3.4 \(\int \frac{1}{\sqrt{-1+x^2} (-4+3 x^2)^2} \, dx\)

Optimal. Leaf size=43 \[ \frac{3 \sqrt{x^2-1} x}{8 \left (4-3 x^2\right )}+\frac{5}{16} \tanh ^{-1}\left (\frac{x}{2 \sqrt{x^2-1}}\right ) \]

[Out]

(3*x*Sqrt[-1 + x^2])/(8*(4 - 3*x^2)) + (5*ArcTanh[x/(2*Sqrt[-1 + x^2])])/16

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Rubi [A]  time = 0.0123968, antiderivative size = 43, 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 = {382, 377, 207} \[ \frac{3 \sqrt{x^2-1} x}{8 \left (4-3 x^2\right )}+\frac{5}{16} \tanh ^{-1}\left (\frac{x}{2 \sqrt{x^2-1}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[-1 + x^2]*(-4 + 3*x^2)^2),x]

[Out]

(3*x*Sqrt[-1 + x^2])/(8*(4 - 3*x^2)) + (5*ArcTanh[x/(2*Sqrt[-1 + x^2])])/16

Rule 382

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(b*x*(a + b*x^n)^(p + 1)*(
c + d*x^n)^(q + 1))/(a*n*(p + 1)*(b*c - a*d)), x] + Dist[(b*c + n*(p + 1)*(b*c - a*d))/(a*n*(p + 1)*(b*c - a*d
)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n, q}, x] && NeQ[b*c - a*d, 0] && EqQ[
n*(p + q + 2) + 1, 0] && (LtQ[p, -1] ||  !LtQ[q, -1]) && NeQ[p, -1]

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 207

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

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{-1+x^2} \left (-4+3 x^2\right )^2} \, dx &=\frac{3 x \sqrt{-1+x^2}}{8 \left (4-3 x^2\right )}-\frac{5}{8} \int \frac{1}{\sqrt{-1+x^2} \left (-4+3 x^2\right )} \, dx\\ &=\frac{3 x \sqrt{-1+x^2}}{8 \left (4-3 x^2\right )}-\frac{5}{8} \operatorname{Subst}\left (\int \frac{1}{-4+x^2} \, dx,x,\frac{x}{\sqrt{-1+x^2}}\right )\\ &=\frac{3 x \sqrt{-1+x^2}}{8 \left (4-3 x^2\right )}+\frac{5}{16} \tanh ^{-1}\left (\frac{x}{2 \sqrt{-1+x^2}}\right )\\ \end{align*}

Mathematica [C]  time = 3.39706, size = 167, normalized size = 3.88 \[ -\frac{x \sqrt{x^2-1} \left (\frac{8 x^2 \left (x^2-1\right ) \, _2F_1\left (2,3;\frac{7}{2};\frac{x^2}{4-3 x^2}\right )}{45 x^2-60}-\frac{x^2 \left (2 x^2-3\right ) \sqrt{\frac{x^2-1}{3 x^2-4}} \left (2 \sqrt{\frac{x^2-x^4}{\left (4-3 x^2\right )^2}}-\sin ^{-1}\left (\sqrt{\frac{x^2}{4-3 x^2}}\right )\right )}{4 \left (\frac{x^2}{4-3 x^2}\right )^{5/2} \left (x^2-1\right )}\right )}{16 \left (1-\frac{3 x^2}{4}\right )^2} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/(Sqrt[-1 + x^2]*(-4 + 3*x^2)^2),x]

[Out]

-(x*Sqrt[-1 + x^2]*(-(x^2*(-3 + 2*x^2)*Sqrt[(-1 + x^2)/(-4 + 3*x^2)]*(2*Sqrt[(x^2 - x^4)/(4 - 3*x^2)^2] - ArcS
in[Sqrt[x^2/(4 - 3*x^2)]]))/(4*(x^2/(4 - 3*x^2))^(5/2)*(-1 + x^2)) + (8*x^2*(-1 + x^2)*Hypergeometric2F1[2, 3,
7/2, x^2/(4 - 3*x^2)])/(-60 + 45*x^2)))/(16*(1 - (3*x^2)/4)^2)

