Optimal. Leaf size=326 \[ -\frac{3 b^2 (d e-c f) \text{PolyLog}\left (2,1-\frac{2}{-c-d x+1}\right ) \left (a+b \coth ^{-1}(c+d x)\right )}{d^2}+\frac{3 b^3 (d e-c f) \text{PolyLog}\left (3,1-\frac{2}{-c-d x+1}\right )}{2 d^2}-\frac{3 b^3 f \text{PolyLog}\left (2,-\frac{c+d x+1}{-c-d x+1}\right )}{2 d^2}-\frac{3 b^2 f \log \left (\frac{2}{-c-d x+1}\right ) \left (a+b \coth ^{-1}(c+d x)\right )}{d^2}-\frac{\left (\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^3}{2 d^2 f}+\frac{(d e-c f) \left (a+b \coth ^{-1}(c+d x)\right )^3}{d^2}-\frac{3 b (d e-c f) \log \left (\frac{2}{-c-d x+1}\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2}{d^2}+\frac{3 b f \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 d^2}+\frac{3 b f (c+d x) \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 d^2}+\frac{(e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right )^3}{2 f} \]
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Rubi [A] time = 0.720792, antiderivative size = 325, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 11, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.611, Rules used = {6112, 5929, 5911, 5985, 5919, 2402, 2315, 6049, 5949, 6059, 6610} \[ -\frac{3 b^2 (d e-c f) \text{PolyLog}\left (2,1-\frac{2}{-c-d x+1}\right ) \left (a+b \coth ^{-1}(c+d x)\right )}{d^2}+\frac{3 b^3 (d e-c f) \text{PolyLog}\left (3,1-\frac{2}{-c-d x+1}\right )}{2 d^2}-\frac{3 b^3 f \text{PolyLog}\left (2,-\frac{c+d x+1}{-c-d x+1}\right )}{2 d^2}-\frac{3 b^2 f \log \left (\frac{2}{-c-d x+1}\right ) \left (a+b \coth ^{-1}(c+d x)\right )}{d^2}+\frac{\left (-\frac{\left (c^2+1\right ) f}{d}+2 c e-\frac{d e^2}{f}\right ) \left (a+b \coth ^{-1}(c+d x)\right )^3}{2 d}+\frac{(d e-c f) \left (a+b \coth ^{-1}(c+d x)\right )^3}{d^2}-\frac{3 b (d e-c f) \log \left (\frac{2}{-c-d x+1}\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2}{d^2}+\frac{3 b f \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 d^2}+\frac{3 b f (c+d x) \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 d^2}+\frac{(e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right )^3}{2 f} \]
Antiderivative was successfully verified.
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Rule 6112
Rule 5929
Rule 5911
Rule 5985
Rule 5919
Rule 2402
Rule 2315
Rule 6049
Rule 5949
Rule 6059
Rule 6610
Rubi steps
\begin{align*} \int (e+f x) \left (a+b \coth ^{-1}(c+d x)\right )^3 \, dx &=\frac{\operatorname{Subst}\left (\int \left (\frac{d e-c f}{d}+\frac{f x}{d}\right ) \left (a+b \coth ^{-1}(x)\right )^3 \, dx,x,c+d x\right )}{d}\\ &=\frac{(e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right )^3}{2 f}-\frac{(3 b) \operatorname{Subst}\left (\int \left (-\frac{f^2 \left (a+b \coth ^{-1}(x)\right )^2}{d^2}+\frac{\left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2+2 f (d e-c f) x\right ) \left (a+b \coth ^{-1}(x)\right )^2}{d^2 \left (1-x^2\right )}\right ) \, dx,x,c+d x\right )}{2 f}\\ &=\frac{(e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right )^3}{2 f}-\frac{(3 b) \operatorname{Subst}\left (\int \frac{\left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2+2 