Integrand size = 25, antiderivative size = 78 \[ \int \frac {\sqrt [4]{-1+x^4} \left (-1+x^8\right )}{x^6 \left (1+x^8\right )} \, dx=\frac {\left (1-x^4\right ) \sqrt [4]{-1+x^4}}{5 x^5}+\frac {1}{4} \text {RootSum}\left [2-2 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x) \text {$\#$1}+\log \left (\sqrt [4]{-1+x^4}-x \text {$\#$1}\right ) \text {$\#$1}}{-1+\text {$\#$1}^4}\&\right ] \]
Time = 0.26 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.95 \[ \int \frac {\sqrt [4]{-1+x^4} \left (-1+x^8\right )}{x^6 \left (1+x^8\right )} \, dx=\frac {-4 \left (-1+x^4\right )^{5/4}+5 x^5 \text {RootSum}\left [2-2 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x) \text {$\#$1}+\log \left (\sqrt [4]{-1+x^4}-x \text {$\#$1}\right ) \text {$\#$1}}{-1+\text {$\#$1}^4}\&\right ]}{20 x^5} \]
(-4*(-1 + x^4)^(5/4) + 5*x^5*RootSum[2 - 2*#1^4 + #1^8 & , (-(Log[x]*#1) + Log[(-1 + x^4)^(1/4) - x*#1]*#1)/(-1 + #1^4) & ])/(20*x^5)
Result contains complex when optimal does not.
Time = 1.85 (sec) , antiderivative size = 522, normalized size of antiderivative = 6.69, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {1388, 7276, 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.
| \(\displaystyle \int \frac {\sqrt [4]{x^4-1} \left (x^8-1\right )}{x^6 \left (x^8+1\right )} \, dx\) |
| \(\Big \downarrow \) 1388 |
| \(\displaystyle \int \frac {\left (x^4-1\right )^{5/4} \left (x^4+1\right )}{x^6 \left (x^8+1\right )}dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (\frac {\left (x^4-1\right )^{5/4}}{x^6}+\frac {\left (x^4-1\right )^{5/4}}{x^2}+\frac {\left (-x^4-1\right ) \left (x^4-1\right )^{5/4} x^2}{x^8+1}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt [4]{x^4-1} x^3 \operatorname {AppellF1}\left (\frac {3}{4},-\frac {5}{4},1,\frac {7}{4},x^4,-i x^4\right )}{6 \sqrt [4]{1-x^4}}+\frac {\sqrt [4]{x^4-1} x^3 \operatorname {AppellF1}\left (\frac {3}{4},-\frac {5}{4},1,\frac {7}{4},x^4,i x^4\right )}{6 \sqrt [4]{1-x^4}}+\frac {\sqrt [4]{x^4-1} x^3 \operatorname {AppellF1}\left (\frac {3}{4},-\frac {1}{4},1,\frac {7}{4},x^4,-i x^4\right )}{6 \sqrt [4]{1-x^4}}+\frac {\sqrt [4]{x^4-1} x^3 \operatorname {AppellF1}\left (\frac {3}{4},-\frac {1}{4},1,\frac {7}{4},x^4,i x^4\right )}{6 \sqrt [4]{1-x^4}}-\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{x^4-1}}\right )+\frac {1}{8} (1-i)^{5/4} \arctan \left (\frac {\sqrt [4]{1-i} x}{\sqrt [4]{x^4-1}}\right )+\frac {\arctan \left (\frac {\sqrt [4]{1-i} x}{\sqrt [4]{x^4-1}}\right )}{4 (1-i)^{3/4}}+\frac {1}{8} (1+i)^{5/4} \arctan \left (\frac {\sqrt [4]{1+i} x}{\sqrt [4]{x^4-1}}\right )+\frac {\arctan \left (\frac {\sqrt [4]{1+i} x}{\sqrt [4]{x^4-1}}\right )}{4 (1+i)^{3/4}}+\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{x^4-1}}\right )-\frac {1}{8} (1-i)^{5/4} \text {arctanh}\left (\frac {\sqrt [4]{1-i} x}{\sqrt [4]{x^4-1}}\right )-\frac {\text {arctanh}\left (\frac {\sqrt [4]{1-i} x}{\sqrt [4]{x^4-1}}\right )}{4 (1-i)^{3/4}}-\frac {1}{8} (1+i)^{5/4} \text {arctanh}\left (\frac {\sqrt [4]{1+i} x}{\sqrt [4]{x^4-1}}\right )-\frac {\text {arctanh}\left (\frac {\sqrt [4]{1+i} x}{\sqrt [4]{x^4-1}}\right )}{4 (1+i)^{3/4}}-\frac {\left (x^4-1\right )^{5/4}}{x}-\frac {\sqrt [4]{x^4-1}}{x}-\frac {\left (x^4-1\right )^{5/4}}{5 x^5}+\sqrt [4]{x^4-1} x^3\) |
-((-1 + x^4)^(1/4)/x) + x^3*(-1 + x^4)^(1/4) - (-1 + x^4)^(5/4)/(5*x^5) - (-1 + x^4)^(5/4)/x + (x^3*(-1 + x^4)^(1/4)*AppellF1[3/4, -5/4, 1, 7/4, x^4 , (-I)*x^4])/(6*(1 - x^4)^(1/4)) + (x^3*(-1 + x^4)^(1/4)*AppellF1[3/4, -5/ 4, 1, 7/4, x^4, I*x^4])/(6*(1 - x^4)^(1/4)) + (x^3*(-1 + x^4)^(1/4)*Appell F1[3/4, -1/4, 1, 7/4, x^4, (-I)*x^4])/(6*(1 - x^4)^(1/4)) + (x^3*(-1 + x^4 )^(1/4)*AppellF1[3/4, -1/4, 1, 7/4, x^4, I*x^4])/(6*(1 - x^4)^(1/4)) - Arc Tan[x/(-1 + x^4)^(1/4)]/2 + ArcTan[((1 - I)^(1/4)*x)/(-1 + x^4)^(1/4)]/(4* (1 - I)^(3/4)) + ((1 - I)^(5/4)*ArcTan[((1 - I)^(1/4)*x)/(-1 + x^4)^(1/4)] )/8 + ArcTan[((1 + I)^(1/4)*x)/(-1 + x^4)^(1/4)]/(4*(1 + I)^(3/4)) + ((1 + I)^(5/4)*ArcTan[((1 + I)^(1/4)*x)/(-1 + x^4)^(1/4)])/8 + ArcTanh[x/(-1 + x^4)^(1/4)]/2 - ArcTanh[((1 - I)^(1/4)*x)/(-1 + x^4)^(1/4)]/(4*(1 - I)^(3/ 4)) - ((1 - I)^(5/4)*ArcTanh[((1 - I)^(1/4)*x)/(-1 + x^4)^(1/4)])/8 - ArcT anh[((1 + I)^(1/4)*x)/(-1 + x^4)^(1/4)]/(4*(1 + I)^(3/4)) - ((1 + I)^(5/4) *ArcTanh[((1 + I)^(1/4)*x)/(-1 + x^4)^(1/4)])/8
3.11.37.3.1 Defintions of rubi rules used
Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[u*(d + e*x^n)^(p + q)*(a/d + (c/e)*x^n)^p, x] /; FreeQ[{a, c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 + a*e^2, 0] && (Integer Q[p] || (GtQ[a, 0] && GtQ[d, 0]))
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Result contains higher order function than in optimal. Order 9 vs. order 1.
