Optimal. Leaf size=35 \[ \frac {\sqrt {c x-1} \log \left (a+b \cosh ^{-1}(c x)\right )}{b c \sqrt {1-c x}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.22, antiderivative size = 48, normalized size of antiderivative = 1.37, number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {5713, 5674} \[ \frac {\sqrt {c x-1} \sqrt {c x+1} \log \left (a+b \cosh ^{-1}(c x)\right )}{b c \sqrt {1-c^2 x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5674
Rule 5713
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )} \, dx &=\frac {\left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )} \, dx}{\sqrt {1-c^2 x^2}}\\ &=\frac {\sqrt {-1+c x} \sqrt {1+c x} \log \left (a+b \cosh ^{-1}(c x)\right )}{b c \sqrt {1-c^2 x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.11, size = 54, normalized size = 1.54 \[ \frac {\sqrt {\frac {c x-1}{c x+1}} (c x+1) \log \left (a+b \cosh ^{-1}(c x)\right )}{b c \sqrt {-((c x-1) (c x+1))}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.49, size = 65, normalized size = 1.86 \[ -\frac {\sqrt {c^{2} x^{2} - 1} \sqrt {-c^{2} x^{2} + 1} \log \left (\frac {b \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + a}{b}\right )}{b c^{3} x^{2} - b c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-c^{2} x^{2} + 1} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.18, size = 55, normalized size = 1.57 \[ -\frac {\sqrt {-c^{2} x^{2}+1}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )}{c \left (c^{2} x^{2}-1\right ) b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-c^{2} x^{2} + 1} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {1}{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,\sqrt {1-c^2\,x^2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {- \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________