3.138 \(\int d^x x^2 \sin (x) \, dx\)

Optimal. Leaf size=162 \[ \frac{x^2 d^x \log (d) \sin (x)}{\log ^2(d)+1}-\frac{x^2 d^x \cos (x)}{\log ^2(d)+1}-\frac{2 x d^x \log ^2(d) \sin (x)}{\left (\log ^2(d)+1\right )^2}+\frac{2 x d^x \sin (x)}{\left (\log ^2(d)+1\right )^2}-\frac{6 d^x \log (d) \sin (x)}{\left (\log ^2(d)+1\right )^3}+\frac{2 d^x \log ^3(d) \sin (x)}{\left (\log ^2(d)+1\right )^3}+\frac{4 x d^x \log (d) \cos (x)}{\left (\log ^2(d)+1\right )^2}-\frac{6 d^x \log ^2(d) \cos (x)}{\left (\log ^2(d)+1\right )^3}+\frac{2 d^x \cos (x)}{\left (\log ^2(d)+1\right )^3} \]

[Out]

(2*d^x*Cos[x])/(1 + Log[d]^2)^3 - (6*d^x*Cos[x]*Log[d]^2)/(1 + Log[d]^2)^3 + (4*
d^x*x*Cos[x]*Log[d])/(1 + Log[d]^2)^2 - (d^x*x^2*Cos[x])/(1 + Log[d]^2) - (6*d^x
*Log[d]*Sin[x])/(1 + Log[d]^2)^3 + (2*d^x*Log[d]^3*Sin[x])/(1 + Log[d]^2)^3 + (2
*d^x*x*Sin[x])/(1 + Log[d]^2)^2 - (2*d^x*x*Log[d]^2*Sin[x])/(1 + Log[d]^2)^2 + (
d^x*x^2*Log[d]*Sin[x])/(1 + Log[d]^2)

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Rubi [A]  time = 0.282436, antiderivative size = 162, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 5, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.556 \[ \frac{x^2 d^x \log (d) \sin (x)}{\log ^2(d)+1}-\frac{x^2 d^x \cos (x)}{\log ^2(d)+1}-\frac{2 x d^x \log ^2(d) \sin (x)}{\left (\log ^2(d)+1\right )^2}+\frac{2 x d^x \sin (x)}{\left (\log ^2(d)+1\right )^2}-\frac{6 d^x \log (d) \sin (x)}{\left (\log ^2(d)+1\right )^3}+\frac{2 d^x \log ^3(d) \sin (x)}{\left (\log ^2(d)+1\right )^3}+\frac{4 x d^x \log (d) \cos (x)}{\left (\log ^2(d)+1\right )^2}-\frac{6 d^x \log ^2(d) \cos (x)}{\left (\log ^2(d)+1\right )^3}+\frac{2 d^x \cos (x)}{\left (\log ^2(d)+1\right )^3} \]

Antiderivative was successfully verified.

[In]  Int[d^x*x^2*Sin[x],x]

[Out]

(2*d^x*Cos[x])/(1 + Log[d]^2)^3 - (6*d^x*Cos[x]*Log[d]^2)/(1 + Log[d]^2)^3 + (4*
d^x*x*Cos[x]*Log[d])/(1 + Log[d]^2)^2 - (d^x*x^2*Cos[x])/(1 + Log[d]^2) - (6*d^x
*Log[d]*Sin[x])/(1 + Log[d]^2)^3 + (2*d^x*Log[d]^3*Sin[x])/(1 + Log[d]^2)^3 + (2
*d^x*x*Sin[x])/(1 + Log[d]^2)^2 - (2*d^x*x*Log[d]^2*Sin[x])/(1 + Log[d]^2)^2 + (
d^x*x^2*Log[d]*Sin[x])/(1 + Log[d]^2)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{d^{x} x^{2} \log{\left (d \right )} \sin{\left (x \right )}}{\log{\left (d \right )}^{2} + 1} - \frac{d^{x} x^{2} \cos{\left (x \right )}}{\log{\left (d \right )}^{2} + 1} - 2 \int x \left (\frac{d^{x} \log{\left (d \right )} \sin{\left (x \right )}}{\log{\left (d \right )}^{2} + 1} - \frac{d^{x} \cos{\left (x \right )}}{\log{\left (d \right )}^{2} + 1}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(d**x*x**2*sin(x),x)

