3.9.99 \(\int \frac {1}{(1+\frac {2 x}{1+x^2})^{3/2}} \, dx\) [899]

Optimal. Leaf size=144 \[ \frac {3 (2+x)}{2 \sqrt {1+\frac {2 x}{1+x^2}}}-\frac {1+x^2}{2 (1+x) \sqrt {1+\frac {2 x}{1+x^2}}}-\frac {3 (1+x) \sinh ^{-1}(x)}{\sqrt {1+x^2} \sqrt {1+\frac {2 x}{1+x^2}}}-\frac {9 (1+x) \tanh ^{-1}\left (\frac {1-x}{\sqrt {2} \sqrt {1+x^2}}\right )}{2 \sqrt {2} \sqrt {1+x^2} \sqrt {1+\frac {2 x}{1+x^2}}} \]

[Out]

3/2*(2+x)/(1+2*x/(x^2+1))^(1/2)+1/2*(-x^2-1)/(1+x)/(1+2*x/(x^2+1))^(1/2)-3*(1+x)*arcsinh(x)/(x^2+1)^(1/2)/(1+2
*x/(x^2+1))^(1/2)-9/4*(1+x)*arctanh(1/2*(1-x)*2^(1/2)/(x^2+1)^(1/2))*2^(1/2)/(x^2+1)^(1/2)/(1+2*x/(x^2+1))^(1/
2)

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Rubi [A]
time = 0.06, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6855, 984, 747, 827, 858, 221, 739, 212} \begin {gather*} \frac {3 (x+2)}{2 \sqrt {\frac {2 x}{x^2+1}+1}}-\frac {x^2+1}{2 (x+1) \sqrt {\frac {2 x}{x^2+1}+1}}-\frac {3 (x+1) \sinh ^{-1}(x)}{\sqrt {x^2+1} \sqrt {\frac {2 x}{x^2+1}+1}}-\frac {9 (x+1) \tanh ^{-1}\left (\frac {1-x}{\sqrt {2} \sqrt {x^2+1}}\right )}{2 \sqrt {2} \sqrt {x^2+1} \sqrt {\frac {2 x}{x^2+1}+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + (2*x)/(1 + x^2))^(-3/2),x]

[Out]

(3*(2 + x))/(2*Sqrt[1 + (2*x)/(1 + x^2)]) - (1 + x^2)/(2*(1 + x)*Sqrt[1 + (2*x)/(1 + x^2)]) - (3*(1 + x)*ArcSi
nh[x])/(Sqrt[1 + x^2]*Sqrt[1 + (2*x)/(1 + x^2)]) - (9*(1 + x)*ArcTanh[(1 - x)/(Sqrt[2]*Sqrt[1 + x^2])])/(2*Sqr
t[2]*Sqrt[1 + x^2]*Sqrt[1 + (2*x)/(1 + x^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 221

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt[a])]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rule 739

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rule 747

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*((a + c*x^2)^p/(e
*(m + 1))), x] - Dist[2*c*(p/(e*(m + 1))), Int[x*(d + e*x)^(m + 1)*(a + c*x^2)^(p - 1), x], x] /; FreeQ[{a, c,
d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && GtQ[p, 0] && (IntegerQ[p] || LtQ[m, -1]) && NeQ[m, -1] &&  !ILtQ[m +
2*p + 1, 0] && IntQuadraticQ[a, 0, c, d, e, m, p, x]

Rule 827

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m
+ 1)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) + e*g*(m + 1)*x)*((a + c*x^2)^p/(e^2*(m + 1)*(m + 2*p + 2))), x] + Di
st[p/(e^2*(m + 1)*(m + 2*p + 2)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^(p - 1)*Simp[g*(2*a*e + 2*a*e*m) + (g*(2*c
*d + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && NeQ[c*d^2 + a*e^2,
0] && RationalQ[p] && p > 0 && (LtQ[m, -1] || EqQ[p, 1] || (IntegerQ[p] &&  !RationalQ[m])) && NeQ[m, -1] &&
!ILtQ[m + 2*p + 1, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 858

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[g/e, Int[(d
+ e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a,
c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !IGtQ[m, 0]

Rule 984

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((d_) + (f_.)*(x_)^2)^(q_.), x_Symbol] :> Dist[(a + b*x + c*x^2)^F
racPart[p]/((4*c)^IntPart[p]*(b + 2*c*x)^(2*FracPart[p])), Int[(b + 2*c*x)^(2*p)*(d + f*x^2)^q, x], x] /; Free
Q[{a, b, c, d, f, p, q}, x] && EqQ[b^2 - 4*a*c, 0] &&  !IntegerQ[p]

