3.7.18 \(\int \frac {\sqrt {x}}{\sqrt {1+x} (1+x^2)} \, dx\) [618]

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

[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 = 0.03, 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 = {924, 95, 214} \begin {gather*} -\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 ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 95

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 214

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

Rule 924

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\\ &=-\text {Subst}\left (\int \frac {1}{i-(1+i) x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {1+x}}\right )+\text {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 [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 0.07, size = 59, normalized size = 0.91 \begin {gather*} -\text {RootSum}\left [16+32 \text {$\#$1}+16 \text {$\#$1}^2+\text {$\#$1}^4\&,\frac {\log \left (-2 x+2 \sqrt {x} \sqrt {1+x}+\text {$\#$1}\right ) \text {$\#$1}^2}{8+8 \text {$\#$1}+\text {$\#$1}^3}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-RootSum[16 + 32*#1 + 16*#1^2 + #1^4 & , (Log[-2*x + 2*Sqrt[x]*Sqrt[1 + x] + #1]*#1^2)/(8 + 8*#1 + #1^3) & ]

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(304\) vs. \(2(45)=90\).
time = 0.18, size = 305, normalized size = 4.69

method	result	size
default	\(\frac {\sqrt {\frac {x \left (1+x \right )}{\left (\sqrt {2}-1+x \right )^{2}}}\, \left (\sqrt {2}-1+x \right ) \left (\sqrt {-2+2 \sqrt {2}}\, \arctan \left (\frac {\sqrt {\frac {\left (3 \sqrt {2}-4\right ) x \left (1+x \right ) \left (4+3 \sqrt {2}\right )}{\left (\sqrt {2}-1+x \right )^{2}}}\, \sqrt {-2+2 \sqrt {2}}\, \left (3+2 \sqrt {2}\right ) \left (\sqrt {2}+1-x \right ) \left (3 \sqrt {2}-4\right ) \left (\sqrt {2}-1+x \right )}{4 x \left (1+x \right )}\right ) \sqrt {1+\sqrt {2}}\, \sqrt {2}-2 \sqrt {-2+2 \sqrt {2}}\, \arctan \left (\frac {\sqrt {\frac {\left (3 \sqrt {2}-4\right ) x \left (1+x \right ) \left (4+3 \sqrt {2}\right )}{\left (\sqrt {2}-1+x \right )^{2}}}\, \sqrt {-2+2 \sqrt {2}}\, \left (3+2 \sqrt {2}\right ) \left (\sqrt {2}+1-x \right ) \left (3 \sqrt {2}-4\right ) \left (\sqrt {2}-1+x \right )}{4 x \left (1+x \right )}\right ) \sqrt {1+\sqrt {2}}+4 \arctanh \left (\frac {\sqrt {2}\, \sqrt {\frac {x \left (1+x \right )}{\left (\sqrt {2}-1+x \right )^{2}}}}{\sqrt {1+\sqrt {2}}}\right ) \sqrt {2}-6 \arctanh \left (\frac {\sqrt {2}\, \sqrt {\frac {x \left (1+x \right )}{\left (\sqrt {2}-1+x \right )^{2}}}}{\sqrt {1+\sqrt {2}}}\right )\right ) \sqrt {2}}{4 \sqrt {x}\, \sqrt {1+x}\, \left (3 \sqrt {2}-4\right ) \sqrt {1+\sqrt {2}}}\)	\(305\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4/x^(1/2)/(1+x)^(1/2)*(x*(1+x)/(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*(1+x)*(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/(1+x))*(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*(1+x
)*(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/(1+x))*(1+2^(1/2))^(1/2)+4*arctanh(2^(1/2)*(x*(1+x)/(2^(1/2)-1+x)^2)^(1/2)/(1+2^(1/2))^(1/2))*2^(1/2)-
6*arctanh(2^(1/2)*(x*(1+x)/(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 \begin {gather*} \text {Failed to integrate} \end {gather*}

