3.2.41 \(\int d^x x^3 \cos (x) \, dx\) [141]

Optimal. Leaf size=260 \[ -\frac {6 d^x \cos (x)}{\left (1+\log ^2(d)\right )^4}+\frac {36 d^x \cos (x) \log ^2(d)}{\left (1+\log ^2(d)\right )^4}-\frac {6 d^x \cos (x) \log ^4(d)}{\left (1+\log ^2(d)\right )^4}-\frac {18 d^x x \cos (x) \log (d)}{\left (1+\log ^2(d)\right )^3}+\frac {6 d^x x \cos (x) \log ^3(d)}{\left (1+\log ^2(d)\right )^3}+\frac {3 d^x x^2 \cos (x)}{\left (1+\log ^2(d)\right )^2}-\frac {3 d^x x^2 \cos (x) \log ^2(d)}{\left (1+\log ^2(d)\right )^2}+\frac {d^x x^3 \cos (x) \log (d)}{1+\log ^2(d)}+\frac {24 d^x \log (d) \sin (x)}{\left (1+\log ^2(d)\right )^4}-\frac {24 d^x \log ^3(d) \sin (x)}{\left (1+\log ^2(d)\right )^4}-\frac {6 d^x x \sin (x)}{\left (1+\log ^2(d)\right )^3}+\frac {18 d^x x \log ^2(d) \sin (x)}{\left (1+\log ^2(d)\right )^3}-\frac {6 d^x x^2 \log (d) \sin (x)}{\left (1+\log ^2(d)\right )^2}+\frac {d^x x^3 \sin (x)}{1+\log ^2(d)} \]

[Out]

-6*d^x*cos(x)/(1+ln(d)^2)^4+36*d^x*cos(x)*ln(d)^2/(1+ln(d)^2)^4-6*d^x*cos(x)*ln(d)^4/(1+ln(d)^2)^4-18*d^x*x*co
s(x)*ln(d)/(1+ln(d)^2)^3+6*d^x*x*cos(x)*ln(d)^3/(1+ln(d)^2)^3+3*d^x*x^2*cos(x)/(1+ln(d)^2)^2-3*d^x*x^2*cos(x)*
ln(d)^2/(1+ln(d)^2)^2+d^x*x^3*cos(x)*ln(d)/(1+ln(d)^2)+24*d^x*ln(d)*sin(x)/(1+ln(d)^2)^4-24*d^x*ln(d)^3*sin(x)
/(1+ln(d)^2)^4-6*d^x*x*sin(x)/(1+ln(d)^2)^3+18*d^x*x*ln(d)^2*sin(x)/(1+ln(d)^2)^3-6*d^x*x^2*ln(d)*sin(x)/(1+ln
(d)^2)^2+d^x*x^3*sin(x)/(1+ln(d)^2)

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Rubi [A]
time = 0.30, antiderivative size = 260, normalized size of antiderivative = 1.00, number of steps used = 25, number of rules used = 5, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {4518, 4554, 14, 4517, 4553} \begin {gather*} \frac {x^3 d^x \sin (x)}{\log ^2(d)+1}+\frac {x^3 d^x \log (d) \cos (x)}{\log ^2(d)+1}-\frac {6 x^2 d^x \log (d) \sin (x)}{\left (\log ^2(d)+1\right )^2}-\frac {3 x^2 d^x \log ^2(d) \cos (x)}{\left (\log ^2(d)+1\right )^2}+\frac {3 x^2 d^x \cos (x)}{\left (\log ^2(d)+1\right )^2}+\frac {18 x d^x \log ^2(d) \sin (x)}{\left (\log ^2(d)+1\right )^3}-\frac {6 x d^x \sin (x)}{\left (\log ^2(d)+1\right )^3}+\frac {24 d^x \log (d) \sin (x)}{\left (\log ^2(d)+1\right )^4}-\frac {24 d^x \log ^3(d) \sin (x)}{\left (\log ^2(d)+1\right )^4}-\frac {18 x d^x \log (d) \cos (x)}{\left (\log ^2(d)+1\right )^3}+\frac {36 d^x \log ^2(d) \cos (x)}{\left (\log ^2(d)+1\right )^4}-\frac {6 d^x \cos (x)}{\left (\log ^2(d)+1\right )^4}-\frac {6 d^x \log ^4(d) \cos (x)}{\left (\log ^2(d)+1\right )^4}+\frac {6 x d^x \log ^3(d) \cos (x)}{\left (\log ^2(d)+1\right )^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[d^x*x^3*Cos[x],x]

[Out]

