∫ √ _x ____√1+x(1+x2 ) dx

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

Optimal. Leaf size=65 \[ -\frac {1}{2} (1-i)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {1-i} \sqrt {x}}{\sqrt {x+1}}\right )-\frac {1}{2} (1+i)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {1+i} \sqrt {x}}{\sqrt {x+1}}\right ) \]

[Out]

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

/2))

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Rubi [A] time = 0.05, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {910, 93, 208} \[ -\frac {1}{2} (1-i)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {1-i} \sqrt {x}}{\sqrt {x+1}}\right )-\frac {1}{2} (1+i)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {1+i} \sqrt {x}}{\sqrt {x+1}}\right ) \]

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

qrt[1 + x]])/2

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin

ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^

(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[

a + b*x, c + d*x]

Rule 208

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

b}, x] && NegQ[a/b]

Rule 910

Int[((d_.) + (e_.)*(x_))^(m_)/(Sqrt[(f_.) + (g_.)*(x_)]*((a_.) + (c_.)*(x_)^2)), x_Symbol] :> Int[ExpandIntegr

and[1/(Sqrt[d + e*x]*Sqrt[f + g*x]), (d + e*x)^(m + 1/2)/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g}, x] &

& NeQ[c*d^2 + a*e^2, 0] && IGtQ[m + 1/2, 0]

Rubi steps

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

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Mathematica [A] time = 0.06, size = 63, normalized size = 0.97 \[ \frac {1}{2} \left (-(-1+i)^{3/2} \tan ^{-1}\left (\sqrt {-1+i} \sqrt {\frac {x}{x+1}}\right )-(1+i)^{3/2} \tanh ^{-1}\left (\sqrt {1+i} \sqrt {\frac {x}{x+1}}\right )\right ) \]

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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fricas [B] time = 0.75, size = 744, normalized size = 11.45 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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

- 2) - 2^(1/4)*(sqrt(2)*(x + 1) - 2*x))*sqrt(2*sqrt(2) + 4) + 4*x + 4*sqrt(2) + 4) - 1/2*2^(1/4)*sqrt(2*sqrt(

2) + 4)*arctan(1/7*(sqrt(2)*(5*sqrt(2) + 6) + 8*sqrt(2) + 4)*sqrt(x + 1)*sqrt(x) - 1/7*sqrt(2)*(sqrt(2)*(5*x +

1) + 6*x + 4) - 1/28*sqrt(-8*sqrt(x + 1)*x^(3/2) + 8*x^2 - 2*(2^(1/4)*sqrt(x + 1)*sqrt(x)*(sqrt(2) - 2) - 2^(

1/4)*(sqrt(2)*(x + 1) - 2*x))*sqrt(2*sqrt(2) + 4) + 4*x + 4*sqrt(2) + 4)*(2*sqrt(2)*(5*sqrt(2) + 6) - (2^(3/4)

*(3*sqrt(2) + 5) + 2*2^(1/4)*(sqrt(2) + 4))*sqrt(2*sqrt(2) + 4) + 16*sqrt(2) + 8) - 1/7*sqrt(2)*(8*x + 3) - 1/

14*((2^(3/4)*(3*sqrt(2) + 5) + 2*2^(1/4)*(sqrt(2) + 4))*sqrt(x + 1)*sqrt(x) - 2^(3/4)*(sqrt(2)*(3*x + 2) + 5*x

+ 1) - 2*2^(1/4)*(sqrt(2)*(x + 3) + 4*x - 2))*sqrt(2*sqrt(2) + 4) - 4/7*x - 5/7) - 1/2*2^(1/4)*sqrt(2*sqrt(2)

+ 4)*arctan(-1/7*(sqrt(2)*(5*sqrt(2) + 6) + 8*sqrt(2) + 4)*sqrt(x + 1)*sqrt(x) + 1/7*sqrt(2)*(sqrt(2)*(5*x +

1) + 6*x + 4) + 1/28*sqrt(-8*sqrt(x + 1)*x^(3/2) + 8*x^2 + 2*(2^(1/4)*sqrt(x + 1)*sqrt(x)*(sqrt(2) - 2) - 2^(1

/4)*(sqrt(2)*(x + 1) - 2*x))*sqrt(2*sqrt(2) + 4) + 4*x + 4*sqrt(2) + 4)*(2*sqrt(2)*(5*sqrt(2) + 6) + (2^(3/4)*

