Optimal. Leaf size=393 \[ \frac {1}{9} a^8 c^4 x^9 \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{11/2}-\frac {7}{72} a^7 c^4 x^8 \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{11/2}+\frac {5}{72} a^6 c^4 x^7 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{11/2}-\frac {5}{144} a^5 c^4 x^6 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{11/2}+\frac {1}{144} a^4 c^4 x^5 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{9/2}+\frac {1}{64} a^3 c^4 x^4 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}+\frac {7}{192} a^2 c^4 x^3 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}+\frac {35}{384} a c^4 x^2 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}+\frac {35}{128} c^4 x \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}+\frac {35 c^4 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )}{128 a} \]
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Rubi [A] time = 0.34, antiderivative size = 393, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6191, 6195, 94, 92, 208} \[ \frac {1}{9} a^8 c^4 x^9 \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{11/2}-\frac {7}{72} a^7 c^4 x^8 \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{11/2}+\frac {5}{72} a^6 c^4 x^7 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{11/2}-\frac {5}{144} a^5 c^4 x^6 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{11/2}+\frac {1}{144} a^4 c^4 x^5 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{9/2}+\frac {1}{64} a^3 c^4 x^4 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}+\frac {7}{192} a^2 c^4 x^3 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}+\frac {35}{384} a c^4 x^2 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}+\frac {35}{128} c^4 x \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}+\frac {35 c^4 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )}{128 a} \]
Antiderivative was successfully verified.
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Rule 92
Rule 94
Rule 208
Rule 6191
Rule 6195
Rubi steps
\begin {align*} \int e^{\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^4 \, dx &=\left (a^8 c^4\right ) \int e^{\coth ^{-1}(a x)} \left (1-\frac {1}{a^2 x^2}\right )^4 x^8 \, dx\\ &=-\left (\left (a^8 c^4\right ) \operatorname {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{7/2} \left (1+\frac {x}{a}\right )^{9/2}}{x^{10}} \, dx,x,\frac {1}{x}\right )\right )\\ &=\frac {1}{9} a^8 c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{11/2} x^9+\frac {1}{9} \left (7 a^7 c^4\right ) \operatorname {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{5/2} \left (1+\frac {x}{a}\right )^{9/2}}{x^9} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {7}{72} a^7 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{11/2} x^8+\frac {1}{9} a^8 c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{11/2} x^9-\frac {1}{72} \left (35 a^6 c^4\right ) \operatorname {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{3/2} \left (1+\frac {x}{a}\right )^{9/2}}{x^8} \, dx,x,\frac {1}{x}\right )\\ &=\frac {5}{72} a^6 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{11/2} x^7-\frac {7}{72} a^7 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{11/2} x^8+\frac {1}{9} a^8 c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{11/2} x^9+\frac {1}{24} \left (5 a^5 c^4\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{9/2}}{x^7} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {5}{144} a^5 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{11/2} x^6+\frac {5}{72} a^6 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{11/2} x^7-\frac {7}{72} a^7 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{11/2} x^8+\frac {1}{9} a^8 c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{11/2} x^9-\frac {1}{144} \left (5 a^4 c^4\right ) \operatorname {Subst}\left (\int \frac {\left (1+\frac {x}{a}\right )^{9/2}}{x^6 \sqrt {1-\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{144} a^4 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2} x^5-\frac {5}{144} a^5 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{11/2} x^6+\frac {5}{72} a^6 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{11/2} x^7-\frac {7}{72} a^7 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{11/2} x^8+\frac {1}{9} a^8 c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{11/2} x^9-\frac {1}{16} \left (a^3 c^4\right ) \operatorname {Subst}\left (\int \frac {\left (1+\frac {x}{a}\right )^{7/2}}{x^5 \sqrt {1-\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{64} a^3 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2} x^4+\frac {1}{144} a^4 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2} x^5-\frac {5}{144} a^5 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{11/2} x^6+\frac {5}{72} a^6 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{11/2} x^7-\frac {7}{72} a^7 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{11/2} x^8+\frac {1}{9} a^8 c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{11/2} x^9-\frac {1}{64} \left (7 a^2 c^4\right ) \operatorname {Subst}\left (\int \frac {\left (1+\frac {x}{a}\right )^{5/2}}{x^4 \sqrt {1-\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {7}{192} a^2 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2} x^3+\frac {1}{64} a^3 