3.6.17 \(\int \frac {-16-16 e^4-8 x^2}{400+1384 x+797 x^2-692 x^3+100 x^4+e^8 (400-200 x+25 x^2)+e^4 (800+1184 x-746 x^2+100 x^3)} \, dx\) [517]

Optimal. Leaf size=21 \[ \log \left (25+\frac {4 x}{(-4+x) \left (1+e^4+2 x\right )}\right ) \]

[Out]

ln(25+4*x/(x-4)/(1+2*x+exp(4)))

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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(43\) vs. \(2(21)=42\).
time = 0.07, antiderivative size = 43, normalized size of antiderivative = 2.05, number of steps used = 3, number of rules used = 2, integrand size = 68, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {2099, 642} \begin {gather*} \log \left (-50 x^2+\left (171-25 e^4\right ) x+100 \left (1+e^4\right )\right )-\log (4-x)-\log \left (2 x+e^4+1\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-16 - 16*E^4 - 8*x^2)/(400 + 1384*x + 797*x^2 - 692*x^3 + 100*x^4 + E^8*(400 - 200*x + 25*x^2) + E^4*(800
+ 1184*x - 746*x^2 + 100*x^3)),x]

[Out]

-Log[4 - x] - Log[1 + E^4 + 2*x] + Log[100*(1 + E^4) + (171 - 25*E^4)*x - 50*x^2]

Rule 642

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

Rule 2099

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}{4-x}-\frac {2}{1+e^4+2 x}+\frac {171-25 e^4-100 x}{100 \left (1+e^4\right )+\left (171-25 e^4\right ) x-50 x^2}\right ) \, dx\\ &=-\log (4-x)-\log \left (1+e^4+2 x\right )+\int \frac {171-25 e^4-100 x}{100 \left (1+e^4\right )+\left (171-25 e^4\right ) x-50 x^2} \, dx\\ &=-\log (4-x)-\log \left (1+e^4+2 x\right )+\log \left (100 \left (1+e^4\right )+\left (171-25 e^4\right ) x-50 x^2\right )\\ \end {aligned} \end {gather*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(53\) vs. \(2(21)=42\).
time = 0.03, size = 53, normalized size = 2.52 \begin {gather*} -8 \left (\frac {1}{8} \log \left (9+e^4+2 (-4+x)\right )-\frac {1}{8} \log \left (16+229 (-4+x)+25 e^4 (-4+x)+50 (-4+x)^2\right )+\frac {1}{8} \log (-4+x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-16 - 16*E^4 - 8*x^2)/(400 + 1384*x + 797*x^2 - 692*x^3 + 100*x^4 + E^8*(400 - 200*x + 25*x^2) + E^
4*(800 + 1184*x - 746*x^2 + 100*x^3)),x]

[Out]

-8*(Log[9 + E^4 + 2*(-4 + x)]/8 - Log[16 + 229*(-4 + x) + 25*E^4*(-4 + x) + 50*(-4 + x)^2]/8 + Log[-4 + x]/8)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.16, size = 104, normalized size = 4.95

method	result	size
norman	\(-\ln \left (x -4\right )-\ln \left (1+2 x +{\mathrm e}^{4}\right )+\ln \left (25 x \,{\mathrm e}^{4}+50 x^{2}-100 \,{\mathrm e}^{4}-171 x -100\right )\)	\(38\)
risch	\(-\ln \left (-2 x^{2}+\left (-{\mathrm e}^{4}+7\right ) x +4 \,{\mathrm e}^{4}+4\right )+\ln \left (-50 x^{2}+\left (-25 \,{\mathrm e}^{4}+171\right ) x +100 \,{\mathrm e}^{4}+100\right )\)	\(44\)
default	\(\left (\munderset {\textit {\_R} =\RootOf \left (100 \textit {\_Z}^{3}+\left (100 \,{\mathrm e}^{4}-292\right ) \textit {\_Z}^{2}+\left (-346 \,{\mathrm e}^{4}+25 \,{\mathrm e}^{8}-371\right ) \textit {\_Z} -200 \,{\mathrm e}^{4}-100 \,{\mathrm e}^{8}-100\right )}{\sum }\frac {\left (100 \textit {\_R} \,{\mathrm e}^{4}+100 \textit {\_R}^{2}+54 \,{\mathrm e}^{4}+25 \,{\mathrm e}^{8}+100 \textit {\_R} +29\right ) \ln \left (x -\textit {\_R} \right )}{200 \textit {\_R} \,{\mathrm e}^{4}+300 \textit {\_R}^{2}-346 \,{\mathrm e}^{4}+25 \,{\mathrm e}^{8}-584 \textit {\_R} -371}\right )-\ln \left (x -4\right )\)	\(104\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-16*exp(4)-8*x^2-16)/((25*x^2-200*x+400)*exp(4)^2+(100*x^3-746*x^2+1184*x+800)*exp(4)+100*x^4-692*x^3+797
*x^2+1384*x+400),x,method=_RETURNVERBOSE)

[Out]

sum((100*_R*exp(4)+100*_R^2+54*exp(4)+25*exp(8)+100*_R+29)/(200*_R*exp(4)+300*_R^2-346*exp(4)+25*exp(8)-584*_R
-371)*ln(x-_R),_R=RootOf(100*_Z^3+(100*exp(4)-292)*_Z^2+(-346*exp(4)+25*exp(8)-371)*_Z-200*exp(4)-100*exp(8)-1
00))-ln(x-4)

