Optimal. Leaf size=97 \[ \frac {2 \left (1-a^2 x^2\right )^{5/2}}{315 a c^4 (1-a x)^5}+\frac {2 \left (1-a^2 x^2\right )^{5/2}}{63 a c^4 (1-a x)^6}+\frac {\left (1-a^2 x^2\right )^{5/2}}{9 a c^4 (1-a x)^7} \]
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Rubi [A] time = 0.07, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6127, 659, 651} \[ \frac {2 \left (1-a^2 x^2\right )^{5/2}}{315 a c^4 (1-a x)^5}+\frac {2 \left (1-a^2 x^2\right )^{5/2}}{63 a c^4 (1-a x)^6}+\frac {\left (1-a^2 x^2\right )^{5/2}}{9 a c^4 (1-a x)^7} \]
Antiderivative was successfully verified.
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Rule 651
Rule 659
Rule 6127
Rubi steps
\begin {align*} \int \frac {e^{3 \tanh ^{-1}(a x)}}{(c-a c x)^4} \, dx &=c^3 \int \frac {\left (1-a^2 x^2\right )^{3/2}}{(c-a c x)^7} \, dx\\ &=\frac {\left (1-a^2 x^2\right )^{5/2}}{9 a c^4 (1-a x)^7}+\frac {1}{9} \left (2 c^2\right ) \int \frac {\left (1-a^2 x^2\right )^{3/2}}{(c-a c x)^6} \, dx\\ &=\frac {\left (1-a^2 x^2\right )^{5/2}}{9 a c^4 (1-a x)^7}+\frac {2 \left (1-a^2 x^2\right )^{5/2}}{63 a c^4 (1-a x)^6}+\frac {1}{63} (2 c) \int \frac {\left (1-a^2 x^2\right )^{3/2}}{(c-a c x)^5} \, dx\\ &=\frac {\left (1-a^2 x^2\right )^{5/2}}{9 a c^4 (1-a x)^7}+\frac {2 \left (1-a^2 x^2\right )^{5/2}}{63 a c^4 (1-a x)^6}+\frac {2 \left (1-a^2 x^2\right )^{5/2}}{315 a c^4 (1-a x)^5}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 43, normalized size = 0.44 \[ \frac {(a x+1)^{5/2} \left (2 a^2 x^2-14 a x+47\right )}{315 a c^4 (1-a x)^{9/2}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.65, size = 145, normalized size = 1.49 \[ \frac {47 \, a^{5} x^{5} - 235 \, a^{4} x^{4} + 470 \, a^{3} x^{3} - 470 \, a^{2} x^{2} + 235 \, a x - {\left (2 \, a^{4} x^{4} - 10 \, a^{3} x^{3} + 21 \, a^{2} x^{2} + 80 \, a x + 47\right )} \sqrt {-a^{2} x^{2} + 1} - 47}{315 \, {\left (a^{6} c^{4} x^{5} - 5 \, a^{5} c^{4} x^{4} + 10 \, a^{4} c^{4} x^{3} - 10 \, a^{3} c^{4} x^{2} + 5 \, a^{2} c^{4} x - a c^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.23, size = 253, normalized size = 2.61 \[ -\frac {2 \, {\left (\frac {108 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}}{a^{2} x} - \frac {1062 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2}}{a^{4} x^{2}} + \frac {1638 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3}}{a^{6} x^{3}} - \frac {3402 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4}}{a^{8} x^{4}} + \frac {2520 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{5}}{a^{10} x^{5}} - \frac {2310 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{6}}{a^{12} x^{6}} + \frac {630 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{7}}{a^{14} x^{7}} - \frac {315 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{8}}{a^{16} x^{8}} - 47\right )}}{315 \, c^{4} {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} - 1\right )}^{9} {\left | a \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 49, normalized size = 0.51 \[ -\frac {\left (2 a^{2} x^{2}-14 a x +47\right ) \left (a x +1\right )^{4}}{315 \left (a x -1\right )^{3} c^{4} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.35, size = 327, normalized size = 3.37 \[ \frac {8}{9 \, {\left (\sqrt {-a^{2} x^{2} + 1} a^{5} c^{4} x^{4} - 4 \, \sqrt {-a^{2} x^{2} + 1} a^{4} c^{4} x^{3} + 6 \, \sqrt {-a^{2} x^{2} + 