Optimal. Leaf size=130 \[ -\frac {b^2 c^2}{3 x}+\frac {1}{3} b^2 c^3 \tanh ^{-1}(c x)-\frac {b c \left (a+b \tanh ^{-1}(c x)\right )}{3 x^2}+\frac {1}{3} c^3 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{3 x^3}+\frac {2}{3} b c^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac {2}{1+c x}\right )-\frac {1}{3} b^2 c^3 \text {PolyLog}\left (2,-1+\frac {2}{1+c x}\right ) \]
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Rubi [A]
time = 0.17, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6037, 6129,
331, 212, 6135, 6079, 2497} \begin {gather*} \frac {1}{3} c^3 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {2}{3} b c^3 \log \left (2-\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{3 x^3}-\frac {b c \left (a+b \tanh ^{-1}(c x)\right )}{3 x^2}-\frac {1}{3} b^2 c^3 \text {Li}_2\left (\frac {2}{c x+1}-1\right )+\frac {1}{3} b^2 c^3 \tanh ^{-1}(c x)-\frac {b^2 c^2}{3 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 331
Rule 2497
Rule 6037
Rule 6079
Rule 6129
Rule 6135
Rubi steps
\begin {align*} \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{x^4} \, dx &=-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{3 x^3}+\frac {1}{3} (2 b c) \int \frac {a+b \tanh ^{-1}(c x)}{x^3 \left (1-c^2 x^2\right )} \, dx\\ &=-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{3 x^3}+\frac {1}{3} (2 b c) \int \frac {a+b \tanh ^{-1}(c x)}{x^3} \, dx+\frac {1}{3} \left (2 b c^3\right ) \int \frac {a+b \tanh ^{-1}(c x)}{x \left (1-c^2 x^2\right )} \, dx\\ &=-\frac {b c \left (a+b \tanh ^{-1}(c x)\right )}{3 x^2}+\frac {1}{3} c^3 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{3 x^3}+\frac {1}{3} \left (b^2 c^2\right ) \int \frac {1}{x^2 \left (1-c^2 x^2\right )} \, dx+\frac {1}{3} \left (2 b c^3\right ) \int \frac {a+b \tanh ^{-1}(c x)}{x (1+c x)} \, dx\\ &=-\frac {b^2 c^2}{3 x}-\frac {b c \left (a+b \tanh ^{-1}(c x)\right )}{3 x^2}+\frac {1}{3} c^3 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{3 x^3}+\frac {2}{3} b c^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac {2}{1+c x}\right )+\frac {1}{3} \left (b^2 c^4\right ) \int \frac {1}{1-c^2 x^2} \, dx-\frac {1}{3} \left (2 b^2 c^4\right ) \int \frac {\log \left (2-\frac {2}{1+c x}\right )}{1-c^2 x^2} \, dx\\ &=-\frac {b^2 c^2}{3 x}+\frac {1}{3} b^2 c^3 \tanh ^{-1}(c x)-\frac {b c \left (a+b \tanh ^{-1}(c x)\right )}{3 x^2}+\frac {1}{3} c^3 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{3 x^3}+\frac {2}{3} b c^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac {2}{1+c x}\right )-\frac {1}{3} b^2 c^3 \text {Li}_2\left (-1+\frac {2}{1+c x}\right )\\ \end {align*}
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Mathematica [A]
time = 0.25, size = 145, normalized size = 1.12 \begin {gather*} -\frac {a^2+a b c x+b^2 c^2 x^2+b^2 \left (1-c^3 x^3\right ) \tanh ^{-1}(c x)^2+b \tanh ^{-1}(c x) \left (2 a+b c x-b c^3 x^3-2 b c^3 x^3 \log \left (1-e^{-2 \tanh ^{-1}(c x)}\right )\right )-2 a b c^3 x^3 \log (c x)+a b c^3 x^3 \log \left (1-c^2 x^2\right )+b^2 c^3 x^3 \text {PolyLog}\left (2,e^{-2 \tanh ^{-1}(c x)}\right )}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(304\) vs.
