Integrand size = 18, antiderivative size = 81 \[ \int \frac {x^{11}}{a+b x^4+c x^8} \, dx=\frac {x^4}{4 c}-\frac {\left (b^2-2 a c\right ) \text {arctanh}\left (\frac {b+2 c x^4}{\sqrt {b^2-4 a c}}\right )}{4 c^2 \sqrt {b^2-4 a c}}-\frac {b \log \left (a+b x^4+c x^8\right )}{8 c^2} \]
1/4*x^4/c-1/8*b*ln(c*x^8+b*x^4+a)/c^2-1/4*(-2*a*c+b^2)*arctanh((2*c*x^4+b) /(-4*a*c+b^2)^(1/2))/c^2/(-4*a*c+b^2)^(1/2)
Time = 0.04 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.96 \[ \int \frac {x^{11}}{a+b x^4+c x^8} \, dx=\frac {2 c x^4+\frac {2 \left (b^2-2 a c\right ) \arctan \left (\frac {b+2 c x^4}{\sqrt {-b^2+4 a c}}\right )}{\sqrt {-b^2+4 a c}}-b \log \left (a+b x^4+c x^8\right )}{8 c^2} \]
(2*c*x^4 + (2*(b^2 - 2*a*c)*ArcTan[(b + 2*c*x^4)/Sqrt[-b^2 + 4*a*c]])/Sqrt [-b^2 + 4*a*c] - b*Log[a + b*x^4 + c*x^8])/(8*c^2)
Time = 0.25 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.99, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1693, 1143, 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.
\(\displaystyle \int \frac {x^{11}}{a+b x^4+c x^8} \, dx\) |
\(\Big \downarrow \) 1693 |
\(\displaystyle \frac {1}{4} \int \frac {x^8}{c x^8+b x^4+a}dx^4\) |
\(\Big \downarrow \) 1143 |
\(\displaystyle \frac {1}{4} \int \left (\frac {1}{c}-\frac {b x^4+a}{c \left (c x^8+b x^4+a\right )}\right )dx^4\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{4} \left (-\frac {\left (b^2-2 a c\right ) \text {arctanh}\left (\frac {b+2 c x^4}{\sqrt {b^2-4 a c}}\right )}{c^2 \sqrt {b^2-4 a c}}-\frac {b \log \left (a+b x^4+c x^8\right )}{2 c^2}+\frac {x^4}{c}\right )\) |
(x^4/c - ((b^2 - 2*a*c)*ArcTanh[(b + 2*c*x^4)/Sqrt[b^2 - 4*a*c]])/(c^2*Sqr t[b^2 - 4*a*c]) - (b*Log[a + b*x^4 + c*x^8])/(2*c^2))/4
3.4.10.3.1 Defintions of rubi rules used
Int[((d_.) + (e_.)*(x_))^(m_)/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[m, 1]
Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol ] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[n2, 2*n] && IntegerQ [Simplify[(m + 1)/n]]
Time = 0.08 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.02
method | result | size |
default | \(\frac {x^{4}}{4 c}+\frac {-\frac {b \ln \left (c \,x^{8}+b \,x^{4}+a \right )}{2 c}+\frac {2 \left (-a +\frac {b^{2}}{2 c}\right ) \arctan \left (\frac {2 c \,x^{4}+b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{4 c}\) | \(83\) |
risch | \(\frac {x^{4}}{4 c}-\frac {\ln \left (\left (-8 a^{2} c^{2}+6 a \,b^{2} c -b^{4}+\sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c -b^{2}\right )^{2}}\, b \right ) x^{4}+2 \sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c -b^{2}\right )^{2}}\, a \right ) a b}{2 c \left (4 a c -b^{2}\right )}+\frac {\ln \left (\left (-8 a^{2} c^{2}+6 a \,b^{2} c -b^{4}+\sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c -b^{2}\right )^{2}}\, b \right ) x^{4}+2 \sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c -b^{2}\right )^{2}}\, a \right ) b^{3}}{8 c^{2} \left (4 a