3.24.6 \(\int \frac {e^{16 x^8} (-839808-209952 x)+e^{16 x^8} (10917504 x^7+6718464 x^8+839808 x^9) \log (13+8 x+x^2)+e^{12 x^8} (279936+69984 x) \log ^4(13+8 x+x^2)+e^{12 x^8} (-3639168 x^7-2239488 x^8-279936 x^9) \log ^5(13+8 x+x^2)+e^{8 x^8} (-31104-7776 x) \log ^8(13+8 x+x^2)+e^{8 x^8} (404352 x^7+248832 x^8+31104 x^9) \log ^9(13+8 x+x^2)+e^{4 x^8} (1152+288 x) \log ^{12}(13+8 x+x^2)+e^{4 x^8} (-14976 x^7-9216 x^8-1152 x^9) \log ^{13}(13+8 x+x^2)}{(13+8 x+x^2) \log ^{17}(13+8 x+x^2)} \, dx\)

Optimal. Leaf size=23 \[ \left (-1+\frac {9 e^{4 x^8}}{\log ^4\left (-3+(4+x)^2\right )}\right )^4 \]

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Rubi [A]  time = 2.24, antiderivative size = 24, normalized size of antiderivative = 1.04, number of steps used = 4, number of rules used = 4, integrand size = 246, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.016, Rules used = {6688, 12, 6712, 32} \begin {gather*} \left (1-\frac {9 e^{4 x^8}}{\log ^4\left (x^2+8 x+13\right )}\right )^4 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(16*x^8)*(-839808 - 209952*x) + E^(16*x^8)*(10917504*x^7 + 6718464*x^8 + 839808*x^9)*Log[13 + 8*x + x^2
] + E^(12*x^8)*(279936 + 69984*x)*Log[13 + 8*x + x^2]^4 + E^(12*x^8)*(-3639168*x^7 - 2239488*x^8 - 279936*x^9)
*Log[13 + 8*x + x^2]^5 + E^(8*x^8)*(-31104 - 7776*x)*Log[13 + 8*x + x^2]^8 + E^(8*x^8)*(404352*x^7 + 248832*x^
8 + 31104*x^9)*Log[13 + 8*x + x^2]^9 + E^(4*x^8)*(1152 + 288*x)*Log[13 + 8*x + x^2]^12 + E^(4*x^8)*(-14976*x^7
- 9216*x^8 - 1152*x^9)*Log[13 + 8*x + x^2]^13)/((13 + 8*x + x^2)*Log[13 + 8*x + x^2]^17),x]

[Out]

