Integrand size = 9, antiderivative size = 162 \[ \int d^x x^2 \sin (x) \, dx=\frac {2 d^x \cos (x)}{\left (1+\log ^2(d)\right )^3}-\frac {6 d^x \cos (x) \log ^2(d)}{\left (1+\log ^2(d)\right )^3}+\frac {4 d^x x \cos (x) \log (d)}{\left (1+\log ^2(d)\right )^2}-\frac {d^x x^2 \cos (x)}{1+\log ^2(d)}-\frac {6 d^x \log (d) \sin (x)}{\left (1+\log ^2(d)\right )^3}+\frac {2 d^x \log ^3(d) \sin (x)}{\left (1+\log ^2(d)\right )^3}+\frac {2 d^x x \sin (x)}{\left (1+\log ^2(d)\right )^2}-\frac {2 d^x x \log ^2(d) \sin (x)}{\left (1+\log ^2(d)\right )^2}+\frac {d^x x^2 \log (d) \sin (x)}{1+\log ^2(d)} \]
2*d^x*cos(x)/(1+ln(d)^2)^3-6*d^x*cos(x)*ln(d)^2/(1+ln(d)^2)^3+4*d^x*x*cos( x)*ln(d)/(1+ln(d)^2)^2-d^x*x^2*cos(x)/(1+ln(d)^2)-6*d^x*ln(d)*sin(x)/(1+ln (d)^2)^3+2*d^x*ln(d)^3*sin(x)/(1+ln(d)^2)^3+2*d^x*x*sin(x)/(1+ln(d)^2)^2-2 *d^x*x*ln(d)^2*sin(x)/(1+ln(d)^2)^2+d^x*x^2*ln(d)*sin(x)/(1+ln(d)^2)
Time = 0.06 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.58 \[ \int d^x x^2 \sin (x) \, dx=\frac {d^x \left (-\cos (x) \left (-2+x^2-4 x \log (d)+2 \left (3+x^2\right ) \log ^2(d)-4 x \log ^3(d)+x^2 \log ^4(d)\right )+\left (2 x+\left (-6+x^2\right ) \log (d)+2 \left (1+x^2\right ) \log ^3(d)-2 x \log ^4(d)+x^2 \log ^5(d)\right ) \sin (x)\right )}{\left (1+\log ^2(d)\right )^3} \]
(d^x*(-(Cos[x]*(-2 + x^2 - 4*x*Log[d] + 2*(3 + x^2)*Log[d]^2 - 4*x*Log[d]^ 3 + x^2*Log[d]^4)) + (2*x + (-6 + x^2)*Log[d] + 2*(1 + x^2)*Log[d]^3 - 2*x *Log[d]^4 + x^2*Log[d]^5)*Sin[x]))/(1 + Log[d]^2)^3
Time = 0.39 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.00, 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.
| \(\displaystyle \int x^2 d^x \sin (x) \, dx\) |
| \(\Big \downarrow \) 4968 |
| \(\displaystyle -2 \int -x \left (\frac {d^x \cos (x)}{\log ^2(d)+1}-\frac {d^x \log (d) \sin (x)}{\log ^2(d)+1}\right )dx+\frac {x^2 d^x \log (d) \sin (x)}{\log ^2(d)+1}-\frac {x^2 d^x \cos (x)}{\log ^2(d)+1}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 2 \int x \left (\frac {d^x \cos (x)}{\log ^2(d)+1}-\frac {d^x \log (d) \sin (x)}{\log ^2(d)+1}\right )dx+\frac {x^2 d^x \log (d) \sin (x)}{\log ^2(d)+1}-\frac {x^2 d^x \cos (x)}{\log ^2(d)+1}\) |
| \(\Big \downarrow \) 2010 |
| \(\displaystyle 2 \int \left (\frac {d^x x \cos (x)}{\log ^2(d)+1}-\frac {d^x x \log (d) \sin (x)}{\log ^2(d)+1}\right )dx+\frac {x^2 d^x \log (d) \sin (x)}{\log ^2(d)+1}-\frac {x^2 d^x \cos (x)}{\log ^2(d)+1}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {x^2 d^x \log (d) \sin (x)}{\log ^2(d)+1}-\frac {x^2 d^x \cos (x)}{\log ^2(d)+1}+2 \left (-\frac {x d^x \log ^2(d) \sin (x)}{\left (\log ^2(d)+1\right )^2}+\frac {x d^x \sin (x)}{\left (\log ^2(d)+1\right )^2}-\frac {3 d^x \log (d) \sin (x)}{\left (\log ^2(d)+1\right )^3}+\frac {d^x \log ^3(d) \sin (x)}{\left (\log ^2(d)+1\right )^3}+\frac {2 x d^x \log (d) \cos (x)}{\left (\log ^2(d)+1\right )^2}-\frac {3 d^x \log ^2(d) \cos (x)}{\left (\log ^2(d)+1\right )^3}+\frac {d^x \cos (x)}{\left (\log ^2(d)+1\right )^3}\right )\) |
