Optimal. Leaf size=119 \[ -\frac {4 c \sqrt {1-a^2 x^2}}{a \sqrt {c-a c x}}-\frac {2 c^2 \left (1-a^2 x^2\right )^{3/2}}{3 a (c-a c x)^{3/2}}+\frac {4 \sqrt {2} \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {1-a^2 x^2}}{\sqrt {2} \sqrt {c-a c x}}\right )}{a} \]
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Rubi [A]
time = 0.08, antiderivative size = 119, 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 = {6262, 679, 675,
214} \begin {gather*} -\frac {2 c^2 \left (1-a^2 x^2\right )^{3/2}}{3 a (c-a c x)^{3/2}}-\frac {4 c \sqrt {1-a^2 x^2}}{a \sqrt {c-a c x}}+\frac {4 \sqrt {2} \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {1-a^2 x^2}}{\sqrt {2} \sqrt {c-a c x}}\right )}{a} \end {gather*}
Antiderivative was successfully verified.
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Rule 214
Rule 675
Rule 679
Rule 6262
Rubi steps
\begin {align*} \int e^{3 \tanh ^{-1}(a x)} \sqrt {c-a c x} \, dx &=c^3 \int \frac {\left (1-a^2 x^2\right )^{3/2}}{(c-a c x)^{5/2}} \, dx\\ &=-\frac {2 c^2 \left (1-a^2 x^2\right )^{3/2}}{3 a (c-a c x)^{3/2}}+\left (2 c^2\right ) \int \frac {\sqrt {1-a^2 x^2}}{(c-a c x)^{3/2}} \, dx\\ &=-\frac {4 c \sqrt {1-a^2 x^2}}{a \sqrt {c-a c x}}-\frac {2 c^2 \left (1-a^2 x^2\right )^{3/2}}{3 a (c-a c x)^{3/2}}+(4 c) \int \frac {1}{\sqrt {c-a c x} \sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {4 c \sqrt {1-a^2 x^2}}{a \sqrt {c-a c x}}-\frac {2 c^2 \left (1-a^2 x^2\right )^{3/2}}{3 a (c-a c x)^{3/2}}-\left (8 a c^2\right ) \text {Subst}\left (\int \frac {1}{-2 a^2 c+a^2 c^2 x^2} \, dx,x,\frac {\sqrt {1-a^2 x^2}}{\sqrt {c-a c x}}\right )\\ &=-\frac {4 c \sqrt {1-a^2 x^2}}{a \sqrt {c-a c x}}-\frac {2 c^2 \left (1-a^2 x^2\right )^{3/2}}{3 a (c-a c x)^{3/2}}+\frac {4 \sqrt {2} \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {1-a^2 x^2}}{\sqrt {2} \sqrt {c-a c x}}\right )}{a}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 67, normalized size = 0.56 \begin {gather*} -\frac {2 \sqrt {c-a c x} \left (\sqrt {1+a x} (7+a x)-6 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {1+a x}}{\sqrt {2}}\right )\right )}{3 a \sqrt {1-a x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.00, size = 95, normalized size = 0.80
method | result | size |
default | \(-\frac {2 \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}\, \left (6 \sqrt {c}\, \sqrt {2}\, \arctanh \left (\frac {\sqrt {\left (a x +1\right ) c}\, \sqrt {2}}{2 \sqrt {c}}\right )-\sqrt {\left (a x +1\right ) c}\, a x -7 \sqrt {\left (a x +1\right ) c}\right )}{3 \left (a x -1\right ) \sqrt {\left (a x +1\right ) c}\, a}\) | \(95\) |
risch | \(\frac {2 \left (a x +7\right ) \left (a x +1\right ) \sqrt {-\frac {\left (-a^{2} x^{2}+1\right ) c}{a x -1}}\, \left (a x -1\right ) c}{3 a \sqrt {\left (a x +1\right ) c}\, \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}}-\frac {4 \sqrt {2}\, \sqrt {c}\, \arctanh \left (\frac {\sqrt {c x a +c}\, \sqrt {2}}{2 \sqrt {c}}\right ) \sqrt {-\frac {\left (-a^{2} x^{2}+1\right ) c}{a x -1}}\, \left (a x -1\right )}{a \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}}\) | \(153\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 220, normalized size = 1.85 \begin {gather*} \left [\frac {2 \, {\left (3 \, \sqrt {2} {\left (a x - 1\right )} \sqrt {c} \log \left (-\frac {a^{2} c x^{2} + 2 \, a c x - 2 \, \sqrt {2} \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {c} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right ) + \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} {\left (a x + 7\right )}\right )}}{3 \, {\left (a^{2} x - a\right )}}, \frac {2 \, {\left (6 \, \sqrt {2} {\left (a x - 1\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {2} \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {-c}}{a^{2} c x^{2} - c}\right ) + \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} {\left (a x + 7\right )}\right )}}{3 \, {\left (a^{2} x - a\right )}}\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 )^{3}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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\,c\,x}\,{\left (a\,x+1\right )}^3}{{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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