Optimal. Leaf size=142 \[ \frac {3 b \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2}{c^2}+\frac {3 b \sqrt {x} \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2}{c}-\frac {\left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^3}{c^2}+x \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^3-\frac {6 b^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2}{1-c \sqrt {x}}\right )}{c^2}-\frac {3 b^3 \text {PolyLog}\left (2,1-\frac {2}{1-c \sqrt {x}}\right )}{c^2} \]
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Rubi [A]
time = 0.20, antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 9, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {6027, 6037,
6127, 6021, 6131, 6055, 2449, 2352, 6095} \begin {gather*} -\frac {6 b^2 \log \left (\frac {2}{1-c \sqrt {x}}\right ) \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{c^2}+\frac {3 b \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2}{c^2}-\frac {\left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^3}{c^2}+\frac {3 b \sqrt {x} \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2}{c}+x \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^3-\frac {3 b^3 \text {Li}_2\left (1-\frac {2}{1-c \sqrt {x}}\right )}{c^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 2352
Rule 2449
Rule 6021
Rule 6027
Rule 6037
Rule 6055
Rule 6095
Rule 6127
Rule 6131
Rubi steps
\begin {align*} \int \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^3 \, dx &=\int \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^3 \, dx\\ \end {align*}
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Mathematica [A]
time = 0.18, size = 201, normalized size = 1.42 \begin {gather*} \frac {6 b^2 \left (-1+c \sqrt {x}\right ) \left (a+b+a c \sqrt {x}\right ) \tanh ^{-1}\left (c \sqrt {x}\right )^2+2 b^3 \left (-1+c^2 x\right ) \tanh ^{-1}\left (c \sqrt {x}\right )^3+6 b \tanh ^{-1}\left (c \sqrt {x}\right ) \left (2 a b c \sqrt {x}+a^2 c^2 x-2 b^2 \log \left (1+e^{-2 \tanh ^{-1}\left (c \sqrt {x}\right )}\right )\right )+a \left (6 a b c \sqrt {x}+2 a^2 c^2 x+3 a b \log \left (1-c \sqrt {x}\right )-3 a b \log \left (1+c \sqrt {x}\right )+6 b^2 \log \left (1-c^2 x\right )\right )+6 b^3 \text {PolyLog}\left (2,-e^{-2 \tanh ^{-1}\left (c \sqrt {x}\right )}\right )}{2 c^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 6.80, size = 5972, normalized size = 42.06
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(5972\) |
default | \(\text {Expression too large to display}\) | \(5972\) |
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 [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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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 )^{3}\, dx \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 = -0.01 \begin {gather*} \int {\left (a+b\,\mathrm {atanh}\left (c\,\sqrt {x}\right )\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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