[Out]

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

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Mathematica [A]  time = 0.110228, size = 94, normalized size = 0.58 \[ \frac{d^x \left (\sin (x) \left (x^2 \log ^5(d)+2 \left (x^2+1\right ) \log ^3(d)+\left (x^2-6\right ) \log (d)-2 x \log ^4(d)+2 x\right )-\cos (x) \left (x^2 \log ^4(d)+2 \left (x^2+3\right ) \log ^2(d)-4 x \log ^3(d)-4 x \log (d)+x^2-2\right )\right )}{\left (\log ^2(d)+1\right )^3} \]

Antiderivative was successfully verified.

[In]  Integrate[d^x*x^2*Sin[x],x]

[Out]

(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

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Maple [A]  time = 0.024, size = 225, normalized size = 1.4 \[{1 \left ({\frac{{x}^{2}{{\rm e}^{x\ln \left ( d \right ) }}}{1+ \left ( \ln \left ( d \right ) \right ) ^{2}} \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}}-{\frac{{x}^{2}{{\rm e}^{x\ln \left ( d \right ) }}}{1+ \left ( \ln \left ( d \right ) \right ) ^{2}}}-2\,{\frac{ \left ( 3\, \left ( \ln \left ( d \right ) \right ) ^{2}-1 \right ){{\rm e}^{x\ln \left ( d \right ) }}}{ \left ( 1+ \left ( \ln \left ( d \right ) \right ) ^{2} \right ) ^{3}}}+4\,{\frac{x\ln \left ( d \right ){{\rm e}^{x\ln \left ( d \right ) }}}{ \left ( 1+ \left ( \ln \left ( d \right ) \right ) ^{2} \right ) ^{2}}}+2\,{\frac{ \left ( 3\, \left ( \ln \left ( d \right ) \right ) ^{2}-1 \right ){{\rm e}^{x\ln \left ( d \right ) }} \left ( \tan \left ( x/2 \right ) \right ) ^{2}}{ \left ( 1+ \left ( \ln \left ( d \right ) \right ) ^{2} \right ) ^{3}}}-4\,{\frac{x\ln \left ( d \right ){{\rm e}^{x\ln \left ( d \right ) }} \left ( \tan \left ( x/2 \right ) \right ) ^{2}}{ \left ( 1+ \left ( \ln \left ( d \right ) \right ) ^{2} \right ) ^{2}}}-4\,{\frac{ \left ( \left ( \ln \left ( d \right ) \right ) ^{2}-1 \right ) x{{\rm e}^{x\ln \left ( d \right ) }}\tan \left ( x/2 \right ) }{ \left ( 1+ \left ( \ln \left ( d \right ) \right ) ^{2} \right ) ^{2}}}+2\,{\frac{\ln \left ( d \right ){x}^{2}{{\rm e}^{x\ln \left ( d \right ) }}\tan \left ( x/2 \right ) }{1+ \left ( \ln \left ( d \right ) \right ) ^{2}}}+4\,{\frac{\ln \left ( d \right ) \left ( \left ( \ln \left ( d \right ) \right ) ^{2}-3 \right ){{\rm e}^{x\ln \left ( d \right ) }}\tan \left ( x/2 \right ) }{ \left ( 1+ \left ( \ln \left ( d \right ) \right ) ^{2} \right ) ^{3}}} \right ) \left ( 1+ \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2} \right ) ^{-1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(d^x*x^2*sin(x),x)

