Integrand size = 37, antiderivative size = 610 \[ \int \frac {1}{432+648 x+216 x^2+972 x^3+1008 x^4+54 x^5+2 x^6+3 x^7} \, dx=-\frac {(-1)^{2/3} \left (166-27 (-2)^{2/3} \sqrt [3]{3}+6 \sqrt [3]{-2} 3^{2/3}\right ) \arctan \left (\frac {3 (-2)^{2/3} \sqrt [3]{3}+2 x}{\sqrt {6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{648\ 2^{5/6} \sqrt [6]{3} \left (1-\sqrt [3]{-1}\right ) \left (1+\sqrt [3]{-1}\right )^2 \left (29-9 (-2)^{2/3} \sqrt [3]{3}\right ) \sqrt {4+3 \sqrt [3]{-2} 3^{2/3}}}+\frac {(-1)^{2/3} \left (3 (-6)^{2/3}-27 \sqrt [3]{-3}-83 \sqrt [3]{2}\right ) \arctan \left (\frac {\sqrt [6]{2} \left (3 \sqrt [3]{-3}-\sqrt [3]{2} x\right )}{\sqrt {3 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{648 \sqrt [6]{6} \left (1+\sqrt [3]{-1}\right )^2 \sqrt {4-3 (-3)^{2/3} \sqrt [3]{2}} \left (29+9 \sqrt [3]{-3} 2^{2/3}\right )}+\frac {\left (83 \sqrt [3]{2}-27 \sqrt [3]{3}-3\ 6^{2/3}\right ) \text {arctanh}\left (\frac {\sqrt [6]{2} \left (3 \sqrt [3]{3}+\sqrt [3]{2} x\right )}{\sqrt {3 \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )}}\right )}{648 \sqrt [6]{6} \left (1-\sqrt [3]{-1}\right ) \left (1+\sqrt [3]{-1}\right )^2 \left (29-9\ 2^{2/3} \sqrt [3]{3}\right ) \sqrt {-4+3 \sqrt [3]{2} 3^{2/3}}}+\frac {243 \log (2+3 x)}{125128}-\frac {\left ((-6)^{2/3}+18 \sqrt [3]{-3}+2 \sqrt [3]{2}\right ) \log \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}{864 \sqrt [3]{3} \left (1+\sqrt [3]{-1}\right )^2 \left (29+9 \sqrt [3]{-3} 2^{2/3}\right )}+\frac {(-1)^{2/3} \left (9 \sqrt [3]{-6}+(-2)^{2/3}+3^{2/3}\right ) \log \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}{648 \sqrt [3]{6} \left (58+9\ 2^{2/3} \sqrt [3]{3}-9 i 2^{2/3} 3^{5/6}\right )}-\frac {\left (54-6^{2/3} \left (2^{2/3}+3^{2/3}\right )\right ) \log \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}{7776 \left (29-9\ 2^{2/3} \sqrt [3]{3}\right )} \] Output:
-1/3888*(-1)^(2/3)*(166-27*(-2)^(2/3)*3^(1/3)+6*(-2)^(1/3)*3^(2/3))*arctan ((3*(-2)^(2/3)*3^(1/3)+2*x)/(24+18*(-2)^(1/3)*3^(2/3))^(1/2))*2^(1/6)*3^(5 /6)/(1-(-1)^(1/3))/(1+(-1)^(1/3))^2/(29-9*(-2)^(2/3)*3^(1/3))/(4+3*(-2)^(1 /3)*3^(2/3))^(1/2)+1/3888*(-1)^(2/3)*(3*(-6)^(2/3)-27*(-3)^(1/3)-83*2^(1/3 ))*arctan(2^(1/6)*(3*(-3)^(1/3)-2^(1/3)*x)/(12-9*(-3)^(2/3)*2^(1/3))^(1/2) )*6^(5/6)/(1+(-1)^(1/3))^2/(4-3*(-3)^(2/3)*2^(1/3))^(1/2)/(29+9*(-3)^(1/3) *2^(2/3))+1/3888*(83*2^(1/3)-27*3^(1/3)-3*6^(2/3))*arctanh(2^(1/6)*(3*3^(1 /3)+2^(1/3)*x)/(-12+9*2^(1/3)*3^(2/3))^(1/2))*6^(5/6)/(1-(-1)^(1/3))/(1+(- 1)^(1/3))^2/(29-9*2^(2/3)*3^(1/3))/(-4+3*2^(1/3)*3^(2/3))^(1/2)+243/125128 *ln(2+3*x)-1/2592*((-6)^(2/3)+18*(-3)^(1/3)+2*2^(1/3))*ln(6-3*(-3)^(1/3)*2 ^(2/3)*x+x^2)*3^(2/3)/(1+(-1)^(1/3))^2/(29+9*(-3)^(1/3)*2^(2/3))+1/3888*(- 1)^(2/3)*(9*(-6)^(1/3)+(-2)^(2/3)+3^(2/3))*ln(6+3*(-2)^(2/3)*3^(1/3)*x+x^2 )*6^(2/3)/(58+9*2^(2/3)*3^(1/3)-9*I*2^(2/3)*3^(5/6))-(54-6^(2/3)*(2^(2/3)+ 3^(2/3)))*ln(6+3*2^(2/3)*3^(1/3)*x+x^2)/(225504-69984*2^(2/3)*3^(1/3))
