Optimal. Leaf size=108 \[ -\frac {2 a \tanh ^{-1}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right )}{c^{3/2}}-\frac {\sqrt {c-a^2 c x^2}}{c^2 x}+\frac {2 a (a x+1)}{3 \left (c-a^2 c x^2\right )^{3/2}}+\frac {a (7 a x+6)}{3 c \sqrt {c-a^2 c x^2}} \]
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Rubi [A] time = 0.34, antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6151, 1805, 807, 266, 63, 208} \[ -\frac {\sqrt {c-a^2 c x^2}}{c^2 x}-\frac {2 a \tanh ^{-1}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right )}{c^{3/2}}+\frac {2 a (a x+1)}{3 \left (c-a^2 c x^2\right )^{3/2}}+\frac {a (7 a x+6)}{3 c \sqrt {c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 266
Rule 807
Rule 1805
Rule 6151
Rubi steps
\begin {align*} \int \frac {e^{2 \tanh ^{-1}(a x)}}{x^2 \left (c-a^2 c x^2\right )^{3/2}} \, dx &=c \int \frac {(1+a x)^2}{x^2 \left (c-a^2 c x^2\right )^{5/2}} \, dx\\ &=\frac {2 a (1+a x)}{3 \left (c-a^2 c x^2\right )^{3/2}}-\frac {1}{3} \int \frac {-3-6 a x-4 a^2 x^2}{x^2 \left (c-a^2 c x^2\right )^{3/2}} \, dx\\ &=\frac {2 a (1+a x)}{3 \left (c-a^2 c x^2\right )^{3/2}}+\frac {a (6+7 a x)}{3 c \sqrt {c-a^2 c x^2}}+\frac {\int \frac {3+6 a x}{x^2 \sqrt {c-a^2 c x^2}} \, dx}{3 c}\\ &=\frac {2 a (1+a x)}{3 \left (c-a^2 c x^2\right )^{3/2}}+\frac {a (6+7 a x)}{3 c \sqrt {c-a^2 c x^2}}-\frac {\sqrt {c-a^2 c x^2}}{c^2 x}+\frac {(2 a) \int \frac {1}{x \sqrt {c-a^2 c x^2}} \, dx}{c}\\ &=\frac {2 a (1+a x)}{3 \left (c-a^2 c x^2\right )^{3/2}}+\frac {a (6+7 a x)}{3 c \sqrt {c-a^2 c x^2}}-\frac {\sqrt {c-a^2 c x^2}}{c^2 x}+\frac {a \operatorname {Subst}\left (\int \frac {1}{x \sqrt {c-a^2 c x}} \, dx,x,x^2\right )}{c}\\ &=\frac {2 a (1+a x)}{3 \left (c-a^2 c x^2\right )^{3/2}}+\frac {a (6+7 a x)}{3 c \sqrt {c-a^2 c x^2}}-\frac {\sqrt {c-a^2 c x^2}}{c^2 x}-\frac {2 \operatorname {Subst}\left (\int \frac {1}{\frac {1}{a^2}-\frac {x^2}{a^2 c}} \, dx,x,\sqrt {c-a^2 c x^2}\right )}{a c^2}\\ &=\frac {2 a (1+a x)}{3 \left (c-a^2 c x^2\right )^{3/2}}+\frac {a (6+7 a x)}{3 c \sqrt {c-a^2 c x^2}}-\frac {\sqrt {c-a^2 c x^2}}{c^2 x}-\frac {2 a \tanh ^{-1}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right )}{c^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 89, normalized size = 0.82 \[ -\frac {2 a \log \left (\sqrt {c} \sqrt {c-a^2 c x^2}+c\right )}{c^{3/2}}+\frac {\left (-10 a^2 x^2+14 a x-3\right ) \sqrt {c-a^2 c x^2}}{3 c^2 x (a x-1)^2}+\frac {2 a \log (x)}{c^{3/2}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.68, size = 238, normalized size = 2.20 \[ \left [\frac {3 \, {\left (a^{3} x^{3} - 2 \, a^{2} x^{2} + a x\right )} \sqrt {c} \log \left (-\frac {a^{2} c x^{2} + 2 \, \sqrt {-a^{2} c x^{2} + c} \sqrt {c} - 2 \, c}{x^{2}}\right ) - \sqrt {-a^{2} c x^{2} + c} {\left (10 \, a^{2} x^{2} - 14 \, a x + 3\right )}}{3 \, {\left (a^{2} c^{2} x^{3} - 2 \, a c^{2} x^{2} + c^{2} x\right )}}, -\frac {6 \, {\left (a^{3} x^{3} - 2 \, a^{2} x^{2} + a x\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} \sqrt {-c}}{a^{2} c x^{2} - c}\right ) + \sqrt {-a^{2} c x^{2} + c} {\left (10 \, a^{2} x^{2} - 14 \, a x + 3\right )}}{3 \, {\left (a^{2} c^{2} x^{3} - 2 \, a c^{2} x^{2} + c^{2} x\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.45, size = 100, normalized size = 0.93 \[ -2 \, a^{2} \sqrt {-c} c {\left (\frac {2 \, {\left | a \right |} \arctan \left (-\frac {\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}}{\sqrt {-c}}\right )}{a^{3} c^{3}} - \frac {1}{{\left ({\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{2} - c\right )} a^{2} c^{2}}\right )} {\left | a \right |} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 178, normalized size = 1.65 \[ \frac {2 a}{c \sqrt {-a^{2} c \,x^{2}+c}}-\frac {2 a \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {-a^{2} c \,x^{2}+c}}{x}\right )}{c^{\frac {3}{2}}}-\frac {1}{c x \sqrt {-a^{2} c \,x^{2}+c}}+\frac {2 a^{2} x}{c \sqrt {-a^{2} c \,x^{2}+c}}-\frac {2}{3 c \left (x -\frac {1}{a}\right ) \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 a c \left (x -\frac {1}{a}\right )}}+\frac {4 a^{2} x}{3 c \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 a c \left (x -\frac {1}{a}\right )}} \]
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 \[ -\int \frac {{\left (a x + 1\right )}^{2}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} {\left (a^{2} x^{2} - 1\right )} x^{2}}\,{d x} \]
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 \[ -\int \frac {{\left (a\,x+1\right )}^2}{x^2\,{\left (c-a^2\,c\,x^2\right )}^{3/2}\,\left (a^2\,x^2-1\right )} \,d x \]
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 \[ - \int \frac {a x}{- a^{3} c x^{5} \sqrt {- a^{2} c x^{2} + c} + a^{2} c x^{4} \sqrt {- a^{2} c x^{2} + c} + a c x^{3} \sqrt {- a^{2} c x^{2} + c} - c x^{2} \sqrt {- a^{2} c x^{2} + c}}\, dx - \int \frac {1}{- a^{3} c x^{5} \sqrt {- a^{2} c x^{2} + c} + a^{2} c x^{4} \sqrt {- a^{2} c x^{2} + c} + a c x^{3} \sqrt {- a^{2} c x^{2} + c} - c x^{2} \sqrt {- a^{2} c x^{2} + c}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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