Optimal. Leaf size=200 \[ -\frac {b^2 c^2 \left (a+b \tanh ^{-1}(c x)\right )}{x}+\frac {1}{2} b c^3 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {b c \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x^2}+\frac {1}{3} c^3 \left (a+b \tanh ^{-1}(c x)\right )^3-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{3 x^3}+b^3 c^3 \log (x)-\frac {1}{2} b^3 c^3 \log \left (1-c^2 x^2\right )+b c^3 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (2-\frac {2}{1+c x}\right )-b^2 c^3 \left (a+b \tanh ^{-1}(c x)\right ) \text {PolyLog}\left (2,-1+\frac {2}{1+c x}\right )-\frac {1}{2} b^3 c^3 \text {PolyLog}\left (3,-1+\frac {2}{1+c x}\right ) \]
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Rubi [A]
time = 0.34, antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps
used = 14, number of rules used = 11, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.786, Rules used = {6037, 6129,
272, 36, 29, 31, 6095, 6135, 6079, 6203, 6745} \begin {gather*} -b^2 c^3 \text {Li}_2\left (\frac {2}{c x+1}-1\right ) \left (a+b \tanh ^{-1}(c x)\right )-\frac {b^2 c^2 \left (a+b \tanh ^{-1}(c x)\right )}{x}+\frac {1}{3} c^3 \left (a+b \tanh ^{-1}(c x)\right )^3+\frac {1}{2} b c^3 \left (a+b \tanh ^{-1}(c x)\right )^2+b c^3 \log \left (2-\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{3 x^3}-\frac {b c \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x^2}-\frac {1}{2} b^3 c^3 \text {Li}_3\left (\frac {2}{c x+1}-1\right )+b^3 c^3 \log (x)-\frac {1}{2} b^3 c^3 \log \left (1-c^2 x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 272
Rule 6037
Rule 6079
Rule 6095
Rule 6129
Rule 6135
Rule 6203
Rule 6745
Rubi steps
\begin {align*} \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{x^4} \, dx &=-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{3 x^3}+(b c) \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{x^3 \left (1-c^2 x^2\right )} \, dx\\ &=-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{3 x^3}+(b c) \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{x^3} \, dx+\left (b c^3\right ) \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{x \left (1-c^2 x^2\right )} \, dx\\ &=-\frac {b c \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x^2}+\frac {1}{3} c^3 \left (a+b \tanh ^{-1}(c x)\right )^3-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{3 x^3}+\left (b^2 c^2\right ) \int \frac {a+b \tanh ^{-1}(c x)}{x^2 \left (1-c^2 x^2\right )} \, dx+\left (b c^3\right ) \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{x (1+c x)} \, dx\\ &=-\frac {b c \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x^2}+\frac {1}{3} c^3 \left (a+b \tanh ^{-1}(c x)\right )^3-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{3 x^3}+b c^3 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (2-\frac {2}{1+c x}\right )+\left (b^2 c^2\right ) \int \frac {a+b \tanh ^{-1}(c x)}{x^2} \, dx+\left (b^2 c^4\right ) \int \frac {a+b \tanh ^{-1}(c x)}{1-c^2 x^2} \, dx-\left (2 b^2 c^4\right ) \int \frac {\left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac {2}{1+c x}\right )}{1-c^2 x^2} \, dx\\ &=-\frac {b^2 c^2 \left (a+b \tanh ^{-1}(c x)\right )}{x}+\frac {1}{2} b c^3 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {b c \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x^2}+\frac {1}{3} c^3 \left (a+b \tanh ^{-1}(c x)\right )^3-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{3 x^3}+b c^3 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (2-\frac {2}{1+c x}\right )-b^2 c^3 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (-1+\frac {2}{1+c x}\right )+\left (b^3 c^3\right ) \int \frac {1}{x \left (1-c^2 x^2\right )} \, dx+\left (b^3 c^4\right ) \int \frac {\text {Li}_2\left (-1+\frac {2}{1+c x}\right )}{1-c^2 x^2} \, dx\\ &=-\frac {b^2 c^2 \left (a+b \tanh ^{-1}(c x)\right )}{x}+\frac {1}{2} b c^3 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {b c \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x^2}+\frac {1}{3} c^3 \left (a+b \tanh ^{-1}(c x)\right )^3-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{3 x^3}+b c^3 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (2-\frac {2}{1+c x}\right )-b^2 c^3 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (-1+\frac {2}{1+c x}\right )-\frac {1}{2} b^3 c^3 \text {Li}_3\left (-1+\frac {2}{1+c x}\right )+\frac {1}{2} \left (b^3 c^3\right ) \text {Subst}\left (\int \frac {1}{x \left (1-c^2 x\right )} \, dx,x,x^2\right )\\ &=-\frac {b^2 c^2 \left (a+b \tanh ^{-1}(c x)\right )}{x}+\frac {1}{2} b c^3 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {b c \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x^2}+\frac {1}{3} c^3 \left (a+b \tanh ^{-1}(c x)\right )^3-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{3 x^3}+b c^3 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (2-\frac {2}{1+c x}\right )-b^2 c^3 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (-1+\frac {2}{1+c x}\right )-\frac {1}{2} b^3 c^3 \text {Li}_3\left (-1+\frac {2}{1+c x}\right )+\frac {1}{2} \left (b^3 c^3\right ) \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )+\frac {1}{2} \left (b^3 c^5\right ) \text {Subst}\left (\int \frac {1}{1-c^2 x} \, dx,x,x^2\right )\\ &=-\frac {b^2 c^2 \left (a+b \tanh ^{-1}(c x)\right )}{x}+\frac {1}{2} b c^3 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {b c \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x^2}+\frac {1}{3} c^3 \left (a+b \tanh ^{-1}(c x)\right )^3-\frac {\left (a+b \tanh ^{-1}(c x)\right )^3}{3 x^3}+b^3 c^3 \log (x)-\frac {1}{2} b^3 c^3 \log \left (1-c^2 x^2\right )+b c^3 \left (a+b \tanh ^{-1}(c x)\right )^2 \log \left (2-\frac {2}{1+c x}\right )-b^2 c^3 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (-1+\frac {2}{1+c x}\right )-\frac {1}{2} b^3 c^3 \text {Li}_3\left (-1+\frac {2}{1+c x}\right )\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.64, size = 323, normalized size = 1.62 \begin {gather*} -\frac {8 a^3+12 a^2 b c x+24 a^2 b \tanh ^{-1}(c x)-24 a^2 b c^3 x^3 \log (x)+12 a^2 b c^3 x^3 \log \left (1-c^2 x^2\right )+24 a b^2 \left (c^2 x^2+\left (1-c^3 x^3\right ) \tanh ^{-1}(c x)^2-c x \tanh ^{-1}(c x) \left (-1+c^2 x^2+2 c^2 x^2 \log \left (1-e^{-2 \tanh ^{-1}(c x)}\right )\right )+c^3 x^3 \text {PolyLog}\left (2,e^{-2 \tanh ^{-1}(c x)}\right )\right )+b^3 \left (-i c^3 \pi ^3 x^3+24 c^2 x^2 \tanh ^{-1}(c x)+12 c x \tanh ^{-1}(c x)^2-12 c^3 x^3 \tanh ^{-1}(c x)^2+8 \tanh ^{-1}(c x)^3+8 c^3 x^3 \tanh ^{-1}(c x)^3-24 c^3 x^3 \tanh ^{-1}(c x)^2 \log \left (1-e^{2 \tanh ^{-1}(c x)}\right )-24 c^3 x^3 \log \left (\frac {c x}{\sqrt {1-c^2 x^2}}\right )-24 c^3 x^3 \tanh ^{-1}(c x) \text {PolyLog}\left (2,e^{2 \tanh ^{-1}(c x)}\right )+12 c^3 x^3 \text {PolyLog}\left (3,e^{2 \tanh ^{-1}(c x)}\right )\right )}{24 x^3} \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 = 2.55, size = 1716, normalized size = 8.58
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(1716\) |
default | \(\text {Expression too large to display}\) | \(1716\) |
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 \frac {\left (a + b \operatorname {atanh}{\left (c x \right )}\right )^{3}}{x^{4}}\, 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 \frac {{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^3}{x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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