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.03 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.23 \[ \int \frac {1}{432+648 x+216 x^2+972 x^3+1008 x^4+54 x^5+2 x^6+3 x^7} \, dx=\frac {243 \log (2+3 x)}{125128}-\frac {\text {RootSum}\left [216+108 \text {$\#$1}^2+324 \text {$\#$1}^3+18 \text {$\#$1}^4+\text {$\#$1}^6\&,\frac {16168 \log (x-\text {$\#$1})-24252 \log (x-\text {$\#$1}) \text {$\#$1}+75744 \log (x-\text {$\#$1}) \text {$\#$1}^2+4482 \log (x-\text {$\#$1}) \text {$\#$1}^3-162 \log (x-\text {$\#$1}) \text {$\#$1}^4+243 \log (x-\text {$\#$1}) \text {$\#$1}^5}{36 \text {$\#$1}+162 \text {$\#$1}^2+12 \text {$\#$1}^3+\text {$\#$1}^5}\&\right ]}{750768} \] Input:
Integrate[(432 + 648*x + 216*x^2 + 972*x^3 + 1008*x^4 + 54*x^5 + 2*x^6 + 3 *x^7)^(-1),x]
Output:
(243*Log[2 + 3*x])/125128 - RootSum[216 + 108*#1^2 + 324*#1^3 + 18*#1^4 + #1^6 & , (16168*Log[x - #1] - 24252*Log[x - #1]*#1 + 75744*Log[x - #1]*#1^ 2 + 4482*Log[x - #1]*#1^3 - 162*Log[x - #1]*#1^4 + 243*Log[x - #1]*#1^5)/( 36*#1 + 162*#1^2 + 12*#1^3 + #1^5) & ]/750768
Time = 2.76 (sec) , antiderivative size = 474, normalized size of antiderivative = 0.78, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {2462, 2009}
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.
\(\displaystyle \int \frac {1}{3 x^7+2 x^6+54 x^5+1008 x^4+972 x^3+216 x^2+648 x+432} \, dx\) |
\(\Big \downarrow \) 2462 |
\(\displaystyle \int \left (\frac {-243 x^5+162 x^4-4482 x^3-75744 x^2+24252 x-16168}{125128 \left (x^6+18 x^4+324 x^3+108 x^2+216\right )}+\frac {729}{125128 (3 x+2)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\left (17703 \sqrt [3]{-6}-51983 (-2)^{2/3}-7752\ 3^{2/3}\right ) \arctan \left (\frac {2 x+3 (-2)^{2/3} \sqrt [3]{3}}{\sqrt {6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{30406104 \sqrt [6]{3} \sqrt {8+9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}}}-\frac {(-1)^{2/3} \left (3876 (-6)^{2/3}-17703 \sqrt [3]{-3}+51983 \sqrt [3]{2}\right ) \arctan \left (\frac {\sqrt [6]{2} \left (3 \sqrt [3]{-3}-\sqrt [3]{2} x\right )}{\sqrt {3 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{10135368 \sqrt [6]{6} \left (1+\sqrt [3]{-1}\right )^2 \sqrt {4-3 (-3)^{2/3} \sqrt [3]{2}}}+\frac {\left (51983 \sqrt [3]{2}+17703 \sqrt [3]{3}+3876\ 6^{2/3}\right ) \text {arctanh}\left (\frac {\sqrt [6]{2} \left (\sqrt [3]{2} x+3 \sqrt [3]{3}\right )}{\sqrt {3 \left (3 \sqrt [3]{2} 3^{2/3}-4\right )}}\right )}{30406104 \sqrt [6]{6} \sqrt {3 \sqrt [3]{2} 3^{2/3}-4}}+\frac {(-1)^{2/3} \left (10599+6561 (-3)^{2/3} \sqrt [3]{2}-2750 \sqrt [3]{-3} 2^{2/3}\right ) \log \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}{6756912 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^2}-\frac {\left (39366+3^{2/3} \left (3533 (-6)^{2/3}-5500 \sqrt [3]{-2}\right )\right ) \log \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}{121624416}-\frac {\left (39366+6^{2/3} \left (2750\ 2^{2/3}+3533\ 3^{2/3}\right )\right ) \log \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )}{121624416}+\frac {243 \log (3 x+2)}{125128}\) |
Input:
Int[(432 + 648*x + 216*x^2 + 972*x^3 + 1008*x^4 + 54*x^5 + 2*x^6 + 3*x^7)^ (-1),x]
Output:
((17703*(-6)^(1/3) - 51983*(-2)^(2/3) - 7752*3^(2/3))*ArcTan[(3*(-2)^(2/3) *3^(1/3) + 2*x)/Sqrt[6*(4 + 3*(-2)^(1/3)*3^(2/3))]])/(30406104*3^(1/6)*Sqr t[8 + (9*I)*2^(1/3)*3^(1/6) + 3*2^(1/3)*3^(2/3)]) - ((-1)^(2/3)*(3876*(-6) ^(2/3) - 17703*(-3)^(1/3) + 51983*2^(1/3))*ArcTan[(2^(1/6)*(3*(-3)^(1/3) - 2^(1/3)*x))/Sqrt[3*(4 - 3*(-3)^(2/3)*2^(1/3))]])/(10135368*6^(1/6)*(1 + ( -1)^(1/3))^2*Sqrt[4 - 3*(-3)^(2/3)*2^(1/3)]) + ((51983*2^(1/3) + 17703*3^( 1/3) + 3876*6^(2/3))*ArcTanh[(2^(1/6)*(3*3^(1/3) + 2^(1/3)*x))/Sqrt[3*(-4 + 3*2^(1/3)*3^(2/3))]])/(30406104*6^(1/6)*Sqrt[-4 + 3*2^(1/3)*3^(2/3)]) + (243*Log[2 + 3*x])/125128 + ((-1)^(2/3)*(10599 + 6561*(-3)^(2/3)*2^(1/3) - 2750*(-3)^(1/3)*2^(2/3))*Log[6 - 3*(-3)^(1/3)*2^(2/3)*x + x^2])/(6756912* 2^(1/3)*3^(2/3)*(1 + (-1)^(1/3))^2) - ((39366 + 3^(2/3)*(3533*(-6)^(2/3) - 5500*(-2)^(1/3)))*Log[6 + 3*(-2)^(2/3)*3^(1/3)*x + x^2])/121624416 - ((39 366 + 6^(2/3)*(2750*2^(2/3) + 3533*3^(2/3)))*Log[6 + 3*2^(2/3)*3^(1/3)*x + x^2])/121624416
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u*Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && GtQ [Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0 ] && RationalFunctionQ[u, x]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.06 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.13
method | result | size |
risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (79206024 \textit {\_Z}^{6}+299024136 \textit {\_Z}^{5}+290189628 \textit {\_Z}^{4}+93944556 \textit {\_Z}^{3}-764154 \textit {\_Z}^{2}+1530 \textit {\_Z} -1\right )}{\sum }\textit {\_R} \ln \left (-133339913862864047026951648912056 \textit {\_R}^{5}-503455034796500794783556665488168 \textit {\_R}^{4}-488751107538106925077309379679144 \textit {\_R}^{3}-158373383880682539034481169501120 \textit {\_R}^{2}+1215540060612927053513490370296 \textit {\_R} +58850476781569891989955805 x -1531118504941184606171339196\right )\right )}{1944}+\frac {243 \ln \left (2+3 x \right )}{125128}\) | \(77\) |
default | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )}{\sum }\frac {\left (-243 \textit {\_R}^{5}+162 \textit {\_R}^{4}-4482 \textit {\_R}^{3}-75744 \textit {\_R}^{2}+24252 \textit {\_R} -16168\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5}+12 \textit {\_R}^{3}+162 \textit {\_R}^{2}+36 \textit {\_R}}\right )}{750768}+\frac {243 \ln \left (2+3 x \right )}{125128}\) | \(87\) |
Input:
int(1/(3*x^7+2*x^6+54*x^5+1008*x^4+972*x^3+216*x^2+648*x+432),x,method=_RE TURNVERBOSE)
Output:
1/1944*sum(_R*ln(-133339913862864047026951648912056*_R^5-50345503479650079 4783556665488168*_R^4-488751107538106925077309379679144*_R^3-1583733838806 82539034481169501120*_R^2+1215540060612927053513490370296*_R+5885047678156 9891989955805*x-1531118504941184606171339196),_R=RootOf(79206024*_Z^6+2990 24136*_Z^5+290189628*_Z^4+93944556*_Z^3-764154*_Z^2+1530*_Z-1))+243/125128 *ln(2+3*x)