1} a^{3} c^{4} x^{2} - 4 \, \sqrt {-a^{2} x^{2} + 1} a^{2} c^{4} x + \sqrt {-a^{2} x^{2} + 1} a c^{4}\right )}} + \frac {68}{63 \, {\left (\sqrt {-a^{2} x^{2} + 1} a^{4} c^{4} x^{3} - 3 \, \sqrt {-a^{2} x^{2} + 1} a^{3} c^{4} x^{2} + 3 \, \sqrt {-a^{2} x^{2} + 1} a^{2} c^{4} x - \sqrt {-a^{2} x^{2} + 1} a c^{4}\right )}} + \frac {106}{315 \, {\left (\sqrt {-a^{2} x^{2} + 1} a^{3} c^{4} x^{2} - 2 \, \sqrt {-a^{2} x^{2} + 1} a^{2} c^{4} x + \sqrt {-a^{2} x^{2} + 1} a c^{4}\right )}} - \frac {1}{315 \, {\left (\sqrt {-a^{2} x^{2} + 1} a^{2} c^{4} x - \sqrt {-a^{2} x^{2} + 1} a c^{4}\right )}} + \frac {2 \, x}{315 \, \sqrt {-a^{2} x^{2} + 1} c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.85, size = 492, normalized size = 5.07 \[ \frac {\sqrt {1-a^2\,x^2}\,\left (\frac {12\,a^4}{35\,c^4\,{\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )}^3}-\frac {8\,a^4}{35\,c^4\,\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )}+\frac {4\,a^5}{7\,c^4\,{\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )}^4\,\sqrt {-a^2}}+\frac {8\,a^7}{35\,c^4\,{\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )}^2\,{\left (-a^2\right )}^{3/2}}\right )}{a^4\,\sqrt {-a^2}}+\frac {\sqrt {1-a^2\,x^2}\,\left (\frac {32\,a^5}{315\,c^4\,\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )}-\frac {16\,a^5}{105\,c^4\,{\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )}^3}+\frac {4\,a^5}{9\,c^4\,{\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )}^5}-\frac {16\,a^2\,{\left (-a^2\right )}^{3/2}}{63\,c^4\,{\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )}^4}+\frac {32\,a^6}{315\,c^4\,{\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )}^2\,\sqrt {-a^2}}\right )}{a^5\,\sqrt {-a^2}}+\frac {\sqrt {1-a^2\,x^2}\,\left (\frac {2\,a^3}{15\,c^4\,\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )}-\frac {a^3}{5\,c^4\,{\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )}^3}+\frac {2\,a^4}{15\,c^4\,{\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )}^2\,\sqrt {-a^2}}\right )}{a^3\,\sqrt {-a^2}} \]
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 \[ \frac {\int \frac {3 a x}{- a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} + 4 a^{5} x^{5} \sqrt {- a^{2} x^{2} + 1} - 5 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} + 5 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} - 4 a x \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {3 a^{2} x^{2}}{- a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} + 4 a^{5} x^{5} \sqrt {- a^{2} x^{2} + 1} - 5 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} + 5 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} - 4 a x \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a^{3} x^{3}}{- a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} + 4 a^{5} x^{5} \sqrt {- a^{2} x^{2} + 1} - 5 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} + 5 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} - 4 a x \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {1}{- a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} + 4 a^{5} x^{5} \sqrt {- a^{2} x^{2} + 1} - 5 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} + 5 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} - 4 a x \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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