Optimal. Leaf size=139 \[ \frac {\sqrt {1-a^2 x^2}}{2 a^3 c (1-a x) \sqrt {c-a^2 c x^2}}+\frac {3 \sqrt {1-a^2 x^2} \log (1-a x)}{4 a^3 c \sqrt {c-a^2 c x^2}}+\frac {\sqrt {1-a^2 x^2} \log (1+a x)}{4 a^3 c \sqrt {c-a^2 c x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.16, antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {6288, 6285, 90}
\begin {gather*} \frac {\sqrt {1-a^2 x^2}}{2 a^3 c (1-a x) \sqrt {c-a^2 c x^2}}+\frac {3 \sqrt {1-a^2 x^2} \log (1-a x)}{4 a^3 c \sqrt {c-a^2 c x^2}}+\frac {\sqrt {1-a^2 x^2} \log (a x+1)}{4 a^3 c \sqrt {c-a^2 c x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 90
Rule 6285
Rule 6288
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)} x^2}{\left (c-a^2 c x^2\right )^{3/2}} \, dx &=\frac {\sqrt {1-a^2 x^2} \int \frac {e^{\tanh ^{-1}(a x)} x^2}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{c \sqrt {c-a^2 c x^2}}\\ &=\frac {\sqrt {1-a^2 x^2} \int \frac {x^2}{(1-a x)^2 (1+a x)} \, dx}{c \sqrt {c-a^2 c x^2}}\\ &=\frac {\sqrt {1-a^2 x^2} \int \left (\frac {1}{2 a^2 (-1+a x)^2}+\frac {3}{4 a^2 (-1+a x)}+\frac {1}{4 a^2 (1+a x)}\right ) \, dx}{c \sqrt {c-a^2 c x^2}}\\ &=\frac {\sqrt {1-a^2 x^2}}{2 a^3 c (1-a x) \sqrt {c-a^2 c x^2}}+\frac {3 \sqrt {1-a^2 x^2} \log (1-a x)}{4 a^3 c \sqrt {c-a^2 c x^2}}+\frac {\sqrt {1-a^2 x^2} \log (1+a x)}{4 a^3 c \sqrt {c-a^2 c x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.04, size = 76, normalized size = 0.55 \begin {gather*} \frac {\sqrt {1-a^2 x^2} \left (\frac {1}{2 a^3 (1-a x)}+\frac {3 \log (1-a x)}{4 a^3}+\frac {\log (1+a x)}{4 a^3}\right )}{c \sqrt {c-a^2 c x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.06, size = 90, normalized size = 0.65
method | result | size |
default | \(-\frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (\ln \left (a x +1\right ) a x +3 \ln \left (a x -1\right ) a x -\ln \left (a x +1\right )-3 \ln \left (a x -1\right )-2\right )}{4 \left (a^{2} x^{2}-1\right ) c^{2} a^{3} \left (a x -1\right )}\) | \(90\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} \left (a x + 1\right )}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^2\,\left (a\,x+1\right )}{{\left (c-a^2\,c\,x^2\right )}^{3/2}\,\sqrt {1-a^2\,x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________