Optimal. Leaf size=295 \[ \frac {3 a \sqrt {1-a^2 x^2}}{4 c^2 (1-a x) \sqrt {c-a^2 c x^2}}-\frac {a \sqrt {1-a^2 x^2}}{8 c^2 (a x+1) \sqrt {c-a^2 c x^2}}+\frac {a \sqrt {1-a^2 x^2}}{8 c^2 (1-a x)^2 \sqrt {c-a^2 c x^2}}-\frac {\sqrt {1-a^2 x^2}}{c^2 x \sqrt {c-a^2 c x^2}}+\frac {a \sqrt {1-a^2 x^2} \log (x)}{c^2 \sqrt {c-a^2 c x^2}}-\frac {23 a \sqrt {1-a^2 x^2} \log (1-a x)}{16 c^2 \sqrt {c-a^2 c x^2}}+\frac {7 a \sqrt {1-a^2 x^2} \log (a x+1)}{16 c^2 \sqrt {c-a^2 c x^2}} \]
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Rubi [A] time = 0.25, antiderivative size = 295, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {6153, 6150, 88} \[ \frac {3 a \sqrt {1-a^2 x^2}}{4 c^2 (1-a x) \sqrt {c-a^2 c x^2}}-\frac {a \sqrt {1-a^2 x^2}}{8 c^2 (a x+1) \sqrt {c-a^2 c x^2}}+\frac {a \sqrt {1-a^2 x^2}}{8 c^2 (1-a x)^2 \sqrt {c-a^2 c x^2}}-\frac {\sqrt {1-a^2 x^2}}{c^2 x \sqrt {c-a^2 c x^2}}+\frac {a \sqrt {1-a^2 x^2} \log (x)}{c^2 \sqrt {c-a^2 c x^2}}-\frac {23 a \sqrt {1-a^2 x^2} \log (1-a x)}{16 c^2 \sqrt {c-a^2 c x^2}}+\frac {7 a \sqrt {1-a^2 x^2} \log (a x+1)}{16 c^2 \sqrt {c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Rule 88
Rule 6150
Rule 6153
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)}}{x^2 \left (c-a^2 c x^2\right )^{5/2}} \, dx &=\frac {\sqrt {1-a^2 x^2} \int \frac {e^{\tanh ^{-1}(a x)}}{x^2 \left (1-a^2 x^2\right )^{5/2}} \, dx}{c^2 \sqrt {c-a^2 c x^2}}\\ &=\frac {\sqrt {1-a^2 x^2} \int \frac {1}{x^2 (1-a x)^3 (1+a x)^2} \, dx}{c^2 \sqrt {c-a^2 c x^2}}\\ &=\frac {\sqrt {1-a^2 x^2} \int \left (\frac {1}{x^2}+\frac {a}{x}-\frac {a^2}{4 (-1+a x)^3}+\frac {3 a^2}{4 (-1+a x)^2}-\frac {23 a^2}{16 (-1+a x)}+\frac {a^2}{8 (1+a x)^2}+\frac {7 a^2}{16 (1+a x)}\right ) \, dx}{c^2 \sqrt {c-a^2 c x^2}}\\ &=-\frac {\sqrt {1-a^2 x^2}}{c^2 x \sqrt {c-a^2 c x^2}}+\frac {a \sqrt {1-a^2 x^2}}{8 c^2 (1-a x)^2 \sqrt {c-a^2 c x^2}}+\frac {3 a \sqrt {1-a^2 x^2}}{4 c^2 (1-a x) \sqrt {c-a^2 c x^2}}-\frac {a \sqrt {1-a^2 x^2}}{8 c^2 (1+a x) \sqrt {c-a^2 c x^2}}+\frac {a \sqrt {1-a^2 x^2} \log (x)}{c^2 \sqrt {c-a^2 c x^2}}-\frac {23 a \sqrt {1-a^2 x^2} \log (1-a x)}{16 c^2 \sqrt {c-a^2 c x^2}}+\frac {7 a \sqrt {1-a^2 x^2} \log (1+a x)}{16 c^2 \sqrt {c-a^2 c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 97, normalized size = 0.33 \[ \frac {\sqrt {1-a^2 x^2} \left (\frac {12 a}{1-a x}-\frac {2 a}{a x+1}+\frac {2 a}{(a x-1)^2}+16 a \log (x)-23 a \log (1-a x)+7 a \log (a x+1)-\frac {16}{x}\right )}{16 c^2 \sqrt {c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.76, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1}}{a^{7} c^{3} x^{9} - a^{6} c^{3} x^{8} - 3 \, a^{5} c^{3} x^{7} + 3 \, a^{4} c^{3} x^{6} + 3 \, a^{3} c^{3} x^{5} - 3 \, a^{2} c^{3} x^{4} - a c^{3} x^{3} + c^{3} x^{2}}, x\right ) \]
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 \[ \int \frac {a x + 1}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \sqrt {-a^{2} x^{2} + 1} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 222, normalized size = 0.75 \[ -\frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (16 a^{4} \ln \relax (x ) x^{4}-23 \ln \left (a x -1\right ) x^{4} a^{4}+7 \ln \left (a x +1\right ) x^{4} a^{4}-16 a^{3} \ln \relax (x ) x^{3}+23 \ln \left (a x -1\right ) x^{3} a^{3}-7 a^{3} x^{3} \ln \left (a x +1\right )-30 x^{3} a^{3}-16 a^{2} \ln \relax (x ) x^{2}+23 \ln \left (a x -1\right ) x^{2} a^{2}-7 \ln \left (a x +1\right ) x^{2} a^{2}+22 a^{2} x^{2}+16 a \ln \relax (x ) x -23 \ln \left (a x -1\right ) x a +7 a x \ln \left (a x +1\right )+28 a x -16\right )}{16 \left (a^{2} x^{2}-1\right ) c^{3} \left (a x -1\right )^{2} \left (a x +1\right ) x} \]
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 \[ \int \frac {a x + 1}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \sqrt {-a^{2} x^{2} + 1} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {a\,x+1}{x^2\,{\left (c-a^2\,c\,x^2\right )}^{5/2}\,\sqrt {1-a^2\,x^2}} \,d x \]
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 \[ \int \frac {a x + 1}{x^{2} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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