c -b^{2}\right )}+\frac {\ln \left (\left (-8 a^{2} c^{2}+6 a \,b^{2} c -b^{4}+\sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c -b^{2}\right )^{2}}\, b \right ) x^{4}+2 \sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c -b^{2}\right )^{2}}\, a \right ) \sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c -b^{2}\right )^{2}}}{8 c^{2} \left (4 a c -b^{2}\right )}-\frac {\ln \left (\left (-8 a^{2} c^{2}+6 a \,b^{2} c -b^{4}-\sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c -b^{2}\right )^{2}}\, b \right ) x^{4}-2 \sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c -b^{2}\right )^{2}}\, a \right ) a b}{2 c \left (4 a c -b^{2}\right )}+\frac {\ln \left (\left (-8 a^{2} c^{2}+6 a \,b^{2} c -b^{4}-\sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c -b^{2}\right )^{2}}\, b \right ) x^{4}-2 \sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c -b^{2}\right )^{2}}\, a \right ) b^{3}}{8 c^{2} \left (4 a c -b^{2}\right )}-\frac {\ln \left (\left (-8 a^{2} c^{2}+6 a \,b^{2} c -b^{4}-\sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c -b^{2}\right )^{2}}\, b \right ) x^{4}-2 \sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c -b^{2}\right )^{2}}\, a \right ) \sqrt {-\left (4 a c -b^{2}\right ) \left (2 a c -b^{2}\right )^{2}}}{8 c^{2} \left (4 a c -b^{2}\right )}\) | \(681\) |
1/4*x^4/c+1/4/c*(-1/2*b/c*ln(c*x^8+b*x^4+a)+2*(-a+1/2/c*b^2)/(4*a*c-b^2)^( 1/2)*arctan((2*c*x^4+b)/(4*a*c-b^2)^(1/2)))
Time = 0.28 (sec) , antiderivative size = 254, normalized size of antiderivative = 3.14 \[ \int \frac {x^{11}}{a+b x^4+c x^8} \, dx=\left [\frac {2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} x^{4} - {\left (b^{2} - 2 \, a c\right )} \sqrt {b^{2} - 4 \, a c} \log \left (\frac {2 \, c^{2} x^{8} + 2 \, b c x^{4} + b^{2} - 2 \, a c + {\left (2 \, c x^{4} + b\right )} \sqrt {b^{2} - 4 \, a c}}{c x^{8} + b x^{4} + a}\right ) - {\left (b^{3} - 4 \, a b c\right )} \log \left (c x^{8} + b x^{4} + a\right )}{8 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )}}, \frac {2 \, {\left (b^{2} c - 4 \, a c^{2}\right )} x^{4} - 2 \, {\left (b^{2} - 2 \, a c\right )} \sqrt {-b^{2} + 4 \, a c} \arctan \left (-\frac {{\left (2 \, c x^{4} + b\right )} \sqrt {-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) - {\left (b^{3} - 4 \, a b c\right )} \log \left (c x^{8} + b x^{4} + a\right )}{8 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )}}\right ] \]
[1/8*(2*(b^2*c - 4*a*c^2)*x^4 - (b^2 - 2*a*c)*sqrt(b^2 - 4*a*c)*log((2*c^2 *x^8 + 2*b*c*x^4 + b^2 - 2*a*c + (2*c*x^4 + b)*sqrt(b^2 - 4*a*c))/(c*x^8 + b*x^4 + a)) - (b^3 - 4*a*b*c)*log(c*x^8 + b*x^4 + a))/(b^2*c^2 - 4*a*c^3) , 1/8*(2*(b^2*c - 4*a*c^2)*x^4 - 2*(b^2 - 2*a*c)*sqrt(-b^2 + 4*a*c)*arctan (-(2*c*x^4 + b)*sqrt(-b^2 + 4*a*c)/(b^2 - 4*a*c)) - (b^3 - 4*a*b*c)*log(c* x^8 + b*x^4 + a))/(b^2*c^2 - 4*a*c^3)]
Leaf count of result is larger than twice the leaf count of optimal. 316 vs. \(2 (73) = 146\).
