3.136 \(\int d^x x \sin (x) \, dx\)

Optimal. Leaf size=84 \[ \frac{x d^x \log (d) \sin (x)}{\log ^2(d)+1}-\frac{d^x \log ^2(d) \sin (x)}{\left (\log ^2(d)+1\right )^2}+\frac{d^x \sin (x)}{\left (\log ^2(d)+1\right )^2}-\frac{x d^x \cos (x)}{\log ^2(d)+1}+\frac{2 d^x \log (d) \cos (x)}{\left (\log ^2(d)+1\right )^2} \]

[Out]

(2*d^x*Cos[x]*Log[d])/(1 + Log[d]^2)^2 - (d^x*x*Cos[x])/(1 + Log[d]^2) + (d^x*Si
n[x])/(1 + Log[d]^2)^2 - (d^x*Log[d]^2*Sin[x])/(1 + Log[d]^2)^2 + (d^x*x*Log[d]*
Sin[x])/(1 + Log[d]^2)

_______________________________________________________________________________________

Rubi [A]  time = 0.0826123, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429 \[ \frac{x d^x \log (d) \sin (x)}{\log ^2(d)+1}-\frac{d^x \log ^2(d) \sin (x)}{\left (\log ^2(d)+1\right )^2}+\frac{d^x \sin (x)}{\left (\log ^2(d)+1\right )^2}-\frac{x d^x \cos (x)}{\log ^2(d)+1}+\frac{2 d^x \log (d) \cos (x)}{\left (\log ^2(d)+1\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[d^x*x*Sin[x],x]

[Out]

(2*d^x*Cos[x]*Log[d])/(1 + Log[d]^2)^2 - (d^x*x*Cos[x])/(1 + Log[d]^2) + (d^x*Si
n[x])/(1 + Log[d]^2)^2 - (d^x*Log[d]^2*Sin[x])/(1 + Log[d]^2)^2 + (d^x*x*Log[d]*
Sin[x])/(1 + Log[d]^2)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 6.05047, size = 88, normalized size = 1.05 \[ \frac{d^{x} x \log{\left (d \right )} \sin{\left (x \right )}}{\log{\left (d \right )}^{2} + 1} - \frac{d^{x} x \cos{\left (x \right )}}{\log{\left (d \right )}^{2} + 1} - \frac{d^{x} \log{\left (d \right )}^{2} \sin{\left (x \right )}}{\left (\log{\left (d \right )}^{2} + 1\right )^{2}} + \frac{2 d^{x} \log{\left (d \right )} \cos{\left (x \right )}}{\left (\log{\left (d \right )}^{2} + 1\right )^{2}} + \frac{d^{x} \sin{\left (x \right )}}{\left (\log{\left (d \right )}^{2} + 1\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(d**x*x*sin(x),x)

[Out]

d**x*x*log(d)*sin(x)/(log(d)**2 + 1) - d**x*x*cos(x)/(log(d)**2 + 1) - d**x*log(
d)**2*sin(x)/(log(d)**2 + 1)**2 + 2*d**x*log(d)*cos(x)/(log(d)**2 + 1)**2 + d**x
*sin(x)/(log(d)**2 + 1)**2

_______________________________________________________________________________________

Mathematica [A]  time = 0.06012, size = 50, normalized size = 0.6 \[ \frac{d^x \left (\sin (x) \left (x \log ^3(d)+x \log (d)-\log ^2(d)+1\right )-\cos (x) \left (x \log ^2(d)-2 \log (d)+x\right )\right )}{\left (\log ^2(d)+1\right )^2} \]

Antiderivative was successfully verified.