Timed out. \[ \int \frac {1}{432+648 x+216 x^2+972 x^3+1008 x^4+54 x^5+2 x^6+3 x^7} \, dx=\text {Timed out} \] Input:
integrate(1/(3*x^7+2*x^6+54*x^5+1008*x^4+972*x^3+216*x^2+648*x+432),x, alg orithm="fricas")
Output:
Timed out
Time = 0.42 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.14 \[ \int \frac {1}{432+648 x+216 x^2+972 x^3+1008 x^4+54 x^5+2 x^6+3 x^7} \, dx=\frac {243 \log {\left (x + \frac {2}{3} \right )}}{125128} + \operatorname {RootSum} {\left (4274996628704706944191954944 t^{6} + 8302092103887569428414464 t^{5} + 4144450225120841023488 t^{4} + 690176868966549504 t^{3} - 2887841890944 t^{2} + 2974320 t - 1, \left ( t \mapsto t \log {\left (\frac {186549983792253452567960080749168217215089045324444510832167064251139373466237534208 t^{6}}{5758416557677247627715162336355749658352447806959442529575} + \frac {43725890271624606138420190534428681871298998400865601422889352424228841586688 t^{5}}{5758416557677247627715162336355749658352447806959442529575} - \frac {522702975661454177912290454139420738467736654309175195594878675614610395234304 t^{4}}{5758416557677247627715162336355749658352447806959442529575} - \frac {321224079878547372471171470938499880967240473547842617172308647169720205312 t^{3}}{5758416557677247627715162336355749658352447806959442529575} - \frac {58689624023686849959854698156402758358057666085552221408970250486091008 t^{2}}{5758416557677247627715162336355749658352447806959442529575} + \frac {9253847552744655196468299940585461581206375367903425714731382944 t}{230336662307089905108606493454229986334097912278377701183} + x - \frac {449582741794983102129276747942184631844470423549131141927666}{17275249673031742883145487009067248975057343420878327588725} \right )} \right )\right )} \] Input:
integrate(1/(3*x**7+2*x**6+54*x**5+1008*x**4+972*x**3+216*x**2+648*x+432), x)
Output:
243*log(x + 2/3)/125128 + RootSum(4274996628704706944191954944*_t**6 + 830 2092103887569428414464*_t**5 + 4144450225120841023488*_t**4 + 690176868966 549504*_t**3 - 2887841890944*_t**2 + 2974320*_t - 1, Lambda(_t, _t*log(186 54998379225345256796008074916821721508904532444451083216706425113937346623 7534208*_t**6/5758416557677247627715162336355749658352447806959442529575 + 4372589027162460613842019053442868187129899840086560142288935242422884158 6688*_t**5/5758416557677247627715162336355749658352447806959442529575 - 52 27029756614541779122904541394207384677366543091751955948786756146103952343 04*_t**4/5758416557677247627715162336355749658352447806959442529575 - 3212 24079878547372471171470938499880967240473547842617172308647169720205312*_t **3/5758416557677247627715162336355749658352447806959442529575 - 586896240 23686849959854698156402758358057666085552221408970250486091008*_t**2/57584 16557677247627715162336355749658352447806959442529575 + 925384755274465519 6468299940585461581206375367903425714731382944*_t/230336662307089905108606 493454229986334097912278377701183 + x - 4495827417949831021292767479421846 31844470423549131141927666/17275249673031742883145487009067248975057343420 878327588725)))
