Optimal. Leaf size=84 \[ \frac {(a x+1)^2}{a^2 \sqrt {c-a^2 c x^2}}+\frac {2 \sqrt {c-a^2 c x^2}}{a^2 c}-\frac {2 \tan ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{a^2 \sqrt {c}} \]
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Rubi [A] time = 0.12, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6151, 789, 641, 217, 203} \[ \frac {(a x+1)^2}{a^2 \sqrt {c-a^2 c x^2}}+\frac {2 \sqrt {c-a^2 c x^2}}{a^2 c}-\frac {2 \tan ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{a^2 \sqrt {c}} \]
Antiderivative was successfully verified.
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Rule 203
Rule 217
Rule 641
Rule 789
Rule 6151
Rubi steps
\begin {align*} \int \frac {e^{2 \tanh ^{-1}(a x)} x}{\sqrt {c-a^2 c x^2}} \, dx &=c \int \frac {x (1+a x)^2}{\left (c-a^2 c x^2\right )^{3/2}} \, dx\\ &=\frac {(1+a x)^2}{a^2 \sqrt {c-a^2 c x^2}}-\frac {2 \int \frac {1+a x}{\sqrt {c-a^2 c x^2}} \, dx}{a}\\ &=\frac {(1+a x)^2}{a^2 \sqrt {c-a^2 c x^2}}+\frac {2 \sqrt {c-a^2 c x^2}}{a^2 c}-\frac {2 \int \frac {1}{\sqrt {c-a^2 c x^2}} \, dx}{a}\\ &=\frac {(1+a x)^2}{a^2 \sqrt {c-a^2 c x^2}}+\frac {2 \sqrt {c-a^2 c x^2}}{a^2 c}-\frac {2 \operatorname {Subst}\left (\int \frac {1}{1+a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c-a^2 c x^2}}\right )}{a}\\ &=\frac {(1+a x)^2}{a^2 \sqrt {c-a^2 c x^2}}+\frac {2 \sqrt {c-a^2 c x^2}}{a^2 c}-\frac {2 \tan ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{a^2 \sqrt {c}}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 78, normalized size = 0.93 \[ \frac {\frac {(a x-3) \sqrt {c-a^2 c x^2}}{a x-1}+2 \sqrt {c} \tan ^{-1}\left (\frac {a x \sqrt {c-a^2 c x^2}}{\sqrt {c} \left (a^2 x^2-1\right )}\right )}{a^2 c} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.79, size = 166, normalized size = 1.98 \[ \left [-\frac {{\left (a x - 1\right )} \sqrt {-c} \log \left (2 \, a^{2} c x^{2} + 2 \, \sqrt {-a^{2} c x^{2} + c} a \sqrt {-c} x - c\right ) - \sqrt {-a^{2} c x^{2} + c} {\left (a x - 3\right )}}{a^{3} c x - a^{2} c}, \frac {2 \, {\left (a x - 1\right )} \sqrt {c} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} a \sqrt {c} x}{a^{2} c x^{2} - c}\right ) + \sqrt {-a^{2} c x^{2} + c} {\left (a x - 3\right )}}{a^{3} c x - a^{2} c}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {undef} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 103, normalized size = 1.23 \[ \frac {\sqrt {-a^{2} c \,x^{2}+c}}{a^{2} c}-\frac {2 \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{a \sqrt {a^{2} c}}-\frac {2 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 a c \left (x -\frac {1}{a}\right )}}{a^{3} c \left (x -\frac {1}{a}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 67, normalized size = 0.80 \[ -a {\left (\frac {2 \, \sqrt {-a^{2} c x^{2} + c}}{a^{4} c x - a^{3} c} + \frac {2 \, \arcsin \left (a x\right )}{a^{3} \sqrt {c}} - \frac {\sqrt {-a^{2} c x^{2} + c}}{a^{3} c}\right )} \]
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 \[ \int -\frac {x\,{\left (a\,x+1\right )}^2}{\sqrt {c-a^2\,c\,x^2}\,\left (a^2\,x^2-1\right )} \,d x \]
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 \[ - \int \frac {x}{a x \sqrt {- a^{2} c x^{2} + c} - \sqrt {- a^{2} c x^{2} + c}}\, dx - \int \frac {a x^{2}}{a x \sqrt {- a^{2} c x^{2} + c} - \sqrt {- a^{2} c x^{2} + c}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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