3.139 \(\int d^x x^2 \cos (x) \, dx\)

Optimal. Leaf size=161 \[ \frac{x^2 d^x \sin (x)}{\log ^2(d)+1}+\frac{x^2 d^x \log (d) \cos (x)}{\log ^2(d)+1}-\frac{4 x d^x \log (d) \sin (x)}{\left (\log ^2(d)+1\right )^2}+\frac{6 d^x \log ^2(d) \sin (x)}{\left (\log ^2(d)+1\right )^3}-\frac{2 d^x \sin (x)}{\left (\log ^2(d)+1\right )^3}-\frac{2 x d^x \log ^2(d) \cos (x)}{\left (\log ^2(d)+1\right )^2}+\frac{2 x d^x \cos (x)}{\left (\log ^2(d)+1\right )^2}+\frac{2 d^x \log ^3(d) \cos (x)}{\left (\log ^2(d)+1\right )^3}-\frac{6 d^x \log (d) \cos (x)}{\left (\log ^2(d)+1\right )^3} \]

[Out]

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

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Rubi [A]  time = 0.170726, antiderivative size = 161, 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, Rules used = {4433, 4466, 14, 4432, 4465} \[ \frac{x^2 d^x \sin (x)}{\log ^2(d)+1}+\frac{x^2 d^x \log (d) \cos (x)}{\log ^2(d)+1}-\frac{4 x d^x \log (d) \sin (x)}{\left (\log ^2(d)+1\right )^2}+\frac{6 d^x \log ^2(d) \sin (x)}{\left (\log ^2(d)+1\right )^3}-\frac{2 d^x \sin (x)}{\left (\log ^2(d)+1\right )^3}-\frac{2 x d^x \log ^2(d) \cos (x)}{\left (\log ^2(d)+1\right )^2}+\frac{2 x d^x \cos (x)}{\left (\log ^2(d)+1\right )^2}+\frac{2 d^x \log ^3(d) \cos (x)}{\left (\log ^2(d)+1\right )^3}-\frac{6 d^x \log (d) \cos (x)}{\left (\log ^2(d)+1\right )^3} \]

Antiderivative was successfully verified.

[In]

Int[d^x*x^2*Cos[x],x]

[Out]

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

Rule 4433

Int[Cos[(d_.) + (e_.)*(x_)]*(F_)^((c_.)*((a_.) + (b_.)*(x_))), x_Symbol] :> Simp[(b*c*Log[F]*F^(c*(a + b*x))*C
os[d + e*x])/(e^2 + b^2*c^2*Log[F]^2), x] + Simp[(e*F^(c*(a + b*x))*Sin[d + e*x])/(e^2 + b^2*c^2*Log[F]^2), x]
/; FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2 + b^2*c^2*Log[F]^2, 0]

Rule 4466

Int[Cos[(d_.) + (e_.)*(x_)]^(n_.)*(F_)^((c_.)*((a_.) + (b_.)*(x_)))*((f_.)*(x_))^(m_.), x_Symbol] :> Module[{u
= IntHide[F^(c*(a + b*x))*Cos[d + e*x]^n, x]}, Dist[(f*x)^m, u, x] - Dist[f*m, Int[(f*x)^(m - 1)*u, x], x]] /
; FreeQ[{F, a, b, c, d, e, f}, x] && IGtQ[n, 0] && GtQ[m, 0]

Rule 14

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]]

Rule 4432

Int[(F_)^((c_.)*((a_.) + (b_.)*(x_)))*Sin[(d_.) + (e_.)*(x_)], x_Symbol] :> Simp[(b*c*Log[F]*F^(c*(a + b*x))*S
in[d + e*x])/(e^2 + b^2*c^2*Log[F]^2), x] - Simp[(e*F^(c*(a + b*x))*Cos[d + e*x])/(e^2 + b^2*c^2*Log[F]^2), x]
/; FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2 + b^2*c^2*Log[F]^2, 0]

