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

Optimal. Leaf size=86 \[ \frac{\left (\sqrt{a} d+\sqrt{b} c\right ) \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{3/4}}+\frac{\left (\sqrt{b} c-\sqrt{a} d\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{3/4}} \]

[Out]

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

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Rubi [A]  time = 0.0401515, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {1167, 205, 208} \[ \frac{\left (\sqrt{a} d+\sqrt{b} c\right ) \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{3/4}}+\frac{\left (\sqrt{b} c-\sqrt{a} d\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{3/4}} \]

Antiderivative was successfully verified.

[In]

Int[(c - d*x^2)/(a - b*x^4),x]

[Out]

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

Rule 1167

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

Rule 205

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

Rule 208

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

Rubi steps

\begin{align*} \int \frac{c-d x^2}{a-b x^4} \, dx &=\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}{\sqrt{a} \sqrt{b}-b x^2} \, dx\\ &=\frac{\left (\sqrt{b} c+\sqrt{a} d\right ) \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{3/4}}+\frac{\left (\sqrt{b} c-\sqrt{a} d\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{3/4}}\\ \end{align*}

Mathematica [A]  time = 0.0224051, size = 95, normalized size = 1.1 \[ \frac{2 \left (\sqrt{a} d+\sqrt{b} c\right ) \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )-\left (\sqrt{b} c-\sqrt{a} d\right ) \left (\log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right )-\log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right )\right )}{4 a^{3/4} b^{3/4}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c - d*x^2)/(a - b*x^4),x]

[Out]

(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] - L
og[a^(1/4) + b^(1/4)*x]))/(4*a^(3/4)*b^(3/4))

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Maple [B]  time = 0.043, size = 122, normalized size = 1.4 \begin{align*}{\frac{c}{4\,a}\sqrt [4]{{\frac{a}{b}}}\ln \left ({ \left ( x+\sqrt [4]{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}} \right ) ^{-1}} \right ) }+{\frac{c}{2\,a}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ) }+{\frac{d}{2\,b}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-{\frac{d}{4\,b}\ln \left ({ \left ( x+\sqrt [4]{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.44033, size = 1527, normalized size = 17.76 \begin{align*} \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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)*s
qrt((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)))

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Sympy [A]  time = 0.789761, size = 110, normalized size = 1.28 \begin{align*} \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 )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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))))

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Giac [B]  time = 1.13848, size = 347, normalized size = 4.03 \begin{align*} \frac{\sqrt{2}{\left (\left (-a b^{3}\right )^{\frac{1}{4}} b^{2} c - \left (-a b^{3}\right )^{\frac{3}{4}} 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 \, a b^{3}} + \frac{\sqrt{2}{\left (\left (-a b^{3}\right )^{\frac{1}{4}} b^{2} c - \left (-a b^{3}\right )^{\frac{3}{4}} 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 \, a b^{3}} + \frac{\sqrt{2}{\left (\left (-a b^{3}\right )^{\frac{1}{4}} b^{2} c + \left (-a b^{3}\right )^{\frac{3}{4}} d\right )} \log \left (x^{2} + \sqrt{2} x \left (-\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{-\frac{a}{b}}\right )}{8 \, a b^{3}} - \frac{\sqrt{2}{\left (\left (-a b^{3}\right )^{\frac{1}{4}} b^{2} c + \left (-a b^{3}\right )^{\frac{3}{4}} d\right )} \log \left (x^{2} - \sqrt{2} x \left (-\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{-\frac{a}{b}}\right )}{8 \, a b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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