Optimal. Leaf size=393 \[ \frac {1}{9} a^8 c^4 x^9 \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{13/2}-\frac {5}{72} a^7 c^4 x^8 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{13/2}+\frac {5}{168} a^6 c^4 x^7 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{13/2}-\frac {5 a^5 c^4 x^6 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{11/2}}{1008}-\frac {11 a^4 c^4 x^5 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{9/2}}{1008}-\frac {11}{448} a^3 c^4 x^4 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}-\frac {11}{192} a^2 c^4 x^3 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}-\frac {55}{384} a c^4 x^2 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}-\frac {55}{128} c^4 x \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}-\frac {55 c^4 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )}{128 a} \]
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Rubi [A] time = 0.33, antiderivative size = 393, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {6191, 6195, 94, 92, 208} \[ \frac {1}{9} a^8 c^4 x^9 \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{13/2}-\frac {5}{72} a^7 c^4 x^8 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{13/2}+\frac {5}{168} a^6 c^4 x^7 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{13/2}-\frac {5 a^5 c^4 x^6 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{11/2}}{1008}-\frac {11 a^4 c^4 x^5 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{9/2}}{1008}-\frac {11}{448} a^3 c^4 x^4 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}-\frac {11}{192} a^2 c^4 x^3 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}-\frac {55}{384} a c^4 x^2 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}-\frac {55}{128} c^4 x \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}-\frac {55 c^4 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )}{128 a} \]
Antiderivative was successfully verified.
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Rule 92
Rule 94
Rule 208
Rule 6191
Rule 6195
Rubi steps
\begin {align*} \int e^{3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^4 \, dx &=\left (a^8 c^4\right ) \int e^{3 \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 )^{5/2} \left (1+\frac {x}{a}\right )^{11/2}}{x^{10}} \, dx,x,\frac {1}{x}\right )\right )\\ &=\frac {1}{9} a^8 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{13/2} x^9+\frac {1}{9} \left (5 a^7 c^4\right ) \operatorname {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{3/2} \left (1+\frac {x}{a}\right )^{11/2}}{x^9} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {5}{72} a^7 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{13/2} x^8+\frac {1}{9} a^8 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{13/2} x^9-\frac {1}{24} \left (5 a^6 c^4\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{11/2}}{x^8} \, dx,x,\frac {1}{x}\right )\\ &=\frac {5}{168} a^6 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{13/2} x^7-\frac {5}{72} a^7 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{13/2} x^8+\frac {1}{9} a^8 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{13/2} x^9+\frac {1}{168} \left (5 a^5 c^4\right ) \operatorname {Subst}\left (\int \frac {\left (1+\frac {x}{a}\right )^{11/2}}{x^7 \sqrt {1-\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {5 a^5 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{11/2} x^6}{1008}+\frac {5}{168} a^6 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{13/2} x^7-\frac {5}{72} a^7 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{13/2} x^8+\frac {1}{9} a^8 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{13/2} x^9+\frac {\left (55 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 )}{1008}\\ &=-\frac {11 a^4 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2} x^5}{1008}-\frac {5 a^5 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{11/2} x^6}{1008}+\frac {5}{168} a^6 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{13/2} x^7-\frac {5}{72} a^7 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{13/2} x^8+\frac {1}{9} a^8 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{13/2} x^9+\frac {1}{112} \left (11 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 {11}{448} a^3 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2} x^4-\frac {11 a^4 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2} x^5}{1008}-\frac {5 a^5 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{11/2} x^6}{1008}+\frac {5}{168} a^6 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{13/2} x^7-\frac {5}{72} a^7 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{13/2} x^8+\frac {1}{9} a^8 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{13/2} x^9+\frac {1}{64} \left (11 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 {11}{192} a^2 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2} x^3-\frac {11}{448} a^3 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2} x^4-\frac {11 a^4 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2} x^5}{1008}-\frac {5 a^5 