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

Optimal. Leaf size=129 \[ \frac{7 \sqrt{x}}{16 \left (x^2+1\right )}+\frac{\sqrt{x}}{4 \left (x^2+1\right )^2}-\frac{21 \log \left (x-\sqrt{2} \sqrt{x}+1\right )}{64 \sqrt{2}}+\frac{21 \log \left (x+\sqrt{2} \sqrt{x}+1\right )}{64 \sqrt{2}}-\frac{21 \tan ^{-1}\left (1-\sqrt{2} \sqrt{x}\right )}{32 \sqrt{2}}+\frac{21 \tan ^{-1}\left (\sqrt{2} \sqrt{x}+1\right )}{32 \sqrt{2}} \]

[Out]

Sqrt[x]/(4*(1 + x^2)^2) + (7*Sqrt[x])/(16*(1 + x^2)) - (21*ArcTan[1 - Sqrt[2]*Sqrt[x]])/(32*Sqrt[2]) + (21*Arc
Tan[1 + Sqrt[2]*Sqrt[x]])/(32*Sqrt[2]) - (21*Log[1 - Sqrt[2]*Sqrt[x] + x])/(64*Sqrt[2]) + (21*Log[1 + Sqrt[2]*
Sqrt[x] + x])/(64*Sqrt[2])

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Rubi [A]  time = 0.0682552, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.615, Rules used = {290, 329, 211, 1165, 628, 1162, 617, 204} \[ \frac{7 \sqrt{x}}{16 \left (x^2+1\right )}+\frac{\sqrt{x}}{4 \left (x^2+1\right )^2}-\frac{21 \log \left (x-\sqrt{2} \sqrt{x}+1\right )}{64 \sqrt{2}}+\frac{21 \log \left (x+\sqrt{2} \sqrt{x}+1\right )}{64 \sqrt{2}}-\frac{21 \tan ^{-1}\left (1-\sqrt{2} \sqrt{x}\right )}{32 \sqrt{2}}+\frac{21 \tan ^{-1}\left (\sqrt{2} \sqrt{x}+1\right )}{32 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[x]*(1 + x^2)^3),x]

[Out]

Sqrt[x]/(4*(1 + x^2)^2) + (7*Sqrt[x])/(16*(1 + x^2)) - (21*ArcTan[1 - Sqrt[2]*Sqrt[x]])/(32*Sqrt[2]) + (21*Arc
Tan[1 + Sqrt[2]*Sqrt[x]])/(32*Sqrt[2]) - (21*Log[1 - Sqrt[2]*Sqrt[x] + x])/(64*Sqrt[2]) + (21*Log[1 + Sqrt[2]*
Sqrt[x] + x])/(64*Sqrt[2])

Rule 290

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(
a*c*n*(p + 1)), x] + Dist[(m + n*(p + 1) + 1)/(a*n*(p + 1)), Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[
{a, b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 329

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/c^n)^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 211

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Di
st[1/(2*r), Int[(r - s*x^2)/(a + b*x^4), x], x] + Dist[1/(2*r), Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[
{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b
]]))