(1 - (9*E^(4*x^8))/Log[13 + 8*x + x^2]^4)^4

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 6712

Int[(u_)*(v_)^(r_.)*((a_.)*(v_)^(p_.) + (b_.)*(w_)^(q_.))^(m_.), x_Symbol] :> With[{c = Simplify[u/(p*w*D[v, x
] - q*v*D[w, x])]}, -Dist[c*q, Subst[Int[(a + b*x^q)^m, x], x, v^(m*p + r + 1)*w], x] /; FreeQ[c, x]] /; FreeQ
[{a, b, m, p, q, r}, x] && EqQ[p + q*(m*p + r + 1), 0] && IntegerQ[q] && IntegerQ[m]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {288 e^{4 x^8} \left (-4-x+4 x^7 \left (13+8 x+x^2\right ) \log \left (13+8 x+x^2\right )\right ) \left (9 e^{4 x^8}-\log ^4\left (13+8 x+x^2\right )\right )^3}{\left (13+8 x+x^2\right ) \log ^{17}\left (13+8 x+x^2\right )} \, dx\\ &=288 \int \frac {e^{4 x^8} \left (-4-x+4 x^7 \left (13+8 x+x^2\right ) \log \left (13+8 x+x^2\right )\right ) \left (9 e^{4 x^8}-\log ^4\left (13+8 x+x^2\right )\right )^3}{\left (13+8 x+x^2\right ) \log ^{17}\left (13+8 x+x^2\right )} \, dx\\ &=36 \operatorname {Subst}\left (\int (-1+9 x)^3 \, dx,x,\frac {e^{4 x^8}}{\log ^4\left (13+8 x+x^2\right )}\right )\\ &=\left (1-\frac {9 e^{4 x^8}}{\log ^4\left (13+8 x+x^2\right )}\right )^4\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.08, size = 83, normalized size = 3.61 \begin {gather*} \frac {9 e^{4 x^8} \left (729 e^{12 x^8}-324 e^{8 x^8} \log ^4\left (13+8 x+x^2\right )+54 e^{4 x^8} \log ^8\left (13+8 x+x^2\right )-4 \log ^{12}\left (13+8 x+x^2\right )\right )}{\log ^{16}\left (13+8 x+x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(16*x^8)*(-839808 - 209952*x) + E^(16*x^8)*(10917504*x^7 + 6718464*x^8 + 839808*x^9)*Log[13 + 8*x
+ x^2] + E^(12*x^8)*(279936 + 69984*x)*Log[13 + 8*x + x^2]^4 + E^(12*x^8)*(-3639168*x^7 - 2239488*x^8 - 27993
6*x^9)*Log[13 + 8*x + x^2]^5 + E^(8*x^8)*(-31104 - 7776*x)*Log[13 + 8*x + x^2]^8 + E^(8*x^8)*(404352*x^7 + 248
832*x^8 + 31104*x^9)*Log[13 + 8*x + x^2]^9 + E^(4*x^8)*(1152 + 288*x)*Log[13 + 8*x + x^2]^12 + E^(4*x^8)*(-149
76*x^7 - 9216*x^8 - 1152*x^9)*Log[13 + 8*x + x^2]^13)/((13 + 8*x + x^2)*Log[13 + 8*x + x^2]^17),x]

[Out]

(9*E^(4*x^8)*(729*E^(12*x^8) - 324*E^(8*x^8)*Log[13 + 8*x + x^2]^4 + 54*E^(4*x^8)*Log[13 + 8*x + x^2]^8 - 4*Lo
g[13 + 8*x + x^2]^12))/Log[13 + 8*x + x^2]^16

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fricas [B]  time = 0.82, size = 79, normalized size = 3.43 \begin {gather*} -\frac {9 \, {\left (4 \, e^{\left (4 \, x^{8}\right )} \log \left (x^{2} + 8 \, x + 13\right )^{12} - 54 \, e^{\left (8 \, x^{8}\right )} \log \left (x^{2} + 8 \, x + 13\right )^{8} + 324 \, e^{\left (12 \, x^{8}\right )} \log \left (x^{2} + 8 \, x + 13\right )^{4} - 729 \, e^{\left (16 \, x^{8}\right )}\right )}}{\log \left (x^{2} + 8 \, x + 13\right )^{16}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-1152*x^9-9216*x^8-14976*x^7)*exp(x^8)^4*log(x^2+8*x+13)^13+(288*x+1152)*exp(x^8)^4*log(x^2+8*x+13
)^12+(31104*x^9+248832*x^8+404352*x^7)*exp(x^8)^8*log(x^2+8*x+13)^9+(-7776*x-31104)*exp(x^8)^8*log(x^2+8*x+13)
^8+(-279936*x^9-2239488*x^8-3639168*x^7)*exp(x^8)^12*log(x^2+8*x+13)^5+(69984*x+279936)*exp(x^8)^12*log(x^2+8*
x+13)^4+(839808*x^9+6718464*x^8+10917504*x^7)*exp(x^8)^16*log(x^2+8*x+13)+(-209952*x-839808)*exp(x^8)^16)/(x^2
+8*x+13)/log(x^2+8*x+13)^17,x, algorithm="fricas")

[Out]

-9*(4*e^(4*x^8)*log(x^2 + 8*x + 13)^12 - 54*e^(8*x^8)*log(x^2 + 8*x + 13)^8 + 324*e^(12*x^8)*log(x^2 + 8*x + 1
3)^4 - 729*e^(16*x^8))/log(x^2 + 8*x + 13)^16

