Optimal. Leaf size=61 \[ \frac {1}{2} c^2 x \sqrt {1-a^2 x^2}+\frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{3 a}+\frac {c^2 \text {ArcSin}(a x)}{2 a} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.03, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6262, 655, 201,
222} \begin {gather*} \frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{3 a}+\frac {1}{2} c^2 x \sqrt {1-a^2 x^2}+\frac {c^2 \text {ArcSin}(a x)}{2 a} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 201
Rule 222
Rule 655
Rule 6262
Rubi steps
\begin {align*} \int e^{\tanh ^{-1}(a x)} (c-a c x)^2 \, dx &=c \int (c-a c x) \sqrt {1-a^2 x^2} \, dx\\ &=\frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{3 a}+c^2 \int \sqrt {1-a^2 x^2} \, dx\\ &=\frac {1}{2} c^2 x \sqrt {1-a^2 x^2}+\frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{3 a}+\frac {1}{2} c^2 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx\\ &=\frac {1}{2} c^2 x \sqrt {1-a^2 x^2}+\frac {c^2 \left (1-a^2 x^2\right )^{3/2}}{3 a}+\frac {c^2 \sin ^{-1}(a x)}{2 a}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.05, size = 59, normalized size = 0.97 \begin {gather*} -\frac {c^2 \left (\sqrt {1-a^2 x^2} \left (-2-3 a x+2 a^2 x^2\right )+6 \text {ArcSin}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{6 a} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(143\) vs.
\(2(51)=102\).
time = 0.00, size = 144, normalized size = 2.36
method | result | size |
risch | \(\frac {\left (2 a^{2} x^{2}-3 a x -2\right ) \left (a^{2} x^{2}-1\right ) c^{2}}{6 a \sqrt {-a^{2} x^{2}+1}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right ) c^{2}}{2 \sqrt {a^{2}}}\) | \(75\) |
default | \(c^{2} \left (a^{3} \left (-\frac {x^{2} \sqrt {-a^{2} x^{2}+1}}{3 a^{2}}-\frac {2 \sqrt {-a^{2} x^{2}+1}}{3 a^{4}}\right )-a^{2} \left (-\frac {x \sqrt {-a^{2} x^{2}+1}}{2 a^{2}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a^{2} \sqrt {a^{2}}}\right )+\frac {\sqrt {-a^{2} x^{2}+1}}{a}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}\right )\) | \(144\) |
meijerg | \(\frac {c^{2} \left (\frac {4 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (4 a^{2} x^{2}+8\right ) \sqrt {-a^{2} x^{2}+1}}{6}\right )}{2 a \sqrt {\pi }}+\frac {c^{2} \left (-\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {3}{2}} \sqrt {-a^{2} x^{2}+1}}{a^{2}}+\frac {\sqrt {\pi }\, \left (-a^{2}\right )^{\frac {3}{2}} \arcsin \left (a x \right )}{a^{3}}\right )}{2 \sqrt {\pi }\, \sqrt {-a^{2}}}+\frac {c^{2} \left (-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {-a^{2} x^{2}+1}\right )}{2 a \sqrt {\pi }}+\frac {c^{2} \arcsin \left (a x \right )}{a}\) | \(153\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.47, size = 72, normalized size = 1.18 \begin {gather*} -\frac {1}{3} \, \sqrt {-a^{2} x^{2} + 1} a c^{2} x^{2} + \frac {1}{2} \, \sqrt {-a^{2} x^{2} + 1} c^{2} x + \frac {c^{2} \arcsin \left (a x\right )}{2 \, a} + \frac {\sqrt {-a^{2} x^{2} + 1} c^{2}}{3 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.38, size = 70, normalized size = 1.15 \begin {gather*} -\frac {6 \, c^{2} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (2 \, a^{2} c^{2} x^{2} - 3 \, a c^{2} x - 2 \, c^{2}\right )} \sqrt {-a^{2} x^{2} + 1}}{6 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 102 vs.
\(2 (48) = 96\).
time = 3.05, size = 102, normalized size = 1.67 \begin {gather*} \begin {cases} \frac {c^{2} \sqrt {- a^{2} x^{2} + 1} - c^{2} \left (\begin {cases} - \frac {a x \sqrt {- a^{2} x^{2} + 1}}{2} + \frac {\operatorname {asin}{\left (a x \right )}}{2} & \text {for}\: a x > -1 \wedge a x < 1 \end {cases}\right ) + c^{2} \left (\begin {cases} \frac {\left (- a^{2} x^{2} + 1\right )^{\frac {3}{2}}}{3} - \sqrt {- a^{2} x^{2} + 1} & \text {for}\: a x > -1 \wedge a x < 1 \end {cases}\right ) + c^{2} \operatorname {asin}{\left (a x \right )}}{a} & \text {for}\: a \neq 0 \\c^{2} x & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.42, size = 54, normalized size = 0.89 \begin {gather*} \frac {c^{2} \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{2 \, {\left | a \right |}} - \frac {1}{6} \, \sqrt {-a^{2} x^{2} + 1} {\left ({\left (2 \, a c^{2} x - 3 \, c^{2}\right )} x - \frac {2 \, c^{2}}{a}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.00, size = 82, normalized size = 1.34 \begin {gather*} \frac {c^2\,x\,\sqrt {1-a^2\,x^2}}{2}+\frac {c^2\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{2\,\sqrt {-a^2}}+\frac {c^2\,\sqrt {1-a^2\,x^2}}{3\,a}-\frac {a\,c^2\,x^2\,\sqrt {1-a^2\,x^2}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________