(3*sqrt(2) + 5) + 2*2^(1/4)*(sqrt(2) + 4))*sqrt(2*sqrt(2) + 4) + 16*sqrt(2) + 8) + 1/7*sqrt(2)*(8*x + 3) - 1/1

4*((2^(3/4)*(3*sqrt(2) + 5) + 2*2^(1/4)*(sqrt(2) + 4))*sqrt(x + 1)*sqrt(x) - 2^(3/4)*(sqrt(2)*(3*x + 2) + 5*x

+ 1) - 2*2^(1/4)*(sqrt(2)*(x + 3) + 4*x - 2))*sqrt(2*sqrt(2) + 4) + 4/7*x + 5/7)

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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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maple [B] time = 0.17, size = 305, normalized size = 4.69 \[ \frac {\sqrt {\frac {\left (x +1\right ) x}{\left (x +\sqrt {2}-1\right )^{2}}}\, \left (x +\sqrt {2}-1\right ) \left (4 \sqrt {2}\, \arctanh \left (\frac {\sqrt {2}\, \sqrt {\frac {\left (x +1\right ) x}{\left (x +\sqrt {2}-1\right )^{2}}}}{\sqrt {1+\sqrt {2}}}\right )-6 \arctanh \left (\frac {\sqrt {2}\, \sqrt {\frac {\left (x +1\right ) x}{\left (x +\sqrt {2}-1\right )^{2}}}}{\sqrt {1+\sqrt {2}}}\right )+\sqrt {-2+2 \sqrt {2}}\, \sqrt {1+\sqrt {2}}\, \sqrt {2}\, \arctan \left (\frac {\sqrt {\frac {\left (3 \sqrt {2}-4\right ) \left (x +1\right ) \left (4+3 \sqrt {2}\right ) x}{\left (x +\sqrt {2}-1\right )^{2}}}\, \sqrt {-2+2 \sqrt {2}}\, \left (3+2 \sqrt {2}\right ) \left (-x +\sqrt {2}+1\right ) \left (3 \sqrt {2}-4\right ) \left (x +\sqrt {2}-1\right )}{4 \left (x +1\right ) x}\right )-2 \sqrt {-2+2 \sqrt {2}}\, \sqrt {1+\sqrt {2}}\, \arctan \left (\frac {\sqrt {\frac {\left (3 \sqrt {2}-4\right ) \left (x +1\right ) \left (4+3 \sqrt {2}\right ) x}{\left (x +\sqrt {2}-1\right )^{2}}}\, \sqrt {-2+2 \sqrt {2}}\, \left (3+2 \sqrt {2}\right ) \left (-x +\sqrt {2}+1\right ) \left (3 \sqrt {2}-4\right ) \left (x +\sqrt {2}-1\right )}{4 \left (x +1\right ) x}\right )\right ) \sqrt {2}}{4 \sqrt {x +1}\, \left (3 \sqrt {2}-4\right ) \sqrt {1+\sqrt {2}}\, \sqrt {x}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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

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

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

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

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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {x}}{{\left (x^{2} + 1\right )} \sqrt {x + 1}}\,{d x} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(x)/((x^2 + 1)*sqrt(x + 1)), x)

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mupad [B] time = 8.49, size = 1610, normalized size = 24.77 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- atan(((((- 2^(1/2)/16 - 1/16)^(1/2) - (2^(1/2)/16 - 1/16)^(1/2))*(((28454158336*x^(1/2))/((x + 1)^(1/2) - 1)

+ ((- 2^(1/2)/16 - 1/16)^(1/2) - (2^(1/2)/16 - 1/16)^(1/2))*(((112742891520*x^(1/2))/((x + 1)^(1/2) - 1) - ((

531502202880*x)/((x + 1)^(1/2) - 1)^2 - 241591910400)*((- 2^(1/2)/16 - 1/16)^(1/2) - (2^(1/2)/16 - 1/16)^(1/2)

))*((- 2^(1/2)/16 - 1/16)^(1/2) - (2^(1/2)/16 - 1/16)^(1/2)) - (12079595520*x)/((x + 1)^(1/2) - 1)^2 + 6845104

1280))*((- 2^(1/2)/16 - 1/16)^(1/2) - (2^(1/2)/16 - 1/16)^(1/2)) + (13555990528*x)/((x + 1)^(1/2) - 1)^2 + 952