f (d e-c f) x\right ) \left (a+b \coth ^{-1}(x)\right )^2}{1-x^2} \, dx,x,c+d x\right )}{2 d^2 f}+\frac{(3 b f) \operatorname{Subst}\left (\int \left (a+b \coth ^{-1}(x)\right )^2 \, dx,x,c+d x\right )}{2 d^2}\\ &=\frac{3 b f (c+d x) \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 d^2}+\frac{(e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right )^3}{2 f}-\frac{(3 b) \operatorname{Subst}\left (\int \left (\frac{d^2 e^2 \left (1+\frac{f \left (-2 c d e+f+c^2 f\right )}{d^2 e^2}\right ) \left (a+b \coth ^{-1}(x)\right )^2}{1-x^2}-\frac{2 f (-d e+c f) x \left (a+b \coth ^{-1}(x)\right )^2}{1-x^2}\right ) \, dx,x,c+d x\right )}{2 d^2 f}-\frac{\left (3 b^2 f\right ) \operatorname{Subst}\left (\int \frac{x \left (a+b \coth ^{-1}(x)\right )}{1-x^2} \, dx,x,c+d x\right )}{d^2}\\ &=\frac{3 b f \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 d^2}+\frac{3 b f (c+d x) \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 d^2}+\frac{(e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right )^3}{2 f}-\frac{\left (3 b^2 f\right ) \operatorname{Subst}\left (\int \frac{a+b \coth ^{-1}(x)}{1-x} \, dx,x,c+d x\right )}{d^2}-\frac{(3 b (d e-c f)) \operatorname{Subst}\left (\int \frac{x \left (a+b \coth ^{-1}(x)\right )^2}{1-x^2} \, dx,x,c+d x\right )}{d^2}-\frac{\left (3 b \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \coth ^{-1}(x)\right )^2}{1-x^2} \, dx,x,c+d x\right )}{2 d^2 f}\\ &=\frac{3 b f \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 d^2}+\frac{3 b f (c+d x) \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 d^2}+\frac{(d e-c f) \left (a+b \coth ^{-1}(c+d x)\right )^3}{d^2}-\frac{\left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^3}{2 d^2 f}+\frac{(e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right )^3}{2 f}-\frac{3 b^2 f \left (a+b \coth ^{-1}(c+d x)\right ) \log \left (\frac{2}{1-c-d x}\right )}{d^2}+\frac{\left (3 b^3 f\right ) \operatorname{Subst}\left (\int \frac{\log \left (\frac{2}{1-x}\right )}{1-x^2} \, dx,x,c+d x\right )}{d^2}-\frac{(3 b (d e-c f)) \operatorname{Subst}\left (\int \frac{\left (a+b \coth ^{-1}(x)\right )^2}{1-x} \, dx,x,c+d x\right )}{d^2}\\ &=\frac{3 b f \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 d^2}+\frac{3 b f (c+d x) \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 d^2}+\frac{(d e-c f) \left (a+b \coth ^{-1}(c+d x)\right )^3}{d^2}-\frac{\left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^3}{2 d^2 f}+\frac{(e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right )^3}{2 f}-\frac{3 b^2 f \left (a+b \coth ^{-1}(c+d x)\right ) \log \left (\frac{2}{1-c-d x}\right )}{d^2}-\frac{3 b (d e-c f) \left (a+b \coth ^{-1}(c+d x)\right )^2 \log \left (\frac{2}{1-c-d x}\right )}{d^2}-\frac{\left (3 b^3 f\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-c-d x}\right )}{d^2}+\frac{\left (6 b^2 (d e-c f)\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \coth ^{-1}(x)\right ) \log \left (\frac{2}{1-x}\right )}{1-x^2} \, dx,x,c+d x\right )}{d^2}\\ &=\frac{3 b f \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 d^2}+\frac{3 