Integrand size = 26, antiderivative size = 148 \[ \int \frac {-1+x^4+7 x^5+x^9}{-7+6 x^4+x^8} \, dx=\frac {x^2}{2}-\frac {\arctan \left (1-\frac {\sqrt {2} x}{\sqrt [4]{7}}\right )}{2 \sqrt {2} 7^{3/4}}+\frac {\arctan \left (1+\frac {\sqrt {2} x}{\sqrt [4]{7}}\right )}{2 \sqrt {2} 7^{3/4}}-\frac {\text {arctanh}\left (x^2\right )}{2}-\frac {\log \left (\sqrt {7}-\sqrt {2} \sqrt [4]{7} x+x^2\right )}{4 \sqrt {2} 7^{3/4}}+\frac {\log \left (\sqrt {7}+\sqrt {2} \sqrt [4]{7} x+x^2\right )}{4 \sqrt {2} 7^{3/4}} \]
1/2*x^2-1/2*arctanh(x^2)+1/28*arctan(-1+1/7*x*2^(1/2)*7^(3/4))*7^(1/4)*2^( 1/2)+1/28*arctan(1+1/7*x*2^(1/2)*7^(3/4))*7^(1/4)*2^(1/2)-1/56*ln(x^2-7^(1 /4)*x*2^(1/2)+7^(1/2))*7^(1/4)*2^(1/2)+1/56*ln(x^2+7^(1/4)*x*2^(1/2)+7^(1/ 2))*7^(1/4)*2^(1/2)
Time = 0.05 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.07 \[ \int \frac {-1+x^4+7 x^5+x^9}{-7+6 x^4+x^8} \, dx=\frac {1}{56} \left (28 x^2-2 \sqrt {2} \sqrt [4]{7} \arctan \left (1-\frac {\sqrt {2} x}{\sqrt [4]{7}}\right )+2 \sqrt {2} \sqrt [4]{7} \arctan \left (1+\frac {\sqrt {2} x}{\sqrt [4]{7}}\right )+14 \log (1-x)+14 \log (1+x)-14 \log \left (1+x^2\right )-\sqrt {2} \sqrt [4]{7} \log \left (7-\sqrt {2} 7^{3/4} x+\sqrt {7} x^2\right )+\sqrt {2} \sqrt [4]{7} \log \left (7+\sqrt {2} 7^{3/4} x+\sqrt {7} x^2\right )\right ) \]
(28*x^2 - 2*Sqrt[2]*7^(1/4)*ArcTan[1 - (Sqrt[2]*x)/7^(1/4)] + 2*Sqrt[2]*7^ (1/4)*ArcTan[1 + (Sqrt[2]*x)/7^(1/4)] + 14*Log[1 - x] + 14*Log[1 + x] - 14 *Log[1 + x^2] - Sqrt[2]*7^(1/4)*Log[7 - Sqrt[2]*7^(3/4)*x + Sqrt[7]*x^2] + Sqrt[2]*7^(1/4)*Log[7 + Sqrt[2]*7^(3/4)*x + Sqrt[7]*x^2])/56
Time = 0.34 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2322, 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.
