Optimal. Leaf size=396 \[ -\frac {b^2 c^2 d^3}{3 x}+\frac {1}{3} b^2 c^3 d^3 \tanh ^{-1}(c x)-\frac {b c d^3 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^2}-\frac {3 b c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{x}+\frac {29}{6} c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 x^3}-\frac {3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x^2}-\frac {3 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{x}+2 c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1-c x}\right )+3 b^2 c^3 d^3 \log (x)-\frac {3}{2} b^2 c^3 d^3 \log \left (1-c^2 x^2\right )+\frac {20}{3} b c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac {2}{1+c x}\right )-b c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \text {PolyLog}\left (2,1-\frac {2}{1-c x}\right )+b c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \text {PolyLog}\left (2,-1+\frac {2}{1-c x}\right )-\frac {10}{3} b^2 c^3 d^3 \text {PolyLog}\left (2,-1+\frac {2}{1+c x}\right )+\frac {1}{2} b^2 c^3 d^3 \text {PolyLog}\left (3,1-\frac {2}{1-c x}\right )-\frac {1}{2} b^2 c^3 d^3 \text {PolyLog}\left (3,-1+\frac {2}{1-c x}\right ) \]
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Rubi [A]
time = 0.68, antiderivative size = 396, normalized size of antiderivative = 1.00, number of steps
used = 28, number of rules used = 17, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.773, Rules used = {6087,
6037, 6129, 331, 212, 6135, 6079, 2497, 272, 36, 29, 31, 6095, 6033, 6199, 6205, 6745}
\begin {gather*} -b c^3 d^3 \text {Li}_2\left (1-\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )+b c^3 d^3 \text {Li}_2\left (\frac {2}{1-c x}-1\right ) \left (a+b \tanh ^{-1}(c x)\right )+\frac {29}{6} c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2+2 c^3 d^3 \tanh ^{-1}\left (1-\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2+\frac {20}{3} b c^3 d^3 \log \left (2-\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )-\frac {3 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{x}-\frac {3 b c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{x}-\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 x^3}-\frac {3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x^2}-\frac {b c d^3 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^2}-\frac {10}{3} b^2 c^3 d^3 \text {Li}_2\left (\frac {2}{c x+1}-1\right )+\frac {1}{2} b^2 c^3 d^3 \text {Li}_3\left (1-\frac {2}{1-c x}\right )-\frac {1}{2} b^2 c^3 d^3 \text {Li}_3\left (\frac {2}{1-c x}-1\right )+3 b^2 c^3 d^3 \log (x)+\frac {1}{3} b^2 c^3 d^3 \tanh ^{-1}(c x)-\frac {b^2 c^2 d^3}{3 x}-\frac {3}{2} b^2 c^3 d^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 212
Rule 272
Rule 331
Rule 2497
Rule 6033
Rule 6037
Rule 6079
Rule 6087
Rule 6095
Rule 6129
Rule 6135
Rule 6199
Rule 6205
Rule 6745
Rubi steps
\begin {align*} \int \frac {(d+c d x)^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{x^4} \, dx &=\int \left (\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{x^4}+\frac {3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{x^3}+\frac {3 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{x^2}+\frac {c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{x}\right ) \, dx\\ &=d^3 \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{x^4} \, dx+\left (3 c d^3\right ) \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{x^3} \, dx+\left (3 c^2 d^3\right ) \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{x^2} \, dx+\left (c^3 d^3\right ) \int \frac {\left (a+b \tanh ^{-1}(c x)\right )^2}{x} \, dx\\ &=-\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 x^3}-\frac {3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x^2}-\frac {3 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{x}+2 c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1-c x}\right )+\frac {1}{3} \left (2 b c d^3\right ) \int \frac {a+b \tanh ^{-1}(c x)}{x^3 \left (1-c^2 x^2\right )} \, dx+\left (3 b c^2 d^3\right ) \int \frac {a+b \tanh ^{-1}(c x)}{x^2 \left (1-c^2 x^2\right )} \, dx+\left (6 b c^3 d^3\right ) \int \frac {a+b \tanh ^{-1}(c x)}{x \left (1-c^2 x^2\right )} \, dx-\left (4 b c^4 d^3\right ) \int \frac {\left (a+b \tanh ^{-1}(c x)\right ) \tanh ^{-1}\left (1-\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx\\ &=3 c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 x^3}-\frac {3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x^2}-\frac {3 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{x}+2 c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1-c x}\right )+\frac {1}{3} \left (2 b c d^3\right ) \int \frac {a+b \tanh ^{-1}(c x)}{x^3} \, dx+\left (3 b c^2 d^3\right ) \int \frac {a+b \tanh ^{-1}(c x)}{x^2} \, dx+\frac {1}{3} \left (2 b c^3 d^3\right ) \int \frac {a+b \tanh ^{-1}(c x)}{x \left (1-c^2 x^2\right )} \, dx+\left (6 b c^3 d^3\right ) \int \frac {a+b \tanh ^{-1}(c x)}{x (1+c x)} \, dx+\left (2 b c^4 d^3\right ) \int \frac {\left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx-\left (2 b c^4 d^3\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+\left (3 b c^4 d^3\right ) \int \frac {a+b \tanh ^{-1}(c x)}{1-c^2 x^2} \, dx\\ &=-\frac {b c d^3 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^2}-\frac {3 b c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{x}+\frac {29}{6} c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 x^3}-\frac {3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x^2}-\frac {3 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{x}+2 c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1-c x}\right )+6 b c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac {2}{1+c x}\right )-b c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-c x}\right )+b c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (-1+\frac {2}{1-c x}\right )+\frac {1}{3} \left (b^2 c^2 d^3\right ) \int \frac {1}{x^2 \left (1-c^2 x^2\right )} \, dx+\frac {1}{3} \left (2 b c^3 d^3\right ) \int \frac {a+b \tanh ^{-1}(c x)}{x (1+c x)} \, dx+\left (3 b^2 c^3 d^3\right ) \int \frac {1}{x \left (1-c^2 x^2\right )} \, dx+\left (b^2 c^4 d^3\right ) \int \frac {\text {Li}_2\left (1-\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx-\left (b^2 c^4 d^3\right ) \int \frac {\text {Li}_2\left (-1+\frac {2}{1-c x}\right )}{1-c^2 x^2} \, dx-\left (6 b^2 c^4 d^3\right ) \int \frac {\log \left (2-\frac {2}{1+c x}\right )}{1-c^2 x^2} \, dx\\ &=-\frac {b^2 c^2 d^3}{3 x}-\frac {b c d^3 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^2}-\frac {3 b c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{x}+\frac {29}{6} c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 x^3}-\frac {3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x^2}-\frac {3 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{x}+2 c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1-c x}\right )+\frac {20}{3} b c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac {2}{1+c x}\right )-b c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-c x}\right )+b c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (-1+\frac {2}{1-c x}\right )-3 b^2 c^3 d^3 \text {Li}_2\left (-1+\frac {2}{1+c x}\right )+\frac {1}{2} b^2 c^3 d^3 \text {Li}_3\left (1-\frac {2}{1-c x}\right )-\frac {1}{2} b^2 c^3 d^3 \text {Li}_3\left (-1+\frac {2}{1-c x}\right )+\frac {1}{2} \left (3 b^2 c^3 d^3\right ) \text {Subst}\left (\int \frac {1}{x \left (1-c^2 x\right )} \, dx,x,x^2\right )+\frac {1}{3} \left (b^2 c^4 d^3\right ) \int \frac {1}{1-c^2 x^2} \, dx-\frac {1}{3} \left (2 b^2 c^4 d^3\right ) \int \frac {\log \left (2-\frac {2}{1+c x}\right )}{1-c^2 x^2} \, dx\\ &=-\frac {b^2 c^2 d^3}{3 x}+\frac {1}{3} b^2 c^3 d^3 \tanh ^{-1}(c x)-\frac {b c d^3 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^2}-\frac {3 b c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{x}+\frac {29}{6} c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 x^3}-\frac {3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x^2}-\frac {3 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{x}+2 c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1-c x}\right )+\frac {20}{3} b c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac {2}{1+c x}\right )-b c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-c x}\right )+b c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (-1+\frac {2}{1-c x}\right )-\frac {10}{3} b^2 