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2} x^4+\frac {1}{144} a^4 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2} x^5-\frac {5}{144} a^5 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{11/2} x^6+\frac {5}{72} a^6 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{11/2} x^7-\frac {7}{72} a^7 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{11/2} x^8+\frac {1}{9} a^8 c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{11/2} x^9-\frac {1}{192} \left (35 a c^4\right ) \operatorname {Subst}\left (\int \frac {\left (1+\frac {x}{a}\right )^{3/2}}{x^3 \sqrt {1-\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {35}{384} a c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2} x^2+\frac {7}{192} a^2 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2} x^3+\frac {1}{64} a^3 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2} x^4+\frac {1}{144} a^4 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2} x^5-\frac {5}{144} a^5 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{11/2} x^6+\frac {5}{72} a^6 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{11/2} x^7-\frac {7}{72} a^7 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{11/2} x^8+\frac {1}{9} a^8 c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{11/2} x^9-\frac {1}{128} \left (35 c^4\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {x}{a}}}{x^2 \sqrt {1-\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {35}{128} c^4 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x+\frac {35}{384} a c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2} x^2+\frac {7}{192} a^2 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2} x^3+\frac {1}{64} a^3 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2} x^4+\frac {1}{144} a^4 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2} x^5-\frac {5}{144} a^5 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{11/2} x^6+\frac {5}{72} a^6 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{11/2} x^7-\frac {7}{72} a^7 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{11/2} x^8+\frac {1}{9} a^8 c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{11/2} x^9-\frac {\left (35 c^4\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{128 a}\\ &=\frac {35}{128} c^4 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x+\frac {35}{384} a c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2} x^2+\frac {7}{192} a^2 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2} x^3+\frac {1}{64} a^3 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2} x^4+\frac {1}{144} a^4 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2} x^5-\frac {5}{144} a^5 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{11/2} x^6+\frac {5}{72} a^6 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{11/2} x^7-\frac {7}{72} a^7 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{11/2} x^8+\frac {1}{9} a^8 c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{11/2} x^9+\frac {\left (35 c^4\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {1}{a}-\frac {x^2}{a}} \, dx,x,\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{128 a^2}\\ &=\frac {35}{128} c^4 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x+\frac {35}{384} a c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2} x^2+\frac {7}{192} a^2 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2} x^3+\frac {1}{64} a^3 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2} x^4+\frac {1}{144} a^4 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2} x^5-\frac {5}{144} a^5 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{11/2} x^6+\frac {5}{72} a^6 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{11/2} x^7-\frac {7}{72} a^7 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{11/2} x^8+\frac {1}{9} a^8 c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{11/2} x^9+\frac {35 c^4 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{128 a}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 111, normalized size = 0.28 \[ \frac {c^4 \left (315 \log \left (x \left (\sqrt {1-\frac {1}{a^2 x^2}}+1\right )\right )+a x \sqrt {1-\frac {1}{a^2 x^2}} \left (128 a^8 x^8+144 a^7 x^7-512 a^6 x^6-600 a^5 x^5+768 a^4 x^4+978 a^3 x^3-512 a^2 x^2-837 a x+128\right )\right )}{1152 a} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.55, size = 169, normalized size = 0.43 \[ \frac {315 \, c^{4} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 315 \, c^{4} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (128 \, a^{9} c^{4} x^{9} + 272 \, a^{8} c^{4} x^{8} - 368 \, a^{7} c^{4} x^{7} - 1112 \, a^{6} c^{4} x^{6} + 168 \, a^{5} c^{4} x^{5} + 1746 \, a^{4} c^{4} x^{4} + 466 \, a^{3} c^{4} x^{3} - 1349 \, a^{2} c^{4} x^{2} - 709 \, a c^{4} x + 128 \, c^{4}\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{1152 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 340, normalized size = 0.87 \[ \frac {1}{1152} \, a c^{4} {\left (\frac {315 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac {315 \, \log \left ({\left | \sqrt {\frac {a x - 1}{a x + 1}} - 1 \right |}\right )}{a^{2}} - \frac {2 \, {\left (\frac {2730 \, {\left (a x - 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{a x + 1} - \frac {10458 \, {\left (a x - 1\right )}^{2} \sqrt {\frac {a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{2}} + \frac {23202 \, {\left (a x - 1\right )}^{3} \sqrt {\frac {a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{3}} + \frac {32768 \, {\left (a x - 1\right )}^{4} \sqrt {\frac {a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{4}} - \frac {23202 \, {\left (a x - 1\right )}^{5} \sqrt {\frac {a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{5}} + \frac {10458 \, {\left (a x - 1\right )}^{6} \sqrt {\frac {a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{6}} - \frac {2730 \, {\left (a x - 1\right )}^{7} \sqrt {\frac {a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{7}} + \frac {315 \, {\left (a x - 1\right )}^{8} \sqrt {\frac {a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{8}} - 315 \, \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{a^{2} {\left (\frac {a x - 1}{a x + 1} - 1\right )}^{9}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 279, normalized size = 0.71 \[ -\frac {\left (a x -1\right ) c^{4} \left (-128 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x^{6} a^{6}-144 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x^{5} a^{5}+384 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x^{4} a^{4}+456 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x^{3} a^{3}-384 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x^{2} a^{2}-522 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x a +384 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}-256 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}+315 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, x a -315 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a \right )}{1152 a \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 415, normalized size = 1.06 \[ \frac {1}{1152} \, {\left (\frac {315 \, c^{4} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac {315 \, c^{4} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}} - \frac {2 \, {\left (315 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {17}{2}} - 2730 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {15}{2}} + 10458 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {13}{2}} - 23202 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {11}{2}} + 32768 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {9}{2}} + 23202 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} - 10458 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} + 2730 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} - 315 \, c^{4} \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{\frac {9 \, {\left (a x - 1\right )} a^{2}}{a x + 1} - \frac {36 \, {\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} + \frac {84 \, {\left (a x - 1\right )}^{3} a^{2}}{{\left (a x + 1\right )}^{3}} - \frac {126 \, {\left (a x - 1\right )}^{4} a^{2}}{{\left (a x + 1\right )}^{4}} + \frac {126 \, {\left (a x - 1\right )}^{5} a^{2}}{{\left (a x + 1\right )}^{5}} - \frac {84 \, {\left (a x - 1\right )}^{6} a^{2}}{{\left (a x + 1\right )}^{6}} + \frac {36 \, {\left (a x - 1\right )}^{7} a^{2}}{{\left (a x + 1\right )}^{7}} - \frac {9 \, {\left (a x - 1\right )}^{8} a^{2}}{{\left (a x + 1\right )}^{8}} + \frac {{\left (a x - 1\right )}^{9} a^{2}}{{\left (a x + 1\right )}^{9}} - a^{2}}\right )} a \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.37, size = 362, normalized size = 0.92 \[ \frac {\frac {455\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{96}-\frac {35\,c^4\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{64}-\frac {581\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{32}+\frac {1289\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}}{32}+\frac {512\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{9/2}}{9}-\frac {1289\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{11/2}}{32}+\frac {581\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{13/2}}{32}-\frac {455\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{15/2}}{96}+\frac {35\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{17/2}}{64}}{a-\frac {9\,a\,\left (a\,x-1\right )}{a\,x+1}+\frac {36\,a\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {84\,a\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}+\frac {126\,a\,{\left (a\,x-1\right )}^4}{{\left (a\,x+1\right )}^4}-\frac {126\,a\,{\left (a\,x-1\right )}^5}{{\left (a\,x+1\right )}^5}+\frac {84\,a\,{\left (a\,x-1\right )}^6}{{\left (a\,x+1\right )}^6}-\frac {36\,a\,{\left (a\,x-1\right )}^7}{{\left (a\,x+1\right )}^7}+\frac {9\,a\,{\left (a\,x-1\right )}^8}{{\left (a\,x+1\right )}^8}-\frac {a\,{\left (a\,x-1\right )}^9}{{\left (a\,x+1\right )}^9}}+\frac {35\,c^4\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{64\,a} \]
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 \[ c^{4} \left (\int \left (- \frac {4 a^{2} x^{2}}{\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\right )\, dx + \int \frac {6 a^{4} x^{4}}{\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\, dx + \int \left (- \frac {4 a^{6} x^{6}}{\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\right )\, dx + \int \frac {a^{8} x^{8}}{\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\, dx + \int \frac {1}{\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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