Optimal. Leaf size=47 \[ \frac {b x \sqrt {-1+c x} \sqrt {1+c x}}{2 c^2}+\frac {\left (b+2 a c^2\right ) \cosh ^{-1}(c x)}{2 c^3} \]
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Rubi [A]
time = 0.03, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {397, 54}
\begin {gather*} \frac {\left (2 a c^2+b\right ) \cosh ^{-1}(c x)}{2 c^3}+\frac {b x \sqrt {c x-1} \sqrt {c x+1}}{2 c^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 54
Rule 397
Rubi steps
\begin {align*} \int \frac {a+b x^2}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx &=\frac {b x \sqrt {-1+c x} \sqrt {1+c x}}{2 c^2}-\frac {\left (-b-2 a c^2\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{2 c^2}\\ &=\frac {b x \sqrt {-1+c x} \sqrt {1+c x}}{2 c^2}+\frac {\left (b+2 a c^2\right ) \cosh ^{-1}(c x)}{2 c^3}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 58, normalized size = 1.23 \begin {gather*} \frac {b c x \sqrt {-1+c x} \sqrt {1+c x}+2 \left (b+2 a c^2\right ) \tanh ^{-1}\left (\sqrt {\frac {-1+c x}{1+c x}}\right )}{2 c^3} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.30, size = 103, normalized size = 2.19
method | result | size |
default | \(\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (\sqrt {c^{2} x^{2}-1}\, \mathrm {csgn}\left (c \right ) c b x +2 \ln \left (\left (\sqrt {c^{2} x^{2}-1}\, \mathrm {csgn}\left (c \right )+c x \right ) \mathrm {csgn}\left (c \right )\right ) a \,c^{2}+\ln \left (\left (\sqrt {c^{2} x^{2}-1}\, \mathrm {csgn}\left (c \right )+c x \right ) \mathrm {csgn}\left (c \right )\right ) b \right ) \mathrm {csgn}\left (c \right )}{2 c^{3} \sqrt {c^{2} x^{2}-1}}\) | \(103\) |
risch | \(\frac {b x \sqrt {c x -1}\, \sqrt {c x +1}}{2 c^{2}}+\frac {\left (\frac {\ln \left (\frac {c^{2} x}{\sqrt {c^{2}}}+\sqrt {c^{2} x^{2}-1}\right ) a}{\sqrt {c^{2}}}+\frac {\ln \left (\frac {c^{2} x}{\sqrt {c^{2}}}+\sqrt {c^{2} x^{2}-1}\right ) b}{2 c^{2} \sqrt {c^{2}}}\right ) \sqrt {\left (c x +1\right ) \left (c x -1\right )}}{\sqrt {c x -1}\, \sqrt {c x +1}}\) | \(116\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 74, normalized size = 1.57 \begin {gather*} \frac {a \log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c} + \frac {\sqrt {c^{2} x^{2} - 1} b x}{2 \, c^{2}} + \frac {b \log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{2 \, c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.97, size = 55, normalized size = 1.17 \begin {gather*} \frac {\sqrt {c x + 1} \sqrt {c x - 1} b c x - {\left (2 \, a c^{2} + b\right )} \log \left (-c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )}{2 \, c^{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 [A]
time = 0.69, size = 69, normalized size = 1.47 \begin {gather*} \frac {\sqrt {c x + 1} \sqrt {c x - 1} {\left (\frac {{\left (c x + 1\right )} b}{c^{2}} - \frac {b}{c^{2}}\right )} - \frac {2 \, {\left (2 \, a c^{2} + b\right )} \log \left (\sqrt {c x + 1} - \sqrt {c x - 1}\right )}{c^{2}}}{2 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 12.69, size = 293, normalized size = 6.23 \begin {gather*} -\frac {\frac {14\,b\,{\left (\sqrt {c\,x-1}-\mathrm {i}\right )}^3}{{\left (\sqrt {c\,x+1}-1\right )}^3}+\frac {14\,b\,{\left (\sqrt {c\,x-1}-\mathrm {i}\right )}^5}{{\left (\sqrt {c\,x+1}-1\right )}^5}+\frac {2\,b\,{\left (\sqrt {c\,x-1}-\mathrm {i}\right )}^7}{{\left (\sqrt {c\,x+1}-1\right )}^7}+\frac {2\,b\,\left (\sqrt {c\,x-1}-\mathrm {i}\right )}{\sqrt {c\,x+1}-1}}{c^3-\frac {4\,c^3\,{\left (\sqrt {c\,x-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {c\,x+1}-1\right )}^2}+\frac {6\,c^3\,{\left (\sqrt {c\,x-1}-\mathrm {i}\right )}^4}{{\left (\sqrt {c\,x+1}-1\right )}^4}-\frac {4\,c^3\,{\left (\sqrt {c\,x-1}-\mathrm {i}\right )}^6}{{\left (\sqrt {c\,x+1}-1\right )}^6}+\frac {c^3\,{\left (\sqrt {c\,x-1}-\mathrm {i}\right )}^8}{{\left (\sqrt {c\,x+1}-1\right )}^8}}+\frac {2\,b\,\mathrm {atanh}\left (\frac {\sqrt {c\,x-1}-\mathrm {i}}{\sqrt {c\,x+1}-1}\right )}{c^3}-\frac {4\,a\,\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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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