b f (c+d x) \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 d^2}+\frac{(d e-c f) \left (a+b \coth ^{-1}(c+d x)\right )^3}{d^2}-\frac{\left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^3}{2 d^2 f}+\frac{(e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right )^3}{2 f}-\frac{3 b^2 f \left (a+b \coth ^{-1}(c+d x)\right ) \log \left (\frac{2}{1-c-d x}\right )}{d^2}-\frac{3 b (d e-c f) \left (a+b \coth ^{-1}(c+d x)\right )^2 \log \left (\frac{2}{1-c-d x}\right )}{d^2}-\frac{3 b^3 f \text{Li}_2\left (1-\frac{2}{1-c-d x}\right )}{2 d^2}-\frac{3 b^2 (d e-c f) \left (a+b \coth ^{-1}(c+d x)\right ) \text{Li}_2\left (1-\frac{2}{1-c-d x}\right )}{d^2}+\frac{\left (3 b^3 (d e-c f)\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (1-\frac{2}{1-x}\right )}{1-x^2} \, dx,x,c+d x\right )}{d^2}\\ &=\frac{3 b f \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 d^2}+\frac{3 b f (c+d x) \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 d^2}+\frac{(d e-c f) \left (a+b \coth ^{-1}(c+d x)\right )^3}{d^2}-\frac{\left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^3}{2 d^2 f}+\frac{(e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right )^3}{2 f}-\frac{3 b^2 f \left (a+b \coth ^{-1}(c+d x)\right ) \log \left (\frac{2}{1-c-d x}\right )}{d^2}-\frac{3 b (d e-c f) \left (a+b \coth ^{-1}(c+d x)\right )^2 \log \left (\frac{2}{1-c-d x}\right )}{d^2}-\frac{3 b^3 f \text{Li}_2\left (1-\frac{2}{1-c-d x}\right )}{2 d^2}-\frac{3 b^2 (d e-c f) \left (a+b \coth ^{-1}(c+d x)\right ) \text{Li}_2\left (1-\frac{2}{1-c-d x}\right )}{d^2}+\frac{3 b^3 (d e-c f) \text{Li}_3\left (1-\frac{2}{1-c-d x}\right )}{2 d^2}\\ \end{align*}
Mathematica [C] time = 1.31426, size = 600, normalized size = 1.84 \[ \frac{12 a b^2 d e \left (\text{PolyLog}\left (2,e^{-2 \coth ^{-1}(c+d x)}\right )+\coth ^{-1}(c+d x) \left ((c+d x-1) \coth ^{-1}(c+d x)-2 \log \left (1-e^{-2 \coth ^{-1}(c+d x)}\right )\right )\right )-12 a b^2 c f \left (\text{PolyLog}\left (2,e^{-2 \coth ^{-1}(c+d x)}\right )+\coth ^{-1}(c+d x) \left ((c+d x-1) \coth ^{-1}(c+d x)-2 \log \left (1-e^{-2 \coth ^{-1}(c+d x)}\right )\right )\right )+2 b^3 f \left (3 \text{PolyLog}\left (2,e^{-2 \coth ^{-1}(c+d x)}\right )+\coth ^{-1}(c+d x) \left (\left (c^2+2 c d x+d^2 x^2-1\right ) \coth ^{-1}(c+d x)^2+3 (c+d x-1) \coth ^{-1}(c+d x)-6 \log \left (1-e^{-2 \coth ^{-1}(c+d x)}\right )\right )\right )+4 b^3 d e \left (-3 \coth ^{-1}(c+d x) \text{PolyLog}\left (2,e^{2 \coth ^{-1}(c+d x)}\right )+\frac{3}{2} \text{PolyLog}\left (3,e^{2 \coth ^{-1}(c+d x)}\right )+(c+d x) \coth ^{-1}(c+d x)^3+\coth ^{-1}(c+d x)^3-3 \coth ^{-1}(c+d x)^2 \log \left (1-e^{2 \coth ^{-1}(c+d x)}\right )-\frac{i \pi ^3}{8}\right )-4 b^3 c f \left (-3 \coth ^{-1}(c+d x) \text{PolyLog}\left (2,e^{2 \coth ^{-1}(c+d x)}\right )+\frac{3}{2} \text{PolyLog}\left (3,e^{2 \coth ^{-1}(c+d x)}\right )+(c+d x) \coth ^{-1}(c+d x)^3+\coth ^{-1}(c+d x)^3-3 \coth ^{-1}(c+d x)^2 \log \left (1-e^{2 \coth ^{-1}(c+d x)}\right )-\frac{i \pi ^3}{8}\right )+2 a^2 (c+d x) (-2 a c f+2 a d e+3 b f)+3 a^2 b (-2 c f+2 d e+f) \log (-c-d x+1)+3 a^2 b (2 