Optimal. Leaf size=125 \[ -\frac {(a x+1)^2}{5 a c^5 \left (1-a^2 x^2\right )^{5/2}}+\frac {22 (a x+1)}{15 a c^5 \left (1-a^2 x^2\right )^{3/2}}-\frac {\sqrt {1-a^2 x^2}}{a c^5}-\frac {2 (23 a x+30)}{15 a c^5 \sqrt {1-a^2 x^2}}+\frac {2 \sin ^{-1}(a x)}{a c^5} \]
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Rubi [A] time = 0.32, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {6131, 6128, 852, 1635, 1814, 641, 216} \[ -\frac {(a x+1)^2}{5 a c^5 \left (1-a^2 x^2\right )^{5/2}}+\frac {22 (a x+1)}{15 a c^5 \left (1-a^2 x^2\right )^{3/2}}-\frac {\sqrt {1-a^2 x^2}}{a c^5}-\frac {2 (23 a x+30)}{15 a c^5 \sqrt {1-a^2 x^2}}+\frac {2 \sin ^{-1}(a x)}{a c^5} \]
Antiderivative was successfully verified.
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Rule 216
Rule 641
Rule 852
Rule 1635
Rule 1814
Rule 6128
Rule 6131
Rubi steps
\begin {align*} \int \frac {e^{-3 \tanh ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^5} \, dx &=-\frac {a^5 \int \frac {e^{-3 \tanh ^{-1}(a x)} x^5}{(1-a x)^5} \, dx}{c^5}\\ &=-\frac {a^5 \int \frac {x^5}{(1-a x)^2 \left (1-a^2 x^2\right )^{3/2}} \, dx}{c^5}\\ &=-\frac {a^5 \int \frac {x^5 (1+a x)^2}{\left (1-a^2 x^2\right )^{7/2}} \, dx}{c^5}\\ &=-\frac {(1+a x)^2}{5 a c^5 \left (1-a^2 x^2\right )^{5/2}}+\frac {a^5 \int \frac {(1+a x) \left (\frac {2}{a^5}+\frac {5 x}{a^4}+\frac {5 x^2}{a^3}+\frac {5 x^3}{a^2}+\frac {5 x^4}{a}\right )}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{5 c^5}\\ &=-\frac {(1+a x)^2}{5 a c^5 \left (1-a^2 x^2\right )^{5/2}}+\frac {22 (1+a x)}{15 a c^5 \left (1-a^2 x^2\right )^{3/2}}-\frac {a^5 \int \frac {\frac {16}{a^5}+\frac {45 x}{a^4}+\frac {30 x^2}{a^3}+\frac {15 x^3}{a^2}}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{15 c^5}\\ &=-\frac {(1+a x)^2}{5 a c^5 \left (1-a^2 x^2\right )^{5/2}}+\frac {22 (1+a x)}{15 a c^5 \left (1-a^2 x^2\right )^{3/2}}-\frac {2 (30+23 a x)}{15 a c^5 \sqrt {1-a^2 x^2}}+\frac {a^5 \int \frac {\frac {30}{a^5}+\frac {15 x}{a^4}}{\sqrt {1-a^2 x^2}} \, dx}{15 c^5}\\ &=-\frac {(1+a x)^2}{5 a c^5 \left (1-a^2 x^2\right )^{5/2}}+\frac {22 (1+a x)}{15 a c^5 \left (1-a^2 x^2\right )^{3/2}}-\frac {2 (30+23 a x)}{15 a c^5 \sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{a c^5}+\frac {2 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{c^5}\\ &=-\frac {(1+a x)^2}{5 a c^5 \left (1-a^2 x^2\right )^{5/2}}+\frac {22 (1+a x)}{15 a c^5 \left (1-a^2 x^2\right )^{3/2}}-\frac {2 (30+23 a x)}{15 a c^5 \sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{a c^5}+\frac {2 \sin ^{-1}(a x)}{a c^5}\\ \end {align*}
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Mathematica [A] time = 0.24, size = 76, normalized size = 0.61 \[ \frac {\frac {\sqrt {1-a^2 x^2} \left (-15 a^4 x^4+76 a^3 x^3-32 a^2 x^2-82 a x+56\right )}{(a x-1)^3 (a x+1)}+30 \sin ^{-1}(a x)}{15 a c^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 151, normalized size = 1.21 \[ -\frac {56 \, a^{4} x^{4} - 112 \, a^{3} x^{3} + 112 \, a x + 60 \, {\left (a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a x - 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (15 \, a^{4} x^{4} - 76 \, a^{3} x^{3} + 32 \, a^{2} x^{2} + 82 \, a x - 56\right )} \sqrt {-a^{2} x^{2} + 1} - 56}{15 \, {\left (a^{5} c^{5} x^{4} - 2 \, a^{4} c^{5} x^{3} + 2 \, a^{2} c^{5} x - a c^{5}\right )}} \]
