3.140 \(\int d^x x^3 \sin (x) \, dx\)

Optimal. Leaf size=261 \[ \frac{x^3 d^x \log (d) \sin (x)}{\log ^2(d)+1}-\frac{x^3 d^x \cos (x)}{\log ^2(d)+1}+\frac{3 x^2 d^x \sin (x)}{\left (\log ^2(d)+1\right )^2}-\frac{3 x^2 d^x \log ^2(d) \sin (x)}{\left (\log ^2(d)+1\right )^2}+\frac{6 x^2 d^x \log (d) \cos (x)}{\left (\log ^2(d)+1\right )^2}-\frac{18 x d^x \log (d) \sin (x)}{\left (\log ^2(d)+1\right )^3}+\frac{36 d^x \log ^2(d) \sin (x)}{\left (\log ^2(d)+1\right )^4}-\frac{6 d^x \sin (x)}{\left (\log ^2(d)+1\right )^4}-\frac{6 d^x \log ^4(d) \sin (x)}{\left (\log ^2(d)+1\right )^4}+\frac{6 x d^x \log ^3(d) \sin (x)}{\left (\log ^2(d)+1\right )^3}-\frac{18 x d^x \log ^2(d) \cos (x)}{\left (\log ^2(d)+1\right )^3}+\frac{6 x d^x \cos (x)}{\left (\log ^2(d)+1\right )^3}-\frac{24 d^x \log (d) \cos (x)}{\left (\log ^2(d)+1\right )^4}+\frac{24 d^x \log ^3(d) \cos (x)}{\left (\log ^2(d)+1\right )^4} \]

[Out]

(-24*d^x*Cos[x]*Log[d])/(1 + Log[d]^2)^4 + (24*d^x*Cos[x]*Log[d]^3)/(1 + Log[d]^
2)^4 + (6*d^x*x*Cos[x])/(1 + Log[d]^2)^3 - (18*d^x*x*Cos[x]*Log[d]^2)/(1 + Log[d
]^2)^3 + (6*d^x*x^2*Cos[x]*Log[d])/(1 + Log[d]^2)^2 - (d^x*x^3*Cos[x])/(1 + Log[
d]^2) - (6*d^x*Sin[x])/(1 + Log[d]^2)^4 + (36*d^x*Log[d]^2*Sin[x])/(1 + Log[d]^2
)^4 - (6*d^x*Log[d]^4*Sin[x])/(1 + Log[d]^2)^4 - (18*d^x*x*Log[d]*Sin[x])/(1 + L
og[d]^2)^3 + (6*d^x*x*Log[d]^3*Sin[x])/(1 + Log[d]^2)^3 + (3*d^x*x^2*Sin[x])/(1
+ Log[d]^2)^2 - (3*d^x*x^2*Log[d]^2*Sin[x])/(1 + Log[d]^2)^2 + (d^x*x^3*Log[d]*S
in[x])/(1 + Log[d]^2)

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Rubi [A]  time = 0.679133, antiderivative size = 261, normalized size of antiderivative = 1., number of steps used = 25, number of rules used = 5, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.556 \[ \frac{x^3 d^x \log (d) \sin (x)}{\log ^2(d)+1}-\frac{x^3 d^x \cos (x)}{\log ^2(d)+1}+\frac{3 x^2 d^x \sin (x)}{\left (\log ^2(d)+1\right )^2}-\frac{3 x^2 d^x \log ^2(d) \sin (x)}{\left (\log ^2(d)+1\right )^2}+\frac{6 x^2 d^x \log (d) \cos (x)}{\left (\log ^2(d)+1\right )^2}-\frac{18 x d^x \log (d) \sin (x)}{\left (\log ^2(d)+1\right )^3}+\frac{36 d^x \log ^2(d) \sin (x)}{\left (\log ^2(d)+1\right )^4}-\frac{6 d^x \sin (x)}{\left (\log ^2(d)+1\right )^4}-\frac{6 d^x \log ^4(d) \sin (x)}{\left (\log ^2(d)+1\right )^4}+\frac{6 x d^x \log ^3(d) \sin (x)}{\left (\log ^2(d)+1\right )^3}-\frac{18 x d^x \log ^2(d) \cos (x)}{\left (\log ^2(d)+1\right )^3}+\frac{6 x d^x \cos (x)}{\left (\log ^2(d)+1\right )^3}-\frac{24 d^x \log (d) \cos (x)}{\left (\log ^2(d)+1\right )^4}+\frac{24 d^x \log ^3(d) \cos (x)}{\left (\log ^2(d)+1\right )^4} \]

Antiderivative was successfully verified.

