Optimal. Leaf size=55 \[ \frac {1}{2} c x \sqrt {1-a^2 x^2}-\frac {c \left (1-a^2 x^2\right )^{3/2}}{3 a}+\frac {c \text {ArcSin}(a x)}{2 a} \]
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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 = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6273, 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.
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Rule 201
Rule 222
Rule 655
Rule 6273
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*}
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Mathematica [A]
time = 0.05, size = 57, normalized size = 1.04 \begin {gather*} \frac {c \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.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(142\) vs.
\(2(45)=90\).
time = 0.05, size = 143, normalized size = 2.60
method | result | size |
risch | \(-\frac {\left (2 a^{2} x^{2}+3 a x -2\right ) \left (a^{2} x^{2}-1\right ) c}{6 a \sqrt {-a^{2} x^{2}+1}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right ) c}{2 \sqrt {a^{2}}}\) | \(71\) |
default | \(-c \left (a^{3} \left (-\frac {x^{2} \sqrt {-a^{2} x^{2}+1}}{3 a^{2}}-\frac {2 \sqrt {-a^{2} x^{2}+1}}{3 a^{4}}\right )+a^{2} \left (-\frac {x \sqrt {-a^{2} x^{2}+1}}{2 a^{2}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a^{2} \sqrt {a^{2}}}\right )+\frac {\sqrt {-a^{2} x^{2}+1}}{a}-\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}\right )\) | \(143\) |
meijerg | \(-\frac {c \left (\frac {4 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (4 a^{2} x^{2}+8\right ) \sqrt {-a^{2} x^{2}+1}}{6}\right )}{2 a \sqrt {\pi }}-\frac {c \left (-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {-a^{2} x^{2}+1}\right )}{2 a \sqrt {\pi }}+\frac {c \left (-\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {3}{2}} \sqrt {-a^{2} x^{2}+1}}{a^{2}}+\frac {\sqrt {\pi }\, \left (-a^{2}\right )^{\frac {3}{2}} \arcsin \left (a x \right )}{a^{3}}\right )}{2 \sqrt {\pi }\, \sqrt {-a^{2}}}+\frac {c \arcsin \left (a x \right )}{a}\) | \(145\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 64, normalized size = 1.16 \begin {gather*} \frac {1}{3} \, \sqrt {-a^{2} x^{2} + 1} a c x^{2} + \frac {1}{2} \, \sqrt {-a^{2} x^{2} + 1} c x + \frac {c \arcsin \left (a x\right )}{2 \, a} - \frac {\sqrt {-a^{2} x^{2} + 1} c}{3 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 63, normalized size = 1.15 \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.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 94 vs.
\(2 (42) = 84\).
time = 2.74, size = 94, normalized size = 1.71 \begin {gather*} \begin {cases} \frac {- c \sqrt {- a^{2} x^{2} + 1} - c \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 \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 \operatorname {asin}{\left (a x \right )}}{a} & \text {for}\: a \neq 0 \\c x & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, 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.
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Mupad [B]
time = 0.88, size = 80, normalized size = 1.45 \begin {gather*} \frac {\sqrt {1-a^2\,x^2}\,\left (\frac {a\,c}{3\,\sqrt {-a^2}}+\frac {c\,x\,\sqrt {-a^2}}{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.
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