-((d^x*x^2*Cos[x])/(1 + Log[d]^2)) + (d^x*x^2*Log[d]*Sin[x])/(1 + Log[d]^2 ) + 2*((d^x*Cos[x])/(1 + Log[d]^2)^3 - (3*d^x*Cos[x]*Log[d]^2)/(1 + Log[d] ^2)^3 + (2*d^x*x*Cos[x]*Log[d])/(1 + Log[d]^2)^2 - (3*d^x*Log[d]*Sin[x])/( 1 + Log[d]^2)^3 + (d^x*Log[d]^3*Sin[x])/(1 + Log[d]^2)^3 + (d^x*x*Sin[x])/ (1 + Log[d]^2)^2 - (d^x*x*Log[d]^2*Sin[x])/(1 + Log[d]^2)^2)
3.2.38.3.1 Defintions of rubi rules used
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 = 0.29 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.63
| method | result | size |
| risch | \(-\frac {i \left (2+\ln \left (d \right )^{2} x^{2}+2 i \ln \left (d \right ) x^{2}-x^{2}-2 x \ln \left (d \right )-2 i x \right ) d^{x} {\mathrm e}^{i x}}{2 \left (\ln \left (d \right )+i\right )^{3}}+\frac {i \left (2-2 x \ln \left (d \right )+2 i x +\ln \left (d \right )^{2} x^{2}-2 i \ln \left (d \right ) x^{2}-x^{2}\right ) d^{x} {\mathrm e}^{-i x}}{2 \left (\ln \left (d \right )-i\right )^{3}}\) | \(102\) |
| parallelrisch | \(\frac {d^{x} \left (\ln \left (d \right )^{5} x^{2} \sin \left (x \right )+\left (-x^{2} \cos \left (x \right )-2 x \sin \left (x \right )\right ) \ln \left (d \right )^{4}+\left (2 x^{2} \sin \left (x \right )+4 x \cos \left (x \right )+2 \sin \left (x \right )\right ) \ln \left (d \right )^{3}+\left (-2 x^{2}-6\right ) \cos \left (x \right ) \ln \left (d \right )^{2}+\left (x^{2} \sin \left (x \right )+4 x \cos \left (x \right )-6 \sin \left (x \right )\right ) \ln \left (d \right )-x^{2} \cos \left (x \right )+2 x \sin \left (x \right )+2 \cos \left (x \right )\right )}{\left (1+\ln \left (d \right )^{2}\right )^{3}}\) | \(113\) |
| norman | \(\frac {\frac {x^{2} {\mathrm e}^{x \ln \left (d \right )} \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{1+\ln \left (d \right )^{2}}-\frac {x^{2} {\mathrm e}^{x \ln \left (d \right )}}{1+\ln \left (d \right )^{2}}-\frac {2 \left (3 \ln \left (d \right )^{2}-1\right ) {\mathrm e}^{x \ln \left (d \right )}}{\left (1+\ln \left (d \right )^{2}\right )^{3}}+\frac {4 \ln \left (d \right ) x \,{\mathrm e}^{x \ln \left (d \right )}}{\left (1+\ln \left (d \right )^{2}\right )^{2}}+\frac {2 \left (3 \ln \left (d \right )^{2}-1\right ) {\mathrm e}^{x \ln \left (d \right )} \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{\left (1+\ln \left (d \right )^{2}\right )^{3}}-\frac {4 \ln \left (d \right ) x \,{\mathrm e}^{x \ln \left (d \right )} \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{\left (1+\ln \left (d \right )^{2}\right )^{2}}-\frac {4 \left (\ln \left (d \right )^{2}-1\right ) x \,{\mathrm e}^{x \ln \left (d \right )} \tan \left (\frac {x}{2}\right )}{\left (1+\ln \left (d \right )^{2}\right )^{2}}+\frac {2 \ln \left (d \right ) x^{2} {\mathrm e}^{x \ln \left (d \right )} \tan \left (\frac {x}{2}\right )}{1+\ln \left (d \right )^{2}}+\frac {4 \ln \left (d \right ) \left (\ln \left (d \right )^{2}-3\right ) {\mathrm e}^{x \ln \left (d \right )} \tan \left (\frac {x}{2}\right )}{\left (1+\ln \left (d \right )^{2}\right )^{3}}}{1+\tan ^{2}\left (\frac {x}{2}\right )}\) | \(225\) |