Rule 6855

Int[(u_.)*((a_.) + (b_.)*(v_)^(n_)*(x_)^(m_.))^(p_), x_Symbol] :> Dist[(a + b*x^m*v^n)^FracPart[p]/(v^(n*FracP
art[p])*(b*x^m + a/v^n)^FracPart[p]), Int[u*v^(n*p)*(b*x^m + a/v^n)^p, x], x] /; FreeQ[{a, b, m, p}, x] &&  !I
ntegerQ[p] && ILtQ[n, 0] && BinomialQ[v, x]

Rubi steps

\begin {align*} \int \frac {1}{\left (1+\frac {2 x}{1+x^2}\right )^{3/2}} \, dx &=\frac {\sqrt {1+2 x+x^2} \int \frac {\left (1+x^2\right )^{3/2}}{\left (1+2 x+x^2\right )^{3/2}} \, dx}{\sqrt {1+x^2} \sqrt {1+\frac {2 x}{1+x^2}}}\\ &=\frac {(4 (2+2 x)) \int \frac {\left (1+x^2\right )^{3/2}}{(2+2 x)^3} \, dx}{\sqrt {1+x^2} \sqrt {1+\frac {2 x}{1+x^2}}}\\ &=-\frac {1+x^2}{2 (1+x) \sqrt {1+\frac {2 x}{1+x^2}}}+\frac {(3 (2+2 x)) \int \frac {x \sqrt {1+x^2}}{(2+2 x)^2} \, dx}{\sqrt {1+x^2} \sqrt {1+\frac {2 x}{1+x^2}}}\\ &=\frac {3 (2+x)}{2 \sqrt {1+\frac {2 x}{1+x^2}}}-\frac {1+x^2}{2 (1+x) \sqrt {1+\frac {2 x}{1+x^2}}}-\frac {(3 (2+2 x)) \int \frac {-4+8 x}{(2+2 x) \sqrt {1+x^2}} \, dx}{8 \sqrt {1+x^2} \sqrt {1+\frac {2 x}{1+x^2}}}\\ &=\frac {3 (2+x)}{2 \sqrt {1+\frac {2 x}{1+x^2}}}-\frac {1+x^2}{2 (1+x) \sqrt {1+\frac {2 x}{1+x^2}}}-\frac {(3 (2+2 x)) \int \frac {1}{\sqrt {1+x^2}} \, dx}{2 \sqrt {1+x^2} \sqrt {1+\frac {2 x}{1+x^2}}}+\frac {(9 (2+2 x)) \int \frac {1}{(2+2 x) \sqrt {1+x^2}} \, dx}{2 \sqrt {1+x^2} \sqrt {1+\frac {2 x}{1+x^2}}}\\ &=\frac {3 (2+x)}{2 \sqrt {1+\frac {2 x}{1+x^2}}}-\frac {1+x^2}{2 (1+x) \sqrt {1+\frac {2 x}{1+x^2}}}-\frac {3 (1+x) \sinh ^{-1}(x)}{\sqrt {1+x^2} \sqrt {1+\frac {2 x}{1+x^2}}}-\frac {(9 (2+2 x)) \text {Subst}\left (\int \frac {1}{8-x^2} \, dx,x,\frac {2-2 x}{\sqrt {1+x^2}}\right )}{2 \sqrt {1+x^2} \sqrt {1+\frac {2 x}{1+x^2}}}\\ &=\frac {3 (2+x)}{2 \sqrt {1+\frac {2 x}{1+x^2}}}-\frac {1+x^2}{2 (1+x) \sqrt {1+\frac {2 x}{1+x^2}}}-\frac {3 (1+x) \sinh ^{-1}(x)}{\sqrt {1+x^2} \sqrt {1+\frac {2 x}{1+x^2}}}-\frac {9 (1+x) \tanh ^{-1}\left (\frac {1-x}{\sqrt {2} \sqrt {1+x^2}}\right )}{2 \sqrt {2} \sqrt {1+x^2} \sqrt {1+\frac {2 x}{1+x^2}}}\\ \end {align*}

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Mathematica [A]
time = 0.30, size = 106, normalized size = 0.74 \begin {gather*} \frac {(1+x) \left (\sqrt {1+x^2} \left (5+9 x+2 x^2\right )-6 (1+x)^2 \tanh ^{-1}\left (\frac {x}{\sqrt {1+x^2}}\right )+9 \sqrt {2} (1+x)^2 \tanh ^{-1}\left (\frac {1+x-\sqrt {1+x^2}}{\sqrt {2}}\right )\right )}{2 \left (\frac {(1+x)^2}{1+x^2}\right )^{3/2} \left (1+x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((1 + x)*(Sqrt[1 + x^2]*(5 + 9*x + 2*x^2) - 6*(1 + x)^2*ArcTanh[x/Sqrt[1 + x^2]] + 9*Sqrt[2]*(1 + x)^2*ArcTanh
[(1 + x - Sqrt[1 + x^2])/Sqrt[2]]))/(2*((1 + x)^2/(1 + x^2))^(3/2)*(1 + x^2)^(3/2))