Time = 2.70 (sec) , antiderivative size = 316, normalized size of antiderivative = 3.90 \[ \int \frac {x^{11}}{a+b x^4+c x^8} \, dx=\left (- \frac {b}{8 c^{2}} - \frac {\sqrt {- 4 a c + b^{2}} \cdot \left (2 a c - b^{2}\right )}{8 c^{2} \cdot \left (4 a c - b^{2}\right )}\right ) \log {\left (x^{4} + \frac {- a b - 16 a c^{2} \left (- \frac {b}{8 c^{2}} - \frac {\sqrt {- 4 a c + b^{2}} \cdot \left (2 a c - b^{2}\right )}{8 c^{2} \cdot \left (4 a c - b^{2}\right )}\right ) + 4 b^{2} c \left (- \frac {b}{8 c^{2}} - \frac {\sqrt {- 4 a c + b^{2}} \cdot \left (2 a c - b^{2}\right )}{8 c^{2} \cdot \left (4 a c - b^{2}\right )}\right )}{2 a c - b^{2}} \right )} + \left (- \frac {b}{8 c^{2}} + \frac {\sqrt {- 4 a c + b^{2}} \cdot \left (2 a c - b^{2}\right )}{8 c^{2} \cdot \left (4 a c - b^{2}\right )}\right ) \log {\left (x^{4} + \frac {- a b - 16 a c^{2} \left (- \frac {b}{8 c^{2}} + \frac {\sqrt {- 4 a c + b^{2}} \cdot \left (2 a c - b^{2}\right )}{8 c^{2} \cdot \left (4 a c - b^{2}\right )}\right ) + 4 b^{2} c \left (- \frac {b}{8 c^{2}} + \frac {\sqrt {- 4 a c + b^{2}} \cdot \left (2 a c - b^{2}\right )}{8 c^{2} \cdot \left (4 a c - b^{2}\right )}\right )}{2 a c - b^{2}} \right )} + \frac {x^{4}}{4 c} \]
(-b/(8*c**2) - sqrt(-4*a*c + b**2)*(2*a*c - b**2)/(8*c**2*(4*a*c - b**2))) *log(x**4 + (-a*b - 16*a*c**2*(-b/(8*c**2) - sqrt(-4*a*c + b**2)*(2*a*c - b**2)/(8*c**2*(4*a*c - b**2))) + 4*b**2*c*(-b/(8*c**2) - sqrt(-4*a*c + b** 2)*(2*a*c - b**2)/(8*c**2*(4*a*c - b**2))))/(2*a*c - b**2)) + (-b/(8*c**2) + sqrt(-4*a*c + b**2)*(2*a*c - b**2)/(8*c**2*(4*a*c - b**2)))*log(x**4 + (-a*b - 16*a*c**2*(-b/(8*c**2) + sqrt(-4*a*c + b**2)*(2*a*c - b**2)/(8*c** 2*(4*a*c - b**2))) + 4*b**2*c*(-b/(8*c**2) + sqrt(-4*a*c + b**2)*(2*a*c - b**2)/(8*c**2*(4*a*c - b**2))))/(2*a*c - b**2)) + x**4/(4*c)
Exception generated. \[ \int \frac {x^{11}}{a+b x^4+c x^8} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` for more deta
