Optimal. Leaf size=168 \[ \frac {\left (1-a^2 x^2\right )^{3/2}}{a c^4 (1-a x)^2}+\frac {184 \left (1-a^2 x^2\right )^{3/2}}{105 a c^4 (1-a x)^3}-\frac {26 \left (1-a^2 x^2\right )^{3/2}}{35 a c^4 (1-a x)^4}+\frac {\left (1-a^2 x^2\right )^{3/2}}{7 a c^4 (1-a x)^5}-\frac {10 \sqrt {1-a^2 x^2}}{a c^4 (1-a x)}+\frac {5 \sin ^{-1}(a x)}{a c^4} \]
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Rubi [A] time = 0.35, antiderivative size = 168, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6131, 6128, 1639, 1637, 659, 651, 663, 216} \[ \frac {\left (1-a^2 x^2\right )^{3/2}}{a c^4 (1-a x)^2}+\frac {184 \left (1-a^2 x^2\right )^{3/2}}{105 a c^4 (1-a x)^3}-\frac {26 \left (1-a^2 x^2\right )^{3/2}}{35 a c^4 (1-a x)^4}+\frac {\left (1-a^2 x^2\right )^{3/2}}{7 a c^4 (1-a x)^5}-\frac {10 \sqrt {1-a^2 x^2}}{a c^4 (1-a x)}+\frac {5 \sin ^{-1}(a x)}{a c^4} \]
Antiderivative was successfully verified.
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Rule 216
Rule 651
Rule 659
Rule 663
Rule 1637
Rule 1639
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 \sqrt {1-a^2 x^2}}{(1-a x)^5} \, dx}{c^4}\\ &=\frac {\left (1-a^2 x^2\right )^{3/2}}{a c^4 (1-a x)^2}-\frac {\int \frac {\sqrt {1-a^2 x^2} \left (2 a^2-7 a^3 x+9 a^4 x^2-5 a^5 x^3\right )}{(1-a x)^5} \, dx}{a^2 c^4}\\ &=\frac {\left (1-a^2 x^2\right )^{3/2}}{a c^4 (1-a x)^2}-\frac {\int \left (\frac {a^2 \sqrt {1-a^2 x^2}}{(-1+a x)^5}+\frac {4 a^2 \sqrt {1-a^2 x^2}}{(-1+a x)^4}+\frac {6 a^2 \sqrt {1-a^2 x^2}}{(-1+a x)^3}+\frac {5 a^2 \sqrt {1-a^2 x^2}}{(-1+a x)^2}\right ) \, dx}{a^2 c^4}\\ &=\frac {\left (1-a^2 x^2\right )^{3/2}}{a c^4 (1-a x)^2}-\frac {\int \frac {\sqrt {1-a^2 x^2}}{(-1+a x)^5} \, dx}{c^4}-\frac {4 \int \frac {\sqrt {1-a^2 x^2}}{(-1+a x)^4} \, dx}{c^4}-\frac {5 \int \frac {\sqrt {1-a^2 x^2}}{(-1+a x)^2} \, dx}{c^4}-\frac {6 \int \frac {\sqrt {1-a^2 x^2}}{(-1+a x)^3} \, dx}{c^4}\\ &=-\frac {10 \sqrt {1-a^2 x^2}}{a c^4 (1-a x)}+\frac {\left (1-a^2 x^2\right )^{3/2}}{7 a c^4 (1-a x)^5}-\frac {4 \left (1-a^2 x^2\right )^{3/2}}{5 a c^4 (1-a x)^4}+\frac {2 \left (1-a^2 x^2\right )^{3/2}}{a c^4 (1-a x)^3}+\frac {\left (1-a^2 x^2\right )^{3/2}}{a c^4 (1-a x)^2}+\frac {2 \int \frac {\sqrt {1-a^2 x^2}}{(-1+a x)^4} \, dx}{7 c^4}+\frac {4 \int \frac {\sqrt {1-a^2 x^2}}{(-1+a x)^3} \, dx}{5 c^4}+\frac {5 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{c^4}\\ &=-\frac {10 \sqrt {1-a^2 x^2}}{a c^4 (1-a x)}+\frac {\left (1-a^2 x^2\right )^{3/2}}{7 a c^4 (1-a x)^5}-\frac {26 \left (1-a^2 x^2\right )^{3/2}}{35 a c^4 (1-a x)^4}+\frac {26 \left (1-a^2 x^2\right )^{3/2}}{15 a c^4 (1-a x)^3}+\frac {\left (1-a^2 x^2\right )^{3/2}}{a c^4 (1-a x)^2}+\frac {5 \sin ^{-1}(a x)}{a c^4}-\frac {2 \int \frac {\sqrt {1-a^2 x^2}}{(-1+a x)^3} \, dx}{35 c^4}\\ &=-\frac {10 \sqrt {1-a^2 x^2}}{a c^4 (1-a x)}+\frac {\left (1-a^2 x^2\right )^{3/2}}{7 a c^4 (1-a x)^5}-\frac {26 \left (1-a^2 x^2\right )^{3/2}}{35 a c^4 (1-a x)^4}+\frac {184 \left (1-a^2 x^2\right )^{3/2}}{105 a c^4 (1-a x)^3}+\frac {\left (1-a^2 x^2\right )^{3/2}}{a c^4 (1-a x)^2}+\frac {5 \sin ^{-1}(a x)}{a c^4}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 