Optimal. Leaf size=94 \[ -\frac {(a x+1)^2}{3 a c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {8 (a x+1)}{3 a c^3 \sqrt {1-a^2 x^2}}+\frac {\sqrt {1-a^2 x^2}}{a c^3}-\frac {2 \sin ^{-1}(a x)}{a c^3} \]
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Rubi [A] time = 0.26, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {6131, 6128, 852, 1635, 641, 216} \[ -\frac {(a x+1)^2}{3 a c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {8 (a x+1)}{3 a c^3 \sqrt {1-a^2 x^2}}+\frac {\sqrt {1-a^2 x^2}}{a c^3}-\frac {2 \sin ^{-1}(a x)}{a c^3} \]
Antiderivative was successfully verified.
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Rule 216
Rule 641
Rule 852
Rule 1635
Rule 6128
Rule 6131
Rubi steps
\begin {align*} \int \frac {e^{-\tanh ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx &=-\frac {a^3 \int \frac {e^{-\tanh ^{-1}(a x)} x^3}{(1-a x)^3} \, dx}{c^3}\\ &=-\frac {a^3 \int \frac {x^3}{(1-a x)^2 \sqrt {1-a^2 x^2}} \, dx}{c^3}\\ &=-\frac {a^3 \int \frac {x^3 (1+a x)^2}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{c^3}\\ &=-\frac {(1+a x)^2}{3 a c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {a^3 \int \frac {(1+a x) \left (\frac {2}{a^3}+\frac {3 x}{a^2}+\frac {3 x^2}{a}\right )}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{3 c^3}\\ &=-\frac {(1+a x)^2}{3 a c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {8 (1+a x)}{3 a c^3 \sqrt {1-a^2 x^2}}-\frac {a^3 \int \frac {\frac {6}{a^3}+\frac {3 x}{a^2}}{\sqrt {1-a^2 x^2}} \, dx}{3 c^3}\\ &=-\frac {(1+a x)^2}{3 a c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {8 (1+a x)}{3 a c^3 \sqrt {1-a^2 x^2}}+\frac {\sqrt {1-a^2 x^2}}{a c^3}-\frac {2 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{c^3}\\ &=-\frac {(1+a x)^2}{3 a c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {8 (1+a x)}{3 a c^3 \sqrt {1-a^2 x^2}}+\frac {\sqrt {1-a^2 x^2}}{a c^3}-\frac {2 \sin ^{-1}(a x)}{a c^3}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 53, normalized size = 0.56 \[ \frac {\frac {\sqrt {1-a^2 x^2} \left (3 a^2 x^2-14 a x+10\right )}{(a x-1)^2}-6 \sin ^{-1}(a x)}{3 a c^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.64, size = 107, normalized size = 1.14 \[ \frac {10 \, a^{2} x^{2} - 20 \, a x + 12 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (3 \, a^{2} x^{2} - 14 \, a x + 10\right )} \sqrt {-a^{2} x^{2} + 1} + 10}{3 \, {\left (a^{3} c^{3} x^{2} - 2 \, a^{2} c^{3} x + a c^{3}\right )}} \]
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 = 0.05, size = 242, normalized size = 2.57 \[ \frac {5 \left (-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )\right )^{\frac {3}{2}}}{4 a^{3} c^{3} \left (x -\frac {1}{a}\right )^{2}}+\frac {17 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{8 a \,c^{3}}-\frac {17 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}\right )}{8 c^{3} \sqrt {a^{2}}}+\frac {\left (-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )\right )^{\frac {3}{2}}}{6 a^{4} c^{3} \left (x -\frac {1}{a}\right )^{3}}+\frac {\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{8 a \,c^{3}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{8 c^{3} \sqrt {a^{2}}} \]
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 \[ \int \frac {\sqrt {-a^{2} x^{2} + 1}}{{\left (a x + 1\right )} {\left (c - \frac {c}{a x}\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 139, normalized size = 1.48 \[ \frac {\sqrt {1-a^2\,x^2}}{a\,c^3}-\frac {2\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{c^3\,\sqrt {-a^2}}-\frac {a\,\sqrt {1-a^2\,x^2}}{3\,\left (a^4\,c^3\,x^2-2\,a^3\,c^3\,x+a^2\,c^3\right )}+\frac {8\,\sqrt {1-a^2\,x^2}}{3\,\sqrt {-a^2}\,\left (c^3\,x\,\sqrt {-a^2}-\frac {c^3\,\sqrt {-a^2}}{a}\right )} \]
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 \[ \frac {a^{3} \int \frac {x^{3} \sqrt {- a^{2} x^{2} + 1}}{a^{4} x^{4} - 2 a^{3} x^{3} + 2 a x - 1}\, dx}{c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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