Optimal. Leaf size=138 \[ \frac {\sin ^{-1}(a x)}{a^4 c^4}+\frac {86 \left (1-a^2 x^2\right )^{3/2}}{105 a^4 c^4 (1-a x)^3}-\frac {19 \left (1-a^2 x^2\right )^{3/2}}{35 a^4 c^4 (1-a x)^4}+\frac {\left (1-a^2 x^2\right )^{3/2}}{7 a^4 c^4 (1-a x)^5}-\frac {2 \sqrt {1-a^2 x^2}}{a^4 c^4 (1-a x)} \]
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Rubi [A] time = 0.27, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {6128, 1637, 659, 651, 663, 216} \[ \frac {86 \left (1-a^2 x^2\right )^{3/2}}{105 a^4 c^4 (1-a x)^3}-\frac {19 \left (1-a^2 x^2\right )^{3/2}}{35 a^4 c^4 (1-a x)^4}+\frac {\left (1-a^2 x^2\right )^{3/2}}{7 a^4 c^4 (1-a x)^5}-\frac {2 \sqrt {1-a^2 x^2}}{a^4 c^4 (1-a x)}+\frac {\sin ^{-1}(a x)}{a^4 c^4} \]
Antiderivative was successfully verified.
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Rule 216
Rule 651
Rule 659
Rule 663
Rule 1637
Rule 6128
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)} x^3}{(c-a c x)^4} \, dx &=c \int \frac {x^3 \sqrt {1-a^2 x^2}}{(c-a c x)^5} \, dx\\ &=c \int \left (-\frac {\sqrt {1-a^2 x^2}}{a^3 c^5 (-1+a x)^5}-\frac {3 \sqrt {1-a^2 x^2}}{a^3 c^5 (-1+a x)^4}-\frac {3 \sqrt {1-a^2 x^2}}{a^3 c^5 (-1+a x)^3}-\frac {\sqrt {1-a^2 x^2}}{a^3 c^5 (-1+a x)^2}\right ) \, dx\\ &=-\frac {\int \frac {\sqrt {1-a^2 x^2}}{(-1+a x)^5} \, dx}{a^3 c^4}-\frac {\int \frac {\sqrt {1-a^2 x^2}}{(-1+a x)^2} \, dx}{a^3 c^4}-\frac {3 \int \frac {\sqrt {1-a^2 x^2}}{(-1+a x)^4} \, dx}{a^3 c^4}-\frac {3 \int \frac {\sqrt {1-a^2 x^2}}{(-1+a x)^3} \, dx}{a^3 c^4}\\ &=-\frac {2 \sqrt {1-a^2 x^2}}{a^4 c^4 (1-a x)}+\frac {\left (1-a^2 x^2\right )^{3/2}}{7 a^4 c^4 (1-a x)^5}-\frac {3 \left (1-a^2 x^2\right )^{3/2}}{5 a^4 c^4 (1-a x)^4}+\frac {\left (1-a^2 x^2\right )^{3/2}}{a^4 c^4 (1-a x)^3}+\frac {2 \int \frac {\sqrt {1-a^2 x^2}}{(-1+a x)^4} \, dx}{7 a^3 c^4}+\frac {3 \int \frac {\sqrt {1-a^2 x^2}}{(-1+a x)^3} \, dx}{5 a^3 c^4}+\frac {\int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{a^3 c^4}\\ &=-\frac {2 \sqrt {1-a^2 x^2}}{a^4 c^4 (1-a x)}+\frac {\left (1-a^2 x^2\right )^{3/2}}{7 a^4 c^4 (1-a x)^5}-\frac {19 \left (1-a^2 x^2\right )^{3/2}}{35 a^4 c^4 (1-a x)^4}+\frac {4 \left (1-a^2 x^2\right )^{3/2}}{5 a^4 c^4 (1-a x)^3}+\frac {\sin ^{-1}(a x)}{a^4 c^4}-\frac {2 \int \frac {\sqrt {1-a^2 x^2}}{(-1+a x)^3} \, dx}{35 a^3 c^4}\\ &=-\frac {2 \sqrt {1-a^2 x^2}}{a^4 c^4 (1-a x)}+\frac {\left (1-a^2 x^2\right )^{3/2}}{7 a^4 c^4 (1-a x)^5}-\frac {19 \left (1-a^2 x^2\right )^{3/2}}{35 a^4 c^4 (1-a x)^4}+\frac {86 \left (1-a^2 x^2\right )^{3/2}}{105 a^4 c^4 (1-a x)^3}+\frac {\sin ^{-1}(a x)}{a^4 c^4}\\ \end {align*}
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Mathematica [A] time = 0.25, size = 94, normalized size = 0.68 \[ \frac {\sqrt {a x+1} \left (\sqrt {1-a^2 x^2} \left (296 a^3 x^3-659 a^2 x^2+559 a x-166\right )+105 (a x-1)^4 \sin ^{-1}(a x)\right )}{105 a^4 c^4 (1-a x)^{7/2} \sqrt {1-a^2 x^2}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.60, size = 172, normalized size = 1.25 \[ -\frac {166 \, a^{4} x^{4} - 664 \, a^{3} x^{3} + 996 \, a^{2} x^{2} - 664 \, a x + 210 \, {\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - {\left (296 \, a^{3} x^{3} - 659 \, a^{2} x^{2} + 559 \, a x - 166\right )} \sqrt {-a^{2} x^{2} + 1} + 166}{105 \, {\left (a^{8} c^{4} x^{4} - 4 \, a^{7} c^{4} x^{3} + 6 \, a^{6} c^{4} x^{2} - 4 \, a^{5} c^{4} x + a^{4} c^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 