Optimal. Leaf size=356 \[ -\frac{2 b^2 d^2 \text{PolyLog}\left (2,1-\frac{2}{1-c x}\right )}{5 c^4}+\frac{1}{6} c^2 d^2 x^6 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{2 b d^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )}{5 c^2}+\frac{5 a b d^2 x}{6 c^3}-\frac{d^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{60 c^4}-\frac{4 b d^2 \log \left (\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{5 c^4}+\frac{2}{5} c d^2 x^5 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{15} b c d^2 x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{4} d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{5} b d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac{5 b d^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )}{18 c}+\frac{31 b^2 d^2 x^2}{180 c^2}+\frac{53 b^2 d^2 \log \left (1-c^2 x^2\right )}{90 c^4}+\frac{3 b^2 d^2 x}{5 c^3}+\frac{5 b^2 d^2 x \tanh ^{-1}(c x)}{6 c^3}-\frac{3 b^2 d^2 \tanh ^{-1}(c x)}{5 c^4}+\frac{b^2 d^2 x^3}{15 c}+\frac{1}{60} b^2 d^2 x^4 \]
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Rubi [A] time = 1.02416, antiderivative size = 356, normalized size of antiderivative = 1., number of steps used = 43, number of rules used = 15, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.682, Rules used = {5940, 5916, 5980, 266, 43, 5910, 260, 5948, 302, 206, 321, 5984, 5918, 2402, 2315} \[ -\frac{2 b^2 d^2 \text{PolyLog}\left (2,1-\frac{2}{1-c x}\right )}{5 c^4}+\frac{1}{6} c^2 d^2 x^6 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{2 b d^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )}{5 c^2}+\frac{5 a b d^2 x}{6 c^3}-\frac{d^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{60 c^4}-\frac{4 b d^2 \log \left (\frac{2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{5 c^4}+\frac{2}{5} c d^2 x^5 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{15} b c d^2 x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{4} d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{5} b d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac{5 b d^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )}{18 c}+\frac{31 b^2 d^2 x^2}{180 c^2}+\frac{53 b^2 d^2 \log \left (1-c^2 x^2\right )}{90 c^4}+\frac{3 b^2 d^2 x}{5 c^3}+\frac{5 b^2 d^2 x \tanh ^{-1}(c x)}{6 c^3}-\frac{3 b^2 d^2 \tanh ^{-1}(c x)}{5 c^4}+\frac{b^2 d^2 x^3}{15 c}+\frac{1}{60} b^2 d^2 x^4 \]
Antiderivative was successfully verified.
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Rule 5940
Rule 5916
Rule 5980
Rule 266
Rule 43
Rule 5910
Rule 260
Rule 5948
Rule 302
Rule 206
Rule 321
Rule 5984
Rule 5918
Rule 2402
Rule 2315
Rubi steps
\begin{align*} \int x^3 (d+c d x)^2 \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx &=\int \left (d^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )^2+2 c d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )^2+c^2 d^2 x^5 \left (a+b \tanh ^{-1}(c x)\right )^2\right ) \, dx\\ &=d^2 \int x^3 \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx+\left (2 c d^2\right ) \int x^4 \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx+\left (c^2 d^2\right ) \int x^5 \left (a+b \tanh ^{-1}(c x)\right )^2 \, dx\\ &=\frac{1}{4} d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{2}{5} c d^2 x^5 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{6} c^2 d^2 x^6 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac{1}{2} \left (b c d^2\right ) \int \frac{x^4 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx-\frac{1}{5} \left (4 b c^2 d^2\right ) \int \frac{x^5 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx-\frac{1}{3} \left (b c^3 d^2\right ) \int \frac{x^6 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx\\ &=\frac{1}{4} d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{2}{5} c d^2 x^5 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{6} c^2 d^2 x^6 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{5} \left (4 b d^2\right ) \int x^3 \left (a+b \tanh ^{-1}(c x)\right ) \, dx-\frac{1}{5} \left (4 b d^2\right ) \int \frac{x^3 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx+\frac{\left (b d^2\right ) \int x^2 \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{2 c}-\frac{\left (b d^2\right ) \int \frac{x^2 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{2 c}+\frac{1}{3} \left (b c d^2\right ) \int x^4 \left (a+b \tanh ^{-1}(c