(-6*d^x*Cos[x])/(1 + Log[d]^2)^4 + (36*d^x*Cos[x]*Log[d]^2)/(1 + Log[d]^2)^4 - (6*d^x*Cos[x]*Log[d]^4)/(1 + Lo
g[d]^2)^4 - (18*d^x*x*Cos[x]*Log[d])/(1 + Log[d]^2)^3 + (6*d^x*x*Cos[x]*Log[d]^3)/(1 + Log[d]^2)^3 + (3*d^x*x^
2*Cos[x])/(1 + Log[d]^2)^2 - (3*d^x*x^2*Cos[x]*Log[d]^2)/(1 + Log[d]^2)^2 + (d^x*x^3*Cos[x]*Log[d])/(1 + Log[d
]^2) + (24*d^x*Log[d]*Sin[x])/(1 + Log[d]^2)^4 - (24*d^x*Log[d]^3*Sin[x])/(1 + Log[d]^2)^4 - (6*d^x*x*Sin[x])/
(1 + Log[d]^2)^3 + (18*d^x*x*Log[d]^2*Sin[x])/(1 + Log[d]^2)^3 - (6*d^x*x^2*Log[d]*Sin[x])/(1 + Log[d]^2)^2 +
(d^x*x^3*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^3 \cos (x) \, dx &=\frac {d^x x^3 \cos (x) \log (d)}{1+\log ^2(d)}+\frac {d^x x^3 \sin (x)}{1+\log ^2(d)}-3 \int x^2 \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^3 \cos (x) \log (d)}{1+\log ^2(d)}+\frac {d^x x^3 \sin (x)}{1+\log ^2(d)}-3 \int \left (\frac {d^x x^2 \cos (x) \log (d)}{1+\log ^2(d)}+\frac {d^x x^2 \sin (x)}{1+\log ^2(d)}\right ) \, dx\\ &=\frac {d^x x^3 \cos (x) \log (d)}{1+\log ^2(d)}+\frac {d^x x^3 \sin (x)}{1+\log ^2(d)}-\frac {3 \int d^x x^2 \sin (x) \, dx}{1+\log ^2(d)}-\frac {(3 \log (d)) \int d^x x^2 \cos (x) \, dx}{1+\log ^2(d)}\\ &=\frac {3 d^x x^2 \cos (x)}{\left (1+\log ^2(d)\right )^2}-\frac {3 d^x x^2 \cos (x) \log ^2(d)}{\left (1+\log ^2(d)\right )^2}+\frac {d^x x^3 \cos (x) \log (d)}{1+\log ^2(d)}-\frac {6 d^x x^2 \log (d) \sin (x)}{\left (1+\log ^2(d)\right )^2}+\frac {d^x x^3 \sin (x)}{1+\log ^2(d)}+\frac {6 \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}{1+\log ^2(d)}+\frac {(6 \log (d)) \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}{1+\log ^2(d)}\\ &=\frac {3 d^x x^2 \cos (x)}{\left (1+\log ^2(d)\right )^2}-\frac {3 d^x x^2 \cos (x) \log ^2(d)}{\left (1+\log ^2(d)\right )^2}+\frac {d^x x^3 \cos (x) \log (d)}{1+\log ^2(d)}-\frac {6 d^x x^2 \log (d) \sin (x)}{\left (1+\log ^2(d)\right )^2}+\frac {d^x x^3 \sin (x)}{1+\log ^2(d)}+\frac {6 \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}{1+\log ^2(d)}+\frac {(6 \log (d)) \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}{1+\log ^2(d)}\\ &=\frac {3 d^x x^2 \cos (x)}{\left (1+\log ^2(d)\right )^2}-\frac {3 d^x x^2 \cos (x) \log ^2(d)}{\left (1+\log ^2(d)\right )^2}+\frac {d^x x^3 \cos (x) \log (d)}{1+\log ^2(d)}-\frac {6 d^x x^2 \log (d) \sin (x)}{\left (1+\log ^2(d)\right )^2}+\frac {d^x x^3 \sin (x)}{1+\log ^2(d)}-\frac {6 \int d^x x \cos (x) \, dx}{\left (1+\log ^2(d)\right )^2}+2 \frac {(6 \log (d)) \int d^x x \sin (x) \, dx}{\left (1+\log ^2(d)\right )^2}+\frac {\left (6 \log ^2(d)\right ) \int d^x x \cos (x) \, dx}{\left (1+\log ^2(d)\right )^2}\\ &=-\frac {6 d^x x \cos (x) \log (d)}{\left (1+\log ^2(d)\right )^3}+\frac {6 d^x x \cos (x) \log ^3(d)}{\left (1+\log ^2(d)\right )^3}+\frac {3 d^x x^2 \cos (x)}{\left (1+\log ^2(d)\right )^2}-\frac {3 d^x x^2 \cos (x) \log ^2(d)}{\left (1+\log ^2(d)\right )^2}+\frac {d^x x^3 \cos (x) \log (d)}{1+\log ^2(d)}-\frac {6 d^x x \sin (x)}{\left (1+\log ^2(d)\right )^3}+\frac {6 d^x x \log ^2(d) \sin (x)}{\left (1+\log ^2(d)\right )^3}-\frac {6 