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

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

[Out]

5/16*arctanh(1/2*x/(x^2-1)^(1/2))+3/8*x*(x^2-1)^(1/2)/(-3*x^2+4)

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Rubi [A]
time = 0.01, antiderivative size = 43, normalized size of antiderivative = 1.00, 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 = {390, 385, 213} \begin {gather*} \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 ) \end {gather*}

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 213

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

Rule 385

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 390

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] && Eq
Q[n*(p + q + 2) + 1, 0] && (LtQ[p, -1] ||  !LtQ[q, -1]) && NeQ[p, -1]

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} \text {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*}

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Mathematica [A]
time = 0.08, size = 71, normalized size = 1.65 \begin {gather*} -\frac {3 x \sqrt {-1+x^2}}{8 \left (-4+3 x^2\right )}+\frac {5}{32} \log \left (2-x^2+x \sqrt {-1+x^2}\right )-\frac {5}{32} \log \left (2-3 x^2+3 x \sqrt {-1+x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*x*Sqrt[-1 + x^2])/(8*(-4 + 3*x^2)) + (5*Log[2 - x^2 + x*Sqrt[-1 + x^2]])/32 - (5*Log[2 - 3*x^2 + 3*x*Sqrt[
-1 + x^2]])/32

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Mathics [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {cought exception: maximum recursion depth exceeded} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

cought exception: maximum recursion depth exceeded

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(171\) vs. \(2(33)=66\).
time = 0.18, size = 172, normalized size = 4.00

method	result	size
trager	\(-\frac {3 x \sqrt {x^{2}-1}}{8 \left (3 x^{2}-4\right )}-\frac {5 \ln \left (-\frac {4 \sqrt {x^{2}-1}\, x -5 x^{2}+4}{3 x^{2}-4}\right )}{32}\)	\(52\)
risch	\(-\frac {3 x \sqrt {x^{2}-1}}{8 \left (3 x^{2}-4\right )}+\frac {5 \arctanh \left (\frac {3 \left (\frac {2}{3}+\frac {4 \sqrt {3}\, \left (x -\frac {2 \sqrt {3}}{3}\right )}{3}\right ) \sqrt {3}}{2 \sqrt {9 \left (x -\frac {2 \sqrt {3}}{3}\right )^{2}+12 \sqrt {3}\, \left (x -\frac {2 \sqrt {3}}{3}\right )+3}}\right )}{32}-\frac {5 \arctanh \left (\frac {3 \left (\frac {2}{3}-\frac {4 \sqrt {3}\, \left (x +\frac {2 \sqrt {3}}{3}\right )}{3}\right ) \sqrt {3}}{2 \sqrt {9 \left (x +\frac {2 \sqrt {3}}{3}\right )^{2}-12 \sqrt {3}\, \left (x +\frac {2 \sqrt {3}}{3}\right )+3}}\right )}{32}\)	\(119\)
default	\(-\frac {5 \arctanh \left (\frac {3 \left (\frac {2}{3}-\frac {4 \sqrt {3}\, \left (x +\frac {2 \sqrt {3}}{3}\right )}{3}\right ) \sqrt {3}}{2 \sqrt {9 \left (x +\frac {2 \sqrt {3}}{3}\right )^{2}-12 \sqrt {3}\, \left (x +\frac {2 \sqrt {3}}{3}\right )+3}}\right )}{32}-\frac {\sqrt {\left (x -\frac {2 \sqrt {3}}{3}\right )^{2}+\frac {4 \sqrt {3}\, \left (x -\frac {2 \sqrt {3}}{3}\right )}{3}+\frac {1}{3}}}{16 \left (x -\frac {2 \sqrt {3}}{3}\right )}+\frac {5 \arctanh \left (\frac {3 \left (\frac {2}{3}+\frac {4 \sqrt {3}\, \left (x -\frac {2 \sqrt {3}}{3}\right )}{3}\right ) \sqrt {3}}{2 \sqrt {9 \left (x -\frac {2 \sqrt {3}}{3}\right )^{2}+12 \sqrt {3}\, \left (x -\frac {2 \sqrt {3}}{3}\right )+3}}\right )}{32}-\frac {\sqrt {\left (x +\frac {2 \sqrt {3}}{3}\right )^{2}-\frac {4 \sqrt {3}\, \left (x +\frac {2 \sqrt {3}}{3}\right )}{3}+\frac {1}{3}}}{16 \left (x +\frac {2 \sqrt {3}}{3}\right )}\)	\(172\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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))-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)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (33) = 66\).
time = 0.32, size = 80, normalized size = 1.86 \begin {gather*} -\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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {\left (x - 1\right ) \left (x + 1\right )} \left (3 x^{2} - 4\right )^{2}}\, dx \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (33) = 66\).
time = 0.02, size = 108, normalized size = 2.51 \begin {gather*} -8 \left (-\frac {5}{256} \ln \left |\left (\sqrt {x^{2}-1}-x\right )^{2}-3\right |+\frac {5}{256} \ln \left |3 \left (\sqrt {x^{2}-1}-x\right )^{2}-1\right |+\frac {-5 \left (\sqrt {x^{2}-1}-x\right )^{2}+3}{32 \left (3 \left (\sqrt {x^{2}-1}-x\right )^{4}-10 \left (\sqrt {x^{2}-1}-x\right )^{2}+3\right )}\right ) \end {gather*}

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]

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))

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{\sqrt {x^2-1}\,{\left (3\,x^2-4\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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