Optimal. Leaf size=125 \[ \frac {(a x+1)^3}{5 a c^4 \left (1-a^2 x^2\right )^{5/2}}-\frac {6 (a x+1)^2}{5 a c^4 \left (1-a^2 x^2\right )^{3/2}}+\frac {24 (a x+1)}{5 a c^4 \sqrt {1-a^2 x^2}}+\frac {\sqrt {1-a^2 x^2}}{a c^4}-\frac {3 \sin ^{-1}(a x)}{a c^4} \]
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Rubi [A] time = 0.35, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 8, 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} \[ \frac {(a x+1)^3}{5 a c^4 \left (1-a^2 x^2\right )^{5/2}}-\frac {6 (a x+1)^2}{5 a c^4 \left (1-a^2 x^2\right )^{3/2}}+\frac {24 (a x+1)}{5 a c^4 \sqrt {1-a^2 x^2}}+\frac {\sqrt {1-a^2 x^2}}{a c^4}-\frac {3 \sin ^{-1}(a x)}{a c^4} \]
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 )^4} \, dx &=\frac {a^4 \int \frac {e^{-\tanh ^{-1}(a x)} x^4}{(1-a x)^4} \, dx}{c^4}\\ &=\frac {a^4 \int \frac {x^4}{(1-a x)^3 \sqrt {1-a^2 x^2}} \, dx}{c^4}\\ &=\frac {a^4 \int \frac {x^4 (1+a x)^3}{\left (1-a^2 x^2\right )^{7/2}} \, dx}{c^4}\\ &=\frac {(1+a x)^3}{5 a c^4 \left (1-a^2 x^2\right )^{5/2}}-\frac {a^4 \int \frac {(1+a x)^2 \left (\frac {3}{a^4}+\frac {5 x}{a^3}+\frac {5 x^2}{a^2}+\frac {5 x^3}{a}\right )}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{5 c^4}\\ &=\frac {(1+a x)^3}{5 a c^4 \left (1-a^2 x^2\right )^{5/2}}-\frac {6 (1+a x)^2}{5 a c^4 \left (1-a^2 x^2\right )^{3/2}}+\frac {a^4 \int \frac {(1+a x) \left (\frac {27}{a^4}+\frac {30 x}{a^3}+\frac {15 x^2}{a^2}\right )}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{15 c^4}\\ &=\frac {(1+a x)^3}{5 a c^4 \left (1-a^2 x^2\right )^{5/2}}-\frac {6 (1+a x)^2}{5 a c^4 \left (1-a^2 x^2\right )^{3/2}}+\frac {24 (1+a x)}{5 a c^4 \sqrt {1-a^2 x^2}}-\frac {a^4 \int \frac {\frac {45}{a^4}+\frac {15 x}{a^3}}{\sqrt {1-a^2 x^2}} \, dx}{15 c^4}\\ &=\frac {(1+a x)^3}{5 a c^4 \left (1-a^2 x^2\right )^{5/2}}-\frac {6 (1+a x)^2}{5 a c^4 \left (1-a^2 x^2\right )^{3/2}}+\frac {24 (1+a x)}{5 a c^4 \sqrt {1-a^2 x^2}}+\frac {\sqrt {1-a^2 x^2}}{a c^4}-\frac {3 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{c^4}\\ &=\frac {(1+a x)^3}{5 a c^4 \left (1-a^2 x^2\right )^{5/2}}-\frac {6 (1+a x)^2}{5 a c^4 \left (1-a^2 x^2\right )^{3/2}}+\frac {24 (1+a x)}{5 a c^4 \sqrt {1-a^2 x^2}}+\frac {\sqrt {1-a^2 x^2}}{a c^4}-\frac {3 \sin ^{-1}(a x)}{a c^4}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 61, normalized size = 0.49 \[ \frac {\frac {\sqrt {1-a^2 x^2} \left (5 a^3 x^3-39 a^2 x^2+57 a x-24\right )}{(a x-1)^3}-15 \sin ^{-1}(a x)}{5 a c^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 143, normalized size = 1.14 \[ \frac {24 \, a^{3} x^{3} - 72 \, a^{2} x^{2} + 72 \, a x + 30 \, {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (5 \, a^{3} x^{3} - 39 \, a^{2} x^{2} + 57 \, a x - 24\right )} \sqrt {-a^{2} x^{2} + 1} - 24}{5 \, {\left (a^{4} c^{4} x^{3} - 3 \, a^{3} c^{4} x^{2} + 3 \, a^{2} c^{4} x - a c^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 180, normalized size = 1.44 \[ -\frac {3 \, \arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{c^{4} {\left | a \right |}} + \frac {\sqrt {-a^{2} x^{2} + 1}}{a c^{4}} - \frac {2 \, {\left (\frac {80 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}}{a^{2} x} - \frac {120 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2}}{a^{4} x^{2}} + \frac {70 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3}}{a^{6} x^{3}} - \frac {15 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4}}{a^{8} x^{4}} - 19\right )}}{5 \, c^{4} {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} - 1\right )}^{5} {\left | a \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 286, normalized size = 2.29 \[ \frac {17 \left (-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )\right )^{\frac {3}{2}}}{8 a^{3} c^{4} \left (x -\frac {1}{a}\right )^{2}}+\frac {49 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{16 a \,c^{4}}-\frac {49 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}\right )}{16 c^{4} \sqrt {a^{2}}}+\frac {11 \left (-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )\right )^{\frac {3}{2}}}{20 a^{4} c^{4} \left (x -\frac {1}{a}\right )^{3}}+\frac {\left (-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )\right )^{\frac {3}{2}}}{10 a^{5} c^{4} \left (x -\frac {1}{a}\right )^{4}}+\frac {\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{16 a \,c^{4}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{16 c^{4} \sqrt {a^{2}}} \]
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 \[ \int \frac {\sqrt {-a^{2} x^{2} + 1}}{{\left (a x + 1\right )} {\left (c - \frac {c}{a x}\right )}^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.84, size = 272, normalized size = 2.18 \[ \frac {2\,a^4\,\sqrt {1-a^2\,x^2}}{15\,\left (a^7\,c^4\,x^2-2\,a^6\,c^4\,x+a^5\,c^4\right )}-\frac {3\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{c^4\,\sqrt {-a^2}}-\frac {4\,a\,\sqrt {1-a^2\,x^2}}{3\,\left (a^4\,c^4\,x^2-2\,a^3\,c^4\,x+a^2\,c^4\right )}+\frac {\sqrt {1-a^2\,x^2}}{a\,c^4}+\frac {24\,\sqrt {1-a^2\,x^2}}{5\,\sqrt {-a^2}\,\left (c^4\,x\,\sqrt {-a^2}-\frac {c^4\,\sqrt {-a^2}}{a}\right )}+\frac {\sqrt {1-a^2\,x^2}}{5\,\sqrt {-a^2}\,\left (3\,c^4\,x\,\sqrt {-a^2}-\frac {c^4\,\sqrt {-a^2}}{a}+a^2\,c^4\,x^3\,\sqrt {-a^2}-3\,a\,c^4\,x^2\,\sqrt {-a^2}\right )} \]
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 \[ \frac {a^{4} \int \frac {x^{4} \sqrt {- a^{2} x^{2} + 1}}{a^{5} x^{5} - 3 a^{4} x^{4} + 2 a^{3} x^{3} + 2 a^{2} x^{2} - 3 a x + 1}\, dx}{c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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