3.2.38 \(\int d^x x^2 \sin (x) \, dx\) [138]

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

[Out]

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)

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Rubi [A]
time = 0.11, antiderivative size = 162, normalized size of antiderivative = 1.00, 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 = {4517, 4553, 14, 4518, 4554} \begin {gather*} \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} \end {gather*}

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*Lo
g[d]*Sin[x])/(1 + Log[d]^2)

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 4517

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 4518

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 4553

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]

Rule 4554

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]

Rubi steps

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

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Mathematica [A]
time = 0.06, size = 94, normalized size = 0.58 \begin {gather*} \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} \end {gather*}

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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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 4.97, size = 549, normalized size = 3.39 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\left (\frac {I x \text {Cos}\left [x\right ]}{4}+\frac {x \text {Sin}\left [x\right ]}{4}-\frac {x^2 \text {Cos}\left [x\right ]}{4}+\frac {I x^2 \text {Sin}\left [x\right ]}{4}-\frac {I x^3 \text {Cos}\left [x\right ]}{6}+\frac {x^3 \text {Sin}\left [x\right ]}{6}+\frac {\text {Cos}\left [x\right ]}{4}\right ) E^{-I x},d\text {==}E^{-I}\right \},\left \{\left (\left (-\frac {I}{4}\right ) x \text {Cos}\left [x\right ]+\frac {x \text {Sin}\left [x\right ]}{4}-\frac {x^2 \text {Cos}\left [x\right ]}{4}-\frac {I x^2 \text {Sin}\left [x\right ]}{4}+\frac {I x^3 \text {Cos}\left [x\right ]}{6}+\frac {x^3 \text {Sin}\left [x\right ]}{6}+\frac {\text {Cos}\left [x\right ]}{4}\right ) E^{I x},d\text {==}E^I\right \}\right \},\frac {-2 x d^x \text {Log}\left [d\right ]^4 \text {Sin}\left [x\right ]}{1+\text {Log}\left [d\right ]^6+3 \text {Log}\left [d\right ]^2+3 \text {Log}\left [d\right ]^4}+\frac {2 x d^x \text {Sin}\left [x\right ]}{1+\text {Log}\left [d\right ]^6+3 \text {Log}\left [d\right ]^2+3 \text {Log}\left [d\right ]^4}+\frac {4 x \text {Cos}\left [x\right ] \text {Log}\left [d\right ] d^x}{1+\text {Log}\left [d\right ]^6+3 \text {Log}\left [d\right ]^2+3 \text {Log}\left [d\right ]^4}+\frac {4 x \text {Cos}\left [x\right ] d^x \text {Log}\left [d\right ]^3}{1+\text {Log}\left [d\right ]^6+3 \text {Log}\left [d\right ]^2+3 \text {Log}\left [d\right ]^4}-\frac {2 x^2 \text {Cos}\left [x\right ] d^x \text {Log}\left [d\right ]^2}{1+\text {Log}\left [d\right ]^6+3 \text {Log}\left [d\right ]^2+3 \text {Log}\left [d\right ]^4}-\frac {x^2 \text {Cos}\left [x\right ] d^x \text {Log}\left [d\right ]^4}{1+\text {Log}\left [d\right ]^6+3 \text {Log}\left [d\right ]^2+3 \text {Log}\left [d\right ]^4}-\frac {x^2 \text {Cos}\left [x\right ] d^x}{1+\text {Log}\left [d\right ]^6+3 \text {Log}\left [d\right ]^2+3 \text {Log}\left [d\right ]^4}+\frac {2 x^2 d^x \text {Log}\left [d\right ]^3 \text {Sin}\left [x\right ]}{1+\text {Log}\left [d\right ]^6+3 \text {Log}\left [d\right ]^2+3 \text {Log}\left [d\right ]^4}+\frac {x^2 \text {Log}\left [d\right ] d^x \text {Sin}\left [x\right ]}{1+\text {Log}\left [d\right ]^6+3 \text {Log}\left [d\right ]^2+3 \text {Log}\left [d\right ]^4}+\frac {x^2 d^x \text {Log}\left [d\right ]^5 \text {Sin}\left [x\right ]}{1+\text {Log}\left [d\right ]^6+3 \text {Log}\left [d\right ]^2+3 \text {Log}\left [d\right ]^4}-\frac {6 \text {Cos}\left [x\right ] d^x \text {Log}\left [d\right ]^2}{1+\text {Log}\left [d\right ]^6+3 \text {Log}\left [d\right ]^2+3 \text {Log}\left [d\right ]^4}-\frac {6 \text {Log}\left [d\right ] d^x \text {Sin}\left [x\right ]}{1+\text {Log}\left [d\right ]^6+3 \text {Log}\left [d\right ]^2+3 \text {Log}\left [d\right ]^4}+\frac {2 \text {Cos}\left [x\right ] d^x}{1+\text {Log}\left [d\right ]^6+3 \text {Log}\left [d\right ]^2+3 \text {Log}\left [d\right ]^4}+\frac {2 d^x \text {Log}\left [d\right ]^3 \text {Sin}\left [x\right ]}{1+\text {Log}\left [d\right ]^6+3 \text {Log}\left [d\right ]^2+3 \text {Log}\left [d\right ]^4}\right ] \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[x^2*d^x*Sin[x],x]')