[In]  Int[d^x*x^3*Sin[x],x]

[Out]

(-24*d^x*Cos[x]*Log[d])/(1 + Log[d]^2)^4 + (24*d^x*Cos[x]*Log[d]^3)/(1 + Log[d]^
2)^4 + (6*d^x*x*Cos[x])/(1 + Log[d]^2)^3 - (18*d^x*x*Cos[x]*Log[d]^2)/(1 + Log[d
]^2)^3 + (6*d^x*x^2*Cos[x]*Log[d])/(1 + Log[d]^2)^2 - (d^x*x^3*Cos[x])/(1 + Log[
d]^2) - (6*d^x*Sin[x])/(1 + Log[d]^2)^4 + (36*d^x*Log[d]^2*Sin[x])/(1 + Log[d]^2
)^4 - (6*d^x*Log[d]^4*Sin[x])/(1 + Log[d]^2)^4 - (18*d^x*x*Log[d]*Sin[x])/(1 + L
og[d]^2)^3 + (6*d^x*x*Log[d]^3*Sin[x])/(1 + Log[d]^2)^3 + (3*d^x*x^2*Sin[x])/(1
+ Log[d]^2)^2 - (3*d^x*x^2*Log[d]^2*Sin[x])/(1 + Log[d]^2)^2 + (d^x*x^3*Log[d]*S
in[x])/(1 + Log[d]^2)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{d^{x} x^{3} \log{\left (d \right )} \sin{\left (x \right )}}{\log{\left (d \right )}^{2} + 1} - \frac{d^{x} x^{3} \cos{\left (x \right )}}{\log{\left (d \right )}^{2} + 1} - 3 \int x^{2} \left (\frac{d^{x} \log{\left (d \right )} \sin{\left (x \right )}}{\log{\left (d \right )}^{2} + 1} - \frac{d^{x} \cos{\left (x \right )}}{\log{\left (d \right )}^{2} + 1}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(d**x*x**3*sin(x),x)

[Out]

d**x*x**3*log(d)*sin(x)/(log(d)**2 + 1) - d**x*x**3*cos(x)/(log(d)**2 + 1) - 3*I
ntegral(x**2*(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 = 0.210981, size = 169, normalized size = 0.65 \[ \frac{d^x \left (\sin (x) \left (x^3 \log ^7(d)-3 x^2 \log ^6(d)+3 x \left (x^2+2\right ) \log ^5(d)-3 \left (x^2+2\right ) \log ^4(d)+3 x \left (x^2-4\right ) \log ^3(d)+3 \left (x^2+12\right ) \log ^2(d)+x \left (x^2-18\right ) \log (d)+3 \left (x^2-2\right )\right )-\cos (x) \left (x^3 \log ^6(d)-6 x^2 \log ^5(d)+3 x \left (x^2+6\right ) \log ^4(d)-12 \left (x^2+2\right ) \log ^3(d)+3 x \left (x^2+4\right ) \log ^2(d)-6 \left (x^2-4\right ) \log (d)+x \left (x^2-6\right )\right )\right )}{\left (\log ^2(d)+1\right )^4} \]

Antiderivative was successfully verified.