Rule 4465

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]}, Dist[(f*x)^m, u, x] - Dist[f*m, Int[(f*x)^(m - 1)*u, x], x]] /
; FreeQ[{F, a, b, c, d, e, f}, x] && IGtQ[n, 0] && GtQ[m, 0]

Rubi steps

\begin{align*} \int d^x x^2 \cos (x) \, dx &=\frac{d^x x^2 \cos (x) \log (d)}{1+\log ^2(d)}+\frac{d^x x^2 \sin (x)}{1+\log ^2(d)}-2 \int x \left (\frac{d^x \cos (x) \log (d)}{1+\log ^2(d)}+\frac{d^x \sin (x)}{1+\log ^2(d)}\right ) \, dx\\ &=\frac{d^x x^2 \cos (x) \log (d)}{1+\log ^2(d)}+\frac{d^x x^2 \sin (x)}{1+\log ^2(d)}-2 \int \left (\frac{d^x x \cos (x) \log (d)}{1+\log ^2(d)}+\frac{d^x x \sin (x)}{1+\log ^2(d)}\right ) \, dx\\ &=\frac{d^x x^2 \cos (x) \log (d)}{1+\log ^2(d)}+\frac{d^x x^2 \sin (x)}{1+\log ^2(d)}-\frac{2 \int d^x x \sin (x) \, dx}{1+\log ^2(d)}-\frac{(2 \log (d)) \int d^x x \cos (x) \, dx}{1+\log ^2(d)}\\ &=\frac{2 d^x x \cos (x)}{\left (1+\log ^2(d)\right )^2}-\frac{2 d^x x \cos (x) \log ^2(d)}{\left (1+\log ^2(d)\right )^2}+\frac{d^x x^2 \cos (x) \log (d)}{1+\log ^2(d)}-\frac{4 d^x x \log (d) \sin (x)}{\left (1+\log ^2(d)\right )^2}+\frac{d^x x^2 \sin (x)}{1+\log ^2(d)}+\frac{2 \int \left (-\frac{d^x \cos (x)}{1+\log ^2(d)}+\frac{d^x \log (d) \sin (x)}{1+\log ^2(d)}\right ) \, dx}{1+\log ^2(d)}+\frac{(2 \log (d)) \int \left (\frac{d^x \cos (x) \log (d)}{1+\log ^2(d)}+\frac{d^x \sin (x)}{1+\log ^2(d)}\right ) \, dx}{1+\log ^2(d)}\\ &=\frac{2 d^x x \cos (x)}{\left (1+\log ^2(d)\right )^2}-\frac{2 d^x x \cos (x) \log ^2(d)}{\left (1+\log ^2(d)\right )^2}+\frac{d^x x^2 \cos (x) \log (d)}{1+\log ^2(d)}-\frac{4 d^x x \log (d) \sin (x)}{\left (1+\log ^2(d)\right )^2}+\frac{d^x x^2 \sin (x)}{1+\log ^2(d)}-\frac{2 \int d^x \cos (x) \, dx}{\left (1+\log ^2(d)\right )^2}+2 \frac{(2 \log (d)) \int d^x \sin (x) \, dx}{\left (1+\log ^2(d)\right )^2}+\frac{\left (2 \log ^2(d)\right ) \int d^x \cos (x) \, dx}{\left (1+\log ^2(d)\right )^2}\\ &=-\frac{2 d^x \cos (x) \log (d)}{\left (1+\log ^2(d)\right )^3}+\frac{2 d^x \cos (x) \log ^3(d)}{\left (1+\log ^2(d)\right )^3}+\frac{2 d^x x \cos (x)}{\left (1+\log ^2(d)\right )^2}-\frac{2 d^x x \cos (x) \log ^2(d)}{\left (1+\log ^2(d)\right )^2}+\frac{d^x x^2 \cos (x) \log (d)}{1+\log ^2(d)}-\frac{2 d^x \sin (x)}{\left (1+\log ^2(d)\right )^3}+\frac{2 d^x \log ^2(d) \sin (x)}{\left (1+\log ^2(d)\right )^3}-\frac{4 d^x x \log (d) \sin (x)}{\left (1+\log ^2(d)\right )^2}+\frac{d^x x^2 \sin (x)}{1+\log ^2(d)}+2 \left (-\frac{2 d^x \cos (x) \log (d)}{\left (1+\log ^2(d)\right )^3}+\frac{2 d^x \log ^2(d) \sin (x)}{\left (1+\log ^2(d)\right )^3}\right )\\ \end{align*}