[Out]

Piecewise[{{(I / 4 x Cos[x] + x Sin[x] / 4 - x ^ 2 Cos[x] / 4 + I / 4 x ^ 2 Sin[x] - I / 6 x ^ 3 Cos[x] + x ^
3 Sin[x] / 6 + Cos[x] / 4) E ^ (-I x), d == E ^ (-I)}, {((-I / 4) x Cos[x] + x Sin[x] / 4 - x ^ 2 Cos[x] / 4 -
I / 4 x ^ 2 Sin[x] + I / 6 x ^ 3 Cos[x] + x ^ 3 Sin[x] / 6 + Cos[x] / 4) E ^ (I x), d == E ^ I}}, -2 x d ^ x
Log[d] ^ 4 Sin[x] / (1 + Log[d] ^ 6 + 3 Log[d] ^ 2 + 3 Log[d] ^ 4) + 2 x d ^ x Sin[x] / (1 + Log[d] ^ 6 + 3 Lo
g[d] ^ 2 + 3 Log[d] ^ 4) + 4 x Cos[x] Log[d] d ^ x / (1 + Log[d] ^ 6 + 3 Log[d] ^ 2 + 3 Log[d] ^ 4) + 4 x Cos[
x] d ^ x Log[d] ^ 3 / (1 + Log[d] ^ 6 + 3 Log[d] ^ 2 + 3 Log[d] ^ 4) - 2 x ^ 2 Cos[x] d ^ x Log[d] ^ 2 / (1 +
Log[d] ^ 6 + 3 Log[d] ^ 2 + 3 Log[d] ^ 4) - x ^ 2 Cos[x] d ^ x Log[d] ^ 4 / (1 + Log[d] ^ 6 + 3 Log[d] ^ 2 + 3
Log[d] ^ 4) - x ^ 2 Cos[x] d ^ x / (1 + Log[d] ^ 6 + 3 Log[d] ^ 2 + 3 Log[d] ^ 4) + 2 x ^ 2 d ^ x Log[d] ^ 3
Sin[x] / (1 + Log[d] ^ 6 + 3 Log[d] ^ 2 + 3 Log[d] ^ 4) + x ^ 2 Log[d] d ^ x Sin[x] / (1 + Log[d] ^ 6 + 3 Log[
d] ^ 2 + 3 Log[d] ^ 4) + x ^ 2 d ^ x Log[d] ^ 5 Sin[x] / (1 + Log[d] ^ 6 + 3 Log[d] ^ 2 + 3 Log[d] ^ 4) - 6 Co
s[x] d ^ x Log[d] ^ 2 / (1 + Log[d] ^ 6 + 3 Log[d] ^ 2 + 3 Log[d] ^ 4) - 6 Log[d] d ^ x Sin[x] / (1 + Log[d] ^
6 + 3 Log[d] ^ 2 + 3 Log[d] ^ 4) + 2 Cos[x] d ^ x / (1 + Log[d] ^ 6 + 3 Log[d] ^ 2 + 3 Log[d] ^ 4) + 2 d ^ x
Log[d] ^ 3 Sin[x] / (1 + Log[d] ^ 6 + 3 Log[d] ^ 2 + 3 Log[d] ^ 4)]

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Maple [C] Result contains complex when optimal does not.
time = 0.06, size = 102, normalized size = 0.63