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Maxima [A]
time = 0.28, size = 37, normalized size = 1.76 \begin {gather*} \log \left (50 \, x^{2} + x {\left (25 \, e^{4} - 171\right )} - 100 \, e^{4} - 100\right ) - \log \left (2 \, x + e^{4} + 1\right ) - \log \left (x - 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-16*exp(4)-8*x^2-16)/((25*x^2-200*x+400)*exp(4)^2+(100*x^3-746*x^2+1184*x+800)*exp(4)+100*x^4-692*x
^3+797*x^2+1384*x+400),x, algorithm="maxima")

[Out]

log(50*x^2 + x*(25*e^4 - 171) - 100*e^4 - 100) - log(2*x + e^4 + 1) - log(x - 4)

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Fricas [A]
time = 0.32, size = 38, normalized size = 1.81 \begin {gather*} \log \left (50 \, x^{2} + 25 \, {\left (x - 4\right )} e^{4} - 171 \, x - 100\right ) - \log \left (2 \, x^{2} + {\left (x - 4\right )} e^{4} - 7 \, x - 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-16*exp(4)-8*x^2-16)/((25*x^2-200*x+400)*exp(4)^2+(100*x^3-746*x^2+1184*x+800)*exp(4)+100*x^4-692*x
^3+797*x^2+1384*x+400),x, algorithm="fricas")

[Out]

log(50*x^2 + 25*(x - 4)*e^4 - 171*x - 100) - log(2*x^2 + (x - 4)*e^4 - 7*x - 4)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 42 vs. \(2 (17) = 34\).
time = 0.98, size = 42, normalized size = 2.00 \begin {gather*} - \log {\left (x^{2} + x \left (- \frac {7}{2} + \frac {e^{4}}{2}\right ) - 2 e^{4} - 2 \right )} + \log {\left (x^{2} + x \left (- \frac {171}{50} + \frac {e^{4}}{2}\right ) - 2 e^{4} - 2 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-16*exp(4)-8*x**2-16)/((25*x**2-200*x+400)*exp(4)**2+(100*x**3-746*x**2+1184*x+800)*exp(4)+100*x**4
-692*x**3+797*x**2+1384*x+400),x)

[Out]

-log(x**2 + x*(-7/2 + exp(4)/2) - 2*exp(4) - 2) + log(x**2 + x*(-171/50 + exp(4)/2) - 2*exp(4) - 2)

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Giac [A]
time = 0.40, size = 40, normalized size = 1.90 \begin {gather*} \log \left ({\left | 50 \, x^{2} + 25 \, x e^{4} - 171 \, x - 100 \, e^{4} - 100 \right |}\right ) - \log \left ({\left | 2 \, x + e^{4} + 1 \right |}\right ) - \log \left ({\left | x - 4 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-16*exp(4)-8*x^2-16)/((25*x^2-200*x+400)*exp(4)^2+(100*x^3-746*x^2+1184*x+800)*exp(4)+100*x^4-692*x
^3+797*x^2+1384*x+400),x, algorithm="giac")

[Out]

log(abs(50*x^2 + 25*x*e^4 - 171*x - 100*e^4 - 100)) - log(abs(2*x + e^4 + 1)) - log(abs(x - 4))

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Mupad [B]
time = 0.79, size = 97, normalized size = 4.62 \begin {gather*} -\mathrm {atan}\left (\frac {{\mathrm {e}}^4\,3788800{}\mathrm {i}-{\mathrm {e}}^8\,640000{}\mathrm {i}+x^2\,\left (160000\,{\mathrm {e}}^4-1107200\right )\,2{}\mathrm {i}+x\,\left (40000\,{\mathrm {e}}^8-1833600\,{\mathrm {e}}^4+635200\right )\,2{}\mathrm {i}+4428800{}\mathrm {i}}{392211200\,{\mathrm {e}}^4+76640000\,{\mathrm {e}}^8+4000000\,{\mathrm {e}}^{12}-2\,x^2\,\left (18160000\,{\mathrm {e}}^4+1000000\,{\mathrm {e}}^8+79899200\right )+2\,x\,\left (22893600\,{\mathrm {e}}^4-5620000\,{\mathrm {e}}^8-500000\,{\mathrm {e}}^{12}[phone]\right )+12800\,x^2+319571200}\right )\,2{}\mathrm {i} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(16*exp(4) + 8*x^2 + 16)/(1384*x + exp(8)*(25*x^2 - 200*x + 400) + exp(4)*(1184*x - 746*x^2 + 100*x^3 + 8
00) + 797*x^2 - 692*x^3 + 100*x^4 + 400),x)

[Out]

-atan((exp(4)*3788800i - exp(8)*640000i + x^2*(160000*exp(4) - 1107200)*2i + x*(40000*exp(8) - 1833600*exp(4)
+ 635200)*2i + 4428800i)/(392211200*exp(4) + 76640000*exp(8) + 4000000*exp(12) - 2*x^2*(18160000*exp(4) + 1000
000*exp(8) + 79899200) + 2*x*(22893600*exp(4) - 5620000*exp(8) - 500000*exp(12) + 276384800) + 12800*x^2 + 319
571200))*2i

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