d^x x^2 \log (d) \sin (x)}{\left (1+\log ^2(d)\right )^2}+\frac {d^x x^3 \sin (x)}{1+\log ^2(d)}+\frac {6 \int \left (\frac {d^x \cos (x) \log (d)}{1+\log ^2(d)}+\frac {d^x \sin (x)}{1+\log ^2(d)}\right ) \, dx}{\left (1+\log ^2(d)\right )^2}+2 \left (-\frac {6 d^x x \cos (x) \log (d)}{\left (1+\log ^2(d)\right )^3}+\frac {6 d^x x \log ^2(d) \sin (x)}{\left (1+\log ^2(d)\right )^3}-\frac {(6 \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}{\left (1+\log ^2(d)\right )^2}\right )-\frac {\left (6 \log ^2(d)\right ) \int \left (\frac {d^x \cos (x) \log (d)}{1+\log ^2(d)}+\frac {d^x \sin (x)}{1+\log ^2(d)}\right ) \, dx}{\left (1+\log ^2(d)\right )^2}\\ &=-\frac {6 d^x x \cos (x) \log (d)}{\left (1+\log ^2(d)\right )^3}+\frac {6 d^x x \cos (x) \log ^3(d)}{\left (1+\log ^2(d)\right )^3}+\frac {3 d^x x^2 \cos (x)}{\left (1+\log ^2(d)\right )^2}-\frac {3 d^x x^2 \cos (x) \log ^2(d)}{\left (1+\log ^2(d)\right )^2}+\frac {d^x x^3 \cos (x) \log (d)}{1+\log ^2(d)}-\frac {6 d^x x \sin (x)}{\left (1+\log ^2(d)\right )^3}+\frac {6 d^x x \log ^2(d) \sin (x)}{\left (1+\log ^2(d)\right )^3}-\frac {6 d^x x^2 \log (d) \sin (x)}{\left (1+\log ^2(d)\right )^2}+\frac {d^x x^3 \sin (x)}{1+\log ^2(d)}+\frac {6 \int d^x \sin (x) \, dx}{\left (1+\log ^2(d)\right )^3}+\frac {(6 \log (d)) \int d^x \cos (x) \, dx}{\left (1+\log ^2(d)\right )^3}+2 \left (-\frac {6 d^x x \cos (x) \log (d)}{\left (1+\log ^2(d)\right )^3}+\frac {6 d^x x \log ^2(d) \sin (x)}{\left (1+\log ^2(d)\right )^3}+\frac {(6 \log (d)) \int d^x \cos (x) \, dx}{\left (1+\log ^2(d)\right )^3}-\frac {\left (6 \log ^2(d)\right ) \int d^x \sin (x) \, dx}{\left (1+\log ^2(d)\right )^3}\right )-\frac {\left (6 \log ^2(d)\right ) \int d^x \sin (x) \, dx}{\left (1+\log ^2(d)\right )^3}-\frac {\left (6 \log ^3(d)\right ) \int d^x \cos (x) \, dx}{\left (1+\log ^2(d)\right )^3}\\ &=-\frac {6 d^x \cos (x)}{\left (1+\log ^2(d)\right )^4}+\frac {12 d^x \cos (x) \log ^2(d)}{\left (1+\log ^2(d)\right )^4}-\frac {6 d^x \cos (x) \log ^4(d)}{\left (1+\log ^2(d)\right )^4}-\frac {6 d^x x \cos (x) \log (d)}{\left (1+\log ^2(d)\right )^3}+\frac {6 d^x x \cos (x) \log ^3(d)}{\left (1+\log ^2(d)\right )^3}+\frac {3 d^x x^2 \cos (x)}{\left (1+\log ^2(d)\right )^2}-\frac {3 d^x x^2 \cos (x) \log ^2(d)}{\left (1+\log ^2(d)\right )^2}+\frac {d^x x^3 \cos (x) \log (d)}{1+\log ^2(d)}+\frac {12 d^x \log (d) \sin (x)}{\left (1+\log ^2(d)\right )^4}-\frac {12 d^x \log ^3(d) \sin (x)}{\left (1+\log ^2(d)\right )^4}-\frac {6 d^x x \sin (x)}{\left (1+\log ^2(d)\right )^3}+\frac {6 d^x x \log ^2(d) \sin (x)}{\left (1+\log ^2(d)\right )^3}-\frac {6 d^x x^2 \log (d) \sin (x)}{\left (1+\log ^2(d)\right )^2}+\frac {d^x x^3 \sin (x)}{1+\log ^2(d)}+2 \left (\frac {12 d^x \cos (x) \log ^2(d)}{\left (1+\log ^2(d)\right )^4}-\frac {6 d^x x \cos (x) \log (d)}{\left (1+\log ^2(d)\right )^3}+\frac {6 d^x \log (d) \sin (x)}{\left (1+\log ^2(d)\right )^4}-\frac {6 d^x \log ^3(d) \sin (x)}{\left (1+\log ^2(d)\right )^4}+\frac {6 d^x x \log ^2(d) \sin (x)}{\left (1+\log ^2(d)\right )^3}\right )\\ \end {align*}