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giac [B]  time = 0.35, size = 79, normalized size = 3.43 \begin {gather*} -\frac {9 \, {\left (4 \, e^{\left (4 \, x^{8}\right )} \log \left (x^{2} + 8 \, x + 13\right )^{12} - 54 \, e^{\left (8 \, x^{8}\right )} \log \left (x^{2} + 8 \, x + 13\right )^{8} + 324 \, e^{\left (12 \, x^{8}\right )} \log \left (x^{2} + 8 \, x + 13\right )^{4} - 729 \, e^{\left (16 \, x^{8}\right )}\right )}}{\log \left (x^{2} + 8 \, x + 13\right )^{16}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-1152*x^9-9216*x^8-14976*x^7)*exp(x^8)^4*log(x^2+8*x+13)^13+(288*x+1152)*exp(x^8)^4*log(x^2+8*x+13
)^12+(31104*x^9+248832*x^8+404352*x^7)*exp(x^8)^8*log(x^2+8*x+13)^9+(-7776*x-31104)*exp(x^8)^8*log(x^2+8*x+13)
^8+(-279936*x^9-2239488*x^8-3639168*x^7)*exp(x^8)^12*log(x^2+8*x+13)^5+(69984*x+279936)*exp(x^8)^12*log(x^2+8*
x+13)^4+(839808*x^9+6718464*x^8+10917504*x^7)*exp(x^8)^16*log(x^2+8*x+13)+(-209952*x-839808)*exp(x^8)^16)/(x^2
+8*x+13)/log(x^2+8*x+13)^17,x, algorithm="giac")

[Out]

-9*(4*e^(4*x^8)*log(x^2 + 8*x + 13)^12 - 54*e^(8*x^8)*log(x^2 + 8*x + 13)^8 + 324*e^(12*x^8)*log(x^2 + 8*x + 1
3)^4 - 729*e^(16*x^8))/log(x^2 + 8*x + 13)^16

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maple [B]  time = 0.69, size = 80, normalized size = 3.48

method	result	size
risch	\(\frac {9 \,{\mathrm e}^{4 x^{8}} \left (729 \,{\mathrm e}^{12 x^{8}}-324 \,{\mathrm e}^{8 x^{8}} \ln \left (x^{2}+8 x +13\right )^{4}+54 \,{\mathrm e}^{4 x^{8}} \ln \left (x^{2}+8 x +13\right )^{8}-4 \ln \left (x^{2}+8 x +13\right )^{12}\right )}{\ln \left (x^{2}+8 x +13\right )^{16}}\)	\(80\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-1152*x^9-9216*x^8-14976*x^7)*exp(x^8)^4*ln(x^2+8*x+13)^13+(288*x+1152)*exp(x^8)^4*ln(x^2+8*x+13)^12+(31
104*x^9+248832*x^8+404352*x^7)*exp(x^8)^8*ln(x^2+8*x+13)^9+(-7776*x-31104)*exp(x^8)^8*ln(x^2+8*x+13)^8+(-27993
6*x^9-2239488*x^8-3639168*x^7)*exp(x^8)^12*ln(x^2+8*x+13)^5+(69984*x+279936)*exp(x^8)^12*ln(x^2+8*x+13)^4+(839
808*x^9+6718464*x^8+10917504*x^7)*exp(x^8)^16*ln(x^2+8*x+13)+(-209952*x-839808)*exp(x^8)^16)/(x^2+8*x+13)/ln(x
^2+8*x+13)^17,x,method=_RETURNVERBOSE)

[Out]

9*exp(4*x^8)*(729*exp(12*x^8)-324*exp(8*x^8)*ln(x^2+8*x+13)^4+54*exp(4*x^8)*ln(x^2+8*x+13)^8-4*ln(x^2+8*x+13)^
12)/ln(x^2+8*x+13)^16

