Optimal. Leaf size=74 \[ \frac {a x+1}{3 c^2 \left (1-a^2 x^2\right )^{3/2}}+\frac {2 a x+3}{3 c^2 \sqrt {1-a^2 x^2}}-\frac {\tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{c^2} \]
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Rubi [A] time = 0.12, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {6148, 823, 12, 266, 63, 208} \[ \frac {a x+1}{3 c^2 \left (1-a^2 x^2\right )^{3/2}}+\frac {2 a x+3}{3 c^2 \sqrt {1-a^2 x^2}}-\frac {\tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{c^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 63
Rule 208
Rule 266
Rule 823
Rule 6148
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)}}{x \left (c-a^2 c x^2\right )^2} \, dx &=\frac {\int \frac {1+a x}{x \left (1-a^2 x^2\right )^{5/2}} \, dx}{c^2}\\ &=\frac {1+a x}{3 c^2 \left (1-a^2 x^2\right )^{3/2}}+\frac {\int \frac {3 a^2+2 a^3 x}{x \left (1-a^2 x^2\right )^{3/2}} \, dx}{3 a^2 c^2}\\ &=\frac {1+a x}{3 c^2 \left (1-a^2 x^2\right )^{3/2}}+\frac {3+2 a x}{3 c^2 \sqrt {1-a^2 x^2}}+\frac {\int \frac {3 a^4}{x \sqrt {1-a^2 x^2}} \, dx}{3 a^4 c^2}\\ &=\frac {1+a x}{3 c^2 \left (1-a^2 x^2\right )^{3/2}}+\frac {3+2 a x}{3 c^2 \sqrt {1-a^2 x^2}}+\frac {\int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx}{c^2}\\ &=\frac {1+a x}{3 c^2 \left (1-a^2 x^2\right )^{3/2}}+\frac {3+2 a x}{3 c^2 \sqrt {1-a^2 x^2}}+\frac {\operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )}{2 c^2}\\ &=\frac {1+a x}{3 c^2 \left (1-a^2 x^2\right )^{3/2}}+\frac {3+2 a x}{3 c^2 \sqrt {1-a^2 x^2}}-\frac {\operatorname {Subst}\left (\int \frac {1}{\frac {1}{a^2}-\frac {x^2}{a^2}} \, dx,x,\sqrt {1-a^2 x^2}\right )}{a^2 c^2}\\ &=\frac {1+a x}{3 c^2 \left (1-a^2 x^2\right )^{3/2}}+\frac {3+2 a x}{3 c^2 \sqrt {1-a^2 x^2}}-\frac {\tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{c^2}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 77, normalized size = 1.04 \[ \frac {2 a^2 x^2-3 (a x-1) \sqrt {1-a^2 x^2} \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )+a x-4}{3 c^2 (a x-1) \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.74, size = 127, normalized size = 1.72 \[ \frac {4 \, a^{3} x^{3} - 4 \, a^{2} x^{2} - 4 \, a x + 3 \, {\left (a^{3} x^{3} - a^{2} x^{2} - a x + 1\right )} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - {\left (2 \, a^{2} x^{2} + a x - 4\right )} \sqrt {-a^{2} x^{2} + 1} + 4}{3 \, {\left (a^{3} c^{2} x^{3} - a^{2} c^{2} x^{2} - a c^{2} x + c^{2}\right )}} \]
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 \[ \int \frac {a x + 1}{{\left (a^{2} c x^{2} - c\right )}^{2} \sqrt {-a^{2} x^{2} + 1} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 182, normalized size = 2.46 \[ \frac {-\arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )+\frac {\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}}{2 a}-\frac {3 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{4 a \left (x -\frac {1}{a}\right )}+\frac {\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{4 a \left (x +\frac {1}{a}\right )}}{c^{2}} \]
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 \[ \int \frac {a x + 1}{{\left (a^{2} c x^{2} - c\right )}^{2} \sqrt {-a^{2} x^{2} + 1} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.91, size = 173, normalized size = 2.34 \[ \frac {a^2\,\sqrt {1-a^2\,x^2}}{6\,\left (a^4\,c^2\,x^2-2\,a^3\,c^2\,x+a^2\,c^2\right )}-\frac {a\,\sqrt {1-a^2\,x^2}}{4\,\sqrt {-a^2}\,\left (c^2\,x\,\sqrt {-a^2}+\frac {c^2\,\sqrt {-a^2}}{a}\right )}+\frac {11\,a\,\sqrt {1-a^2\,x^2}}{12\,\sqrt {-a^2}\,\left (c^2\,x\,\sqrt {-a^2}-\frac {c^2\,\sqrt {-a^2}}{a}\right )}+\frac {\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{c^2} \]
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 \[ \frac {\int \frac {a}{a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {1}{a^{4} x^{5} \sqrt {- a^{2} x^{2} + 1} - 2 a^{2} x^{3} \sqrt {- a^{2} x^{2} + 1} + x \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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