\(2(116)=232\).
time = 0.09, size = 305, normalized size = 2.35
method | result | size |
derivativedivides | \(c^{3} \left (-\frac {a^{2}}{3 c^{3} x^{3}}-\frac {b^{2} \arctanh \left (c x \right )^{2}}{3 c^{3} x^{3}}-\frac {b^{2} \arctanh \left (c x \right ) \ln \left (c x -1\right )}{3}-\frac {b^{2} \arctanh \left (c x \right )}{3 c^{2} x^{2}}+\frac {2 b^{2} \ln \left (c x \right ) \arctanh \left (c x \right )}{3}-\frac {b^{2} \arctanh \left (c x \right ) \ln \left (c x +1\right )}{3}-\frac {b^{2} \ln \left (c x -1\right )}{6}+\frac {b^{2} \ln \left (c x +1\right )}{6}-\frac {b^{2}}{3 c x}+\frac {b^{2} \dilog \left (\frac {c x}{2}+\frac {1}{2}\right )}{3}+\frac {b^{2} \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{6}-\frac {b^{2} \ln \left (c x -1\right )^{2}}{12}-\frac {b^{2} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (c x +1\right )}{6}+\frac {b^{2} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{6}+\frac {b^{2} \ln \left (c x +1\right )^{2}}{12}-\frac {b^{2} \dilog \left (c x \right )}{3}-\frac {b^{2} \dilog \left (c x +1\right )}{3}-\frac {b^{2} \ln \left (c x \right ) \ln \left (c x +1\right )}{3}-\frac {2 a b \arctanh \left (c x \right )}{3 c^{3} x^{3}}-\frac {a b \ln \left (c x -1\right )}{3}-\frac {a b}{3 c^{2} x^{2}}+\frac {2 a b \ln \left (c x \right )}{3}-\frac {a b \ln \left (c x +1\right )}{3}\right )\) | \(305\) |
default | \(c^{3} \left (-\frac {a^{2}}{3 c^{3} x^{3}}-\frac {b^{2} \arctanh \left (c x \right )^{2}}{3 c^{3} x^{3}}-\frac {b^{2} \arctanh \left (c x \right ) \ln \left (c x -1\right )}{3}-\frac {b^{2} \arctanh \left (c x \right )}{3 c^{2} x^{2}}+\frac {2 b^{2} \ln \left (c x \right ) \arctanh \left (c x \right )}{3}-\frac {b^{2} \arctanh \left (c x \right ) \ln \left (c x +1\right )}{3}-\frac {b^{2} \ln \left (c x -1\right )}{6}+\frac {b^{2} \ln \left (c x +1\right )}{6}-\frac {b^{2}}{3 c x}+\frac {b^{2} \dilog \left (\frac {c x}{2}+\frac {1}{2}\right )}{3}+\frac {b^{2} \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{6}-\frac {b^{2} \ln \left (c x -1\right )^{2}}{12}-\frac {b^{2} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (c x +1\right )}{6}+\frac {b^{2} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{6}+\frac {b^{2} \ln \left (c x +1\right )^{2}}{12}-\frac {b^{2} \dilog \left (c x \right )}{3}-\frac {b^{2} \dilog \left (c x +1\right )}{3}-\frac {b^{2} \ln \left (c x \right ) \ln \left (c x +1\right )}{3}-\frac {2 a b \arctanh \left (c x \right )}{3 c^{3} x^{3}}-\frac {a b \ln \left (c x -1\right )}{3}-\frac {a b}{3 c^{2} x^{2}}+\frac {2 a b \ln \left (c x \right )}{3}-\frac {a b \ln \left (c x +1\right )}{3}\right )\) | \(305\) |
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 )^{2}}{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 = -0.01 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^2}{x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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