Optimal. Leaf size=144 \[ \frac {x \sqrt {\frac {1}{a x}+1}}{c \left (1-\frac {1}{a x}\right )^{3/2}}-\frac {14 \sqrt {\frac {1}{a x}+1}}{3 a c \sqrt {1-\frac {1}{a x}}}-\frac {5 \sqrt {\frac {1}{a x}+1}}{3 a c \left (1-\frac {1}{a x}\right )^{3/2}}+\frac {3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )}{a c} \]
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Rubi [A] time = 0.10, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {6194, 99, 152, 12, 92, 208} \[ \frac {x \sqrt {\frac {1}{a x}+1}}{c \left (1-\frac {1}{a x}\right )^{3/2}}-\frac {14 \sqrt {\frac {1}{a x}+1}}{3 a c \sqrt {1-\frac {1}{a x}}}-\frac {5 \sqrt {\frac {1}{a x}+1}}{3 a c \left (1-\frac {1}{a x}\right )^{3/2}}+\frac {3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )}{a c} \]
Antiderivative was successfully verified.
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Rule 12
Rule 92
Rule 99
Rule 152
Rule 208
Rule 6194
Rubi steps
\begin {align*} \int \frac {e^{3 \coth ^{-1}(a x)}}{c-\frac {c}{a^2 x^2}} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {x}{a}}}{x^2 \left (1-\frac {x}{a}\right )^{5/2}} \, dx,x,\frac {1}{x}\right )}{c}\\ &=\frac {\sqrt {1+\frac {1}{a x}} x}{c \left (1-\frac {1}{a x}\right )^{3/2}}-\frac {\operatorname {Subst}\left (\int \frac {\frac {3}{a}+\frac {2 x}{a^2}}{x \left (1-\frac {x}{a}\right )^{5/2} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{c}\\ &=-\frac {5 \sqrt {1+\frac {1}{a x}}}{3 a c \left (1-\frac {1}{a x}\right )^{3/2}}+\frac {\sqrt {1+\frac {1}{a x}} x}{c \left (1-\frac {1}{a x}\right )^{3/2}}+\frac {a \operatorname {Subst}\left (\int \frac {-\frac {9}{a^2}-\frac {5 x}{a^3}}{x \left (1-\frac {x}{a}\right )^{3/2} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{3 c}\\ &=-\frac {5 \sqrt {1+\frac {1}{a x}}}{3 a c \left (1-\frac {1}{a x}\right )^{3/2}}-\frac {14 \sqrt {1+\frac {1}{a x}}}{3 a c \sqrt {1-\frac {1}{a x}}}+\frac {\sqrt {1+\frac {1}{a x}} x}{c \left (1-\frac {1}{a x}\right )^{3/2}}-\frac {a^2 \operatorname {Subst}\left (\int \frac {9}{a^3 x \sqrt {1-\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{3 c}\\ &=-\frac {5 \sqrt {1+\frac {1}{a x}}}{3 a c \left (1-\frac {1}{a x}\right )^{3/2}}-\frac {14 \sqrt {1+\frac {1}{a x}}}{3 a c \sqrt {1-\frac {1}{a x}}}+\frac {\sqrt {1+\frac {1}{a x}} x}{c \left (1-\frac {1}{a x}\right )^{3/2}}-\frac {3 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{a c}\\ &=-\frac {5 \sqrt {1+\frac {1}{a x}}}{3 a c \left (1-\frac {1}{a x}\right )^{3/2}}-\frac {14 \sqrt {1+\frac {1}{a x}}}{3 a c \sqrt {1-\frac {1}{a x}}}+\frac {\sqrt {1+\frac {1}{a x}} x}{c \left (1-\frac {1}{a x}\right )^{3/2}}+\frac {3 \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 c}\\ &=-\frac {5 \sqrt {1+\frac {1}{a x}}}{3 a c \left (1-\frac {1}{a x}\right )^{3/2}}-\frac {14 \sqrt {1+\frac {1}{a x}}}{3 a c \sqrt {1-\frac {1}{a x}}}+\frac {\sqrt {1+\frac {1}{a x}} x}{c \left (1-\frac {1}{a x}\right )^{3/2}}+\frac {3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{a c}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 69, normalized size = 0.48 \[ \frac {\frac {x \sqrt {1-\frac {1}{a^2 x^2}} \left (3 a^2 x^2-19 a x+14\right )}{(a x-1)^2}+\frac {9 \log \left (x \left (\sqrt {1-\frac {1}{a^2 x^2}}+1\right )\right )}{a}}{3 c} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.55, size = 128, normalized size = 0.89 \[ \frac {9 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 9 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (3 \, a^{3} x^{3} - 16 \, a^{2} x^{2} - 5 \, a x + 14\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{3 \, {\left (a^{3} c x^{2} - 2 \, a^{2} c x + a c\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 148, normalized size = 1.03 \[ \frac {1}{3} \, a {\left (\frac {9 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c} - \frac {9 \, \log \left ({\left | \sqrt {\frac {a x - 1}{a x + 1}} - 1 \right |}\right )}{a^{2} c} - \frac {{\left (a x + 1\right )} {\left (\frac {12 \, {\left (a x - 1\right )}}{a x + 1} + 1\right )}}{{\left (a x - 1\right )} a^{2} c \sqrt {\frac {a x - 1}{a x + 1}}} - \frac {6 \, \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c {\left (\frac {a x - 1}{a x + 1} - 1\right )}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 346, normalized size = 2.40 \[ \frac {9 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x^{3} a^{4}+9 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, x^{3} a^{3}-27 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x^{2} a^{3}-6 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} x a -27 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, x^{2} a^{2}+27 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x \,a^{2}+5 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}+27 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, x a -9 a \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right )-9 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{3 a \left (a x -1\right ) \sqrt {a^{2}}\, c \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \left (a x +1\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 133, normalized size = 0.92 \[ \frac {1}{3} \, a {\left (\frac {\frac {11 \, {\left (a x - 1\right )}}{a x + 1} - \frac {18 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + 1}{a^{2} c \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} - a^{2} c \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} + \frac {9 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c} - \frac {9 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2} c}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.35, size = 100, normalized size = 0.69 \[ \frac {6\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a\,c}-\frac {\frac {11\,\left (a\,x-1\right )}{3\,\left (a\,x+1\right )}-\frac {6\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}+\frac {1}{3}}{a\,c\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}-a\,c\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}} \]
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 \[ \frac {a^{2} \int \frac {x^{2}}{\frac {a^{3} x^{3} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {a^{2} x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} + \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\, dx}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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