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Mathematica [A]
time = 0.09, size = 168, normalized size = 0.65 \begin {gather*} \frac {d^x \left (\cos (x) \left (3 \left (-2+x^2\right )+x \left (-18+x^2\right ) \log (d)+3 \left (12+x^2\right ) \log ^2(d)+3 x \left (-4+x^2\right ) \log ^3(d)-3 \left (2+x^2\right ) \log ^4(d)+3 x \left (2+x^2\right ) \log ^5(d)-3 x^2 \log ^6(d)+x^3 \log ^7(d)\right )+\left (x \left (-6+x^2\right )-6 \left (-4+x^2\right ) \log (d)+3 x \left (4+x^2\right ) \log ^2(d)-12 \left (2+x^2\right ) \log ^3(d)+3 x \left (6+x^2\right ) \log ^4(d)-6 x^2 \log ^5(d)+x^3 \log ^6(d)\right ) \sin (x)\right )}{\left (1+\log ^2(d)\right )^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[d^x*x^3*Cos[x],x]

[Out]

(d^x*(Cos[x]*(3*(-2 + x^2) + x*(-18 + x^2)*Log[d] + 3*(12 + x^2)*Log[d]^2 + 3*x*(-4 + x^2)*Log[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*Log[d]^7) + (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)*Sin[
x]))/(1 + Log[d]^2)^4

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Maple [C] Result contains complex when optimal does not.
time = 0.07, size = 164, normalized size = 0.63

method	result	size
risch	\(\frac {\left (-6+\ln \left (d \right )^{3} x^{3}+3 i \ln \left (d \right )^{2} x^{3}-3 \ln \left (d \right ) x^{3}-i x^{3}+6 x \ln \left (d \right )+6 i x -3 \ln \left (d \right )^{2} x^{2}-6 i \ln \left (d \right ) x^{2}+3 x^{2}\right ) d^{x} {\mathrm e}^{i x}}{2 \left (\ln \left (d \right )+i\right )^{4}}+\frac {\left (-6+6 x \ln \left (d \right )-6 i x -3 \ln \left (d \right )^{2} x^{2}+6 i \ln \left (d \right ) x^{2}+3 x^{2}+\ln \left (d \right )^{3} x^{3}-3 i \ln \left (d \right )^{2} x^{3}-3 \ln \left (d \right ) x^{3}+i x^{3}\right ) d^{x} {\mathrm e}^{-i x}}{2 \left (\ln \left (d \right )-i\right )^{4}}\)	\(164\)
norman	\(\frac {\frac {\ln \left (d \right ) x^{3} {\mathrm e}^{x \ln \left (d \right )}}{1+\ln \left (d \right )^{2}}+\frac {2 x^{3} {\mathrm e}^{x \ln \left (d \right )} \tan \left (\frac {x}{2}\right )}{1+\ln \left (d \right )^{2}}-\frac {3 \left (\ln \left (d \right )^{2}-1\right ) x^{2} {\mathrm e}^{x \ln \left (d \right )}}{\ln \left (d \right )^{4}+2 \ln \left (d \right )^{2}+1}-\frac {6 \left (\ln \left (d \right )^{4}-6 \ln \left (d \right )^{2}+1\right ) {\mathrm e}^{x \ln \left (d \right )}}{\left (\ln \left (d \right )^{6}+3 \ln \left (d \right )^{4}+3 \ln \left (d \right )^{2}+1\right ) \left (1+\ln \left (d \right )^{2}\right )}-\frac {\ln \left (d \right ) x^{3} {\mathrm e}^{x \ln \left (d \right )} \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{1+\ln \left (d \right )^{2}}+\frac {3 \left (\ln \left (d \right )^{2}-1\right ) x^{2} {\mathrm e}^{x \ln \left (d \right )} \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{\ln \left (d \right )^{4}+2 \ln \left (d \right )^{2}+1}+\frac {6 \left (\ln \left (d \right )^{4}-6 \ln \left (d \right )^{2}+1\right ) {\mathrm e}^{x \ln \left (d \right )} \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{\left (\ln \left (d \right )^{6}+3 \ln \left (d \right )^{4}+3 \ln \left (d \right )^{2}+1\right ) \left (1+\ln \left (d \right )^{2}\right )}-\frac {12 \ln \left (d \right ) x^{2} {\mathrm e}^{x \ln \left (d \right )} \tan \left (\frac {x}{2}\right )}{\ln \left (d \right )^{4}+2 \ln \left (d \right )^{2}+1}+\frac {12 \left (3 \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 ) \left (\ln \left (d \right )^{4}+2 \ln \left (d \right )^{2}+1\right )}+\frac {6 \ln \left (d \right ) \left (\ln \left (d \right )^{2}-3\right ) x \,{\mathrm e}^{x \ln \left (d \right )}}{\left (1+\ln \left (d \right )^{2}\right ) \left (\ln \left (d \right )^{4}+2 \ln \left (d \right )^{2}+1\right )}-\frac {48 \ln \left (d \right ) \left (\ln \left (d \right )^{2}-1\right ) {\mathrm e}^{x \ln \left (d \right )} \tan \left (\frac {x}{2}\right )}{\left (\ln \left (d \right )^{4}+2 \ln \left (d \right )^{2}+1\right ) \left (1+\ln \left (d \right )^{2}\right )^{2}}-\frac {6 \ln \left (d \right ) \left (\ln \left (d \right )^{2}-3\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 ) \left (\ln \left (d \right )^{4}+2 \ln \left (d \right )^{2}+1\right )}}{1+\tan ^{2}\left (\frac {x}{2}\right )}\)	\(441\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(d^x*x^3*cos(x),x,method=_RETURNVERBOSE)

[Out]

1/2*(-6+ln(d)^3*x^3+3*I*ln(d)^2*x^3-3*ln(d)*x^3-I*x^3+6*x*ln(d)+6*I*x-3*ln(d)^2*x^2-6*I*ln(d)*x^2+3*x^2)*d^x/(
ln(d)+I)^4*exp(I*x)+1/2*(-6+6*x*ln(d)-6*I*x-3*ln(d)^2*x^2+6*I*ln(d)*x^2+3*x^2+ln(d)^3*x^3-3*I*ln(d)^2*x^3-3*ln
(d)*x^3+I*x^3)*d^x/(ln(d)-I)^4*exp(-I*x)

