Optimal. Leaf size=74 \[ -\frac {\sqrt {1-a^2 x^2}}{a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}+\frac {\left (1-a^2 x^2\right )^{3/2}}{3 a^3}+\frac {\text {ArcSin}(a x)}{2 a^3} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.04, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6259, 811, 655,
201, 222} \begin {gather*} \frac {\text {ArcSin}(a x)}{2 a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}+\frac {\left (1-a^2 x^2\right )^{3/2}}{3 a^3}-\frac {\sqrt {1-a^2 x^2}}{a^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 201
Rule 222
Rule 655
Rule 811
Rule 6259
Rubi steps
\begin {align*} \int e^{\tanh ^{-1}(a x)} x^2 \, dx &=\int \frac {x^2 (1+a x)}{\sqrt {1-a^2 x^2}} \, dx\\ &=\frac {\int \frac {1+a x}{\sqrt {1-a^2 x^2}} \, dx}{a^2}-\frac {\int (1+a x) \sqrt {1-a^2 x^2} \, dx}{a^2}\\ &=-\frac {\sqrt {1-a^2 x^2}}{a^3}+\frac {\left (1-a^2 x^2\right )^{3/2}}{3 a^3}+\frac {\int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{a^2}-\frac {\int \sqrt {1-a^2 x^2} \, dx}{a^2}\\ &=-\frac {\sqrt {1-a^2 x^2}}{a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}+\frac {\left (1-a^2 x^2\right )^{3/2}}{3 a^3}+\frac {\sin ^{-1}(a x)}{a^3}-\frac {\int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{2 a^2}\\ &=-\frac {\sqrt {1-a^2 x^2}}{a^3}-\frac {x \sqrt {1-a^2 x^2}}{2 a^2}+\frac {\left (1-a^2 x^2\right )^{3/2}}{3 a^3}+\frac {\sin ^{-1}(a x)}{2 a^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.05, size = 44, normalized size = 0.59 \begin {gather*} \frac {-\sqrt {1-a^2 x^2} \left (4+3 a x+2 a^2 x^2\right )+3 \text {ArcSin}(a x)}{6 a^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.40, size = 90, normalized size = 1.22
method | result | size |
risch | \(\frac {\left (2 a^{2} x^{2}+3 a x +4\right ) \left (a^{2} x^{2}-1\right )}{6 a^{3} \sqrt {-a^{2} x^{2}+1}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a^{2} \sqrt {a^{2}}}\) | \(72\) |
default | \(a \left (-\frac {x^{2} \sqrt {-a^{2} x^{2}+1}}{3 a^{2}}-\frac {2 \sqrt {-a^{2} x^{2}+1}}{3 a^{4}}\right )-\frac {x \sqrt {-a^{2} x^{2}+1}}{2 a^{2}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a^{2} \sqrt {a^{2}}}\) | \(90\) |
meijerg | \(\frac {\frac {4 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (4 a^{2} x^{2}+8\right ) \sqrt {-a^{2} x^{2}+1}}{6}}{2 a^{3} \sqrt {\pi }}-\frac {-\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {3}{2}} \sqrt {-a^{2} x^{2}+1}}{a^{2}}+\frac {\sqrt {\pi }\, \left (-a^{2}\right )^{\frac {3}{2}} \arcsin \left (a x \right )}{a^{3}}}{2 a^{2} \sqrt {\pi }\, \sqrt {-a^{2}}}\) | \(105\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.46, size = 65, normalized size = 0.88 \begin {gather*} -\frac {\sqrt {-a^{2} x^{2} + 1} x^{2}}{3 \, a} - \frac {\sqrt {-a^{2} x^{2} + 1} x}{2 \, a^{2}} + \frac {\arcsin \left (a x\right )}{2 \, a^{3}} - \frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{3 \, a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.38, size = 57, normalized size = 0.77 \begin {gather*} -\frac {{\left (2 \, a^{2} x^{2} + 3 \, a x + 4\right )} \sqrt {-a^{2} x^{2} + 1} + 6 \, \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right )}{6 \, a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 2.02, size = 133, normalized size = 1.80 \begin {gather*} a \left (\begin {cases} - \frac {x^{2} \sqrt {- a^{2} x^{2} + 1}}{3 a^{2}} - \frac {2 \sqrt {- a^{2} x^{2} + 1}}{3 a^{4}} & \text {for}\: a \neq 0 \\\frac {x^{4}}{4} & \text {otherwise} \end {cases}\right ) + \begin {cases} - \frac {i x \sqrt {a^{2} x^{2} - 1}}{2 a^{2}} - \frac {i \operatorname {acosh}{\left (a x \right )}}{2 a^{3}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {x^{3}}{2 \sqrt {- a^{2} x^{2} + 1}} - \frac {x}{2 a^{2} \sqrt {- a^{2} x^{2} + 1}} + \frac {\operatorname {asin}{\left (a x \right )}}{2 a^{3}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.05, size = 82, normalized size = 1.11 \begin {gather*} \frac {\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{2\,a^2\,\sqrt {-a^2}}+\frac {\sqrt {1-a^2\,x^2}\,\left (\frac {2}{3\,a\,\sqrt {-a^2}}+\frac {a\,x^2}{3\,\sqrt {-a^2}}-\frac {x\,\sqrt {-a^2}}{2\,a^2}\right )}{\sqrt {-a^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________