69, normalized size = 0.41 \[ \frac {\frac {\sqrt {1-a^2 x^2} \left (-105 a^4 x^4+1444 a^3 x^3-3256 a^2 x^2+2771 a x-824\right )}{(a x-1)^4}+525 \sin ^{-1}(a x)}{105 a c^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 177, normalized size = 1.05 \[ -\frac {824 \, a^{4} x^{4} - 3296 \, a^{3} x^{3} + 4944 \, a^{2} x^{2} - 3296 \, a x + 1050 \, {\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (105 \, a^{4} x^{4} - 1444 \, a^{3} x^{3} + 3256 \, a^{2} x^{2} - 2771 \, a x + 824\right )} \sqrt {-a^{2} x^{2} + 1} + 824}{105 \, {\left (a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} + 6 \, a^{3} c^{4} x^{2} - 4 \, a^{2} c^{4} x + a c^{4}\right )}} \]
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 [A] time = 0.05, size = 228, normalized size = 1.36 \[ -\frac {\sqrt {-a^{2} x^{2}+1}}{a \,c^{4}}+\frac {5 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{c^{4} \sqrt {a^{2}}}+\frac {446 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{105 a^{3} c^{4} \left (x -\frac {1}{a}\right )^{2}}+\frac {1024 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{105 a^{2} c^{4} \left (x -\frac {1}{a}\right )}+\frac {57 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{35 a^{4} c^{4} \left (x -\frac {1}{a}\right )^{3}}+\frac {2 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{7 a^{5} c^{4} \left (x -\frac {1}{a}\right )^{4}} \]
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 \[ \int \frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1} {\left (c - \frac {c}{a x}\right )}^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.83, size = 389, normalized size = 2.32 \[ \frac {16\,a\,\sqrt {1-a^2\,x^2}}{3\,\left (a^4\,c^4\,x^2-2\,a^3\,c^4\,x+a^2\,c^4\right )}+\frac {5\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{c^4\,\sqrt {-a^2}}+\frac {4\,a^3\,\sqrt {1-a^2\,x^2}}{35\,\left (a^6\,c^4\,x^2-2\,a^5\,c^4\,x+a^4\,c^4\right )}-\frac {6\,a^4\,\sqrt {1-a^2\,x^2}}{5\,\left (a^7\,c^4\,x^2-2\,a^6\,c^4\,x+a^5\,c^4\right )}-\frac {\sqrt {1-a^2\,x^2}}{a\,c^4}+\frac {2\,a\,\sqrt {1-a^2\,x^2}}{7\,\left (a^6\,c^4\,x^4-4\,a^5\,c^4\,x^3+6\,a^4\,c^4\,x^2-4\,a^3\,c^4\,x+a^2\,c^4\right )}-\frac {1024\,\sqrt {1-a^2\,x^2}}{105\,\sqrt {-a^2}\,\left (c^4\,x\,\sqrt {-a^2}-\frac {c^4\,\sqrt {-a^2}}{a}\right )}-\frac {57\,\sqrt {1-a^2\,x^2}}{35\,\sqrt {-a^2}\,\left (3\,c^4\,x\,\sqrt {-a^2}-\frac {c^4\,\sqrt {-a^2}}{a}+a^2\,c^4\,x^3\,\sqrt {-a^2}-3\,a\,c^4\,x^2\,\sqrt {-a^2}\right )} \]
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 \[ \frac {a^{4} \left (\int \frac {x^{4}}{a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 4 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} + 6 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} - 4 a x \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a x^{5}}{a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 4 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} + 6 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} - 4 a x \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx\right )}{c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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