\[ \int \frac {1}{432+648 x+216 x^2+972 x^3+1008 x^4+54 x^5+2 x^6+3 x^7} \, dx=\int { \frac {1}{3 \, x^{7} + 2 \, x^{6} + 54 \, x^{5} + 1008 \, x^{4} + 972 \, x^{3} + 216 \, x^{2} + 648 \, x + 432} \,d x } \] Input:
integrate(1/(3*x^7+2*x^6+54*x^5+1008*x^4+972*x^3+216*x^2+648*x+432),x, alg orithm="maxima")
Output:
-1/125128*integrate((243*x^5 - 162*x^4 + 4482*x^3 + 75744*x^2 - 24252*x + 16168)/(x^6 + 18*x^4 + 324*x^3 + 108*x^2 + 216), x) + 243/125128*log(3*x + 2)
\[ \int \frac {1}{432+648 x+216 x^2+972 x^3+1008 x^4+54 x^5+2 x^6+3 x^7} \, dx=\int { \frac {1}{3 \, x^{7} + 2 \, x^{6} + 54 \, x^{5} + 1008 \, x^{4} + 972 \, x^{3} + 216 \, x^{2} + 648 \, x + 432} \,d x } \] Input:
integrate(1/(3*x^7+2*x^6+54*x^5+1008*x^4+972*x^3+216*x^2+648*x+432),x, alg orithm="giac")
Output:
integrate(1/(3*x^7 + 2*x^6 + 54*x^5 + 1008*x^4 + 972*x^3 + 216*x^2 + 648*x + 432), x)
Time = 22.02 (sec) , antiderivative size = 467, normalized size of antiderivative = 0.77 \[ \int \frac {1}{432+648 x+216 x^2+972 x^3+1008 x^4+54 x^5+2 x^6+3 x^7} \, dx=\text {Too large to display} \] Input:
int(1/(648*x + 216*x^2 + 972*x^3 + 1008*x^4 + 54*x^5 + 2*x^6 + 3*x^7 + 432 ),x)
Output:
(243*log(x + 2/3))/125128 + symsum(log((2*root(z^6 + (243*z^5)/125128 + (2 98549*z^4)/307953021312 + (869857*z^3)/5387946060874752 - (4717*z^2)/69827 78094893678592 + (85*z)/122170685548259800645632 - 1/427499662870470694419 1954944, z, k))/729 + (7*root(z^6 + (243*z^5)/125128 + (298549*z^4)/307953 021312 + (869857*z^3)/5387946060874752 - (4717*z^2)/6982778094893678592 + (85*z)/122170685548259800645632 - 1/4274996628704706944191954944, z, k)*x) /243 - (1445769776*root(z^6 + (243*z^5)/125128 + (298549*z^4)/307953021312 + (869857*z^3)/5387946060874752 - (4717*z^2)/6982778094893678592 + (85*z) /122170685548259800645632 - 1/4274996628704706944191954944, z, k)^2*x)/196 83 + (1605207476480*root(z^6 + (243*z^5)/125128 + (298549*z^4)/30795302131 2 + (869857*z^3)/5387946060874752 - (4717*z^2)/6982778094893678592 + (85*z )/122170685548259800645632 - 1/4274996628704706944191954944, z, k)^3*x)/27 - 11385414082473984*root(z^6 + (243*z^5)/125128 + (298549*z^4)/3079530213 12 + (869857*z^3)/5387946060874752 - (4717*z^2)/6982778094893678592 + (85* z)/122170685548259800645632 - 1/4274996628704706944191954944, z, k)^4*x - 46778486686192041984*root(z^6 + (243*z^5)/125128 + (298549*z^4)/3079530213 12 + (869857*z^3)/5387946060874752 - (4717*z^2)/6982778094893678592 + (85* z)/122170685548259800645632 - 1/4274996628704706944191954944, z, k)^5*x - 50765351266770267930624*root(z^6 + (243*z^5)/125128 + (298549*z^4)/3079530 21312 + (869857*z^3)/5387946060874752 - (4717*z^2)/6982778094893678592 ...
\[ \int \frac {1}{432+648 x+216 x^2+972 x^3+1008 x^4+54 x^5+2 x^6+3 x^7} \, dx=\int \frac {1}{3 x^{7}+2 x^{6}+54 x^{5}+1008 x^{4}+972 x^{3}+216 x^{2}+648 x +432}d x \] Input:
int(1/(3*x^7+2*x^6+54*x^5+1008*x^4+972*x^3+216*x^2+648*x+432),x)
Output:
int(1/(3*x**7 + 2*x**6 + 54*x**5 + 1008*x**4 + 972*x**3 + 216*x**2 + 648*x + 432),x)