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Maple [A]
time = 0.16, size = 217, normalized size = 1.51

method	result	size
risch	\(\frac {2 x^{4}+9 x^{3}+7 x^{2}+9 x +5}{2 \left (1+x \right ) \left (x^{2}+1\right ) \sqrt {\frac {\left (1+x \right )^{2}}{x^{2}+1}}}+\frac {\left (-3 \arcsinh \left (x \right )-\frac {9 \sqrt {2}\, \arctanh \left (\frac {\left (2-2 x \right ) \sqrt {2}}{4 \sqrt {\left (1+x \right )^{2}-2 x}}\right )}{4}\right ) \left (1+x \right )}{\sqrt {\frac {\left (1+x \right )^{2}}{x^{2}+1}}\, \sqrt {x^{2}+1}}\)	\(109\)
trager	\(\frac {\left (x^{2}+1\right ) \left (2 x^{2}+9 x +5\right ) \sqrt {-\frac {-x^{2}-2 x -1}{x^{2}+1}}}{2 \left (1+x \right )^{3}}+3 \ln \left (-\frac {\sqrt {-\frac {-x^{2}-2 x -1}{x^{2}+1}}\, x^{2}-x^{2}+\sqrt {-\frac {-x^{2}-2 x -1}{x^{2}+1}}-x}{1+x}\right )+\frac {9 \RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}-2\right ) x^{2}+2 \sqrt {-\frac {-x^{2}-2 x -1}{x^{2}+1}}\, x^{2}-\RootOf \left (\textit {\_Z}^{2}-2\right )+2 \sqrt {-\frac {-x^{2}-2 x -1}{x^{2}+1}}}{\left (1+x \right )^{2}}\right )}{4}\)	\(193\)
default	\(-\frac {\left (1+x \right ) \left (-\left (x^{2}+1\right )^{\frac {5}{2}} x +\left (x^{2}+1\right )^{\frac {3}{2}} x^{3}+\left (x^{2}+1\right )^{\frac {5}{2}}-\left (x^{2}+1\right )^{\frac {3}{2}} x^{2}-18 \arctanh \left (\frac {\left (-1+x \right ) \sqrt {2}}{2 \sqrt {x^{2}+1}}\right ) \sqrt {2}\, x^{2}-5 x \left (x^{2}+1\right )^{\frac {3}{2}}+6 \sqrt {x^{2}+1}\, x^{3}+24 \arcsinh \left (x \right ) x^{2}-36 \arctanh \left (\frac {\left (-1+x \right ) \sqrt {2}}{2 \sqrt {x^{2}+1}}\right ) \sqrt {2}\, x -3 \left (x^{2}+1\right )^{\frac {3}{2}}-6 \sqrt {x^{2}+1}\, x^{2}+48 \arcsinh \left (x \right ) x -18 \sqrt {2}\, \arctanh \left (\frac {\left (-1+x \right ) \sqrt {2}}{2 \sqrt {x^{2}+1}}\right )-30 x \sqrt {x^{2}+1}+24 \arcsinh \left (x \right )-18 \sqrt {x^{2}+1}\right )}{8 \left (\frac {x^{2}+2 x +1}{x^{2}+1}\right )^{\frac {3}{2}} \left (x^{2}+1\right )^{\frac {3}{2}}}\)	\(217\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8/((x^2+2*x+1)/(x^2+1))^(3/2)*(1+x)*(-(x^2+1)^(5/2)*x+(x^2+1)^(3/2)*x^3+(x^2+1)^(5/2)-(x^2+1)^(3/2)*x^2-18*
arctanh(1/2*(-1+x)*2^(1/2)/(x^2+1)^(1/2))*2^(1/2)*x^2-5*x*(x^2+1)^(3/2)+6*(x^2+1)^(1/2)*x^3+24*arcsinh(x)*x^2-
36*arctanh(1/2*(-1+x)*2^(1/2)/(x^2+1)^(1/2))*2^(1/2)*x-3*(x^2+1)^(3/2)-6*(x^2+1)^(1/2)*x^2+48*arcsinh(x)*x-18*
2^(1/2)*arctanh(1/2*(-1+x)*2^(1/2)/(x^2+1)^(1/2))-30*x*(x^2+1)^(1/2)+24*arcsinh(x)-18*(x^2+1)^(1/2))/(x^2+1)^(
3/2)

