Integrand size = 16, antiderivative size = 110 \[ \int \frac {c+d x}{\left (a-b x^4\right )^2} \, dx=\frac {x (c+d x)}{4 a \left (a-b x^4\right )}+\frac {3 c \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 a^{7/4} \sqrt [4]{b}}+\frac {3 c \text {arctanh}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 a^{7/4} \sqrt [4]{b}}+\frac {d \text {arctanh}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{4 a^{3/2} \sqrt {b}} \]
1/4*x*(d*x+c)/a/(-b*x^4+a)+3/8*c*arctan(b^(1/4)*x/a^(1/4))/a^(7/4)/b^(1/4) +3/8*c*arctanh(b^(1/4)*x/a^(1/4))/a^(7/4)/b^(1/4)+1/4*d*arctanh(x^2*b^(1/2 )/a^(1/2))/a^(3/2)/b^(1/2)
Time = 0.15 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.53 \[ \int \frac {c+d x}{\left (a-b x^4\right )^2} \, dx=\frac {\frac {4 a x (c+d x)}{a-b x^4}+\frac {6 \sqrt [4]{a} c \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{\sqrt [4]{b}}-\frac {\left (3 \sqrt [4]{a} \sqrt [4]{b} c+2 \sqrt {a} d\right ) \log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right )}{\sqrt {b}}+\frac {\left (3 \sqrt [4]{a} \sqrt [4]{b} c-2 \sqrt {a} d\right ) \log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right )}{\sqrt {b}}+\frac {2 \sqrt {a} d \log \left (\sqrt {a}+\sqrt {b} x^2\right )}{\sqrt {b}}}{16 a^2} \]
((4*a*x*(c + d*x))/(a - b*x^4) + (6*a^(1/4)*c*ArcTan[(b^(1/4)*x)/a^(1/4)]) /b^(1/4) - ((3*a^(1/4)*b^(1/4)*c + 2*Sqrt[a]*d)*Log[a^(1/4) - b^(1/4)*x])/ Sqrt[b] + ((3*a^(1/4)*b^(1/4)*c - 2*Sqrt[a]*d)*Log[a^(1/4) + b^(1/4)*x])/S qrt[b] + (2*Sqrt[a]*d*Log[Sqrt[a] + Sqrt[b]*x^2])/Sqrt[b])/(16*a^2)
Time = 0.28 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.05, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2394, 25, 2415, 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.
\(\displaystyle \int \frac {c+d x}{\left (a-b x^4\right )^2} \, dx\) |
\(\Big \downarrow \) 2394 |
\(\displaystyle \frac {x (c+d x)}{4 a \left (a-b x^4\right )}-\frac {\int -\frac {3 c+2 d x}{a-b x^4}dx}{4 a}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int \frac {3 c+2 d x}{a-b x^4}dx}{4 a}+\frac {x (c+d x)}{4 a \left (a-b x^4\right )}\) |
\(\Big \downarrow \) 2415 |
\(\displaystyle \frac {\int \left (\frac {3 c}{a-b x^4}+\frac {2 d x}{a-b x^4}\right )dx}{4 a}+\frac {x (c+d x)}{4 a \left (a-b x^4\right )}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\frac {3 c \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} \sqrt [4]{b}}+\frac {3 c \text {arctanh}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} \sqrt [4]{b}}+\frac {d \text {arctanh}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b}}}{4 a}+\frac {x (c+d x)}{4 a \left (a-b x^4\right )}\) |
(x*(c + d*x))/(4*a*(a - b*x^4)) + ((3*c*ArcTan[(b^(1/4)*x)/a^(1/4)])/(2*a^ (3/4)*b^(1/4)) + (3*c*ArcTanh[(b^(1/4)*x)/a^(1/4)])/(2*a^(3/4)*b^(1/4)) + (d*ArcTanh[(Sqrt[b]*x^2)/Sqrt[a]])/(Sqrt[a]*Sqrt[b]))/(4*a)
3.2.17.3.1 Defintions of rubi rules used
Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(-x)*Pq*((a + b *x^n)^(p + 1)/(a*n*(p + 1))), x] + Simp[1/(a*n*(p + 1)) Int[ExpandToSum[n *(p + 1)*Pq + D[x*Pq, x], x]*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x ] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1] && LtQ[Expon[Pq, x], n - 1]
Int[(Pq_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = Sum[x^ii*((Coeff [Pq, x, ii] + Coeff[Pq, x, n/2 + ii]*x^(n/2))/(a + b*x^n)), {ii, 0, n/2 - 1 }]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n/2, 0] && Expon[Pq, x] < n
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.50 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.63
method | result | size |
risch | \(\frac {\frac {d \,x^{2}}{4 a}+\frac {c x}{4 a}}{-b \,x^{4}+a}-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4} b -a \right )}{\sum }\frac {\left (2 \textit {\_R} d +3 c \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}}{16 b a}\) | \(69\) |
default | \(c \left (\frac {x}{4 a \left (-b \,x^{4}+a \right )}+\frac {3 \left (\frac {a}{b}\right )^{\frac {1}{4}} \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )\right )}{16 a^{2}}\right )+d \left (\frac {x^{2}}{4 a \left (-b \,x^{4}+a \right )}+\frac {\ln \left (\frac {a +x^{2} \sqrt {a b}}{a -x^{2} \sqrt {a b}}\right )}{8 a \sqrt {a b}}\right )\) | \(128\) |
(1/4*d/a*x^2+1/4*c/a*x)/(-b*x^4+a)-1/16/b/a*sum((2*_R*d+3*c)/_R^3*ln(x-_R) ,_R=RootOf(_Z^4*b-a))
Result contains complex when optimal does not.