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 = 0.06, size = 468, normalized size = 3.74 \[ -\frac {49 \left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{384 a \,c^{5}}+\frac {31 \left (-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )\right )^{\frac {5}{2}}}{48 a^{4} c^{5} \left (x -\frac {1}{a}\right )^{3}}-\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{32 a^{4} c^{5} \left (x +\frac {1}{a}\right )^{3}}-\frac {9 \left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{64 a^{3} c^{5} \left (x +\frac {1}{a}\right )^{2}}-\frac {49 \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}\, x}{256 c^{5}}-\frac {49 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{256 c^{5} \sqrt {a^{2}}}-\frac {139 \left (-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )\right )^{\frac {5}{2}}}{96 a^{3} c^{5} \left (x -\frac {1}{a}\right )^{2}}+\frac {561 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}\, x}{256 c^{5}}+\frac {561 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}\right )}{256 c^{5} \sqrt {a^{2}}}+\frac {\left (-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )\right )^{\frac {5}{2}}}{40 a^{6} c^{5} \left (x -\frac {1}{a}\right )^{5}}+\frac {7 \left (-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )\right )^{\frac {5}{2}}}{48 a^{5} c^{5} \left (x -\frac {1}{a}\right )^{4}}-\frac {187 \left (-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )\right )^{\frac {3}{2}}}{128 a \,c^{5}} \]
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 \[ \int \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (a x + 1\right )}^{3} {\left (c - \frac {c}{a x}\right )}^{5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.91, size = 275, normalized size = 2.20 \[ \frac {41\,a\,\sqrt {1-a^2\,x^2}}{60\,\left (a^4\,c^5\,x^2-2\,a^3\,c^5\,x+a^2\,c^5\right )}+\frac {2\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{c^5\,\sqrt {-a^2}}-\frac {\sqrt {1-a^2\,x^2}}{a\,c^5}+\frac {\sqrt {1-a^2\,x^2}}{8\,\sqrt {-a^2}\,\left (c^5\,x\,\sqrt {-a^2}+\frac {c^5\,\sqrt {-a^2}}{a}\right )}-\frac {383\,\sqrt {1-a^2\,x^2}}{120\,\sqrt {-a^2}\,\left (c^5\,x\,\sqrt {-a^2}-\frac {c^5\,\sqrt {-a^2}}{a}\right )}-\frac {\sqrt {1-a^2\,x^2}}{10\,\sqrt {-a^2}\,\left (3\,c^5\,x\,\sqrt {-a^2}-\frac {c^5\,\sqrt {-a^2}}{a}+a^2\,c^5\,x^3\,\sqrt {-a^2}-3\,a\,c^5\,x^2\,\sqrt {-a^2}\right )} \]
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 \[ \frac {a^{5} \left (\int \frac {x^{5} \sqrt {- a^{2} x^{2} + 1}}{a^{8} x^{8} - 2 a^{7} x^{7} - 2 a^{6} x^{6} + 6 a^{5} x^{5} - 6 a^{3} x^{3} + 2 a^{2} x^{2} + 2 a x - 1}\, dx + \int \left (- \frac {a^{2} x^{7} \sqrt {- a^{2} x^{2} + 1}}{a^{8} x^{8} - 2 a^{7} x^{7} - 2 a^{6} x^{6} + 6 a^{5} x^{5} - 6 a^{3} x^{3} + 2 a^{2} x^{2} + 2 a x - 1}\right )\, dx\right )}{c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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