Optimal. Leaf size=39 \[ \frac {b \sqrt {x}}{c}+a x-\frac {b \tanh ^{-1}\left (c \sqrt {x}\right )}{c^2}+b x \tanh ^{-1}\left (c \sqrt {x}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6021, 52, 65,
212} \begin {gather*} a x-\frac {b \tanh ^{-1}\left (c \sqrt {x}\right )}{c^2}+\frac {b \sqrt {x}}{c}+b x \tanh ^{-1}\left (c \sqrt {x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 212
Rule 6021
Rubi steps
\begin {align*} \int \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right ) \, dx &=a x+b \int \tanh ^{-1}\left (c \sqrt {x}\right ) \, dx\\ &=a x+b x \tanh ^{-1}\left (c \sqrt {x}\right )-\frac {1}{2} (b c) \int \frac {\sqrt {x}}{1-c^2 x} \, dx\\ &=\frac {b \sqrt {x}}{c}+a x+b x \tanh ^{-1}\left (c \sqrt {x}\right )-\frac {b \int \frac {1}{\sqrt {x} \left (1-c^2 x\right )} \, dx}{2 c}\\ &=\frac {b \sqrt {x}}{c}+a x+b x \tanh ^{-1}\left (c \sqrt {x}\right )-\frac {b \text {Subst}\left (\int \frac {1}{1-c^2 x^2} \, dx,x,\sqrt {x}\right )}{c}\\ &=\frac {b \sqrt {x}}{c}+a x-\frac {b \tanh ^{-1}\left (c \sqrt {x}\right )}{c^2}+b x \tanh ^{-1}\left (c \sqrt {x}\right )\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 42, normalized size = 1.08 \begin {gather*} a x+b x \tanh ^{-1}\left (c \sqrt {x}\right )-b c \left (-\frac {\sqrt {x}}{c^2}+\frac {\tanh ^{-1}\left (c \sqrt {x}\right )}{c^3}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 50, normalized size = 1.28
method | result | size |
default | \(a x +b x \arctanh \left (c \sqrt {x}\right )+\frac {b \sqrt {x}}{c}+\frac {b \ln \left (c \sqrt {x}-1\right )}{2 c^{2}}-\frac {b \ln \left (1+c \sqrt {x}\right )}{2 c^{2}}\) | \(50\) |
derivativedivides | \(\frac {a \,c^{2} x +b \,c^{2} x \arctanh \left (c \sqrt {x}\right )+b c \sqrt {x}+\frac {b \ln \left (c \sqrt {x}-1\right )}{2}-\frac {b \ln \left (1+c \sqrt {x}\right )}{2}}{c^{2}}\) | \(56\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 53, normalized size = 1.36 \begin {gather*} \frac {1}{2} \, {\left (c {\left (\frac {2 \, \sqrt {x}}{c^{2}} - \frac {\log \left (c \sqrt {x} + 1\right )}{c^{3}} + \frac {\log \left (c \sqrt {x} - 1\right )}{c^{3}}\right )} + 2 \, x \operatorname {artanh}\left (c \sqrt {x}\right )\right )} b + a x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.47, size = 56, normalized size = 1.44 \begin {gather*} \frac {2 \, a c^{2} x + 2 \, b c \sqrt {x} + {\left (b c^{2} x - b\right )} \log \left (-\frac {c^{2} x + 2 \, c \sqrt {x} + 1}{c^{2} x - 1}\right )}{2 \, c^{2}} \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 \left (a + b \operatorname {atanh}{\left (c \sqrt {x} \right )}\right )\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 174 vs.
\(2 (33) = 66\).
time = 0.48, size = 174, normalized size = 4.46 \begin {gather*} 2 \, b c {\left (\frac {1}{c^{3} {\left (\frac {c \sqrt {x} + 1}{c \sqrt {x} - 1} - 1\right )}} + \frac {{\left (c \sqrt {x} + 1\right )} \log \left (-\frac {\frac {c {\left (\frac {c \sqrt {x} + 1}{c \sqrt {x} - 1} + 1\right )}}{\frac {{\left (c \sqrt {x} + 1\right )} c}{c \sqrt {x} - 1} - c} + 1}{\frac {c {\left (\frac {c \sqrt {x} + 1}{c \sqrt {x} - 1} + 1\right )}}{\frac {{\left (c \sqrt {x} + 1\right )} c}{c \sqrt {x} - 1} - c} - 1}\right )}{{\left (c \sqrt {x} - 1\right )} c^{3} {\left (\frac {c \sqrt {x} + 1}{c \sqrt {x} - 1} - 1\right )}^{2}}\right )} + a x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.90, size = 32, normalized size = 0.82 \begin {gather*} a\,x+b\,x\,\mathrm {atanh}\left (c\,\sqrt {x}\right )-\frac {b\,\left (\mathrm {atanh}\left (c\,\sqrt {x}\right )-c\,\sqrt {x}\right )}{c^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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