3.238 \(\int \log (1+x+\sqrt {1+x}) \, dx\)

Optimal. Leaf size=32 \[ -x+\sqrt {x+1}+x \log \left (x+\sqrt {x+1}+1\right )+\frac {1}{2} \log (x+1) \]

[Out]

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

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Rubi [A]  time = 0.21, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2548} \[ -x+\sqrt {x+1}+x \log \left (x+\sqrt {x+1}+1\right )+\frac {1}{2} \log (x+1) \]

Antiderivative was successfully verified.

[In]

Int[Log[1 + x + Sqrt[1 + x]],x]

[Out]

-x + Sqrt[1 + x] + Log[1 + x]/2 + x*Log[1 + x + Sqrt[1 + x]]

Rule 2548

Int[Log[u_], x_Symbol] :> Simp[x*Log[u], x] - Int[SimplifyIntegrand[(x*D[u, x])/u, x], x] /; InverseFunctionFr
eeQ[u, x]

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 38, normalized size = 1.19 \[ -x+\sqrt {x+1}-\log \left (\sqrt {x+1}+1\right )+(x+1) \log \left (x+\sqrt {x+1}+1\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Log[1 + x + Sqrt[1 + x]],x]

[Out]

-x + Sqrt[1 + x] - Log[1 + Sqrt[1 + x]] + (1 + x)*Log[1 + x + Sqrt[1 + x]]

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fricas [A]  time = 0.46, size = 38, normalized size = 1.19 \[ {\left (x - 1\right )} \log \left (x + \sqrt {x + 1} + 1\right ) - x + \sqrt {x + 1} + \log \left (\sqrt {x + 1} + 1\right ) + 2 \, \log \left (\sqrt {x + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.18, size = 33, normalized size = 1.03 \[ {\left (x + 1\right )} \log \left (x + \sqrt {x + 1} + 1\right ) - x + \sqrt {x + 1} - \log \left (\sqrt {x + 1} + 1\right ) - 1 \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x + 1)*log(x + sqrt(x + 1) + 1) - x + sqrt(x + 1) - log(sqrt(x + 1) + 1) - 1

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maple [A]  time = 0.07, size = 34, normalized size = 1.06 \[ -x -\ln \left (1+\sqrt {x +1}\right )+\left (x +1\right ) \ln \left (x +1+\sqrt {x +1}\right )-1+\sqrt {x +1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.63, size = 33, normalized size = 1.03 \[ {\left (x + 1\right )} \log \left (x + \sqrt {x + 1} + 1\right ) - x + \sqrt {x + 1} - \log \left (\sqrt {x + 1} + 1\right ) - 1 \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x + 1)*log(x + sqrt(x + 1) + 1) - x + sqrt(x + 1) - log(sqrt(x + 1) + 1) - 1

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mupad [B]  time = 0.11, size = 26, normalized size = 0.81 \[ \ln \left (\sqrt {x+1}\right )-x+\sqrt {x+1}+x\,\ln \left (x+\sqrt {x+1}+1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(x + (x + 1)^(1/2) + 1),x)

[Out]

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

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sympy [B]  time = 0.89, size = 184, normalized size = 5.75 \[ \frac {x \sqrt {x + 1} \log {\left (x + \sqrt {x + 1} + 1 \right )}}{\sqrt {x + 1} + 1} - \frac {x \sqrt {x + 1}}{\sqrt {x + 1} + 1} + \frac {x \log {\left (x + \sqrt {x + 1} + 1 \right )}}{\sqrt {x + 1} + 1} - \frac {\sqrt {x + 1} \log {\left (\sqrt {x + 1} + 1 \right )}}{\sqrt {x + 1} + 1} + \frac {\sqrt {x + 1} \log {\left (x + \sqrt {x + 1} + 1 \right )}}{\sqrt {x + 1} + 1} - \frac {\sqrt {x + 1}}{\sqrt {x + 1} + 1} - \frac {\log {\left (\sqrt {x + 1} + 1 \right )}}{\sqrt {x + 1} + 1} + \frac {\log {\left (x + \sqrt {x + 1} + 1 \right )}}{\sqrt {x + 1} + 1} - \frac {1}{\sqrt {x + 1} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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