Optimal. Leaf size=48 \[ \frac {x \tanh ^{-1}(a x)}{c \sqrt {c-a^2 c x^2}}-\frac {1}{a c \sqrt {c-a^2 c x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {5958} \[ \frac {x \tanh ^{-1}(a x)}{c \sqrt {c-a^2 c x^2}}-\frac {1}{a c \sqrt {c-a^2 c x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5958
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(a x)}{\left (c-a^2 c x^2\right )^{3/2}} \, dx &=-\frac {1}{a c \sqrt {c-a^2 c x^2}}+\frac {x \tanh ^{-1}(a x)}{c \sqrt {c-a^2 c x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.06, size = 43, normalized size = 0.90 \[ \frac {\sqrt {c-a^2 c x^2} \left (1-a x \tanh ^{-1}(a x)\right )}{a c^2 \left (a^2 x^2-1\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.52, size = 54, normalized size = 1.12 \[ -\frac {\sqrt {-a^{2} c x^{2} + c} {\left (a x \log \left (-\frac {a x + 1}{a x - 1}\right ) - 2\right )}}{2 \, {\left (a^{3} c^{2} x^{2} - a c^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.69, size = 70, normalized size = 1.46 \[ -\frac {\sqrt {-a^{2} c x^{2} + c} x \log \left (-\frac {a x + 1}{a x - 1}\right )}{2 \, {\left (a^{2} c x^{2} - c\right )} c} - \frac {1}{\sqrt {-a^{2} c x^{2} + c} a c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.47, size = 74, normalized size = 1.54 \[ -\frac {\left (\arctanh \left (a x \right )-1\right ) \sqrt {-\left (a x -1\right ) \left (a x +1\right ) c}}{2 a \left (a x -1\right ) c^{2}}-\frac {\left (\arctanh \left (a x \right )+1\right ) \sqrt {-\left (a x -1\right ) \left (a x +1\right ) c}}{2 a \left (a x +1\right ) c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.43, size = 90, normalized size = 1.88 \[ -\frac {a^{2} {\left (\frac {\sqrt {-a^{2} c x^{2} + c}}{a^{4} c x + a^{3} c} - \frac {\sqrt {-a^{2} c x^{2} + c}}{a^{4} c x - a^{3} c}\right )}}{2 \, c} + \frac {x \operatorname {artanh}\left (a x\right )}{\sqrt {-a^{2} c x^{2} + c} c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\mathrm {atanh}\left (a\,x\right )}{{\left (c-a^2\,c\,x^2\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {atanh}{\left (a x \right )}}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________