-1/2*I*(2+ln(d)^2*x^2+2*I*ln(d)*x^2-x^2-2*x*ln(d)-2*I*x)*d^x/(ln(d)+I)^3*e xp(I*x)+1/2*I*(2-2*x*ln(d)+2*I*x+ln(d)^2*x^2-2*I*ln(d)*x^2-x^2)*d^x/(ln(d) -I)^3*exp(-I*x)
Time = 0.26 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.71 \[ \int d^x x^2 \sin (x) \, dx=-\frac {{\left (x^{2} \cos \left (x\right ) \log \left (d\right )^{4} - 4 \, x \cos \left (x\right ) \log \left (d\right )^{3} + 2 \, {\left (x^{2} + 3\right )} \cos \left (x\right ) \log \left (d\right )^{2} - 4 \, x \cos \left (x\right ) \log \left (d\right ) + {\left (x^{2} - 2\right )} \cos \left (x\right ) - {\left (x^{2} \log \left (d\right )^{5} - 2 \, x \log \left (d\right )^{4} + 2 \, {\left (x^{2} + 1\right )} \log \left (d\right )^{3} + {\left (x^{2} - 6\right )} \log \left (d\right ) + 2 \, x\right )} \sin \left (x\right )\right )} d^{x}}{\log \left (d\right )^{6} + 3 \, \log \left (d\right )^{4} + 3 \, \log \left (d\right )^{2} + 1} \]
-(x^2*cos(x)*log(d)^4 - 4*x*cos(x)*log(d)^3 + 2*(x^2 + 3)*cos(x)*log(d)^2 - 4*x*cos(x)*log(d) + (x^2 - 2)*cos(x) - (x^2*log(d)^5 - 2*x*log(d)^4 + 2* (x^2 + 1)*log(d)^3 + (x^2 - 6)*log(d) + 2*x)*sin(x))*d^x/(log(d)^6 + 3*log (d)^4 + 3*log(d)^2 + 1)
Result contains complex when optimal does not.
Time = 1.28 (sec) , antiderivative size = 665, normalized size of antiderivative = 4.10 \[ \int d^x x^2 \sin (x) \, dx =\text {Too large to display} \]
Piecewise((x**3*exp(-I*x)*sin(x)/6 - I*x**3*exp(-I*x)*cos(x)/6 + I*x**2*ex p(-I*x)*sin(x)/4 - x**2*exp(-I*x)*cos(x)/4 + x*exp(-I*x)*sin(x)/4 + I*x*ex p(-I*x)*cos(x)/4 + exp(-I*x)*cos(x)/4, Eq(d, exp(-I))), (x**3*exp(I*x)*sin (x)/6 + I*x**3*exp(I*x)*cos(x)/6 - I*x**2*exp(I*x)*sin(x)/4 - x**2*exp(I*x )*cos(x)/4 + x*exp(I*x)*sin(x)/4 - I*x*exp(I*x)*cos(x)/4 + exp(I*x)*cos(x) /4, Eq(d, exp(I))), (d**x*x**2*log(d)**5*sin(x)/(log(d)**6 + 3*log(d)**4 + 3*log(d)**2 + 1) - d**x*x**2*log(d)**4*cos(x)/(log(d)**6 + 3*log(d)**4 + 3*log(d)**2 + 1) + 2*d**x*x**2*log(d)**3*sin(x)/(log(d)**6 + 3*log(d)**4 + 3*log(d)**2 + 1) - 2*d**x*x**2*log(d)**2*cos(x)/(log(d)**6 + 3*log(d)**4 + 3*log(d)**2 + 1) + d**x*x**2*log(d)*sin(x)/(log(d)**6 + 3*log(d)**4 + 3* log(d)**2 + 1) - d**x*x**2*cos(x)/(log(d)**6 + 3*log(d)**4 + 3*log(d)**2 + 1) - 2*d**x*x*log(d)**4*sin(x)/(log(d)**6 + 3*log(d)**4 + 3*log(d)**2 + 1 ) + 4*d**x*x*log(d)**3*cos(x)/(log(d)**6 + 3*log(d)**4 + 3*log(d)**2 + 1) + 4*d**x*x*log(d)*cos(x)/(log(d)**6 + 3*log(d)**4 + 3*log(d)**2 + 1) + 2*d **x*x*sin(x)/(log(d)**6 + 3*log(d)**4 + 3*log(d)**2 + 1) + 2*d**x*log(d)** 3*sin(x)/(log(d)**6 + 3*log(d)**4 + 3*log(d)**2 + 1) - 6*d**x*log(d)**2*co s(x)/(log(d)**6 + 3*log(d)**4 + 3*log(d)**2 + 1) - 6*d**x*log(d)*sin(x)/(l og(d)**6 + 3*log(d)**4 + 3*log(d)**2 + 1) + 2*d**x*cos(x)/(log(d)**6 + 3*l og(d)**4 + 3*log(d)**2 + 1), True))