Rule 1165

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1162

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

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

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{x} \left (1+x^2\right )^3} \, dx &=\frac{\sqrt{x}}{4 \left (1+x^2\right )^2}+\frac{7}{8} \int \frac{1}{\sqrt{x} \left (1+x^2\right )^2} \, dx\\ &=\frac{\sqrt{x}}{4 \left (1+x^2\right )^2}+\frac{7 \sqrt{x}}{16 \left (1+x^2\right )}+\frac{21}{32} \int \frac{1}{\sqrt{x} \left (1+x^2\right )} \, dx\\ &=\frac{\sqrt{x}}{4 \left (1+x^2\right )^2}+\frac{7 \sqrt{x}}{16 \left (1+x^2\right )}+\frac{21}{16} \operatorname{Subst}\left (\int \frac{1}{1+x^4} \, dx,x,\sqrt{x}\right )\\ &=\frac{\sqrt{x}}{4 \left (1+x^2\right )^2}+\frac{7 \sqrt{x}}{16 \left (1+x^2\right )}+\frac{21}{32} \operatorname{Subst}\left (\int \frac{1-x^2}{1+x^4} \, dx,x,\sqrt{x}\right )+\frac{21}{32} \operatorname{Subst}\left (\int \frac{1+x^2}{1+x^4} \, dx,x,\sqrt{x}\right )\\ &=\frac{\sqrt{x}}{4 \left (1+x^2\right )^2}+\frac{7 \sqrt{x}}{16 \left (1+x^2\right )}+\frac{21}{64} \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{2} x+x^2} \, dx,x,\sqrt{x}\right )+\frac{21}{64} \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{2} x+x^2} \, dx,x,\sqrt{x}\right )-\frac{21 \operatorname{Subst}\left (\int \frac{\sqrt{2}+2 x}{-1-\sqrt{2} x-x^2} \, dx,x,\sqrt{x}\right )}{64 \sqrt{2}}-\frac{21 \operatorname{Subst}\left (\int \frac{\sqrt{2}-2 x}{-1+\sqrt{2} x-x^2} \, dx,x,\sqrt{x}\right )}{64 \sqrt{2}}\\ &=\frac{\sqrt{x}}{4 \left (1+x^2\right )^2}+\frac{7 \sqrt{x}}{16 \left (1+x^2\right )}-\frac{21 \log \left (1-\sqrt{2} \sqrt{x}+x\right )}{64 \sqrt{2}}+\frac{21 \log \left (1+\sqrt{2} \sqrt{x}+x\right )}{64 \sqrt{2}}+\frac{21 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt{2} \sqrt{x}\right )}{32 \sqrt{2}}-\frac{21 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\sqrt{2} \sqrt{x}\right )}{32 \sqrt{2}}\\ &=\frac{\sqrt{x}}{4 \left (1+x^2\right )^2}+\frac{7 \sqrt{x}}{16 \left (1+x^2\right )}-\frac{21 \tan ^{-1}\left (1-\sqrt{2} \sqrt{x}\right )}{32 \sqrt{2}}+\frac{21 \tan ^{-1}\left (1+\sqrt{2} \sqrt{x}\right )}{32 \sqrt{2}}-\frac{21 \log \left (1-\sqrt{2} \sqrt{x}+x\right )}{64 \sqrt{2}}+\frac{21 \log \left (1+\sqrt{2} \sqrt{x}+x\right )}{64 \sqrt{2}}\\ \end{align*}

Mathematica [A]  time = 0.0313471, size = 121, normalized size = 0.94 \[ \frac{1}{128} \left (\frac{56 \sqrt{x}}{x^2+1}+\frac{32 \sqrt{x}}{\left (x^2+1\right )^2}-21 \sqrt{2} \log \left (x-\sqrt{2} \sqrt{x}+1\right )+21 \sqrt{2} \log \left (x+\sqrt{2} \sqrt{x}+1\right )-42 \sqrt{2} \tan ^{-1}\left (1-\sqrt{2} \sqrt{x}\right )+42 \sqrt{2} \tan ^{-1}\left (\sqrt{2} \sqrt{x}+1\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[x]*(1 + x^2)^3),x]

[Out]

((32*Sqrt[x])/(1 + x^2)^2 + (56*Sqrt[x])/(1 + x^2) - 42*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[x]] + 42*Sqrt[2]*ArcTa
n[1 + Sqrt[2]*Sqrt[x]] - 21*Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[x] + x] + 21*Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[x] + x])/12
8

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Maple [A]  time = 0.004, size = 86, normalized size = 0.7 \begin{align*}{\frac{1}{4\, \left ({x}^{2}+1 \right ) ^{2}}\sqrt{x}}+{\frac{7}{16\,{x}^{2}+16}\sqrt{x}}+{\frac{21\,\sqrt{2}}{64}\arctan \left ( 1+\sqrt{2}\sqrt{x} \right ) }+{\frac{21\,\sqrt{2}}{64}\arctan \left ( -1+\sqrt{2}\sqrt{x} \right ) }+{\frac{21\,\sqrt{2}}{128}\ln \left ({ \left ( 1+x+\sqrt{2}\sqrt{x} \right ) \left ( 1+x-\sqrt{2}\sqrt{x} \right ) ^{-1}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2+1)^3/x^(1/2),x)

