\(\int \frac {c+d x^2}{a-b x^4} \, dx\) [203]

Optimal result

Integrand size = 18, antiderivative size = 85 \[ \int \frac {c+d x^2}{a-b x^4} \, dx=\frac {\left (c-\frac {\sqrt {a} d}{\sqrt {b}}\right ) \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} \sqrt [4]{b}}+\frac {\left (\sqrt {b} c+\sqrt {a} d\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{3/4}} \] Output:

1/2*(c-a^(1/2)*d/b^(1/2))*arctan(b^(1/4)*x/a^(1/4))/a^(3/4)/b^(1/4)+1/2*(b 
^(1/2)*c+a^(1/2)*d)*arctanh(b^(1/4)*x/a^(1/4))/a^(3/4)/b^(3/4)
 

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.12 \[ \int \frac {c+d x^2}{a-b x^4} \, dx=\frac {2 \left (\sqrt {b} c-\sqrt {a} d\right ) \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )-\left (\sqrt {b} c+\sqrt {a} d\right ) \left (\log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right )-\log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right )\right )}{4 a^{3/4} b^{3/4}} \] Input:

Integrate[(c + d*x^2)/(a - b*x^4),x]
 

Output:

(2*(Sqrt[b]*c - Sqrt[a]*d)*ArcTan[(b^(1/4)*x)/a^(1/4)] - (Sqrt[b]*c + Sqrt 
[a]*d)*(Log[a^(1/4) - b^(1/4)*x] - Log[a^(1/4) + b^(1/4)*x]))/(4*a^(3/4)*b 
^(3/4))
 

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1481, 218, 221}

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.

\(\displaystyle \int \frac {c+d x^2}{a-b x^4} \, dx\)

\(\Big \downarrow \) 1481

\(\displaystyle \frac {1}{2} \left (\frac {\sqrt {b} c}{\sqrt {a}}+d\right ) \int \frac {1}{\sqrt {a} \sqrt {b}-b x^2}dx-\frac {1}{2} \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right ) \int \frac {1}{-b x^2-\sqrt {a} \sqrt {b}}dx\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {1}{2} \left (\frac {\sqrt {b} c}{\sqrt {a}}+d\right ) \int \frac {1}{\sqrt {a} \sqrt {b}-b x^2}dx+\frac {\arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right )}{2 \sqrt [4]{a} b^{3/4}}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\frac {\sqrt {b} c}{\sqrt {a}}-d\right )}{2 \sqrt [4]{a} b^{3/4}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\frac {\sqrt {b} c}{\sqrt {a}}+d\right )}{2 \sqrt [4]{a} b^{3/4}}\)

Input:

Int[(c + d*x^2)/(a - b*x^4),x]
 

Output:

(((Sqrt[b]*c)/Sqrt[a] - d)*ArcTan[(b^(1/4)*x)/a^(1/4)])/(2*a^(1/4)*b^(3/4) 
) + (((Sqrt[b]*c)/Sqrt[a] + d)*ArcTanh[(b^(1/4)*x)/a^(1/4)])/(2*a^(1/4)*b^ 
(3/4))
 

Defintions of rubi rules used

rule 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 1481

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
(-a)*c, 2]}, Simp[(e/2 + c*(d/(2*q)))   Int[1/(-q + c*x^2), x], x] + Simp[( 
e/2 - c*(d/(2*q)))   Int[1/(q + c*x^2), x], x]] /; FreeQ[{a, c, d, e}, x] & 
& NeQ[c*d^2 - a*e^2, 0] && PosQ[(-a)*c]
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.10 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.42

method	result	size
risch	\(-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4} b -a \right )}{\sum }\frac {\left (\textit {\_R}^{2} d +c \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}}{4 b}\)	\(36\)
default	\(\frac {c \left (\frac {a}{b}\right )^{\frac {1}{4}} \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )\right )}{4 a}-\frac {d \left (2 \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )-\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )\right )}{4 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}\)	\(104\)

Input:

int((d*x^2+c)/(-b*x^4+a),x,method=_RETURNVERBOSE)
 

Output:

-1/4/b*sum((_R^2*d+c)/_R^3*ln(x-_R),_R=RootOf(_Z^4*b-a))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 755 vs. \(2 (57) = 114\).

