Integrand size = 22, antiderivative size = 89 \[ \int \frac {1}{a+8 x-8 x^2+4 x^3-x^4} \, dx=-\frac {\arctan \left (\frac {-1+x}{\sqrt {1-\sqrt {4+a}}}\right )}{2 \sqrt {4+a} \sqrt {1-\sqrt {4+a}}}+\frac {\arctan \left (\frac {-1+x}{\sqrt {1+\sqrt {4+a}}}\right )}{2 \sqrt {4+a} \sqrt {1+\sqrt {4+a}}} \]
-1/2*arctan((-1+x)/(1-(4+a)^(1/2))^(1/2))/(4+a)^(1/2)/(1-(4+a)^(1/2))^(1/2 )+1/2*arctan((-1+x)/(1+(4+a)^(1/2))^(1/2))/(4+a)^(1/2)/(1+(4+a)^(1/2))^(1/ 2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.02 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.64 \[ \int \frac {1}{a+8 x-8 x^2+4 x^3-x^4} \, dx=-\frac {1}{4} \text {RootSum}\left [a+8 \text {$\#$1}-8 \text {$\#$1}^2+4 \text {$\#$1}^3-\text {$\#$1}^4\&,\frac {\log (x-\text {$\#$1})}{-2+4 \text {$\#$1}-3 \text {$\#$1}^2+\text {$\#$1}^3}\&\right ] \]
-1/4*RootSum[a + 8*#1 - 8*#1^2 + 4*#1^3 - #1^4 & , Log[x - #1]/(-2 + 4*#1 - 3*#1^2 + #1^3) & ]
Time = 0.23 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {2458, 1406, 217}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {1}{a-x^4+4 x^3-8 x^2+8 x} \, dx\) |
\(\Big \downarrow \) 2458 |
\(\displaystyle \int \frac {1}{a-(x-1)^4-2 (x-1)^2+3}d(x-1)\) |
\(\Big \downarrow \) 1406 |
\(\displaystyle \frac {\int \frac {1}{-(x-1)^2+\sqrt {a+4}-1}d(x-1)}{2 \sqrt {a+4}}-\frac {\int \frac {1}{-(x-1)^2-\sqrt {a+4}-1}d(x-1)}{2 \sqrt {a+4}}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {\arctan \left (\frac {x-1}{\sqrt {\sqrt {a+4}+1}}\right )}{2 \sqrt {a+4} \sqrt {\sqrt {a+4}+1}}-\frac {\arctan \left (\frac {x-1}{\sqrt {1-\sqrt {a+4}}}\right )}{2 \sqrt {a+4} \sqrt {1-\sqrt {a+4}}}\) |
-1/2*ArcTan[(-1 + x)/Sqrt[1 - Sqrt[4 + a]]]/(Sqrt[4 + a]*Sqrt[1 - Sqrt[4 + a]]) + ArcTan[(-1 + x)/Sqrt[1 + Sqrt[4 + a]]]/(2*Sqrt[4 + a]*Sqrt[1 + Sqr t[4 + a]])
3.2.20.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(-1), x_Symbol] :> With[{q = Rt[b^ 2 - 4*a*c, 2]}, Simp[c/q Int[1/(b/2 - q/2 + c*x^2), x], x] - Simp[c/q I nt[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c , 0] && PosQ[b^2 - 4*a*c]
Int[(Pn_)^(p_.), x_Symbol] :> With[{S = Coeff[Pn, x, Expon[Pn, x] - 1]/(Exp on[Pn, x]*Coeff[Pn, x, Expon[Pn, x]])}, Subst[Int[ExpandToSum[Pn /. x -> x - S, x]^p, x], x, x + S] /; BinomialQ[Pn /. x -> x - S, x] || (IntegerQ[Exp on[Pn, x]/2] && TrinomialQ[Pn /. x -> x - S, x])] /; FreeQ[p, x] && PolyQ[P n, x] && GtQ[Expon[Pn, x], 2] && NeQ[Coeff[Pn, x, Expon[Pn, x] - 1], 0]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.07 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.57
method | result | size |
default | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-4 \textit {\_Z}^{3}+8 \textit {\_Z}^{2}-8 \textit {\_Z} -a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{-\textit {\_R}^{3}+3 \textit {\_R}^{2}-4 \textit {\_R} +2}\right )}{4}\) | \(51\) |
risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-4 \textit {\_Z}^{3}+8 \textit {\_Z}^{2}-8 \textit {\_Z} -a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{-\textit {\_R}^{3}+3 \textit {\_R}^{2}-4 \textit {\_R} +2}\right )}{4}\) | \(51\) |
Leaf count of result is larger than twice the leaf count of optimal. 457 vs. \(2 (65) = 130\).
