Optimal. Leaf size=189 \[ -\frac {(1-a x)^3}{9 a c^4 \left (1-a^2 x^2\right )^{9/2}}+\frac {22 (1-a x)^2}{21 a c^4 \left (1-a^2 x^2\right )^{7/2}}-\frac {478 (1-a x)}{105 a c^4 \left (1-a^2 x^2\right )^{5/2}}-\frac {\sqrt {1-a^2 x^2}}{a c^4}-\frac {4 (630-431 a x)}{315 a c^4 \sqrt {1-a^2 x^2}}+\frac {2 (1155-829 a x)}{315 a c^4 \left (1-a^2 x^2\right )^{3/2}}-\frac {3 \sin ^{-1}(a x)}{a c^4} \]
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Rubi [A] time = 0.64, antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {6157, 6149, 1635, 1814, 641, 216} \[ -\frac {(1-a x)^3}{9 a c^4 \left (1-a^2 x^2\right )^{9/2}}+\frac {22 (1-a x)^2}{21 a c^4 \left (1-a^2 x^2\right )^{7/2}}-\frac {478 (1-a x)}{105 a c^4 \left (1-a^2 x^2\right )^{5/2}}-\frac {\sqrt {1-a^2 x^2}}{a c^4}-\frac {4 (630-431 a x)}{315 a c^4 \sqrt {1-a^2 x^2}}+\frac {2 (1155-829 a x)}{315 a c^4 \left (1-a^2 x^2\right )^{3/2}}-\frac {3 \sin ^{-1}(a x)}{a c^4} \]
Antiderivative was successfully verified.
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Rule 216
Rule 641
Rule 1635
Rule 1814
Rule 6149
Rule 6157
Rubi steps
\begin {align*} \int \frac {e^{-3 \tanh ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^4} \, dx &=\frac {a^8 \int \frac {e^{-3 \tanh ^{-1}(a x)} x^8}{\left (1-a^2 x^2\right )^4} \, dx}{c^4}\\ &=\frac {a^8 \int \frac {x^8 (1-a x)^3}{\left (1-a^2 x^2\right )^{11/2}} \, dx}{c^4}\\ &=-\frac {(1-a x)^3}{9 a c^4 \left (1-a^2 x^2\right )^{9/2}}-\frac {a^8 \int \frac {(1-a x)^2 \left (\frac {3}{a^8}-\frac {9 x}{a^7}+\frac {9 x^2}{a^6}-\frac {9 x^3}{a^5}+\frac {9 x^4}{a^4}-\frac {9 x^5}{a^3}+\frac {9 x^6}{a^2}-\frac {9 x^7}{a}\right )}{\left (1-a^2 x^2\right )^{9/2}} \, dx}{9 c^4}\\ &=-\frac {(1-a x)^3}{9 a c^4 \left (1-a^2 x^2\right )^{9/2}}+\frac {22 (1-a x)^2}{21 a c^4 \left (1-a^2 x^2\right )^{7/2}}+\frac {a^8 \int \frac {(1-a x) \left (\frac {111}{a^8}-\frac {378 x}{a^7}+\frac {315 x^2}{a^6}-\frac {252 x^3}{a^5}+\frac {189 x^4}{a^4}-\frac {126 x^5}{a^3}+\frac {63 x^6}{a^2}\right )}{\left (1-a^2 x^2\right )^{7/2}} \, dx}{63 c^4}\\ &=-\frac {(1-a x)^3}{9 a c^4 \left (1-a^2 x^2\right )^{9/2}}+\frac {22 (1-a x)^2}{21 a c^4 \left (1-a^2 x^2\right )^{7/2}}-\frac {478 (1-a x)}{105 a c^4 \left (1-a^2 x^2\right )^{5/2}}-\frac {a^8 \int \frac {\frac {879}{a^8}-\frac {4725 x}{a^7}+\frac {3150 x^2}{a^6}-\frac {1890 x^3}{a^5}+\frac {945 x^4}{a^4}-\frac {315 x^5}{a^3}}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{315 c^4}\\ &=-\frac {(1-a x)^3}{9 a c^4 \left (1-a^2 x^2\right )^{9/2}}+\frac {22 (1-a x)^2}{21 a c^4 \left (1-a^2 x^2\right )^{7/2}}-\frac {478 (1-a x)}{105 a c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac {2 (1155-829 a x)}{315 a c^4 \left (1-a^2 x^2\right )^{3/2}}+\frac {a^8 \int \frac {\frac {2337}{a^8}-\frac {6615 x}{a^7}+\frac {2835 x^2}{a^6}-\frac {945 x^3}{a^5}}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{945 c^4}\\ &=-\frac {(1-a x)^3}{9 a c^4 \left (1-a^2 x^2\right )^{9/2}}+\frac {22 (1-a x)^2}{21 a c^4 \left (1-a^2 x^2\right )^{7/2}}-\frac {478 (1-a x)}{105 a c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac {2 (1155-829 a x)}{315 a c^4 \left (1-a^2 x^2\right )^{3/2}}-\frac {4 (630-431 a x)}{315 a c^4 \sqrt {1-a^2 x^2}}-\frac {a^8 \int \frac {\frac {2835}{a^8}-\frac {945 x}{a^7}}{\sqrt {1-a^2 x^2}} \, dx}{945 c^4}\\ &=-\frac {(1-a x)^3}{9 a c^4 \left (1-a^2 x^2\right )^{9/2}}+\frac {22 (1-a x)^2}{21 a c^4 \left (1-a^2 x^2\right )^{7/2}}-\frac {478 (1-a x)}{105 a c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac {2 (1155-829 a x)}{315 a c^4 \left (1-a^2 x^2\right )^{3/2}}-\frac {4 (630-431 a x)}{315 a c^4 \sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{a c^4}-\frac {3 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{c^4}\\ &=-\frac {(1-a x)^3}{9 a c^4 \left (1-a^2 x^2\right )^{9/2}}+\frac {22 (1-a x)^2}{21 a c^4 \left (1-a^2 x^2\right )^{7/2}}-\frac {478 (1-a x)}{105 a c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac {2 (1155-829 a x)}{315 a c^4 \left (1-a^2 x^2\right )^{3/2}}-\frac {4 (630-431 a x)}{315 a c^4 \sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{a c^4}-\frac {3 \sin ^{-1}(a x)}{a c^4}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 122, normalized size = 0.65 \[ \frac {315 a^7 x^7+2669 a^6 x^6+2967 a^5 x^5-4029 a^4 x^4-7399 a^3 x^3-339 a^2 x^2-945 (a x-1) (a x+1)^4 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)+4047 a x+1664}{315 a (a x-1) \sqrt {1-a^2 x^2} (a c x+c)^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 278, normalized size = 1.47 \[ -\frac {1664 \, a^{7} x^{7} + 4992 \, a^{6} x^{6} + 1664 \, a^{5} x^{5} - 8320 \, a^{4} x^{4} - 8320 \, a^{3} x^{3} + 1664 \, a^{2} x^{2} + 4992 \, a x - 1890 \, {\left (a^{7} x^{7} + 3 \, a^{6} x^{6} + a^{5} x^{5} - 5 \, a^{4} x^{4} - 5 \, a^{3} x^{3} + a^{2} x^{2} + 3 \, a x + 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (315 \, a^{7} x^{7} + 2669 \, a^{6} x^{6} + 2967 \, a^{5} x^{5} - 4029 \, a^{4} x^{4} - 7399 \, a^{3} x^{3} - 339 \, a^{2} x^{2} + 4047 \, a x + 1664\right )} \sqrt {-a^{2} x^{2} + 1} + 1664}{315 \, {\left (a^{8} c^{4} x^{7} + 3 \, a^{7} c^{4} x^{6} + a^{6} c^{4} x^{5} - 5 \, a^{5} c^{4} x^{4} - 5 \, a^{4} c^{4} x^{3} + a^{3} c^{4} x^{2} + 3 \, a^{2} c^{4} x + a c^{4}\right )}} \]
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 \[ \int \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (a x + 1\right )}^{3} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.08, size = 576, normalized size = 3.05 \[ -\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{144 a^{8} c^{4} \left (x +\frac {1}{a}\right )^{7}}+\frac {13 \left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{252 a^{7} c^{4} \left (x +\frac {1}{a}\right )^{6}}-\frac {1629 \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 )}{512 c^{4} \sqrt {a^{2}}}-\frac {811 \left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{384 a^{3} c^{4} \left (x +\frac {1}{a}\right )^{2}}-\frac {1629 \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}\, x}{512 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}}}{384 a^{5} c^{4} \left (x -\frac {1}{a}\right )^{4}}+\frac {29 \left (-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )\right )^{\frac {5}{2}}}{768 a^{4} c^{4} \left (x -\frac {1}{a}\right )^{3}}-\frac {25 \left (-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )\right )^{\frac {5}{2}}}{192 a^{3} c^{4} \left (x -\frac {1}{a}\right )^{2}}+\frac {93 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}\, x}{512 c^{4}}+\frac {93 \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 )}{512 c^{4} \sqrt {a^{2}}}-\frac {1723 \left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{10080 a^{6} c^{4} \left (x +\frac {1}{a}\right )^{5}}+\frac {35 \left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{96 a^{5} c^{4} \left (x +\frac {1}{a}\right )^{4}}-\frac {769 \left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {5}{2}}}{768 a^{4} c^{4} \left (x +\frac {1}{a}\right )^{3}}-\frac {31 \left (-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )\right )^{\frac {3}{2}}}{256 a \,c^{4}}-\frac {543 \left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{256 a \,c^{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 \[ \int \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (a x + 1\right )}^{3} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.79, size = 671, normalized size = 3.55 \[ \frac {a\,\sqrt {1-a^2\,x^2}}{96\,\left (a^4\,c^4\,x^2-2\,a^3\,c^4\,x+a^2\,c^4\right )}+\frac {67\,a\,\sqrt {1-a^2\,x^2}}{24\,\left (a^4\,c^4\,x^2+2\,a^3\,c^4\,x+a^2\,c^4\right )}+\frac {\sqrt {1-a^2\,x^2}}{36\,\sqrt {-a^2}\,\left (5\,c^4\,x\,\sqrt {-a^2}+\frac {c^4\,\sqrt {-a^2}}{a}+10\,a^2\,c^4\,x^3\,\sqrt {-a^2}+5\,a^3\,c^4\,x^4\,\sqrt {-a^2}+a^4\,c^4\,x^5\,\sqrt {-a^2}+10\,a\,c^4\,x^2\,\sqrt {-a^2}\right )}-\frac {3\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{c^4\,\sqrt {-a^2}}+\frac {a^3\,\sqrt {1-a^2\,x^2}}{10\,\left (a^6\,c^4\,x^2+2\,a^5\,c^4\,x+a^4\,c^4\right )}-\frac {1759\,a^8\,\sqrt {1-a^2\,x^2}}{2520\,\left (a^{11}\,c^4\,x^2+2\,a^{10}\,c^4\,x+a^9\,c^4\right )}-\frac {\sqrt {1-a^2\,x^2}}{a\,c^4}+\frac {a\,\sqrt {1-a^2\,x^2}}{4\,\left (a^6\,c^4\,x^4+4\,a^5\,c^4\,x^3+6\,a^4\,c^4\,x^2+4\,a^3\,c^4\,x+a^2\,c^4\right )}+\frac {113591\,\sqrt {1-a^2\,x^2}}{20160\,\sqrt {-a^2}\,\left (c^4\,x\,\sqrt {-a^2}+\frac {c^4\,\sqrt {-a^2}}{a}\right )}-\frac {31\,\sqrt {1-a^2\,x^2}}{192\,\sqrt {-a^2}\,\left (c^4\,x\,\sqrt {-a^2}-\frac {c^4\,\sqrt {-a^2}}{a}\right )}+\frac {1507\,\sqrt {1-a^2\,x^2}}{1680\,\sqrt {-a^2}\,\left (3\,c^4\,x\,\sqrt {-a^2}+\frac {c^4\,\sqrt {-a^2}}{a}+a^2\,c^4\,x^3\,\sqrt {-a^2}+3\,a\,c^4\,x^2\,\sqrt {-a^2}\right )}-\frac {a^{10}\,\sqrt {1-a^2\,x^2}}{63\,\left (a^{15}\,c^4\,x^4+4\,a^{14}\,c^4\,x^3+6\,a^{13}\,c^4\,x^2+4\,a^{12}\,c^4\,x+a^{11}\,c^4\right )} \]
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 \[ \frac {a^{8} \left (\int \frac {x^{8} \sqrt {- a^{2} x^{2} + 1}}{a^{11} x^{11} + 3 a^{10} x^{10} - a^{9} x^{9} - 11 a^{8} x^{8} - 6 a^{7} x^{7} + 14 a^{6} x^{6} + 14 a^{5} x^{5} - 6 a^{4} x^{4} - 11 a^{3} x^{3} - a^{2} x^{2} + 3 a x + 1}\, dx + \int \left (- \frac {a^{2} x^{10} \sqrt {- a^{2} x^{2} + 1}}{a^{11} x^{11} + 3 a^{10} x^{10} - a^{9} x^{9} - 11 a^{8} x^{8} - 6 a^{7} x^{7} + 14 a^{6} x^{6} + 14 a^{5} x^{5} - 6 a^{4} x^{4} - 11 a^{3} x^{3} - a^{2} x^{2} + 3 a x + 1}\right )\, dx\right )}{c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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