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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 744 vs. \(2 (37) = 74\).
time = 2.72, size = 744, normalized size = 11.45 \begin {gather*} \frac {1}{8} \cdot 2^{\frac {1}{4}} \sqrt {2 \, \sqrt {2} + 4} {\left (\sqrt {2} - 1\right )} \log \left (-8 \, \sqrt {x + 1} x^{\frac {3}{2}} + 8 \, x^{2} + 2 \, {\left (2^{\frac {1}{4}} \sqrt {x + 1} \sqrt {x} {\left (\sqrt {2} - 2\right )} - 2^{\frac {1}{4}} {\left (\sqrt {2} {\left (x + 1\right )} - 2 \, x\right )}\right )} \sqrt {2 \, \sqrt {2} + 4} + 4 \, x + 4 \, \sqrt {2} + 4\right ) - \frac {1}{8} \cdot 2^{\frac {1}{4}} \sqrt {2 \, \sqrt {2} + 4} {\left (\sqrt {2} - 1\right )} \log \left (-8 \, \sqrt {x + 1} x^{\frac {3}{2}} + 8 \, x^{2} - 2 \, {\left (2^{\frac {1}{4}} \sqrt {x + 1} \sqrt {x} {\left (\sqrt {2} - 2\right )} - 2^{\frac {1}{4}} {\left (\sqrt {2} {\left (x + 1\right )} - 2 \, x\right )}\right )} \sqrt {2 \, \sqrt {2} + 4} + 4 \, x + 4 \, \sqrt {2} + 4\right ) - \frac {1}{2} \cdot 2^{\frac {1}{4}} \sqrt {2 \, \sqrt {2} + 4} \arctan \left (\frac {1}{7} \, {\left (\sqrt {2} {\left (5 \, \sqrt {2} + 6\right )} + 8 \, \sqrt {2} + 4\right )} \sqrt {x + 1} \sqrt {x} - \frac {1}{7} \, \sqrt {2} {\left (\sqrt {2} {\left (5 \, x + 1\right )} + 6 \, x + 4\right )} - \frac {1}{28} \, \sqrt {-8 \, \sqrt {x + 1} x^{\frac {3}{2}} + 8 \, x^{2} - 2 \, {\left (2^{\frac {1}{4}} \sqrt {x + 1} \sqrt {x} {\left (\sqrt {2} - 2\right )} - 2^{\frac {1}{4}} {\left (\sqrt {2} {\left (x + 1\right )} - 2 \, x\right )}\right )} \sqrt {2 \, \sqrt {2} + 4} + 4 \, x + 4 \, \sqrt {2} + 4} {\left (2 \, \sqrt {2} {\left (5 \, \sqrt {2} + 6\right )} - {\left (2^{\frac {3}{4}} {\left (3 \, \sqrt {2} + 5\right )} + 2 \cdot 2^{\frac {1}{4}} {\left (\sqrt {2} + 4\right )}\right )} \sqrt {2 \, \sqrt {2} + 4} + 16 \, \sqrt {2} + 8\right )} - \frac {1}{7} \, \sqrt {2} {\left (8 \, x + 3\right )} - \frac {1}{14} \, {\left ({\left (2^{\frac {3}{4}} {\left (3 \, \sqrt {2} + 5\right )} + 2 \cdot 2^{\frac {1}{4}} {\left (\sqrt {2} + 4\right )}\right )} \sqrt {x + 1} \sqrt {x} - 2^{\frac {3}{4}} {\left (\sqrt {2} {\left (3 \, x + 2\right )} + 5 \, x + 1\right )} - 2 \cdot 2^{\frac {1}{4}} {\left (\sqrt {2} {\left (x + 3\right )} + 4 \, x - 2\right )}\right )} \sqrt {2 \, \sqrt {2} + 4} - \frac {4}{7} \, x - \frac {5}{7}\right ) - \frac {1}{2} \cdot 2^{\frac {1}{4}} \sqrt {2 \, \sqrt {2} + 4} \arctan \left (-\frac {1}{7} \, {\left (\sqrt {2} {\left (5 \, \sqrt {2} + 6\right )} + 8 \, \sqrt {2} + 4\right )} \sqrt {x + 1} \sqrt {x} + \frac {1}{7} \, \sqrt {2} {\left (\sqrt {2} {\left (5 \, x + 1\right )} + 6 \, x + 4\right )} + \frac {1}{28} \, \sqrt {-8 \, \sqrt {x + 1} x^{\frac {3}{2}} + 8 \, x^{2} + 2 \, {\left (2^{\frac {1}{4}} \sqrt {x + 1} \sqrt {x} {\left (\sqrt {2} - 2\right )} - 2^{\frac {1}{4}} {\left (\sqrt {2} {\left (x + 1\right )} - 2 \, x\right )}\right )} \sqrt {2 \, \sqrt {2} + 4} + 4 \, x + 4 \, \sqrt {2} + 4} {\left (2 \, \sqrt {2} {\left (5 \, \sqrt {2} + 6\right )} + {\left (2^{\frac {3}{4}} {\left (3 \, \sqrt {2} + 5\right )} + 2 \cdot 2^{\frac {1}{4}} {\left (\sqrt {2} + 4\right )}\right )} \sqrt {2 \, \sqrt {2} + 4} + 16 \, \sqrt {2} + 8\right )} + \frac {1}{7} \, \sqrt {2} {\left (8 \, x + 3\right )} - \frac {1}{14} \, {\left ({\left (2^{\frac {3}{4}} {\left (3 \, \sqrt {2} + 5\right )} + 2 \cdot 2^{\frac {1}{4}} {\left (\sqrt {2} + 4\right )}\right )} \sqrt {x + 1} \sqrt {x} - 2^{\frac {3}{4}} {\left (\sqrt {2} {\left (3 \, x + 2\right )} + 5 \, x + 1\right )} - 2 \cdot 2^{\frac {1}{4}} {\left (\sqrt {2} {\left (x + 3\right )} + 4 \, x - 2\right )}\right )} \sqrt {2 \, \sqrt {2} + 4} + \frac {4}{7} \, x + \frac {5}{7}\right ) \end {gather*}

