Optimal. Leaf size=55 \[ \frac {1}{2} c x \sqrt {1-a^2 x^2}+\frac {c \left (1-a^2 x^2\right )^{3/2}}{3 a}+\frac {c \text {ArcSin}(a x)}{2 a} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6274, 655, 201,
222} \begin {gather*} \frac {c \left (1-a^2 x^2\right )^{3/2}}{3 a}+\frac {1}{2} c x \sqrt {1-a^2 x^2}+\frac {c \text {ArcSin}(a x)}{2 a} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 201
Rule 222
Rule 655
Rule 6274
Rubi steps
\begin {align*} \int e^{-\tanh ^{-1}(a x)} \left (c-a^2 c x^2\right ) \, dx &=c \int (1-a x) \sqrt {1-a^2 x^2} \, dx\\ &=\frac {c \left (1-a^2 x^2\right )^{3/2}}{3 a}+c \int \sqrt {1-a^2 x^2} \, dx\\ &=\frac {1}{2} c x \sqrt {1-a^2 x^2}+\frac {c \left (1-a^2 x^2\right )^{3/2}}{3 a}+\frac {1}{2} c \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx\\ &=\frac {1}{2} c x \sqrt {1-a^2 x^2}+\frac {c \left (1-a^2 x^2\right )^{3/2}}{3 a}+\frac {c \sin ^{-1}(a x)}{2 a}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.06, size = 57, normalized size = 1.04 \begin {gather*} \frac {c \left (\left (2+3 a x-2 a^2 x^2\right ) \sqrt {1-a^2 x^2}-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 [A]
time = 0.06, size = 64, normalized size = 1.16
method | result | size |
default | \(-c \left (-\frac {\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}{3 a}-\frac {x \sqrt {-a^{2} x^{2}+1}}{2}-\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 \sqrt {a^{2}}}\right )\) | \(64\) |
risch | \(\frac {\left (2 a^{2} x^{2}-3 a x -2\right ) \left (a^{2} x^{2}-1\right ) c}{6 a \sqrt {-a^{2} x^{2}+1}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right ) c}{2 \sqrt {a^{2}}}\) | \(71\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.45, size = 45, normalized size = 0.82 \begin {gather*} \frac {1}{2} \, \sqrt {-a^{2} x^{2} + 1} c x + \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} c}{3 \, a} + \frac {c \arcsin \left (a x\right )}{2 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.38, size = 62, normalized size = 1.13 \begin {gather*} -\frac {6 \, c \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (2 \, a^{2} c x^{2} - 3 \, a c x - 2 \, c\right )} \sqrt {-a^{2} x^{2} + 1}}{6 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] Result contains complex when optimal does not.
time = 2.33, size = 107, normalized size = 1.95 \begin {gather*} - a c \left (\begin {cases} \frac {x^{2}}{2} & \text {for}\: a^{2} = 0 \\- \frac {\left (- a^{2} x^{2} + 1\right )^{\frac {3}{2}}}{3 a^{2}} & \text {otherwise} \end {cases}\right ) + c \left (\begin {cases} \frac {i x \sqrt {a^{2} x^{2} - 1}}{2} - \frac {i \operatorname {acosh}{\left (a x \right )}}{2 a} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\- \frac {a^{2} x^{3}}{2 \sqrt {- a^{2} x^{2} + 1}} + \frac {x}{2 \sqrt {- a^{2} x^{2} + 1}} + \frac {\operatorname {asin}{\left (a x \right )}}{2 a} & \text {otherwise} \end {cases}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.41, size = 46, normalized size = 0.84 \begin {gather*} \frac {c \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 x - 3 \, c\right )} x - \frac {2 \, c}{a}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.91, size = 80, normalized size = 1.45 \begin {gather*} \frac {\sqrt {1-a^2\,x^2}\,\left (\frac {c\,x\,\sqrt {-a^2}}{2}-\frac {a\,c}{3\,\sqrt {-a^2}}+\frac {a^3\,c\,x^2}{3\,\sqrt {-a^2}}\right )}{\sqrt {-a^2}}+\frac {c\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{2\,\sqrt {-a^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________