Optimal. Leaf size=393 \[ \frac {1}{9} a^8 c^4 x^9 \left (1-\frac {1}{a x}\right )^{11/2} \left (\frac {1}{a x}+1\right )^{7/2}-\frac {11}{72} a^7 c^4 x^8 \left (1-\frac {1}{a x}\right )^{9/2} \left (\frac {1}{a x}+1\right )^{7/2}+\frac {11}{56} a^6 c^4 x^7 \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{7/2}-\frac {11}{48} a^5 c^4 x^6 \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{7/2}+\frac {11}{48} a^4 c^4 x^5 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{7/2}-\frac {11}{64} a^3 c^4 x^4 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}+\frac {11}{192} a^2 c^4 x^3 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}+\frac {55}{384} a c^4 x^2 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}+\frac {55}{128} c^4 x \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}+\frac {55 c^4 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )}{128 a} \]
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Rubi [A] time = 0.34, 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} \[ \frac {1}{9} a^8 c^4 x^9 \left (1-\frac {1}{a x}\right )^{11/2} \left (\frac {1}{a x}+1\right )^{7/2}-\frac {11}{72} a^7 c^4 x^8 \left (1-\frac {1}{a x}\right )^{9/2} \left (\frac {1}{a x}+1\right )^{7/2}+\frac {11}{56} a^6 c^4 x^7 \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{7/2}-\frac {11}{48} a^5 c^4 x^6 \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{7/2}+\frac {11}{48} a^4 c^4 x^5 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{7/2}-\frac {11}{64} a^3 c^4 x^4 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}+\frac {11}{192} a^2 c^4 x^3 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}+\frac {55}{384} a c^4 x^2 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}+\frac {55}{128} c^4 x \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}+\frac {55 c^4 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )}{128 a} \]
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 )^{11/2} \left (1+\frac {x}{a}\right )^{5/2}}{x^{10}} \, dx,x,\frac {1}{x}\right )\right )\\ &=\frac {1}{9} a^8 c^4 \left (1-\frac {1}{a x}\right )^{11/2} \left (1+\frac {1}{a x}\right )^{7/2} x^9+\frac {1}{9} \left (11 a^7 c^4\right ) \operatorname {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{9/2} \left (1+\frac {x}{a}\right )^{5/2}}{x^9} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {11}{72} a^7 c^4 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{7/2} x^8+\frac {1}{9} a^8 c^4 \left (1-\frac {1}{a x}\right )^{11/2} \left (1+\frac {1}{a x}\right )^{7/2} x^9-\frac {1}{8} \left (11 a^6 c^4\right ) \operatorname {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{7/2} \left (1+\frac {x}{a}\right )^{5/2}}{x^8} \, dx,x,\frac {1}{x}\right )\\ &=\frac {11}{56} a^6 c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{7/2} x^7-\frac {11}{72} a^7 c^4 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{7/2} x^8+\frac {1}{9} a^8 c^4 \left (1-\frac {1}{a x}\right )^{11/2} \left (1+\frac {1}{a x}\right )^{7/2} x^9+\frac {1}{8} \left (11 a^5 c^4\right ) \operatorname {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{5/2} \left (1+\frac {x}{a}\right )^{5/2}}{x^7} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {11}{48} a^5 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{7/2} x^6+\frac {11}{56} a^6 c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{7/2} x^7-\frac {11}{72} a^7 c^4 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{7/2} x^8+\frac {1}{9} a^8 c^4 \left (1-\frac {1}{a x}\right )^{11/2} \left (1+\frac {1}{a x}\right )^{7/2} x^9-\frac {1}{48} \left (55 a^4 c^4\right ) \operatorname {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{3/2} \left (1+\frac {x}{a}\right )^{5/2}}{x^6} \, dx,x,\frac {1}{x}\right )\\ &=\frac {11}{48} a^4 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{7/2} x^5-\frac {11}{48} a^5 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{7/2} x^6+\frac {11}{56} a^6 c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{7/2} x^7-\frac {11}{72} a^7 c^4 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{7/2} x^8+\frac {1}{9} a^8 c^4 \left (1-\frac {1}{a x}\right )^{11/2} \left (1+\frac {1}{a x}\right )^{7/2} x^9+\frac {1}{16} \left (11 a^3 c^4\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{5/2}}{x^5} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {11}{64} a^3 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2} x^4+\frac {11}{48} a^4 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{7/2} x^5-\frac {11}{48} a^5 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{7/2} x^6+\frac {11}{56} a^6 c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{7/2} x^7-\frac {11}{72} a^7 c^4 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{7/2} x^8+\frac {1}{9} a^8 c^4 \left (1-\frac {1}{a x}\right )^{11/2} \left (1+\frac {1}{a x}\right )^{7/2} 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}{64} a^3 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2} x^4+\frac {11}{48} a^4 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{7/2} x^5-\frac {11}{48} a^5 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{7/2} x^6+\frac {11}{56} a^6 c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{7/2} x^7-\frac {11}{72} a^7 c^4 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{7/2} x^8+\frac {1}{9} a^8 c^4 \left (1-\frac {1}{a x}\right )^{11/2} \left (1+\frac {1}{a x}\right )^{7/2} 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}{64} a^3 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2} x^4+\frac {11}{48} a^4 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{7/2} x^5-\frac {11}{48} a^5 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{7/2} x^6+\frac {11}{56} a^6 c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{7/2} x^7-\frac {11}{72} a^7 c^4 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{7/2} x^8+\frac {1}{9} a^8 c^4 \left (1-\frac {1}{a x}\right )^{11/2} \left (1+\frac {1}{a x}\right )^{7/2} 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}{64} a^3 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2} x^4+\frac {11}{48} a^4 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{7/2} x^5-\frac {11}{48} a^5 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{7/2} x^6+\frac {11}{56} a^6 c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{7/2} x^7-\frac {11}{72} a^7 c^4 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{7/2} x^8+\frac {1}{9} a^8 c^4 \left (1-\frac {1}{a x}\right )^{11/2} \left (1+\frac {1}{a x}\right )^{7/2} 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}{64} a^3 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2} x^4+\frac {11}{48} a^4 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{7/2} x^5-\frac {11}{48} a^5 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{7/2} x^6+\frac {11}{56} a^6 c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{7/2} x^7-\frac {11}{72} a^7 c^4 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{7/2} x^8+\frac {1}{9} a^8 c^4 \left (1-\frac {1}{a x}\right )^{11/2} \left (1+\frac {1}{a x}\right )^{7/2} 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}{64} a^3 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2} x^4+\frac {11}{48} a^4 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{7/2} x^5-\frac {11}{48} a^5 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{7/2} x^6+\frac {11}{56} a^6 c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{7/2} x^7-\frac {11}{72} a^7 c^4 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{7/2} x^8+\frac {1}{9} a^8 c^4 \left (1-\frac {1}{a x}\right )^{11/2} \left (1+\frac {1}{a x}\right )^{7/2} 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 = 0.21, size = 111, normalized size = 0.28 \[ \frac {c^4 \left (3465 \log \left (x \left (\sqrt {1-\frac {1}{a^2 x^2}}+1\right )\right )+a x \sqrt {1-\frac {1}{a^2 x^2}} \left (896 a^8 x^8-3024 a^7 x^7+1024 a^6 x^6+7224 a^5 x^5-8448 a^4 x^4-3066 a^3 x^3+10240 a^2 x^2-4599 a x-3712\right )\right )}{8064 a} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.94, size = 169, normalized size = 0.43 \[ \frac {3465 \, c^{4} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 3465 \, c^{4} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (896 \, a^{9} c^{4} x^{9} - 2128 \, a^{8} c^{4} x^{8} - 2000 \, a^{7} c^{4} x^{7} + 8248 \, a^{6} c^{4} x^{6} - 1224 \, a^{5} c^{4} x^{5} - 11514 \, a^{4} c^{4} x^{4} + 7174 \, a^{3} c^{4} x^{3} + 5641 \, a^{2} c^{4} x^{2} - 8311 \, a c^{4} x - 3712 \, c^{4}\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{8064 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 198, normalized size = 0.50 \[ -\frac {55 \, c^{4} \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right ) \mathrm {sgn}\left (a x + 1\right )}{128 \, {\left | a \right |}} - \frac {1}{8064} \, \sqrt {a^{2} x^{2} - 1} {\left (\frac {3712 \, c^{4} \mathrm {sgn}\left (a x + 1\right )}{a} + {\left (4599 \, c^{4} \mathrm {sgn}\left (a x + 1\right ) - 2 \, {\left (5120 \, a c^{4} \mathrm {sgn}\left (a x + 1\right ) - {\left (1533 \, a^{2} c^{4} \mathrm {sgn}\left (a x + 1\right ) + 4 \, {\left (1056 \, a^{3} c^{4} \mathrm {sgn}\left (a x + 1\right ) - {\left (903 \, a^{4} c^{4} \mathrm {sgn}\left (a x + 1\right ) + 2 \, {\left (64 \, a^{5} c^{4} \mathrm {sgn}\left (a x + 1\right ) + 7 \, {\left (8 \, a^{7} c^{4} x \mathrm {sgn}\left (a x + 1\right ) - 27 \, a^{6} c^{4} \mathrm {sgn}\left (a x + 1\right )\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 288, normalized size = 0.73 \[ \frac {\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right )^{2} c^{4} \left (896 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x^{6} a^{6}-3024 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x^{5} a^{5}+1920 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x^{4} a^{4}+4200 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x^{3} a^{3}-6528 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x^{2} a^{2}+1134 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x a +8064 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}-4352 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}-3465 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, x a +3465 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a \right )}{8064 a \left (a x -1\right ) \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.39, size = 415, normalized size = 1.06 \[ \frac {1}{8064} \, {\left (\frac {3465 \, c^{4} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac {3465 \, c^{4} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}} - \frac {2 \, {\left (3465 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {17}{2}} - 30030 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {15}{2}} + 115038 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {13}{2}} + 334602 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {11}{2}} - 360448 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {9}{2}} + 255222 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} - 115038 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} + 30030 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} - 3465 \, c^{4} \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{\frac {9 \, {\left (a x - 1\right )} a^{2}}{a x + 1} - \frac {36 \, {\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} + \frac {84 \, {\left (a x - 1\right )}^{3} a^{2}}{{\left (a x + 1\right )}^{3}} - \frac {126 \, {\left (a x - 1\right )}^{4} a^{2}}{{\left (a x + 1\right )}^{4}} + \frac {126 \, {\left (a x - 1\right )}^{5} a^{2}}{{\left (a x + 1\right )}^{5}} - \frac {84 \, {\left (a x - 1\right )}^{6} a^{2}}{{\left (a x + 1\right )}^{6}} + \frac {36 \, {\left (a x - 1\right )}^{7} a^{2}}{{\left (a x + 1\right )}^{7}} - \frac {9 \, {\left (a x - 1\right )}^{8} a^{2}}{{\left (a x + 1\right )}^{8}} + \frac {{\left (a x - 1\right )}^{9} a^{2}}{{\left (a x + 1\right )}^{9}} - a^{2}}\right )} a \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.17, size = 362, normalized size = 0.92 \[ \frac {\frac {715\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{96}-\frac {55\,c^4\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{64}-\frac {913\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{32}+\frac {14179\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}}{224}-\frac {5632\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{9/2}}{63}+\frac {18589\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{11/2}}{224}+\frac {913\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{13/2}}{32}-\frac {715\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{15/2}}{96}+\frac {55\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{17/2}}{64}}{a-\frac {9\,a\,\left (a\,x-1\right )}{a\,x+1}+\frac {36\,a\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {84\,a\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}+\frac {126\,a\,{\left (a\,x-1\right )}^4}{{\left (a\,x+1\right )}^4}-\frac {126\,a\,{\left (a\,x-1\right )}^5}{{\left (a\,x+1\right )}^5}+\frac {84\,a\,{\left (a\,x-1\right )}^6}{{\left (a\,x+1\right )}^6}-\frac {36\,a\,{\left (a\,x-1\right )}^7}{{\left (a\,x+1\right )}^7}+\frac {9\,a\,{\left (a\,x-1\right )}^8}{{\left (a\,x+1\right )}^8}-\frac {a\,{\left (a\,x-1\right )}^9}{{\left (a\,x+1\right )}^9}}+\frac {55\,c^4\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{64\,a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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