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

[Out]

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

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Fricas [A]
time = 0.33, size = 205, normalized size = 1.42 \begin {gather*} \frac {10 \, x^{3} + 9 \, \sqrt {2} {\left (x^{3} + 3 \, x^{2} + 3 \, x + 1\right )} \log \left (-\frac {x^{2} + \sqrt {2} {\left (x^{2} - 1\right )} + {\left (2 \, x^{2} + \sqrt {2} {\left (x^{2} + 1\right )} + 2\right )} \sqrt {\frac {x^{2} + 2 \, x + 1}{x^{2} + 1}} - 1}{x^{2} + 2 \, x + 1}\right ) + 30 \, x^{2} + 12 \, {\left (x^{3} + 3 \, x^{2} + 3 \, x + 1\right )} \log \left (-\frac {x^{2} - {\left (x^{2} + 1\right )} \sqrt {\frac {x^{2} + 2 \, x + 1}{x^{2} + 1}} + x}{x + 1}\right ) + 2 \, {\left (2 \, x^{4} + 9 \, x^{3} + 7 \, x^{2} + 9 \, x + 5\right )} \sqrt {\frac {x^{2} + 2 \, x + 1}{x^{2} + 1}} + 30 \, x + 10}{4 \, {\left (x^{3} + 3 \, x^{2} + 3 \, x + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(10*x^3 + 9*sqrt(2)*(x^3 + 3*x^2 + 3*x + 1)*log(-(x^2 + sqrt(2)*(x^2 - 1) + (2*x^2 + sqrt(2)*(x^2 + 1) + 2
)*sqrt((x^2 + 2*x + 1)/(x^2 + 1)) - 1)/(x^2 + 2*x + 1)) + 30*x^2 + 12*(x^3 + 3*x^2 + 3*x + 1)*log(-(x^2 - (x^2
+ 1)*sqrt((x^2 + 2*x + 1)/(x^2 + 1)) + x)/(x + 1)) + 2*(2*x^4 + 9*x^3 + 7*x^2 + 9*x + 5)*sqrt((x^2 + 2*x + 1)
/(x^2 + 1)) + 30*x + 10)/(x^3 + 3*x^2 + 3*x + 1)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (\frac {2 x}{x^{2} + 1} + 1\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+2*x/(x**2+1))**(3/2),x)

[Out]

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

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Giac [A]
time = 5.33, size = 170, normalized size = 1.18 \begin {gather*} \frac {9 \, \sqrt {2} \log \left (\frac {{\left | -2 \, x - 2 \, \sqrt {2} + 2 \, \sqrt {x^{2} + 1} - 2 \right |}}{{\left | -2 \, x + 2 \, \sqrt {2} + 2 \, \sqrt {x^{2} + 1} - 2 \right |}}\right )}{4 \, \mathrm {sgn}\left (x + 1\right )} + \frac {3 \, \log \left (-x + \sqrt {x^{2} + 1}\right )}{\mathrm {sgn}\left (x + 1\right )} + \frac {\sqrt {x^{2} + 1}}{\mathrm {sgn}\left (x + 1\right )} + \frac {7 \, {\left (x - \sqrt {x^{2} + 1}\right )}^{3} + 5 \, {\left (x - \sqrt {x^{2} + 1}\right )}^{2} - 13 \, x + 13 \, \sqrt {x^{2} + 1} + 5}{{\left ({\left (x - \sqrt {x^{2} + 1}\right )}^{2} + 2 \, x - 2 \, \sqrt {x^{2} + 1} - 1\right )}^{2} \mathrm {sgn}\left (x + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

9/4*sqrt(2)*log(abs(-2*x - 2*sqrt(2) + 2*sqrt(x^2 + 1) - 2)/abs(-2*x + 2*sqrt(2) + 2*sqrt(x^2 + 1) - 2))/sgn(x
+ 1) + 3*log(-x + sqrt(x^2 + 1))/sgn(x + 1) + sqrt(x^2 + 1)/sgn(x + 1) + (7*(x - sqrt(x^2 + 1))^3 + 5*(x - sq
rt(x^2 + 1))^2 - 13*x + 13*sqrt(x^2 + 1) + 5)/(((x - sqrt(x^2 + 1))^2 + 2*x - 2*sqrt(x^2 + 1) - 1)^2*sgn(x + 1
))

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{{\left (\frac {2\,x}{x^2+1}+1\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((2*x)/(x^2 + 1) + 1)^(3/2),x)

[Out]

int(1/((2*x)/(x^2 + 1) + 1)^(3/2), x)

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