Optimal. Leaf size=3 \[ \text {Chi}\left (\cosh ^{-1}(x)\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.12, antiderivative size = 3, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {5781, 3301} \[ \text {Chi}\left (\cosh ^{-1}(x)\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3301
Rule 5781
Rubi steps
\begin {align*} \int \frac {x}{\sqrt {-1+x} \sqrt {1+x} \cosh ^{-1}(x)} \, dx &=\operatorname {Subst}\left (\int \frac {\cosh (x)}{x} \, dx,x,\cosh ^{-1}(x)\right )\\ &=\text {Chi}\left (\cosh ^{-1}(x)\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 0.06, size = 19, normalized size = 6.33 \[ \frac {1}{2} (x-1) \text {Chi}\left (\cosh ^{-1}(x)\right ) \text {csch}^2\left (\frac {1}{2} \cosh ^{-1}(x)\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {x + 1} \sqrt {x - 1} x}{{\left (x^{2} - 1\right )} \operatorname {arcosh}\relax (x)}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{\sqrt {x + 1} \sqrt {x - 1} \operatorname {arcosh}\relax (x)}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.42, size = 78, normalized size = 26.00 \[ -\frac {\sqrt {-2+2 x}\, \sqrt {2+2 x}\, \sqrt {-1+x}\, \sqrt {1+x}\, \Ei \left (1, \mathrm {arccosh}\relax (x )\right )}{4 \left (x^{2}-1\right )}-\frac {\sqrt {-2+2 x}\, \sqrt {2+2 x}\, \sqrt {-1+x}\, \sqrt {1+x}\, \Ei \left (1, -\mathrm {arccosh}\relax (x )\right )}{4 \left (x^{2}-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{\sqrt {x + 1} \sqrt {x - 1} \operatorname {arcosh}\relax (x)}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.33 \[ \int \frac {x}{\mathrm {acosh}\relax (x)\,\sqrt {x-1}\,\sqrt {x+1}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{\sqrt {x - 1} \sqrt {x + 1} \operatorname {acosh}{\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________