x)\right ) \, dx-\frac{1}{3} \left (b c d^2\right ) \int \frac{x^4 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx\\ &=\frac{b d^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )}{6 c}+\frac{1}{5} b d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{15} b c d^2 x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{4} d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{2}{5} c d^2 x^5 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{6} c^2 d^2 x^6 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac{1}{6} \left (b^2 d^2\right ) \int \frac{x^3}{1-c^2 x^2} \, dx+\frac{\left (b d^2\right ) \int \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{2 c^3}-\frac{\left (b d^2\right ) \int \frac{a+b \tanh ^{-1}(c x)}{1-c^2 x^2} \, dx}{2 c^3}+\frac{\left (4 b d^2\right ) \int x \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{5 c^2}-\frac{\left (4 b d^2\right ) \int \frac{x \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{5 c^2}+\frac{\left (b d^2\right ) \int x^2 \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{3 c}-\frac{\left (b d^2\right ) \int \frac{x^2 \left (a+b \tanh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{3 c}-\frac{1}{5} \left (b^2 c d^2\right ) \int \frac{x^4}{1-c^2 x^2} \, dx-\frac{1}{15} \left (b^2 c^2 d^2\right ) \int \frac{x^5}{1-c^2 x^2} \, dx\\ &=\frac{a b d^2 x}{2 c^3}+\frac{2 b d^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )}{5 c^2}+\frac{5 b d^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )}{18 c}+\frac{1}{5} b d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{15} b c d^2 x^5 \left (a+b \tanh ^{-1}(c x)\right )+\frac{3 d^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{20 c^4}+\frac{1}{4} d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{2}{5} c d^2 x^5 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{6} c^2 d^2 x^6 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac{1}{12} \left (b^2 d^2\right ) \operatorname{Subst}\left (\int \frac{x}{1-c^2 x} \, dx,x,x^2\right )-\frac{1}{9} \left (b^2 d^2\right ) \int \frac{x^3}{1-c^2 x^2} \, dx+\frac{\left (b d^2\right ) \int \left (a+b \tanh ^{-1}(c x)\right ) \, dx}{3 c^3}-\frac{\left (b d^2\right ) \int \frac{a+b \tanh ^{-1}(c x)}{1-c^2 x^2} \, dx}{3 c^3}-\frac{\left (4 b d^2\right ) \int \frac{a+b \tanh ^{-1}(c x)}{1-c x} \, dx}{5 c^3}+\frac{\left (b^2 d^2\right ) \int \tanh ^{-1}(c x) \, dx}{2 c^3}-\frac{\left (2 b^2 d^2\right ) \int \frac{x^2}{1-c^2 x^2} \, dx}{5 c}-\frac{1}{5} \left (b^2 c d^2\right ) \int \left (-\frac{1}{c^4}-\frac{x^2}{c^2}+\frac{1}{c^4 \left (1-c^2 x^2\right )}\right ) \, dx-\frac{1}{30} \left (b^2 c^2 d^2\right ) \operatorname{Subst}\left (\int \frac{x^2}{1-c^2 x} \, dx,x,x^2\right )\\ &=\frac{5 a b d^2 x}{6 c^3}+\frac{3 b^2 d^2 x}{5 c^3}+\frac{b^2 d^2 x^3}{15 c}+\frac{b^2 d^2 x \tanh ^{-1}(c x)}{2 c^3}+\frac{2 b d^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )}{5 c^2}+\frac{5 b d^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )}{18 c}+\frac{1}{5} b d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{15} b c d^2 x^5 \left (a+b \tanh ^{-1}(c x)\right )-\frac{d^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{60 c^4}+\frac{1}{4} d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{2}{5} c d^2 x^5 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{6} c^2 d^2 x^6 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac{4 b d^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1-c x}\right )}{5 c^4}-\frac{1}{18} \left (b^2 d^2\right ) \operatorname{Subst}\left (\int \frac{x}{1-c^2 x} \, dx,x,x^2\right )-\frac{1}{12} \left (b^2 d^2\right ) \operatorname{Subst}\left (\int \left (-\frac{1}{c^2}-\frac{1}{c^2 \left (-1+c^2 x\right )}\right ) \, dx,x,x^2\right )-\frac{\left (b^2 d^2\right ) \int \frac{1}{1-c^2 x^2} \, dx}{5 c^3}+\frac{\left (b^2 d^2\right ) \int \tanh ^{-1}(c x) \, dx}{3 c^3}-\frac{\left (2 b^2 d^2\right ) \int \frac{1}{1-c^2 x^2} \, dx}{5 c^3}+\frac{\left (4 b^2 d^2\right ) \int \frac{\log \left (\frac{2}{1-c x}\right )}{1-c^2 x^2} \, dx}{5 c^3}-\frac{\left (b^2 d^2\right ) \int \frac{x}{1-c^2 x^2} \, dx}{2 c^2}-\frac{1}{30} \left (b^2 c^2 d^2\right ) \operatorname{Subst}\left (\int \left (-\frac{1}{c^4}-\frac{x}{c^2}-\frac{1}{c^4 \left (-1+c^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=\frac{5 a b d^2 x}{6 c^3}+\frac{3 b^2 