Time = 0.25 (sec) , antiderivative size = 457, normalized size of antiderivative = 5.13 \[ \int \frac {1}{a+8 x-8 x^2+4 x^3-x^4} \, dx=\frac {1}{4} \, \sqrt {\frac {\frac {a^{2} + 7 \, a + 12}{\sqrt {a^{3} + 10 \, a^{2} + 33 \, a + 36}} + 1}{a^{2} + 7 \, a + 12}} \log \left ({\left (a - \frac {a^{2} + 7 \, a + 12}{\sqrt {a^{3} + 10 \, a^{2} + 33 \, a + 36}} + 4\right )} \sqrt {\frac {\frac {a^{2} + 7 \, a + 12}{\sqrt {a^{3} + 10 \, a^{2} + 33 \, a + 36}} + 1}{a^{2} + 7 \, a + 12}} + x - 1\right ) - \frac {1}{4} \, \sqrt {\frac {\frac {a^{2} + 7 \, a + 12}{\sqrt {a^{3} + 10 \, a^{2} + 33 \, a + 36}} + 1}{a^{2} + 7 \, a + 12}} \log \left (-{\left (a - \frac {a^{2} + 7 \, a + 12}{\sqrt {a^{3} + 10 \, a^{2} + 33 \, a + 36}} + 4\right )} \sqrt {\frac {\frac {a^{2} + 7 \, a + 12}{\sqrt {a^{3} + 10 \, a^{2} + 33 \, a + 36}} + 1}{a^{2} + 7 \, a + 12}} + x - 1\right ) + \frac {1}{4} \, \sqrt {-\frac {\frac {a^{2} + 7 \, a + 12}{\sqrt {a^{3} + 10 \, a^{2} + 33 \, a + 36}} - 1}{a^{2} + 7 \, a + 12}} \log \left ({\left (a + \frac {a^{2} + 7 \, a + 12}{\sqrt {a^{3} + 10 \, a^{2} + 33 \, a + 36}} + 4\right )} \sqrt {-\frac {\frac {a^{2} + 7 \, a + 12}{\sqrt {a^{3} + 10 \, a^{2} + 33 \, a + 36}} - 1}{a^{2} + 7 \, a + 12}} + x - 1\right ) - \frac {1}{4} \, \sqrt {-\frac {\frac {a^{2} + 7 \, a + 12}{\sqrt {a^{3} + 10 \, a^{2} + 33 \, a + 36}} - 1}{a^{2} + 7 \, a + 12}} \log \left (-{\left (a + \frac {a^{2} + 7 \, a + 12}{\sqrt {a^{3} + 10 \, a^{2} + 33 \, a + 36}} + 4\right )} \sqrt {-\frac {\frac {a^{2} + 7 \, a + 12}{\sqrt {a^{3} + 10 \, a^{2} + 33 \, a + 36}} - 1}{a^{2} + 7 \, a + 12}} + x - 1\right ) \]