Mathematica [A]  time = 0.073773, size = 93, normalized size = 0.58 \[ \frac{d^x \left (\sin (x) \left (x^2 \log ^4(d)+2 \left (x^2+3\right ) \log ^2(d)-4 x \log ^3(d)-4 x \log (d)+x^2-2\right )+\cos (x) \left (x^2 \log ^5(d)+2 \left (x^2+1\right ) \log ^3(d)+\left (x^2-6\right ) \log (d)-2 x \log ^4(d)+2 x\right )\right )}{\left (\log ^2(d)+1\right )^3} \]

Antiderivative was successfully verified.

[In]

Integrate[d^x*x^2*Cos[x],x]

[Out]

(d^x*(Cos[x]*(2*x + (-6 + x^2)*Log[d] + 2*(1 + x^2)*Log[d]^3 - 2*x*Log[d]^4 + x^2*Log[d]^5) + (-2 + x^2 - 4*x*
Log[d] + 2*(3 + x^2)*Log[d]^2 - 4*x*Log[d]^3 + x^2*Log[d]^4)*Sin[x]))/(1 + Log[d]^2)^3

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Maple [A]  time = 0.028, size = 231, normalized size = 1.4 \begin{align*}{ \left ({\frac{\ln \left ( d \right ){x}^{2}{{\rm e}^{x\ln \left ( d \right ) }}}{1+ \left ( \ln \left ( d \right ) \right ) ^{2}}}+2\,{\frac{{x}^{2}{{\rm e}^{x\ln \left ( d \right ) }}\tan \left ( x/2 \right ) }{1+ \left ( \ln \left ( d \right ) \right ) ^{2}}}-2\,{\frac{ \left ( \left ( \ln \left ( d \right ) \right ) ^{2}-1 \right ) x{{\rm e}^{x\ln \left ( d \right ) }}}{ \left ( 1+ \left ( \ln \left ( d \right ) \right ) ^{2} \right ) ^{2}}}+4\,{\frac{ \left ( 3\, \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 ) ^{3}}}+2\,{\frac{\ln \left ( d \right ) \left ( \left ( \ln \left ( d \right ) \right ) ^{2}-3 \right ){{\rm e}^{x\ln \left ( d \right ) }}}{ \left ( 1+ \left ( \ln \left ( d \right ) \right ) ^{2} \right ) ^{3}}}-8\,{\frac{x\ln \left ( d \right ){{\rm e}^{x\ln \left ( d \right ) }}\tan \left ( x/2 \right ) }{ \left ( 1+ \left ( \ln \left ( d \right ) \right ) ^{2} \right ) ^{2}}}-{\frac{\ln \left ( d \right ){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}}+2\,{\frac{ \left ( \left ( \ln \left ( d \right ) \right ) ^{2}-1 \right ) x{{\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{\ln \left ( d \right ) \left ( \left ( \ln \left ( d \right ) \right ) ^{2}-3 \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}}} \right ) \left ( \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}+1 \right ) ^{-1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(d^x*x^2*cos(x),x)

[Out]

