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

Optimal. Leaf size=71 \[ -\frac {3 \sqrt {x^2-1} x}{8 \left (3 x^2-4\right )}+\frac {5}{32} \log \left (-x^2+\sqrt {x^2-1} x+2\right )-\frac {5}{32} \log \left (-3 x^2+3 \sqrt {x^2-1} x+2\right ) \]

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Rubi [A]  time = 0.02, antiderivative size = 43, normalized size of antiderivative = 0.61, 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} \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 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])

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 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]

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*}

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Mathematica [C]  time = 3.25, size = 167, normalized size = 2.35 \begin {gather*} -\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} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/16*(x*Sqrt[-1 + x^2]*(-1/4*(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] - ArcSin[Sqrt[x^2/(4 - 3*x^2)]]))/((x^2/(4 - 3*x^2))^(5/2)*(-1 + x^2)) + (8*x^2*(-1 + x^2)*Hypergeometric2F
1[2, 3, 7/2, x^2/(4 - 3*x^2)])/(-60 + 45*x^2)))/(1 - (3*x^2)/4)^2

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IntegrateAlgebraic [A]  time = 0.14, size = 71, normalized size = 1.00 \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]

IntegrateAlgebraic[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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fricas [A]  time = 0.43, size = 80, normalized size = 1.13 \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/(x^2-1)^(1/2)/(3*x^2-4)^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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giac [A]  time = 0.32, size = 94, normalized size = 1.32 \begin {gather*} \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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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maple [A]  time = 0.24, size = 52, normalized size = 0.73

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/8*x/(3*x^2-4)*(x^2-1)^(1/2)-5/32*ln(-(4*x*(x^2-1)^(1/2)-5*x^2+4)/(3*x^2-4))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (3 \, x^{2} - 4\right )}^{2} \sqrt {x^{2} - 1}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \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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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/(x**2-1)**(1/2)/(3*x**2-4)**2,x)

[Out]

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

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