Optimal. Leaf size=33 \[ \frac {a \sqrt {c x-1} \sqrt {c x+1}}{x}+\frac {b \cosh ^{-1}(c x)}{c} \]
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Rubi [A] time = 0.06, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {454, 52} \[ \frac {a \sqrt {c x-1} \sqrt {c x+1}}{x}+\frac {b \cosh ^{-1}(c x)}{c} \]
Antiderivative was successfully verified.
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Rule 52
Rule 454
Rubi steps
\begin {align*} \int \frac {a+b x^2}{x^2 \sqrt {-1+c x} \sqrt {1+c x}} \, dx &=\frac {a \sqrt {-1+c x} \sqrt {1+c x}}{x}+b \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=\frac {a \sqrt {-1+c x} \sqrt {1+c x}}{x}+\frac {b \cosh ^{-1}(c x)}{c}\\ \end {align*}
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Mathematica [B] time = 0.03, size = 73, normalized size = 2.21 \[ \frac {\sqrt {c^2 x^2-1} \left (\frac {a \sqrt {c^2 x^2-1}}{x}+\frac {b \tanh ^{-1}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right )}{c}\right )}{\sqrt {c x-1} \sqrt {c x+1}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.18, size = 56, normalized size = 1.70 \[ \frac {a c^{2} x + \sqrt {c x + 1} \sqrt {c x - 1} a c - b x \log \left (-c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )}{c x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 58, normalized size = 1.76 \[ \frac {\frac {16 \, a c^{2}}{{\left (\sqrt {c x + 1} - \sqrt {c x - 1}\right )}^{4} + 4} - b \log \left ({\left (\sqrt {c x + 1} - \sqrt {c x - 1}\right )}^{4}\right )}{2 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.07, size = 77, normalized size = 2.33 \[ \frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (\sqrt {c^{2} x^{2}-1}\, a c \,\mathrm {csgn}\relax (c )+b x \ln \left (\left (c x +\sqrt {c^{2} x^{2}-1}\, \mathrm {csgn}\relax (c )\right ) \mathrm {csgn}\relax (c )\right )\right ) \mathrm {csgn}\relax (c )}{\sqrt {c^{2} x^{2}-1}\, c x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.18, size = 44, normalized size = 1.33 \[ \frac {b \log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c} + \frac {\sqrt {c^{2} x^{2} - 1} a}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.59, size = 61, normalized size = 1.85 \[ \frac {a\,\sqrt {c\,x-1}\,\sqrt {c\,x+1}}{x}-\frac {4\,b\,\mathrm {atan}\left (\frac {c\,\left (\sqrt {c\,x-1}-\mathrm {i}\right )}{\left (\sqrt {c\,x+1}-1\right )\,\sqrt {-c^2}}\right )}{\sqrt {-c^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 35.17, size = 148, normalized size = 4.48 \[ - \frac {a c {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {5}{4}, \frac {7}{4}, 1 & \frac {3}{2}, \frac {3}{2}, 2 \\1, \frac {5}{4}, \frac {3}{2}, \frac {7}{4}, 2 & 0 \end {matrix} \middle | {\frac {1}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}}} - \frac {i a c {G_{6, 6}^{2, 6}\left (\begin {matrix} \frac {1}{2}, \frac {3}{4}, 1, \frac {5}{4}, \frac {3}{2}, 1 & \\\frac {3}{4}, \frac {5}{4} & \frac {1}{2}, 1, 1, 0 \end {matrix} \middle | {\frac {e^{2 i \pi }}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}}} + \frac {b {G_{6, 6}^{6, 2}\left (\begin {matrix} \frac {1}{4}, \frac {3}{4} & \frac {1}{2}, \frac {1}{2}, 1, 1 \\0, \frac {1}{4}, \frac {1}{2}, \frac {3}{4}, 1, 0 & \end {matrix} \middle | {\frac {1}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} c} - \frac {i b {G_{6, 6}^{2, 6}\left (\begin {matrix} - \frac {1}{2}, - \frac {1}{4}, 0, \frac {1}{4}, \frac {1}{2}, 1 & \\- \frac {1}{4}, \frac {1}{4} & - \frac {1}{2}, 0, 0, 0 \end {matrix} \middle | {\frac {e^{2 i \pi }}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} c} \]
Verification of antiderivative is not currently implemented for this CAS.
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