Optimal. Leaf size=117 \[ -\frac {3 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )}{a c^{3/2}}-\frac {2 x \sqrt {c-\frac {c}{a x}}}{c^2 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {3 x \sqrt {1-\frac {1}{a^2 x^2}}}{c \sqrt {c-\frac {c}{a x}}} \]
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Rubi [A] time = 0.22, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {6177, 869, 873, 875, 208} \[ -\frac {2 x \sqrt {c-\frac {c}{a x}}}{c^2 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {3 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )}{a c^{3/2}}+\frac {3 x \sqrt {1-\frac {1}{a^2 x^2}}}{c \sqrt {c-\frac {c}{a x}}} \]
Antiderivative was successfully verified.
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Rule 208
Rule 869
Rule 873
Rule 875
Rule 6177
Rubi steps
\begin {align*} \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^{3/2}} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {\left (c-\frac {c x}{a}\right )^{3/2}}{x^2 \left (1-\frac {x^2}{a^2}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{c^3}\\ &=-\frac {2 \sqrt {c-\frac {c}{a x}} x}{c^2 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {3 \operatorname {Subst}\left (\int \frac {\sqrt {c-\frac {c x}{a}}}{x^2 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{c^2}\\ &=\frac {3 \sqrt {1-\frac {1}{a^2 x^2}} x}{c \sqrt {c-\frac {c}{a x}}}-\frac {2 \sqrt {c-\frac {c}{a x}} x}{c^2 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {3 \operatorname {Subst}\left (\int \frac {\sqrt {c-\frac {c x}{a}}}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{2 a c^2}\\ &=\frac {3 \sqrt {1-\frac {1}{a^2 x^2}} x}{c \sqrt {c-\frac {c}{a x}}}-\frac {2 \sqrt {c-\frac {c}{a x}} x}{c^2 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {3 \operatorname {Subst}\left (\int \frac {1}{-\frac {c}{a^2}+\frac {c^2 x^2}{a^2}} \, dx,x,\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )}{a^3}\\ &=\frac {3 \sqrt {1-\frac {1}{a^2 x^2}} x}{c \sqrt {c-\frac {c}{a x}}}-\frac {2 \sqrt {c-\frac {c}{a x}} x}{c^2 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {3 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {1-\frac {1}{a^2 x^2}}}{\sqrt {c-\frac {c}{a x}}}\right )}{a c^{3/2}}\\ \end {align*}
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Mathematica [C] time = 0.05, size = 64, normalized size = 0.55 \[ \frac {2 \left (1-\frac {1}{a x}\right )^{3/2} \, _2F_1\left (-\frac {1}{2},2;\frac {1}{2};1+\frac {1}{a x}\right )}{a \sqrt {\frac {1}{a x}+1} \left (c-\frac {c}{a x}\right )^{3/2}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.67, size = 311, normalized size = 2.66 \[ \left [\frac {3 \, {\left (a x - 1\right )} \sqrt {c} \log \left (-\frac {8 \, a^{3} c x^{3} - 7 \, a c x - 4 \, {\left (2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + a x\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}} - c}{a x - 1}\right ) + 4 \, {\left (a^{2} x^{2} + 3 \, a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{4 \, {\left (a^{2} c^{2} x - a c^{2}\right )}}, \frac {3 \, {\left (a x - 1\right )} \sqrt {-c} \arctan \left (\frac {2 \, {\left (a^{2} x^{2} + a x\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) + 2 \, {\left (a^{2} x^{2} + 3 \, a x\right )} \sqrt {\frac {a x - 1}{a x + 1}} \sqrt {\frac {a c x - c}{a x}}}{2 \, {\left (a^{2} c^{2} x - a c^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 149, normalized size = 1.27 \[ -\frac {\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) \sqrt {\frac {c \left (a x -1\right )}{a x}}\, x \left (-2 a^{\frac {3}{2}} x \sqrt {\left (a x +1\right ) x}+3 \ln \left (\frac {2 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right ) x a -6 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}+3 \ln \left (\frac {2 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right )\right )}{2 \left (a x -1\right )^{2} \sqrt {a}\, c^{2} \sqrt {\left (a x +1\right ) x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{{\left (c - \frac {c}{a x}\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{{\left (c-\frac {c}{a\,x}\right )}^{3/2}} \,d x \]
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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