[Out]

1/4*x^(1/2)/(x^2+1)^2+7/16*x^(1/2)/(x^2+1)+21/64*arctan(1+2^(1/2)*x^(1/2))*2^(1/2)+21/64*arctan(-1+2^(1/2)*x^(
1/2))*2^(1/2)+21/128*2^(1/2)*ln((1+x+2^(1/2)*x^(1/2))/(1+x-2^(1/2)*x^(1/2)))

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Maxima [A]  time = 2.96133, size = 134, normalized size = 1.04 \begin{align*} \frac{21}{64} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, \sqrt{x}\right )}\right ) + \frac{21}{64} \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, \sqrt{x}\right )}\right ) + \frac{21}{128} \, \sqrt{2} \log \left (\sqrt{2} \sqrt{x} + x + 1\right ) - \frac{21}{128} \, \sqrt{2} \log \left (-\sqrt{2} \sqrt{x} + x + 1\right ) + \frac{7 \, x^{\frac{5}{2}} + 11 \, \sqrt{x}}{16 \,{\left (x^{4} + 2 \, x^{2} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

21/64*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(x))) + 21/64*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*sqrt(
x))) + 21/128*sqrt(2)*log(sqrt(2)*sqrt(x) + x + 1) - 21/128*sqrt(2)*log(-sqrt(2)*sqrt(x) + x + 1) + 1/16*(7*x^
(5/2) + 11*sqrt(x))/(x^4 + 2*x^2 + 1)

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Fricas [A]  time = 1.47195, size = 522, normalized size = 4.05 \begin{align*} -\frac{84 \, \sqrt{2}{\left (x^{4} + 2 \, x^{2} + 1\right )} \arctan \left (\sqrt{2} \sqrt{\sqrt{2} \sqrt{x} + x + 1} - \sqrt{2} \sqrt{x} - 1\right ) + 84 \, \sqrt{2}{\left (x^{4} + 2 \, x^{2} + 1\right )} \arctan \left (\frac{1}{2} \, \sqrt{2} \sqrt{-4 \, \sqrt{2} \sqrt{x} + 4 \, x + 4} - \sqrt{2} \sqrt{x} + 1\right ) - 21 \, \sqrt{2}{\left (x^{4} + 2 \, x^{2} + 1\right )} \log \left (4 \, \sqrt{2} \sqrt{x} + 4 \, x + 4\right ) + 21 \, \sqrt{2}{\left (x^{4} + 2 \, x^{2} + 1\right )} \log \left (-4 \, \sqrt{2} \sqrt{x} + 4 \, x + 4\right ) - 8 \,{\left (7 \, x^{2} + 11\right )} \sqrt{x}}{128 \,{\left (x^{4} + 2 \, x^{2} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/128*(84*sqrt(2)*(x^4 + 2*x^2 + 1)*arctan(sqrt(2)*sqrt(sqrt(2)*sqrt(x) + x + 1) - sqrt(2)*sqrt(x) - 1) + 84*
sqrt(2)*(x^4 + 2*x^2 + 1)*arctan(1/2*sqrt(2)*sqrt(-4*sqrt(2)*sqrt(x) + 4*x + 4) - sqrt(2)*sqrt(x) + 1) - 21*sq
rt(2)*(x^4 + 2*x^2 + 1)*log(4*sqrt(2)*sqrt(x) + 4*x + 4) + 21*sqrt(2)*(x^4 + 2*x^2 + 1)*log(-4*sqrt(2)*sqrt(x)
+ 4*x + 4) - 8*(7*x^2 + 11)*sqrt(x))/(x^4 + 2*x^2 + 1)

