Optimal. Leaf size=297 \[ -\frac {\sqrt {1-a^2 x^2}}{3 c x^3 \sqrt {c-a^2 c x^2}}-\frac {a \sqrt {1-a^2 x^2}}{2 c x^2 \sqrt {c-a^2 c x^2}}-\frac {2 a^2 \sqrt {1-a^2 x^2}}{c x \sqrt {c-a^2 c x^2}}+\frac {a^3 \sqrt {1-a^2 x^2}}{2 c (1-a x) \sqrt {c-a^2 c x^2}}+\frac {2 a^3 \sqrt {1-a^2 x^2} \log (x)}{c \sqrt {c-a^2 c x^2}}-\frac {9 a^3 \sqrt {1-a^2 x^2} \log (1-a x)}{4 c \sqrt {c-a^2 c x^2}}+\frac {a^3 \sqrt {1-a^2 x^2} \log (1+a x)}{4 c \sqrt {c-a^2 c x^2}} \]
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Rubi [A]
time = 0.18, antiderivative size = 297, 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 = {6288, 6285, 90}
\begin {gather*} -\frac {2 a^2 \sqrt {1-a^2 x^2}}{c x \sqrt {c-a^2 c x^2}}-\frac {a \sqrt {1-a^2 x^2}}{2 c x^2 \sqrt {c-a^2 c x^2}}-\frac {\sqrt {1-a^2 x^2}}{3 c x^3 \sqrt {c-a^2 c x^2}}+\frac {a^3 \sqrt {1-a^2 x^2}}{2 c (1-a x) \sqrt {c-a^2 c x^2}}+\frac {2 a^3 \sqrt {1-a^2 x^2} \log (x)}{c \sqrt {c-a^2 c x^2}}-\frac {9 a^3 \sqrt {1-a^2 x^2} \log (1-a x)}{4 c \sqrt {c-a^2 c x^2}}+\frac {a^3 \sqrt {1-a^2 x^2} \log (a x+1)}{4 c \sqrt {c-a^2 c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 90
Rule 6285
Rule 6288
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)}}{x^4 \left (c-a^2 c x^2\right )^{3/2}} \, dx &=\frac {\sqrt {1-a^2 x^2} \int \frac {e^{\tanh ^{-1}(a x)}}{x^4 \left (1-a^2 x^2\right )^{3/2}} \, dx}{c \sqrt {c-a^2 c x^2}}\\ &=\frac {\sqrt {1-a^2 x^2} \int \frac {1}{x^4 (1-a x)^2 (1+a x)} \, dx}{c \sqrt {c-a^2 c x^2}}\\ &=\frac {\sqrt {1-a^2 x^2} \int \left (\frac {1}{x^4}+\frac {a}{x^3}+\frac {2 a^2}{x^2}+\frac {2 a^3}{x}+\frac {a^4}{2 (-1+a x)^2}-\frac {9 a^4}{4 (-1+a x)}+\frac {a^4}{4 (1+a x)}\right ) \, dx}{c \sqrt {c-a^2 c x^2}}\\ &=-\frac {\sqrt {1-a^2 x^2}}{3 c x^3 \sqrt {c-a^2 c x^2}}-\frac {a \sqrt {1-a^2 x^2}}{2 c x^2 \sqrt {c-a^2 c x^2}}-\frac {2 a^2 \sqrt {1-a^2 x^2}}{c x \sqrt {c-a^2 c x^2}}+\frac {a^3 \sqrt {1-a^2 x^2}}{2 c (1-a x) \sqrt {c-a^2 c x^2}}+\frac {2 a^3 \sqrt {1-a^2 x^2} \log (x)}{c \sqrt {c-a^2 c x^2}}-\frac {9 a^3 \sqrt {1-a^2 x^2} \log (1-a x)}{4 c \sqrt {c-a^2 c x^2}}+\frac {a^3 \sqrt {1-a^2 x^2} \log (1+a x)}{4 c \sqrt {c-a^2 c x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 99, normalized size = 0.33 \begin {gather*} \frac {\sqrt {1-a^2 x^2} \left (-\frac {4}{x^3}-\frac {6 a}{x^2}-\frac {24 a^2}{x}+\frac {6 a^3}{1-a x}+24 a^3 \log (x)-27 a^3 \log (1-a x)+3 a^3 \log (1+a x)\right )}{12 c \sqrt {c-a^2 c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 151, normalized size = 0.51
method | result | size |
default | \(-\frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (3 \ln \left (a x +1\right ) a^{4} x^{4}+24 a^{4} \ln \left (x \right ) x^{4}-27 \ln \left (a x -1\right ) a^{4} x^{4}-3 \ln \left (a x +1\right ) a^{3} x^{3}-24 a^{3} \ln \left (x \right ) x^{3}+27 \ln \left (a x -1\right ) a^{3} x^{3}-30 a^{3} x^{3}+18 a^{2} x^{2}+2 a x +4\right )}{12 \left (a^{2} x^{2}-1\right ) c^{2} x^{3} \left (a x -1\right )}\) | \(151\) |
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 [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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a x + 1}{x^{4} \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 {3}{2}}}\, dx \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 = -0.00 \begin {gather*} \int \frac {a\,x+1}{x^4\,{\left (c-a^2\,c\,x^2\right )}^{3/2}\,\sqrt {1-a^2\,x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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