[In]  Integrate[d^x*x*Sin[x],x]

[Out]

(d^x*(-(Cos[x]*(x - 2*Log[d] + x*Log[d]^2)) + (1 + x*Log[d] - Log[d]^2 + x*Log[d
]^3)*Sin[x]))/(1 + Log[d]^2)^2

_______________________________________________________________________________________

Maple [A]  time = 0.016, size = 137, normalized size = 1.6 \[{1 \left ({\frac{x{{\rm e}^{x\ln \left ( d \right ) }}}{1+ \left ( \ln \left ( d \right ) \right ) ^{2}} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}}+2\,{\frac{\ln \left ( d \right ){{\rm e}^{x\ln \left ( d \right ) }}}{ \left ( 1+ \left ( \ln \left ( d \right ) \right ) ^{2} \right ) ^{2}}}-{\frac{x{{\rm e}^{x\ln \left ( d \right ) }}}{1+ \left ( \ln \left ( d \right ) \right ) ^{2}}}-2\,{\frac{\ln \left ( d \right ){{\rm e}^{x\ln \left ( d \right ) }} \left ( \tan \left ( x/2 \right ) \right ) ^{2}}{ \left ( 1+ \left ( \ln \left ( d \right ) \right ) ^{2} \right ) ^{2}}}-2\,{\frac{ \left ( \left ( \ln \left ( d \right ) \right ) ^{2}-1 \right ){{\rm e}^{x\ln \left ( d \right ) }}\tan \left ( x/2 \right ) }{ \left ( 1+ \left ( \ln \left ( d \right ) \right ) ^{2} \right ) ^{2}}}+2\,{\frac{x\ln \left ( d \right ){{\rm e}^{x\ln \left ( d \right ) }}\tan \left ( x/2 \right ) }{1+ \left ( \ln \left ( d \right ) \right ) ^{2}}} \right ) \left ( 1+ \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2} \right ) ^{-1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(d^x*x*sin(x),x)

[Out]

(1/(1+ln(d)^2)*x*exp(x*ln(d))*tan(1/2*x)^2+2/(1+ln(d)^2)^2*ln(d)*exp(x*ln(d))-1/
(1+ln(d)^2)*x*exp(x*ln(d))-2/(1+ln(d)^2)^2*ln(d)*exp(x*ln(d))*tan(1/2*x)^2-2*(ln
(d)^2-1)/(1+ln(d)^2)^2*exp(x*ln(d))*tan(1/2*x)+2*ln(d)/(1+ln(d)^2)*x*exp(x*ln(d)
)*tan(1/2*x))/(1+tan(1/2*x)^2)

_______________________________________________________________________________________

Maxima [A]  time = 1.44978, size = 81, normalized size = 0.96 \[ -\frac{{\left ({\left (\log \left (d\right )^{2} + 1\right )} x - 2 \, \log \left (d\right )\right )} d^{x} \cos \left (x\right ) -{\left ({\left (\log \left (d\right )^{3} + \log \left (d\right )\right )} x - \log \left (d\right )^{2} + 1\right )} d^{x} \sin \left (x\right )}{\log \left (d\right )^{4} + 2 \, \log \left (d\right )^{2} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(d^x*x*sin(x),x, algorithm="maxima")

[Out]

-(((log(d)^2 + 1)*x - 2*log(d))*d^x*cos(x) - ((log(d)^3 + log(d))*x - log(d)^2 +
1)*d^x*sin(x))/(log(d)^4 + 2*log(d)^2 + 1)

_______________________________________________________________________________________

Fricas [A]  time = 0.250232, size = 81, normalized size = 0.96 \[ -\frac{{\left (x \cos \left (x\right ) \log \left (d\right )^{2} + x \cos \left (x\right ) - 2 \, \cos \left (x\right ) \log \left (d\right ) -{\left (x \log \left (d\right )^{3} + x \log \left (d\right ) - \log \left (d\right )^{2} + 1\right )} \sin \left (x\right )\right )} d^{x}}{\log \left (d\right )^{4} + 2 \, \log \left (d\right )^{2} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(d^x*x*sin(x),x, algorithm="fricas")

[Out]

