Integrand size = 60, antiderivative size = 117 \[ \int \frac {3 \sqrt {7}+6 \sqrt {11}+\left (-121 \sqrt {105}-98 \sqrt {165}\right ) x^4}{-12 \sqrt {15}+10200 x^4-118580 \sqrt {15} x^8} \, dx=-\frac {\arctan \left (\sqrt [8]{\frac {5}{3}} \sqrt {7} x\right )}{8\ 3^{3/8} 5^{5/8}}-\frac {\arctan \left (\sqrt [8]{\frac {5}{3}} \sqrt {11} x\right )}{4\ 3^{3/8} 5^{5/8}}-\frac {\text {arctanh}\left (\sqrt [8]{\frac {5}{3}} \sqrt {7} x\right )}{8\ 3^{3/8} 5^{5/8}}-\frac {\text {arctanh}\left (\sqrt [8]{\frac {5}{3}} \sqrt {11} x\right )}{4\ 3^{3/8} 5^{5/8}} \] Output:
-1/120*arctan(1/3*5^(1/8)*3^(7/8)*7^(1/2)*x)*3^(5/8)*5^(3/8)-1/60*arctan(1 /3*5^(1/8)*3^(7/8)*11^(1/2)*x)*3^(5/8)*5^(3/8)-1/120*arctanh(1/3*5^(1/8)*3 ^(7/8)*7^(1/2)*x)*3^(5/8)*5^(3/8)-1/60*arctanh(1/3*5^(1/8)*3^(7/8)*11^(1/2 )*x)*3^(5/8)*5^(3/8)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.12 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.01 \[ \int \frac {3 \sqrt {7}+6 \sqrt {11}+\left (-121 \sqrt {105}-98 \sqrt {165}\right ) x^4}{-12 \sqrt {15}+10200 x^4-118580 \sqrt {15} x^8} \, dx=\frac {1}{160} \text {RootSum}\left [3 \sqrt {15}-2550 \text {$\#$1}^4+29645 \sqrt {15} \text {$\#$1}^8\&,\frac {-3 \sqrt {7} \log (x-\text {$\#$1})-6 \sqrt {11} \log (x-\text {$\#$1})+121 \sqrt {105} \log (x-\text {$\#$1}) \text {$\#$1}^4+98 \sqrt {165} \log (x-\text {$\#$1}) \text {$\#$1}^4}{-255 \text {$\#$1}^3+5929 \sqrt {15} \text {$\#$1}^7}\&\right ] \] Input:
Integrate[(3*Sqrt[7] + 6*Sqrt[11] + (-121*Sqrt[105] - 98*Sqrt[165])*x^4)/( -12*Sqrt[15] + 10200*x^4 - 118580*Sqrt[15]*x^8),x]
Output:
RootSum[3*Sqrt[15] - 2550*#1^4 + 29645*Sqrt[15]*#1^8 & , (-3*Sqrt[7]*Log[x - #1] - 6*Sqrt[11]*Log[x - #1] + 121*Sqrt[105]*Log[x - #1]*#1^4 + 98*Sqrt [165]*Log[x - #1]*#1^4)/(-255*#1^3 + 5929*Sqrt[15]*#1^7) & ]/160
Time = 0.54 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.31, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {1752, 756, 216, 219}
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.
\(\displaystyle \int \frac {\left (-121 \sqrt {105}-98 \sqrt {165}\right ) x^4+6 \sqrt {11}+3 \sqrt {7}}{-118580 \sqrt {15} x^8+10200 x^4-12 \sqrt {15}} \, dx\) |
\(\Big \downarrow \) 1752 |
\(\displaystyle -98 \sqrt {165} \int \frac {1}{2940-118580 \sqrt {15} x^4}dx-121 \sqrt {105} \int \frac {1}{7260-118580 \sqrt {15} x^4}dx\) |
\(\Big \downarrow \) 756 |
\(\displaystyle -121 \sqrt {105} \left (\frac {\int \frac {1}{\sqrt [4]{3}-7 \sqrt [4]{5} x^2}dx}{4840\ 3^{3/4}}+\frac {\int \frac {1}{7 \sqrt [4]{5} x^2+\sqrt [4]{3}}dx}{4840\ 3^{3/4}}\right )-98 \sqrt {165} \left (\frac {\int \frac {1}{\sqrt [4]{3}-11 \sqrt [4]{5} x^2}dx}{1960\ 3^{3/4}}+\frac {\int \frac {1}{11 \sqrt [4]{5} x^2+\sqrt [4]{3}}dx}{1960\ 3^{3/4}}\right )\) |
\(\Big \downarrow \) 216 |
