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

Optimal. Leaf size=12 \[ \log \left (a+\sqrt{x^2+1}\right ) \]

[Out]

Log[a + Sqrt[1 + x^2]]

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Rubi [A]  time = 0.0468336, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {2155, 31} \[ \log \left (a+\sqrt{x^2+1}\right ) \]

Antiderivative was successfully verified.

[In]

Int[x/(1 + x^2 + a*Sqrt[1 + x^2]),x]

[Out]

Log[a + Sqrt[1 + x^2]]

Rule 2155

Int[(x_)^(m_.)/((c_) + (d_.)*(x_)^(n_) + (e_.)*Sqrt[(a_) + (b_.)*(x_)^(n_)]), x_Symbol] :> Dist[1/n, Subst[Int
[x^((m + 1)/n - 1)/(c + d*x + e*Sqrt[a + b*x]), x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && EqQ[b*c
- a*d, 0] && IntegerQ[(m + 1)/n]

Rule 31

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

Rubi steps

\begin{align*} \int \frac{x}{1+x^2+a \sqrt{1+x^2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{1+x+a \sqrt{1+x}} \, dx,x,x^2\right )\\ &=\operatorname{Subst}\left (\int \frac{1}{a+x} \, dx,x,\sqrt{1+x^2}\right )\\ &=\log \left (a+\sqrt{1+x^2}\right )\\ \end{align*}

Mathematica [A]  time = 0.0262214, size = 12, normalized size = 1. \[ \log \left (a+\sqrt{x^2+1}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Log[a + Sqrt[1 + x^2]]

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Maple [B]  time = 0.04, size = 328, normalized size = 27.3 \begin{align*}{\frac{1}{a}\sqrt{{x}^{2}+1}}-{\frac{1}{2\,a}\sqrt{ \left ( x+\sqrt{ \left ( 1+a \right ) \left ( a-1 \right ) } \right ) ^{2}-2\,\sqrt{ \left ( 1+a \right ) \left ( a-1 \right ) } \left ( x+\sqrt{ \left ( 1+a \right ) \left ( a-1 \right ) } \right ) +{a}^{2}}}+{\frac{a}{2}\ln \left ({ \left ( 2\,{a}^{2}-2\,\sqrt{ \left ( 1+a \right ) \left ( a-1 \right ) } \left ( x+\sqrt{ \left ( 1+a \right ) \left ( a-1 \right ) } \right ) +2\,\sqrt{{a}^{2}}\sqrt{ \left ( x+\sqrt{ \left ( 1+a \right ) \left ( a-1 \right ) } \right ) ^{2}-2\,\sqrt{ \left ( 1+a \right ) \left ( a-1 \right ) } \left ( x+\sqrt{ \left ( 1+a \right ) \left ( a-1 \right ) } \right ) +{a}^{2}} \right ) \left ( x+\sqrt{ \left ( 1+a \right ) \left ( a-1 \right ) } \right ) ^{-1}} \right ){\frac{1}{\sqrt{{a}^{2}}}}}-{\frac{1}{2\,a}\sqrt{ \left ( x-\sqrt{ \left ( 1+a \right ) \left ( a-1 \right ) } \right ) ^{2}+2\,\sqrt{ \left ( 1+a \right ) \left ( a-1 \right ) } \left ( x-\sqrt{ \left ( 1+a \right ) \left ( a-1 \right ) } \right ) +{a}^{2}}}+{\frac{a}{2}\ln \left ({ \left ( 2\,{a}^{2}+2\,\sqrt{ \left ( 1+a \right ) \left ( a-1 \right ) } \left ( x-\sqrt{ \left ( 1+a \right ) \left ( a-1 \right ) } \right ) +2\,\sqrt{{a}^{2}}\sqrt{ \left ( x-\sqrt{ \left ( 1+a \right ) \left ( a-1 \right ) } \right ) ^{2}+2\,\sqrt{ \left ( 1+a \right ) \left ( a-1 \right ) } \left ( x-\sqrt{ \left ( 1+a \right ) \left ( a-1 \right ) } \right ) +{a}^{2}} \right ) \left ( x-\sqrt{ \left ( 1+a \right ) \left ( a-1 \right ) } \right ) ^{-1}} \right ){\frac{1}{\sqrt{{a}^{2}}}}}+{\frac{\ln \left ( -{a}^{2}+{x}^{2}+1 \right ) }{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/a*(x^2+1)^(1/2)-1/2/a*((x+((1+a)*(a-1))^(1/2))^2-2*((1+a)*(a-1))^(1/2)*(x+((1+a)*(a-1))^(1/2))+a^2)^(1/2)+1/
2*a/(a^2)^(1/2)*ln((2*a^2-2*((1+a)*(a-1))^(1/2)*(x+((1+a)*(a-1))^(1/2))+2*(a^2)^(1/2)*((x+((1+a)*(a-1))^(1/2))
^2-2*((1+a)*(a-1))^(1/2)*(x+((1+a)*(a-1))^(1/2))+a^2)^(1/2))/(x+((1+a)*(a-1))^(1/2)))-1/2/a*((x-((1+a)*(a-1))^
(1/2))^2+2*((1+a)*(a-1))^(1/2)*(x-((1+a)*(a-1))^(1/2))+a^2)^(1/2)+1/2*a/(a^2)^(1/2)*ln((2*a^2+2*((1+a)*(a-1))^
(1/2)*(x-((1+a)*(a-1))^(1/2))+2*(a^2)^(1/2)*((x-((1+a)*(a-1))^(1/2))^2+2*((1+a)*(a-1))^(1/2)*(x-((1+a)*(a-1))^
(1/2))+a^2)^(1/2))/(x-((1+a)*(a-1))^(1/2)))+1/2*ln(-a^2+x^2+1)

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Maxima [A]  time = 0.945527, size = 14, normalized size = 1.17 \begin{align*} \log \left (a + \sqrt{x^{2} + 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(a + sqrt(x^2 + 1))

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Fricas [B]  time = 1.97612, size = 167, normalized size = 13.92 \begin{align*} \frac{1}{2} \, \log \left (-a^{2} + x^{2} + 1\right ) - \frac{1}{2} \, \log \left (a x + x^{2} - \sqrt{x^{2} + 1}{\left (a + x\right )} + 1\right ) + \frac{1}{2} \, \log \left (-a x + x^{2} + \sqrt{x^{2} + 1}{\left (a - x\right )} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*log(-a^2 + x^2 + 1) - 1/2*log(a*x + x^2 - sqrt(x^2 + 1)*(a + x) + 1) + 1/2*log(-a*x + x^2 + sqrt(x^2 + 1)*
(a - x) + 1)

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Sympy [B]  time = 1.71292, size = 53, normalized size = 4.42 \begin{align*} - \frac{a \left (- \frac{\log{\left (2 a + 2 \sqrt{x^{2} + 1} \right )}}{a} + \frac{\log{\left (- 2 \sqrt{x^{2} + 1} \right )}}{a}\right )}{2} + \frac{\log{\left (a \sqrt{x^{2} + 1} + x^{2} + 1 \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a*(-log(2*a + 2*sqrt(x**2 + 1))/a + log(-2*sqrt(x**2 + 1))/a)/2 + log(a*sqrt(x**2 + 1) + x**2 + 1)/2

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Giac [A]  time = 1.06548, size = 15, normalized size = 1.25 \begin{align*} \log \left ({\left | a + \sqrt{x^{2} + 1} \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(abs(a + sqrt(x^2 + 1)))