-(x*cos(x)*log(d)^2 + x*cos(x) - 2*cos(x)*log(d) - (x*log(d)^3 + x*log(d) - log(
d)^2 + 1)*sin(x))*d^x/(log(d)^4 + 2*log(d)^2 + 1)

_______________________________________________________________________________________

Sympy [A]  time = 5.81825, size = 308, normalized size = 3.67 \[ \begin{cases} \frac{x^{2} e^{- i x} \sin{\left (x \right )}}{4} - \frac{i x^{2} e^{- i x} \cos{\left (x \right )}}{4} + \frac{i x e^{- i x} \sin{\left (x \right )}}{4} - \frac{x e^{- i x} \cos{\left (x \right )}}{4} + \frac{i e^{- i x} \cos{\left (x \right )}}{4} & \text{for}\: d = e^{- i} \\\frac{x^{2} e^{i x} \sin{\left (x \right )}}{4} + \frac{i x^{2} e^{i x} \cos{\left (x \right )}}{4} - \frac{i x e^{i x} \sin{\left (x \right )}}{4} - \frac{x e^{i x} \cos{\left (x \right )}}{4} - \frac{i e^{i x} \cos{\left (x \right )}}{4} & \text{for}\: d = e^{i} \\\frac{d^{x} x \log{\left (d \right )}^{3} \sin{\left (x \right )}}{\log{\left (d \right )}^{4} + 2 \log{\left (d \right )}^{2} + 1} - \frac{d^{x} x \log{\left (d \right )}^{2} \cos{\left (x \right )}}{\log{\left (d \right )}^{4} + 2 \log{\left (d \right )}^{2} + 1} + \frac{d^{x} x \log{\left (d \right )} \sin{\left (x \right )}}{\log{\left (d \right )}^{4} + 2 \log{\left (d \right )}^{2} + 1} - \frac{d^{x} x \cos{\left (x \right )}}{\log{\left (d \right )}^{4} + 2 \log{\left (d \right )}^{2} + 1} - \frac{d^{x} \log{\left (d \right )}^{2} \sin{\left (x \right )}}{\log{\left (d \right )}^{4} + 2 \log{\left (d \right )}^{2} + 1} + \frac{2 d^{x} \log{\left (d \right )} \cos{\left (x \right )}}{\log{\left (d \right )}^{4} + 2 \log{\left (d \right )}^{2} + 1} + \frac{d^{x} \sin{\left (x \right )}}{\log{\left (d \right )}^{4} + 2 \log{\left (d \right )}^{2} + 1} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(d**x*x*sin(x),x)

[Out]

Piecewise((x**2*exp(-I*x)*sin(x)/4 - I*x**2*exp(-I*x)*cos(x)/4 + I*x*exp(-I*x)*s
in(x)/4 - x*exp(-I*x)*cos(x)/4 + I*exp(-I*x)*cos(x)/4, Eq(d, exp(-I))), (x**2*ex
p(I*x)*sin(x)/4 + I*x**2*exp(I*x)*cos(x)/4 - I*x*exp(I*x)*sin(x)/4 - x*exp(I*x)*
cos(x)/4 - I*exp(I*x)*cos(x)/4, Eq(d, exp(I))), (d**x*x*log(d)**3*sin(x)/(log(d)
**4 + 2*log(d)**2 + 1) - d**x*x*log(d)**2*cos(x)/(log(d)**4 + 2*log(d)**2 + 1) +
d**x*x*log(d)*sin(x)/(log(d)**4 + 2*log(d)**2 + 1) - d**x*x*cos(x)/(log(d)**4 +
2*log(d)**2 + 1) - d**x*log(d)**2*sin(x)/(log(d)**4 + 2*log(d)**2 + 1) + 2*d**x
*log(d)*cos(x)/(log(d)**4 + 2*log(d)**2 + 1) + d**x*sin(x)/(log(d)**4 + 2*log(d)
**2 + 1), True))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.220464, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(d^x*x*sin(x),x, algorithm="giac")

[Out]

Done