method	result	size
risch	\(-\frac {i \left (2+\ln \left (d \right )^{2} x^{2}+2 i \ln \left (d \right ) x^{2}-x^{2}-2 x \ln \left (d \right )-2 i x \right ) d^{x} {\mathrm e}^{i x}}{2 \left (\ln \left (d \right )+i\right )^{3}}+\frac {i \left (2-2 x \ln \left (d \right )+2 i x +\ln \left (d \right )^{2} x^{2}-2 i \ln \left (d \right ) x^{2}-x^{2}\right ) d^{x} {\mathrm e}^{-i x}}{2 \left (\ln \left (d \right )-i\right )^{3}}\)	\(102\)
norman	\(\frac {\frac {x^{2} {\mathrm e}^{x \ln \left (d \right )} \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{1+\ln \left (d \right )^{2}}-\frac {x^{2} {\mathrm e}^{x \ln \left (d \right )}}{1+\ln \left (d \right )^{2}}-\frac {2 \left (3 \ln \left (d \right )^{2}-1\right ) {\mathrm e}^{x \ln \left (d \right )}}{\left (1+\ln \left (d \right )^{2}\right )^{3}}+\frac {4 \ln \left (d \right ) x \,{\mathrm e}^{x \ln \left (d \right )}}{\left (1+\ln \left (d \right )^{2}\right )^{2}}+\frac {2 \left (3 \ln \left (d \right )^{2}-1\right ) {\mathrm e}^{x \ln \left (d \right )} \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{\left (1+\ln \left (d \right )^{2}\right )^{3}}-\frac {4 \ln \left (d \right ) x \,{\mathrm e}^{x \ln \left (d \right )} \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{\left (1+\ln \left (d \right )^{2}\right )^{2}}+\frac {2 \ln \left (d \right ) x^{2} {\mathrm e}^{x \ln \left (d \right )} \tan \left (\frac {x}{2}\right )}{1+\ln \left (d \right )^{2}}-\frac {4 \left (\ln \left (d \right )^{2}-1\right ) x \,{\mathrm e}^{x \ln \left (d \right )} \tan \left (\frac {x}{2}\right )}{\left (1+\ln \left (d \right )^{2}\right )^{2}}+\frac {4 \ln \left (d \right ) \left (\ln \left (d \right )^{2}-3\right ) {\mathrm e}^{x \ln \left (d \right )} \tan \left (\frac {x}{2}\right )}{\left (1+\ln \left (d \right )^{2}\right )^{3}}}{1+\tan ^{2}\left (\frac {x}{2}\right )}\)	\(225\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(d^x*x^2*sin(x),x,method=_RETURNVERBOSE)

[Out]

-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*exp(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)

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Maxima [A]
time = 0.32, size = 107, normalized size = 0.66 \begin {gather*} -\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} \end {gather*}

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 = 0.31, size = 115, normalized size = 0.71 \begin {gather*} -\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} \end {gather*}

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

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Sympy [C] Result contains complex when optimal does not.
time = 1.06, size = 665, normalized size = 4.10 \begin {gather*} \begin {cases} \frac {x^{3} e^{- i x} \sin {\left (x \right )}}{6} - \frac {i x^{3} e^{- i x} \cos {\left (x \right )}}{6} + \frac {i x^{2} e^{- i x} \sin {\left (x \right )}}{4} - \frac {x^{2} e^{- i x} \cos {\left (x \right )}}{4} + \frac {x e^{- i x} \sin {\left (x \right )}}{4} + \frac {i x e^{- i x} \cos {\left (x \right )}}{4} + \frac {e^{- i x} \cos {\left (x \right )}}{4} & \text {for}\: d = e^{- i} \\\frac {x^{3} e^{i x} \sin {\left (x \right )}}{6} + \frac {i x^{3} e^{i x} \cos {\left (x \right )}}{6} - \frac {i x^{2} e^{i x} \sin {\left (x \right )}}{4} - \frac {x^{2} e^{i x} \cos {\left (x \right )}}{4} + \frac {x e^{i x} \sin {\left (x \right )}}{4} - \frac {i x e^{i x} \cos {\left (x \right )}}{4} + \frac {e^{i x} \cos {\left (x \right )}}{4} & \text {for}\: d = e^{i} \\\frac {d^{x} x^{2} \log {\left (d \right )}^{5} \sin {\left (x \right )}}{\log {\left (d \right )}^{6} + 3 \log {\left (d \right )}^{4} + 3 \log {\left (d \right )}^{2} + 1} - \frac {d^{x} x^{2} \log {\left (d \right )}^{4} \cos {\left (x \right )}}{\log {\left (d \right )}^{6} + 3 \log {\left (d \right )}^{4} + 3 \log {\left (d \right )}^{2} + 1} + \frac {2 d^{x} x^{2} \log {\left (d \right )}^{3} \sin {\left (x \right )}}{\log {\left (d \right )}^{6} + 3 \log {\left (d \right )}^{4} + 3 \log {\left (d \right )}^{2} + 1} - \frac {2 d^{x} x^{2} \log {\left (d \right )}^{2} \cos {\left (x \right )}}{\log {\left (d \right )}^{6} + 3 \log {\left (d \right )}^{4} + 3 \log {\left (d \right )}^{2} + 1} + \frac {d^{x} x^{2} \log {\left (d \right )} \sin {\left (x \right )}}{\log {\left (d \right )}^{6} + 3 \log {\left (d \right )}^{4} + 3 \log {\left (d \right )}^{2} + 1} - \frac {d^{x} x^{2} \cos {\left (x \right )}}{\log {\left (d \right )}^{6} + 3 \log {\left (d \right )}^{4} + 3 \log {\left (d \right )}^{2} + 1} - \frac {2 d^{x} x \log {\left (d \right )}^{4} \sin {\left (x \right )}}{\log {\left (d \right )}^{6} + 3 \log {\left (d \right )}^{4} + 3 \log {\left (d \right )}^{2} + 1} + \frac {4 d^{x} x \log {\left (d \right )}^{3} \cos {\left (x \right )}}{\log {\left (d \right )}^{6} + 3 \log {\left (d \right )}^{4} + 3 \log {\left (d \right )}^{2} + 1} + \frac {4 d^{x} x \log {\left (d \right )} \cos {\left (x \right )}}{\log {\left (d \right )}^{6} + 3 \log {\left (d \right )}^{4} + 3 \log {\left (d \right )}^{2} + 1} + \frac {2 d^{x} x \sin {\left (x \right )}}{\log {\left (d \right )}^{6} + 3 \log {\left (d \right )}^{4} + 3 \log {\left (d \right )}^{2} + 1} + \frac {2 d^{x} \log {\left (d \right )}^{3} \sin {\left (x \right )}}{\log {\left (d \right )}^{6} + 3 \log {\left (d \right )}^{4} + 3 \log {\left (d \right )}^{2} + 1} - \frac {6 d^{x} \log {\left (d \right )}^{2} \cos {\left (x \right )}}{\log {\left (d \right )}^{6} + 3 \log {\left (d \right )}^{4} + 3 \log {\left (d \right )}^{2} + 1} - \frac {6 d^{x} \log {\left (d \right )} \sin {\left (x \right )}}{\log {\left (d \right )}^{6} + 3 \log {\left (d \right )}^{4} + 3 \log {\left (d \right )}^{2} + 1} + \frac {2 d^{x} \cos {\left (x \right )}}{\log {\left (d \right )}^{6} + 3 \log {\left (d \right )}^{4} + 3 \log {\left (d \right )}^{2} + 1} & \text {otherwise} \end {cases} \end {gather*}

