Optimal. Leaf size=82 \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right )}{c^{3/2}}+\frac {2 (a x+1)}{3 \left (c-a^2 c x^2\right )^{3/2}}+\frac {4 a x+3}{3 c \sqrt {c-a^2 c x^2}} \]
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Rubi [A] time = 0.27, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {6151, 1805, 823, 12, 266, 63, 208} \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right )}{c^{3/2}}+\frac {2 (a x+1)}{3 \left (c-a^2 c x^2\right )^{3/2}}+\frac {4 a x+3}{3 c \sqrt {c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 63
Rule 208
Rule 266
Rule 823
Rule 1805
Rule 6151
Rubi steps
\begin {align*} \int \frac {e^{2 \tanh ^{-1}(a x)}}{x \left (c-a^2 c x^2\right )^{3/2}} \, dx &=c \int \frac {(1+a x)^2}{x \left (c-a^2 c x^2\right )^{5/2}} \, dx\\ &=\frac {2 (1+a x)}{3 \left (c-a^2 c x^2\right )^{3/2}}-\frac {1}{3} \int \frac {-3-4 a x}{x \left (c-a^2 c x^2\right )^{3/2}} \, dx\\ &=\frac {2 (1+a x)}{3 \left (c-a^2 c x^2\right )^{3/2}}+\frac {3+4 a x}{3 c \sqrt {c-a^2 c x^2}}-\frac {\int -\frac {3 a^2 c^2}{x \sqrt {c-a^2 c x^2}} \, dx}{3 a^2 c^3}\\ &=\frac {2 (1+a x)}{3 \left (c-a^2 c x^2\right )^{3/2}}+\frac {3+4 a x}{3 c \sqrt {c-a^2 c x^2}}+\frac {\int \frac {1}{x \sqrt {c-a^2 c x^2}} \, dx}{c}\\ &=\frac {2 (1+a x)}{3 \left (c-a^2 c x^2\right )^{3/2}}+\frac {3+4 a x}{3 c \sqrt {c-a^2 c x^2}}+\frac {\operatorname {Subst}\left (\int \frac {1}{x \sqrt {c-a^2 c x}} \, dx,x,x^2\right )}{2 c}\\ &=\frac {2 (1+a x)}{3 \left (c-a^2 c x^2\right )^{3/2}}+\frac {3+4 a x}{3 c \sqrt {c-a^2 c x^2}}-\frac {\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^2 c^2}\\ &=\frac {2 (1+a x)}{3 \left (c-a^2 c x^2\right )^{3/2}}+\frac {3+4 a x}{3 c \sqrt {c-a^2 c x^2}}-\frac {\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 = 0.12, size = 75, normalized size = 0.91 \[ -\frac {\log \left (\sqrt {c} \sqrt {c-a^2 c x^2}+c\right )}{c^{3/2}}+\frac {(5-4 a x) \sqrt {c-a^2 c x^2}}{3 c^2 (a x-1)^2}+\frac {\log (x)}{c^{3/2}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.51, size = 202, normalized size = 2.46 \[ \left [\frac {3 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt {c} \log \left (-\frac {a^{2} c x^{2} + 2 \, \sqrt {-a^{2} c x^{2} + c} \sqrt {c} - 2 \, c}{x^{2}}\right ) - 2 \, \sqrt {-a^{2} c x^{2} + c} {\left (4 \, a x - 5\right )}}{6 \, {\left (a^{2} c^{2} x^{2} - 2 \, a c^{2} x + c^{2}\right )}}, -\frac {3 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} \sqrt {-c}}{a^{2} c x^{2} - c}\right ) + \sqrt {-a^{2} c x^{2} + c} {\left (4 \, a x - 5\right )}}{3 \, {\left (a^{2} c^{2} x^{2} - 2 \, a c^{2} x + c^{2}\right )}}\right ] \]
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 [B] time = 0.04, size = 152, normalized size = 1.85 \[ \frac {1}{c \sqrt {-a^{2} c \,x^{2}+c}}-\frac {\ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {-a^{2} c \,x^{2}+c}}{x}\right )}{c^{\frac {3}{2}}}-\frac {2}{3 a c \left (x -\frac {1}{a}\right ) \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 a c \left (x -\frac {1}{a}\right )}}-\frac {2 \left (-2 \left (x -\frac {1}{a}\right ) a^{2} c -2 a c \right )}{3 a \,c^{2} \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 a c \left (x -\frac {1}{a}\right )}} \]
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 \[ -\int \frac {{\left (a x + 1\right )}^{2}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} {\left (a^{2} x^{2} - 1\right )} x}\,{d x} \]
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 \[ -\int \frac {{\left (a\,x+1\right )}^2}{x\,{\left (c-a^2\,c\,x^2\right )}^{3/2}\,\left (a^2\,x^2-1\right )} \,d x \]
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 \[ - \int \frac {a x}{- a^{3} c x^{4} \sqrt {- a^{2} c x^{2} + c} + a^{2} c x^{3} \sqrt {- a^{2} c x^{2} + c} + a c x^{2} \sqrt {- a^{2} c x^{2} + c} - c x \sqrt {- a^{2} c x^{2} + c}}\, dx - \int \frac {1}{- a^{3} c x^{4} \sqrt {- a^{2} c x^{2} + c} + a^{2} c x^{3} \sqrt {- a^{2} c x^{2} + c} + a c x^{2} \sqrt {- a^{2} c x^{2} + c} - c x \sqrt {- a^{2} c x^{2} + c}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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