(1/(1+ln(d)^2)*ln(d)*x^2*exp(x*ln(d))+2/(1+ln(d)^2)*x^2*exp(x*ln(d))*tan(1/2*x)-2*(ln(d)^2-1)/(1+ln(d)^2)^2*x*
exp(x*ln(d))+4*(3*ln(d)^2-1)/(1+ln(d)^2)^3*exp(x*ln(d))*tan(1/2*x)+2*ln(d)*(ln(d)^2-3)/(1+ln(d)^2)^3*exp(x*ln(
d))-8/(1+ln(d)^2)^2*ln(d)*x*exp(x*ln(d))*tan(1/2*x)-1/(1+ln(d)^2)*ln(d)*x^2*exp(x*ln(d))*tan(1/2*x)^2+2*(ln(d)
^2-1)/(1+ln(d)^2)^2*x*exp(x*ln(d))*tan(1/2*x)^2-2*ln(d)*(ln(d)^2-3)/(1+ln(d)^2)^3*exp(x*ln(d))*tan(1/2*x)^2)/(
tan(1/2*x)^2+1)

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Maxima [A]  time = 1.0198, size = 142, normalized size = 0.88 \begin{align*} \frac{{\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} \cos \left (x\right ) +{\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} \sin \left (x\right )}{\log \left (d\right )^{6} + 3 \, \log \left (d\right )^{4} + 3 \, \log \left (d\right )^{2} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(d^x*x^2*cos(x),x, algorithm="maxima")

[Out]

(((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*cos(x) + ((log(d)^4 +
2*log(d)^2 + 1)*x^2 - 4*(log(d)^3 + log(d))*x + 6*log(d)^2 - 2)*d^x*sin(x))/(log(d)^6 + 3*log(d)^4 + 3*log(d)
^2 + 1)

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Fricas [A]  time = 1.78349, size = 329, normalized size = 2.04 \begin{align*} \frac{{\left (x^{2} \cos \left (x\right ) \log \left (d\right )^{5} - 2 \, x \cos \left (x\right ) \log \left (d\right )^{4} + 2 \,{\left (x^{2} + 1\right )} \cos \left (x\right ) \log \left (d\right )^{3} +{\left (x^{2} - 6\right )} \cos \left (x\right ) \log \left (d\right ) + 2 \, x \cos \left (x\right ) +{\left (x^{2} \log \left (d\right )^{4} - 4 \, x \log \left (d\right )^{3} + 2 \,{\left (x^{2} + 3\right )} \log \left (d\right )^{2} + x^{2} - 4 \, x \log \left (d\right ) - 2\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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(d^x*x^2*cos(x),x, algorithm="fricas")

[Out]

(x^2*cos(x)*log(d)^5 - 2*x*cos(x)*log(d)^4 + 2*(x^2 + 1)*cos(x)*log(d)^3 + (x^2 - 6)*cos(x)*log(d) + 2*x*cos(x
) + (x^2*log(d)^4 - 4*x*log(d)^3 + 2*(x^2 + 3)*log(d)^2 + x^2 - 4*x*log(d) - 2)*sin(x))*d^x/(log(d)^6 + 3*log(
d)^4 + 3*log(d)^2 + 1)

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Sympy [B]  time = 9.79313, size = 665, normalized size = 4.13 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(d**x*x**2*cos(x),x)

[Out]

