Optimal. Leaf size=130 \[ -\frac {x (5-8 a x)}{5 c^3 \sqrt {1-a^2 x^2}}+\frac {16 \sqrt {1-a^2 x^2}}{5 a c^3}+\frac {a^2 x^3 (5-6 a x)}{15 c^3 \left (1-a^2 x^2\right )^{3/2}}-\frac {a^4 x^5 (1-a x)}{5 c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac {\sin ^{-1}(a x)}{a c^3} \]
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Rubi [A] time = 0.20, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 7, 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} \[ -\frac {a^4 x^5 (1-a x)}{5 c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac {a^2 x^3 (5-6 a x)}{15 c^3 \left (1-a^2 x^2\right )^{3/2}}-\frac {x (5-8 a x)}{5 c^3 \sqrt {1-a^2 x^2}}+\frac {16 \sqrt {1-a^2 x^2}}{5 a c^3}+\frac {\sin ^{-1}(a x)}{a c^3} \]
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 )^3} \, dx &=-\frac {a^6 \int \frac {e^{-\tanh ^{-1}(a x)} x^6}{\left (1-a^2 x^2\right )^3} \, dx}{c^3}\\ &=-\frac {a^6 \int \frac {x^6 (1-a x)}{\left (1-a^2 x^2\right )^{7/2}} \, dx}{c^3}\\ &=-\frac {a^4 x^5 (1-a x)}{5 c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac {a^4 \int \frac {x^4 (5-6 a x)}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{5 c^3}\\ &=-\frac {a^4 x^5 (1-a x)}{5 c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac {a^2 x^3 (5-6 a x)}{15 c^3 \left (1-a^2 x^2\right )^{3/2}}-\frac {a^2 \int \frac {x^2 (15-24 a x)}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{15 c^3}\\ &=-\frac {a^4 x^5 (1-a x)}{5 c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac {a^2 x^3 (5-6 a x)}{15 c^3 \left (1-a^2 x^2\right )^{3/2}}-\frac {x (5-8 a x)}{5 c^3 \sqrt {1-a^2 x^2}}+\frac {\int \frac {15-48 a x}{\sqrt {1-a^2 x^2}} \, dx}{15 c^3}\\ &=-\frac {a^4 x^5 (1-a x)}{5 c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac {a^2 x^3 (5-6 a x)}{15 c^3 \left (1-a^2 x^2\right )^{3/2}}-\frac {x (5-8 a x)}{5 c^3 \sqrt {1-a^2 x^2}}+\frac {16 \sqrt {1-a^2 x^2}}{5 a c^3}+\frac {\int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{c^3}\\ &=-\frac {a^4 x^5 (1-a x)}{5 c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac {a^2 x^3 (5-6 a x)}{15 c^3 \left (1-a^2 x^2\right )^{3/2}}-\frac {x (5-8 a x)}{5 c^3 \sqrt {1-a^2 x^2}}+\frac {16 \sqrt {1-a^2 x^2}}{5 a c^3}+\frac {\sin ^{-1}(a x)}{a c^3}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 108, normalized size = 0.83 \[ \frac {-15 a^5 x^5-38 a^4 x^4+52 a^3 x^3+87 a^2 x^2+15 (a x-1) (a x+1)^2 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)-33 a x-48}{15 a c^3 (a x-1) (a x+1)^2 \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 208, normalized size = 1.60 \[ \frac {48 \, a^{5} x^{5} + 48 \, a^{4} x^{4} - 96 \, a^{3} x^{3} - 96 \, a^{2} x^{2} + 48 \, a x - 30 \, {\left (a^{5} x^{5} + a^{4} x^{4} - 2 \, a^{3} x^{3} - 2 \, a^{2} x^{2} + a x + 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (15 \, a^{5} x^{5} + 38 \, a^{4} x^{4} - 52 \, a^{3} x^{3} - 87 \, a^{2} x^{2} + 33 \, a x + 48\right )} \sqrt {-a^{2} x^{2} + 1} + 48}{15 \, {\left (a^{6} c^{3} x^{5} + a^{5} c^{3} x^{4} - 2 \, a^{4} c^{3} x^{3} - 2 \, a^{3} c^{3} x^{2} + a^{2} c^{3} x + a c^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-a^{2} x^{2} + 1}}{{\left (a x + 1\right )} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 356, normalized size = 2.74 \[ \frac {\left (-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )\right )^{\frac {3}{2}}}{4 a^{3} c^{3} \left (x -\frac {1}{a}\right )^{2}}+\frac {19 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{32 a \,c^{3}}-\frac {19 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}\right )}{32 c^{3} \sqrt {a^{2}}}+\frac {\left (-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )\right )^{\frac {3}{2}}}{48 a^{4} c^{3} \left (x -\frac {1}{a}\right )^{3}}+\frac {15 \left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{16 a^{3} c^{3} \left (x +\frac {1}{a}\right )^{2}}+\frac {51 \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{32 a \,c^{3}}+\frac {51 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{32 c^{3} \sqrt {a^{2}}}-\frac {43 \left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{240 a^{4} c^{3} \left (x +\frac {1}{a}\right )^{3}}+\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{40 a^{5} c^{3} \left (x +\frac {1}{a}\right )^{4}} \]
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 \[ \int \frac {\sqrt {-a^{2} x^{2} + 1}}{{\left (a x + 1\right )} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.28, size = 365, normalized size = 2.81 \[ \frac {\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{c^3\,\sqrt {-a^2}}-\frac {5\,a\,\sqrt {1-a^2\,x^2}}{12\,\left (a^4\,c^3\,x^2+2\,a^3\,c^3\,x+a^2\,c^3\right )}-\frac {a\,\sqrt {1-a^2\,x^2}}{24\,\left (a^4\,c^3\,x^2-2\,a^3\,c^3\,x+a^2\,c^3\right )}+\frac {a^6\,\sqrt {1-a^2\,x^2}}{30\,\left (a^9\,c^3\,x^2+2\,a^8\,c^3\,x+a^7\,c^3\right )}+\frac {\sqrt {1-a^2\,x^2}}{a\,c^3}-\frac {493\,\sqrt {1-a^2\,x^2}}{240\,\sqrt {-a^2}\,\left (c^3\,x\,\sqrt {-a^2}+\frac {c^3\,\sqrt {-a^2}}{a}\right )}+\frac {25\,\sqrt {1-a^2\,x^2}}{48\,\sqrt {-a^2}\,\left (c^3\,x\,\sqrt {-a^2}-\frac {c^3\,\sqrt {-a^2}}{a}\right )}-\frac {\sqrt {1-a^2\,x^2}}{20\,\sqrt {-a^2}\,\left (3\,c^3\,x\,\sqrt {-a^2}+\frac {c^3\,\sqrt {-a^2}}{a}+a^2\,c^3\,x^3\,\sqrt {-a^2}+3\,a\,c^3\,x^2\,\sqrt {-a^2}\right )} \]
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 \[ \frac {a^{6} \int \frac {x^{6} \sqrt {- a^{2} x^{2} + 1}}{a^{7} x^{7} + a^{6} x^{6} - 3 a^{5} x^{5} - 3 a^{4} x^{4} + 3 a^{3} x^{3} + 3 a^{2} x^{2} - a x - 1}\, dx}{c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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