Optimal. Leaf size=96 \[ -\frac {x (4 a x+3)}{3 c^4 \sqrt {1-a^2 x^2}}-\frac {8 \sqrt {1-a^2 x^2}}{3 a c^4}+\frac {a^2 x^3 (a x+1)}{3 c^4 \left (1-a^2 x^2\right )^{3/2}}+\frac {\sin ^{-1}(a x)}{a c^4} \]
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Rubi [A] time = 0.16, antiderivative size = 96, 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, 850, 819, 641, 216} \[ \frac {a^2 x^3 (a x+1)}{3 c^4 \left (1-a^2 x^2\right )^{3/2}}-\frac {x (4 a x+3)}{3 c^4 \sqrt {1-a^2 x^2}}-\frac {8 \sqrt {1-a^2 x^2}}{3 a c^4}+\frac {\sin ^{-1}(a x)}{a c^4} \]
Antiderivative was successfully verified.
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Rule 216
Rule 641
Rule 819
Rule 850
Rule 6128
Rule 6131
Rubi steps
\begin {align*} \int \frac {e^{-3 \tanh ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx &=\frac {a^4 \int \frac {e^{-3 \tanh ^{-1}(a x)} x^4}{(1-a x)^4} \, dx}{c^4}\\ &=\frac {a^4 \int \frac {x^4}{(1-a x) \left (1-a^2 x^2\right )^{3/2}} \, dx}{c^4}\\ &=\frac {a^4 \int \frac {x^4 (1+a x)}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{c^4}\\ &=\frac {a^2 x^3 (1+a x)}{3 c^4 \left (1-a^2 x^2\right )^{3/2}}-\frac {a^2 \int \frac {x^2 (3+4 a x)}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{3 c^4}\\ &=\frac {a^2 x^3 (1+a x)}{3 c^4 \left (1-a^2 x^2\right )^{3/2}}-\frac {x (3+4 a x)}{3 c^4 \sqrt {1-a^2 x^2}}+\frac {\int \frac {3+8 a x}{\sqrt {1-a^2 x^2}} \, dx}{3 c^4}\\ &=\frac {a^2 x^3 (1+a x)}{3 c^4 \left (1-a^2 x^2\right )^{3/2}}-\frac {x (3+4 a x)}{3 c^4 \sqrt {1-a^2 x^2}}-\frac {8 \sqrt {1-a^2 x^2}}{3 a c^4}+\frac {\int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{c^4}\\ &=\frac {a^2 x^3 (1+a x)}{3 c^4 \left (1-a^2 x^2\right )^{3/2}}-\frac {x (3+4 a x)}{3 c^4 \sqrt {1-a^2 x^2}}-\frac {8 \sqrt {1-a^2 x^2}}{3 a c^4}+\frac {\sin ^{-1}(a x)}{a c^4}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 68, normalized size = 0.71 \[ \frac {\frac {\sqrt {1-a^2 x^2} \left (-3 a^3 x^3+7 a^2 x^2+5 a x-8\right )}{(a x-1)^2 (a x+1)}+3 \sin ^{-1}(a x)}{3 a c^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.85, size = 142, normalized size = 1.48 \[ -\frac {8 \, a^{3} x^{3} - 8 \, a^{2} x^{2} - 8 \, a x + 6 \, {\left (a^{3} x^{3} - a^{2} x^{2} - a x + 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (3 \, a^{3} x^{3} - 7 \, a^{2} x^{2} - 5 \, a x + 8\right )} \sqrt {-a^{2} x^{2} + 1} + 8}{3 \, {\left (a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} - a^{2} c^{4} x + a c^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (a x + 1\right )}^{3} {\left (c - \frac {c}{a x}\right )}^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 424, normalized size = 4.42 \[ -\frac {23 \left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{96 a \,c^{4}}-\frac {29 \left (-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )\right )^{\frac {3}{2}}}{32 a \,c^{4}}+\frac {\left (-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )\right )^{\frac {5}{2}}}{24 a^{5} c^{4} \left (x -\frac {1}{a}\right )^{4}}+\frac {17 \left (-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )\right )^{\frac {5}{2}}}{48 a^{4} c^{4} \left (x -\frac {1}{a}\right )^{3}}-\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{16 a^{4} c^{4} \left (x +\frac {1}{a}\right )^{3}}-\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{4 a^{3} c^{4} \left (x +\frac {1}{a}\right )^{2}}-\frac {23 \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}\, x}{64 c^{4}}-\frac {23 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{64 c^{4} \sqrt {a^{2}}}-\frac {43 \left (-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )\right )^{\frac {5}{2}}}{48 a^{3} c^{4} \left (x -\frac {1}{a}\right )^{2}}+\frac {87 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}\, x}{64 c^{4}}+\frac {87 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}\right )}{64 c^{4} \sqrt {a^{2}}} \]
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 \[ \int \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (a x + 1\right )}^{3} {\left (c - \frac {c}{a x}\right )}^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 188, normalized size = 1.96 \[ \frac {a\,\sqrt {1-a^2\,x^2}}{6\,\left (a^4\,c^4\,x^2-2\,a^3\,c^4\,x+a^2\,c^4\right )}+\frac {\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{c^4\,\sqrt {-a^2}}-\frac {\sqrt {1-a^2\,x^2}}{a\,c^4}+\frac {\sqrt {1-a^2\,x^2}}{4\,\sqrt {-a^2}\,\left (c^4\,x\,\sqrt {-a^2}+\frac {c^4\,\sqrt {-a^2}}{a}\right )}-\frac {19\,\sqrt {1-a^2\,x^2}}{12\,\sqrt {-a^2}\,\left (c^4\,x\,\sqrt {-a^2}-\frac {c^4\,\sqrt {-a^2}}{a}\right )} \]
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 \[ \frac {a^{4} \left (\int \frac {x^{4} \sqrt {- a^{2} x^{2} + 1}}{a^{7} x^{7} - a^{6} x^{6} - 3 a^{5} x^{5} + 3 a^{4} x^{4} + 3 a^{3} x^{3} - 3 a^{2} x^{2} - a x + 1}\, dx + \int \left (- \frac {a^{2} x^{6} \sqrt {- a^{2} x^{2} + 1}}{a^{7} x^{7} - a^{6} x^{6} - 3 a^{5} x^{5} + 3 a^{4} x^{4} + 3 a^{3} x^{3} - 3 a^{2} x^{2} - a x + 1}\right )\, dx\right )}{c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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