\(\displaystyle \int \frac {x^9+7 x^5+x^4-1}{x^8+6 x^4-7} \, dx\) |
\(\Big \downarrow \) 2322 |
\(\displaystyle \int \left (\frac {x^4-1}{x^8+6 x^4-7}+\frac {x \left (x^8+7 x^4\right )}{x^8+6 x^4-7}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\arctan \left (1-\frac {\sqrt {2} x}{\sqrt [4]{7}}\right )}{2 \sqrt {2} 7^{3/4}}+\frac {\arctan \left (\frac {\sqrt {2} x}{\sqrt [4]{7}}+1\right )}{2 \sqrt {2} 7^{3/4}}-\frac {\text {arctanh}\left (x^2\right )}{2}+\frac {x^2}{2}-\frac {\log \left (x^2-\sqrt {2} \sqrt [4]{7} x+\sqrt {7}\right )}{4 \sqrt {2} 7^{3/4}}+\frac {\log \left (x^2+\sqrt {2} \sqrt [4]{7} x+\sqrt {7}\right )}{4 \sqrt {2} 7^{3/4}}\) |
x^2/2 - ArcTan[1 - (Sqrt[2]*x)/7^(1/4)]/(2*Sqrt[2]*7^(3/4)) + ArcTan[1 + ( Sqrt[2]*x)/7^(1/4)]/(2*Sqrt[2]*7^(3/4)) - ArcTanh[x^2]/2 - Log[Sqrt[7] - S qrt[2]*7^(1/4)*x + x^2]/(4*Sqrt[2]*7^(3/4)) + Log[Sqrt[7] + Sqrt[2]*7^(1/4 )*x + x^2]/(4*Sqrt[2]*7^(3/4))
3.4.67.3.1 Defintions of rubi rules used
Int[(Pq_)*((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^(p_), x_Symbol] :> Mo dule[{q = Expon[Pq, x], j, k}, Int[Sum[x^j*Sum[Coeff[Pq, x, j + k*n]*x^(k*n ), {k, 0, (q - j)/n + 1}]*(a + b*x^n + c*x^(2*n))^p, {j, 0, n - 1}], x]] /; FreeQ[{a, b, c, p}, x] && EqQ[n2, 2*n] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && !PolyQ[Pq, x^n]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.13 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.30
method | result | size |
risch | \(\frac {x^{2}}{2}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (343 \textit {\_Z}^{4}+1\right )}{\sum }\textit {\_R} \ln \left (x +7 \textit {\_R} \right )\right )}{4}+\frac {\ln \left (x^{2}-1\right )}{4}-\frac {\ln \left (x^{2}+1\right )}{4}\) | \(44\) |
default | \(\frac {x^{2}}{2}+\frac {\ln \left (x +1\right )}{4}+\frac {7^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+7^{\frac {1}{4}} x \sqrt {2}+\sqrt {7}}{x^{2}-7^{\frac {1}{4}} x \sqrt {2}+\sqrt {7}}\right )+2 \arctan \left (1+\frac {x \sqrt {2}\, 7^{\frac {3}{4}}}{7}\right )+2 \arctan \left (-1+\frac {x \sqrt {2}\, 7^{\frac {3}{4}}}{7}\right )\right )}{56}-\frac {\ln \left (x^{2}+1\right )}{4}+\frac {\ln \left (x -1\right )}{4}\) | \(99\) |
Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.72 \[ \int \frac {-1+x^4+7 x^5+x^9}{-7+6 x^4+x^8} \, dx=\left (\frac {1}{2744} i + \frac {1}{2744}\right ) \cdot 343^{\frac {3}{4}} \sqrt {2} \log \left (\left (i + 1\right ) \cdot 343^{\frac {3}{4}} \sqrt {2} + 98 \, x\right ) - \left (\frac {1}{2744} i - \frac {1}{2744}\right ) \cdot 343^{\frac {3}{4}} \sqrt {2} \log \left (-\left (i - 1\right ) \cdot 343^{\frac {3}{4}} \sqrt {2} + 98 \, x\right ) + \left (\frac {1}{2744} i - \frac {1}{2744}\right ) \cdot 343^{\frac {3}{4}} \sqrt {2} \log \left (\left (i - 1\right ) \cdot 343^{\frac {3}{4}} \sqrt {2} + 98 \, x\right ) - \left (\frac {1}{2744} i + \frac {1}{2744}\right ) \cdot 343^{\frac {3}{4}} \sqrt {2} \log \left (-\left (i + 1\right ) \cdot 343^{\frac {3}{4}} \sqrt {2} + 98 \, x\right ) + \frac {1}{2} \, x^{2} - \frac {1}{4} \, \log \left (x^{2} + 1\right ) + \frac {1}{4} \, \log \left (x^{2} - 1\right ) \]
