Optimal. Leaf size=342 \[ \frac {8 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{9/2}}{7 a}+\frac {47 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{9/2}}{42 a}+c^4 x \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{9/2}+\frac {61 c^4 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{9/2}}{70 a}-\frac {131 c^4 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}}{280 a}-\frac {91 c^4 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}}{120 a}-\frac {67 c^4 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}}{48 a}-\frac {51 c^4 \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}{16 a}+\frac {35 c^4 \csc ^{-1}(a x)}{16 a}+\frac {c^4 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )}{a} \]
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Rubi [A] time = 0.25, antiderivative size = 342, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 8, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6194, 97, 154, 157, 41, 216, 92, 208} \[ \frac {8 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{9/2}}{7 a}+\frac {47 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{9/2}}{42 a}+c^4 x \left (1-\frac {1}{a x}\right )^{7/2} \left (\frac {1}{a x}+1\right )^{9/2}+\frac {61 c^4 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{9/2}}{70 a}-\frac {131 c^4 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}}{280 a}-\frac {91 c^4 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}}{120 a}-\frac {67 c^4 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}}{48 a}-\frac {51 c^4 \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}{16 a}+\frac {35 c^4 \csc ^{-1}(a x)}{16 a}+\frac {c^4 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )}{a} \]
Antiderivative was successfully verified.
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Rule 41
Rule 92
Rule 97
Rule 154
Rule 157
Rule 208
Rule 216
Rule 6194
Rubi steps
\begin {align*} \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^4 \, dx &=-\left (c^4 \operatorname {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{7/2} \left (1+\frac {x}{a}\right )^{9/2}}{x^2} \, dx,x,\frac {1}{x}\right )\right )\\ &=c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{9/2} x-c^4 \operatorname {Subst}\left (\int \frac {\left (\frac {1}{a}-\frac {8 x}{a^2}\right ) \left (1-\frac {x}{a}\right )^{5/2} \left (1+\frac {x}{a}\right )^{7/2}}{x} \, dx,x,\frac {1}{x}\right )\\ &=\frac {8 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{9/2}}{7 a}+c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{9/2} x-\frac {1}{7} \left (a c^4\right ) \operatorname {Subst}\left (\int \frac {\left (\frac {7}{a^2}-\frac {47 x}{a^3}\right ) \left (1-\frac {x}{a}\right )^{3/2} \left (1+\frac {x}{a}\right )^{7/2}}{x} \, dx,x,\frac {1}{x}\right )\\ &=\frac {47 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}{42 a}+\frac {8 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{9/2}}{7 a}+c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{9/2} x-\frac {1}{42} \left (a^2 c^4\right ) \operatorname {Subst}\left (\int \frac {\left (\frac {42}{a^3}-\frac {183 x}{a^4}\right ) \sqrt {1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{7/2}}{x} \, dx,x,\frac {1}{x}\right )\\ &=\frac {61 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2}}{70 a}+\frac {47 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}{42 a}+\frac {8 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{9/2}}{7 a}+c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{9/2} x-\frac {1}{210} \left (a^3 c^4\right ) \operatorname {Subst}\left (\int \frac {\left (\frac {210}{a^4}-\frac {393 x}{a^5}\right ) \left (1+\frac {x}{a}\right )^{7/2}}{x \sqrt {1-\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {131 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}{280 a}+\frac {61 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2}}{70 a}+\frac {47 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}{42 a}+\frac {8 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{9/2}}{7 a}+c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{9/2} x+\frac {1}{840} \left (a^4 c^4\right ) \operatorname {Subst}\left (\int \frac {\left (-\frac {840}{a^5}+\frac {1911 x}{a^6}\right ) \left (1+\frac {x}{a}\right )^{5/2}}{x \sqrt {1-\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {91 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}{120 a}-\frac {131 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}{280 a}+\frac {61 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2}}{70 a}+\frac {47 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}{42 a}+\frac {8 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{9/2}}{7 a}+c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{9/2} x-\frac {\left (a^5 c^4\right ) \operatorname {Subst}\left (\int \frac {\left (\frac {2520}{a^6}-\frac {7035 x}{a^7}\right ) \left (1+\frac {x}{a}\right )^{3/2}}{x \sqrt {1-\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{2520}\\ &=-\frac {67 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}{48 a}-\frac {91 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}{120 