Time = 0.20 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.66 \[ \int d^x x^2 \sin (x) \, dx=-\frac {{\left ({\left (\log \left (d\right )^{4} + 2 \, \log \left (d\right )^{2} + 1\right )} x^{2} - 4 \, {\left (\log \left (d\right )^{3} + \log \left (d\right )\right )} x + 6 \, \log \left (d\right )^{2} - 2\right )} d^{x} \cos \left (x\right ) - {\left ({\left (\log \left (d\right )^{5} + 2 \, \log \left (d\right )^{3} + \log \left (d\right )\right )} x^{2} + 2 \, \log \left (d\right )^{3} - 2 \, {\left (\log \left (d\right )^{4} - 1\right )} x - 6 \, \log \left (d\right )\right )} d^{x} \sin \left (x\right )}{\log \left (d\right )^{6} + 3 \, \log \left (d\right )^{4} + 3 \, \log \left (d\right )^{2} + 1} \]
-(((log(d)^4 + 2*log(d)^2 + 1)*x^2 - 4*(log(d)^3 + log(d))*x + 6*log(d)^2 - 2)*d^x*cos(x) - ((log(d)^5 + 2*log(d)^3 + log(d))*x^2 + 2*log(d)^3 - 2*( log(d)^4 - 1)*x - 6*log(d))*d^x*sin(x))/(log(d)^6 + 3*log(d)^4 + 3*log(d)^ 2 + 1)
Result contains complex when optimal does not.
Time = 0.33 (sec) , antiderivative size = 2631, normalized size of antiderivative = 16.24 \[ \int d^x x^2 \sin (x) \, dx=\text {Too large to display} \]
-1/2*(((3*pi - pi^3*sgn(d) + 3*pi*log(abs(d))^2*sgn(d) + pi^3 - 3*pi*log(a bs(d))^2 + 3*pi^2*sgn(d) - 3*pi^2 + 6*log(abs(d))^2 - 3*pi*sgn(d) - 2)*(pi ^2*x^2*sgn(d) - pi^2*x^2 + 2*x^2*log(abs(d))^2 - 2*pi*x^2*sgn(d) + 2*pi*x^ 2 - 2*x^2 - 4*x*log(abs(d)) + 4)/((3*pi - pi^3*sgn(d) + 3*pi*log(abs(d))^2 *sgn(d) + pi^3 - 3*pi*log(abs(d))^2 + 3*pi^2*sgn(d) - 3*pi^2 + 6*log(abs(d ))^2 - 3*pi*sgn(d) - 2)^2 + (3*pi^2*log(abs(d))*sgn(d) - 3*pi^2*log(abs(d) ) + 2*log(abs(d))^3 - 6*pi*log(abs(d))*sgn(d) + 6*pi*log(abs(d)) - 6*log(a bs(d)))^2) - 2*(pi*x^2*log(abs(d))*sgn(d) - pi*x^2*log(abs(d)) + 2*x^2*log (abs(d)) - pi*x*sgn(d) + pi*x - 2*x)*(3*pi^2*log(abs(d))*sgn(d) - 3*pi^2*l og(abs(d)) + 2*log(abs(d))^3 - 6*pi*log(abs(d))*sgn(d) + 6*pi*log(abs(d)) - 6*log(abs(d)))/((3*pi - pi^3*sgn(d) + 3*pi*log(abs(d))^2*sgn(d) + pi^3 - 3*pi*log(abs(d))^2 + 3*pi^2*sgn(d) - 3*pi^2 + 6*log(abs(d))^2 - 3*pi*sgn( d) - 2)^2 + (3*pi^2*log(abs(d))*sgn(d) - 3*pi^2*log(abs(d)) + 2*log(abs(d) )^3 - 6*pi*log(abs(d))*sgn(d) + 6*pi*log(abs(d)) - 6*log(abs(d)))^2))*cos( 1/2*pi*x*sgn(d) - 1/2*pi*x + x) - (2*(3*pi - pi^3*sgn(d) + 3*pi*log(abs(d) )^2*sgn(d) + pi^3 - 3*pi*log(abs(d))^2 + 3*pi^2*sgn(d) - 3*pi^2 + 6*log(ab s(d))^2 - 3*pi*sgn(d) - 2)*(pi*x^2*log(abs(d))*sgn(d) - pi*x^2*log(abs(d)) + 2*x^2*log(abs(d)) - pi*x*sgn(d) + pi*x - 2*x)/((3*pi - pi^3*sgn(d) + 3* pi*log(abs(d))^2*sgn(d) + pi^3 - 3*pi*log(abs(d))^2 + 3*pi^2*sgn(d) - 3*pi ^2 + 6*log(abs(d))^2 - 3*pi*sgn(d) - 2)^2 + (3*pi^2*log(abs(d))*sgn(d) ...