1/4*sqrt(((a^2 + 7*a + 12)/sqrt(a^3 + 10*a^2 + 33*a + 36) + 1)/(a^2 + 7*a + 12))*log((a - (a^2 + 7*a + 12)/sqrt(a^3 + 10*a^2 + 33*a + 36) + 4)*sqrt( ((a^2 + 7*a + 12)/sqrt(a^3 + 10*a^2 + 33*a + 36) + 1)/(a^2 + 7*a + 12)) + x - 1) - 1/4*sqrt(((a^2 + 7*a + 12)/sqrt(a^3 + 10*a^2 + 33*a + 36) + 1)/(a ^2 + 7*a + 12))*log(-(a - (a^2 + 7*a + 12)/sqrt(a^3 + 10*a^2 + 33*a + 36) + 4)*sqrt(((a^2 + 7*a + 12)/sqrt(a^3 + 10*a^2 + 33*a + 36) + 1)/(a^2 + 7*a + 12)) + x - 1) + 1/4*sqrt(-((a^2 + 7*a + 12)/sqrt(a^3 + 10*a^2 + 33*a + 36) - 1)/(a^2 + 7*a + 12))*log((a + (a^2 + 7*a + 12)/sqrt(a^3 + 10*a^2 + 3 3*a + 36) + 4)*sqrt(-((a^2 + 7*a + 12)/sqrt(a^3 + 10*a^2 + 33*a + 36) - 1) /(a^2 + 7*a + 12)) + x - 1) - 1/4*sqrt(-((a^2 + 7*a + 12)/sqrt(a^3 + 10*a^ 2 + 33*a + 36) - 1)/(a^2 + 7*a + 12))*log(-(a + (a^2 + 7*a + 12)/sqrt(a^3 + 10*a^2 + 33*a + 36) + 4)*sqrt(-((a^2 + 7*a + 12)/sqrt(a^3 + 10*a^2 + 33* a + 36) - 1)/(a^2 + 7*a + 12)) + x - 1)
Time = 0.54 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.74 \[ \int \frac {1}{a+8 x-8 x^2+4 x^3-x^4} \, dx=- \operatorname {RootSum} {\left (t^{4} \cdot \left (256 a^{3} + 2816 a^{2} + 10240 a + 12288\right ) + t^{2} \left (- 32 a - 128\right ) - 1, \left ( t \mapsto t \log {\left (64 t^{3} a^{2} + 448 t^{3} a + 768 t^{3} - 4 t a - 20 t + x - 1 \right )} \right )\right )} \]
-RootSum(_t**4*(256*a**3 + 2816*a**2 + 10240*a + 12288) + _t**2*(-32*a - 1 28) - 1, Lambda(_t, _t*log(64*_t**3*a**2 + 448*_t**3*a + 768*_t**3 - 4*_t* a - 20*_t + x - 1)))
\[ \int \frac {1}{a+8 x-8 x^2+4 x^3-x^4} \, dx=\int { -\frac {1}{x^{4} - 4 \, x^{3} + 8 \, x^{2} - a - 8 \, x} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 2669 vs. \(2 (65) = 130\).
