Optimal. Leaf size=234 \[ \frac {b^3 \sqrt {x}}{2 c^3}-\frac {b^3 \tanh ^{-1}\left (c \sqrt {x}\right )}{2 c^4}+\frac {b^2 x \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{2 c^2}+\frac {2 b \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2}{c^4}+\frac {3 b \sqrt {x} \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2}{2 c^3}+\frac {b x^{3/2} \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2}{2 c}-\frac {\left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^3}{2 c^4}+\frac {1}{2} x^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^3-\frac {4 b^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right ) \log \left (\frac {2}{1-c \sqrt {x}}\right )}{c^4}-\frac {2 b^3 \text {PolyLog}\left (2,1-\frac {2}{1-c \sqrt {x}}\right )}{c^4} \]
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Rubi [A]
time = 0.42, antiderivative size = 234, normalized size of antiderivative = 1.00, number of steps
used = 19, number of rules used = 11, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.688, Rules used = {6039, 6037,
6127, 327, 212, 6131, 6055, 2449, 2352, 6021, 6095} \begin {gather*} -\frac {4 b^2 \log \left (\frac {2}{1-c \sqrt {x}}\right ) \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{c^4}+\frac {b^2 x \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )}{2 c^2}+\frac {2 b \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2}{c^4}-\frac {\left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^3}{2 c^4}+\frac {3 b \sqrt {x} \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2}{2 c^3}+\frac {b x^{3/2} \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^2}{2 c}+\frac {1}{2} x^2 \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^3-\frac {2 b^3 \text {Li}_2\left (1-\frac {2}{1-c \sqrt {x}}\right )}{c^4}-\frac {b^3 \tanh ^{-1}\left (c \sqrt {x}\right )}{2 c^4}+\frac {b^3 \sqrt {x}}{2 c^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 327
Rule 2352
Rule 2449
Rule 6021
Rule 6037
Rule 6039
Rule 6055
Rule 6095
Rule 6127
Rule 6131
Rubi steps
\begin {align*} \int x \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^3 \, dx &=\int x \left (a+b \tanh ^{-1}\left (c \sqrt {x}\right )\right )^3 \, dx\\ \end {align*}
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Mathematica [A]
time = 0.38, size = 285, normalized size = 1.22 \begin {gather*} \frac {-2 a b^2+6 a^2 b c \sqrt {x}+2 b^3 c \sqrt {x}+2 a b^2 c^2 x+2 a^2 b c^3 x^{3/2}+2 a^3 c^4 x^2+2 b^2 \left (b \left (-4+3 c \sqrt {x}+c^3 x^{3/2}\right )+3 a \left (-1+c^4 x^2\right )\right ) \tanh ^{-1}\left (c \sqrt {x}\right )^2+2 b^3 \left (-1+c^4 x^2\right ) \tanh ^{-1}\left (c \sqrt {x}\right )^3+2 b \tanh ^{-1}\left (c \sqrt {x}\right ) \left (3 a^2 c^4 x^2+b^2 \left (-1+c^2 x\right )+2 a b c \sqrt {x} \left (3+c^2 x\right )-8 b^2 \log \left (1+e^{-2 \tanh ^{-1}\left (c \sqrt {x}\right )}\right )\right )+3 a^2 b \log \left (1-c \sqrt {x}\right )-3 a^2 b \log \left (1+c \sqrt {x}\right )+8 a b^2 \log \left (1-c^2 x\right )+8 b^3 \text {PolyLog}\left (2,-e^{-2 \tanh ^{-1}\left (c \sqrt {x}\right )}\right )}{4 c^4} \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 = 7.33, size = 1250, normalized size = 5.34
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(1250\) |
default | \(\text {Expression too large to display}\) | \(1250\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1184 vs.
\(2 (191) = 382\).
time = 0.69, size = 1184, normalized size = 5.06 \begin {gather*} \frac {1}{2} \, a^{3} x^{2} - \frac {1}{32} \, a b^{2} c {\left (\frac {3 \, c^{3} x^{2} + 10 \, c x - 2 \, {\left (3 \, c^{3} x^{2} + 4 \, c^{2} x^{\frac {3}{2}} + 6 \, c x + 12 \, \sqrt {x}\right )} \log \left (c \sqrt {x} + 1\right )}{c^{4}} - \frac {14 \, \log \left (c \sqrt {x} + 1\right )}{c^{5}} - \frac {14 \, \log \left (c \sqrt {x} - 1\right )}{c^{5}}\right )} - \frac {1}{16} \, {\left (12 \, x^{2} \log \left (c \sqrt {x} + 1\right ) - c {\left (\frac {3 \, c^{3} x^{2} - 4 \, c^{2} x^{\frac {3}{2}} + 6 \, c x - 12 \, \sqrt {x}}{c^{4}} + \frac {12 \, \log \left (c \sqrt {x} + 1\right )}{c^{5}}\right )}\right )} a b^{2} \log \left (-c \sqrt {x} + 1\right ) + \frac {1}{16} \, {\left (12 \, x^{2} \log \left (c \sqrt {x} + 1\right ) - c {\left (\frac {3 \, c^{3} x^{2} - 4 \, c^{2} x^{\frac {3}{2}} + 6 \, c x - 12 \, \sqrt {x}}{c^{4}} + \frac {12 \, \log \left (c \sqrt {x} + 1\right )}{c^{5}}\right )}\right )} a^{2} b - \frac {1}{16} \, {\left (12 \, x^{2} \log \left (-c \sqrt {x} + 1\right ) - c {\left (\frac {3 \, c^{3} x^{2} + 4 \, c^{2} x^{\frac {3}{2}} + 6 \, c x + 12 \, \sqrt {x}}{c^{4}} + \frac {12 \, \log \left (c \sqrt {x} - 1\right )}{c^{5}}\right )}\right )} a^{2} b + \frac {{\left (9 \, {\left (8 \, \log \left (-c \sqrt {x} + 1\right )^{2} - 4 \, \log \left (-c \sqrt {x} + 1\right ) + 1\right )} {\left (c \sqrt {x} - 1\right )}^{4} + 32 \, {\left (9 \, \log \left (-c \sqrt {x} + 1\right )^{2} - 6 \, \log \left (-c \sqrt {x} + 1\right ) + 2\right )} {\left (c \sqrt {x} - 1\right )}^{3} + 216 \, {\left (2 \, \log \left (-c \sqrt {x} + 1\right )^{2} - 2 \, \log \left (-c \sqrt {x} + 1\right ) + 1\right )} {\left (c \sqrt {x} - 1\right )}^{2} + 288 \, {\left (\log \left (-c \sqrt {x} + 1\right )^{2} - 2 \, \log \left (-c \sqrt {x} + 1\right ) + 2\right )} {\left (c \sqrt {x} - 1\right )}\right )} a b^{2}}{192 \, c^{4}} - \frac {{\left (9 \, {\left (32 \, \log \left (-c \sqrt {x} + 1\right )^{3} - 24 \, \log \left (-c \sqrt {x} + 1\right )^{2} + 12 \, \log \left (-c \sqrt {x} + 1\right ) - 3\right )} {\left (c \sqrt {x} - 1\right )}^{4} + 128 \, {\left (9 \, \log \left (-c \sqrt {x} + 1\right )^{3} - 9 \, \log \left (-c \sqrt {x} + 1\right )^{2} + 6 \, \log \left (-c \sqrt {x} + 1\right ) - 2\right )} {\left (c \sqrt {x} - 1\right )}^{3} + 432 \, {\left (4 \, \log \left (-c \sqrt {x} + 1\right )^{3} - 6 \, \log \left (-c \sqrt {x} + 1\right )^{2} + 6 \, \log \left (-c \sqrt {x} + 1\right ) - 3\right )} {\left (c \sqrt {x} - 1\right )}^{2} + 1152 \, {\left (\log \left (-c \sqrt {x} + 1\right )^{3} - 3 \, \log \left (-c \sqrt {x} + 1\right )^{2} + 6 \, \log \left (-c \sqrt {x} + 1\right ) - 6\right )} {\left (c \sqrt {x} - 1\right )}\right )} b^{3}}{4608 \, c^{4}} + \frac {2 \, {\left (\log \left (c \sqrt {x} + 1\right ) \log \left (-\frac {1}{2} \, c \sqrt {x} + \frac {1}{2}\right ) + {\rm Li}_2\left (\frac {1}{2} \, c \sqrt {x} + \frac {1}{2}\right )\right )} b^{3}}{c^{4}} - \frac {319 \, b^{3} \log \left (c \sqrt {x} - 1\right )}{384 \, c^{4}} + \frac {{\left (25 \, a b^{2} - 4 \, b^{3}\right )} \log \left (c \sqrt {x} + 1\right )}{16 \, c^{4}} + \frac {288 \, {\left (b^{3} c^{4} x^{2} - b^{3}\right )} \log \left (c \sqrt {x} + 1\right )^{3} + 27 \, {\left (8 \, a b^{2} c^{4} - b^{3} c^{4}\right )} x^{2} + 576 \, {\left (3 \, a b^{2} c^{4} x^{2} + b^{3} c^{3} x^{\frac {3}{2}} + 3 \, b^{3} c \sqrt {x} - 3 \, a b^{2} + 4 \, b^{3}\right )} \log \left (c \sqrt {x} + 1\right )^{2} - 72 \, {\left (3 \, b^{3} c^{4} x^{2} - 4 \, b^{3} c^{3} x^{\frac {3}{2}} + 6 \, b^{3} c^{2} x - 12 \, b^{3} c \sqrt {x} + 7 \, b^{3} - 12 \, {\left (b^{3} c^{4} x^{2} - b^{3}\right )} \log \left (c \sqrt {x} + 1\right )\right )} \log \left (-c \sqrt {x} + 1\right )^{2} - 4 \, {\left (168 \, a b^{2} c^{3} + 37 \, b^{3} c^{3}\right )} x^{\frac {3}{2}} + 6 \, {\left (312 \, a b^{2} c^{2} - 115 \, b^{3} c^{2}\right )} x - 288 \, {\left (3 \, a b^{2} c^{4} x^{2} - 4 \, a b^{2} c^{3} x^{\frac {3}{2}} - 12 \, a b^{2} c \sqrt {x} + 2 \, {\left (3 \, a b^{2} c^{2} - 2 \, b^{3} c^{2}\right )} x\right )} \log \left (c \sqrt {x} + 1\right ) + 12 \, {\left (9 \, b^{3} c^{4} x^{2} + 28 \, b^{3} c^{3} x^{\frac {3}{2}} - 18 \, b^{3} c^{2} x + 300 \, b^{3} c \sqrt {x} - 72 \, {\left (b^{3} c^{4} x^{2} - b^{3}\right )} \log \left (c \sqrt {x} + 1\right )^{2} - 96 \, {\left (b^{3} c^{3} x^{\frac {3}{2}} + 3 \, b^{3} c \sqrt {x} + 4 \, b^{3}\right )} \log \left (c \sqrt {x} + 1\right )\right )} \log \left (-c \sqrt {x} + 1\right ) - 12 \, {\left (600 \, a b^{2} c + 223 \, b^{3} c\right )} \sqrt {x}}{4608 \, c^{4}} \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 x \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 = -0.00 \begin {gather*} \int x\,{\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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