Time = 0.39 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.82 \[ \int d^x x^2 \sin (x) \, dx=\frac {d^x\,\left (2\,\cos \left (x\right )-x^2\,\cos \left (x\right )+2\,x\,\sin \left (x\right )\right )+d^x\,{\ln \left (d\right )}^3\,\left (2\,\sin \left (x\right )+2\,x^2\,\sin \left (x\right )+4\,x\,\cos \left (x\right )\right )-d^x\,{\ln \left (d\right )}^2\,\left (6\,\cos \left (x\right )+2\,x^2\,\cos \left (x\right )\right )+d^x\,\ln \left (d\right )\,\left (x^2\,\sin \left (x\right )-6\,\sin \left (x\right )+4\,x\,\cos \left (x\right )\right )-d^x\,{\ln \left (d\right )}^4\,\left (x^2\,\cos \left (x\right )+2\,x\,\sin \left (x\right )\right )+d^x\,x^2\,{\ln \left (d\right )}^5\,\sin \left (x\right )}{{\left ({\ln \left (d\right )}^2+1\right )}^3} \]
(d^x*(2*cos(x) - x^2*cos(x) + 2*x*sin(x)) + d^x*log(d)^3*(2*sin(x) + 2*x^2 *sin(x) + 4*x*cos(x)) - d^x*log(d)^2*(6*cos(x) + 2*x^2*cos(x)) + d^x*log(d )*(x^2*sin(x) - 6*sin(x) + 4*x*cos(x)) - d^x*log(d)^4*(x^2*cos(x) + 2*x*si n(x)) + d^x*x^2*log(d)^5*sin(x))/(log(d)^2 + 1)^3
Time = 0.00 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.86 \[ \int d^x x^2 \sin (x) \, dx=\frac {d^{x} \left (-\cos \left (x \right ) \mathrm {log}\left (d \right )^{4} x^{2}+4 \cos \left (x \right ) \mathrm {log}\left (d \right )^{3} x -2 \cos \left (x \right ) \mathrm {log}\left (d \right )^{2} x^{2}-6 \cos \left (x \right ) \mathrm {log}\left (d \right )^{2}+4 \cos \left (x \right ) \mathrm {log}\left (d \right ) x -\cos \left (x \right ) x^{2}+2 \cos \left (x \right )+\mathrm {log}\left (d \right )^{5} \sin \left (x \right ) x^{2}-2 \mathrm {log}\left (d \right )^{4} \sin \left (x \right ) x +2 \mathrm {log}\left (d \right )^{3} \sin \left (x \right ) x^{2}+2 \mathrm {log}\left (d \right )^{3} \sin \left (x \right )+\mathrm {log}\left (d \right ) \sin \left (x \right ) x^{2}-6 \,\mathrm {log}\left (d \right ) \sin \left (x \right )+2 \sin \left (x \right ) x \right )}{\mathrm {log}\left (d \right )^{6}+3 \mathrm {log}\left (d \right )^{4}+3 \mathrm {log}\left (d \right )^{2}+1} \]
(d**x*( - cos(x)*log(d)**4*x**2 + 4*cos(x)*log(d)**3*x - 2*cos(x)*log(d)** 2*x**2 - 6*cos(x)*log(d)**2 + 4*cos(x)*log(d)*x - cos(x)*x**2 + 2*cos(x) + log(d)**5*sin(x)*x**2 - 2*log(d)**4*sin(x)*x + 2*log(d)**3*sin(x)*x**2 + 2*log(d)**3*sin(x) + log(d)*sin(x)*x**2 - 6*log(d)*sin(x) + 2*sin(x)*x))/( log(d)**6 + 3*log(d)**4 + 3*log(d)**2 + 1)