a}-\frac {131 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}{280 a}+\frac {61 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2}}{70 a}+\frac {47 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}{42 a}+\frac {8 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{9/2}}{7 a}+c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{9/2} x+\frac {\left (a^6 c^4\right ) \operatorname {Subst}\left (\int \frac {\left (-\frac {5040}{a^7}+\frac {16065 x}{a^8}\right ) \sqrt {1+\frac {x}{a}}}{x \sqrt {1-\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{5040}\\ &=-\frac {51 c^4 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}{16 a}-\frac {67 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}{48 a}-\frac {91 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}{120 a}-\frac {131 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}{280 a}+\frac {61 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2}}{70 a}+\frac {47 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}{42 a}+\frac {8 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{9/2}}{7 a}+c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{9/2} x-\frac {\left (a^7 c^4\right ) \operatorname {Subst}\left (\int \frac {\frac {5040}{a^8}-\frac {11025 x}{a^9}}{x \sqrt {1-\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{5040}\\ &=-\frac {51 c^4 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}{16 a}-\frac {67 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}{48 a}-\frac {91 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}{120 a}-\frac {131 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}{280 a}+\frac {61 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2}}{70 a}+\frac {47 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}{42 a}+\frac {8 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{9/2}}{7 a}+c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{9/2} x+\frac {\left (35 c^4\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{16 a^2}-\frac {c^4 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{a}\\ &=-\frac {51 c^4 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}{16 a}-\frac {67 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}{48 a}-\frac {91 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}{120 a}-\frac {131 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}{280 a}+\frac {61 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2}}{70 a}+\frac {47 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}{42 a}+\frac {8 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{9/2}}{7 a}+c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{9/2} x+\frac {c^4 \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 )}{a^2}+\frac {\left (35 c^4\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{16 a^2}\\ &=-\frac {51 c^4 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}{16 a}-\frac {67 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}{48 a}-\frac {91 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}{120 a}-\frac {131 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}{280 a}+\frac {61 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2}}{70 a}+\frac {47 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{9/2}}{42 a}+\frac {8 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{9/2}}{7 a}+c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{9/2} x+\frac {35 c^4 \csc ^{-1}(a x)}{16 a}+\frac {c^4 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{a}\\ \end {align*}
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Mathematica [A] time = 0.32, size = 120, normalized size = 0.35 \[ \frac {c^4 \left (3675 a^6 \sin ^{-1}\left (\frac {1}{a x}\right )+1680 a^6 \log \left (x \left (\sqrt {1-\frac {1}{a^2 x^2}}+1\right )\right )+\frac {\sqrt {1-\frac {1}{a^2 x^2}} \left (1680 a^7 x^7-2816 a^6 x^6+3045 a^5 x^5+1952 a^4 x^4-1330 a^3 x^3-1056 a^2 x^2+280 a x+240\right )}{x^6}\right )}{1680 a^7} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.63, size = 201, normalized size = 0.59 \[ -\frac {7350 \, a^{7} c^{4} x^{7} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) - 1680 \, a^{7} c^{4} x^{7} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) + 1680 \, a^{7} c^{4} x^{7} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) - {\left (1680 \, a^{8} c^{4} x^{8} - 1136 \, a^{7} c^{4} x^{7} + 229 \, a^{6} c^{4} x^{6} + 4997 \, a^{5} c^{4} x^{5} + 622 \, a^{4} c^{4} x^{4} - 2386 \, a^{3} c^{4} x^{3} - 776 \, a^{2} c^{4} x^{2} + 520 \, a c^{4} x + 240 \, c^{4}\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{1680 \, a^{8} x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 335, normalized size = 0.98 \[ -\frac {1}{840} \, a c^{4} {\left (\frac {3675 \, \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right )}{a^{2}} - \frac {840 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} + \frac {840 \, \log \left ({\left | \sqrt {\frac {a x - 1}{a x + 1}} - 1 \right |}\right )}{a^{2}} + \frac {1680 \, \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} {\left (\frac {a x - 1}{a x + 1} - 1\right )}} + \frac {\frac {1260 \, {\left (a x - 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{a x + 1} + \frac {18921 \, {\left (a x - 1\right )}^{2} \sqrt {\frac {a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{2}} + \frac {73152 \, {\left (a x - 1\right )}^{3} \sqrt {\frac {a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{3}} + \frac {60151 \, {\left (a x - 1\right )}^{4} \sqrt {\frac {a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{4}} + \frac {23380 \, {\left (a x - 1\right )}^{5} \sqrt {\frac {a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{5}} + \frac {3675 \, {\left (a x - 1\right )}^{6} \sqrt {\frac {a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{6}} - 315 \, \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} {\left (\frac {a x - 1}{a x + 1} + 1\right )}^{7}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 320, normalized size = 0.94 \[ \frac {\left (a x -1\right ) c^{4} \left (-1680 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, x^{8} a^{8}+1680 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x^{6} a^{6}+3675 a^{7} x^{7} \sqrt {a^{2}}\, \sqrt {a^{2} x^{2}-1}+3675 a^{7} x^{7} \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )+1680 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x^{7} a^{8}-1995 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x^{5} a^{5}-1136 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x^{4} a^{4}+1050 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x^{3} a^{3}+816 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x^{2} a^{2}-280 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x a -240 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\right )}{1680 \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{8} x^{7} \sqrt {a^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 380, normalized size = 1.11 \[ -\frac {1}{840} \, {\left (\frac {3675 \, c^{4} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right )}{a^{2}} - \frac {840 \, c^{4} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} + \frac {840 \, c^{4} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}} - \frac {5355 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {15}{2}} + 31465 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {13}{2}} + 72051 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {11}{2}} + 71801 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {9}{2}} + 4569 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} + 17619 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} + 10185 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + 1995 \, c^{4} \sqrt {\frac {a x - 1}{a x + 1}}}{\frac {6 \, {\left (a x - 1\right )} a^{2}}{a x + 1} + \frac {14 \, {\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} + \frac {14 \, {\left (a x - 1\right )}^{3} a^{2}}{{\left (a x + 1\right )}^{3}} - \frac {14 \, {\left (a x - 1\right )}^{5} a^{2}}{{\left (a x + 1\right )}^{5}} - \frac {14 \, {\left (a x - 1\right )}^{6} a^{2}}{{\left (a x + 1\right )}^{6}} - \frac {6 \, {\left (a x - 1\right )}^{7} a^{2}}{{\left (a x + 1\right )}^{7}} - \frac {{\left (a x - 1\right )}^{8} a^{2}}{{\left (a x + 1\right )}^{8}} + a^{2}}\right )} a \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.42, size = 332, normalized size = 0.97 \[ \frac {\frac {19\,c^4\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{8}+\frac {97\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{8}+\frac {839\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{40}+\frac {1523\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}}{280}+\frac {71801\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{9/2}}{840}+\frac {3431\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{11/2}}{40}+\frac {899\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{13/2}}{24}+\frac {51\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{15/2}}{8}}{a+\frac {6\,a\,\left (a\,x-1\right )}{a\,x+1}+\frac {14\,a\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}+\frac {14\,a\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}-\frac {14\,a\,{\left (a\,x-1\right )}^5}{{\left (a\,x+1\right )}^5}-\frac {14\,a\,{\left (a\,x-1\right )}^6}{{\left (a\,x+1\right )}^6}-\frac {6\,a\,{\left (a\,x-1\right )}^7}{{\left (a\,x+1\right )}^7}-\frac {a\,{\left (a\,x-1\right )}^8}{{\left (a\,x+1\right )}^8}}-\frac {35\,c^4\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{8\,a}+\frac {2\,c^4\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a} \]
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 \[ \frac {c^{4} \left (\int \frac {a^{8}}{\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\, dx + \int \frac {1}{x^{8} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\, dx + \int \left (- \frac {4 a^{2}}{x^{6} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\right )\, dx + \int \frac {6 a^{4}}{x^{4} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\, dx + \int \left (- \frac {4 a^{6}}{x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\right )\, dx\right )}{a^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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