Piecewise((I*x**3*exp(-I*x)*sin(x)/6 + x**3*exp(-I*x)*cos(x)/6 + x**2*exp(-I*x)*sin(x)/4 + I*x**2*exp(-I*x)*co
s(x)/4 - I*x*exp(-I*x)*sin(x)/4 + x*exp(-I*x)*cos(x)/4 - exp(-I*x)*sin(x)/4, Eq(d, exp(-I))), (-I*x**3*exp(I*x
)*sin(x)/6 + x**3*exp(I*x)*cos(x)/6 + x**2*exp(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 - exp(I*x)*sin(x)/4, Eq(d, exp(I))), (d**x*x**2*log(d)**5*cos(x)/(log(d)**6 + 3*log(d)
**4 + 3*log(d)**2 + 1) + d**x*x**2*log(d)**4*sin(x)/(log(d)**6 + 3*log(d)**4 + 3*log(d)**2 + 1) + 2*d**x*x**2*
log(d)**3*cos(x)/(log(d)**6 + 3*log(d)**4 + 3*log(d)**2 + 1) + 2*d**x*x**2*log(d)**2*sin(x)/(log(d)**6 + 3*log
(d)**4 + 3*log(d)**2 + 1) + d**x*x**2*log(d)*cos(x)/(log(d)**6 + 3*log(d)**4 + 3*log(d)**2 + 1) + d**x*x**2*si
n(x)/(log(d)**6 + 3*log(d)**4 + 3*log(d)**2 + 1) - 2*d**x*x*log(d)**4*cos(x)/(log(d)**6 + 3*log(d)**4 + 3*log(
d)**2 + 1) - 4*d**x*x*log(d)**3*sin(x)/(log(d)**6 + 3*log(d)**4 + 3*log(d)**2 + 1) - 4*d**x*x*log(d)*sin(x)/(l
og(d)**6 + 3*log(d)**4 + 3*log(d)**2 + 1) + 2*d**x*x*cos(x)/(log(d)**6 + 3*log(d)**4 + 3*log(d)**2 + 1) + 2*d*
*x*log(d)**3*cos(x)/(log(d)**6 + 3*log(d)**4 + 3*log(d)**2 + 1) + 6*d**x*log(d)**2*sin(x)/(log(d)**6 + 3*log(d
)**4 + 3*log(d)**2 + 1) - 6*d**x*log(d)*cos(x)/(log(d)**6 + 3*log(d)**4 + 3*log(d)**2 + 1) - 2*d**x*sin(x)/(lo
g(d)**6 + 3*log(d)**4 + 3*log(d)**2 + 1), True))

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Giac [C]  time = 1.20871, size = 3552, normalized size = 22.06 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(d^x*x^2*cos(x),x, algorithm="giac")

[Out]

