 - 6*sin(x)))/( log(d)**8 + 4*log(d)**6 + 6*log(d)**4 + 4*log(d)**2 + 1)