d^2 x}{5 c^3}+\frac{7 b^2 d^2 x^2}{60 c^2}+\frac{b^2 d^2 x^3}{15 c}+\frac{1}{60} b^2 d^2 x^4-\frac{3 b^2 d^2 \tanh ^{-1}(c x)}{5 c^4}+\frac{5 b^2 d^2 x \tanh ^{-1}(c x)}{6 c^3}+\frac{2 b d^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )}{5 c^2}+\frac{5 b d^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )}{18 c}+\frac{1}{5} b d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{15} b c d^2 x^5 \left (a+b \tanh ^{-1}(c x)\right )-\frac{d^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{60 c^4}+\frac{1}{4} d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{2}{5} c d^2 x^5 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{6} c^2 d^2 x^6 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac{4 b d^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1-c x}\right )}{5 c^4}+\frac{11 b^2 d^2 \log \left (1-c^2 x^2\right )}{30 c^4}-\frac{1}{18} \left (b^2 d^2\right ) \operatorname{Subst}\left (\int \left (-\frac{1}{c^2}-\frac{1}{c^2 \left (-1+c^2 x\right )}\right ) \, dx,x,x^2\right )-\frac{\left (4 b^2 d^2\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-c x}\right )}{5 c^4}-\frac{\left (b^2 d^2\right ) \int \frac{x}{1-c^2 x^2} \, dx}{3 c^2}\\ &=\frac{5 a b d^2 x}{6 c^3}+\frac{3 b^2 d^2 x}{5 c^3}+\frac{31 b^2 d^2 x^2}{180 c^2}+\frac{b^2 d^2 x^3}{15 c}+\frac{1}{60} b^2 d^2 x^4-\frac{3 b^2 d^2 \tanh ^{-1}(c x)}{5 c^4}+\frac{5 b^2 d^2 x \tanh ^{-1}(c x)}{6 c^3}+\frac{2 b d^2 x^2 \left (a+b \tanh ^{-1}(c x)\right )}{5 c^2}+\frac{5 b d^2 x^3 \left (a+b \tanh ^{-1}(c x)\right )}{18 c}+\frac{1}{5} b d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )+\frac{1}{15} b c d^2 x^5 \left (a+b \tanh ^{-1}(c x)\right )-\frac{d^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{60 c^4}+\frac{1}{4} d^2 x^4 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{2}{5} c d^2 x^5 \left (a+b \tanh ^{-1}(c x)\right )^2+\frac{1}{6} c^2 d^2 x^6 \left (a+b \tanh ^{-1}(c x)\right )^2-\frac{4 b d^2 \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac{2}{1-c x}\right )}{5 c^4}+\frac{53 b^2 d^2 \log \left (1-c^2 x^2\right )}{90 c^4}-\frac{2 b^2 d^2 \text{Li}_2\left (1-\frac{2}{1-c x}\right )}{5 c^4}\\ \end{align*}
Mathematica [A] time = 1.04655, size = 329, normalized size = 0.92 \[ \frac{d^2 \left (72 b^2 \text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c x)}\right )+30 a^2 c^6 x^6+72 a^2 c^5 x^5+45 a^2 c^4 x^4+12 a b c^5 x^5+36 a b c^4 x^4+50 a b c^3 x^3+72 a b c^2 x^2+72 a b \log \left (c^2 x^2-1\right )+2 b \tanh ^{-1}(c x) \left (3 a c^4 x^4 \left (10 c^2 x^2+24 c x+15\right )+b \left (6 c^5 x^5+18 c^4 x^4+25 c^3 x^3+36 c^2 x^2+75 c x-54\right )-72 b \log \left (e^{-2 \tanh ^{-1}(c x)}+1\right )\right )+150 a b c x+75 a b \log (1-c x)-75 a b \log (c x+1)-108 a b+3 b^2 c^4 x^4+12 b^2 c^3 x^3+31 b^2 c^2 x^2+106 b^2 \log \left (1-c^2 x^2\right )+3 b^2 \left (10 c^6 x^6+24 c^5 x^5+15 c^4 x^4-49\right ) \tanh ^{-1}(c x)^2+108 b^2 c x-34 b^2\right )}{180 c^4} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.054, size = 569, normalized size = 1.6 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 2.24754, size = 1034, normalized size = 2.9 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (a^{2} c^{2} d^{2} x^{5} + 2 \, a^{2} c d^{2} x^{4} + a^{2} d^{2} x^{3} +{\left (b^{2} c^{2} d^{2} x^{5} + 2 \, b^{2} c d^{2} x^{4} + b^{2} d^{2} x^{3}\right )} \operatorname{artanh}\left (c x\right )^{2} + 2 \,{\left (a b c^{2} d^{2} x^{5} + 2 \, a b c d^{2} x^{4} + a b d^{2} x^{3}\right )} \operatorname{artanh}\left (c x\right ), x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} d^{2} \left (\int a^{2} x^{3}\, dx + \int 2 a^{2} c x^{4}\, dx + \int a^{2} c^{2} x^{5}\, dx + \int b^{2} x^{3} \operatorname{atanh}^{2}{\left (c x \right )}\, dx + \int 2 a b x^{3} \operatorname{atanh}{\left (c x \right )}\, dx + \int 2 b^{2} c x^{4} \operatorname{atanh}^{2}{\left (c x \right )}\, dx + \int b^{2} c^{2} x^{5} \operatorname{atanh}^{2}{\left (c x \right )}\, dx + \int 4 a b c x^{4} \operatorname{atanh}{\left (c x \right )}\, dx + \int 2 a b c^{2} x^{5} \operatorname{atanh}{\left (c x \right )}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (c d x + d\right )}^{2}{\left (b \operatorname{artanh}\left (c x\right ) + a\right )}^{2} x^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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