Optimal. Leaf size=166 \[ -\frac{3 b^3 \text{PolyLog}\left (2,\frac{2}{c+d x+1}-1\right )}{2 d e^3}+\frac{3 b^2 \log \left (2-\frac{2}{c+d x+1}\right ) \left (a+b \tanh ^{-1}(c+d x)\right )}{d e^3}-\frac{3 b \left (a+b \tanh ^{-1}(c+d x)\right )^2}{2 d e^3 (c+d x)}+\frac{3 b \left (a+b \tanh ^{-1}(c+d x)\right )^2}{2 d e^3}-\frac{\left (a+b \tanh ^{-1}(c+d x)\right )^3}{2 d e^3 (c+d x)^2}+\frac{\left (a+b \tanh ^{-1}(c+d x)\right )^3}{2 d e^3} \]
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Rubi [A] time = 0.33502, antiderivative size = 166, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.348, Rules used = {6107, 12, 5916, 5982, 5988, 5932, 2447, 5948} \[ -\frac{3 b^3 \text{PolyLog}\left (2,\frac{2}{c+d x+1}-1\right )}{2 d e^3}+\frac{3 b^2 \log \left (2-\frac{2}{c+d x+1}\right ) \left (a+b \tanh ^{-1}(c+d x)\right )}{d e^3}-\frac{3 b \left (a+b \tanh ^{-1}(c+d x)\right )^2}{2 d e^3 (c+d x)}+\frac{3 b \left (a+b \tanh ^{-1}(c+d x)\right )^2}{2 d e^3}-\frac{\left (a+b \tanh ^{-1}(c+d x)\right )^3}{2 d e^3 (c+d x)^2}+\frac{\left (a+b \tanh ^{-1}(c+d x)\right )^3}{2 d e^3} \]
Antiderivative was successfully verified.
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Rule 6107
Rule 12
Rule 5916
Rule 5982
Rule 5988
Rule 5932
Rule 2447
Rule 5948
Rubi steps
\begin{align*} \int \frac{\left (a+b \tanh ^{-1}(c+d x)\right )^3}{(c e+d e x)^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (a+b \tanh ^{-1}(x)\right )^3}{e^3 x^3} \, dx,x,c+d x\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\left (a+b \tanh ^{-1}(x)\right )^3}{x^3} \, dx,x,c+d x\right )}{d e^3}\\ &=-\frac{\left (a+b \tanh ^{-1}(c+d x)\right )^3}{2 d e^3 (c+d x)^2}+\frac{(3 b) \operatorname{Subst}\left (\int \frac{\left (a+b \tanh ^{-1}(x)\right )^2}{x^2 \left (1-x^2\right )} \, dx,x,c+d x\right )}{2 d e^3}\\ &=-\frac{\left (a+b \tanh ^{-1}(c+d x)\right )^3}{2 d e^3 (c+d x)^2}+\frac{(3 b) \operatorname{Subst}\left (\int \frac{\left (a+b \tanh ^{-1}(x)\right )^2}{x^2} \, dx,x,c+d x\right )}{2 d e^3}+\frac{(3 b) \operatorname{Subst}\left (\int \frac{\left (a+b \tanh ^{-1}(x)\right )^2}{1-x^2} \, dx,x,c+d x\right )}{2 d e^3}\\ &=-\frac{3 b \left (a+b \tanh ^{-1}(c+d x)\right )^2}{2 d e^3 (c+d x)}+\frac{\left (a+b \tanh ^{-1}(c+d x)\right )^3}{2 d e^3}-\frac{\left (a+b \tanh ^{-1}(c+d x)\right )^3}{2 d e^3 (c+d x)^2}+\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{a+b \tanh ^{-1}(x)}{x \left (1-x^2\right )} \, dx,x,c+d x\right )}{d e^3}\\ &=\frac{3 b \left (a+b \tanh ^{-1}(c+d x)\right )^2}{2 d e^3}-\frac{3 b \left (a+b \tanh ^{-1}(c+d x)\right )^2}{2 d e^3 (c+d x)}+\frac{\left (a+b \tanh ^{-1}(c+d x)\right )^3}{2 d e^3}-\frac{\left (a+b \tanh ^{-1}(c+d x)\right )^3}{2 d e^3 (c+d x)^2}+\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{a+b \tanh ^{-1}(x)}{x (1+x)} \, dx,x,c+d x\right )}{d e^3}\\ &=\frac{3 b \left (a+b \tanh ^{-1}(c+d x)\right )^2}{2 d e^3}-\frac{3 b \left (a+b \tanh ^{-1}(c+d x)\right )^2}{2 d e^3 (c+d x)}+\frac{\left (a+b \tanh ^{-1}(c+d x)\right )^3}{2 d e^3}-\frac{\left (a+b \tanh ^{-1}(c+d x)\right )^3}{2 d e^3 (c+d x)^2}+\frac{3 b^2 \left (a+b \tanh ^{-1}(c+d x)\right ) \log \left (2-\frac{2}{1+c+d x}\right )}{d e^3}-\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (2-\frac{2}{1+x}\right )}{1-x^2} \, dx,x,c+d x\right )}{d e^3}\\ &=\frac{3 b \left (a+b \tanh ^{-1}(c+d x)\right )^2}{2 d e^3}-\frac{3 b \left (a+b \tanh ^{-1}(c+d x)\right )^2}{2 d e^3 (c+d x)}+\frac{\left (a+b \tanh ^{-1}(c+d x)\right )^3}{2 d e^3}-\frac{\left (a+b \tanh ^{-1}(c+d x)\right )^3}{2 d e^3 (c+d x)^2}+\frac{3 b^2 \left (a+b \tanh ^{-1}(c+d x)\right ) \log \left (2-\frac{2}{1+c+d x}\right )}{d e^3}-\frac{3 b^3 \text{Li}_2\left (-1+\frac{2}{1+c+d x}\right )}{2 d e^3}\\ \end{align*}
Mathematica [C] time = 1.12085, size = 335, normalized size = 2.02 \[ \frac{-12 b^3 (c+d x)^2 \text{PolyLog}\left (2,e^{-2 \tanh ^{-1}(c+d x)}\right )-12 a^2 b c-12 a^2 b d x-4 a^3+12 b \tanh ^{-1}(c+d x) \left (a \left (a \left (c^2+2 c d x+d^2 x^2-1\right )-2 b (c+d x)\right )+2 b^2 (c+d x)^2 \log \left (1-e^{-2 \tanh ^{-1}(c+d x)}\right )\right )+24 a b^2 c^2 \log \left (\frac{c+d x}{\sqrt{1-(c+d x)^2}}\right )+24 a b^2 d^2 x^2 \log \left (\frac{c+d x}{\sqrt{1-(c+d x)^2}}\right )+48 a b^2 c d x \log \left (\frac{c+d x}{\sqrt{1-(c+d x)^2}}\right )+12 b^2 (c+d x-1) \tanh ^{-1}(c+d x)^2 (a (c+d x+1)+b (c+d x))+4 b^3 \left (c^2+2 c d x+d^2 x^2-1\right ) \tanh ^{-1}(c+d x)^3+2 i \pi ^3 b^3 c^2 d x+i \pi ^3 b^3 c^3+i \pi ^3 b^3 c d^2 x^2}{8 d e^3 (c+d x)^2} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.437, size = 5796, normalized size = 34.9 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \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 (\frac{b^{3} \operatorname{artanh}\left (d x + c\right )^{3} + 3 \, a b^{2} \operatorname{artanh}\left (d x + c\right )^{2} + 3 \, a^{2} b \operatorname{artanh}\left (d x + c\right ) + a^{3}}{d^{3} e^{3} x^{3} + 3 \, c d^{2} e^{3} x^{2} + 3 \, c^{2} d e^{3} x + c^{3} e^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 \frac{{\left (b \operatorname{artanh}\left (d x + c\right ) + a\right )}^{3}}{{\left (d e x + c e\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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