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Maxima [A]
time = 1.59, size = 184, normalized size = 0.71 \begin {gather*} \frac {{\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} \cos \left (x\right ) + {\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} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(d^x*x^3*cos(x),x, algorithm="maxima")

[Out]

(((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*cos(x) + ((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*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 = 0.52, size = 202, normalized size = 0.78 \begin {gather*} \frac {{\left (x^{3} \cos \left (x\right ) \log \left (d\right )^{7} - 3 \, x^{2} \cos \left (x\right ) \log \left (d\right )^{6} + 3 \, {\left (x^{3} + 2 \, x\right )} \cos \left (x\right ) \log \left (d\right )^{5} - 3 \, {\left (x^{2} + 2\right )} \cos \left (x\right ) \log \left (d\right )^{4} + 3 \, {\left (x^{3} - 4 \, x\right )} \cos \left (x\right ) \log \left (d\right )^{3} + 3 \, {\left (x^{2} + 12\right )} \cos \left (x\right ) \log \left (d\right )^{2} + {\left (x^{3} - 18 \, x\right )} \cos \left (x\right ) \log \left (d\right ) + 3 \, {\left (x^{2} - 2\right )} \cos \left (x\right ) + {\left (x^{3} \log \left (d\right )^{6} - 6 \, x^{2} \log \left (d\right )^{5} + 3 \, {\left (x^{3} + 6 \, x\right )} \log \left (d\right )^{4} - 12 \, {\left (x^{2} + 2\right )} \log \left (d\right )^{3} + x^{3} + 3 \, {\left (x^{3} + 4 \, x\right )} \log \left (d\right )^{2} - 6 \, {\left (x^{2} - 4\right )} \log \left (d\right ) - 6 \, x\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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(d^x*x^3*cos(x),x, algorithm="fricas")

[Out]

(x^3*cos(x)*log(d)^7 - 3*x^2*cos(x)*log(d)^6 + 3*(x^3 + 2*x)*cos(x)*log(d)^5 - 3*(x^2 + 2)*cos(x)*log(d)^4 + 3
*(x^3 - 4*x)*cos(x)*log(d)^3 + 3*(x^2 + 12)*cos(x)*log(d)^2 + (x^3 - 18*x)*cos(x)*log(d) + 3*(x^2 - 2)*cos(x)
+ (x^3*log(d)^6 - 6*x^2*log(d)^5 + 3*(x^3 + 6*x)*log(d)^4 - 12*(x^2 + 2)*log(d)^3 + x^3 + 3*(x^3 + 4*x)*log(d)
^2 - 6*(x^2 - 4)*log(d) - 6*x)*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 [C] Result contains complex when optimal does not.
time = 2.14, size = 1355, normalized size = 5.21 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(d**x*x**3*cos(x),x)

[Out]

Piecewise((I*x**4*exp(-I*x)*sin(x)/8 + x**4*exp(-I*x)*cos(x)/8 + x**3*exp(-I*x)*sin(x)/4 + I*x**3*exp(-I*x)*co
s(x)/4 - 3*I*x**2*exp(-I*x)*sin(x)/8 + 3*x**2*exp(-I*x)*cos(x)/8 - 3*x*exp(-I*x)*sin(x)/8 - 3*I*x*exp(-I*x)*co
s(x)/8 + 3*I*exp(-I*x)*sin(x)/8, Eq(d, exp(-I))), (-I*x**4*exp(I*x)*sin(x)/8 + x**4*exp(I*x)*cos(x)/8 + x**3*e
xp(I*x)*sin(x)/4 - I*x**3*exp(I*x)*cos(x)/4 + 3*I*x**2*exp(I*x)*sin(x)/8 + 3*x**2*exp(I*x)*cos(x)/8 - 3*x*exp(
I*x)*sin(x)/8 + 3*I*x*exp(I*x)*cos(x)/8 - 3*I*exp(I*x)*sin(x)/8, Eq(d, exp(I))), (d**x*x**3*log(d)**7*cos(x)/(
log(d)**8 + 4*log(d)**6 + 6*log(d)**4 + 4*log(d)**2 + 1) + d**x*x**3*log(d)**6*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)**5*cos(x)/(log(d)**8 + 4*log(d)**6 + 6*log(d)**4 + 4*lo
g(d)**2 + 1) + 3*d**x*x**3*log(d)**4*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)**3*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)**2*sin(x
)/(log(d)**8 + 4*log(d)**6 + 6*log(d)**4 + 4*log(d)**2 + 1) + d**x*x**3*log(d)*cos(x)/(log(d)**8 + 4*log(d)**6
+ 6*log(d)**4 + 4*log(d)**2 + 1) + d**x*x**3*sin(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*cos(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*sin(x)/(log(d)**8 + 4*log(d)**6 + 6*log(d)**4 + 4*log(d)**2 + 1) - 3*d**x*x**2*log(d)**4*cos(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*sin(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)**2*cos(x)/(log(d)**8 + 4*log(d)**6 + 6*log(d)**4 + 4*log(d)**
2 + 1) - 6*d**x*x**2*log(d)*sin(x)/(log(d)**8 + 4*log(d)**6 + 6*log(d)**4 + 4*log(d)**2 + 1) + 3*d**x*x**2*cos
(x)/(log(d)**8 + 4*log(d)**6 + 6*log(d)**4 + 4*log(d)**2 + 1) + 6*d**x*x*log(d)**5*cos(x)/(log(d)**8 + 4*log(d
)**6 + 6*log(d)**4 + 4*log(d)**2 + 1) + 18*d**x*x*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*log(d)**3*cos(x)/(log(d)**8 + 4*log(d)**6 + 6*log(d)**4 + 4*log(d)**2 + 1) + 12*d**
x*x*log(d)**2*sin(x)/(log(d)**8 + 4*log(d)**6 + 6*log(d)**4 + 4*log(d)**2 + 1) - 18*d**x*x*log(d)*cos(x)/(log(
d)**8 + 4*log(d)**6 + 6*log(d)**4 + 4*log(d)**2 + 1) - 6*d**x*x*sin(x)/(log(d)**8 + 4*log(d)**6 + 6*log(d)**4
+ 4*log(d)**2 + 1) - 6*d**x*log(d)**4*cos(x)/(log(d)**8 + 4*log(d)**6 + 6*log(d)**4 + 4*log(d)**2 + 1) - 24*d*
*x*log(d)**3*sin(x)/(log(d)**8 + 4*log(d)**6 + 6*log(d)**4 + 4*log(d)**2 + 1) + 36*d**x*log(d)**2*cos(x)/(log(
d)**8 + 4*log(d)**6 + 6*log(d)**4 + 4*log(d)**2 + 1) + 24*d**x*log(d)*sin(x)/(log(d)**8 + 4*log(d)**6 + 6*log(
d)**4 + 4*log(d)**2 + 1) - 6*d**x*cos(x)/(log(d)**8 + 4*log(d)**6 + 6*log(d)**4 + 4*log(d)**2 + 1), True))