Time = 0.08 (sec) , antiderivative size = 755, normalized size of antiderivative = 8.88 \[ \int \frac {c+d x^2}{a-b x^4} \, dx=\frac {1}{4} \, \sqrt {\frac {a b \sqrt {\frac {b^{2} c^{4} + 2 \, a b c^{2} d^{2} + a^{2} d^{4}}{a^{3} b^{3}}} + 2 \, c d}{a b}} \log \left (-{\left (b^{2} c^{4} - a^{2} d^{4}\right )} x + {\left (a^{3} b^{2} d \sqrt {\frac {b^{2} c^{4} + 2 \, a b c^{2} d^{2} + a^{2} d^{4}}{a^{3} b^{3}}} - a b^{2} c^{3} - a^{2} b c d^{2}\right )} \sqrt {\frac {a b \sqrt {\frac {b^{2} c^{4} + 2 \, a b c^{2} d^{2} + a^{2} d^{4}}{a^{3} b^{3}}} + 2 \, c d}{a b}}\right ) - \frac {1}{4} \, \sqrt {\frac {a b \sqrt {\frac {b^{2} c^{4} + 2 \, a b c^{2} d^{2} + a^{2} d^{4}}{a^{3} b^{3}}} + 2 \, c d}{a b}} \log \left (-{\left (b^{2} c^{4} - a^{2} d^{4}\right )} x - {\left (a^{3} b^{2} d \sqrt {\frac {b^{2} c^{4} + 2 \, a b c^{2} d^{2} + a^{2} d^{4}}{a^{3} b^{3}}} - a b^{2} c^{3} - a^{2} b c d^{2}\right )} \sqrt {\frac {a b \sqrt {\frac {b^{2} c^{4} + 2 \, a b c^{2} d^{2} + a^{2} d^{4}}{a^{3} b^{3}}} + 2 \, c d}{a b}}\right ) - \frac {1}{4} \, \sqrt {-\frac {a b \sqrt {\frac {b^{2} c^{4} + 2 \, a b c^{2} d^{2} + a^{2} d^{4}}{a^{3} b^{3}}} - 2 \, c d}{a b}} \log \left (-{\left (b^{2} c^{4} - a^{2} d^{4}\right )} x + {\left (a^{3} b^{2} d \sqrt {\frac {b^{2} c^{4} + 2 \, a b c^{2} d^{2} + a^{2} d^{4}}{a^{3} b^{3}}} + a b^{2} c^{3} + a^{2} b c d^{2}\right )} \sqrt {-\frac {a b \sqrt {\frac {b^{2} c^{4} + 2 \, a b c^{2} d^{2} + a^{2} d^{4}}{a^{3} b^{3}}} - 2 \, c d}{a b}}\right ) + \frac {1}{4} \, \sqrt {-\frac {a b \sqrt {\frac {b^{2} c^{4} + 2 \, a b c^{2} d^{2} + a^{2} d^{4}}{a^{3} b^{3}}} - 2 \, c d}{a b}} \log \left (-{\left (b^{2} c^{4} - a^{2} d^{4}\right )} x - {\left (a^{3} b^{2} d \sqrt {\frac {b^{2} c^{4} + 2 \, a b c^{2} d^{2} + a^{2} d^{4}}{a^{3} b^{3}}} + a b^{2} c^{3} + a^{2} b c d^{2}\right )} \sqrt {-\frac {a b \sqrt {\frac {b^{2} c^{4} + 2 \, a b c^{2} d^{2} + a^{2} d^{4}}{a^{3} b^{3}}} - 2 \, c d}{a b}}\right ) \] Input:

integrate((d*x^2+c)/(-b*x^4+a),x, algorithm="fricas")
 

Output:

1/4*sqrt((a*b*sqrt((b^2*c^4 + 2*a*b*c^2*d^2 + a^2*d^4)/(a^3*b^3)) + 2*c*d) 
/(a*b))*log(-(b^2*c^4 - a^2*d^4)*x + (a^3*b^2*d*sqrt((b^2*c^4 + 2*a*b*c^2* 
d^2 + a^2*d^4)/(a^3*b^3)) - a*b^2*c^3 - a^2*b*c*d^2)*sqrt((a*b*sqrt((b^2*c 
^4 + 2*a*b*c^2*d^2 + a^2*d^4)/(a^3*b^3)) + 2*c*d)/(a*b))) - 1/4*sqrt((a*b* 
sqrt((b^2*c^4 + 2*a*b*c^2*d^2 + a^2*d^4)/(a^3*b^3)) + 2*c*d)/(a*b))*log(-( 
b^2*c^4 - a^2*d^4)*x - (a^3*b^2*d*sqrt((b^2*c^4 + 2*a*b*c^2*d^2 + a^2*d^4) 
/(a^3*b^3)) - a*b^2*c^3 - a^2*b*c*d^2)*sqrt((a*b*sqrt((b^2*c^4 + 2*a*b*c^2 
*d^2 + a^2*d^4)/(a^3*b^3)) + 2*c*d)/(a*b))) - 1/4*sqrt(-(a*b*sqrt((b^2*c^4 
 + 2*a*b*c^2*d^2 + a^2*d^4)/(a^3*b^3)) - 2*c*d)/(a*b))*log(-(b^2*c^4 - a^2 
*d^4)*x + (a^3*b^2*d*sqrt((b^2*c^4 + 2*a*b*c^2*d^2 + a^2*d^4)/(a^3*b^3)) + 
 a*b^2*c^3 + a^2*b*c*d^2)*sqrt(-(a*b*sqrt((b^2*c^4 + 2*a*b*c^2*d^2 + a^2*d 
^4)/(a^3*b^3)) - 2*c*d)/(a*b))) + 1/4*sqrt(-(a*b*sqrt((b^2*c^4 + 2*a*b*c^2 
*d^2 + a^2*d^4)/(a^3*b^3)) - 2*c*d)/(a*b))*log(-(b^2*c^4 - a^2*d^4)*x - (a 
^3*b^2*d*sqrt((b^2*c^4 + 2*a*b*c^2*d^2 + a^2*d^4)/(a^3*b^3)) + a*b^2*c^3 + 
 a^2*b*c*d^2)*sqrt(-(a*b*sqrt((b^2*c^4 + 2*a*b*c^2*d^2 + a^2*d^4)/(a^3*b^3 
)) - 2*c*d)/(a*b)))
 

Sympy [A] (verification not implemented)

Time = 0.34 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.29 \[ \int \frac {c+d x^2}{a-b x^4} \, dx=- \operatorname {RootSum} {\left (256 t^{4} a^{3} b^{3} - 64 t^{2} a^{2} b^{2} c d - a^{2} d^{4} + 2 a b c^{2} d^{2} - b^{2} c^{4}, \left ( t \mapsto t \log {\left (x + \frac {- 64 t^{3} a^{3} b^{2} d + 12 t a^{2} b c d^{2} + 4 t a b^{2} c^{3}}{a^{2} d^{4} - b^{2} c^{4}} \right )} \right )\right )} \] Input:

integrate((d*x**2+c)/(-b*x**4+a),x)
 

Output:

-RootSum(256*_t**4*a**3*b**3 - 64*_t**2*a**2*b**2*c*d - a**2*d**4 + 2*a*b* 
c**2*d**2 - b**2*c**4, Lambda(_t, _t*log(x + (-64*_t**3*a**3*b**2*d + 12*_ 
t*a**2*b*c*d**2 + 4*_t*a*b**2*c**3)/(a**2*d**4 - b**2*c**4))))
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.28 \[ \int \frac {c+d x^2}{a-b x^4} \, dx=\frac {{\left (\sqrt {b} c - \sqrt {a} d\right )} \arctan \left (\frac {\sqrt {b} x}{\sqrt {\sqrt {a} \sqrt {b}}}\right )}{2 \, \sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {{\left (\sqrt {b} c + \sqrt {a} d\right )} \log \left (\frac {\sqrt {b} x - \sqrt {\sqrt {a} \sqrt {b}}}{\sqrt {b} x + \sqrt {\sqrt {a} \sqrt {b}}}\right )}{4 \, \sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} \] Input:

integrate((d*x^2+c)/(-b*x^4+a),x, algorithm="maxima")
 

Output:

1/2*(sqrt(b)*c - sqrt(a)*d)*arctan(sqrt(b)*x/sqrt(sqrt(a)*sqrt(b)))/(sqrt( 
a)*sqrt(sqrt(a)*sqrt(b))*sqrt(b)) - 1/4*(sqrt(b)*c + sqrt(a)*d)*log((sqrt( 
b)*x - sqrt(sqrt(a)*sqrt(b)))/(sqrt(b)*x + sqrt(sqrt(a)*sqrt(b))))/(sqrt(a 
)*sqrt(sqrt(a)*sqrt(b))*sqrt(b))
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 230 vs. \(2 (57) = 114\).

