Optimal. Leaf size=105 \[ \frac {a x+1}{3 c^2 x \left (1-a^2 x^2\right )^{3/2}}-\frac {8 \sqrt {1-a^2 x^2}}{3 c^2 x}+\frac {3 a x+4}{3 c^2 x \sqrt {1-a^2 x^2}}-\frac {a \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{c^2} \]
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Rubi [A] time = 0.14, antiderivative size = 105, 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, 807, 266, 63, 208} \[ \frac {a x+1}{3 c^2 x \left (1-a^2 x^2\right )^{3/2}}-\frac {8 \sqrt {1-a^2 x^2}}{3 c^2 x}+\frac {3 a x+4}{3 c^2 x \sqrt {1-a^2 x^2}}-\frac {a \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{c^2} \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 266
Rule 807
Rule 823
Rule 6148
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)}}{x^2 \left (c-a^2 c x^2\right )^2} \, dx &=\frac {\int \frac {1+a x}{x^2 \left (1-a^2 x^2\right )^{5/2}} \, dx}{c^2}\\ &=\frac {1+a x}{3 c^2 x \left (1-a^2 x^2\right )^{3/2}}+\frac {\int \frac {4 a^2+3 a^3 x}{x^2 \left (1-a^2 x^2\right )^{3/2}} \, dx}{3 a^2 c^2}\\ &=\frac {1+a x}{3 c^2 x \left (1-a^2 x^2\right )^{3/2}}+\frac {4+3 a x}{3 c^2 x \sqrt {1-a^2 x^2}}+\frac {\int \frac {8 a^4+3 a^5 x}{x^2 \sqrt {1-a^2 x^2}} \, dx}{3 a^4 c^2}\\ &=\frac {1+a x}{3 c^2 x \left (1-a^2 x^2\right )^{3/2}}+\frac {4+3 a x}{3 c^2 x \sqrt {1-a^2 x^2}}-\frac {8 \sqrt {1-a^2 x^2}}{3 c^2 x}+\frac {a \int \frac {1}{x \sqrt {1-a^2 x^2}} \, dx}{c^2}\\ &=\frac {1+a x}{3 c^2 x \left (1-a^2 x^2\right )^{3/2}}+\frac {4+3 a x}{3 c^2 x \sqrt {1-a^2 x^2}}-\frac {8 \sqrt {1-a^2 x^2}}{3 c^2 x}+\frac {a \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 x \left (1-a^2 x^2\right )^{3/2}}+\frac {4+3 a x}{3 c^2 x \sqrt {1-a^2 x^2}}-\frac {8 \sqrt {1-a^2 x^2}}{3 c^2 x}-\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 c^2}\\ &=\frac {1+a x}{3 c^2 x \left (1-a^2 x^2\right )^{3/2}}+\frac {4+3 a x}{3 c^2 x \sqrt {1-a^2 x^2}}-\frac {8 \sqrt {1-a^2 x^2}}{3 c^2 x}-\frac {a \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{c^2}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 91, normalized size = 0.87 \[ \frac {8 a^3 x^3-5 a^2 x^2-3 a x (a x-1) \sqrt {1-a^2 x^2} \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )-7 a x+3}{3 c^2 x (a x-1) \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.64, size = 153, normalized size = 1.46 \[ \frac {4 \, a^{4} x^{4} - 4 \, a^{3} x^{3} - 4 \, a^{2} x^{2} + 4 \, a x + 3 \, {\left (a^{4} x^{4} - a^{3} x^{3} - a^{2} x^{2} + a x\right )} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - {\left (8 \, a^{3} x^{3} - 5 \, a^{2} x^{2} - 7 \, a x + 3\right )} \sqrt {-a^{2} x^{2} + 1}}{3 \, {\left (a^{3} c^{2} x^{4} - a^{2} c^{2} x^{3} - a c^{2} x^{2} + c^{2} x\right )}} \]
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 \[ \int \frac {a x + 1}{{\left (a^{2} c x^{2} - c\right )}^{2} \sqrt {-a^{2} x^{2} + 1} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 150, normalized size = 1.43 \[ \frac {-a \arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )-\frac {\sqrt {-a^{2} x^{2}+1}}{x}+\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{6 a \left (x -\frac {1}{a}\right )^{2}}-\frac {17 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{12 \left (x -\frac {1}{a}\right )}-\frac {\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{4 \left (x +\frac {1}{a}\right )}}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 144, normalized size = 1.37 \[ -\frac {\frac {3 \, a^{2} \log \left (\sqrt {-a^{2} x^{2} + 1} + 1\right )}{c^{2}} - \frac {3 \, a^{2} \log \left (\sqrt {-a^{2} x^{2} + 1} - 1\right )}{c^{2}} + \frac {2 \, {\left (3 \, {\left (a^{2} x^{2} - 1\right )} a^{2} - a^{2}\right )}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} c^{2}}}{6 \, a} + \frac {8 \, a^{4} x^{4} - 12 \, a^{2} x^{2} + 3}{3 \, {\left (a^{2} c^{2} x^{3} - c^{2} x\right )} \sqrt {a x + 1} \sqrt {-a x + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.90, size = 198, normalized size = 1.89 \[ \frac {a^3\,\sqrt {1-a^2\,x^2}}{6\,\left (a^4\,c^2\,x^2-2\,a^3\,c^2\,x+a^2\,c^2\right )}-\frac {\sqrt {1-a^2\,x^2}}{c^2\,x}+\frac {a^2\,\sqrt {1-a^2\,x^2}}{4\,\sqrt {-a^2}\,\left (c^2\,x\,\sqrt {-a^2}+\frac {c^2\,\sqrt {-a^2}}{a}\right )}+\frac {17\,a^2\,\sqrt {1-a^2\,x^2}}{12\,\sqrt {-a^2}\,\left (c^2\,x\,\sqrt {-a^2}-\frac {c^2\,\sqrt {-a^2}}{a}\right )}+\frac {a\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{c^2} \]
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 \[ \frac {\int \frac {a}{a^{4} x^{5} \sqrt {- a^{2} x^{2} + 1} - 2 a^{2} x^{3} \sqrt {- a^{2} x^{2} + 1} + x \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {1}{a^{4} x^{6} \sqrt {- a^{2} x^{2} + 1} - 2 a^{2} x^{4} \sqrt {- a^{2} x^{2} + 1} + x^{2} \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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