(1/2744*I + 1/2744)*343^(3/4)*sqrt(2)*log((I + 1)*343^(3/4)*sqrt(2) + 98*x ) - (1/2744*I - 1/2744)*343^(3/4)*sqrt(2)*log(-(I - 1)*343^(3/4)*sqrt(2) + 98*x) + (1/2744*I - 1/2744)*343^(3/4)*sqrt(2)*log((I - 1)*343^(3/4)*sqrt( 2) + 98*x) - (1/2744*I + 1/2744)*343^(3/4)*sqrt(2)*log(-(I + 1)*343^(3/4)* sqrt(2) + 98*x) + 1/2*x^2 - 1/4*log(x^2 + 1) + 1/4*log(x^2 - 1)
Time = 0.21 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.99 \[ \int \frac {-1+x^4+7 x^5+x^9}{-7+6 x^4+x^8} \, dx=\frac {x^{2}}{2} + \frac {\log {\left (x^{2} - 1 \right )}}{4} - \frac {\log {\left (x^{2} + 1 \right )}}{4} - \frac {\sqrt {2} \cdot \sqrt [4]{7} \log {\left (x^{2} - \sqrt {2} \cdot \sqrt [4]{7} x + \sqrt {7} \right )}}{56} + \frac {\sqrt {2} \cdot \sqrt [4]{7} \log {\left (x^{2} + \sqrt {2} \cdot \sqrt [4]{7} x + \sqrt {7} \right )}}{56} + \frac {\sqrt {2} \cdot \sqrt [4]{7} \operatorname {atan}{\left (\frac {\sqrt {2} \cdot 7^{\frac {3}{4}} x}{7} - 1 \right )}}{28} + \frac {\sqrt {2} \cdot \sqrt [4]{7} \operatorname {atan}{\left (\frac {\sqrt {2} \cdot 7^{\frac {3}{4}} x}{7} + 1 \right )}}{28} \]
x**2/2 + log(x**2 - 1)/4 - log(x**2 + 1)/4 - sqrt(2)*7**(1/4)*log(x**2 - s qrt(2)*7**(1/4)*x + sqrt(7))/56 + sqrt(2)*7**(1/4)*log(x**2 + sqrt(2)*7**( 1/4)*x + sqrt(7))/56 + sqrt(2)*7**(1/4)*atan(sqrt(2)*7**(3/4)*x/7 - 1)/28 + sqrt(2)*7**(1/4)*atan(sqrt(2)*7**(3/4)*x/7 + 1)/28
Time = 0.27 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.89 \[ \int \frac {-1+x^4+7 x^5+x^9}{-7+6 x^4+x^8} \, dx=\frac {1}{2} \, x^{2} + \frac {1}{28} \cdot 7^{\frac {1}{4}} \sqrt {2} \arctan \left (\frac {1}{14} \cdot 7^{\frac {3}{4}} \sqrt {2} {\left (2 \, x + 7^{\frac {1}{4}} \sqrt {2}\right )}\right ) + \frac {1}{28} \cdot 7^{\frac {1}{4}} \sqrt {2} \arctan \left (\frac {1}{14} \cdot 7^{\frac {3}{4}} \sqrt {2} {\left (2 \, x - 7^{\frac {1}{4}} \sqrt {2}\right )}\right ) + \frac {1}{56} \cdot 7^{\frac {1}{4}} \sqrt {2} \log \left (x^{2} + 7^{\frac {1}{4}} \sqrt {2} x + \sqrt {7}\right ) - \frac {1}{56} \cdot 7^{\frac {1}{4}} \sqrt {2} \log \left (x^{2} - 7^{\frac {1}{4}} \sqrt {2} x + \sqrt {7}\right ) - \frac {1}{4} \, \log \left (x^{2} + 1\right ) + \frac {1}{4} \, \log \left (x + 1\right ) + \frac {1}{4} \, \log \left (x - 1\right ) \]
1/2*x^2 + 1/28*7^(1/4)*sqrt(2)*arctan(1/14*7^(3/4)*sqrt(2)*(2*x + 7^(1/4)* sqrt(2))) + 1/28*7^(1/4)*sqrt(2)*arctan(1/14*7^(3/4)*sqrt(2)*(2*x - 7^(1/4 )*sqrt(2))) + 1/56*7^(1/4)*sqrt(2)*log(x^2 + 7^(1/4)*sqrt(2)*x + sqrt(7)) - 1/56*7^(1/4)*sqrt(2)*log(x^2 - 7^(1/4)*sqrt(2)*x + sqrt(7)) - 1/4*log(x^ 2 + 1) + 1/4*log(x + 1) + 1/4*log(x - 1)