c^3 d^3 \text {Li}_2\left (-1+\frac {2}{1+c x}\right )+\frac {1}{2} b^2 c^3 d^3 \text {Li}_3\left (1-\frac {2}{1-c x}\right )-\frac {1}{2} b^2 c^3 d^3 \text {Li}_3\left (-1+\frac {2}{1-c x}\right )+\frac {1}{2} \left (3 b^2 c^3 d^3\right ) \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )+\frac {1}{2} \left (3 b^2 c^5 d^3\right ) \text {Subst}\left (\int \frac {1}{1-c^2 x} \, dx,x,x^2\right )\\ &=-\frac {b^2 c^2 d^3}{3 x}+\frac {1}{3} b^2 c^3 d^3 \tanh ^{-1}(c x)-\frac {b c d^3 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^2}-\frac {3 b c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{x}+\frac {29}{6} c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{3 x^3}-\frac {3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 x^2}-\frac {3 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2}{x}+2 c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1-c x}\right )+3 b^2 c^3 d^3 \log (x)-\frac {3}{2} b^2 c^3 d^3 \log \left (1-c^2 x^2\right )+\frac {20}{3} b c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (2-\frac {2}{1+c x}\right )-b c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1-c x}\right )+b c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right ) \text {Li}_2\left (-1+\frac {2}{1-c x}\right )-\frac {10}{3} b^2 c^3 d^3 \text {Li}_2\left (-1+\frac {2}{1+c x}\right )+\frac {1}{2} b^2 c^3 d^3 \text {Li}_3\left (1-\frac {2}{1-c x}\right )-\frac {1}{2} b^2 c^3 d^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 = 0.42, size = 569, normalized size = 1.44 \begin {gather*} \frac {d^3 \left (-8 a^2-36 a^2 c x-8 a b c x-72 a^2 c^2 x^2-72 a b c^2 x^2-8 b^2 c^2 x^2+i b^2 c^3 \pi ^3 x^3-16 a b \tanh ^{-1}(c x)-72 a b c x \tanh ^{-1}(c x)-8 b^2 c x \tanh ^{-1}(c x)-144 a b c^2 x^2 \tanh ^{-1}(c x)-72 b^2 c^2 x^2 \tanh ^{-1}(c x)+8 b^2 c^3 x^3 \tanh ^{-1}(c x)-8 b^2 \tanh ^{-1}(c x)^2-36 b^2 c x \tanh ^{-1}(c x)^2-72 b^2 c^2 x^2 \tanh ^{-1}(c x)^2+116 b^2 c^3 x^3 \tanh ^{-1}(c x)^2-16 b^2 c^3 x^3 \tanh ^{-1}(c x)^3+160 b^2 c^3 x^3 \tanh ^{-1}(c x) \log \left (1-e^{-2 \tanh ^{-1}(c x)}\right )-24 b^2 c^3 x^3 \tanh ^{-1}(c x)^2 \log \left (1+e^{-2 \tanh ^{-1}(c x)}\right )+24 b^2 c^3 x^3 \tanh ^{-1}(c x)^2 \log \left (1-e^{2 \tanh ^{-1}(c x)}\right )+24 a^2 c^3 x^3 \log (x)+160 a b c^3 x^3 \log (c x)-36 a b c^3 x^3 \log (1-c x)+36 a b c^3 x^3 \log (1+c x)+72 b^2 c^3 x^3 \log \left (\frac {c x}{\sqrt {1-c^2 x^2}}\right )-80 a b c^3 x^3 \log \left (1-c^2 x^2\right )+24 b^2 c^3 x^3 \tanh ^{-1}(c x) \text {PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c x)}\right )-80 b^2 c^3 x^3 \text {PolyLog}\left (2,e^{-2 \tanh ^{-1}(c x)}\right )+24 b^2 c^3 x^3 \tanh ^{-1}(c x) \text {PolyLog}\left (2,e^{2 \tanh ^{-1}(c x)}\right )-24 a b c^3 x^3 \text {PolyLog}(2,-c x)+24 a b c^3 x^3 \text {PolyLog}(2,c x)+12 b^2 c^3 x^3 \text {PolyLog}\left (3,-e^{-2 \tanh ^{-1}(c x)}\right )-12 b^2 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 = 7.74, size = 1267, normalized size = 3.20
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(1267\) |
default | \(\text {Expression too large to display}\) | \(1267\) |
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*} d^{3} \left (\int \frac {a^{2}}{x^{4}}\, dx + \int \frac {3 a^{2} c}{x^{3}}\, dx + \int \frac {3 a^{2} c^{2}}{x^{2}}\, dx + \int \frac {a^{2} c^{3}}{x}\, dx + \int \frac {b^{2} \operatorname {atanh}^{2}{\left (c x \right )}}{x^{4}}\, dx + \int \frac {2 a b \operatorname {atanh}{\left (c x \right )}}{x^{4}}\, dx + \int \frac {3 b^{2} c \operatorname {atanh}^{2}{\left (c x \right )}}{x^{3}}\, dx + \int \frac {3 b^{2} c^{2} \operatorname {atanh}^{2}{\left (c x \right )}}{x^{2}}\, dx + \int \frac {b^{2} c^{3} \operatorname {atanh}^{2}{\left (c x \right )}}{x}\, dx + \int \frac {6 a b c \operatorname {atanh}{\left (c x \right )}}{x^{3}}\, dx + \int \frac {6 a b c^{2} \operatorname {atanh}{\left (c x \right )}}{x^{2}}\, dx + \int \frac {2 a b c^{3} \operatorname {atanh}{\left (c x \right )}}{x}\, dx\right ) \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 )}^2\,{\left (d+c\,d\,x\right )}^3}{x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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