Optimal. Leaf size=133 \[ -\frac {\sin ^{-1}(a x)}{a^7 c^3}+\frac {x^5 (a x+1)}{5 a^2 c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac {16 \sqrt {1-a^2 x^2}}{5 a^7 c^3}+\frac {x (8 a x+5)}{5 a^6 c^3 \sqrt {1-a^2 x^2}}-\frac {x^3 (6 a x+5)}{15 a^4 c^3 \left (1-a^2 x^2\right )^{3/2}} \]
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Rubi [A] time = 0.15, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {6148, 819, 641, 216} \[ \frac {x^5 (a x+1)}{5 a^2 c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac {x^3 (6 a x+5)}{15 a^4 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {x (8 a x+5)}{5 a^6 c^3 \sqrt {1-a^2 x^2}}+\frac {16 \sqrt {1-a^2 x^2}}{5 a^7 c^3}-\frac {\sin ^{-1}(a x)}{a^7 c^3} \]
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^6}{\left (c-a^2 c x^2\right )^3} \, dx &=\frac {\int \frac {x^6 (1+a x)}{\left (1-a^2 x^2\right )^{7/2}} \, dx}{c^3}\\ &=\frac {x^5 (1+a x)}{5 a^2 c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac {\int \frac {x^4 (5+6 a x)}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{5 a^2 c^3}\\ &=\frac {x^5 (1+a x)}{5 a^2 c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac {x^3 (5+6 a x)}{15 a^4 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {\int \frac {x^2 (15+24 a x)}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{15 a^4 c^3}\\ &=\frac {x^5 (1+a x)}{5 a^2 c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac {x^3 (5+6 a x)}{15 a^4 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {x (5+8 a x)}{5 a^6 c^3 \sqrt {1-a^2 x^2}}-\frac {\int \frac {15+48 a x}{\sqrt {1-a^2 x^2}} \, dx}{15 a^6 c^3}\\ &=\frac {x^5 (1+a x)}{5 a^2 c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac {x^3 (5+6 a x)}{15 a^4 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {x (5+8 a x)}{5 a^6 c^3 \sqrt {1-a^2 x^2}}+\frac {16 \sqrt {1-a^2 x^2}}{5 a^7 c^3}-\frac {\int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{a^6 c^3}\\ &=\frac {x^5 (1+a x)}{5 a^2 c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac {x^3 (5+6 a x)}{15 a^4 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {x (5+8 a x)}{5 a^6 c^3 \sqrt {1-a^2 x^2}}+\frac {16 \sqrt {1-a^2 x^2}}{5 a^7 c^3}-\frac {\sin ^{-1}(a x)}{a^7 c^3}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 108, normalized size = 0.81 \[ \frac {-15 a^5 x^5+38 a^4 x^4+52 a^3 x^3-87 a^2 x^2-15 (a x-1)^2 (a x+1) \sqrt {1-a^2 x^2} \sin ^{-1}(a x)-33 a x+48}{15 a^7 c^3 (a x-1)^2 (a x+1) \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 213, normalized size = 1.60 \[ \frac {48 \, a^{5} x^{5} - 48 \, a^{4} x^{4} - 96 \, a^{3} x^{3} + 96 \, a^{2} x^{2} + 48 \, a x + 30 \, {\left (a^{5} x^{5} - a^{4} x^{4} - 2 \, a^{3} x^{3} + 2 \, a^{2} x^{2} + a x - 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (15 \, a^{5} x^{5} - 38 \, a^{4} x^{4} - 52 \, a^{3} x^{3} + 87 \, a^{2} x^{2} + 33 \, a x - 48\right )} \sqrt {-a^{2} x^{2} + 1} - 48}{15 \, {\left (a^{12} c^{3} x^{5} - a^{11} c^{3} x^{4} - 2 \, a^{10} c^{3} x^{3} + 2 \, a^{9} c^{3} x^{2} + a^{8} c^{3} x - a^{7} c^{3}\right )}} \]
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 \[ \int -\frac {{\left (a x + 1\right )} x^{6}}{{\left (a^{2} c x^{2} - c\right )}^{3} \sqrt {-a^{2} x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 262, normalized size = 1.97 \[ \frac {\sqrt {-a^{2} x^{2}+1}}{a^{7} c^{3}}-\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{c^{3} a^{6} \sqrt {a^{2}}}-\frac {23 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{60 c^{3} a^{9} \left (x -\frac {1}{a}\right )^{2}}-\frac {493 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{240 c^{3} a^{8} \left (x -\frac {1}{a}\right )}-\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{20 c^{3} a^{10} \left (x -\frac {1}{a}\right )^{3}}-\frac {\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{24 c^{3} a^{9} \left (x +\frac {1}{a}\right )^{2}}+\frac {25 \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{48 c^{3} a^{8} \left (x +\frac {1}{a}\right )} \]
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 \[ -\int \frac {{\left (a x + 1\right )} x^{6}}{{\left (a^{2} c x^{2} - c\right )}^{3} \sqrt {-a^{2} x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.92, size = 379, normalized size = 2.85 \[ \frac {a^6\,\sqrt {1-a^2\,x^2}}{30\,\left (a^{15}\,c^3\,x^2-2\,a^{14}\,c^3\,x+a^{13}\,c^3\right )}-\frac {\sqrt {1-a^2\,x^2}}{24\,\left (a^9\,c^3\,x^2+2\,a^8\,c^3\,x+a^7\,c^3\right )}-\frac {\sqrt {1-a^2\,x^2}}{20\,\sqrt {-a^2}\,\left (a^5\,c^3\,\sqrt {-a^2}+3\,a^7\,c^3\,x^2\,\sqrt {-a^2}-a^8\,c^3\,x^3\,\sqrt {-a^2}-3\,a^6\,c^3\,x\,\sqrt {-a^2}\right )}-\frac {5\,\sqrt {1-a^2\,x^2}}{12\,\left (a^9\,c^3\,x^2-2\,a^8\,c^3\,x+a^7\,c^3\right )}-\frac {25\,\sqrt {1-a^2\,x^2}}{48\,\left (a^5\,c^3\,\sqrt {-a^2}+a^6\,c^3\,x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}-\frac {493\,\sqrt {1-a^2\,x^2}}{240\,\left (a^5\,c^3\,\sqrt {-a^2}-a^6\,c^3\,x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}+\frac {\sqrt {1-a^2\,x^2}}{a^7\,c^3}-\frac {\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{a^6\,c^3\,\sqrt {-a^2}} \]
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 \[ \frac {\int \frac {x^{6}}{- a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a x^{7}}{- a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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