Optimal. Leaf size=86 \[ \frac {3 \sqrt {c-a^2 c x^2}}{2 a}+\frac {(1+a x) \sqrt {c-a^2 c x^2}}{2 a}-\frac {3 \sqrt {c} \text {ArcTan}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{2 a} \]
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Rubi [A]
time = 0.08, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6302, 6276,
685, 655, 223, 209} \begin {gather*} -\frac {3 \sqrt {c} \text {ArcTan}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{2 a}+\frac {(a x+1) \sqrt {c-a^2 c x^2}}{2 a}+\frac {3 \sqrt {c-a^2 c x^2}}{2 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 223
Rule 655
Rule 685
Rule 6276
Rule 6302
Rubi steps
\begin {align*} \int e^{2 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2} \, dx &=-\int e^{2 \tanh ^{-1}(a x)} \sqrt {c-a^2 c x^2} \, dx\\ &=-\left (c \int \frac {(1+a x)^2}{\sqrt {c-a^2 c x^2}} \, dx\right )\\ &=\frac {(1+a x) \sqrt {c-a^2 c x^2}}{2 a}-\frac {1}{2} (3 c) \int \frac {1+a x}{\sqrt {c-a^2 c x^2}} \, dx\\ &=\frac {3 \sqrt {c-a^2 c x^2}}{2 a}+\frac {(1+a x) \sqrt {c-a^2 c x^2}}{2 a}-\frac {1}{2} (3 c) \int \frac {1}{\sqrt {c-a^2 c x^2}} \, dx\\ &=\frac {3 \sqrt {c-a^2 c x^2}}{2 a}+\frac {(1+a x) \sqrt {c-a^2 c x^2}}{2 a}-\frac {1}{2} (3 c) \text {Subst}\left (\int \frac {1}{1+a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c-a^2 c x^2}}\right )\\ &=\frac {3 \sqrt {c-a^2 c x^2}}{2 a}+\frac {(1+a x) \sqrt {c-a^2 c x^2}}{2 a}-\frac {3 \sqrt {c} \tan ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{2 a}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 76, normalized size = 0.88 \begin {gather*} \frac {\sqrt {c-a^2 c x^2} \left ((4+a x) \sqrt {1-a^2 x^2}+6 \text {ArcSin}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{2 a \sqrt {1-a^2 x^2}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.20, size = 136, normalized size = 1.58
method | result | size |
risch | \(-\frac {\left (a x +4\right ) \left (a^{2} x^{2}-1\right ) c}{2 a \sqrt {-c \left (a^{2} x^{2}-1\right )}}-\frac {3 c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{2 \sqrt {a^{2} c}}\) | \(69\) |
default | \(\frac {x \sqrt {-a^{2} c \,x^{2}+c}}{2}+\frac {c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{2 \sqrt {a^{2} c}}+\frac {2 \sqrt {-a^{2} c \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a c}-\frac {2 a c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a c}}\right )}{\sqrt {a^{2} c}}}{a}\) | \(136\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 47, normalized size = 0.55 \begin {gather*} \frac {1}{2} \, \sqrt {-a^{2} c x^{2} + c} x - \frac {3 \, \sqrt {c} \arcsin \left (a x\right )}{2 \, a} + \frac {2 \, \sqrt {-a^{2} c x^{2} + c}}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 134, normalized size = 1.56 \begin {gather*} \left [\frac {2 \, \sqrt {-a^{2} c x^{2} + c} {\left (a x + 4\right )} + 3 \, \sqrt {-c} \log \left (2 \, a^{2} c x^{2} - 2 \, \sqrt {-a^{2} c x^{2} + c} a \sqrt {-c} x - c\right )}{4 \, a}, \frac {\sqrt {-a^{2} c x^{2} + c} {\left (a x + 4\right )} + 3 \, \sqrt {c} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} a \sqrt {c} x}{a^{2} c x^{2} - c}\right )}{2 \, a}\right ] \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*} \int \frac {\sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )} \left (a x + 1\right )}{a x - 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 62, normalized size = 0.72 \begin {gather*} \frac {1}{2} \, \sqrt {-a^{2} c x^{2} + c} {\left (x + \frac {4}{a}\right )} + \frac {3 \, c \log \left ({\left | -\sqrt {-a^{2} c} x + \sqrt {-a^{2} c x^{2} + c} \right |}\right )}{2 \, \sqrt {-c} {\left | a \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {c-a^2\,c\,x^2}\,\left (a\,x+1\right )}{a\,x-1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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