d e-(2 c+1) f) \log (c+d x+1)-6 a^2 b (c+d x) \coth ^{-1}(c+d x) (c f-d (2 e+f x))+2 a^3 f (c+d x)^2+12 a b^2 f \left (-\log \left (\frac{1}{(c+d x) \sqrt{1-\frac{1}{(c+d x)^2}}}\right )+\frac{1}{2} \left ((c+d x)^2-1\right ) \coth ^{-1}(c+d x)^2+(c+d x) \coth ^{-1}(c+d x)\right )}{4 d^2} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 1.107, size = 12285, normalized size = 37.7 \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*} \frac{1}{2} \, a^{3} f x^{2} + \frac{3}{4} \,{\left (2 \, x^{2} \operatorname{arcoth}\left (d x + c\right ) + d{\left (\frac{2 \, x}{d^{2}} - \frac{{\left (c^{2} + 2 \, c + 1\right )} \log \left (d x + c + 1\right )}{d^{3}} + \frac{{\left (c^{2} - 2 \, c + 1\right )} \log \left (d x + c - 1\right )}{d^{3}}\right )}\right )} a^{2} b f + a^{3} e x + \frac{3 \,{\left (2 \,{\left (d x + c\right )} \operatorname{arcoth}\left (d x + c\right ) + \log \left (-{\left (d x + c\right )}^{2} + 1\right )\right )} a^{2} b e}{2 \, d} + \frac{{\left (b^{3} d^{2} f x^{2} + 2 \, b^{3} d^{2} e x -{\left (c^{2} f - 2 \,{\left (d e - f\right )} c - 2 \, d e + f\right )} b^{3}\right )} \log \left (d x + c + 1\right )^{3} + 3 \,{\left (2 \, a b^{2} d^{2} f x^{2} + 2 \,{\left (2 \, a b^{2} d^{2} e + b^{3} d f\right )} x -{\left (b^{3} d^{2} f x^{2} + 2 \, b^{3} d^{2} e x -{\left (c^{2} f - 2 \,{\left (d e + f\right )} c + 2 \, d e + f\right )} b^{3}\right )} \log \left (d x + c - 1\right )\right )} \log \left (d x + c + 1\right )^{2}}{16 \, d^{2}} + \int -\frac{{\left (b^{3} d^{2} f x^{2} +{\left (d^{2} e + c d f + d f\right )} b^{3} x +{\left (c d e + d e\right )} b^{3}\right )} \log \left (d x + c - 1\right )^{3} - 6 \,{\left (a b^{2} d^{2} f x^{2} +{\left (d^{2} e + c d f + d f\right )} a b^{2} x +{\left (c d e + d e\right )} a b^{2}\right )} \log \left (d x + c - 1\right )^{2} + 3 \,{\left (2 \, a b^{2} d^{2} f x^{2} -{\left (b^{3} d^{2} f x^{2} +{\left (d^{2} e + c d f + d f\right )} b^{3} x +{\left (c d e + d e\right )} b^{3}\right )} \log \left (d x + c - 1\right )^{2} + 2 \,{\left (2 \, a b^{2} d^{2} e + b^{3} d f\right )} x +{\left (4 \,{\left (c d e + d e\right )} a b^{2} +{\left (c^{2} f - 2 \,{\left (d e + f\right )} c + 2 \, d e + f\right )} b^{3} +{\left (4 \, a b^{2} d^{2} f - b^{3} d^{2} f\right )} x^{2} - 2 \,{\left (b^{3} d^{2} e - 2 \,{\left (d^{2} e + c d f + d f\right )} a b^{2}\right )} x\right )} \log \left (d x + c - 1\right )\right )} \log \left (d x + c + 1\right )}{8 \,{\left (d^{2} x + c d + d\right )}}\,{d x} \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^{3} f x + a^{3} e +{\left (b^{3} f x + b^{3} e\right )} \operatorname{arcoth}\left (d x + c\right )^{3} + 3 \,{\left (a b^{2} f x + a b^{2} e\right )} \operatorname{arcoth}\left (d x + c\right )^{2} + 3 \,{\left (a^{2} b f x + a^{2} b e\right )} \operatorname{arcoth}\left (d x + c\right ), 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{\left (f x + e\right )}{\left (b \operatorname{arcoth}\left (d x + c\right ) + a\right )}^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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