1/2*((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(abs(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*sg
n(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) + (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^2*log(abs(d))*sgn(d) - 3*p
i^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)))/((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) + ((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)*(p
i^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(ab
s(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(ab
s(d))*sgn(d) + 6*pi*log(abs(d)) - 6*log(abs(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*log(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))*sin(1/2*pi*x*sgn(d) - 1/2*pi*x + x))*abs(d)^x + 1/2*((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(abs(d))^2 - 3*pi*sgn(d) + 2)*(pi*x^2*log(abs(d))*s
gn(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) - 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) + (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^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)))/((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*lo
g(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) + ((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*sg
n(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(abs(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*log(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*l
og(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*l
og(abs(d))*sgn(d) - 6*pi*log(abs(d)) - 6*log(abs(d)))^2))*sin(1/2*pi*x*sgn(d) - 1/2*pi*x - x))*abs(d)^x - 1/2*
I*abs(d)^x*((-4*I*pi^2*x^2*sgn(d) + 8*pi*x^2*log(abs(d))*sgn(d) + 4*I*pi^2*x^2 - 8*pi*x^2*log(abs(d)) - 8*I*x^
2*log(abs(d))^2 + 8*I*pi*x^2*sgn(d) - 8*I*pi*x^2 + 16*x^2*log(abs(d)) - 8*pi*x*sgn(d) + 8*pi*x + 8*I*x^2 + 16*
I*x*log(abs(d)) - 16*x - 16*I)*e^(1/2*I*pi*x*sgn(d) - 1/2*I*pi*x + I*x)/(24*I*pi - 8*I*pi^3*sgn(d) + 24*pi^2*l
og(abs(d))*sgn(d) + 24*I*pi*log(abs(d))^2*sgn(d) + 8*I*pi^3 - 24*pi^2*log(abs(d)) - 24*I*pi*log(abs(d))^2 + 16
*log(abs(d))^3 + 24*I*pi^2*sgn(d) - 48*pi*log(abs(d))*sgn(d) - 24*I*pi^2 + 48*pi*log(abs(d)) + 48*I*log(abs(d)
)^2 - 24*I*pi*sgn(d) - 48*log(abs(d)) - 16*I) - (-4*I*pi^2*x^2*sgn(d) - 8*pi*x^2*log(abs(d))*sgn(d) + 4*I*pi^2
*x^2 + 8*pi*x^2*log(abs(d)) - 8*I*x^2*log(abs(d))^2 + 8*I*pi*x^2*sgn(d) - 8*I*pi*x^2 - 16*x^2*log(abs(d)) + 8*
pi*x*sgn(d) - 8*pi*x + 8*I*x^2 + 16*I*x*log(abs(d)) + 16*x - 16*I)*e^(-1/2*I*pi*x*sgn(d) + 1/2*I*pi*x - I*x)/(
-24*I*pi + 8*I*pi^3*sgn(d) + 24*pi^2*log(abs(d))*sgn(d) - 24*I*pi*log(abs(d))^2*sgn(d) - 8*I*pi^3 - 24*pi^2*lo
g(abs(d)) + 24*I*pi*log(abs(d))^2 + 16*log(abs(d))^3 - 24*I*pi^2*sgn(d) - 48*pi*log(abs(d))*sgn(d) + 24*I*pi^2
+ 48*pi*log(abs(d)) - 48*I*log(abs(d))^2 + 24*I*pi*sgn(d) - 48*log(abs(d)) + 16*I)) - 1/2*I*abs(d)^x*((-4*I*p
i^2*x^2*sgn(d) + 8*pi*x^2*log(abs(d))*sgn(d) + 4*I*pi^2*x^2 - 8*pi*x^2*log(abs(d)) - 8*I*x^2*log(abs(d))^2 - 8
*I*pi*x^2*sgn(d) + 8*I*pi*x^2 - 16*x^2*log(abs(d)) - 8*pi*x*sgn(d) + 8*pi*x + 8*I*x^2 + 16*I*x*log(abs(d)) + 1
6*x - 16*I)*e^(1/2*I*pi*x*sgn(d) - 1/2*I*pi*x - I*x)/(24*I*pi - 8*I*pi^3*sgn(d) + 24*pi^2*log(abs(d))*sgn(d) +
24*I*pi*log(abs(d))^2*sgn(d) + 8*I*pi^3 - 24*pi^2*log(abs(d)) - 24*I*pi*log(abs(d))^2 + 16*log(abs(d))^3 - 24
*I*pi^2*sgn(d) + 48*pi*log(abs(d))*sgn(d) + 24*I*pi^2 - 48*pi*log(abs(d)) - 48*I*log(abs(d))^2 - 24*I*pi*sgn(d
) - 48*log(abs(d)) + 16*I) - (-4*I*pi^2*x^2*sgn(d) - 8*pi*x^2*log(abs(d))*sgn(d) + 4*I*pi^2*x^2 + 8*pi*x^2*log
(abs(d)) - 8*I*x^2*log(abs(d))^2 - 8*I*pi*x^2*sgn(d) + 8*I*pi*x^2 + 16*x^2*log(abs(d)) + 8*pi*x*sgn(d) - 8*pi*
x + 8*I*x^2 + 16*I*x*log(abs(d)) - 16*x - 16*I)*e^(-1/2*I*pi*x*sgn(d) + 1/2*I*pi*x + I*x)/(-24*I*pi + 8*I*pi^3
*sgn(d) + 24*pi^2*log(abs(d))*sgn(d) - 24*I*pi*log(abs(d))^2*sgn(d) - 8*I*pi^3 - 24*pi^2*log(abs(d)) + 24*I*pi
*log(abs(d))^2 + 16*log(abs(d))^3 + 24*I*pi^2*sgn(d) + 48*pi*log(abs(d))*sgn(d) - 24*I*pi^2 - 48*pi*log(abs(d)
) + 48*I*log(abs(d))^2 + 24*I*pi*sgn(d) - 48*log(abs(d)) - 16*I))