Optimal. Leaf size=119 \[ -\frac {2 c^2 \left (1-a^2 x^2\right )^{3/2}}{3 a (c-a c x)^{3/2}}-\frac {4 c \sqrt {1-a^2 x^2}}{a \sqrt {c-a c x}}+\frac {4 \sqrt {2} \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {1-a^2 x^2}}{\sqrt {2} \sqrt {c-a c x}}\right )}{a} \]
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Rubi [A] time = 0.11, 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 = {6127, 665, 661, 208} \[ -\frac {2 c^2 \left (1-a^2 x^2\right )^{3/2}}{3 a (c-a c x)^{3/2}}-\frac {4 c \sqrt {1-a^2 x^2}}{a \sqrt {c-a c x}}+\frac {4 \sqrt {2} \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {1-a^2 x^2}}{\sqrt {2} \sqrt {c-a c x}}\right )}{a} \]
Antiderivative was successfully verified.
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Rule 208
Rule 661
Rule 665
Rule 6127
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 ) \operatorname {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 = 0.03, size = 67, normalized size = 0.56 \[ -\frac {2 \sqrt {c-a c x} \left (\sqrt {a x+1} (a x+7)-6 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {a x+1}}{\sqrt {2}}\right )\right )}{3 a \sqrt {1-a x}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.56, size = 220, normalized size = 1.85 \[ \left [\frac {2 \, {\left (3 \, \sqrt {2} {\left (a x - 1\right )} \sqrt {c} \log \left (-\frac {a^{2} c x^{2} + 2 \, a c x - 2 \, \sqrt {2} \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {c} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right ) + \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} {\left (a x + 7\right )}\right )}}{3 \, {\left (a^{2} x - a\right )}}, \frac {2 \, {\left (6 \, \sqrt {2} {\left (a x - 1\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {2} \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} \sqrt {-c}}{a^{2} c x^{2} - c}\right ) + \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c} {\left (a x + 7\right )}\right )}}{3 \, {\left (a^{2} x - a\right )}}\right ] \]
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 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 95, normalized size = 0.80 \[ -\frac {2 \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}\, \left (6 \sqrt {c}\, \sqrt {2}\, \arctanh \left (\frac {\sqrt {c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right )-x a \sqrt {c \left (a x +1\right )}-7 \sqrt {c \left (a x +1\right )}\right )}{3 \left (a x -1\right ) \sqrt {c \left (a x +1\right )}\, a} \]
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 \[ \int \frac {\sqrt {-a c x + c} {\left (a x + 1\right )}^{3}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {\sqrt {c-a\,c\,x}\,{\left (a\,x+1\right )}^3}{{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\sqrt {- c \left (a x - 1\right )} \left (a x + 1\right )^{3}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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