[In]  Integrate[d^x*x^3*Sin[x],x]

[Out]

(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)*L
og[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*L
og[d]^7)*Sin[x]))/(1 + Log[d]^2)^4

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Maple [A]  time = 0.031, size = 437, normalized size = 1.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(d^x*x^3*sin(x),x)

[Out]

(1/(1+ln(d)^2)*x^3*exp(x*ln(d))*tan(1/2*x)^2-1/(1+ln(d)^2)*x^3*exp(x*ln(d))+6*ln
(d)/(ln(d)^4+2*ln(d)^2+1)*x^2*exp(x*ln(d))-6*(ln(d)^2-1)/(ln(d)^4+2*ln(d)^2+1)*x
^2*exp(x*ln(d))*tan(1/2*x)-6*(3*ln(d)^2-1)/(1+ln(d)^2)/(ln(d)^4+2*ln(d)^2+1)*x*e
xp(x*ln(d))-12*(ln(d)^4-6*ln(d)^2+1)/(ln(d)^6+3*ln(d)^4+3*ln(d)^2+1)/(1+ln(d)^2)
*exp(x*ln(d))*tan(1/2*x)+24/(ln(d)^6+3*ln(d)^4+3*ln(d)^2+1)*ln(d)*(ln(d)^2-1)/(1
+ln(d)^2)*exp(x*ln(d))+2*ln(d)/(1+ln(d)^2)*x^3*exp(x*ln(d))*tan(1/2*x)-6*ln(d)/(
ln(d)^4+2*ln(d)^2+1)*x^2*exp(x*ln(d))*tan(1/2*x)^2+6*(3*ln(d)^2-1)/(1+ln(d)^2)/(
ln(d)^4+2*ln(d)^2+1)*x*exp(x*ln(d))*tan(1/2*x)^2-24/(ln(d)^6+3*ln(d)^4+3*ln(d)^2
+1)*ln(d)*(ln(d)^2-1)/(1+ln(d)^2)*exp(x*ln(d))*tan(1/2*x)^2+12*ln(d)*(ln(d)^2-3)
/(1+ln(d)^2)/(ln(d)^4+2*ln(d)^2+1)*x*exp(x*ln(d))*tan(1/2*x))/(1+tan(1/2*x)^2)

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Maxima [A]  time = 1.48924, size = 251, normalized size = 0.96 \[ -\frac{{\left ({\left (\log \left (d\right )^{6} + 3 \, \log \left (d\right )^{4} + 3 \, \log \left (d\right )^{2} + 1\right )} x^{3} - 6 \,{\left (\log \left (d\right )^{5} + 2 \, \log \left (d\right )^{3} + \log \left (d\right )\right )} x^{2} - 24 \, \log \left (d\right )^{3} + 6 \,{\left (3 \, \log \left (d\right )^{4} + 2 \, \log \left (d\right )^{2} - 1\right )} x + 24 \, \log \left (d\right )\right )} d^{x} \cos \left (x\right ) -{\left ({\left (\log \left (d\right )^{7} + 3 \, \log \left (d\right )^{5} + 3 \, \log \left (d\right )^{3} + \log \left (d\right )\right )} x^{3} - 6 \, \log \left (d\right )^{4} - 3 \,{\left (\log \left (d\right )^{6} + \log \left (d\right )^{4} - \log \left (d\right )^{2} - 1\right )} x^{2} + 6 \,{\left (\log \left (d\right )^{5} - 2 \, \log \left (d\right )^{3} - 3 \, \log \left (d\right )\right )} x + 36 \, \log \left (d\right )^{2} - 6\right )} d^{x} \sin \left (x\right )}{\log \left (d\right )^{8} + 4 \, \log \left (d\right )^{6} + 6 \, \log \left (d\right )^{4} + 4 \, \log \left (d\right )^{2} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(d^x*x^3*sin(x),x, algorithm="maxima")

[Out]

-(((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*log(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)

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Fricas [A]  time = 0.249742, size = 274, normalized size = 1.05 \[ -\frac{{\left (x^{3} \cos \left (x\right ) \log \left (d\right )^{6} - 6 \, x^{2} \cos \left (x\right ) \log \left (d\right )^{5} + 3 \,{\left (x^{3} + 6 \, x\right )} \cos \left (x\right ) \log \left (d\right )^{4} - 12 \,{\left (x^{2} + 2\right )} \cos \left (x\right ) \log \left (d\right )^{3} + 3 \,{\left (x^{3} + 4 \, x\right )} \cos \left (x\right ) \log \left (d\right )^{2} - 6 \,{\left (x^{2} - 4\right )} \cos \left (x\right ) \log \left (d\right ) +{\left (x^{3} - 6 \, x\right )} \cos \left (x\right ) -{\left (x^{3} \log \left (d\right )^{7} - 3 \, x^{2} \log \left (d\right )^{6} + 3 \,{\left (x^{3} + 2 \, x\right )} \log \left (d\right )^{5} - 3 \,{\left (x^{2} + 2\right )} \log \left (d\right )^{4} + 3 \,{\left (x^{3} - 4 \, x\right )} \log \left (d\right )^{3} + 3 \,{\left (x^{2} + 12\right )} \log \left (d\right )^{2} + 3 \, x^{2} +{\left (x^{3} - 18 \, x\right )} \log \left (d\right ) - 6\right )} \sin \left (x\right )\right )} d^{x}}{\log \left (d\right )^{8} + 4 \, \log \left (d\right )^{6} + 6 \, \log \left (d\right )^{4} + 4 \, \log \left (d\right )^{2} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(d^x*x^3*sin(x),x, algorithm="fricas")

