Optimal. Leaf size=85 \[ \frac {2 b \text {ArcTan}\left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right )}{\sqrt {c} \sqrt {d}}+\frac {2 \sqrt {d x} \left (a+b \tanh ^{-1}(c x)\right )}{d}-\frac {2 b \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right )}{\sqrt {c} \sqrt {d}} \]
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Rubi [A]
time = 0.04, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {6049, 335, 304,
211, 214} \begin {gather*} \frac {2 \sqrt {d x} \left (a+b \tanh ^{-1}(c x)\right )}{d}+\frac {2 b \text {ArcTan}\left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right )}{\sqrt {c} \sqrt {d}}-\frac {2 b \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right )}{\sqrt {c} \sqrt {d}} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 214
Rule 304
Rule 335
Rule 6049
Rubi steps
\begin {align*} \int \frac {a+b \tanh ^{-1}(c x)}{\sqrt {d x}} \, dx &=\frac {2 \sqrt {d x} \left (a+b \tanh ^{-1}(c x)\right )}{d}-\frac {(2 b c) \int \frac {\sqrt {d x}}{1-c^2 x^2} \, dx}{d}\\ &=\frac {2 \sqrt {d x} \left (a+b \tanh ^{-1}(c x)\right )}{d}-\frac {(4 b c) \text {Subst}\left (\int \frac {x^2}{1-\frac {c^2 x^4}{d^2}} \, dx,x,\sqrt {d x}\right )}{d^2}\\ &=\frac {2 \sqrt {d x} \left (a+b \tanh ^{-1}(c x)\right )}{d}-(2 b) \text {Subst}\left (\int \frac {1}{d-c x^2} \, dx,x,\sqrt {d x}\right )+(2 b) \text {Subst}\left (\int \frac {1}{d+c x^2} \, dx,x,\sqrt {d x}\right )\\ &=\frac {2 b \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right )}{\sqrt {c} \sqrt {d}}+\frac {2 \sqrt {d x} \left (a+b \tanh ^{-1}(c x)\right )}{d}-\frac {2 b \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right )}{\sqrt {c} \sqrt {d}}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 98, normalized size = 1.15 \begin {gather*} \frac {\sqrt {x} \left (2 a \sqrt {c} \sqrt {x}+2 b \text {ArcTan}\left (\sqrt {c} \sqrt {x}\right )+2 b \sqrt {c} \sqrt {x} \tanh ^{-1}(c x)+b \log \left (1-\sqrt {c} \sqrt {x}\right )-b \log \left (1+\sqrt {c} \sqrt {x}\right )\right )}{\sqrt {c} \sqrt {d x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 68, normalized size = 0.80
method | result | size |
derivativedivides | \(\frac {2 \sqrt {d x}\, a +2 b \sqrt {d x}\, \arctanh \left (c x \right )+\frac {2 b d \arctan \left (\frac {c \sqrt {d x}}{\sqrt {d c}}\right )}{\sqrt {d c}}-\frac {2 b d \arctanh \left (\frac {c \sqrt {d x}}{\sqrt {d c}}\right )}{\sqrt {d c}}}{d}\) | \(68\) |
default | \(\frac {2 \sqrt {d x}\, a +2 b \sqrt {d x}\, \arctanh \left (c x \right )+\frac {2 b d \arctan \left (\frac {c \sqrt {d x}}{\sqrt {d c}}\right )}{\sqrt {d c}}-\frac {2 b d \arctanh \left (\frac {c \sqrt {d x}}{\sqrt {d c}}\right )}{\sqrt {d c}}}{d}\) | \(68\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 103, normalized size = 1.21 \begin {gather*} \frac {{\left (2 \, \sqrt {d x} \operatorname {artanh}\left (c x\right ) + \frac {{\left (\frac {2 \, d^{2} \arctan \left (\frac {\sqrt {d x} c}{\sqrt {c d}}\right )}{\sqrt {c d} c} + \frac {d^{2} \log \left (\frac {\sqrt {d x} c - \sqrt {c d}}{\sqrt {d x} c + \sqrt {c d}}\right )}{\sqrt {c d} c}\right )} c}{d}\right )} b + 2 \, \sqrt {d x} a}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 211, normalized size = 2.48 \begin {gather*} \left [-\frac {2 \, \sqrt {c d} b \arctan \left (\frac {\sqrt {c d} \sqrt {d x}}{c d x}\right ) - \sqrt {c d} b \log \left (\frac {c d x - 2 \, \sqrt {c d} \sqrt {d x} + d}{c x - 1}\right ) - {\left (b c \log \left (-\frac {c x + 1}{c x - 1}\right ) + 2 \, a c\right )} \sqrt {d x}}{c d}, \frac {2 \, \sqrt {-c d} b \arctan \left (\frac {\sqrt {-c d} \sqrt {d x}}{c d x}\right ) - \sqrt {-c d} b \log \left (\frac {c d x - 2 \, \sqrt {-c d} \sqrt {d x} - d}{c x + 1}\right ) + {\left (b c \log \left (-\frac {c x + 1}{c x - 1}\right ) + 2 \, a c\right )} \sqrt {d x}}{c d}\right ] \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 \frac {a + b \operatorname {atanh}{\left (c x \right )}}{\sqrt {d x}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.46, size = 88, normalized size = 1.04 \begin {gather*} \frac {{\left (2 \, c d {\left (\frac {\arctan \left (\frac {\sqrt {d x} c}{\sqrt {c d}}\right )}{\sqrt {c d} c} + \frac {\arctan \left (\frac {\sqrt {d x} c}{\sqrt {-c d}}\right )}{\sqrt {-c d} c}\right )} + \sqrt {d x} \log \left (-\frac {c x + 1}{c x - 1}\right )\right )} b + 2 \, \sqrt {d x} a}{d} \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 {a+b\,\mathrm {atanh}\left (c\,x\right )}{\sqrt {d\,x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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