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Giac [C] Result contains complex when optimal does not.
time = 0.90, size = 5065, normalized size = 19.48 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(d^x*x^3*cos(x),x, algorithm="giac")

[Out]

-1/2*(((4*pi + pi^4*sgn(d) - 6*pi^2*log(abs(d))^2*sgn(d) - pi^4 + 6*pi^2*log(abs(d))^2 - 2*log(abs(d))^4 - 4*p
i^3*sgn(d) + 12*pi*log(abs(d))^2*sgn(d) + 4*pi^3 - 12*pi*log(abs(d))^2 + 6*pi^2*sgn(d) - 6*pi^2 + 12*log(abs(d
))^2 - 4*pi*sgn(d) - 2)*(3*pi^2*x^3*log(abs(d))*sgn(d) - 3*pi^2*x^3*log(abs(d)) + 2*x^3*log(abs(d))^3 - 6*pi*x
^3*log(abs(d))*sgn(d) + 6*pi*x^3*log(abs(d)) - 3*pi^2*x^2*sgn(d) + 3*pi^2*x^2 - 6*x^3*log(abs(d)) - 6*x^2*log(
abs(d))^2 + 6*pi*x^2*sgn(d) - 6*pi*x^2 + 6*x^2 + 12*x*log(abs(d)) - 12)/((4*pi + pi^4*sgn(d) - 6*pi^2*log(abs(
d))^2*sgn(d) - pi^4 + 6*pi^2*log(abs(d))^2 - 2*log(abs(d))^4 - 4*pi^3*sgn(d) + 12*pi*log(abs(d))^2*sgn(d) + 4*
pi^3 - 12*pi*log(abs(d))^2 + 6*pi^2*sgn(d) - 6*pi^2 + 12*log(abs(d))^2 - 4*pi*sgn(d) - 2)^2 + 16*(pi^3*log(abs
(d))*sgn(d) - pi*log(abs(d))^3*sgn(d) - pi^3*log(abs(d)) + pi*log(abs(d))^3 - 3*pi^2*log(abs(d))*sgn(d) + 3*pi
^2*log(abs(d)) - 2*log(abs(d))^3 + 3*pi*log(abs(d))*sgn(d) - 3*pi*log(abs(d)) + 2*log(abs(d)))^2) - 4*(pi^3*x^
3*sgn(d) - 3*pi*x^3*log(abs(d))^2*sgn(d) - pi^3*x^3 + 3*pi*x^3*log(abs(d))^2 - 3*pi^2*x^3*sgn(d) + 3*pi^2*x^3
- 6*x^3*log(abs(d))^2 + 3*pi*x^3*sgn(d) + 6*pi*x^2*log(abs(d))*sgn(d) - 3*pi*x^3 - 6*pi*x^2*log(abs(d)) + 2*x^
3 + 12*x^2*log(abs(d)) - 6*pi*x*sgn(d) + 6*pi*x - 12*x)*(pi^3*log(abs(d))*sgn(d) - pi*log(abs(d))^3*sgn(d) - p
i^3*log(abs(d)) + pi*log(abs(d))^3 - 3*pi^2*log(abs(d))*sgn(d) + 3*pi^2*log(abs(d)) - 2*log(abs(d))^3 + 3*pi*l
og(abs(d))*sgn(d) - 3*pi*log(abs(d)) + 2*log(abs(d)))/((4*pi + pi^4*sgn(d) - 6*pi^2*log(abs(d))^2*sgn(d) - pi^
4 + 6*pi^2*log(abs(d))^2 - 2*log(abs(d))^4 - 4*pi^3*sgn(d) + 12*pi*log(abs(d))^2*sgn(d) + 4*pi^3 - 12*pi*log(a
bs(d))^2 + 6*pi^2*sgn(d) - 6*pi^2 + 12*log(abs(d))^2 - 4*pi*sgn(d) - 2)^2 + 16*(pi^3*log(abs(d))*sgn(d) - pi*l
og(abs(d))^3*sgn(d) - pi^3*log(abs(d)) + pi*log(abs(d))^3 - 3*pi^2*log(abs(d))*sgn(d) + 3*pi^2*log(abs(d)) - 2
*log(abs(d))^3 + 3*pi*log(abs(d))*sgn(d) - 3*pi*log(abs(d)) + 2*log(abs(d)))^2))*cos(1/2*pi*x*sgn(d) - 1/2*pi*
x + x) + ((4*pi + pi^4*sgn(d) - 6*pi^2*log(abs(d))^2*sgn(d) - pi^4 + 6*pi^2*log(abs(d))^2 - 2*log(abs(d))^4 -
4*pi^3*sgn(d) + 12*pi*log(abs(d))^2*sgn(d) + 4*pi^3 - 12*pi*log(abs(d))^2 + 6*pi^2*sgn(d) - 6*pi^2 + 12*log(ab
s(d))^2 - 4*pi*sgn(d) - 2)*(pi^3*x^3*sgn(d) - 3*pi*x^3*log(abs(d))^2*sgn(d) - pi^3*x^3 + 3*pi*x^3*log(abs(d))^
2 - 3*pi^2*x^3*sgn(d) + 3*pi^2*x^3 - 6*x^3*log(abs(d))^2 + 3*pi*x^3*sgn(d) + 6*pi*x^2*log(abs(d))*sgn(d) - 3*p
