3.217 \(\int \frac{1}{(a-(a-b) x^4) \sqrt [4]{a+b x^4}} \, dx\)

Optimal. Leaf size=57 \[ \frac{\tan ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{a+b x^4}}\right )}{2 a^{5/4}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{a+b x^4}}\right )}{2 a^{5/4}} \]

[Out]

ArcTan[(a^(1/4)*x)/(a + b*x^4)^(1/4)]/(2*a^(5/4)) + ArcTanh[(a^(1/4)*x)/(a + b*x^4)^(1/4)]/(2*a^(5/4))

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Rubi [A]  time = 0.0223042, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {377, 212, 206, 203} \[ \frac{\tan ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{a+b x^4}}\right )}{2 a^{5/4}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{a+b x^4}}\right )}{2 a^{5/4}} \]

Antiderivative was successfully verified.

[In]

Int[1/((a - (a - b)*x^4)*(a + b*x^4)^(1/4)),x]

[Out]

ArcTan[(a^(1/4)*x)/(a + b*x^4)^(1/4)]/(2*a^(5/4)) + ArcTanh[(a^(1/4)*x)/(a + b*x^4)^(1/4)]/(2*a^(5/4))

Rule 377

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

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 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{1}{\left (a-(a-b) x^4\right ) \sqrt [4]{a+b x^4}} \, dx &=\operatorname{Subst}\left (\int \frac{1}{a-(a b-a (-a+b)) x^4} \, dx,x,\frac{x}{\sqrt [4]{a+b x^4}}\right )\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{1-\sqrt{a} x^2} \, dx,x,\frac{x}{\sqrt [4]{a+b x^4}}\right )}{2 a}+\frac{\operatorname{Subst}\left (\int \frac{1}{1+\sqrt{a} x^2} \, dx,x,\frac{x}{\sqrt [4]{a+b x^4}}\right )}{2 a}\\ &=\frac{\tan ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{a+b x^4}}\right )}{2 a^{5/4}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{a+b x^4}}\right )}{2 a^{5/4}}\\ \end{align*}

Mathematica [A]  time = 0.0199442, size = 48, normalized size = 0.84 \[ \frac{\tan ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{a+b x^4}}\right )+\tanh ^{-1}\left (\frac{\sqrt [4]{a} x}{\sqrt [4]{a+b x^4}}\right )}{2 a^{5/4}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(ArcTan[(a^(1/4)*x)/(a + b*x^4)^(1/4)] + ArcTanh[(a^(1/4)*x)/(a + b*x^4)^(1/4)])/(2*a^(5/4))

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Maple [F]  time = 0.457, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{a- \left ( a-b \right ){x}^{4}}{\frac{1}{\sqrt [4]{b{x}^{4}+a}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{1}{{\left ({\left (a - b\right )} x^{4} - a\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{1}{a x^{4} \sqrt [4]{a + b x^{4}} - a \sqrt [4]{a + b x^{4}} - b x^{4} \sqrt [4]{a + b x^{4}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(1/(a*x**4*(a + b*x**4)**(1/4) - a*(a + b*x**4)**(1/4) - b*x**4*(a + b*x**4)**(1/4)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{1}{{\left ({\left (a - b\right )} x^{4} - a\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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