3.1.7 \(\int \frac {1}{\sqrt {-1+x^2} (1+x^2)^2} \, dx\) [7]

Optimal. Leaf size=48 \[ -\frac {x \sqrt {-1+x^2}}{4 \left (1+x^2\right )}+\frac {3 \tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )}{4 \sqrt {2}} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {390, 385, 212} \begin {gather*} \frac {3 \tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^2-1}}\right )}{4 \sqrt {2}}-\frac {x \sqrt {x^2-1}}{4 \left (x^2+1\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[-1 + x^2]*(1 + x^2)^2),x]

[Out]

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

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[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 (1+x^2\right )^2} \, dx &=-\frac {x \sqrt {-1+x^2}}{4 \left (1+x^2\right )}+\frac {3}{4} \int \frac {1}{\sqrt {-1+x^2} \left (1+x^2\right )} \, dx\\ &=-\frac {x \sqrt {-1+x^2}}{4 \left (1+x^2\right )}+\frac {3}{4} \text {Subst}\left (\int \frac {1}{1-2 x^2} \, dx,x,\frac {x}{\sqrt {-1+x^2}}\right )\\ &=-\frac {x \sqrt {-1+x^2}}{4 \left (1+x^2\right )}+\frac {3 \tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )}{4 \sqrt {2}}\\ \end {align*}

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Mathematica [A]
time = 0.11, size = 55, normalized size = 1.15 \begin {gather*} \frac {1}{8} \left (-\frac {2 x \sqrt {-1+x^2}}{1+x^2}+3 \sqrt {2} \tanh ^{-1}\left (\frac {1+x^2-x \sqrt {-1+x^2}}{\sqrt {2}}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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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]*(x^2 + 1)^2),x]')

[Out]

cought exception: maximum recursion depth exceeded

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Maple [A]
time = 0.14, size = 45, normalized size = 0.94

method	result	size
risch	\(\frac {3 \arctanh \left (\frac {x \sqrt {2}}{\sqrt {x^{2}-1}}\right ) \sqrt {2}}{8}-\frac {x \sqrt {x^{2}-1}}{4 \left (x^{2}+1\right )}\)	\(37\)
default	\(-\frac {x}{8 \sqrt {x^{2}-1}\, \left (\frac {x^{2}}{x^{2}-1}-\frac {1}{2}\right )}+\frac {3 \arctanh \left (\frac {x \sqrt {2}}{\sqrt {x^{2}-1}}\right ) \sqrt {2}}{8}\)	\(45\)
trager	\(-\frac {x \sqrt {x^{2}-1}}{4 \left (x^{2}+1\right )}+\frac {3 \RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (\frac {3 \RootOf \left (\textit {\_Z}^{2}-2\right ) x^{2}+4 \sqrt {x^{2}-1}\, x -\RootOf \left (\textit {\_Z}^{2}-2\right )}{x^{2}+1}\right )}{16}\)	\(66\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2+1)^2/(x^2-1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/8*x/(x^2-1)^(1/2)/(x^2/(x^2-1)-1/2)+3/8*arctanh(x*2^(1/2)/(x^2-1)^(1/2))*2^(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/(x^2+1)^2/(x^2-1)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/((x^2 + 1)^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. 83 vs. \(2 (36) = 72\).
time = 0.31, size = 83, normalized size = 1.73 \begin {gather*} \frac {3 \, \sqrt {2} {\left (x^{2} + 1\right )} \log \left (\frac {9 \, x^{2} + 2 \, \sqrt {2} {\left (3 \, x^{2} - 1\right )} + 2 \, \sqrt {x^{2} - 1} {\left (3 \, \sqrt {2} x + 4 \, x\right )} - 3}{x^{2} + 1}\right ) - 4 \, x^{2} - 4 \, \sqrt {x^{2} - 1} x - 4}{16 \, {\left (x^{2} + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^2+1)^2/(x^2-1)^(1/2),x, algorithm="fricas")

[Out]

1/16*(3*sqrt(2)*(x^2 + 1)*log((9*x^2 + 2*sqrt(2)*(3*x^2 - 1) + 2*sqrt(x^2 - 1)*(3*sqrt(2)*x + 4*x) - 3)/(x^2 +
1)) - 4*x^2 - 4*sqrt(x^2 - 1)*x - 4)/(x^2 + 1)

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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 (x^{2} + 1\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x**2+1)**2/(x**2-1)**(1/2),x)

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 101 vs. \(2 (36) = 72\).
time = 0.00, size = 122, normalized size = 2.54 \begin {gather*} -8 \left (\frac {3 \left (\sqrt {x^{2}-1}-x\right )^{2}+1}{16 \left (\left (\sqrt {x^{2}-1}-x\right )^{4}+6 \left (\sqrt {x^{2}-1}-x\right )^{2}+1\right )}+\frac {3 \ln \left (\frac {2 \left (\sqrt {x^{2}-1}-x\right )^{2}+6-4 \sqrt {2}}{2 \left (\sqrt {x^{2}-1}-x\right )^{2}+6+4 \sqrt {2}}\right )}{64 \sqrt {2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^2+1)^2/(x^2-1)^(1/2),x)

[Out]

-3/16*sqrt(2)*log(((x - sqrt(x^2 - 1))^2 - 2*sqrt(2) + 3)/((x - sqrt(x^2 - 1))^2 + 2*sqrt(2) + 3)) - 1/2*(3*(x
- sqrt(x^2 - 1))^2 + 1)/((x - sqrt(x^2 - 1))^4 + 6*(x - sqrt(x^2 - 1))^2 + 1)

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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 (x^2+1\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((x^2 - 1)^(1/2)*(x^2 + 1)^2),x)

[Out]

int(1/((x^2 - 1)^(1/2)*(x^2 + 1)^2), x)

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