Optimal. Leaf size=343 \[ c^4 x \left (\frac {1}{a x}+1\right )^{7/2} \left (1-\frac {1}{a x}\right )^{9/2}+\frac {8 c^4 \left (\frac {1}{a x}+1\right )^{7/2} \left (1-\frac {1}{a x}\right )^{7/2}}{7 a}+\frac {7 c^4 \left (\frac {1}{a x}+1\right )^{7/2} \left (1-\frac {1}{a x}\right )^{5/2}}{6 a}+\frac {29 c^4 \left (\frac {1}{a x}+1\right )^{7/2} \left (1-\frac {1}{a x}\right )^{3/2}}{30 a}+\frac {19 c^4 \left (\frac {1}{a x}+1\right )^{7/2} \sqrt {1-\frac {1}{a x}}}{40 a}+\frac {7 c^4 \left (\frac {1}{a x}+1\right )^{5/2} \sqrt {1-\frac {1}{a x}}}{40 a}-\frac {c^4 \left (\frac {1}{a x}+1\right )^{3/2} \sqrt {1-\frac {1}{a x}}}{16 a}-\frac {19 c^4 \sqrt {\frac {1}{a x}+1} \sqrt {1-\frac {1}{a x}}}{16 a}+\frac {35 c^4 \csc ^{-1}(a x)}{16 a}-\frac {c^4 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )}{a} \]
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Rubi [A] time = 0.26, antiderivative size = 343, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {6194, 97, 154, 157, 41, 216, 92, 208} \[ c^4 x \left (\frac {1}{a x}+1\right )^{7/2} \left (1-\frac {1}{a x}\right )^{9/2}+\frac {8 c^4 \left (\frac {1}{a x}+1\right )^{7/2} \left (1-\frac {1}{a x}\right )^{7/2}}{7 a}+\frac {7 c^4 \left (\frac {1}{a x}+1\right )^{7/2} \left (1-\frac {1}{a x}\right )^{5/2}}{6 a}+\frac {29 c^4 \left (\frac {1}{a x}+1\right )^{7/2} \left (1-\frac {1}{a x}\right )^{3/2}}{30 a}+\frac {19 c^4 \left (\frac {1}{a x}+1\right )^{7/2} \sqrt {1-\frac {1}{a x}}}{40 a}+\frac {7 c^4 \left (\frac {1}{a x}+1\right )^{5/2} \sqrt {1-\frac {1}{a x}}}{40 a}-\frac {c^4 \left (\frac {1}{a x}+1\right )^{3/2} \sqrt {1-\frac {1}{a x}}}{16 a}-\frac {19 c^4 \sqrt {\frac {1}{a x}+1} \sqrt {1-\frac {1}{a x}}}{16 a}+\frac {35 c^4 \csc ^{-1}(a x)}{16 a}-\frac {c^4 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )}{a} \]
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 )^{9/2} \left (1+\frac {x}{a}\right )^{7/2}}{x^2} \, dx,x,\frac {1}{x}\right )\right )\\ &=c^4 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{7/2} x-c^4 \operatorname {Subst}\left (\int \frac {\left (-\frac {1}{a}-\frac {8 x}{a^2}\right ) \left (1-\frac {x}{a}\right )^{7/2} \left (1+\frac {x}{a}\right )^{5/2}}{x} \, dx,x,\frac {1}{x}\right )\\ &=\frac {8 c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{7/2}}{7 a}+c^4 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{7/2} x-\frac {1}{7} \left (a c^4\right ) \operatorname {Subst}\left (\int \frac {\left (-\frac {7}{a^2}-\frac {49 x}{a^3}\right ) \left (1-\frac {x}{a}\right )^{5/2} \left (1+\frac {x}{a}\right )^{5/2}}{x} \, dx,x,\frac {1}{x}\right )\\ &=\frac {7 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{7/2}}{6 a}+\frac {8 c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{7/2}}{7 a}+c^4 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{7/2} x-\frac {1}{42} \left (a^2 c^4\right ) \operatorname {Subst}\left (\int \frac {\left (-\frac {42}{a^3}-\frac {203 x}{a^4}\right ) \left (1-\frac {x}{a}\right )^{3/2} \left (1+\frac {x}{a}\right )^{5/2}}{x} \, dx,x,\frac {1}{x}\right )\\ &=\frac {29 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{7/2}}{30 a}+\frac {7 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{7/2}}{6 a}+\frac {8 c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{7/2}}{7 a}+c^4 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{7/2} x-\frac {1}{210} \left (a^3 c^4\right ) \operatorname {Subst}\left (\int \frac {\left (-\frac {210}{a^4}-\frac {399 x}{a^5}\right ) \sqrt {1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{5/2}}{x} \, dx,x,\frac {1}{x}\right )\\ &=\frac {19 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}{40 a}+\frac {29 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{7/2}}{30 a}+\frac {7 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{7/2}}{6 a}+\frac {8 c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{7/2}}{7 a}+c^4 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{7/2} x-\frac {1}{840} \left (a^4 c^4\right ) \operatorname {Subst}\left (\int \frac {\left (-\frac {840}{a^5}+\frac {441 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 {7 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}{40 a}+\frac {19 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}{40 a}+\frac {29 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{7/2}}{30 a}+\frac {7 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{7/2}}{6 a}+\frac {8 c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{7/2}}{7 a}+c^4 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{7/2} x+\frac {\left (a^5 c^4\right ) \operatorname {Subst}\left (\int \frac {\left (\frac {2520}{a^6}+\frac {315 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 {c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}{16 a}+\frac {7 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}{40 a}+\frac {19 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}{40 a}+\frac {29 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{7/2}}{30 a}+\frac {7 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{7/2}}{6 a}+\frac {8 c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{7/2}}{7 a}+c^4 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{7/2} x-\frac {\left (a^6 c^4\right ) \operatorname {Subst}\left (\int \frac {\left (-\frac {5040}{a^7}-\frac {5985 x}{a^8}\right ) \sqrt {1+\frac {x}{a}}}{x \sqrt {1-\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{5040}\\ &=-\frac {19 c^4 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}{16 a}-\frac {c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}{16 a}+\frac {7 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}{40 a}+\frac {19 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}{40 a}+\frac {29 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{7/2}}{30 a}+\frac {7 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{7/2}}{6 a}+\frac {8 c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{7/2}}{7 a}+c^4 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{7/2} 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 {19 c^4 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}{16 a}-\frac {c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}{16 a}+\frac {7 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}{40 a}+\frac {19 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}{40 a}+\frac {29 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{7/2}}{30 a}+\frac {7 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{7/2}}{6 a}+\frac {8 c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{7/2}}{7 a}+c^4 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{7/2} 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 {19 c^4 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}{16 a}-\frac {c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}{16 a}+\frac {7 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}{40 a}+\frac {19 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}{40 a}+\frac {29 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{7/2}}{30 a}+\frac {7 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{7/2}}{6 a}+\frac {8 c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{7/2}}{7 a}+c^4 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{7/2} 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 {19 c^4 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}{16 a}-\frac {c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}{16 a}+\frac {7 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}{40 a}+\frac {19 c^4 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}{40 a}+\frac {29 c^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{7/2}}{30 a}+\frac {7 c^4 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{7/2}}{6 a}+\frac {8 c^4 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{7/2}}{7 a}+c^4 \left (1-\frac {1}{a x}\right )^{9/2} \left (1+\frac {1}{a x}\right )^{7/2} 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 = 0.31, 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 = 0.61, size = 201, normalized size = 0.59 \[ -\frac {7350 \, a^{7} c^{4} x^{7} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) + 1680 \, a^{7} c^{4} x^{7} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 1680 \, a^{7} c^{4} x^{7} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) - {\left (1680 \, a^{8} c^{4} x^{8} + 4496 \, a^{7} c^{4} x^{7} + 5861 \, a^{6} c^{4} x^{6} + 1093 \, a^{5} c^{4} x^{5} - 3282 \, a^{4} c^{4} x^{4} - 274 \, a^{3} c^{4} x^{3} + 1336 \, a^{2} c^{4} x^{2} + 40 \, a c^{4} x - 240 \, c^{4}\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{1680 \, a^{8} x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 524, normalized size = 1.53 \[ -\frac {35 \, c^{4} \arctan \left (-x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1}\right ) \mathrm {sgn}\left (a x + 1\right )}{8 \, a} + \frac {c^{4} \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right ) \mathrm {sgn}\left (a x + 1\right )}{{\left | a \right |}} + \frac {\sqrt {a^{2} x^{2} - 1} c^{4} \mathrm {sgn}\left (a x + 1\right )}{a} - \frac {3045 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{13} c^{4} {\left | a \right |} \mathrm {sgn}\left (a x + 1\right ) - 6720 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{12} a c^{4} \mathrm {sgn}\left (a x + 1\right ) + 6860 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{11} c^{4} {\left | a \right |} \mathrm {sgn}\left (a x + 1\right ) - 20160 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{10} a c^{4} \mathrm {sgn}\left (a x + 1\right ) + 9065 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{9} c^{4} {\left | a \right |} \mathrm {sgn}\left (a x + 1\right ) - 49280 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{8} a c^{4} \mathrm {sgn}\left (a x + 1\right ) - 49280 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{6} a c^{4} \mathrm {sgn}\left (a x + 1\right ) - 9065 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{5} c^{4} {\left | a \right |} \mathrm {sgn}\left (a x + 1\right ) - 38976 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{4} a c^{4} \mathrm {sgn}\left (a x + 1\right ) - 6860 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{3} c^{4} {\left | a \right |} \mathrm {sgn}\left (a x + 1\right ) - 12992 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{2} a c^{4} \mathrm {sgn}\left (a x + 1\right ) - 3045 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )} c^{4} {\left | a \right |} \mathrm {sgn}\left (a x + 1\right ) - 2816 \, a c^{4} \mathrm {sgn}\left (a x + 1\right )}{840 \, {\left ({\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{2} + 1\right )}^{7} a {\left | a \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 320, normalized size = 0.93 \[ -\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) c^{4} \left (-1680 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, x^{8} a^{8}+1680 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x^{6} a^{6}-3675 a^{7} x^{7} \sqrt {a^{2}}\, \sqrt {a^{2} x^{2}-1}-3675 a^{7} x^{7} \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )+1680 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x^{7} a^{8}+1995 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x^{5} a^{5}-1136 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x^{4} a^{4}-1050 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x^{3} a^{3}+816 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x^{2} a^{2}+280 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x a -240 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\right )}{1680 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{8} x^{7} \sqrt {a^{2}}} \]
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 \[ -\frac {1}{840} \, {\left (\frac {3675 \, c^{4} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right )}{a^{2}} + \frac {840 \, c^{4} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac {840 \, c^{4} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}} - \frac {1995 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {15}{2}} + 10185 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {13}{2}} + 17619 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {11}{2}} + 4569 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {9}{2}} + 71801 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} + 72051 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} + 31465 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + 5355 \, c^{4} \sqrt {\frac {a x - 1}{a x + 1}}}{\frac {6 \, {\left (a x - 1\right )} a^{2}}{a x + 1} + \frac {14 \, {\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} + \frac {14 \, {\left (a x - 1\right )}^{3} a^{2}}{{\left (a x + 1\right )}^{3}} - \frac {14 \, {\left (a x - 1\right )}^{5} a^{2}}{{\left (a x + 1\right )}^{5}} - \frac {14 \, {\left (a x - 1\right )}^{6} a^{2}}{{\left (a x + 1\right )}^{6}} - \frac {6 \, {\left (a x - 1\right )}^{7} a^{2}}{{\left (a x + 1\right )}^{7}} - \frac {{\left (a x - 1\right )}^{8} a^{2}}{{\left (a x + 1\right )}^{8}} + a^{2}}\right )} a \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.37, size = 332, normalized size = 0.97 \[ \frac {\frac {51\,c^4\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{8}+\frac {899\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{24}+\frac {3431\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{40}+\frac {71801\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}}{840}+\frac {1523\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{9/2}}{280}+\frac {839\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{11/2}}{40}+\frac {97\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{13/2}}{8}+\frac {19\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{15/2}}{8}}{a+\frac {6\,a\,\left (a\,x-1\right )}{a\,x+1}+\frac {14\,a\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}+\frac {14\,a\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}-\frac {14\,a\,{\left (a\,x-1\right )}^5}{{\left (a\,x+1\right )}^5}-\frac {14\,a\,{\left (a\,x-1\right )}^6}{{\left (a\,x+1\right )}^6}-\frac {6\,a\,{\left (a\,x-1\right )}^7}{{\left (a\,x+1\right )}^7}-\frac {a\,{\left (a\,x-1\right )}^8}{{\left (a\,x+1\right )}^8}}-\frac {35\,c^4\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{8\,a}-\frac {2\,c^4\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a} \]
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 \[ \frac {c^{4} \left (\int a^{8} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}\, dx + \int \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{x^{8}}\, dx + \int \left (- \frac {4 a^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{x^{6}}\right )\, dx + \int \frac {6 a^{4} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{x^{4}}\, dx + \int \left (- \frac {4 a^{6} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{x^{2}}\right )\, dx\right )}{a^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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