Optimal. Leaf size=146 \[ \frac {c \sqrt {c-\frac {c}{a^2 x^2}}}{2 a^3 \sqrt {1-\frac {1}{a^2 x^2}} x^2}+\frac {c \sqrt {c-\frac {c}{a^2 x^2}}}{a^2 \sqrt {1-\frac {1}{a^2 x^2}} x}+\frac {c \sqrt {c-\frac {c}{a^2 x^2}} x}{\sqrt {1-\frac {1}{a^2 x^2}}}+\frac {c \sqrt {c-\frac {c}{a^2 x^2}} \log (x)}{a \sqrt {1-\frac {1}{a^2 x^2}}} \]
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Rubi [A]
time = 0.08, antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6332, 6328, 76}
\begin {gather*} \frac {c x \sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {1-\frac {1}{a^2 x^2}}}+\frac {c \sqrt {c-\frac {c}{a^2 x^2}}}{a^2 x \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {c \log (x) \sqrt {c-\frac {c}{a^2 x^2}}}{a \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {c \sqrt {c-\frac {c}{a^2 x^2}}}{2 a^3 x^2 \sqrt {1-\frac {1}{a^2 x^2}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 76
Rule 6328
Rule 6332
Rubi steps
\begin {align*} \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{3/2} \, dx &=\frac {\left (c \sqrt {c-\frac {c}{a^2 x^2}}\right ) \int e^{\coth ^{-1}(a x)} \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \, dx}{\sqrt {1-\frac {1}{a^2 x^2}}}\\ &=\frac {\left (c \sqrt {c-\frac {c}{a^2 x^2}}\right ) \int \frac {(-1+a x) (1+a x)^2}{x^3} \, dx}{a^3 \sqrt {1-\frac {1}{a^2 x^2}}}\\ &=\frac {\left (c \sqrt {c-\frac {c}{a^2 x^2}}\right ) \int \left (a^3-\frac {1}{x^3}-\frac {a}{x^2}+\frac {a^2}{x}\right ) \, dx}{a^3 \sqrt {1-\frac {1}{a^2 x^2}}}\\ &=\frac {c \sqrt {c-\frac {c}{a^2 x^2}}}{2 a^3 \sqrt {1-\frac {1}{a^2 x^2}} x^2}+\frac {c \sqrt {c-\frac {c}{a^2 x^2}}}{a^2 \sqrt {1-\frac {1}{a^2 x^2}} x}+\frac {c \sqrt {c-\frac {c}{a^2 x^2}} x}{\sqrt {1-\frac {1}{a^2 x^2}}}+\frac {c \sqrt {c-\frac {c}{a^2 x^2}} \log (x)}{a \sqrt {1-\frac {1}{a^2 x^2}}}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 64, normalized size = 0.44 \begin {gather*} \frac {\left (c-\frac {c}{a^2 x^2}\right )^{3/2} \left (\frac {3 a^2}{2}+\frac {1}{2 x^2}+\frac {a}{x}+a^3 x+a^2 \log (x)\right )}{a^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.04, size = 80, normalized size = 0.55
method | result | size |
default | \(\frac {\left (2 a^{3} x^{3}+2 a^{2} \ln \left (x \right ) x^{2}+2 a x +1\right ) \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}\right )^{\frac {3}{2}} x}{2 \left (a x +1\right ) \left (a^{2} x^{2}-1\right ) \sqrt {\frac {a x -1}{a x +1}}}\) | \(80\) |
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 = 0.37, size = 42, normalized size = 0.29 \begin {gather*} \frac {{\left (2 \, a^{3} c x^{3} + 2 \, a^{2} c x^{2} \log \left (x\right ) + 2 \, a c x + c\right )} \sqrt {a^{2} c}}{2 \, a^{4} x^{2}} \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 {{\left (c-\frac {c}{a^2\,x^2}\right )}^{3/2}}{\sqrt {\frac {a\,x-1}{a\,x+1}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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