9458688) + (3556769792*x^(1/2))/((x + 1)^(1/2) - 1))*((- 2^(1/2)/16 - 1/16)^(1/2) - (2^(1/2)/16 - 1/16)^(1/2))

*1i - (((- 2^(1/2)/16 - 1/16)^(1/2) - (2^(1/2)/16 - 1/16)^(1/2))*((13555990528*x)/((x + 1)^(1/2) - 1)^2 - ((28

454158336*x^(1/2))/((x + 1)^(1/2) - 1) + ((- 2^(1/2)/16 - 1/16)^(1/2) - (2^(1/2)/16 - 1/16)^(1/2))*(((11274289

1520*x^(1/2))/((x + 1)^(1/2) - 1) + ((531502202880*x)/((x + 1)^(1/2) - 1)^2 - 241591910400)*((- 2^(1/2)/16 - 1

/16)^(1/2) - (2^(1/2)/16 - 1/16)^(1/2)))*((- 2^(1/2)/16 - 1/16)^(1/2) - (2^(1/2)/16 - 1/16)^(1/2)) + (12079595

520*x)/((x + 1)^(1/2) - 1)^2 - 68451041280))*((- 2^(1/2)/16 - 1/16)^(1/2) - (2^(1/2)/16 - 1/16)^(1/2)) + 95294

58688) - (3556769792*x^(1/2))/((x + 1)^(1/2) - 1))*((- 2^(1/2)/16 - 1/16)^(1/2) - (2^(1/2)/16 - 1/16)^(1/2))*1

i)/((((- 2^(1/2)/16 - 1/16)^(1/2) - (2^(1/2)/16 - 1/16)^(1/2))*(((28454158336*x^(1/2))/((x + 1)^(1/2) - 1) + (

(- 2^(1/2)/16 - 1/16)^(1/2) - (2^(1/2)/16 - 1/16)^(1/2))*(((112742891520*x^(1/2))/((x + 1)^(1/2) - 1) - ((5315

02202880*x)/((x + 1)^(1/2) - 1)^2 - 241591910400)*((- 2^(1/2)/16 - 1/16)^(1/2) - (2^(1/2)/16 - 1/16)^(1/2)))*(

(- 2^(1/2)/16 - 1/16)^(1/2) - (2^(1/2)/16 - 1/16)^(1/2)) - (12079595520*x)/((x + 1)^(1/2) - 1)^2 + 68451041280

))*((- 2^(1/2)/16 - 1/16)^(1/2) - (2^(1/2)/16 - 1/16)^(1/2)) + (13555990528*x)/((x + 1)^(1/2) - 1)^2 + 9529458

688) + (3556769792*x^(1/2))/((x + 1)^(1/2) - 1))*((- 2^(1/2)/16 - 1/16)^(1/2) - (2^(1/2)/16 - 1/16)^(1/2)) + (

((- 2^(1/2)/16 - 1/16)^(1/2) - (2^(1/2)/16 - 1/16)^(1/2))*((13555990528*x)/((x + 1)^(1/2) - 1)^2 - ((284541583

36*x^(1/2))/((x + 1)^(1/2) - 1) + ((- 2^(1/2)/16 - 1/16)^(1/2) - (2^(1/2)/16 - 1/16)^(1/2))*(((112742891520*x^

(1/2))/((x + 1)^(1/2) - 1) + ((531502202880*x)/((x + 1)^(1/2) - 1)^2 - 241591910400)*((- 2^(1/2)/16 - 1/16)^(1

/2) - (2^(1/2)/16 - 1/16)^(1/2)))*((- 2^(1/2)/16 - 1/16)^(1/2) - (2^(1/2)/16 - 1/16)^(1/2)) + (12079595520*x)/

((x + 1)^(1/2) - 1)^2 - 68451041280))*((- 2^(1/2)/16 - 1/16)^(1/2) - (2^(1/2)/16 - 1/16)^(1/2)) + 9529458688)

- (3556769792*x^(1/2))/((x + 1)^(1/2) - 1))*((- 2^(1/2)/16 - 1/16)^(1/2) - (2^(1/2)/16 - 1/16)^(1/2)) + (75497