i*x^3 - 6*pi*x^2*log(abs(d)) + 2*x^3 + 12*x^2*log(abs(d)) - 6*pi*x*sgn(d) + 6*pi*x - 12*x)/((4*pi + pi^4*sgn(d
) - 6*pi^2*log(abs(d))^2*sgn(d) - pi^4 + 6*pi^2*log(abs(d))^2 - 2*log(abs(d))^4 - 4*pi^3*sgn(d) + 12*pi*log(ab
s(d))^2*sgn(d) + 4*pi^3 - 12*pi*log(abs(d))^2 + 6*pi^2*sgn(d) - 6*pi^2 + 12*log(abs(d))^2 - 4*pi*sgn(d) - 2)^2
+ 16*(pi^3*log(abs(d))*sgn(d) - pi*log(abs(d))^3*sgn(d) - pi^3*log(abs(d)) + pi*log(abs(d))^3 - 3*pi^2*log(ab
s(d))*sgn(d) + 3*pi^2*log(abs(d)) - 2*log(abs(d))^3 + 3*pi*log(abs(d))*sgn(d) - 3*pi*log(abs(d)) + 2*log(abs(d
)))^2) + 4*(3*pi^2*x^3*log(abs(d))*sgn(d) - 3*pi^2*x^3*log(abs(d)) + 2*x^3*log(abs(d))^3 - 6*pi*x^3*log(abs(d)
)*sgn(d) + 6*pi*x^3*log(abs(d)) - 3*pi^2*x^2*sgn(d) + 3*pi^2*x^2 - 6*x^3*log(abs(d)) - 6*x^2*log(abs(d))^2 + 6
*pi*x^2*sgn(d) - 6*pi*x^2 + 6*x^2 + 12*x*log(abs(d)) - 12)*(pi^3*log(abs(d))*sgn(d) - pi*log(abs(d))^3*sgn(d)
- pi^3*log(abs(d)) + pi*log(abs(d))^3 - 3*pi^2*log(abs(d))*sgn(d) + 3*pi^2*log(abs(d)) - 2*log(abs(d))^3 + 3*p
i*log(abs(d))*sgn(d) - 3*pi*log(abs(d)) + 2*log(abs(d)))/((4*pi + pi^4*sgn(d) - 6*pi^2*log(abs(d))^2*sgn(d) -
pi^4 + 6*pi^2*log(abs(d))^2 - 2*log(abs(d))^4 - 4*pi^3*sgn(d) + 12*pi*log(abs(d))^2*sgn(d) + 4*pi^3 - 12*pi*lo
g(abs(d))^2 + 6*pi^2*sgn(d) - 6*pi^2 + 12*log(abs(d))^2 - 4*pi*sgn(d) - 2)^2 + 16*(pi^3*log(abs(d))*sgn(d) - p
i*log(abs(d))^3*sgn(d) - pi^3*log(abs(d)) + pi*log(abs(d))^3 - 3*pi^2*log(abs(d))*sgn(d) + 3*pi^2*log(abs(d))
- 2*log(abs(d))^3 + 3*pi*log(abs(d))*sgn(d) - 3*pi*log(abs(d)) + 2*log(abs(d)))^2))*sin(1/2*pi*x*sgn(d) - 1/2*
pi*x + x))*abs(d)^x + 1/2*(((4*pi - pi^4*sgn(d) + 6*pi^2*log(abs(d))^2*sgn(d) + pi^4 - 6*pi^2*log(abs(d))^2 +
2*log(abs(d))^4 - 4*pi^3*sgn(d) + 12*pi*log(abs(d))^2*sgn(d) + 4*pi^3 - 12*pi*log(abs(d))^2 - 6*pi^2*sgn(d) +
6*pi^2 - 12*log(abs(d))^2 - 4*pi*sgn(d) + 2)*(3*pi^2*x^3*log(abs(d))*sgn(d) - 3*pi^2*x^3*log(abs(d)) + 2*x^3*l
og(abs(d))^3 + 6*pi*x^3*log(abs(d))*sgn(d) - 6*pi*x^3*log(abs(d)) - 3*pi^2*x^2*sgn(d) + 3*pi^2*x^2 - 6*x^3*log
(abs(d)) - 6*x^2*log(abs(d))^2 - 6*pi*x^2*sgn(d) + 6*pi*x^2 + 6*x^2 + 12*x*log(abs(d)) - 12)/((4*pi - pi^4*sgn
(d) + 6*pi^2*log(abs(d))^2*sgn(d) + pi^4 - 6*pi^2*log(abs(d))^2 + 2*log(abs(d))^4 - 4*pi^3*sgn(d) + 12*pi*log(
abs(d))^2*sgn(d) + 4*pi^3 - 12*pi*log(abs(d))^2 - 6*pi^2*sgn(d) + 6*pi^2 - 12*log(abs(d))^2 - 4*pi*sgn(d) + 2)
^2 + 16*(pi^3*log(abs(d))*sgn(d) - pi*log(abs(d))^3*sgn(d) - pi^3*log(abs(d)) + pi*log(abs(d))^3 + 3*pi^2*log(
abs(d))*sgn(d) - 3*pi^2*log(abs(d)) + 2*log(abs(d))^3 + 3*pi*log(abs(d))*sgn(d) - 3*pi*log(abs(d)) - 2*log(abs
(d)))^2) + 4*(pi^3*x^3*sgn(d) - 3*pi*x^3*log(ab...

