3.1000 \(\int \frac{1}{(a+b x^4) \sqrt{c x^2+d \sqrt{a+b x^4}}} \, dx\)

Optimal. Leaf size=40 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{d \sqrt{a+b x^4}+c x^2}}\right )}{a \sqrt{c}} \]

[Out]

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

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Rubi [A]  time = 0.135462, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.061, Rules used = {2128, 206} \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{d \sqrt{a+b x^4}+c x^2}}\right )}{a \sqrt{c}} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x^4)*Sqrt[c*x^2 + d*Sqrt[a + b*x^4]]),x]

[Out]

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

Rule 2128

Int[1/(((a_) + (b_.)*(x_)^(n_.))*Sqrt[(c_.)*(x_)^2 + (d_.)*((a_) + (b_.)*(x_)^(n_.))^(p_.)]), x_Symbol] :> Dis
t[1/a, Subst[Int[1/(1 - c*x^2), x], x, x/Sqrt[c*x^2 + d*(a + b*x^n)^(2/n)]], x] /; FreeQ[{a, b, c, d, n}, x] &
& EqQ[p, 2/n]

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])

Rubi steps

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

Mathematica [A]  time = 0.469075, size = 50, normalized size = 1.25 \[ \frac{\sqrt{-\frac{1}{c}} \cot ^{-1}\left (\frac{\sqrt{-\frac{1}{c}} \sqrt{d \sqrt{a+b x^4}+c x^2}}{x}\right )}{a} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*x^4)*Sqrt[c*x^2 + d*Sqrt[a + b*x^4]]),x]

[Out]

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

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Maple [F]  time = 0.023, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{b{x}^{4}+a}{\frac{1}{\sqrt{c{x}^{2}+d\sqrt{b{x}^{4}+a}}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x^4+a)/(c*x^2+d*(b*x^4+a)^(1/2))^(1/2),x)

[Out]

int(1/(b*x^4+a)/(c*x^2+d*(b*x^4+a)^(1/2))^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{4} + a\right )} \sqrt{c x^{2} + \sqrt{b x^{4} + a} d}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^4+a)/(c*x^2+d*(b*x^4+a)^(1/2))^(1/2),x, algorithm="maxima")

[Out]

integrate(1/((b*x^4 + a)*sqrt(c*x^2 + sqrt(b*x^4 + a)*d)), 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/(b*x^4+a)/(c*x^2+d*(b*x^4+a)^(1/2))^(1/2),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a + b x^{4}\right ) \sqrt{c x^{2} + d \sqrt{a + b x^{4}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x**4+a)/(c*x**2+d*(b*x**4+a)**(1/2))**(1/2),x)

[Out]

Integral(1/((a + b*x**4)*sqrt(c*x**2 + d*sqrt(a + b*x**4))), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{4} + a\right )} \sqrt{c x^{2} + \sqrt{b x^{4} + a} d}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^4+a)/(c*x^2+d*(b*x^4+a)^(1/2))^(1/2),x, algorithm="giac")

[Out]

integrate(1/((b*x^4 + a)*sqrt(c*x^2 + sqrt(b*x^4 + a)*d)), x)