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

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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 375 vs. \(2 (37) = 74\).
time = 4.03, size = 375, normalized size = 5.77 \begin {gather*} \frac {1}{4} \, {\left (\sqrt {2 \, \sqrt {2} + 2} + \sqrt {2 \, \sqrt {2} - 2}\right )} \arctan \left (\frac {2 \, \left (\frac {1}{2}\right )^{\frac {3}{4}} {\left (\left (\frac {1}{2}\right )^{\frac {1}{4}} \sqrt {\sqrt {2} + 2} + 2 \, \sqrt {-\frac {1}{x + 1} + 1}\right )}}{\sqrt {-\sqrt {2} + 2}}\right ) + \frac {1}{4} \, {\left (\sqrt {2 \, \sqrt {2} + 2} + \sqrt {2 \, \sqrt {2} - 2}\right )} \arctan \left (-\frac {2 \, \left (\frac {1}{2}\right )^{\frac {3}{4}} {\left (\left (\frac {1}{2}\right )^{\frac {1}{4}} \sqrt {\sqrt {2} + 2} - 2 \, \sqrt {-\frac {1}{x + 1} + 1}\right )}}{\sqrt {-\sqrt {2} + 2}}\right ) - \frac {1}{8} \, {\left (\sqrt {2 \, \sqrt {2} + 2} - \sqrt {2 \, \sqrt {2} - 2}\right )} \log \left (\left (\frac {1}{2}\right )^{\frac {1}{4}} \sqrt {\sqrt {2} + 2} \sqrt {-\frac {1}{x + 1} + 1} + \sqrt {\frac {1}{2}} - \frac {1}{x + 1} + 1\right ) + \frac {1}{8} \, {\left (\sqrt {2 \, \sqrt {2} + 2} - \sqrt {2 \, \sqrt {2} - 2}\right )} \log \left (-\left (\frac {1}{2}\right )^{\frac {1}{4}} \sqrt {\sqrt {2} + 2} \sqrt {-\frac {1}{x + 1} + 1} + \sqrt {\frac {1}{2}} - \frac {1}{x + 1} + 1\right ) - \frac {1}{4} \, \sqrt {2 \, \sqrt {2} + 2} \arctan \left (\frac {2 \, \left (\frac {1}{2}\right )^{\frac {3}{4}} {\left (\left (\frac {1}{2}\right )^{\frac {1}{4}} \sqrt {\sqrt {2} + 2} + 2\right )}}{\sqrt {-\sqrt {2} + 2}}\right ) - \frac {1}{4} \, \sqrt {2 \, \sqrt {2} + 2} \arctan \left (-\frac {2 \, \left (\frac {1}{2}\right )^{\frac {3}{4}} {\left (\left (\frac {1}{2}\right )^{\frac {1}{4}} \sqrt {\sqrt {2} + 2} - 2\right )}}{\sqrt {-\sqrt {2} + 2}}\right ) - \frac {1}{8} \, \sqrt {2 \, \sqrt {2} - 2} \log \left (\left (\frac {1}{2}\right )^{\frac {1}{4}} \sqrt {\sqrt {2} + 2} + \sqrt {\frac {1}{2}} + 1\right ) + \frac {1}{8} \, \sqrt {2 \, \sqrt {2} - 2} \log \left (-\left (\frac {1}{2}\right )^{\frac {1}{4}} \sqrt {\sqrt {2} + 2} + \sqrt {\frac {1}{2}} + 1\right ) \end {gather*}

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]

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

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Mupad [B]
time = 8.49, size = 1610, normalized size = 24.77 \begin {gather*} \text {Too large to display} \end {gather*}

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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