Optimal. Leaf size=124 \[ -\frac {(1-a x)^2}{3 a^2 x \left (c-\frac {c}{a^2 x^2}\right )^{3/2}}+\frac {2 (a x+1) (2 a x+5) (1-a x)^2}{3 a^4 x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2}}+\frac {2 (a x+1)^{3/2} (1-a x)^{3/2} \sin ^{-1}(a x)}{a^4 x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \]
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Rubi [A] time = 0.42, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {6167, 6159, 6129, 98, 143, 41, 216} \[ \frac {2 (a x+1) (2 a x+5) (1-a x)^2}{3 a^4 x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2}}-\frac {(1-a x)^2}{3 a^2 x \left (c-\frac {c}{a^2 x^2}\right )^{3/2}}+\frac {2 (a x+1)^{3/2} (1-a x)^{3/2} \sin ^{-1}(a x)}{a^4 x^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 41
Rule 98
Rule 143
Rule 216
Rule 6129
Rule 6159
Rule 6167
Rubi steps
\begin {align*} \int \frac {e^{-2 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \, dx &=-\int \frac {e^{-2 \tanh ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2}} \, dx\\ &=-\frac {\left ((1-a x)^{3/2} (1+a x)^{3/2}\right ) \int \frac {e^{-2 \tanh ^{-1}(a x)} x^3}{(1-a x)^{3/2} (1+a x)^{3/2}} \, dx}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3}\\ &=-\frac {\left ((1-a x)^{3/2} (1+a x)^{3/2}\right ) \int \frac {x^3}{\sqrt {1-a x} (1+a x)^{5/2}} \, dx}{\left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3}\\ &=-\frac {(1-a x)^2}{3 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x}+\frac {\left ((1-a x)^{3/2} (1+a x)^{3/2}\right ) \int \frac {x (2-4 a x)}{\sqrt {1-a x} (1+a x)^{3/2}} \, dx}{3 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3}\\ &=-\frac {(1-a x)^2}{3 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x}+\frac {2 (1-a x)^2 (1+a x) (5+2 a x)}{3 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3}+\frac {\left (2 (1-a x)^{3/2} (1+a x)^{3/2}\right ) \int \frac {1}{\sqrt {1-a x} \sqrt {1+a x}} \, dx}{a^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3}\\ &=-\frac {(1-a x)^2}{3 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x}+\frac {2 (1-a x)^2 (1+a x) (5+2 a x)}{3 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3}+\frac {\left (2 (1-a x)^{3/2} (1+a x)^{3/2}\right ) \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{a^3 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3}\\ &=-\frac {(1-a x)^2}{3 a^2 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x}+\frac {2 (1-a x)^2 (1+a x) (5+2 a x)}{3 a^4 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3}+\frac {2 (1-a x)^{3/2} (1+a x)^{3/2} \sin ^{-1}(a x)}{a^4 \left (c-\frac {c}{a^2 x^2}\right )^{3/2} x^3}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 95, normalized size = 0.77 \[ \frac {3 a^3 x^3+11 a^2 x^2-6 (a x+1) \sqrt {a^2 x^2-1} \log \left (\sqrt {a^2 x^2-1}+a x\right )-4 a x-10}{3 a^2 c x (a x+1) \sqrt {c-\frac {c}{a^2 x^2}}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.59, size = 279, normalized size = 2.25 \[ \left [\frac {3 \, {\left (a^{2} x^{2} + 2 \, a x + 1\right )} \sqrt {c} \log \left (2 \, a^{2} c x^{2} - 2 \, a^{2} \sqrt {c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right ) + {\left (3 \, a^{3} x^{3} + 14 \, a^{2} x^{2} + 10 \, a x\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{3 \, {\left (a^{3} c^{2} x^{2} + 2 \, a^{2} c^{2} x + a c^{2}\right )}}, \frac {6 \, {\left (a^{2} x^{2} + 2 \, a x + 1\right )} \sqrt {-c} \arctan \left (\frac {a^{2} \sqrt {-c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right ) + {\left (3 \, a^{3} x^{3} + 14 \, a^{2} x^{2} + 10 \, a x\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{3 \, {\left (a^{3} c^{2} x^{2} + 2 \, a^{2} c^{2} x + a c^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {a x - 1}{{\left (a x + 1\right )} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 326, normalized size = 2.63 \[ \frac {\left (3 c^{\frac {3}{2}} \sqrt {\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}}\, x^{3} a^{3}+15 x^{2} a^{2} c^{\frac {3}{2}} \sqrt {\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}}-4 c^{\frac {3}{2}} \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, x^{2} a^{2}-6 \ln \left (x \sqrt {c}+\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right ) \sqrt {\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, x \,a^{2} c -4 c^{\frac {3}{2}} \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, x a -6 \ln \left (x \sqrt {c}+\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right ) \sqrt {\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a c -12 c^{\frac {3}{2}} \sqrt {\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}}+2 \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, c^{\frac {3}{2}}\right ) \left (a x -1\right )}{3 \sqrt {\frac {\left (a x -1\right ) \left (a x +1\right ) c}{a^{2}}}\, x^{3} \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}\right )^{\frac {3}{2}} a^{4} c^{\frac {3}{2}}} \]
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 \[ \int \frac {a x - 1}{{\left (a x + 1\right )} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {a\,x-1}{{\left (c-\frac {c}{a^2\,x^2}\right )}^{3/2}\,\left (a\,x+1\right )} \,d x \]
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 \[ \int \frac {a x - 1}{\left (- c \left (-1 + \frac {1}{a x}\right ) \left (1 + \frac {1}{a x}\right )\right )^{\frac {3}{2}} \left (a x + 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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