3.708 \(\int \frac{1}{1+a+(-1+a) x^4} \, dx\)

Optimal. Leaf size=83 \[ \frac{\tan ^{-1}\left (\frac{\sqrt [4]{1-a} x}{\sqrt [4]{a+1}}\right )}{2 \sqrt{a+1} \sqrt [4]{1-a^2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{1-a} x}{\sqrt [4]{a+1}}\right )}{2 \sqrt{a+1} \sqrt [4]{1-a^2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0724358, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {212, 208, 205} \[ \frac{\tan ^{-1}\left (\frac{\sqrt [4]{1-a} x}{\sqrt [4]{a+1}}\right )}{2 \sqrt{a+1} \sqrt [4]{1-a^2}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{1-a} x}{\sqrt [4]{a+1}}\right )}{2 \sqrt{a+1} \sqrt [4]{1-a^2}} \]

Antiderivative was successfully verified.

[In]

Int[(1 + a + (-1 + a)*x^4)^(-1),x]

[Out]

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

Rule 212

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b), 2]
]}, Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&
!GtQ[a/b, 0]

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]

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]

Rubi steps

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

Mathematica [A]  time = 0.0656576, size = 160, normalized size = 1.93 \[ \frac{-\log \left (\sqrt{a-1} x^2-\sqrt{2} \sqrt [4]{a-1} \sqrt [4]{a+1} x+\sqrt{a+1}\right )+\log \left (\sqrt{a-1} x^2+\sqrt{2} \sqrt [4]{a-1} \sqrt [4]{a+1} x+\sqrt{a+1}\right )-2 \tan ^{-1}\left (1-\sqrt{2} \sqrt [4]{\frac{a-1}{a+1}} x\right )+2 \tan ^{-1}\left (\sqrt{2} \sqrt [4]{\frac{a-1}{a+1}} x+1\right )}{4 \sqrt{2} \sqrt [4]{a-1} (a+1)^{3/4}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(1 + a + (-1 + a)*x^4)^(-1),x]

[Out]

(-2*ArcTan[1 - Sqrt[2]*((-1 + a)/(1 + a))^(1/4)*x] + 2*ArcTan[1 + Sqrt[2]*((-1 + a)/(1 + a))^(1/4)*x] - Log[Sq
rt[1 + a] - Sqrt[2]*(-1 + a)^(1/4)*(1 + a)^(1/4)*x + Sqrt[-1 + a]*x^2] + Log[Sqrt[1 + a] + Sqrt[2]*(-1 + a)^(1
/4)*(1 + a)^(1/4)*x + Sqrt[-1 + a]*x^2])/(4*Sqrt[2]*(-1 + a)^(1/4)*(1 + a)^(3/4))

________________________________________________________________________________________

Maple [B]  time = 0.01, size = 170, normalized size = 2.1 \begin{align*}{\frac{\sqrt{2}}{8+8\,a}\sqrt [4]{{\frac{1+a}{a-1}}}\ln \left ({ \left ({x}^{2}+\sqrt [4]{{\frac{1+a}{a-1}}}x\sqrt{2}+\sqrt{{\frac{1+a}{a-1}}} \right ) \left ({x}^{2}-\sqrt [4]{{\frac{1+a}{a-1}}}x\sqrt{2}+\sqrt{{\frac{1+a}{a-1}}} \right ) ^{-1}} \right ) }+{\frac{\sqrt{2}}{4+4\,a}\sqrt [4]{{\frac{1+a}{a-1}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{1+a}{a-1}}}}}}+1 \right ) }+{\frac{\sqrt{2}}{4+4\,a}\sqrt [4]{{\frac{1+a}{a-1}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{1+a}{a-1}}}}}}-1 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1+a+(a-1)*x^4),x)

[Out]

1/8*((1+a)/(a-1))^(1/4)/(1+a)*2^(1/2)*ln((x^2+((1+a)/(a-1))^(1/4)*x*2^(1/2)+((1+a)/(a-1))^(1/2))/(x^2-((1+a)/(
a-1))^(1/4)*x*2^(1/2)+((1+a)/(a-1))^(1/2)))+1/4*((1+a)/(a-1))^(1/4)/(1+a)*2^(1/2)*arctan(2^(1/2)/((1+a)/(a-1))
^(1/4)*x+1)+1/4*((1+a)/(a-1))^(1/4)/(1+a)*2^(1/2)*arctan(2^(1/2)/((1+a)/(a-1))^(1/4)*x-1)