Time = 1.88 (sec) , antiderivative size = 40560, normalized size of antiderivative = 368.73 \[ \int \frac {c+d x}{\left (a-b x^4\right )^2} \, dx=\text {Too large to display} \]
Time = 0.59 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.42 \[ \int \frac {c+d x}{\left (a-b x^4\right )^2} \, dx=\operatorname {RootSum} {\left (65536 t^{4} a^{7} b^{2} - 2048 t^{2} a^{4} b d^{2} + 1152 t a^{2} b c^{2} d + 16 a d^{4} - 81 b c^{4}, \left ( t \mapsto t \log {\left (x + \frac {32768 t^{3} a^{6} b d^{2} + 4608 t^{2} a^{4} b c^{2} d - 512 t a^{3} d^{4} + 1296 t a^{2} b c^{4} + 360 a c^{2} d^{3}}{192 a c d^{4} + 243 b c^{5}} \right )} \right )\right )} + \frac {- c x - d x^{2}}{- 4 a^{2} + 4 a b x^{4}} \]
RootSum(65536*_t**4*a**7*b**2 - 2048*_t**2*a**4*b*d**2 + 1152*_t*a**2*b*c* *2*d + 16*a*d**4 - 81*b*c**4, Lambda(_t, _t*log(x + (32768*_t**3*a**6*b*d* *2 + 4608*_t**2*a**4*b*c**2*d - 512*_t*a**3*d**4 + 1296*_t*a**2*b*c**4 + 3 60*a*c**2*d**3)/(192*a*c*d**4 + 243*b*c**5)))) + (-c*x - d*x**2)/(-4*a**2 + 4*a*b*x**4)
Time = 0.28 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.43 \[ \int \frac {c+d x}{\left (a-b x^4\right )^2} \, dx=-\frac {d x^{2} + c x}{4 \, {\left (a b x^{4} - a^{2}\right )}} + \frac {\frac {6 \, c \arctan \left (\frac {\sqrt {b} x}{\sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, d \log \left (\sqrt {b} x^{2} + \sqrt {a}\right )}{\sqrt {a} \sqrt {b}} - \frac {2 \, d \log \left (\sqrt {b} x^{2} - \sqrt {a}\right )}{\sqrt {a} \sqrt {b}} - \frac {3 \, c \log \left (\frac {\sqrt {b} x - \sqrt {\sqrt {a} \sqrt {b}}}{\sqrt {b} x + \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}}}{16 \, a} \]
-1/4*(d*x^2 + c*x)/(a*b*x^4 - a^2) + 1/16*(6*c*arctan(sqrt(b)*x/sqrt(sqrt( a)*sqrt(b)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))) + 2*d*log(sqrt(b)*x^2 + sqrt( a))/(sqrt(a)*sqrt(b)) - 2*d*log(sqrt(b)*x^2 - sqrt(a))/(sqrt(a)*sqrt(b)) - 3*c*log((sqrt(b)*x - sqrt(sqrt(a)*sqrt(b)))/(sqrt(b)*x + sqrt(sqrt(a)*sqr t(b))))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))))/a
Leaf count of result is larger than twice the leaf count of optimal. 254 vs. \(2 (79) = 158\).