Time = 16.58 (sec) , antiderivative size = 3768, normalized size of antiderivative = 48.31
\[\text {output too large to display}\]
-1/5*(x^8-2*x^4+1)/x^5/(x^4-1)^(3/4)+(1/16*RootOf(_Z^4+1048576*RootOf(8388 608*_Z^8-4096*_Z^4+1)^4-512)*ln(-(8192*x^12*RootOf(_Z^4+1048576*RootOf(838 8608*_Z^8-4096*_Z^4+1)^4-512)^2*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-4096*(x ^12-3*x^8+3*x^4-1)^(1/4)*RootOf(8388608*_Z^8-4096*_Z^4+1)^4*RootOf(_Z^4+10 48576*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-512)^3*x^9-16384*x^8*RootOf(83886 08*_Z^8-4096*_Z^4+1)^4*RootOf(_Z^4+1048576*RootOf(8388608*_Z^8-4096*_Z^4+1 )^4-512)^2-6*RootOf(_Z^4+1048576*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-512)^2 *x^12+8192*(x^12-3*x^8+3*x^4-1)^(1/4)*RootOf(8388608*_Z^8-4096*_Z^4+1)^4*R ootOf(_Z^4+1048576*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-512)^3*x^5+(x^12-3*x ^8+3*x^4-1)^(1/4)*RootOf(_Z^4+1048576*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-5 12)^3*x^9-262144*RootOf(8388608*_Z^8-4096*_Z^4+1)^4*(x^12-3*x^8+3*x^4-1)^( 1/2)*x^6+65536*RootOf(8388608*_Z^8-4096*_Z^4+1)^4*RootOf(_Z^4+1048576*Root Of(8388608*_Z^8-4096*_Z^4+1)^4-512)*(x^12-3*x^8+3*x^4-1)^(3/4)*x^3+8192*x^ 4*RootOf(8388608*_Z^8-4096*_Z^4+1)^4*RootOf(_Z^4+1048576*RootOf(8388608*_Z ^8-4096*_Z^4+1)^4-512)^2+14*x^8*RootOf(_Z^4+1048576*RootOf(8388608*_Z^8-40 96*_Z^4+1)^4-512)^2-4096*(x^12-3*x^8+3*x^4-1)^(1/4)*RootOf(8388608*_Z^8-40 96*_Z^4+1)^4*RootOf(_Z^4+1048576*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-512)^3 *x-2*(x^12-3*x^8+3*x^4-1)^(1/4)*RootOf(_Z^4+1048576*RootOf(8388608*_Z^8-40 96*_Z^4+1)^4-512)^3*x^5+262144*RootOf(8388608*_Z^8-4096*_Z^4+1)^4*(x^12-3* x^8+3*x^4-1)^(1/2)*x^2+128*(x^12-3*x^8+3*x^4-1)^(1/2)*x^6-10*x^4*RootOf...
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 3.71 (sec) , antiderivative size = 863, normalized size of antiderivative = 11.06 \[ \int \frac {\sqrt [4]{-1+x^4} \left (-1+x^8\right )}{x^6 \left (1+x^8\right )} \, dx=\text {Too large to display} \]
-1/40*(5*x^5*sqrt(-sqrt(I + 1))*log(-(4*sqrt(I + 1)*((3*I + 4)*x^7 + (4*I - 3)*x^3)*(x^4 - 1)^(1/4) + 4*((3*I + 4)*x^5 + (4*I - 3)*x)*(x^4 - 1)^(3/4 ) - (sqrt(I + 1)*(-(15*I - 5)*x^8 + (12*I + 16)*x^4 + I - 7) - 4*((4*I - 3 )*x^6 - (3*I + 4)*x^2)*sqrt(x^4 - 1))*sqrt(-sqrt(I + 1)))/(x^8 + 1)) - 5*x ^5*sqrt(-sqrt(I + 1))*log(-(4*sqrt(I + 1)*((3*I + 4)*x^7 + (4*I - 3)*x^3)* (x^4 - 1)^(1/4) + 4*((3*I + 4)*x^5 + (4*I - 3)*x)*(x^4 - 1)^(3/4) - (sqrt( I + 1)*((15*I - 5)*x^8 - (12*I + 16)*x^4 - I + 7) - 4*(-(4*I - 3)*x^6 + (3 *I + 4)*x^2)*sqrt(x^4 - 1))*sqrt(-sqrt(I + 