________________________________________________________________________________________

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(1/(1+a+(-1+a)*x^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 1.81583, size = 549, normalized size = 6.61 \begin{align*} \left (-\frac{1}{a^{4} + 2 \, a^{3} - 2 \, a - 1}\right )^{\frac{1}{4}} \arctan \left (-{\left (a^{3} + a^{2} - a - 1\right )} x \left (-\frac{1}{a^{4} + 2 \, a^{3} - 2 \, a - 1}\right )^{\frac{3}{4}} +{\left (a^{3} + a^{2} - a - 1\right )} \sqrt{x^{2} +{\left (a^{2} + 2 \, a + 1\right )} \sqrt{-\frac{1}{a^{4} + 2 \, a^{3} - 2 \, a - 1}}} \left (-\frac{1}{a^{4} + 2 \, a^{3} - 2 \, a - 1}\right )^{\frac{3}{4}}\right ) + \frac{1}{4} \, \left (-\frac{1}{a^{4} + 2 \, a^{3} - 2 \, a - 1}\right )^{\frac{1}{4}} \log \left ({\left (a + 1\right )} \left (-\frac{1}{a^{4} + 2 \, a^{3} - 2 \, a - 1}\right )^{\frac{1}{4}} + x\right ) - \frac{1}{4} \, \left (-\frac{1}{a^{4} + 2 \, a^{3} - 2 \, a - 1}\right )^{\frac{1}{4}} \log \left (-{\left (a + 1\right )} \left (-\frac{1}{a^{4} + 2 \, a^{3} - 2 \, a - 1}\right )^{\frac{1}{4}} + x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+a+(-1+a)*x^4),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0.351847, size = 32, normalized size = 0.39 \begin{align*} \operatorname{RootSum}{\left (t^{4} \left (256 a^{4} + 512 a^{3} - 512 a - 256\right ) + 1, \left ( t \mapsto t \log{\left (4 t a + 4 t + x \right )} \right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+a+(-1+a)*x**4),x)

[Out]

RootSum(_t**4*(256*a**4 + 512*a**3 - 512*a - 256) + 1, Lambda(_t, _t*log(4*_t*a + 4*_t + x)))

________________________________________________________________________________________

Giac [B]  time = 1.11222, size = 360, normalized size = 4.34 \begin{align*} \frac{{\left (a^{4} - 2 \, a^{3} + 2 \, a - 1\right )}^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (\frac{a + 1}{a - 1}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a + 1}{a - 1}\right )^{\frac{1}{4}}}\right )}{2 \,{\left (\sqrt{2} a^{2} - \sqrt{2}\right )}} + \frac{{\left (a^{4} - 2 \, a^{3} + 2 \, a - 1\right )}^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (\frac{a + 1}{a - 1}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a + 1}{a - 1}\right )^{\frac{1}{4}}}\right )}{2 \,{\left (\sqrt{2} a^{2} - \sqrt{2}\right )}} + \frac{{\left (a^{4} - 2 \, a^{3} + 2 \, a - 1\right )}^{\frac{1}{4}} \log \left (x^{2} + \sqrt{2} x \left (\frac{a + 1}{a - 1}\right )^{\frac{1}{4}} + \sqrt{\frac{a + 1}{a - 1}}\right )}{4 \,{\left (\sqrt{2} a^{2} - \sqrt{2}\right )}} - \frac{{\left (a^{4} - 2 \, a^{3} + 2 \, a - 1\right )}^{\frac{1}{4}} \log \left (x^{2} - \sqrt{2} x \left (\frac{a + 1}{a - 1}\right )^{\frac{1}{4}} + \sqrt{\frac{a + 1}{a - 1}}\right )}{4 \,{\left (\sqrt{2} a^{2} - \sqrt{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+a+(-1+a)*x^4),x, algorithm="giac")

[Out]

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