Optimal. Leaf size=110 \[ -\frac {2 x \sqrt {a x-1}}{3 a^3 \sqrt {1-a x}}-\frac {x^2 \sqrt {1-a^2 x^2} \cosh ^{-1}(a x)}{3 a^2}-\frac {2 \sqrt {1-a^2 x^2} \cosh ^{-1}(a x)}{3 a^4}-\frac {x^3 \sqrt {a x-1}}{9 a \sqrt {1-a x}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.39, antiderivative size = 158, normalized size of antiderivative = 1.44, number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {5798, 5759, 5718, 8, 30} \[ -\frac {x^3 \sqrt {a x-1} \sqrt {a x+1}}{9 a \sqrt {1-a^2 x^2}}-\frac {2 x \sqrt {a x-1} \sqrt {a x+1}}{3 a^3 \sqrt {1-a^2 x^2}}-\frac {x^2 (1-a x) (a x+1) \cosh ^{-1}(a x)}{3 a^2 \sqrt {1-a^2 x^2}}-\frac {2 (1-a x) (a x+1) \cosh ^{-1}(a x)}{3 a^4 \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 30
Rule 5718
Rule 5759
Rule 5798
Rubi steps
\begin {align*} \int \frac {x^3 \cosh ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx &=\frac {\left (\sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {x^3 \cosh ^{-1}(a x)}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{\sqrt {1-a^2 x^2}}\\ &=-\frac {x^2 (1-a x) (1+a x) \cosh ^{-1}(a x)}{3 a^2 \sqrt {1-a^2 x^2}}+\frac {\left (2 \sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {x \cosh ^{-1}(a x)}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{3 a^2 \sqrt {1-a^2 x^2}}-\frac {\left (\sqrt {-1+a x} \sqrt {1+a x}\right ) \int x^2 \, dx}{3 a \sqrt {1-a^2 x^2}}\\ &=-\frac {x^3 \sqrt {-1+a x} \sqrt {1+a x}}{9 a \sqrt {1-a^2 x^2}}-\frac {2 (1-a x) (1+a x) \cosh ^{-1}(a x)}{3 a^4 \sqrt {1-a^2 x^2}}-\frac {x^2 (1-a x) (1+a x) \cosh ^{-1}(a x)}{3 a^2 \sqrt {1-a^2 x^2}}-\frac {\left (2 \sqrt {-1+a x} \sqrt {1+a x}\right ) \int 1 \, dx}{3 a^3 \sqrt {1-a^2 x^2}}\\ &=-\frac {2 x \sqrt {-1+a x} \sqrt {1+a x}}{3 a^3 \sqrt {1-a^2 x^2}}-\frac {x^3 \sqrt {-1+a x} \sqrt {1+a x}}{9 a \sqrt {1-a^2 x^2}}-\frac {2 (1-a x) (1+a x) \cosh ^{-1}(a x)}{3 a^4 \sqrt {1-a^2 x^2}}-\frac {x^2 (1-a x) (1+a x) \cosh ^{-1}(a x)}{3 a^2 \sqrt {1-a^2 x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.13, size = 74, normalized size = 0.67 \[ -\frac {a x \sqrt {a x-1} \sqrt {a x+1} \left (a^2 x^2+6\right )-3 \left (a^4 x^4+a^2 x^2-2\right ) \cosh ^{-1}(a x)}{9 a^4 \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.18, size = 101, normalized size = 0.92 \[ -\frac {3 \, {\left (a^{4} x^{4} + a^{2} x^{2} - 2\right )} \sqrt {-a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) - {\left (a^{3} x^{3} + 6 \, a x\right )} \sqrt {a^{2} x^{2} - 1} \sqrt {-a^{2} x^{2} + 1}}{9 \, {\left (a^{6} x^{2} - a^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.53, size = 311, normalized size = 2.83 \[ -\frac {\sqrt {-a^{2} x^{2}+1}\, \left (4 x^{4} a^{4}-5 a^{2} x^{2}+4 a^{3} x^{3} \sqrt {a x -1}\, \sqrt {a x +1}-3 \sqrt {a x +1}\, \sqrt {a x -1}\, a x +1\right ) \left (-1+3 \,\mathrm {arccosh}\left (a x \right )\right )}{72 a^{4} \left (a^{2} x^{2}-1\right )}-\frac {3 \sqrt {-a^{2} x^{2}+1}\, \left (\sqrt {a x +1}\, \sqrt {a x -1}\, a x +a^{2} x^{2}-1\right ) \left (-1+\mathrm {arccosh}\left (a x \right )\right )}{8 a^{4} \left (a^{2} x^{2}-1\right )}-\frac {3 \sqrt {-a^{2} x^{2}+1}\, \left (a^{2} x^{2}-\sqrt {a x +1}\, \sqrt {a x -1}\, a x -1\right ) \left (1+\mathrm {arccosh}\left (a x \right )\right )}{8 a^{4} \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (4 x^{4} a^{4}-5 a^{2} x^{2}-4 a^{3} x^{3} \sqrt {a x -1}\, \sqrt {a x +1}+3 \sqrt {a x +1}\, \sqrt {a x -1}\, a x +1\right ) \left (1+3 \,\mathrm {arccosh}\left (a x \right )\right )}{72 a^{4} \left (a^{2} x^{2}-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [C] time = 0.90, size = 62, normalized size = 0.56 \[ \frac {1}{9} \, a {\left (\frac {i \, x^{3}}{a^{2}} + \frac {6 i \, x}{a^{4}}\right )} - \frac {1}{3} \, {\left (\frac {\sqrt {-a^{2} x^{2} + 1} x^{2}}{a^{2}} + \frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{a^{4}}\right )} \operatorname {arcosh}\left (a x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^3\,\mathrm {acosh}\left (a\,x\right )}{\sqrt {1-a^2\,x^2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3} \operatorname {acosh}{\left (a x \right )}}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________