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Sympy [B]  time = 14.4115, size = 481, normalized size = 3.73 \begin{align*} \frac{56 x^{\frac{5}{2}}}{128 x^{4} + 256 x^{2} + 128} + \frac{88 \sqrt{x}}{128 x^{4} + 256 x^{2} + 128} - \frac{21 \sqrt{2} x^{4} \log{\left (- 4 \sqrt{2} \sqrt{x} + 4 x + 4 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac{21 \sqrt{2} x^{4} \log{\left (4 \sqrt{2} \sqrt{x} + 4 x + 4 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac{42 \sqrt{2} x^{4} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} - 1 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac{42 \sqrt{2} x^{4} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} + 1 \right )}}{128 x^{4} + 256 x^{2} + 128} - \frac{42 \sqrt{2} x^{2} \log{\left (- 4 \sqrt{2} \sqrt{x} + 4 x + 4 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac{42 \sqrt{2} x^{2} \log{\left (4 \sqrt{2} \sqrt{x} + 4 x + 4 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac{84 \sqrt{2} x^{2} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} - 1 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac{84 \sqrt{2} x^{2} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} + 1 \right )}}{128 x^{4} + 256 x^{2} + 128} - \frac{21 \sqrt{2} \log{\left (- 4 \sqrt{2} \sqrt{x} + 4 x + 4 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac{21 \sqrt{2} \log{\left (4 \sqrt{2} \sqrt{x} + 4 x + 4 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac{42 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} - 1 \right )}}{128 x^{4} + 256 x^{2} + 128} + \frac{42 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} \sqrt{x} + 1 \right )}}{128 x^{4} + 256 x^{2} + 128} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

56*x**(5/2)/(128*x**4 + 256*x**2 + 128) + 88*sqrt(x)/(128*x**4 + 256*x**2 + 128) - 21*sqrt(2)*x**4*log(-4*sqrt
(2)*sqrt(x) + 4*x + 4)/(128*x**4 + 256*x**2 + 128) + 21*sqrt(2)*x**4*log(4*sqrt(2)*sqrt(x) + 4*x + 4)/(128*x**
4 + 256*x**2 + 128) + 42*sqrt(2)*x**4*atan(sqrt(2)*sqrt(x) - 1)/(128*x**4 + 256*x**2 + 128) + 42*sqrt(2)*x**4*
atan(sqrt(2)*sqrt(x) + 1)/(128*x**4 + 256*x**2 + 128) - 42*sqrt(2)*x**2*log(-4*sqrt(2)*sqrt(x) + 4*x + 4)/(128
*x**4 + 256*x**2 + 128) + 42*sqrt(2)*x**2*log(4*sqrt(2)*sqrt(x) + 4*x + 4)/(128*x**4 + 256*x**2 + 128) + 84*sq
rt(2)*x**2*atan(sqrt(2)*sqrt(x) - 1)/(128*x**4 + 256*x**2 + 128) + 84*sqrt(2)*x**2*atan(sqrt(2)*sqrt(x) + 1)/(
128*x**4 + 256*x**2 + 128) - 21*sqrt(2)*log(-4*sqrt(2)*sqrt(x) + 4*x + 4)/(128*x**4 + 256*x**2 + 128) + 21*sqr
t(2)*log(4*sqrt(2)*sqrt(x) + 4*x + 4)/(128*x**4 + 256*x**2 + 128) + 42*sqrt(2)*atan(sqrt(2)*sqrt(x) - 1)/(128*
x**4 + 256*x**2 + 128) + 42*sqrt(2)*atan(sqrt(2)*sqrt(x) + 1)/(128*x**4 + 256*x**2 + 128)

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Giac [A]  time = 3.14375, size = 127, normalized size = 0.98 \begin{align*} \frac{21}{64} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, \sqrt{x}\right )}\right ) + \frac{21}{64} \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, \sqrt{x}\right )}\right ) + \frac{21}{128} \, \sqrt{2} \log \left (\sqrt{2} \sqrt{x} + x + 1\right ) - \frac{21}{128} \, \sqrt{2} \log \left (-\sqrt{2} \sqrt{x} + x + 1\right ) + \frac{7 \, x^{\frac{5}{2}} + 11 \, \sqrt{x}}{16 \,{\left (x^{2} + 1\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

21/64*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(x))) + 21/64*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*sqrt(
x))) + 21/128*sqrt(2)*log(sqrt(2)*sqrt(x) + x + 1) - 21/128*sqrt(2)*log(-sqrt(2)*sqrt(x) + x + 1) + 1/16*(7*x^
(5/2) + 11*sqrt(x))/(x^2 + 1)^2