3.31.48 \(\int \frac {720 x^4-840 x^5+245 x^6+e^8 (12+432 x^2-504 x^3+147 x^4)+e^4 (-1152 x^3+1344 x^4-392 x^5)}{144 x^5-168 x^6+49 x^7+e^8 (12 x-7 x^2+144 x^3-168 x^4+49 x^5)+e^4 (-288 x^4+336 x^5-98 x^6)} \, dx\)

Optimal. Leaf size=32 \[ \log \left (x \left (x-\frac {x^2}{e^4}\right )^2-\frac {x}{-x+4 (-3+2 x)}\right ) \]

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Rubi [A]  time = 0.23, antiderivative size = 56, normalized size of antiderivative = 1.75, number of steps used = 3, number of rules used = 2, integrand size = 124, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.016, Rules used = {2074, 1587} \begin {gather*} \log \left (-7 x^5+2 \left (6+7 e^4\right ) x^4-e^4 \left (24+7 e^4\right ) x^3+12 e^8 x^2+e^8\right )-\log (12-7 x)+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(720*x^4 - 840*x^5 + 245*x^6 + E^8*(12 + 432*x^2 - 504*x^3 + 147*x^4) + E^4*(-1152*x^3 + 1344*x^4 - 392*x^
5))/(144*x^5 - 168*x^6 + 49*x^7 + E^8*(12*x - 7*x^2 + 144*x^3 - 168*x^4 + 49*x^5) + E^4*(-288*x^4 + 336*x^5 -
98*x^6)),x]

[Out]

-Log[12 - 7*x] + Log[x] + Log[E^8 + 12*E^8*x^2 - E^4*(24 + 7*E^4)*x^3 + 2*(6 + 7*E^4)*x^4 - 7*x^5]

Rule 1587

Int[(Pp_)/(Qq_), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[(Coeff[Pp, x, p]*Log[RemoveConte
nt[Qq, x]])/(q*Coeff[Qq, x, q]), x] /; EqQ[p, q - 1] && EqQ[Pp, Simplify[(Coeff[Pp, x, p]*D[Qq, x])/(q*Coeff[Q
q, x, q])]]] /; PolyQ[Pp, x] && PolyQ[Qq, x]

Rule 2074

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {1}{x}-\frac {7}{-12+7 x}+\frac {x \left (24 e^8-3 e^4 \left (24+7 e^4\right ) x+8 \left (6+7 e^4\right ) x^2-35 x^3\right )}{e^8+12 e^8 x^2-24 e^4 \left (1+\frac {7 e^4}{24}\right ) x^3+12 \left (1+\frac {7 e^4}{6}\right ) x^4-7 x^5}\right ) \, dx\\ &=-\log (12-7 x)+\log (x)+\int \frac {x \left (24 e^8-3 e^4 \left (24+7 e^4\right ) x+8 \left (6+7 e^4\right ) x^2-35 x^3\right )}{e^8+12 e^8 x^2-24 e^4 \left (1+\frac {7 e^4}{24}\right ) x^3+12 \left (1+\frac {7 e^4}{6}\right ) x^4-7 x^5} \, dx\\ &=-\log (12-7 x)+\log (x)+\log \left (e^8+12 e^8 x^2-e^4 \left (24+7 e^4\right ) x^3+2 \left (6+7 e^4\right ) x^4-7 x^5\right )\\ \end {aligned} \end {gather*}

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Mathematica [C]  time = 0.32, size = 253, normalized size = 7.91 \begin {gather*} -\log (12-7 x)+\log (x)+12 \text {RootSum}\left [-e^8-12 e^8 \text {$\#$1}^2+24 e^4 \text {$\#$1}^3+7 e^8 \text {$\#$1}^3-12 \text {$\#$1}^4-14 e^4 \text {$\#$1}^4+7 \text {$\#$1}^5\&,\frac {e^4 \log (x-\text {$\#$1})-\log (x-\text {$\#$1}) \text {$\#$1}}{-24 e^4+48 \text {$\#$1}+21 e^4 \text {$\#$1}-35 \text {$\#$1}^2}\&\right ]+\text {RootSum}\left [-e^8-12 e^8 \text {$\#$1}^2+24 e^4 \text {$\#$1}^3+7 e^8 \text {$\#$1}^3-12 \text {$\#$1}^4-14 e^4 \text {$\#$1}^4+7 \text {$\#$1}^5\&,\frac {-36 e^4 \log (x-\text {$\#$1})+60 \log (x-\text {$\#$1}) \text {$\#$1}+21 e^4 \log (x-\text {$\#$1}) \text {$\#$1}-35 \log (x-\text {$\#$1}) \text {$\#$1}^2}{-24 e^4+48 \text {$\#$1}+21 e^4 \text {$\#$1}-35 \text {$\#$1}^2}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(720*x^4 - 840*x^5 + 245*x^6 + E^8*(12 + 432*x^2 - 504*x^3 + 147*x^4) + E^4*(-1152*x^3 + 1344*x^4 -
392*x^5))/(144*x^5 - 168*x^6 + 49*x^7 + E^8*(12*x - 7*x^2 + 144*x^3 - 168*x^4 + 49*x^5) + E^4*(-288*x^4 + 336*
x^5 - 98*x^6)),x]

