Optimal. Leaf size=25 \[ \text {Int}\left (\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt {d+e x^2}},x\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt {d+e x^2}} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
Rubi steps
\begin {align*} \int \frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt {d+e x^2}} \, dx &=\int \frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt {d+e x^2}} \, dx\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 11.42, size = 0, normalized size = 0.00 \[ \int \frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt {d+e x^2}} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.72, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} \operatorname {arcosh}\left (c x\right )^{2} + 2 \, a b \operatorname {arcosh}\left (c x\right ) + a^{2}}{\sqrt {e x^{2} + d}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}{\sqrt {e x^{2} + d}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.47, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \,\mathrm {arccosh}\left (c x \right )\right )^{2}}{\sqrt {e \,x^{2}+d}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {a^{2} \operatorname {arsinh}\left (\frac {e x}{\sqrt {d e}}\right )}{\sqrt {e}} + \int \frac {b^{2} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )^{2}}{\sqrt {e x^{2} + d}} + \frac {2 \, a b \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )}{\sqrt {e x^{2} + d}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [A] time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2}{\sqrt {e\,x^2+d}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}{\sqrt {d + e x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________