Time = 0.12 (sec) , antiderivative size = 230, normalized size of antiderivative = 2.71 \[ \int \frac {c+d x^2}{a-b x^4} \, dx=-\frac {\sqrt {2} {\left (b^{2} c + \sqrt {-a b} b d\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (-\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{4 \, \left (-a b^{3}\right )^{\frac {3}{4}}} - \frac {\sqrt {2} {\left (b^{2} c - \sqrt {-a b} b d\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (-\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{4 \, \left (-a b^{3}\right )^{\frac {3}{4}}} - \frac {\sqrt {2} {\left (b^{2} c - \sqrt {-a b} b d\right )} \log \left (x^{2} + \sqrt {2} x \left (-\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {-\frac {a}{b}}\right )}{8 \, \left (-a b^{3}\right )^{\frac {3}{4}}} + \frac {\sqrt {2} {\left (b^{2} c - \sqrt {-a b} b d\right )} \log \left (x^{2} - \sqrt {2} x \left (-\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {-\frac {a}{b}}\right )}{8 \, \left (-a b^{3}\right )^{\frac {3}{4}}} \] Input:

integrate((d*x^2+c)/(-b*x^4+a),x, algorithm="giac")
 

Output:

-1/4*sqrt(2)*(b^2*c + sqrt(-a*b)*b*d)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(- 
a/b)^(1/4))/(-a/b)^(1/4))/(-a*b^3)^(3/4) - 1/4*sqrt(2)*(b^2*c - sqrt(-a*b) 
*b*d)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(-a/b)^(1/4))/(-a/b)^(1/4))/(-a*b^ 
3)^(3/4) - 1/8*sqrt(2)*(b^2*c - sqrt(-a*b)*b*d)*log(x^2 + sqrt(2)*x*(-a/b) 
^(1/4) + sqrt(-a/b))/(-a*b^3)^(3/4) + 1/8*sqrt(2)*(b^2*c - sqrt(-a*b)*b*d) 
*log(x^2 - sqrt(2)*x*(-a/b)^(1/4) + sqrt(-a/b))/(-a*b^3)^(3/4)
 

Mupad [B] (verification not implemented)

Time = 17.40 (sec) , antiderivative size = 579, normalized size of antiderivative = 6.81 \[ \int \frac {c+d x^2}{a-b x^4} \, dx=2\,\mathrm {atanh}\left (\frac {8\,b^3\,c^2\,x\,\sqrt {\frac {c\,d}{8\,a\,b}-\frac {c^2\,\sqrt {a^3\,b^3}}{16\,a^3\,b^2}-\frac {d^2\,\sqrt {a^3\,b^3}}{16\,a^2\,b^3}}}{2\,b^2\,c^2\,d+2\,a\,b\,d^3-\frac {2\,b\,c^3\,\sqrt {a^3\,b^3}}{a^2}-\frac {2\,c\,d^2\,\sqrt {a^3\,b^3}}{a}}+\frac {8\,a\,b^2\,d^2\,x\,\sqrt {\frac {c\,d}{8\,a\,b}-\frac {c^2\,\sqrt {a^3\,b^3}}{16\,a^3\,b^2}-\frac {d^2\,\sqrt {a^3\,b^3}}{16\,a^2\,b^3}}}{2\,b^2\,c^2\,d+2\,a\,b\,d^3-\frac {2\,b\,c^3\,\sqrt {a^3\,b^3}}{a^2}-\frac {2\,c\,d^2\,\sqrt {a^3\,b^3}}{a}}\right )\,\sqrt {-\frac {a\,d^2\,\sqrt {a^3\,b^3}+b\,c^2\,\sqrt {a^3\,b^3}-2\,a^2\,b^2\,c\,d}{16\,a^3\,b^3}}+2\,\mathrm {atanh}\left (\frac {8\,b^3\,c^2\,x\,\sqrt {\frac {c\,d}{8\,a\,b}+\frac {c^2\,\sqrt {a^3\,b^3}}{16\,a^3\,b^2}+\frac {d^2\,\sqrt {a^3\,b^3}}{16\,a^2\,b^3}}}{2\,b^2\,c^2\,d+2\,a\,b\,d^3+\frac {2\,b\,c^3\,\sqrt {a^3\,b^3}}{a^2}+\frac {2\,c\,d^2\,\sqrt {a^3\,b^3}}{a}}+\frac {8\,a\,b^2\,d^2\,x\,\sqrt {\frac {c\,d}{8\,a\,b}+\frac {c^2\,\sqrt {a^3\,b^3}}{16\,a^3\,b^2}+\frac {d^2\,\sqrt {a^3\,b^3}}{16\,a^2\,b^3}}}{2\,b^2\,c^2\,d+2\,a\,b\,d^3+\frac {2\,b\,c^3\,\sqrt {a^3\,b^3}}{a^2}+\frac {2\,c\,d^2\,\sqrt {a^3\,b^3}}{a}}\right )\,\sqrt {\frac {a\,d^2\,\sqrt {a^3\,b^3}+b\,c^2\,\sqrt {a^3\,b^3}+2\,a^2\,b^2\,c\,d}{16\,a^3\,b^3}} \] Input:

int((c + d*x^2)/(a - b*x^4),x)
 