[Out]

-Log[12 - 7*x] + Log[x] + 12*RootSum[-E^8 - 12*E^8*#1^2 + 24*E^4*#1^3 + 7*E^8*#1^3 - 12*#1^4 - 14*E^4*#1^4 + 7
*#1^5 & , (E^4*Log[x - #1] - Log[x - #1]*#1)/(-24*E^4 + 48*#1 + 21*E^4*#1 - 35*#1^2) & ] + RootSum[-E^8 - 12*E
^8*#1^2 + 24*E^4*#1^3 + 7*E^8*#1^3 - 12*#1^4 - 14*E^4*#1^4 + 7*#1^5 & , (-36*E^4*Log[x - #1] + 60*Log[x - #1]*
#1 + 21*E^4*Log[x - #1]*#1 - 35*Log[x - #1]*#1^2)/(-24*E^4 + 48*#1 + 21*E^4*#1 - 35*#1^2) & ]

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fricas [B]  time = 0.57, size = 53, normalized size = 1.66 \begin {gather*} \log \left (7 \, x^{6} - 12 \, x^{5} + {\left (7 \, x^{4} - 12 \, x^{3} - x\right )} e^{8} - 2 \, {\left (7 \, x^{5} - 12 \, x^{4}\right )} e^{4}\right ) - \log \left (7 \, x - 12\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((147*x^4-504*x^3+432*x^2+12)*exp(4)^2+(-392*x^5+1344*x^4-1152*x^3)*exp(4)+245*x^6-840*x^5+720*x^4)/
((49*x^5-168*x^4+144*x^3-7*x^2+12*x)*exp(4)^2+(-98*x^6+336*x^5-288*x^4)*exp(4)+49*x^7-168*x^6+144*x^5),x, algo
rithm="fricas")

[Out]

log(7*x^6 - 12*x^5 + (7*x^4 - 12*x^3 - x)*e^8 - 2*(7*x^5 - 12*x^4)*e^4) - log(7*x - 12)

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giac [B]  time = 0.74, size = 58, normalized size = 1.81 \begin {gather*} \log \left ({\left | 7 \, x^{5} - 14 \, x^{4} e^{4} - 12 \, x^{4} + 7 \, x^{3} e^{8} + 24 \, x^{3} e^{4} - 12 \, x^{2} e^{8} - e^{8} \right |}\right ) - \log \left ({\left | 7 \, x - 12 \right |}\right ) + \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((147*x^4-504*x^3+432*x^2+12)*exp(4)^2+(-392*x^5+1344*x^4-1152*x^3)*exp(4)+245*x^6-840*x^5+720*x^4)/
((49*x^5-168*x^4+144*x^3-7*x^2+12*x)*exp(4)^2+(-98*x^6+336*x^5-288*x^4)*exp(4)+49*x^7-168*x^6+144*x^5),x, algo
rithm="giac")

[Out]

log(abs(7*x^5 - 14*x^4*e^4 - 12*x^4 + 7*x^3*e^8 + 24*x^3*e^4 - 12*x^2*e^8 - e^8)) - log(abs(7*x - 12)) + log(a
bs(x))

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maple [A]  time = 0.15, size = 52, normalized size = 1.62