Time = 2.52 (sec) , antiderivative size = 2669, normalized size of antiderivative = 29.99 \[ \int \frac {1}{a+8 x-8 x^2+4 x^3-x^4} \, dx=\text {Too large to display} \]
-1/4*sqrt(((a + 4)^(3/2) + a + 4)/(a^3 + 11*a^2 + 40*a + 48))*log(abs(sqrt (a + 4)*a^5 + sqrt(a^2 + (a^2 + 7*a + 12)*sqrt(a + 4) + 7*a + 12)*a^4*x + a^5 + sqrt(a^2 + (a^2 + 7*a + 12)*sqrt(a + 4) + 7*a + 12)*sqrt(a + 4)*a^3* x - sqrt(a^2 + (a^2 + 7*a + 12)*sqrt(a + 4) + 7*a + 12)*a^4 + 17*sqrt(a + 4)*a^4 + 14*sqrt(a^2 + (a^2 + 7*a + 12)*sqrt(a + 4) + 7*a + 12)*a^3*x - sq rt(a^2 + (a^2 + 7*a + 12)*sqrt(a + 4) + 7*a + 12)*sqrt(a + 4)*a^3 + 17*a^4 + 10*sqrt(a^2 + (a^2 + 7*a + 12)*sqrt(a + 4) + 7*a + 12)*sqrt(a + 4)*a^2* x - 14*sqrt(a^2 + (a^2 + 7*a + 12)*sqrt(a + 4) + 7*a + 12)*a^3 + 111*sqrt( a + 4)*a^3 + 69*sqrt(a^2 + (a^2 + 7*a + 12)*sqrt(a + 4) + 7*a + 12)*a^2*x - 10*sqrt(a^2 + (a^2 + 7*a + 12)*sqrt(a + 4) + 7*a + 12)*sqrt(a + 4)*a^2 + 111*a^3 + 29*sqrt(a^2 + (a^2 + 7*a + 12)*sqrt(a + 4) + 7*a + 12)*sqrt(a + 4)*a*x - 69*sqrt(a^2 + (a^2 + 7*a + 12)*sqrt(a + 4) + 7*a + 12)*a^2 + 351 *sqrt(a + 4)*a^2 + 144*sqrt(a^2 + (a^2 + 7*a + 12)*sqrt(a + 4) + 7*a + 12) *a*x - 29*sqrt(a^2 + (a^2 + 7*a + 12)*sqrt(a + 4) + 7*a + 12)*sqrt(a + 4)* a + 351*a^2 + 28*sqrt(a^2 + (a^2 + 7*a + 12)*sqrt(a + 4) + 7*a + 12)*sqrt( a + 4)*x - 144*sqrt(a^2 + (a^2 + 7*a + 12)*sqrt(a + 4) + 7*a + 12)*a + 544 *sqrt(a + 4)*a + 112*sqrt(a^2 + (a^2 + 7*a + 12)*sqrt(a + 4) + 7*a + 12)*x - 28*sqrt(a^2 + (a^2 + 7*a + 12)*sqrt(a + 4) + 7*a + 12)*sqrt(a + 4) + 54 4*a - 112*sqrt(a^2 + (a^2 + 7*a + 12)*sqrt(a + 4) + 7*a + 12) + 336*sqrt(a + 4) + 336)) + 1/4*sqrt(((a + 4)^(3/2) + a + 4)/(a^3 + 11*a^2 + 40*a +...
Time = 0.32 (sec) , antiderivative size = 571, normalized size of antiderivative = 6.42 \[ \int \frac {1}{a+8 x-8 x^2+4 x^3-x^4} \, dx=-\mathrm {atan}\left (-\frac {a\,8{}\mathrm {i}-x\,16{}\mathrm {i}+x\,\sqrt {a^3+12\,a^2+48\,a+64}\,1{}\mathrm {i}-a\,x\,8{}\mathrm {i}-\sqrt {a^3+12\,a^2+48\,a+64}\,1{}\mathrm {i}-a^2\,x\,1{}\mathrm {i}+a^2\,1{}\mathrm {i}+16{}\mathrm {i}}{44\,a^2\,\sqrt {\frac {a-\sqrt {a^3+12\,a^2+48\,a+64}+4}{16\,a^3+176\,a^2+640\,a+768}}+4\,a^3\,\sqrt {\frac {a-\sqrt {a^3+12\,a^2+48\,a+64}+4}{16\,a^3+176\,a^2+640\,a+768}}+160\,a\,\sqrt {\frac {a-\sqrt {a^3+12\,a^2+48\,a+64}+4}{16\,a^3+176\,a^2+640\,a+768}}+192\,\sqrt {\frac {a-\sqrt {a^3+12\,a^2+48\,a+64}+4}{16\,a^3+176\,a^2+640\,a+768}}}\right )\,\sqrt {\frac {a-\sqrt {a^3+12\,a^2+48\,a+64}+4}{16\,a^3+176\,a^2+640\,a+768}}\,2{}\mathrm {i}-\mathrm {atan}\left (-\frac {a\,8{}\mathrm {i}-x\,16{}\mathrm {i}-x\,\sqrt {a^3+12\,a^2+48\,a+64}\,1{}\mathrm {i}-a\,x\,8{}\mathrm {i}+\sqrt {a^3+12\,a^2+48\,a+64}\,1{}\mathrm {i}-a^2\,x\,1{}\mathrm {i}+a^2\,1{}\mathrm {i}+16{}\mathrm {i}}{160\,a\,\sqrt {\frac {a+\sqrt {a^3+12\,a^2+48\,a+64}+4}{16\,a^3+176\,a^2+640\,a+768}}+192\,\sqrt {\frac {a+\sqrt {a^3+12\,a^2+48\,a+64}+4}{16\,a^3+176\,a^2+640\,a+768}}+44\,a^2\,\sqrt {\frac {a+\sqrt {a^3+12\,a^2+48\,a+64}+4}{16\,a^3+176\,a^2+640\,a+768}}+4\,a^3\,\sqrt {\frac {a+\sqrt {a^3+12\,a^2+48\,a+64}+4}{16\,a^3+176\,a^2+640\,a+768}}}\right )\,\sqrt {\frac {a+\sqrt {a^3+12\,a^2+48\,a+64}+4}{16\,a^3+176\,a^2+640\,a+768}}\,2{}\mathrm {i} \]
- atan(-(a*8i - x*16i + x*(48*a + 12*a^2 + a^3 + 64)^(1/2)*1i - a*x*8i - ( 48*a + 12*a^2 + a^3 + 64)^(1/2)*1i - a^2*x*1i + a^2*1i + 16i)/(44*a^2*((a - (48*a + 12*a^2 + a^3 + 64)^(1/2) + 4)/(640*a + 176*a^2 + 16*a^3 + 768))^ (1/2) + 4*a^3*((a - (48*a + 12*a^2 + a^3 + 64)^(1/2) + 4)/(640*a + 176*a^2 + 16*a^3 + 768))^(1/2) + 160*a*((a - (48*a + 12*a^2 + a^3 + 64)^(1/2) + 4 )/(640*a + 176*a^2 + 16*a^3 + 768))^(1/2) + 192*((a - (48*a + 12*a^2 + a^3 + 64)^(1/2) + 4)/(640*a + 176*a^2 + 16*a^3 + 768))^(1/2)))*((a - (48*a + 12*a^2 + a^3 + 64)^(1/2) + 4)/(640*a + 176*a^2 + 16*a^3 + 768))^(1/2)*2i - atan(-(a*8i - x*16i - x*(48*a + 12*a^2 + a^3 + 64)^(1/2)*1i - a*x*8i + (4 8*a + 12*a^2 + a^3 + 64)^(1/2)*1i - a^2*x*1i + a^2*1i + 16i)/(160*a*((a + (48*a + 12*a^2 + a^3 + 64)^(1/2) + 4)/(640*a + 176*a^2 + 16*a^3 + 768))^(1 /2) + 192*((a + (48*a + 12*a^2 + a^3 + 64)^(1/2) + 4)/(640*a + 176*a^2 + 1 6*a^3 + 768))^(1/2) + 44*a^2*((a + (48*a + 12*a^2 + a^3 + 64)^(1/2) + 4)/( 640*a + 176*a^2 + 16*a^3 + 768))^(1/2) + 4*a^3*((a + (48*a + 12*a^2 + a^3 + 64)^(1/2) + 4)/(640*a + 176*a^2 + 16*a^3 + 768))^(1/2)))*((a + (48*a + 1 2*a^2 + a^3 + 64)^(1/2) + 4)/(640*a + 176*a^2 + 16*a^3 + 768))^(1/2)*2i