\(\displaystyle -98 \sqrt {165} \left (\frac {\int \frac {1}{\sqrt [4]{3}-11 \sqrt [4]{5} x^2}dx}{1960\ 3^{3/4}}+\frac {\arctan \left (\sqrt [8]{\frac {5}{3}} \sqrt {11} x\right )}{1960\ 3^{7/8} \sqrt [8]{5} \sqrt {11}}\right )-121 \sqrt {105} \left (\frac {\int \frac {1}{\sqrt [4]{3}-7 \sqrt [4]{5} x^2}dx}{4840\ 3^{3/4}}+\frac {\arctan \left (\sqrt [8]{\frac {5}{3}} \sqrt {7} x\right )}{4840\ 3^{7/8} \sqrt [8]{5} \sqrt {7}}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -121 \sqrt {105} \left (\frac {\arctan \left (\sqrt [8]{\frac {5}{3}} \sqrt {7} x\right )}{4840\ 3^{7/8} \sqrt [8]{5} \sqrt {7}}+\frac {\text {arctanh}\left (\sqrt [8]{\frac {5}{3}} \sqrt {7} x\right )}{4840\ 3^{7/8} \sqrt [8]{5} \sqrt {7}}\right )-98 \sqrt {165} \left (\frac {\arctan \left (\sqrt [8]{\frac {5}{3}} \sqrt {11} x\right )}{1960\ 3^{7/8} \sqrt [8]{5} \sqrt {11}}+\frac {\text {arctanh}\left (\sqrt [8]{\frac {5}{3}} \sqrt {11} x\right )}{1960\ 3^{7/8} \sqrt [8]{5} \sqrt {11}}\right )\) |
Input:
Int[(3*Sqrt[7] + 6*Sqrt[11] + (-121*Sqrt[105] - 98*Sqrt[165])*x^4)/(-12*Sq rt[15] + 10200*x^4 - 118580*Sqrt[15]*x^8),x]
Output:
-121*Sqrt[105]*(ArcTan[(5/3)^(1/8)*Sqrt[7]*x]/(4840*3^(7/8)*5^(1/8)*Sqrt[7 ]) + ArcTanh[(5/3)^(1/8)*Sqrt[7]*x]/(4840*3^(7/8)*5^(1/8)*Sqrt[7])) - 98*S qrt[165]*(ArcTan[(5/3)^(1/8)*Sqrt[11]*x]/(1960*3^(7/8)*5^(1/8)*Sqrt[11]) + ArcTanh[(5/3)^(1/8)*Sqrt[11]*x]/(1960*3^(7/8)*5^(1/8)*Sqrt[11]))
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 ]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a) Int[1/(r - s*x^2), x], x] + Simp[r/(2*a) Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ[a /b, 0]
Int[((d_) + (e_.)*(x_)^(n_))/((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_)), x _Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q)) Int[1/(b/2 - q/2 + c*x^n), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q)) I nt[1/(b/2 + q/2 + c*x^n), x], x]] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[n2 , 2*n] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && (PosQ[b^2 - 4*a*c] || !IGtQ[n/2, 0])
Leaf count of result is larger than twice the leaf count of optimal. \(191\) vs. \(2(85)=170\).
Time = 0.62 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.64
method | result | size |
default | \(-\frac {\left (-49 \sqrt {7}\, \sqrt {15}-98 \sqrt {11}\, \sqrt {15}+121 \sqrt {105}+98 \sqrt {165}\right ) \sqrt {35}\, \sqrt {\sqrt {5}\, 15^{\frac {1}{4}}}\, \left (\ln \left (\frac {x +\frac {\sqrt {35}\, \sqrt {\sqrt {5}\, 15^{\frac {1}{4}}}}{35}}{x -\frac {\sqrt {35}\, \sqrt {\sqrt {5}\, 15^{\frac {1}{4}}}}{35}}\right )+2 \arctan \left (\frac {x \sqrt {35}}{\sqrt {\sqrt {5}\, 15^{\frac {1}{4}}}}\right )\right )}{604800}+\frac {\left (-121 \sqrt {7}\, \sqrt {15}-242 \sqrt {11}\, \sqrt {15}+121 \sqrt {105}+98 \sqrt {165}\right ) \sqrt {55}\, \sqrt {\sqrt {5}\, 15^{\frac {1}{4}}}\, \left (\ln \left (\frac {x +\frac {\sqrt {55}\, \sqrt {\sqrt {5}\, 15^{\frac {1}{4}}}}{55}}{x -\frac {\sqrt {55}\, \sqrt {\sqrt {5}\, 15^{\frac {1}{4}}}}{55}}\right )+2 \arctan \left (\frac {x \sqrt {55}}{\sqrt {\sqrt {5}\, 15^{\frac {1}{4}}}}\right )\right )}{950400}\) | \(192\) |
Input:
int((3*7^(1/2)+6*11^(1/2)+(-121*105^(1/2)-98*165^(1/2))*x^4)/(-12*15^(1/2) +10200*x^4-118580*15^(1/2)*x^8),x,method=_RETURNVERBOSE)
Output:
-1/604800*(-49*7^(1/2)*15^(1/2)-98*11^(1/2)*15^(1/2)+121*105^(1/2)+98*165^ (1/2))*35^(1/2)*(5^(1/2)*15^(1/4))^(1/2)*(ln((x+1/35*35^(1/2)*(5^(1/2)*15^ (1/4))^(1/2))/(x-1/35*35^(1/2)*(5^(1/2)*15^(1/4))^(1/2)))+2*arctan(x*35^(1 /2)/(5^(1/2)*15^(1/4))^(1/2)))+1/950400*(-121*7^(1/2)*15^(1/2)-242*11^(1/2 )*15^(1/2)+121*105^(1/2)+98*165^(1/2))*55^(1/2)*(5^(1/2)*15^(1/4))^(1/2)*( ln((x+1/55*55^(1/2)*(5^(1/2)*15^(1/4))^(1/2))/(x-1/55*55^(1/2)*(5^(1/2)*15 ^(1/4))^(1/2)))+2*arctan(x*55^(1/2)/(5^(1/2)*15^(1/4))^(1/2)))
\[ \int \frac {3 \sqrt {7}+6 \sqrt {11}+\left (-121 \sqrt {105}-98 \sqrt {165}\right ) x^4}{-12 \sqrt {15}+10200 x^4-118580 \sqrt {15} x^8} \, dx=\int { \frac {x^{4} {\left (98 \, \sqrt {165} + 121 \, \sqrt {105}\right )} - 6 \, \sqrt {11} - 3 \, \sqrt {7}}{4 \, {\left (29645 \, \sqrt {15} x^{8} - 2550 \, x^{4} + 3 \, \sqrt {15}\right )}} \,d x } \] Input:
integrate((3*7^(1/2)+6*11^(1/2)+(-121*105^(1/2)-98*165^(1/2))*x^4)/(-12*15 ^(1/2)+10200*x^4-118580*15^(1/2)*x^8),x, algorithm="fricas")
Output:
0
Exception generated. \[ \int \frac {3 \sqrt {7}+6 \sqrt {11}+\left (-121 \sqrt {105}-98 \sqrt {165}\right ) x^4}{-12 \sqrt {15}+10200 x^4-118580 \sqrt {15} x^8} \, dx=\text {Exception raised: PolynomialError} \] Input:
integrate((3*7**(1/2)+6*11**(1/2)+(-121*105**(1/2)-98*165**(1/2))*x**4)/(- 12*15**(1/2)+10200*x**4-118580*15**(1/2)*x**8),x)
Output:
Exception raised: PolynomialError >> 1/(8876619644311361811449478210059404 03892090960648942070894847733907374495278002081072424698386397863110524278 65545878613259174768870299313417544282422774969138741079283023024302083445 868952121128899
\[ \int \frac {3 \sqrt {7}+6 \sqrt {11}+\left (-121 \sqrt {105}-98 \sqrt {165}\right ) x^4}{-12 \sqrt {15}+10200 x^4-118580 \sqrt {15} x^8} \, dx=\int { \frac {x^{4} {\left (98 \, \sqrt {165} + 121 \, \sqrt {105}\right )} - 6 \, \sqrt {11} - 3 \, \sqrt {7}}{4 \, {\left (29645 \, \sqrt {15} x^{8} - 2550 \, x^{4} + 3 \, \sqrt {15}\right )}} \,d x } \] Input:
integrate((3*7^(1/2)+6*11^(1/2)+(-121*105^(1/2)-98*165^(1/2))*x^4)/(-12*15 ^(1/2)+10200*x^4-118580*15^(1/2)*x^8),x, algorithm="maxima")
Output:
1/4*integrate((x^4*(98*sqrt(165) + 121*sqrt(105)) - 6*sqrt(11) - 3*sqrt(7) )/(29645*sqrt(15)*x^8 - 2550*x^4 + 3*sqrt(15)), x)
Exception generated. \[ \int \frac {3 \sqrt {7}+6 \sqrt {11}+\left (-121 \sqrt {105}-98 \sqrt {165}\right ) x^4}{-12 \sqrt {15}+10200 x^4-118580 \sqrt {15} x^8} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((3*7^(1/2)+6*11^(1/2)+(-121*105^(1/2)-98*165^(1/2))*x^4)/(-12*15 ^(1/2)+10200*x^4-118580*15^(1/2)*x^8),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro unding error%%%{%%{[1274,0,-1480290,0,427747950,0,-19021101750,0]:[1,0,-11 40,0,3163
Time = 18.53 (sec) , antiderivative size = 1889, normalized size of antiderivative = 16.15 \[ \int \frac {3 \sqrt {7}+6 \sqrt {11}+\left (-121 \sqrt {105}-98 \sqrt {165}\right ) x^4}{-12 \sqrt {15}+10200 x^4-118580 \sqrt {15} x^8} \, dx=\text {Too large to display} \] Input:
int(-(3*7^(1/2) - x^4*(121*105^(1/2) + 98*165^(1/2)) + 6*11^(1/2))/(12*15^ (1/2) + 118580*15^(1/2)*x^8 - 10200*x^4),x)
Output:
atan((((39715937915904*x)/17683290930758754796256905891884705078125 + ((17 *15^(1/2))/147456000 - (5^(1/2)*888446500935303168^(1/2))/5349660268953600 )^(1/4)*((940369969152*7^(1/2)*15^(1/2))/360883488382831730535855222283361 328125 + (117546246144*11^(1/2)*15^(1/2))/14614290025420458509303228009822 0703125 - ((4092730840461606912*15^(1/2)*x)/707331637230350191850276235675 388203125 + ((34665798542819328*7^(1/2))/288706790706265384428684177826689 0625 + (69331597085638656*11^(1/2))/1169143202033636680744258240785765625) *((17*15^(1/2))/147456000 - (5^(1/2)*888446500935303168^(1/2))/53496602689 53600)^(1/4))*((17*15^(1/2))/147456000 - (5^(1/2)*888446500935303168^(1/2) )/5349660268953600)^(3/4)))*((17*15^(1/2))/147456000 - (5^(1/2)*8884465009 35303168^(1/2))/5349660268953600)^(1/4)*1i + ((39715937915904*x)/176832909 30758754796256905891884705078125 - ((17*15^(1/2))/147456000 - (5^(1/2)*888 446500935303168^(1/2))/5349660268953600)^(1/4)*((940369969152*7^(1/2)*15^( 1/2))/360883488382831730535855222283361328125 + (117546246144*11^(1/2)*15^ (1/2))/146142900254204585093032280098220703125 + ((4092730840461606912*15^ (1/2)*x)/707331637230350191850276235675388203125 - ((34665798542819328*7^( 1/2))/2887067907062653844286841778266890625 + (69331597085638656*11^(1/2)) /1169143202033636680744258240785765625)*((17*15^(1/2))/147456000 - (5^(1/2 )*888446500935303168^(1/2))/5349660268953600)^(1/4))*((17*15^(1/2))/147456 000 - (5^(1/2)*888446500935303168^(1/2))/5349660268953600)^(3/4)))*((17...
Time = 0.27 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.91 \[ \int \frac {3 \sqrt {7}+6 \sqrt {11}+\left (-121 \sqrt {105}-98 \sqrt {165}\right ) x^4}{-12 \sqrt {15}+10200 x^4-118580 \sqrt {15} x^8} \, dx=\frac {\sqrt {3}\, 15^{\frac {1}{8}} 5^{\frac {1}{4}} \left (-4 \mathit {atan} \left (\frac {11 \,5^{\frac {1}{4}} x 15^{\frac {7}{8}}}{15 \sqrt {11}}\right )-2 \mathit {atan} \left (\frac {7 \,5^{\frac {1}{4}} x 15^{\frac {7}{8}}}{15 \sqrt {7}}\right )+2 \,\mathrm {log}\left (-15^{\frac {1}{8}} \sqrt {11}+11 \,5^{\frac {1}{4}} x \right )+\mathrm {log}\left (-15^{\frac {1}{8}} \sqrt {7}+7 \,5^{\frac {1}{4}} x \right )-2 \,\mathrm {log}\left (15^{\frac {1}{8}} \sqrt {11}+11 \,5^{\frac {1}{4}} x \right )-\mathrm {log}\left (15^{\frac {1}{8}} \sqrt {7}+7 \,5^{\frac {1}{4}} x \right )\right )}{240} \] Input:
int((3*7^(1/2)+6*11^(1/2)+(-121*105^(1/2)-98*165^(1/2))*x^4)/(-12*15^(1/2) +10200*x^4-118580*15^(1/2)*x^8),x)
Output:
(sqrt(3)*15**(1/8)*5**(1/4)*( - 4*atan((11*5**(1/4)*x)/(15**(1/8)*sqrt(11) )) - 2*atan((7*5**(1/4)*x)/(15**(1/8)*sqrt(7))) + 2*log( - 15**(1/8)*sqrt( 11) + 11*5**(1/4)*x) + log( - 15**(1/8)*sqrt(7) + 7*5**(1/4)*x) - 2*log(15 **(1/8)*sqrt(11) + 11*5**(1/4)*x) - log(15**(1/8)*sqrt(7) + 7*5**(1/4)*x)) )/240