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maxima [B]  time = 2.71, size = 79, normalized size = 3.43 \begin {gather*} -\frac {9 \, {\left (4 \, e^{\left (4 \, x^{8}\right )} \log \left (x^{2} + 8 \, x + 13\right )^{12} - 54 \, e^{\left (8 \, x^{8}\right )} \log \left (x^{2} + 8 \, x + 13\right )^{8} + 324 \, e^{\left (12 \, x^{8}\right )} \log \left (x^{2} + 8 \, x + 13\right )^{4} - 729 \, e^{\left (16 \, x^{8}\right )}\right )}}{\log \left (x^{2} + 8 \, x + 13\right )^{16}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-1152*x^9-9216*x^8-14976*x^7)*exp(x^8)^4*log(x^2+8*x+13)^13+(288*x+1152)*exp(x^8)^4*log(x^2+8*x+13
)^12+(31104*x^9+248832*x^8+404352*x^7)*exp(x^8)^8*log(x^2+8*x+13)^9+(-7776*x-31104)*exp(x^8)^8*log(x^2+8*x+13)
^8+(-279936*x^9-2239488*x^8-3639168*x^7)*exp(x^8)^12*log(x^2+8*x+13)^5+(69984*x+279936)*exp(x^8)^12*log(x^2+8*
x+13)^4+(839808*x^9+6718464*x^8+10917504*x^7)*exp(x^8)^16*log(x^2+8*x+13)+(-209952*x-839808)*exp(x^8)^16)/(x^2
+8*x+13)/log(x^2+8*x+13)^17,x, algorithm="maxima")

[Out]

-9*(4*e^(4*x^8)*log(x^2 + 8*x + 13)^12 - 54*e^(8*x^8)*log(x^2 + 8*x + 13)^8 + 324*e^(12*x^8)*log(x^2 + 8*x + 1
3)^4 - 729*e^(16*x^8))/log(x^2 + 8*x + 13)^16

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mupad [F(-1)]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \text {Hanged} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(16*x^8)*(209952*x + 839808) - exp(4*x^8)*log(8*x + x^2 + 13)^12*(288*x + 1152) + exp(8*x^8)*log(8*x
+ x^2 + 13)^8*(7776*x + 31104) - exp(12*x^8)*log(8*x + x^2 + 13)^4*(69984*x + 279936) - exp(16*x^8)*log(8*x +
x^2 + 13)*(10917504*x^7 + 6718464*x^8 + 839808*x^9) + exp(4*x^8)*log(8*x + x^2 + 13)^13*(14976*x^7 + 9216*x^8
+ 1152*x^9) - exp(8*x^8)*log(8*x + x^2 + 13)^9*(404352*x^7 + 248832*x^8 + 31104*x^9) + exp(12*x^8)*log(8*x + x
^2 + 13)^5*(3639168*x^7 + 2239488*x^8 + 279936*x^9))/(log(8*x + x^2 + 13)^17*(8*x + x^2 + 13)),x)

[Out]

\text{Hanged}

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sympy [B]  time = 0.80, size = 92, normalized size = 4.00 \begin {gather*} \frac {6561 e^{16 x^{8}} \log {\left (x^{2} + 8 x + 13 \right )}^{24} - 2916 e^{12 x^{8}} \log {\left (x^{2} + 8 x + 13 \right )}^{28} + 486 e^{8 x^{8}} \log {\left (x^{2} + 8 x + 13 \right )}^{32} - 36 e^{4 x^{8}} \log {\left (x^{2} + 8 x + 13 \right )}^{36}}{\log {\left (x^{2} + 8 x + 13 \right )}^{40}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-1152*x**9-9216*x**8-14976*x**7)*exp(x**8)**4*ln(x**2+8*x+13)**13+(288*x+1152)*exp(x**8)**4*ln(x**
2+8*x+13)**12+(31104*x**9+248832*x**8+404352*x**7)*exp(x**8)**8*ln(x**2+8*x+13)**9+(-7776*x-31104)*exp(x**8)**
8*ln(x**2+8*x+13)**8+(-279936*x**9-2239488*x**8-3639168*x**7)*exp(x**8)**12*ln(x**2+8*x+13)**5+(69984*x+279936
)*exp(x**8)**12*ln(x**2+8*x+13)**4+(839808*x**9+6718464*x**8+10917504*x**7)*exp(x**8)**16*ln(x**2+8*x+13)+(-20
9952*x-839808)*exp(x**8)**16)/(x**2+8*x+13)/ln(x**2+8*x+13)**17,x)

[Out]

(6561*exp(16*x**8)*log(x**2 + 8*x + 13)**24 - 2916*exp(12*x**8)*log(x**2 + 8*x + 13)**28 + 486*exp(8*x**8)*log
(x**2 + 8*x + 13)**32 - 36*exp(4*x**8)*log(x**2 + 8*x + 13)**36)/log(x**2 + 8*x + 13)**40

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