Time = 1.56 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.93 \[ \int \frac {x^{11}}{a+b x^4+c x^8} \, dx=\frac {x^{4}}{4 \, c} - \frac {b \log \left (c x^{8} + b x^{4} + a\right )}{8 \, c^{2}} + \frac {{\left (b^{2} - 2 \, a c\right )} \arctan \left (\frac {2 \, c x^{4} + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{4 \, \sqrt {-b^{2} + 4 \, a c} c^{2}} \]
1/4*x^4/c - 1/8*b*log(c*x^8 + b*x^4 + a)/c^2 + 1/4*(b^2 - 2*a*c)*arctan((2 *c*x^4 + b)/sqrt(-b^2 + 4*a*c))/(sqrt(-b^2 + 4*a*c)*c^2)
Time = 8.92 (sec) , antiderivative size = 3916, normalized size of antiderivative = 48.35 \[ \int \frac {x^{11}}{a+b x^4+c x^8} \, dx=\text {Too large to display} \]
x^4/(4*c) + (log(a + b*x^4 + c*x^8)*(4*b^3 - 16*a*b*c))/(2*(64*a*c^3 - 16* b^2*c^2)) - (atan((8*c^4*x^4*(((a*c - b^2)*(((((2*a*c - b^2)*((((448*b^4*c ^6 - 384*a*b^2*c^7)/c^4 + (256*b^3*c^4*(4*b^3 - 16*a*b*c))/(64*a*c^3 - 16* b^2*c^2))*(2*a*c - b^2))/(8*c^2*(4*a*c - b^2)^(1/2)) + (32*b^3*c^2*(4*b^3 - 16*a*b*c)*(2*a*c - b^2))/((4*a*c - b^2)^(1/2)*(64*a*c^3 - 16*b^2*c^2)))) /(8*c^2*(4*a*c - b^2)^(1/2)) + (4*b^3*(4*b^3 - 16*a*b*c)*(2*a*c - b^2)^2)/ ((4*a*c - b^2)*(64*a*c^3 - 16*b^2*c^2)))*(2*a*c - b^2))/(8*c^2*(4*a*c - b^ 2)^(1/2)) - ((4*b^3 - 16*a*b*c)*(((4*b^3 - 16*a*b*c)*((((448*b^4*c^6 - 384 *a*b^2*c^7)/c^4 + (256*b^3*c^4*(4*b^3 - 16*a*b*c))/(64*a*c^3 - 16*b^2*c^2) )*(2*a*c - b^2))/(8*c^2*(4*a*c - b^2)^(1/2)) + (32*b^3*c^2*(4*b^3 - 16*a*b *c)*(2*a*c - b^2))/((4*a*c - b^2)^(1/2)*(64*a*c^3 - 16*b^2*c^2))))/(2*(64* a*c^3 - 16*b^2*c^2)) + ((2*a*c - b^2)*((144*b^5*c^4 - 240*a*b^3*c^5 + 96*a ^2*b*c^6)/c^4 + ((4*b^3 - 16*a*b*c)*((448*b^4*c^6 - 384*a*b^2*c^7)/c^4 + ( 256*b^3*c^4*(4*b^3 - 16*a*b*c))/(64*a*c^3 - 16*b^2*c^2)))/(2*(64*a*c^3 - 1 6*b^2*c^2))))/(8*c^2*(4*a*c - b^2)^(1/2))))/(2*(64*a*c^3 - 16*b^2*c^2)) + (((8*a^3*c^5 - 20*b^6*c^2 + 48*a*b^4*c^3 - 36*a^2*b^2*c^4)/c^4 - ((4*b^3 - 16*a*b*c)*((144*b^5*c^4 - 240*a*b^3*c^5 + 96*a^2*b*c^6)/c^4 + ((4*b^3 - 1 6*a*b*c)*((448*b^4*c^6 - 384*a*b^2*c^7)/c^4 + (256*b^3*c^4*(4*b^3 - 16*a*b *c))/(64*a*c^3 - 16*b^2*c^2)))/(2*(64*a*c^3 - 16*b^2*c^2))))/(2*(64*a*c^3 - 16*b^2*c^2)))*(2*a*c - b^2))/(8*c^2*(4*a*c - b^2)^(1/2)) + (b^3*(4*b^...