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Mupad [B]
time = 0.64, size = 232, normalized size = 0.89 \begin {gather*} -\frac {d^x\,\left (6\,\cos \left (x\right )-3\,x^2\,\cos \left (x\right )-x^3\,\sin \left (x\right )+6\,x\,\sin \left (x\right )\right )-d^x\,{\ln \left (d\right )}^5\,\left (3\,x^3\,\cos \left (x\right )-6\,x^2\,\sin \left (x\right )+6\,x\,\cos \left (x\right )\right )+d^x\,{\ln \left (d\right )}^4\,\left (6\,\cos \left (x\right )+3\,x^2\,\cos \left (x\right )-3\,x^3\,\sin \left (x\right )-18\,x\,\sin \left (x\right )\right )+d^x\,{\ln \left (d\right )}^3\,\left (24\,\sin \left (x\right )-3\,x^3\,\cos \left (x\right )+12\,x^2\,\sin \left (x\right )+12\,x\,\cos \left (x\right )\right )-d^x\,{\ln \left (d\right )}^2\,\left (36\,\cos \left (x\right )+3\,x^2\,\cos \left (x\right )+3\,x^3\,\sin \left (x\right )+12\,x\,\sin \left (x\right )\right )+d^x\,{\ln \left (d\right )}^6\,\left (3\,x^2\,\cos \left (x\right )-x^3\,\sin \left (x\right )\right )-d^x\,\ln \left (d\right )\,\left (24\,\sin \left (x\right )+x^3\,\cos \left (x\right )-6\,x^2\,\sin \left (x\right )-18\,x\,\cos \left (x\right )\right )-d^x\,x^3\,{\ln \left (d\right )}^7\,\cos \left (x\right )}{{\left ({\ln \left (d\right )}^2+1\right )}^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(d^x*x^3*cos(x),x)

[Out]

-(d^x*(6*cos(x) - 3*x^2*cos(x) - x^3*sin(x) + 6*x*sin(x)) - d^x*log(d)^5*(3*x^3*cos(x) - 6*x^2*sin(x) + 6*x*co
s(x)) + d^x*log(d)^4*(6*cos(x) + 3*x^2*cos(x) - 3*x^3*sin(x) - 18*x*sin(x)) + d^x*log(d)^3*(24*sin(x) - 3*x^3*
cos(x) + 12*x^2*sin(x) + 12*x*cos(x)) - d^x*log(d)^2*(36*cos(x) + 3*x^2*cos(x) + 3*x^3*sin(x) + 12*x*sin(x)) +
d^x*log(d)^6*(3*x^2*cos(x) - x^3*sin(x)) - d^x*log(d)*(24*sin(x) + x^3*cos(x) - 6*x^2*sin(x) - 18*x*cos(x)) -
d^x*x^3*log(d)^7*cos(x))/(log(d)^2 + 1)^4

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