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{11/2} x^6}{1008}+\frac {5}{168} a^6 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{13/2} x^7-\frac {5}{72} a^7 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{13/2} x^8+\frac {1}{9} a^8 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{13/2} x^9+\frac {1}{192} \left (55 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 {55}{384} a c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2} x^2-\frac {11}{192} a^2 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2} x^3-\frac {11}{448} a^3 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2} x^4-\frac {11 a^4 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2} x^5}{1008}-\frac {5 a^5 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{11/2} x^6}{1008}+\frac {5}{168} a^6 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{13/2} x^7-\frac {5}{72} a^7 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{13/2} x^8+\frac {1}{9} a^8 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{13/2} x^9+\frac {1}{128} \left (55 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 {55}{128} c^4 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x-\frac {55}{384} a c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2} x^2-\frac {11}{192} a^2 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2} x^3-\frac {11}{448} a^3 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2} x^4-\frac {11 a^4 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2} x^5}{1008}-\frac {5 a^5 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{11/2} x^6}{1008}+\frac {5}{168} a^6 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{13/2} x^7-\frac {5}{72} a^7 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{13/2} x^8+\frac {1}{9} a^8 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{13/2} x^9+\frac {\left (55 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 {55}{128} c^4 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x-\frac {55}{384} a c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2} x^2-\frac {11}{192} a^2 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2} x^3-\frac {11}{448} a^3 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2} x^4-\frac {11 a^4 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2} x^5}{1008}-\frac {5 a^5 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{11/2} x^6}{1008}+\frac {5}{168} a^6 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{13/2} x^7-\frac {5}{72} a^7 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{13/2} x^8+\frac {1}{9} a^8 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{13/2} x^9-\frac {\left (55 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 {55}{128} c^4 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x-\frac {55}{384} a c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2} x^2-\frac {11}{192} a^2 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2} x^3-\frac {11}{448} a^3 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2} x^4-\frac {11 a^4 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2} x^5}{1008}-\frac {5 a^5 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{11/2} x^6}{1008}+\frac {5}{168} a^6 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{13/2} x^7-\frac {5}{72} a^7 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{13/2} x^8+\frac {1}{9} a^8 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{13/2} x^9-\frac {55 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 \[ \frac {c^4 \left (a x \sqrt {1-\frac {1}{a^2 x^2}} \left (896 a^8 x^8+3024 a^7 x^7+1024 a^6 x^6-7224 a^5 x^5-8448 a^4 x^4+3066 a^3 x^3+10240 a^2 x^2+4599 a x-3712\right )-3465 \log \left (x \left (\sqrt {1-\frac {1}{a^2 x^2}}+1\right )\right )\right )}{8064 a} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.71, size = 170, normalized size = 0.43 \[ -\frac {3465 \, c^{4} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 3465 \, c^{4} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) - {\left (896 \, a^{9} c^{4} x^{9} + 3920 \, a^{8} c^{4} x^{8} + 4048 \, a^{7} c^{4} x^{7} - 6200 \, a^{6} c^{4} x^{6} - 15672 \, a^{5} c^{4} x^{5} - 5382 \, a^{4} c^{4} x^{4} + 13306 \, a^{3} c^{4} x^{3} + 14839 \, a^{2} c^{4} x^{2} + 887 \, a c^{4} x - 3712 \, c^{4}\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{8064 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 340, normalized size = 0.87 \[ -\frac {1}{8064} \, a c^{4} {\left (\frac {3465 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac {3465 \, \log \left ({\left | \sqrt {\frac {a x - 1}{a x + 1}} - 1 \right |}\right )}{a^{2}} - \frac {2 \, {\left (\frac {30030 \, {\left (a x - 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{a x + 1} - \frac {115038 \, {\left (a x - 1\right )}^{2} \sqrt {\frac {a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{2}} - \frac {334602 \, {\left (a x - 1\right )}^{3} \sqrt {\frac {a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{3}} + \frac {360448 \, {\left (a x - 1\right )}^{4} \sqrt {\frac {a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{4}} - \frac {255222 \, {\left (a x - 1\right )}^{5} \sqrt {\frac {a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{5}} + \frac {115038 \, {\left (a x - 1\right )}^{6} \sqrt {\frac {a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{6}} - \frac {30030 \, {\left (a x - 1\right )}^{7} \sqrt {\frac {a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{7}} + \frac {3465 \, {\left (a x - 1\right )}^{8} \sqrt {\frac {a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{8}} - 3465 \, \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{a^{2} {\left (\frac {a x - 1}{a x + 1} - 1\right )}^{9}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 288, normalized size = 0.73 \[ \frac {\left (a x -1\right )^{2} c^{4} \left (896 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x^{6} a^{6}+3024 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x^{5} a^{5}+1920 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x^{4} a^{4}-4200 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x^{3} a^{3}-6528 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x^{2} a^{2}-1134 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x a +8064 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}-4352 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}+3465 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, x a -3465 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a \right )}{8064 a \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}} \]
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 \[ -\frac {1}{8064} \, {\left (\frac {3465 \, c^{4} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac {3465 \, c^{4} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}} - \frac {2 \, {\left (3465 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {17}{2}} - 30030 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {15}{2}} + 115038 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {13}{2}} - 255222 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {11}{2}} + 360448 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {9}{2}} - 334602 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} - 115038 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} + 30030 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} - 3465 \, c^{4} \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{\frac {9 \, {\left (a x - 1\right )} a^{2}}{a x + 1} - \frac {36 \, {\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} + \frac {84 \, {\left (a x - 1\right )}^{3} a^{2}}{{\left (a x + 1\right )}^{3}} - \frac {126 \, {\left (a x - 1\right )}^{4} a^{2}}{{\left (a x + 1\right )}^{4}} + \frac {126 \, {\left (a x - 1\right )}^{5} a^{2}}{{\left (a x + 1\right )}^{5}} - \frac {84 \, {\left (a x - 1\right )}^{6} a^{2}}{{\left (a x + 1\right )}^{6}} + \frac {36 \, {\left (a x - 1\right )}^{7} a^{2}}{{\left (a x + 1\right )}^{7}} - \frac {9 \, {\left (a x - 1\right )}^{8} a^{2}}{{\left (a x + 1\right )}^{8}} + \frac {{\left (a x - 1\right )}^{9} a^{2}}{{\left (a x + 1\right )}^{9}} - a^{2}}\right )} a \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.21, size = 362, normalized size = 0.92 \[ \frac {\frac {55\,c^4\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{64}-\frac {715\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{96}+\frac {913\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{32}+\frac {18589\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}}{224}-\frac {5632\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{9/2}}{63}+\frac {14179\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{11/2}}{224}-\frac {913\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{13/2}}{32}+\frac {715\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{15/2}}{96}-\frac {55\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{17/2}}{64}}{a-\frac {9\,a\,\left (a\,x-1\right )}{a\,x+1}+\frac {36\,a\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {84\,a\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}+\frac {126\,a\,{\left (a\,x-1\right )}^4}{{\left (a\,x+1\right )}^4}-\frac {126\,a\,{\left (a\,x-1\right )}^5}{{\left (a\,x+1\right )}^5}+\frac {84\,a\,{\left (a\,x-1\right )}^6}{{\left (a\,x+1\right )}^6}-\frac {36\,a\,{\left (a\,x-1\right )}^7}{{\left (a\,x+1\right )}^7}+\frac {9\,a\,{\left (a\,x-1\right )}^8}{{\left (a\,x+1\right )}^8}-\frac {a\,{\left (a\,x-1\right )}^9}{{\left (a\,x+1\right )}^9}}-\frac {55\,c^4\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{64\,a} \]
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 \[ c^{4} \left (\int \left (- \frac {4 a^{2} x^{2}}{\frac {a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\right )\, dx + \int \frac {6 a^{4} x^{4}}{\frac {a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\, dx + \int \left (- \frac {4 a^{6} x^{6}}{\frac {a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\right )\, dx + \int \frac {a^{8} x^{8}}{\frac {a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\, dx + \int \frac {1}{\frac {a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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