220, normalized size = 1.59 \[ \frac {\arcsin \left (a x\right ) \mathrm {sgn}\relax (a)}{a^{3} c^{4} {\left | a \right |}} + \frac {2 \, {\left (\frac {1057 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}}{a^{2} x} - \frac {2751 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2}}{a^{4} x^{2}} + \frac {3640 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3}}{a^{6} x^{3}} - \frac {2170 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4}}{a^{8} x^{4}} + \frac {735 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{5}}{a^{10} x^{5}} - \frac {105 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{6}}{a^{12} x^{6}} - 166\right )}}{105 \, a^{3} c^{4} {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} - 1\right )}^{7} {\left | a \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 210, normalized size = 1.52 \[ \frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{c^{4} a^{3} \sqrt {a^{2}}}+\frac {229 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{105 c^{4} a^{6} \left (x -\frac {1}{a}\right )^{2}}+\frac {296 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{105 c^{4} a^{5} \left (x -\frac {1}{a}\right )}+\frac {2 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{7 c^{4} a^{8} \left (x -\frac {1}{a}\right )^{4}}+\frac {43 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{35 c^{4} a^{7} \left (x -\frac {1}{a}\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.64, size = 208, normalized size = 1.51 \[ \frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{7 \, {\left (a^{8} c^{4} x^{4} - 4 \, a^{7} c^{4} x^{3} + 6 \, a^{6} c^{4} x^{2} - 4 \, a^{5} c^{4} x + a^{4} c^{4}\right )}} + \frac {43 \, \sqrt {-a^{2} x^{2} + 1}}{35 \, {\left (a^{7} c^{4} x^{3} - 3 \, a^{6} c^{4} x^{2} + 3 \, a^{5} c^{4} x - a^{4} c^{4}\right )}} + \frac {229 \, \sqrt {-a^{2} x^{2} + 1}}{105 \, {\left (a^{6} c^{4} x^{2} - 2 \, a^{5} c^{4} x + a^{4} c^{4}\right )}} + \frac {296 \, \sqrt {-a^{2} x^{2} + 1}}{105 \, {\left (a^{5} c^{4} x - a^{4} c^{4}\right )}} + \frac {\arcsin \left (a x\right )}{a^{4} c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 281, normalized size = 2.04 \[ \frac {2\,\sqrt {1-a^2\,x^2}}{7\,\left (a^8\,c^4\,x^4-4\,a^7\,c^4\,x^3+6\,a^6\,c^4\,x^2-4\,a^5\,c^4\,x+a^4\,c^4\right )}+\frac {229\,\sqrt {1-a^2\,x^2}}{105\,\left (a^6\,c^4\,x^2-2\,a^5\,c^4\,x+a^4\,c^4\right )}+\frac {43\,\sqrt {1-a^2\,x^2}}{35\,\sqrt {-a^2}\,\left (a^2\,c^4\,\sqrt {-a^2}+3\,a^4\,c^4\,x^2\,\sqrt {-a^2}-a^5\,c^4\,x^3\,\sqrt {-a^2}-3\,a^3\,c^4\,x\,\sqrt {-a^2}\right )}+\frac {296\,\sqrt {1-a^2\,x^2}}{105\,\left (a^2\,c^4\,\sqrt {-a^2}-a^3\,c^4\,x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}+\frac {\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{a^3\,c^4\,\sqrt {-a^2}} \]
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 \[ \frac {\int \frac {x^{3}}{a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 4 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} + 6 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} - 4 a x \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a x^{4}}{a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 4 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} + 6 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} - 4 a x \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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