Optimal. Leaf size=97 \[ -\frac {x (3-4 a x)}{3 c^2 \sqrt {1-a^2 x^2}}+\frac {8 \sqrt {1-a^2 x^2}}{3 a c^2}+\frac {a^2 x^3 (1-a x)}{3 c^2 \left (1-a^2 x^2\right )^{3/2}}+\frac {\sin ^{-1}(a x)}{a c^2} \]
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Rubi [A] time = 0.16, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {6157, 6149, 819, 641, 216} \[ \frac {a^2 x^3 (1-a x)}{3 c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac {x (3-4 a x)}{3 c^2 \sqrt {1-a^2 x^2}}+\frac {8 \sqrt {1-a^2 x^2}}{3 a c^2}+\frac {\sin ^{-1}(a x)}{a c^2} \]
Antiderivative was successfully verified.
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Rule 216
Rule 641
Rule 819
Rule 6149
Rule 6157
Rubi steps
\begin {align*} \int \frac {e^{-\tanh ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx &=\frac {a^4 \int \frac {e^{-\tanh ^{-1}(a x)} x^4}{\left (1-a^2 x^2\right )^2} \, dx}{c^2}\\ &=\frac {a^4 \int \frac {x^4 (1-a x)}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{c^2}\\ &=\frac {a^2 x^3 (1-a x)}{3 c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac {a^2 \int \frac {x^2 (3-4 a x)}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{3 c^2}\\ &=\frac {a^2 x^3 (1-a x)}{3 c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac {x (3-4 a x)}{3 c^2 \sqrt {1-a^2 x^2}}+\frac {\int \frac {3-8 a x}{\sqrt {1-a^2 x^2}} \, dx}{3 c^2}\\ &=\frac {a^2 x^3 (1-a x)}{3 c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac {x (3-4 a x)}{3 c^2 \sqrt {1-a^2 x^2}}+\frac {8 \sqrt {1-a^2 x^2}}{3 a c^2}+\frac {\int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{c^2}\\ &=\frac {a^2 x^3 (1-a x)}{3 c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac {x (3-4 a x)}{3 c^2 \sqrt {1-a^2 x^2}}+\frac {8 \sqrt {1-a^2 x^2}}{3 a c^2}+\frac {\sin ^{-1}(a x)}{a c^2}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 78, normalized size = 0.80 \[ \frac {-3 a^3 x^3-7 a^2 x^2+3 (a x+1) \sqrt {1-a^2 x^2} \sin ^{-1}(a x)+5 a x+8}{3 a c^2 (a x+1) \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 141, normalized size = 1.45 \[ \frac {8 \, a^{3} x^{3} + 8 \, a^{2} x^{2} - 8 \, a x - 6 \, {\left (a^{3} x^{3} + a^{2} x^{2} - a x - 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (3 \, a^{3} x^{3} + 7 \, a^{2} x^{2} - 5 \, a x - 8\right )} \sqrt {-a^{2} x^{2} + 1} - 8}{3 \, {\left (a^{4} c^{2} x^{3} + a^{3} c^{2} x^{2} - a^{2} c^{2} x - a c^{2}\right )}} \]
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 \[ \int \frac {\sqrt {-a^{2} x^{2} + 1}}{{\left (a x + 1\right )} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 274, normalized size = 2.82 \[ \frac {\left (-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )\right )^{\frac {3}{2}}}{8 a^{3} c^{2} \left (x -\frac {1}{a}\right )^{2}}+\frac {7 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{16 a \,c^{2}}-\frac {7 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}\right )}{16 c^{2} \sqrt {a^{2}}}+\frac {3 \left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{4 a^{3} c^{2} \left (x +\frac {1}{a}\right )^{2}}+\frac {23 \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{16 a \,c^{2}}+\frac {23 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{16 c^{2} \sqrt {a^{2}}}-\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{12 a^{4} c^{2} \left (x +\frac {1}{a}\right )^{3}} \]
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 \[ \int \frac {\sqrt {-a^{2} x^{2} + 1}}{{\left (a x + 1\right )} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.16, size = 188, normalized size = 1.94 \[ \frac {\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{c^2\,\sqrt {-a^2}}-\frac {a\,\sqrt {1-a^2\,x^2}}{6\,\left (a^4\,c^2\,x^2+2\,a^3\,c^2\,x+a^2\,c^2\right )}+\frac {\sqrt {1-a^2\,x^2}}{a\,c^2}-\frac {19\,\sqrt {1-a^2\,x^2}}{12\,\sqrt {-a^2}\,\left (c^2\,x\,\sqrt {-a^2}+\frac {c^2\,\sqrt {-a^2}}{a}\right )}+\frac {\sqrt {1-a^2\,x^2}}{4\,\sqrt {-a^2}\,\left (c^2\,x\,\sqrt {-a^2}-\frac {c^2\,\sqrt {-a^2}}{a}\right )} \]
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 \[ \frac {a^{4} \int \frac {x^{4} \sqrt {- a^{2} x^{2} + 1}}{a^{5} x^{5} + a^{4} x^{4} - 2 a^{3} x^{3} - 2 a^{2} x^{2} + a x + 1}\, dx}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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