Optimal. Leaf size=151 \[ \frac {\sqrt {a x-1} \cosh ^{-1}(a x)^3}{6 a^3 \sqrt {1-a x}}+\frac {\sqrt {a x-1} \cosh ^{-1}(a x)}{4 a^3 \sqrt {1-a x}}-\frac {x \sqrt {1-a^2 x^2} \cosh ^{-1}(a x)^2}{2 a^2}-\frac {x \sqrt {1-a x} \sqrt {a x+1}}{4 a^2}-\frac {x^2 \sqrt {a x-1} \cosh ^{-1}(a x)}{2 a \sqrt {1-a x}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.51, antiderivative size = 207, normalized size of antiderivative = 1.37, number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5798, 5759, 5676, 5662, 90, 52} \[ -\frac {x (1-a x) (a x+1)}{4 a^2 \sqrt {1-a^2 x^2}}+\frac {\sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^3}{6 a^3 \sqrt {1-a^2 x^2}}-\frac {x (1-a x) (a x+1) \cosh ^{-1}(a x)^2}{2 a^2 \sqrt {1-a^2 x^2}}-\frac {x^2 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)}{2 a \sqrt {1-a^2 x^2}}+\frac {\sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)}{4 a^3 \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 52
Rule 90
Rule 5662
Rule 5676
Rule 5759
Rule 5798
Rubi steps
\begin {align*} \int \frac {x^2 \cosh ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx &=\frac {\left (\sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {x^2 \cosh ^{-1}(a x)^2}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{\sqrt {1-a^2 x^2}}\\ &=-\frac {x (1-a x) (1+a x) \cosh ^{-1}(a x)^2}{2 a^2 \sqrt {1-a^2 x^2}}+\frac {\left (\sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {\cosh ^{-1}(a x)^2}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{2 a^2 \sqrt {1-a^2 x^2}}-\frac {\left (\sqrt {-1+a x} \sqrt {1+a x}\right ) \int x \cosh ^{-1}(a x) \, dx}{a \sqrt {1-a^2 x^2}}\\ &=-\frac {x^2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{2 a \sqrt {1-a^2 x^2}}-\frac {x (1-a x) (1+a x) \cosh ^{-1}(a x)^2}{2 a^2 \sqrt {1-a^2 x^2}}+\frac {\sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^3}{6 a^3 \sqrt {1-a^2 x^2}}+\frac {\left (\sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {x^2}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{2 \sqrt {1-a^2 x^2}}\\ &=-\frac {x (1-a x) (1+a x)}{4 a^2 \sqrt {1-a^2 x^2}}-\frac {x^2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{2 a \sqrt {1-a^2 x^2}}-\frac {x (1-a x) (1+a x) \cosh ^{-1}(a x)^2}{2 a^2 \sqrt {1-a^2 x^2}}+\frac {\sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^3}{6 a^3 \sqrt {1-a^2 x^2}}+\frac {\left (\sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {1}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{4 a^2 \sqrt {1-a^2 x^2}}\\ &=-\frac {x (1-a x) (1+a x)}{4 a^2 \sqrt {1-a^2 x^2}}+\frac {\sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{4 a^3 \sqrt {1-a^2 x^2}}-\frac {x^2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{2 a \sqrt {1-a^2 x^2}}-\frac {x (1-a x) (1+a x) \cosh ^{-1}(a x)^2}{2 a^2 \sqrt {1-a^2 x^2}}+\frac {\sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^3}{6 a^3 \sqrt {1-a^2 x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.18, size = 87, normalized size = 0.58 \[ -\frac {\sqrt {-((a x-1) (a x+1))} \left (4 \cosh ^{-1}(a x)^3-6 \cosh \left (2 \cosh ^{-1}(a x)\right ) \cosh ^{-1}(a x)+\left (6 \cosh ^{-1}(a x)^2+3\right ) \sinh \left (2 \cosh ^{-1}(a x)\right )\right )}{24 a^3 \sqrt {\frac {a x-1}{a x+1}} (a x+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-a^{2} x^{2} + 1} x^{2} \operatorname {arcosh}\left (a x\right )^{2}}{a^{2} x^{2} - 1}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2} \operatorname {arcosh}\left (a x\right )^{2}}{\sqrt {-a^{2} x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.46, size = 239, normalized size = 1.58 \[ -\frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \mathrm {arccosh}\left (a x \right )^{3}}{6 a^{3} \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (2 x^{3} a^{3}-2 a x +2 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}-\sqrt {a x -1}\, \sqrt {a x +1}\right ) \left (2 \mathrm {arccosh}\left (a x \right )^{2}-2 \,\mathrm {arccosh}\left (a x \right )+1\right )}{16 a^{3} \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (2 x^{3} a^{3}-2 a x -2 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}+\sqrt {a x -1}\, \sqrt {a x +1}\right ) \left (2 \mathrm {arccosh}\left (a x \right )^{2}+2 \,\mathrm {arccosh}\left (a x \right )+1\right )}{16 a^{3} \left (a^{2} x^{2}-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^2\,{\mathrm {acosh}\left (a\,x\right )}^2}{\sqrt {1-a^2\,x^2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2} \operatorname {acosh}^{2}{\left (a x \right )}}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________