Optimal. Leaf size=193 \[ -\frac {\sqrt {c-a^2 c x^2}}{3 a \sqrt {1-\frac {1}{a^2 x^2}} x^4}+\frac {3 \sqrt {c-a^2 c x^2}}{2 \sqrt {1-\frac {1}{a^2 x^2}} x^3}-\frac {4 a \sqrt {c-a^2 c x^2}}{\sqrt {1-\frac {1}{a^2 x^2}} x^2}-\frac {4 a^2 \sqrt {c-a^2 c x^2} \log (x)}{\sqrt {1-\frac {1}{a^2 x^2}} x}+\frac {4 a^2 \sqrt {c-a^2 c x^2} \log (1+a x)}{\sqrt {1-\frac {1}{a^2 x^2}} x} \]
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Rubi [A]
time = 0.17, antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6327, 6328, 90}
\begin {gather*} -\frac {4 a \sqrt {c-a^2 c x^2}}{x^2 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {4 a^2 \log (x) \sqrt {c-a^2 c x^2}}{x \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {4 a^2 \sqrt {c-a^2 c x^2} \log (a x+1)}{x \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {\sqrt {c-a^2 c x^2}}{3 a x^4 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {3 \sqrt {c-a^2 c x^2}}{2 x^3 \sqrt {1-\frac {1}{a^2 x^2}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 90
Rule 6327
Rule 6328
Rubi steps
\begin {align*} \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {c-a^2 c x^2}}{x^4} \, dx &=\frac {\sqrt {c-a^2 c x^2} \int \frac {e^{-3 \coth ^{-1}(a x)} \sqrt {1-\frac {1}{a^2 x^2}}}{x^3} \, dx}{\sqrt {1-\frac {1}{a^2 x^2}} x}\\ &=\frac {\sqrt {c-a^2 c x^2} \int \frac {(-1+a x)^2}{x^4 (1+a x)} \, dx}{a \sqrt {1-\frac {1}{a^2 x^2}} x}\\ &=\frac {\sqrt {c-a^2 c x^2} \int \left (\frac {1}{x^4}-\frac {3 a}{x^3}+\frac {4 a^2}{x^2}-\frac {4 a^3}{x}+\frac {4 a^4}{1+a x}\right ) \, dx}{a \sqrt {1-\frac {1}{a^2 x^2}} x}\\ &=-\frac {\sqrt {c-a^2 c x^2}}{3 a \sqrt {1-\frac {1}{a^2 x^2}} x^4}+\frac {3 \sqrt {c-a^2 c x^2}}{2 \sqrt {1-\frac {1}{a^2 x^2}} x^3}-\frac {4 a \sqrt {c-a^2 c x^2}}{\sqrt {1-\frac {1}{a^2 x^2}} x^2}-\frac {4 a^2 \sqrt {c-a^2 c x^2} \log (x)}{\sqrt {1-\frac {1}{a^2 x^2}} x}+\frac {4 a^2 \sqrt {c-a^2 c x^2} \log (1+a x)}{\sqrt {1-\frac {1}{a^2 x^2}} x}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 78, normalized size = 0.40 \begin {gather*} \frac {\sqrt {c-a^2 c x^2} \left (-\frac {1}{3 x^3}+\frac {3 a}{2 x^2}-\frac {4 a^2}{x}-4 a^3 \log (x)+4 a^3 \log (1+a x)\right )}{a \sqrt {1-\frac {1}{a^2 x^2}} x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 85, normalized size = 0.44
method | result | size |
default | \(\frac {\left (24 \ln \left (a x +1\right ) a^{3} x^{3}-24 \ln \left (x \right ) a^{3} x^{3}-24 a^{2} x^{2}+9 a x -2\right ) \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (a x +1\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{6 \left (a x -1\right )^{2} x^{3}}\) | \(85\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 98, normalized size = 0.51 \begin {gather*} \frac {24 \, a^{4} \sqrt {-c} x^{3} \log \left (\frac {2 \, a^{3} c x^{2} + 2 \, a^{2} c x - \sqrt {-a^{2} c} {\left (2 \, a x + 1\right )} \sqrt {-c} + a c}{a x^{2} + x}\right ) - {\left (24 \, a^{2} x^{2} - 9 \, a x + 2\right )} \sqrt {-a^{2} c}}{6 \, a x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\sqrt {c-a^2\,c\,x^2}\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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