Time = 0.27 (sec) , antiderivative size = 254, normalized size of antiderivative = 2.31 \[ \int \frac {c+d x}{\left (a-b x^4\right )^2} \, dx=\frac {3 \, \sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} c \log \left (x^{2} + \sqrt {2} x \left (-\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {-\frac {a}{b}}\right )}{32 \, a^{2} b} - \frac {3 \, \sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} c \log \left (x^{2} - \sqrt {2} x \left (-\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {-\frac {a}{b}}\right )}{32 \, a^{2} b} - \frac {d x^{2} + c x}{4 \, {\left (b x^{4} - a\right )} a} + \frac {\sqrt {2} {\left (2 \, \sqrt {2} \sqrt {-a b} b d + 3 \, \left (-a b^{3}\right )^{\frac {1}{4}} b c\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (-\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{16 \, a^{2} b^{2}} + \frac {\sqrt {2} {\left (2 \, \sqrt {2} \sqrt {-a b} b d + 3 \, \left (-a b^{3}\right )^{\frac {1}{4}} b c\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (-\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{16 \, a^{2} b^{2}} \]
3/32*sqrt(2)*(-a*b^3)^(1/4)*c*log(x^2 + sqrt(2)*x*(-a/b)^(1/4) + sqrt(-a/b ))/(a^2*b) - 3/32*sqrt(2)*(-a*b^3)^(1/4)*c*log(x^2 - sqrt(2)*x*(-a/b)^(1/4 ) + sqrt(-a/b))/(a^2*b) - 1/4*(d*x^2 + c*x)/((b*x^4 - a)*a) + 1/16*sqrt(2) *(2*sqrt(2)*sqrt(-a*b)*b*d + 3*(-a*b^3)^(1/4)*b*c)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(-a/b)^(1/4))/(-a/b)^(1/4))/(a^2*b^2) + 1/16*sqrt(2)*(2*sqrt(2) *sqrt(-a*b)*b*d + 3*(-a*b^3)^(1/4)*b*c)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)* (-a/b)^(1/4))/(-a/b)^(1/4))/(a^2*b^2)
Time = 9.19 (sec) , antiderivative size = 283, normalized size of antiderivative = 2.57 \[ \int \frac {c+d x}{\left (a-b x^4\right )^2} \, dx=\left (\sum _{k=1}^4\ln \left (-\frac {b^2\,\left (3\,c\,d^2+2\,d^3\,x+{\mathrm {root}\left (65536\,a^7\,b^2\,z^4-2048\,a^4\,b\,d^2\,z^2+1152\,a^2\,b\,c^2\,d\,z-81\,b\,c^4+16\,a\,d^4,z,k\right )}^2\,a^3\,b\,c\,192-{\mathrm {root}\left (65536\,a^7\,b^2\,z^4-2048\,a^4\,b\,d^2\,z^2+1152\,a^2\,b\,c^2\,d\,z-81\,b\,c^4+16\,a\,d^4,z,k\right )}^2\,a^3\,b\,d\,x\,128+\mathrm {root}\left (65536\,a^7\,b^2\,z^4-2048\,a^4\,b\,d^2\,z^2+1152\,a^2\,b\,c^2\,d\,z-81\,b\,c^4+16\,a\,d^4,z,k\right )\,a\,b\,c^2\,x\,36\right )}{a^3\,16}\right )\,\mathrm {root}\left (65536\,a^7\,b^2\,z^4-2048\,a^4\,b\,d^2\,z^2+1152\,a^2\,b\,c^2\,d\,z-81\,b\,c^4+16\,a\,d^4,z,k\right )\right )+\frac {\frac {d\,x^2}{4\,a}+\frac {c\,x}{4\,a}}{a-b\,x^4} \]
symsum(log(-(b^2*(3*c*d^2 + 2*d^3*x + 192*root(65536*a^7*b^2*z^4 - 2048*a^ 4*b*d^2*z^2 + 1152*a^2*b*c^2*d*z - 81*b*c^4 + 16*a*d^4, z, k)^2*a^3*b*c - 128*root(65536*a^7*b^2*z^4 - 2048*a^4*b*d^2*z^2 + 1152*a^2*b*c^2*d*z - 81* b*c^4 + 16*a*d^4, z, k)^2*a^3*b*d*x + 36*root(65536*a^7*b^2*z^4 - 2048*a^4 *b*d^2*z^2 + 1152*a^2*b*c^2*d*z - 81*b*c^4 + 16*a*d^4, z, k)*a*b*c^2*x))/( 16*a^3))*root(65536*a^7*b^2*z^4 - 2048*a^4*b*d^2*z^2 + 1152*a^2*b*c^2*d*z - 81*b*c^4 + 16*a*d^4, z, k), k, 1, 4) + ((d*x^2)/(4*a) + (c*x)/(4*a))/(a - b*x^4)