47200*x)/((x + 1)^(1/2) - 1)^2 + 503316480))*((- 2^(1/2)/16 - 1/16)^(1/2)*2i - (2^(1/2)/16 - 1/16)^(1/2)*2i) -

atan(((x^(1/2)*(- 2^(1/2)/16 - 1/16)^(1/2)*848i)/((x + 1)^(1/2) - 1) + (x^(1/2)*(2^(1/2)/16 - 1/16)^(1/2)*848

i)/((x + 1)^(1/2) - 1) + (x^(1/2)*(- 2^(1/2)/16 - 1/16)^(3/2)*6784i)/((x + 1)^(1/2) - 1) + (x^(1/2)*(2^(1/2)/1

6 - 1/16)^(3/2)*6784i)/((x + 1)^(1/2) - 1) + (x^(1/2)*(- 2^(1/2)/16 - 1/16)^(5/2)*26880i)/((x + 1)^(1/2) - 1)

+ (x^(1/2)*(2^(1/2)/16 - 1/16)^(5/2)*26880i)/((x + 1)^(1/2) - 1) + (x^(1/2)*(2^(1/2)/16 - 1/16)^2*(- 2^(1/2)/1

6 - 1/16)^(1/2)*134400i)/((x + 1)^(1/2) - 1) + (x^(1/2)*(2^(1/2)/16 - 1/16)^(1/2)*(2^(1/2)/16 + 1/16)^2*134400

i)/((x + 1)^(1/2) - 1) + (x^(1/2)*(2^(1/2)/16 - 1/16)*(- 2^(1/2)/16 - 1/16)^(1/2)*20352i)/((x + 1)^(1/2) - 1)

- (x^(1/2)*(2^(1/2)/16 - 1/16)^(1/2)*(2^(1/2)/16 + 1/16)*20352i)/((x + 1)^(1/2) - 1) + (x^(1/2)*(2^(1/2)/16 -

1/16)*(- 2^(1/2)/16 - 1/16)^(3/2)*268800i)/((x + 1)^(1/2) - 1) - (x^(1/2)*(2^(1/2)/16 - 1/16)^(3/2)*(2^(1/2)/1

6 + 1/16)*268800i)/((x + 1)^(1/2) - 1))/(4544*(2^(1/2)/16 - 1/16)^(1/2)*(- 2^(1/2)/16 - 1/16)^(1/2) + 65280*(2

^(1/2)/16 - 1/16)^(1/2)*(- 2^(1/2)/16 - 1/16)^(3/2) + 65280*(2^(1/2)/16 - 1/16)^(3/2)*(- 2^(1/2)/16 - 1/16)^(1

/2) + 345600*(2^(1/2)/16 - 1/16)^(1/2)*(- 2^(1/2)/16 - 1/16)^(5/2) + 1152000*(2^(1/2)/16 - 1/16)^(3/2)*(- 2^(1

/2)/16 - 1/16)^(3/2) + 345600*(2^(1/2)/16 - 1/16)^(5/2)*(- 2^(1/2)/16 - 1/16)^(1/2) + x/((x + 1)^(1/2) - 1)^2

+ (6464*x*(2^(1/2)/16 - 1/16)^(1/2)*(- 2^(1/2)/16 - 1/16)^(1/2))/((x + 1)^(1/2) - 1)^2 - (11520*x*(2^(1/2)/16

- 1/16)^(1/2)*(- 2^(1/2)/16 - 1/16)^(3/2))/((x + 1)^(1/2) - 1)^2 - (11520*x*(2^(1/2)/16 - 1/16)^(3/2)*(- 2^(1/

2)/16 - 1/16)^(1/2))/((x + 1)^(1/2) - 1)^2 - (760320*x*(2^(1/2)/16 - 1/16)^(1/2)*(- 2^(1/2)/16 - 1/16)^(5/2))/

((x + 1)^(1/2) - 1)^2 - (2534400*x*(2^(1/2)/16 - 1/16)^(3/2)*(- 2^(1/2)/16 - 1/16)^(3/2))/((x + 1)^(1/2) - 1)^

2 - (760320*x*(2^(1/2)/16 - 1/16)^(5/2)*(- 2^(1/2)/16 - 1/16)^(1/2))/((x + 1)^(1/2) - 1)^2 + 1))*((- 2^(1/2)/1

6 - 1/16)^(1/2)*2i + (2^(1/2)/16 - 1/16)^(1/2)*2i)

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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {x}}{\sqrt {x + 1} \left (x^{2} + 1\right )}\, dx \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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