Time = 0.28 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.82 \[ \int \frac {-1+x^4+7 x^5+x^9}{-7+6 x^4+x^8} \, dx=\frac {1}{2} \, x^{2} + \frac {1}{28} \cdot 28^{\frac {1}{4}} \arctan \left (\frac {1}{14} \cdot 7^{\frac {3}{4}} \sqrt {2} {\left (2 \, x + 7^{\frac {1}{4}} \sqrt {2}\right )}\right ) + \frac {1}{28} \cdot 28^{\frac {1}{4}} \arctan \left (\frac {1}{14} \cdot 7^{\frac {3}{4}} \sqrt {2} {\left (2 \, x - 7^{\frac {1}{4}} \sqrt {2}\right )}\right ) + \frac {1}{56} \cdot 28^{\frac {1}{4}} \log \left (x^{2} + 7^{\frac {1}{4}} \sqrt {2} x + \sqrt {7}\right ) - \frac {1}{56} \cdot 28^{\frac {1}{4}} \log \left (x^{2} - 7^{\frac {1}{4}} \sqrt {2} x + \sqrt {7}\right ) - \frac {1}{4} \, \log \left (x^{2} + 1\right ) + \frac {1}{4} \, \log \left ({\left | x + 1 \right |}\right ) + \frac {1}{4} \, \log \left ({\left | x - 1 \right |}\right ) \]
1/2*x^2 + 1/28*28^(1/4)*arctan(1/14*7^(3/4)*sqrt(2)*(2*x + 7^(1/4)*sqrt(2) )) + 1/28*28^(1/4)*arctan(1/14*7^(3/4)*sqrt(2)*(2*x - 7^(1/4)*sqrt(2))) + 1/56*28^(1/4)*log(x^2 + 7^(1/4)*sqrt(2)*x + sqrt(7)) - 1/56*28^(1/4)*log(x ^2 - 7^(1/4)*sqrt(2)*x + sqrt(7)) - 1/4*log(x^2 + 1) + 1/4*log(abs(x + 1)) + 1/4*log(abs(x - 1))
Time = 9.10 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.84 \[ \int \frac {-1+x^4+7 x^5+x^9}{-7+6 x^4+x^8} \, dx=\frac {\mathrm {atan}\left (x^2\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2}+\frac {x^2}{2}+\sqrt {2}\,7^{1/4}\,\mathrm {atan}\left (\frac {\sqrt {2}\,7^{1/4}\,x\,\left (\frac {89653248}{2401}+\frac {89653248}{2401}{}\mathrm {i}\right )}{-\frac {1048576}{49}+\frac {\sqrt {7}\,179306496{}\mathrm {i}}{2401}}+\frac {\sqrt {2}\,7^{3/4}\,x\,\left (-\frac {524288}{343}+\frac {524288}{343}{}\mathrm {i}\right )}{-\frac {1048576}{49}+\frac {\sqrt {7}\,179306496{}\mathrm {i}}{2401}}\right )\,\left (\frac {1}{28}+\frac {1}{28}{}\mathrm {i}\right )+\sqrt {2}\,7^{1/4}\,\mathrm {atan}\left (\frac {\sqrt {2}\,7^{1/4}\,x\,\left (\frac {89653248}{2401}-\frac {89653248}{2401}{}\mathrm {i}\right )}{\frac {1048576}{49}+\frac {\sqrt {7}\,179306496{}\mathrm {i}}{2401}}+\frac {\sqrt {2}\,7^{3/4}\,x\,\left (-\frac {524288}{343}-\frac {524288}{343}{}\mathrm {i}\right )}{\frac {1048576}{49}+\frac {\sqrt {7}\,179306496{}\mathrm {i}}{2401}}\right )\,\left (-\frac {1}{28}+\frac {1}{28}{}\mathrm {i}\right ) \]
(atan(x^2*1i)*1i)/2 + x^2/2 + 2^(1/2)*7^(1/4)*atan((2^(1/2)*7^(1/4)*x*(896 53248/2401 + 89653248i/2401))/((7^(1/2)*179306496i)/2401 - 1048576/49) - ( 2^(1/2)*7^(3/4)*x*(524288/343 - 524288i/343))/((7^(1/2)*179306496i)/2401 - 1048576/49))*(1/28 + 1i/28) - 2^(1/2)*7^(1/4)*atan((2^(1/2)*7^(1/4)*x*(89 653248/2401 - 89653248i/2401))/((7^(1/2)*179306496i)/2401 + 1048576/49) - (2^(1/2)*7^(3/4)*x*(524288/343 + 524288i/343))/((7^(1/2)*179306496i)/2401 + 1048576/49))*(1/28 - 1i/28)