[Out]

(1/(1+ln(d)^2)*x^2*exp(x*ln(d))*tan(1/2*x)^2-1/(1+ln(d)^2)*x^2*exp(x*ln(d))-2*(3
*ln(d)^2-1)/(1+ln(d)^2)^3*exp(x*ln(d))+4/(1+ln(d)^2)^2*ln(d)*x*exp(x*ln(d))+2*(3
*ln(d)^2-1)/(1+ln(d)^2)^3*exp(x*ln(d))*tan(1/2*x)^2-4/(1+ln(d)^2)^2*ln(d)*x*exp(
x*ln(d))*tan(1/2*x)^2-4*(ln(d)^2-1)/(1+ln(d)^2)^2*x*exp(x*ln(d))*tan(1/2*x)+2*ln
(d)/(1+ln(d)^2)*x^2*exp(x*ln(d))*tan(1/2*x)+4*ln(d)*(ln(d)^2-3)/(1+ln(d)^2)^3*ex
p(x*ln(d))*tan(1/2*x))/(1+tan(1/2*x)^2)

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Maxima [A]  time = 1.40864, size = 144, normalized size = 0.89 \[ -\frac{{\left ({\left (\log \left (d\right )^{4} + 2 \, \log \left (d\right )^{2} + 1\right )} x^{2} - 4 \,{\left (\log \left (d\right )^{3} + \log \left (d\right )\right )} x + 6 \, \log \left (d\right )^{2} - 2\right )} d^{x} \cos \left (x\right ) -{\left ({\left (\log \left (d\right )^{5} + 2 \, \log \left (d\right )^{3} + \log \left (d\right )\right )} x^{2} + 2 \, \log \left (d\right )^{3} - 2 \,{\left (\log \left (d\right )^{4} - 1\right )} x - 6 \, \log \left (d\right )\right )} d^{x} \sin \left (x\right )}{\log \left (d\right )^{6} + 3 \, \log \left (d\right )^{4} + 3 \, \log \left (d\right )^{2} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(d^x*x^2*sin(x),x, algorithm="maxima")

[Out]

-(((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)

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Fricas [A]  time = 0.242909, size = 155, normalized size = 0.96 \[ -\frac{{\left (x^{2} \cos \left (x\right ) \log \left (d\right )^{4} - 4 \, x \cos \left (x\right ) \log \left (d\right )^{3} + 2 \,{\left (x^{2} + 3\right )} \cos \left (x\right ) \log \left (d\right )^{2} - 4 \, x \cos \left (x\right ) \log \left (d\right ) +{\left (x^{2} - 2\right )} \cos \left (x\right ) -{\left (x^{2} \log \left (d\right )^{5} - 2 \, x \log \left (d\right )^{4} + 2 \,{\left (x^{2} + 1\right )} \log \left (d\right )^{3} +{\left (x^{2} - 6\right )} \log \left (d\right ) + 2 \, x\right )} \sin \left (x\right )\right )} d^{x}}{\log \left (d\right )^{6} + 3 \, \log \left (d\right )^{4} + 3 \, \log \left (d\right )^{2} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(d^x*x^2*sin(x),x, algorithm="fricas")

[Out]

-(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)*lo
g(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)

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Sympy [A]  time = 16.6294, size = 665, normalized size = 4.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(d**x*x**2*sin(x),x)

[Out]

Piecewise((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))), (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*s
in(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*cos(x)/(log(d)**6
+ 3*log(d)**4 + 3*log(d)**2 + 1) - 6*d**x*log(d)*sin(x)/(log(d)**6 + 3*log(d)**4
+ 3*log(d)**2 + 1) + 2*d**x*cos(x)/(log(d)**6 + 3*log(d)**4 + 3*log(d)**2 + 1),
True))

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GIAC/XCAS [A]  time = 0.229414, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(d^x*x^2*sin(x),x, algorithm="giac")

[Out]

Done