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)*co
s(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)*s
in(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*l
og(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)/(lo
g(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 [C] Result contains complex when optimal does not.
time = 0.00, size = 3343, normalized size = 20.64

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]

-1/2*(((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^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(ab
s(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*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(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*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*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*(((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^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*log(abs(d))^2 - 3*pi*sgn(d) + 2)^2 + (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)))^2))*cos(1/2*pi*x*sg
n(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(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*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*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)))^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(ab
s(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 - 2*(
(-I*pi^2*x^2*sgn(d) + 2*pi*x^2*log(abs(d))*sgn(d) + I*pi^2*x^2 - 2*pi*x^2*log(abs(d)) - 2*I*x^2*log(abs(d))^2
+ 2*I*pi*x^2*sgn(d) - 2*I*pi*x^2 + 4*x^2*log(abs(d)) - 2*pi*x*sgn(d) + 2*pi*x + 2*I*x^2 + 4*I*x*log(abs(d)) -
4*x - 4*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) + (-I*pi^2*x^2*sgn(d) - 2*pi*x^2*log(abs(d))*sgn(d) + I*pi^2*x^2 + 2*pi*x^2*log(abs(
d)) - 2*I*x^2*log(abs(d))^2 + 2*I*pi*x^2*sgn(d) - 2*I*pi*x^2 - 4*x^2*log(abs(d)) + 2*pi*x*sgn(d) - 2*pi*x + 2*
I*x^2 + 4*I*x*log(abs(d)) + 4*x - 4*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))*abs(d)^x - 2*((I*pi^2*x^2*sgn(d) - 2*pi*x^2*log(abs(d
))*sgn(d) - I*pi^2*x^2 + 2*pi*x^2*log(abs(d)) + 2*I*x^2*log(abs(d))^2 + 2*I*pi*x^2*sgn(d) - 2*I*pi*x^2 + 4*x^2
*log(abs(d)) + 2*pi*x*sgn(d) - 2*pi*x - 2*I*x^2 - 4*I*x*log(abs(d)) - 4*x + 4*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) + (I*pi^2*x^2*
sgn(d) + 2*pi*x^2*log(abs(d))*sgn(d) - I*pi^2*x^2 - 2*pi*x^2*log(abs(d)) + 2*I*x^2*log(abs(d))^2 + 2*I*pi*x^2*
sgn(d) - 2*I*pi*x^2 - 4*x^2*log(abs(d)) - 2*pi*x*sgn(d) + 2*pi*x - 2*I*x^2 - 4*I*x*log(abs(d)) + 4*x + 4*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(a
bs(d)) - 16*I))*abs(d)^x

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Mupad [B]
time = 0.39, size = 133, normalized size = 0.82 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(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*
sin(x)) + d^x*x^2*log(d)^5*sin(x))/(log(d)^2 + 1)^3

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