[Out]

-(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)*l
og(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/(log(d)^8 + 4*log(d)^6 + 6*log(d)
^4 + 4*log(d)^2 + 1)

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Sympy [A]  time = 45.4307, size = 1355, normalized size = 5.19 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(d**x*x**3*sin(x),x)

[Out]

Piecewise((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))), (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*log(d)**2 + 1) - d**x*x**3*log(d)**6*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)**5*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)**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*log(d)
**2 + 1) + d**x*x**3*log(d)*sin(x)/(log(d)**8 + 4*log(d)**6 + 6*log(d)**4 + 4*lo
g(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*log(d)**6 + 6*log(d)**4
+ 4*log(d)**2 + 1) + 6*d**x*x**2*log(d)**5*cos(x)/(log(d)**8 + 4*log(d)**6 + 6*l
og(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) + 12*d**x*x**2*log(d)**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)**2*sin(x)/(l
og(d)**8 + 4*log(d)**6 + 6*log(d)**4 + 4*log(d)**2 + 1) + 6*d**x*x**2*log(d)*cos
(x)/(log(d)**8 + 4*log(d)**6 + 6*log(d)**4 + 4*log(d)**2 + 1) + 3*d**x*x**2*sin(
x)/(log(d)**8 + 4*log(d)**6 + 6*log(d)**4 + 4*log(d)**2 + 1) + 6*d**x*x*log(d)**
5*sin(x)/(log(d)**8 + 4*log(d)**6 + 6*log(d)**4 + 4*log(d)**2 + 1) - 18*d**x*x*l
og(d)**4*cos(x)/(log(d)**8 + 4*log(d)**6 + 6*log(d)**4 + 4*log(d)**2 + 1) - 12*d
**x*x*log(d)**3*sin(x)/(log(d)**8 + 4*log(d)**6 + 6*log(d)**4 + 4*log(d)**2 + 1)
- 12*d**x*x*log(d)**2*cos(x)/(log(d)**8 + 4*log(d)**6 + 6*log(d)**4 + 4*log(d)*
*2 + 1) - 18*d**x*x*log(d)*sin(x)/(log(d)**8 + 4*log(d)**6 + 6*log(d)**4 + 4*log
(d)**2 + 1) + 6*d**x*x*cos(x)/(log(d)**8 + 4*log(d)**6 + 6*log(d)**4 + 4*log(d)*
*2 + 1) - 6*d**x*log(d)**4*sin(x)/(log(d)**8 + 4*log(d)**6 + 6*log(d)**4 + 4*log
(d)**2 + 1) + 24*d**x*log(d)**3*cos(x)/(log(d)**8 + 4*log(d)**6 + 6*log(d)**4 +
4*log(d)**2 + 1) + 36*d**x*log(d)**2*sin(x)/(log(d)**8 + 4*log(d)**6 + 6*log(d)*
*4 + 4*log(d)**2 + 1) - 24*d**x*log(d)*cos(x)/(log(d)**8 + 4*log(d)**6 + 6*log(d
)**4 + 4*log(d)**2 + 1) - 6*d**x*sin(x)/(log(d)**8 + 4*log(d)**6 + 6*log(d)**4 +
4*log(d)**2 + 1), True))

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GIAC/XCAS [A]  time = 0.254913, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(d^x*x^3*sin(x),x, algorithm="giac")

[Out]

Done