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Maple [B]  time = 0.047, size = 172, normalized size = 4. \begin{align*} -{\frac{1}{16}\sqrt{ \left ( x+{\frac{2\,\sqrt{3}}{3}} \right ) ^{2}-{\frac{4\,\sqrt{3}}{3} \left ( x+{\frac{2\,\sqrt{3}}{3}} \right ) }+{\frac{1}{3}}} \left ( x+{\frac{2\,\sqrt{3}}{3}} \right ) ^{-1}}-{\frac{5}{32}{\it Artanh} \left ({\frac{3\,\sqrt{3}}{2} \left ({\frac{2}{3}}-{\frac{4\,\sqrt{3}}{3} \left ( x+{\frac{2\,\sqrt{3}}{3}} \right ) } \right ){\frac{1}{\sqrt{9\, \left ( x+2/3\,\sqrt{3} \right ) ^{2}-12\,\sqrt{3} \left ( x+2/3\,\sqrt{3} \right ) +3}}}} \right ) }-{\frac{1}{16}\sqrt{ \left ( x-{\frac{2\,\sqrt{3}}{3}} \right ) ^{2}+{\frac{4\,\sqrt{3}}{3} \left ( x-{\frac{2\,\sqrt{3}}{3}} \right ) }+{\frac{1}{3}}} \left ( x-{\frac{2\,\sqrt{3}}{3}} \right ) ^{-1}}+{\frac{5}{32}{\it Artanh} \left ({\frac{3\,\sqrt{3}}{2} \left ({\frac{2}{3}}+{\frac{4\,\sqrt{3}}{3} \left ( x-{\frac{2\,\sqrt{3}}{3}} \right ) } \right ){\frac{1}{\sqrt{9\, \left ( x-2/3\,\sqrt{3} \right ) ^{2}+12\,\sqrt{3} \left ( x-2/3\,\sqrt{3} \right ) +3}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(3*x^2-4)^2/(x^2-1)^(1/2),x)

[Out]

-1/16/(x+2/3*3^(1/2))*((x+2/3*3^(1/2))^2-4/3*3^(1/2)*(x+2/3*3^(1/2))+1/3)^(1/2)-5/32*arctanh(3/2*(2/3-4/3*3^(1
/2)*(x+2/3*3^(1/2)))*3^(1/2)/(9*(x+2/3*3^(1/2))^2-12*3^(1/2)*(x+2/3*3^(1/2))+3)^(1/2))-1/16/(x-2/3*3^(1/2))*((
x-2/3*3^(1/2))^2+4/3*3^(1/2)*(x-2/3*3^(1/2))+1/3)^(1/2)+5/32*arctanh(3/2*(2/3+4/3*3^(1/2)*(x-2/3*3^(1/2)))*3^(
1/2)/(9*(x-2/3*3^(1/2))^2+12*3^(1/2)*(x-2/3*3^(1/2))+3)^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (3 \, x^{2} - 4\right )}^{2} \sqrt{x^{2} - 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3*x^2-4)^2/(x^2-1)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/((3*x^2 - 4)^2*sqrt(x^2 - 1)), x)

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Fricas [B]  time = 2.04815, size = 205, normalized size = 4.77 \begin{align*} -\frac{12 \, x^{2} + 5 \,{\left (3 \, x^{2} - 4\right )} \log \left (3 \, x^{2} - 3 \, \sqrt{x^{2} - 1} x - 2\right ) - 5 \,{\left (3 \, x^{2} - 4\right )} \log \left (x^{2} - \sqrt{x^{2} - 1} x - 2\right ) + 12 \, \sqrt{x^{2} - 1} x - 16}{32 \,{\left (3 \, x^{2} - 4\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3*x^2-4)^2/(x^2-1)^(1/2),x, algorithm="fricas")

[Out]

-1/32*(12*x^2 + 5*(3*x^2 - 4)*log(3*x^2 - 3*sqrt(x^2 - 1)*x - 2) - 5*(3*x^2 - 4)*log(x^2 - sqrt(x^2 - 1)*x - 2
) + 12*sqrt(x^2 - 1)*x - 16)/(3*x^2 - 4)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{\left (x - 1\right ) \left (x + 1\right )} \left (3 x^{2} - 4\right )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3*x**2-4)**2/(x**2-1)**(1/2),x)

[Out]

Integral(1/(sqrt((x - 1)*(x + 1))*(3*x**2 - 4)**2), x)

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Giac [B]  time = 1.08749, size = 127, normalized size = 2.95 \begin{align*} \frac{5 \,{\left (x - \sqrt{x^{2} - 1}\right )}^{2} - 3}{4 \,{\left (3 \,{\left (x - \sqrt{x^{2} - 1}\right )}^{4} - 10 \,{\left (x - \sqrt{x^{2} - 1}\right )}^{2} + 3\right )}} - \frac{5}{32} \, \log \left ({\left | 3 \,{\left (x - \sqrt{x^{2} - 1}\right )}^{2} - 1 \right |}\right ) + \frac{5}{32} \, \log \left ({\left |{\left (x - \sqrt{x^{2} - 1}\right )}^{2} - 3 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3*x^2-4)^2/(x^2-1)^(1/2),x, algorithm="giac")

[Out]

1/4*(5*(x - sqrt(x^2 - 1))^2 - 3)/(3*(x - sqrt(x^2 - 1))^4 - 10*(x - sqrt(x^2 - 1))^2 + 3) - 5/32*log(abs(3*(x
- sqrt(x^2 - 1))^2 - 1)) + 5/32*log(abs((x - sqrt(x^2 - 1))^2 - 3))