Optimal. Leaf size=234 \[ \frac {c^2 \sqrt {c-\frac {c}{a^2 x^2}}}{4 a^5 \sqrt {1-\frac {1}{a^2 x^2}} x^4}+\frac {c^2 \sqrt {c-\frac {c}{a^2 x^2}}}{a^4 \sqrt {1-\frac {1}{a^2 x^2}} x^3}+\frac {c^2 \sqrt {c-\frac {c}{a^2 x^2}}}{a^3 \sqrt {1-\frac {1}{a^2 x^2}} x^2}-\frac {2 c^2 \sqrt {c-\frac {c}{a^2 x^2}}}{a^2 \sqrt {1-\frac {1}{a^2 x^2}} x}+\frac {c^2 \sqrt {c-\frac {c}{a^2 x^2}} x}{\sqrt {1-\frac {1}{a^2 x^2}}}+\frac {3 c^2 \sqrt {c-\frac {c}{a^2 x^2}} \log (x)}{a \sqrt {1-\frac {1}{a^2 x^2}}} \]
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Rubi [A]
time = 0.10, antiderivative size = 234, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6332, 6328, 76}
\begin {gather*} \frac {c^2 x \sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {1-\frac {1}{a^2 x^2}}}-\frac {2 c^2 \sqrt {c-\frac {c}{a^2 x^2}}}{a^2 x \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {3 c^2 \log (x) \sqrt {c-\frac {c}{a^2 x^2}}}{a \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {c^2 \sqrt {c-\frac {c}{a^2 x^2}}}{4 a^5 x^4 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {c^2 \sqrt {c-\frac {c}{a^2 x^2}}}{a^4 x^3 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {c^2 \sqrt {c-\frac {c}{a^2 x^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^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^{5/2} \, dx &=\frac {\left (c^2 \sqrt {c-\frac {c}{a^2 x^2}}\right ) \int e^{3 \coth ^{-1}(a x)} \left (1-\frac {1}{a^2 x^2}\right )^{5/2} \, dx}{\sqrt {1-\frac {1}{a^2 x^2}}}\\ &=\frac {\left (c^2 \sqrt {c-\frac {c}{a^2 x^2}}\right ) \int \frac {(-1+a x) (1+a x)^4}{x^5} \, dx}{a^5 \sqrt {1-\frac {1}{a^2 x^2}}}\\ &=\frac {\left (c^2 \sqrt {c-\frac {c}{a^2 x^2}}\right ) \int \left (a^5-\frac {1}{x^5}-\frac {3 a}{x^4}-\frac {2 a^2}{x^3}+\frac {2 a^3}{x^2}+\frac {3 a^4}{x}\right ) \, dx}{a^5 \sqrt {1-\frac {1}{a^2 x^2}}}\\ &=\frac {c^2 \sqrt {c-\frac {c}{a^2 x^2}}}{4 a^5 \sqrt {1-\frac {1}{a^2 x^2}} x^4}+\frac {c^2 \sqrt {c-\frac {c}{a^2 x^2}}}{a^4 \sqrt {1-\frac {1}{a^2 x^2}} x^3}+\frac {c^2 \sqrt {c-\frac {c}{a^2 x^2}}}{a^3 \sqrt {1-\frac {1}{a^2 x^2}} x^2}-\frac {2 c^2 \sqrt {c-\frac {c}{a^2 x^2}}}{a^2 \sqrt {1-\frac {1}{a^2 x^2}} x}+\frac {c^2 \sqrt {c-\frac {c}{a^2 x^2}} x}{\sqrt {1-\frac {1}{a^2 x^2}}}+\frac {3 c^2 \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 = 0.04, size = 87, normalized size = 0.37 \begin {gather*} \frac {\left (c-\frac {c}{a^2 x^2}\right )^{5/2} \left (\frac {(1+a x)^5}{4 x^4}+\frac {3}{4} a \left (-\frac {1}{3 x^3}-\frac {2 a}{x^2}-\frac {6 a^2}{x}+a^4 x+4 a^3 \log (x)\right )\right )}{a^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.04, size = 96, normalized size = 0.41
method | result | size |
default | \(\frac {\left (4 a^{5} x^{5}+12 \ln \left (x \right ) a^{4} x^{4}-8 a^{3} x^{3}+4 a^{2} x^{2}+4 a x +1\right ) \left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}\right )^{\frac {5}{2}} x}{4 \left (a x +1\right )^{3} \left (a^{2} x^{2}-1\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}\) | \(96\) |
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 = 0.38, size = 72, normalized size = 0.31 \begin {gather*} \frac {{\left (4 \, a^{5} c^{2} x^{5} + 12 \, a^{4} c^{2} x^{4} \log \left (x\right ) - 8 \, a^{3} c^{2} x^{3} + 4 \, a^{2} c^{2} x^{2} + 4 \, a c^{2} x + c^{2}\right )} \sqrt {a^{2} c}}{4 \, a^{6} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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 {{\left (c-\frac {c}{a^2\,x^2}\right )}^{5/2}}{{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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