Optimal. Leaf size=93 \[ \frac {5 c \sqrt {1-a^2 x^2}}{2 a}+\frac {5 c (1-a x) \sqrt {1-a^2 x^2}}{6 a}+\frac {c (1-a x)^2 \sqrt {1-a^2 x^2}}{3 a}+\frac {5 c \text {ArcSin}(a x)}{2 a} \]
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Rubi [A]
time = 0.04, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6274, 685, 655,
222} \begin {gather*} \frac {c \sqrt {1-a^2 x^2} (1-a x)^2}{3 a}+\frac {5 c \sqrt {1-a^2 x^2} (1-a x)}{6 a}+\frac {5 c \sqrt {1-a^2 x^2}}{2 a}+\frac {5 c \text {ArcSin}(a x)}{2 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 222
Rule 655
Rule 685
Rule 6274
Rubi steps
\begin {align*} \int e^{-3 \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right ) \, dx &=c \int \frac {(1-a x)^3}{\sqrt {1-a^2 x^2}} \, dx\\ &=\frac {c (1-a x)^2 \sqrt {1-a^2 x^2}}{3 a}+\frac {1}{3} (5 c) \int \frac {(1-a x)^2}{\sqrt {1-a^2 x^2}} \, dx\\ &=\frac {5 c (1-a x) \sqrt {1-a^2 x^2}}{6 a}+\frac {c (1-a x)^2 \sqrt {1-a^2 x^2}}{3 a}+\frac {1}{2} (5 c) \int \frac {1-a x}{\sqrt {1-a^2 x^2}} \, dx\\ &=\frac {5 c \sqrt {1-a^2 x^2}}{2 a}+\frac {5 c (1-a x) \sqrt {1-a^2 x^2}}{6 a}+\frac {c (1-a x)^2 \sqrt {1-a^2 x^2}}{3 a}+\frac {1}{2} (5 c) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx\\ &=\frac {5 c \sqrt {1-a^2 x^2}}{2 a}+\frac {5 c (1-a x) \sqrt {1-a^2 x^2}}{6 a}+\frac {c (1-a x)^2 \sqrt {1-a^2 x^2}}{3 a}+\frac {5 c \sin ^{-1}(a x)}{2 a}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 70, normalized size = 0.75 \begin {gather*} \frac {c \left (\frac {\sqrt {1+a x} \left (22-31 a x+11 a^2 x^2-2 a^3 x^3\right )}{\sqrt {1-a x}}-30 \text {ArcSin}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{6 a} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(269\) vs.
\(2(79)=158\).
time = 0.06, size = 270, normalized size = 2.90
method | result | size |
risch | \(-\frac {\left (2 a^{2} x^{2}-9 a x +22\right ) \left (a^{2} x^{2}-1\right ) c}{6 a \sqrt {-a^{2} x^{2}+1}}+\frac {5 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right ) c}{2 \sqrt {a^{2}}}\) | \(71\) |
default | \(-c \left (\frac {\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{3}+a \left (-\frac {\left (-2 a^{2} \left (x +\frac {1}{a}\right )+2 a \right ) \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{4 a^{2}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{2 \sqrt {a^{2}}}\right )}{a}-\frac {2 \left (\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{a \left (x +\frac {1}{a}\right )^{2}}+3 a \left (\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{3}+a \left (-\frac {\left (-2 a^{2} \left (x +\frac {1}{a}\right )+2 a \right ) \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{4 a^{2}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{2 \sqrt {a^{2}}}\right )\right )\right )}{a^{2}}\right )\) | \(270\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.49, size = 122, normalized size = 1.31 \begin {gather*} -\frac {1}{2} \, \sqrt {a^{2} x^{2} + 4 \, a x + 3} c x + \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} c}{a^{2} x + a} - \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} c}{3 \, a} + \frac {i \, c \arcsin \left (a x + 2\right )}{2 \, a} + \frac {3 \, c \arcsin \left (a x\right )}{a} - \frac {\sqrt {a^{2} x^{2} + 4 \, a x + 3} c}{a} + \frac {3 \, \sqrt {-a^{2} x^{2} + 1} c}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 63, normalized size = 0.68 \begin {gather*} -\frac {30 \, c \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - {\left (2 \, a^{2} c x^{2} - 9 \, a c x + 22 \, 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 [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} c \left (\int \frac {\sqrt {- a^{2} x^{2} + 1}}{a^{3} x^{3} + 3 a^{2} x^{2} + 3 a x + 1}\, dx + \int \left (- \frac {2 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1}}{a^{3} x^{3} + 3 a^{2} x^{2} + 3 a x + 1}\right )\, dx + \int \frac {a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1}}{a^{3} x^{3} + 3 a^{2} x^{2} + 3 a x + 1}\, dx\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 46, normalized size = 0.49 \begin {gather*} \frac {5 \, 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 - 9 \, c\right )} x + \frac {22 \, c}{a}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 74, normalized size = 0.80 \begin {gather*} \frac {11\,c\,\sqrt {1-a^2\,x^2}}{3\,a}-\frac {3\,c\,x\,\sqrt {1-a^2\,x^2}}{2}+\frac {5\,c\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{2\,\sqrt {-a^2}}+\frac {a\,c\,x^2\,\sqrt {1-a^2\,x^2}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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