Output:

2*atanh((8*b^3*c^2*x*((c*d)/(8*a*b) - (c^2*(a^3*b^3)^(1/2))/(16*a^3*b^2) - 
 (d^2*(a^3*b^3)^(1/2))/(16*a^2*b^3))^(1/2))/(2*b^2*c^2*d + 2*a*b*d^3 - (2* 
b*c^3*(a^3*b^3)^(1/2))/a^2 - (2*c*d^2*(a^3*b^3)^(1/2))/a) + (8*a*b^2*d^2*x 
*((c*d)/(8*a*b) - (c^2*(a^3*b^3)^(1/2))/(16*a^3*b^2) - (d^2*(a^3*b^3)^(1/2 
))/(16*a^2*b^3))^(1/2))/(2*b^2*c^2*d + 2*a*b*d^3 - (2*b*c^3*(a^3*b^3)^(1/2 
))/a^2 - (2*c*d^2*(a^3*b^3)^(1/2))/a))*(-(a*d^2*(a^3*b^3)^(1/2) + b*c^2*(a 
^3*b^3)^(1/2) - 2*a^2*b^2*c*d)/(16*a^3*b^3))^(1/2) + 2*atanh((8*b^3*c^2*x* 
((c*d)/(8*a*b) + (c^2*(a^3*b^3)^(1/2))/(16*a^3*b^2) + (d^2*(a^3*b^3)^(1/2) 
)/(16*a^2*b^3))^(1/2))/(2*b^2*c^2*d + 2*a*b*d^3 + (2*b*c^3*(a^3*b^3)^(1/2) 
)/a^2 + (2*c*d^2*(a^3*b^3)^(1/2))/a) + (8*a*b^2*d^2*x*((c*d)/(8*a*b) + (c^ 
2*(a^3*b^3)^(1/2))/(16*a^3*b^2) + (d^2*(a^3*b^3)^(1/2))/(16*a^2*b^3))^(1/2 
))/(2*b^2*c^2*d + 2*a*b*d^3 + (2*b*c^3*(a^3*b^3)^(1/2))/a^2 + (2*c*d^2*(a^ 
3*b^3)^(1/2))/a))*((a*d^2*(a^3*b^3)^(1/2) + b*c^2*(a^3*b^3)^(1/2) + 2*a^2* 
b^2*c*d)/(16*a^3*b^3))^(1/2)
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.19 \[ \int \frac {c+d x^2}{a-b x^4} \, dx=\frac {-2 \sqrt {a}\, \mathit {atan} \left (\frac {\sqrt {b}\, x}{b^{\frac {1}{4}} a^{\frac {1}{4}}}\right ) d +2 \sqrt {b}\, \mathit {atan} \left (\frac {\sqrt {b}\, x}{b^{\frac {1}{4}} a^{\frac {1}{4}}}\right ) c -\sqrt {a}\, \mathrm {log}\left (a^{\frac {1}{4}}-b^{\frac {1}{4}} x \right ) d +\sqrt {a}\, \mathrm {log}\left (a^{\frac {1}{4}}+b^{\frac {1}{4}} x \right ) d -\sqrt {b}\, \mathrm {log}\left (a^{\frac {1}{4}}-b^{\frac {1}{4}} x \right ) c +\sqrt {b}\, \mathrm {log}\left (a^{\frac {1}{4}}+b^{\frac {1}{4}} x \right ) c}{4 b^{\frac {3}{4}} a^{\frac {3}{4}}} \] Input:

int((d*x^2+c)/(-b*x^4+a),x)
 

Output:

(b**(1/4)*a**(1/4)*( - 2*sqrt(a)*atan((sqrt(b)*x)/(b**(1/4)*a**(1/4)))*d + 
 2*sqrt(b)*atan((sqrt(b)*x)/(b**(1/4)*a**(1/4)))*c - sqrt(a)*log(a**(1/4) 
- b**(1/4)*x)*d + sqrt(a)*log(a**(1/4) + b**(1/4)*x)*d - sqrt(b)*log(a**(1 
/4) - b**(1/4)*x)*c + sqrt(b)*log(a**(1/4) + b**(1/4)*x)*c))/(4*a*b)