Optimal. Leaf size=99 \[ \frac {\sin ^{-1}(a x)}{a^5 c^2}+\frac {x^3 (a x+1)}{3 a^2 c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac {8 \sqrt {1-a^2 x^2}}{3 a^5 c^2}-\frac {x (4 a x+3)}{3 a^4 c^2 \sqrt {1-a^2 x^2}} \]
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Rubi [A] time = 0.12, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {6148, 819, 641, 216} \[ \frac {x^3 (a x+1)}{3 a^2 c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac {x (4 a x+3)}{3 a^4 c^2 \sqrt {1-a^2 x^2}}-\frac {8 \sqrt {1-a^2 x^2}}{3 a^5 c^2}+\frac {\sin ^{-1}(a x)}{a^5 c^2} \]
Antiderivative was successfully verified.
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Rule 216
Rule 641
Rule 819
Rule 6148
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)} x^4}{\left (c-a^2 c x^2\right )^2} \, dx &=\frac {\int \frac {x^4 (1+a x)}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{c^2}\\ &=\frac {x^3 (1+a x)}{3 a^2 c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac {\int \frac {x^2 (3+4 a x)}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{3 a^2 c^2}\\ &=\frac {x^3 (1+a x)}{3 a^2 c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac {x (3+4 a x)}{3 a^4 c^2 \sqrt {1-a^2 x^2}}+\frac {\int \frac {3+8 a x}{\sqrt {1-a^2 x^2}} \, dx}{3 a^4 c^2}\\ &=\frac {x^3 (1+a x)}{3 a^2 c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac {x (3+4 a x)}{3 a^4 c^2 \sqrt {1-a^2 x^2}}-\frac {8 \sqrt {1-a^2 x^2}}{3 a^5 c^2}+\frac {\int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{a^4 c^2}\\ &=\frac {x^3 (1+a x)}{3 a^2 c^2 \left (1-a^2 x^2\right )^{3/2}}-\frac {x (3+4 a x)}{3 a^4 c^2 \sqrt {1-a^2 x^2}}-\frac {8 \sqrt {1-a^2 x^2}}{3 a^5 c^2}+\frac {\sin ^{-1}(a x)}{a^5 c^2}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 78, normalized size = 0.79 \[ \frac {3 a^3 x^3-7 a^2 x^2+3 (a x-1) \sqrt {1-a^2 x^2} \sin ^{-1}(a x)-5 a x+8}{3 a^5 c^2 (a x-1) \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 144, normalized size = 1.45 \[ -\frac {8 \, a^{3} x^{3} - 8 \, a^{2} x^{2} - 8 \, a x + 6 \, {\left (a^{3} x^{3} - a^{2} x^{2} - a x + 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (3 \, a^{3} x^{3} - 7 \, a^{2} x^{2} - 5 \, a x + 8\right )} \sqrt {-a^{2} x^{2} + 1} + 8}{3 \, {\left (a^{8} c^{2} x^{3} - a^{7} c^{2} x^{2} - a^{6} c^{2} x + a^{5} c^{2}\right )}} \]
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 \[ \int \frac {{\left (a x + 1\right )} x^{4}}{{\left (a^{2} c x^{2} - c\right )}^{2} \sqrt {-a^{2} x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 180, normalized size = 1.82 \[ -\frac {\sqrt {-a^{2} x^{2}+1}}{a^{5} c^{2}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{c^{2} a^{4} \sqrt {a^{2}}}+\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{6 c^{2} a^{7} \left (x -\frac {1}{a}\right )^{2}}+\frac {19 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{12 c^{2} a^{6} \left (x -\frac {1}{a}\right )}-\frac {\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{4 c^{2} a^{6} \left (x +\frac {1}{a}\right )} \]
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 \[ \int \frac {{\left (a x + 1\right )} x^{4}}{{\left (a^{2} c x^{2} - c\right )}^{2} \sqrt {-a^{2} x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 196, normalized size = 1.98 \[ \frac {\sqrt {1-a^2\,x^2}}{6\,\left (a^7\,c^2\,x^2-2\,a^6\,c^2\,x+a^5\,c^2\right )}+\frac {\sqrt {1-a^2\,x^2}}{4\,\left (a^3\,c^2\,\sqrt {-a^2}+a^4\,c^2\,x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}+\frac {19\,\sqrt {1-a^2\,x^2}}{12\,\left (a^3\,c^2\,\sqrt {-a^2}-a^4\,c^2\,x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}-\frac {\sqrt {1-a^2\,x^2}}{a^5\,c^2}+\frac {\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{a^4\,c^2\,\sqrt {-a^2}} \]
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 \[ \frac {\int \frac {x^{4}}{a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a x^{5}}{a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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