1)))/(x^8 + 1)) + 5*x^5*sqrt(-s qrt(-I + 1))*log(-(4*sqrt(-I + 1)*(-(3*I - 4)*x^7 - (4*I + 3)*x^3)*(x^4 - 1)^(1/4) + 4*(-(3*I - 4)*x^5 - (4*I + 3)*x)*(x^4 - 1)^(3/4) - (sqrt(-I + 1 )*((15*I + 5)*x^8 - (12*I - 16)*x^4 - I - 7) - 4*(-(4*I + 3)*x^6 + (3*I - 4)*x^2)*sqrt(x^4 - 1))*sqrt(-sqrt(-I + 1)))/(x^8 + 1)) - 5*x^5*sqrt(-sqrt( -I + 1))*log(-(4*sqrt(-I + 1)*(-(3*I - 4)*x^7 - (4*I + 3)*x^3)*(x^4 - 1)^( 1/4) + 4*(-(3*I - 4)*x^5 - (4*I + 3)*x)*(x^4 - 1)^(3/4) - (sqrt(-I + 1)*(- (15*I + 5)*x^8 + (12*I - 16)*x^4 + I + 7) - 4*((4*I + 3)*x^6 - (3*I - 4)*x ^2)*sqrt(x^4 - 1))*sqrt(-sqrt(-I + 1)))/(x^8 + 1)) - 5*(-I + 1)^(1/4)*x^5* log(-(4*sqrt(-I + 1)*((3*I - 4)*x^7 + (4*I + 3)*x^3)*(x^4 - 1)^(1/4) + 4*( -(3*I - 4)*x^5 - (4*I + 3)*x)*(x^4 - 1)^(3/4) - (-I + 1)^(1/4)*(sqrt(-I + 1)*((15*I + 5)*x^8 - (12*I - 16)*x^4 - I - 7) - 4*((4*I + 3)*x^6 - (3*I - 4)*x^2)*sqrt(x^4 - 1)))/(x^8 + 1)) + 5*(-I + 1)^(1/4)*x^5*log(-(4*sqrt(...
Not integrable
Time = 42.27 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.53 \[ \int \frac {\sqrt [4]{-1+x^4} \left (-1+x^8\right )}{x^6 \left (1+x^8\right )} \, dx=\int \frac {\sqrt [4]{\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \left (x^{4} + 1\right )}{x^{6} \left (x^{8} + 1\right )}\, dx \]
Integral(((x - 1)*(x + 1)*(x**2 + 1))**(1/4)*(x - 1)*(x + 1)*(x**2 + 1)*(x **4 + 1)/(x**6*(x**8 + 1)), x)
Not integrable
Time = 0.31 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.32 \[ \int \frac {\sqrt [4]{-1+x^4} \left (-1+x^8\right )}{x^6 \left (1+x^8\right )} \, dx=\int { \frac {{\left (x^{8} - 1\right )} {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{{\left (x^{8} + 1\right )} x^{6}} \,d x } \]
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.31 (sec) , antiderivative size = 290, normalized size of antiderivative = 3.72 \[ \int \frac {\sqrt [4]{-1+x^4} \left (-1+x^8\right )}{x^6 \left (1+x^8\right )} \, dx=\frac {1}{144115188075855872} i \, \left (8 i + 8\right )^{\frac {63}{4}} \log \left (\left (-281474976710656 i + 281474976710656\right )^{\frac {1}{4}} - \frac {4096 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{144115188075855872} i \, \left (8 i + 8\right )^{\frac {63}{4}} \log \left (-\left (-281474976710656 i + 281474976710656\right )^{\frac {1}{4}} - \frac {4096 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{536870912} \, \left (8 i + 8\right )^{\frac {31}{4}} \log \left (i \, \left (-16777216 i + 16777216\right )^{\frac {1}{4}} - \frac {64 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{536870912} \, \left (8 i + 8\right )^{\frac {31}{4}} \log \left (-i \, \left (-16777216 i + 16777216\right )^{\frac {1}{4}} - \frac {64 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right ) + \frac {{\left (x^{4} - 1\right )}^{\frac {1}{4}} {\left (\frac {1}{x^{4}} - 1\right )}}{5 \, x} - \frac {i \, \left (8 i + 8\right )^{\frac {15}{4}} \log \left (\left (16777216 i + 16777216\right )^{\frac {1}{4}} - \frac {64 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right )}{256 \, {\left (\sqrt {\sqrt {2} + 2} + i \, \sqrt {-\sqrt {2} + 2}\right )}^{7}} + \frac {\left (8 i + 8\right )^{\frac {15}{4}} \log \left (i \, \left (16777216 i + 16777216\right )^{\frac {1}{4}} - \frac {64 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right )}{256 \, {\left (\sqrt {\sqrt {2} + 2} + i \, \sqrt {-\sqrt {2} + 2}\right )}^{7}} - \frac {\left (8 i + 8\right )^{\frac {15}{4}} \log \left (-i \, \left (16777216 i + 16777216\right )^{\frac {1}{4}} - \frac {64 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right )}{256 \, {\left (\sqrt {\sqrt {2} + 2} + i \, \sqrt {-\sqrt {2} + 2}\right )}^{7}} + \frac {i \, \left (8 i + 8\right )^{\frac {15}{4}} \log \left (-\left (16777216 i + 16777216\right )^{\frac {1}{4}} - \frac {64 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right )}{256 \, {\left (\sqrt {\sqrt {2} + 2} + i \, \sqrt {-\sqrt {2} + 2}\right )}^{7}} \]
1/144115188075855872*I*(8*I + 8)^(63/4)*log((-281474976710656*I + 28147497 6710656)^(1/4) - 4096*(x^4 - 1)^(1/4)/x) - 1/144115188075855872*I*(8*I + 8 )^(63/4)*log(-(-281474976710656*I + 281474976710656)^(1/4) - 4096*(x^4 - 1 )^(1/4)/x) - 1/536870912*(8*I + 8)^(31/4)*log(I*(-16777216*I + 16777216)^( 1/4) - 64*(x^4 - 1)^(1/4)/x) + 1/536870912*(8*I + 8)^(31/4)*log(-I*(-16777 216*I + 16777216)^(1/4) - 64*(x^4 - 1)^(1/4)/x) + 1/5*(x^4 - 1)^(1/4)*(1/x ^4 - 1)/x - 1/256*I*(8*I + 8)^(15/4)*log((16777216*I + 16777216)^(1/4) - 6 4*(x^4 - 1)^(1/4)/x)/(sqrt(sqrt(2) + 2) + I*sqrt(-sqrt(2) + 2))^7 + 1/256* (8*I + 8)^(15/4)*log(I*(16777216*I + 16777216)^(1/4) - 64*(x^4 - 1)^(1/4)/ x)/(sqrt(sqrt(2) + 2) + I*sqrt(-sqrt(2) + 2))^7 - 1/256*(8*I + 8)^(15/4)*l og(-I*(16777216*I + 16777216)^(1/4) - 64*(x^4 - 1)^(1/4)/x)/(sqrt(sqrt(2) + 2) + I*sqrt(-sqrt(2) + 2))^7 + 1/256*I*(8*I + 8)^(15/4)*log(-(16777216*I + 16777216)^(1/4) - 64*(x^4 - 1)^(1/4)/x)/(sqrt(sqrt(2) + 2) + I*sqrt(-sq rt(2) + 2))^7
Not integrable
Time = 6.17 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.32 \[ \int \frac {\sqrt [4]{-1+x^4} \left (-1+x^8\right )}{x^6 \left (1+x^8\right )} \, dx=\int \frac {{\left (x^4-1\right )}^{1/4}\,\left (x^8-1\right )}{x^6\,\left (x^8+1\right )} \,d x \]