method	result	size
risch	\(-\ln \left (-7 x +12\right )+\ln \left (7 x^{6}+\left (-14 \,{\mathrm e}^{4}-12\right ) x^{5}+\left (7 \,{\mathrm e}^{8}+24 \,{\mathrm e}^{4}\right ) x^{4}-12 \,{\mathrm e}^{8} x^{3}-x \,{\mathrm e}^{8}\right )\)	\(52\)
default	\(\ln \left (-14 x^{4} {\mathrm e}^{4}+7 x^{5}+7 \,{\mathrm e}^{8} x^{3}+24 x^{3} {\mathrm e}^{4}-12 x^{4}-12 x^{2} {\mathrm e}^{8}-{\mathrm e}^{8}\right )+\ln \relax (x )-\ln \left (7 x -12\right )\)	\(56\)
norman	\(\ln \left (-14 x^{4} {\mathrm e}^{4}+7 x^{5}+7 \,{\mathrm e}^{8} x^{3}+24 x^{3} {\mathrm e}^{4}-12 x^{4}-12 x^{2} {\mathrm e}^{8}-{\mathrm e}^{8}\right )+\ln \relax (x )-\ln \left (7 x -12\right )\)	\(62\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((147*x^4-504*x^3+432*x^2+12)*exp(4)^2+(-392*x^5+1344*x^4-1152*x^3)*exp(4)+245*x^6-840*x^5+720*x^4)/((49*x
^5-168*x^4+144*x^3-7*x^2+12*x)*exp(4)^2+(-98*x^6+336*x^5-288*x^4)*exp(4)+49*x^7-168*x^6+144*x^5),x,method=_RET
URNVERBOSE)

[Out]

-ln(-7*x+12)+ln(7*x^6+(-14*exp(4)-12)*x^5+(7*exp(8)+24*exp(4))*x^4-12*exp(8)*x^3-x*exp(8))

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maxima [B]  time = 0.37, size = 53, normalized size = 1.66 \begin {gather*} \log \left (7 \, x^{5} - 2 \, x^{4} {\left (7 \, e^{4} + 6\right )} + x^{3} {\left (7 \, e^{8} + 24 \, e^{4}\right )} - 12 \, x^{2} e^{8} - e^{8}\right ) - \log \left (7 \, x - 12\right ) + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((147*x^4-504*x^3+432*x^2+12)*exp(4)^2+(-392*x^5+1344*x^4-1152*x^3)*exp(4)+245*x^6-840*x^5+720*x^4)/
((49*x^5-168*x^4+144*x^3-7*x^2+12*x)*exp(4)^2+(-98*x^6+336*x^5-288*x^4)*exp(4)+49*x^7-168*x^6+144*x^5),x, algo
rithm="maxima")

[Out]

log(7*x^5 - 2*x^4*(7*e^4 + 6) + x^3*(7*e^8 + 24*e^4) - 12*x^2*e^8 - e^8) - log(7*x - 12) + log(x)

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mupad [B]  time = 2.32, size = 49, normalized size = 1.53 \begin {gather*} \ln \left (\frac {24\,x^4\,{\mathrm {e}}^4}{7}-\frac {x\,{\mathrm {e}}^8}{7}-2\,x^5\,{\mathrm {e}}^4-\frac {12\,x^3\,{\mathrm {e}}^8}{7}+x^4\,{\mathrm {e}}^8-\frac {12\,x^5}{7}+x^6\right )-\ln \left (x-\frac {12}{7}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(8)*(432*x^2 - 504*x^3 + 147*x^4 + 12) - exp(4)*(1152*x^3 - 1344*x^4 + 392*x^5) + 720*x^4 - 840*x^5 +
245*x^6)/(exp(8)*(12*x - 7*x^2 + 144*x^3 - 168*x^4 + 49*x^5) - exp(4)*(288*x^4 - 336*x^5 + 98*x^6) + 144*x^5 -
168*x^6 + 49*x^7),x)

[Out]

log((24*x^4*exp(4))/7 - (x*exp(8))/7 - 2*x^5*exp(4) - (12*x^3*exp(8))/7 + x^4*exp(8) - (12*x^5)/7 + x^6) - log
(x - 12/7)

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sympy [B]  time = 4.84, size = 54, normalized size = 1.69 \begin {gather*} - \log {\left (7 x - 12 \right )} + \log {\left (x^{6} + x^{5} \left (- 2 e^{4} - \frac {12}{7}\right ) + x^{4} \left (\frac {24 e^{4}}{7} + e^{8}\right ) - \frac {12 x^{3} e^{8}}{7} - \frac {x e^{8}}{7} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((147*x**4-504*x**3+432*x**2+12)*exp(4)**2+(-392*x**5+1344*x**4-1152*x**3)*exp(4)+245*x**6-840*x**5+
720*x**4)/((49*x**5-168*x**4+144*x**3-7*x**2+12*x)*exp(4)**2+(-98*x**6+336*x**5-288*x**4)*exp(4)+49*x**7-168*x
**6+144*x**5),x)

[Out]

-log(7*x - 12) + log(x**6 + x**5*(-2*exp(4) - 12/7) + x**4*(24*exp(4)/7 + exp(8)) - 12*x**3*exp(8)/7 - x*exp(8
)/7)

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