Optimal. Leaf size=374 \[ -\frac{b^2 \left (\left (3 c^2+1\right ) f^2-6 c d e f+3 d^2 e^2\right ) \text{PolyLog}\left (2,-\frac{c+d x+1}{-c-d x+1}\right )}{3 d^3}-\frac{(d e-c f) \left (\left (c^2+3\right ) f^2-2 c d e f+d^2 e^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2}{3 d^3 f}+\frac{\left (\left (3 c^2+1\right ) f^2-6 c d e f+3 d^2 e^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2}{3 d^3}-\frac{2 b \left (\left (3 c^2+1\right ) f^2-6 c d e f+3 d^2 e^2\right ) \log \left (\frac{2}{-c-d x+1}\right ) \left (a+b \coth ^{-1}(c+d x)\right )}{3 d^3}+\frac{2 a b f x (d e-c f)}{d^2}+\frac{b f^2 (c+d x)^2 \left (a+b \coth ^{-1}(c+d x)\right )}{3 d^3}+\frac{(e+f x)^3 \left (a+b \coth ^{-1}(c+d x)\right )^2}{3 f}+\frac{b^2 f (d e-c f) \log \left (1-(c+d x)^2\right )}{d^3}+\frac{2 b^2 f (c+d x) (d e-c f) \coth ^{-1}(c+d x)}{d^3}-\frac{b^2 f^2 \tanh ^{-1}(c+d x)}{3 d^3}+\frac{b^2 f^2 x}{3 d^2} \]
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Rubi [A] time = 0.636536, antiderivative size = 374, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 13, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.65, Rules used = {6112, 5929, 5911, 260, 5917, 321, 206, 6049, 5949, 5985, 5919, 2402, 2315} \[ -\frac{b^2 \left (\left (3 c^2+1\right ) f^2-6 c d e f+3 d^2 e^2\right ) \text{PolyLog}\left (2,-\frac{c+d x+1}{-c-d x+1}\right )}{3 d^3}-\frac{(d e-c f) \left (\left (c^2+3\right ) f^2-2 c d e f+d^2 e^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2}{3 d^3 f}+\frac{\left (\left (3 c^2+1\right ) f^2-6 c d e f+3 d^2 e^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2}{3 d^3}-\frac{2 b \left (\left (3 c^2+1\right ) f^2-6 c d e f+3 d^2 e^2\right ) \log \left (\frac{2}{-c-d x+1}\right ) \left (a+b \coth ^{-1}(c+d x)\right )}{3 d^3}+\frac{2 a b f x (d e-c f)}{d^2}+\frac{b f^2 (c+d x)^2 \left (a+b \coth ^{-1}(c+d x)\right )}{3 d^3}+\frac{(e+f x)^3 \left (a+b \coth ^{-1}(c+d x)\right )^2}{3 f}+\frac{b^2 f (d e-c f) \log \left (1-(c+d x)^2\right )}{d^3}+\frac{2 b^2 f (c+d x) (d e-c f) \coth ^{-1}(c+d x)}{d^3}-\frac{b^2 f^2 \tanh ^{-1}(c+d x)}{3 d^3}+\frac{b^2 f^2 x}{3 d^2} \]
Antiderivative was successfully verified.
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Rule 6112
Rule 5929
Rule 5911
Rule 260
Rule 5917
Rule 321
Rule 206
Rule 6049
Rule 5949
Rule 5985
Rule 5919
Rule 2402
Rule 2315
Rubi steps
\begin{align*} \int (e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right )^2 \, dx &=\frac{\operatorname{Subst}\left (\int \left (\frac{d e-c f}{d}+\frac{f x}{d}\right )^2 \left (a+b \coth ^{-1}(x)\right )^2 \, dx,x,c+d x\right )}{d}\\ &=\frac{(e+f x)^3 \left (a+b \coth ^{-1}(c+d x)\right )^2}{3 f}-\frac{(2 b) \operatorname{Subst}\left (\int \left (-\frac{3 f^2 (d e-c f) \left (a+b \coth ^{-1}(x)\right )}{d^3}-\frac{f^3 x \left (a+b \coth ^{-1}(x)\right )}{d^3}+\frac{\left ((d e-c f) \left (d^2 e^2-2 c d e f+3 f^2+c^2 f^2\right )+f \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) x\right ) \left (a+b \coth ^{-1}(x)\right )}{d^3 \left (1-x^2\right )}\right ) \, dx,x,c+d x\right )}{3 f}\\ &=\frac{(e+f x)^3 \left (a+b \coth ^{-1}(c+d x)\right )^2}{3 f}-\frac{(2 b) \operatorname{Subst}\left (\int \frac{\left ((d e-c f) \left (d^2 e^2-2 c d e f+3 f^2+c^2 f^2\right )+f \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) x\right ) \left (a+b \coth ^{-1}(x)\right )}{1-x^2} \, dx,x,c+d x\right )}{3 d^3 f}+\frac{\left (2 b f^2\right ) \operatorname{Subst}\left (\int x \left (a+b \coth ^{-1}(x)\right ) \, dx,x,c+d x\right )}{3 d^3}+\frac{(2 b f (d e-c f)) \operatorname{Subst}\left (\int \left (a+b \coth ^{-1}(x)\right ) \, dx,x,c+d x\right )}{d^3}\\ &=\frac{2 a b f (d e-c f) x}{d^2}+\frac{b f^2 (c+d x)^2 \left (a+b \coth ^{-1}(c+d x)\right )}{3 d^3}+\frac{(e+f x)^3 \left (a+b \coth ^{-1}(c+d x)\right )^2}{3 f}-\frac{(2 b) \operatorname{Subst}\left (\int \left (\frac{(d e-c f) \left (d^2 e^2-2 c d e f+\left (3+c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(x)\right )}{1-x^2}+\frac{f \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) x \left (a+b \coth ^{-1}(x)\right )}{1-x^2}\right ) \, dx,x,c+d x\right )}{3 d^3 f}-\frac{\left (b^2 f^2\right ) \operatorname{Subst}\left (\int \frac{x^2}{1-x^2} \, dx,x,c+d x\right )}{3 d^3}+\frac{\left (2 b^2 f (d e-c f)\right ) \operatorname{Subst}\left (\int \coth ^{-1}(x) \, dx,x,c+d x\right )}{d^3}\\ &=\frac{b^2 f^2 x}{3 d^2}+\frac{2 a b f (d e-c f) x}{d^2}+\frac{2 b^2 f (d e-c f) (c+d x) \coth ^{-1}(c+d x)}{d^3}+\frac{b f^2 (c+d x)^2 \left (a+b \coth ^{-1}(c+d x)\right )}{3 d^3}+\frac{(e+f x)^3 \left (a+b \coth ^{-1}(c+d x)\right )^2}{3 f}-\frac{\left (b^2 f^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,c+d x\right )}{3 d^3}-\frac{\left (2 b^2 f (d e-c f)\right ) \operatorname{Subst}\left (\int \frac{x}{1-x^2} \, dx,x,c+d x\right )}{d^3}-\frac{\left (2 b (d e-c f) \left (d^2 e^2-2 c d e f+\left (3+c^2\right ) f^2\right )\right ) \operatorname{Subst}\left (\int \frac{a+b \coth ^{-1}(x)}{1-x^2} \, dx,x,c+d x\right )}{3 d^3 f}-\frac{\left (2 b \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right )\right ) \operatorname{Subst}\left (\int \frac{x \left (a+b \coth ^{-1}(x)\right )}{1-x^2} \, dx,x,c+d x\right )}{3 d^3}\\ &=\frac{b^2 f^2 x}{3 d^2}+\frac{2 a b f (d e-c f) x}{d^2}+\frac{2 b^2 f (d e-c f) (c+d x) \coth ^{-1}(c+d x)}{d^3}+\frac{b f^2 (c+d x)^2 \left (a+b \coth ^{-1}(c+d x)\right )}{3 d^3}-\frac{(d e-c f) \left (d^2 e^2-2 c d e f+\left (3+c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2}{3 d^3 f}+\frac{\left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2}{3 d^3}+\frac{(e+f x)^3 \left (a+b \coth ^{-1}(c+d x)\right )^2}{3 f}-\frac{b^2 f^2 \tanh ^{-1}(c+d x)}{3 d^3}+\frac{b^2 f (d e-c f) \log \left (1-(c+d x)^2\right )}{d^3}-\frac{\left (2 b \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right )\right ) \operatorname{Subst}\left (\int \frac{a+b \coth ^{-1}(x)}{1-x} \, dx,x,c+d x\right )}{3 d^3}\\ &=\frac{b^2 f^2 x}{3 d^2}+\frac{2 a b f (d e-c f) x}{d^2}+\frac{2 b^2 f (d e-c f) (c+d x) \coth ^{-1}(c+d x)}{d^3}+\frac{b f^2 (c+d x)^2 \left (a+b \coth ^{-1}(c+d x)\right )}{3 d^3}-\frac{(d e-c f) \left (d^2 e^2-2 c d e f+\left (3+c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2}{3 d^3 f}+\frac{\left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2}{3 d^3}+\frac{(e+f x)^3 \left (a+b \coth ^{-1}(c+d x)\right )^2}{3 f}-\frac{b^2 f^2 \tanh ^{-1}(c+d x)}{3 d^3}-\frac{2 b \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(c+d x)\right ) \log \left (\frac{2}{1-c-d x}\right )}{3 d^3}+\frac{b^2 f (d e-c f) \log \left (1-(c+d x)^2\right )}{d^3}+\frac{\left (2 b^2 \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right )\right ) \operatorname{Subst}\left (\int \frac{\log \left (\frac{2}{1-x}\right )}{1-x^2} \, dx,x,c+d x\right )}{3 d^3}\\ &=\frac{b^2 f^2 x}{3 d^2}+\frac{2 a b f (d e-c f) x}{d^2}+\frac{2 b^2 f (d e-c f) (c+d x) \coth ^{-1}(c+d x)}{d^3}+\frac{b f^2 (c+d x)^2 \left (a+b \coth ^{-1}(c+d x)\right )}{3 d^3}-\frac{(d e-c f) \left (d^2 e^2-2 c d e f+\left (3+c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2}{3 d^3 f}+\frac{\left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2}{3 d^3}+\frac{(e+f x)^3 \left (a+b \coth ^{-1}(c+d x)\right )^2}{3 f}-\frac{b^2 f^2 \tanh ^{-1}(c+d x)}{3 d^3}-\frac{2 b \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(c+d x)\right ) \log \left (\frac{2}{1-c-d x}\right )}{3 d^3}+\frac{b^2 f (d e-c f) \log \left (1-(c+d x)^2\right )}{d^3}-\frac{\left (2 b^2 \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right )\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-c-d x}\right )}{3 d^3}\\ &=\frac{b^2 f^2 x}{3 d^2}+\frac{2 a b f (d e-c f) x}{d^2}+\frac{2 b^2 f (d e-c f) (c+d x) \coth ^{-1}(c+d x)}{d^3}+\frac{b f^2 (c+d x)^2 \left (a+b \coth ^{-1}(c+d x)\right )}{3 d^3}-\frac{(d e-c f) \left (d^2 e^2-2 c d e f+\left (3+c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2}{3 d^3 f}+\frac{\left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2}{3 d^3}+\frac{(e+f x)^3 \left (a+b \coth ^{-1}(c+d x)\right )^2}{3 f}-\frac{b^2 f^2 \tanh ^{-1}(c+d x)}{3 d^3}-\frac{2 b \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(c+d x)\right ) \log \left (\frac{2}{1-c-d x}\right )}{3 d^3}+\frac{b^2 f (d e-c f) \log \left (1-(c+d x)^2\right )}{d^3}-\frac{b^2 \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \text{Li}_2\left (1-\frac{2}{1-c-d x}\right )}{3 d^3}\\ \end{align*}
Mathematica [B] time = 7.23623, size = 1054, normalized size = 2.82 \[ \frac{1}{3} a^2 f^2 x^3+a^2 e f x^2+a^2 e^2 x+\frac{1}{3} a b \left (2 x \left (3 e^2+3 f x e+f^2 x^2\right ) \coth ^{-1}(c+d x)+\frac{d f x (6 d e-4 c f+d f x)-(c-1) \left (3 d^2 e^2-3 (c-1) d f e+(c-1)^2 f^2\right ) \log (-c-d x+1)+(c+1) \left (3 d^2 e^2-3 (c+1) d f e+(c+1)^2 f^2\right ) \log (c+d x+1)}{d^3}\right )+\frac{b^2 e^2 \left (1-(c+d x)^2\right ) \left (\coth ^{-1}(c+d x) \left (-(c+d x) \coth ^{-1}(c+d x)+\coth ^{-1}(c+d x)+2 \log \left (1-e^{-2 \coth ^{-1}(c+d x)}\right )\right )-\text{PolyLog}\left (2,e^{-2 \coth ^{-1}(c+d x)}\right )\right )}{d (c+d x)^2 \left (1-\frac{1}{(c+d x)^2}\right )}-\frac{b^2 e f \left (1-(c+d x)^2\right ) \left (2 c \coth ^{-1}(c+d x)^2+(c+d x)^2 \left (1-\frac{1}{(c+d x)^2}\right ) \coth ^{-1}(c+d x)^2-2 (c+d x) \left (c \coth ^{-1}(c+d x)-1\right ) \coth ^{-1}(c+d x)+4 c \log \left (1-e^{-2 \coth ^{-1}(c+d x)}\right ) \coth ^{-1}(c+d x)-2 \log \left (\frac{1}{(c+d x) \sqrt{1-\frac{1}{(c+d x)^2}}}\right )-2 c \text{PolyLog}\left (2,e^{-2 \coth ^{-1}(c+d x)}\right )\right )}{d^2 (c+d x)^2 \left (1-\frac{1}{(c+d x)^2}\right )}-\frac{b^2 f^2 (c+d x) \sqrt{1-\frac{1}{(c+d x)^2}} \left (1-(c+d x)^2\right ) \left (\frac{9 \coth ^{-1}(c+d x)^2 c^2}{(c+d x) \sqrt{1-\frac{1}{(c+d x)^2}}}+3 \coth ^{-1}(c+d x)^2 \cosh \left (3 \coth ^{-1}(c+d x)\right ) c^2+\frac{18 \coth ^{-1}(c+d x) \log \left (1-e^{-2 \coth ^{-1}(c+d x)}\right ) c^2}{(c+d x) \sqrt{1-\frac{1}{(c+d x)^2}}}-3 \coth ^{-1}(c+d x)^2 \sinh \left (3 \coth ^{-1}(c+d x)\right ) c^2-6 \coth ^{-1}(c+d x) \log \left (1-e^{-2 \coth ^{-1}(c+d x)}\right ) \sinh \left (3 \coth ^{-1}(c+d x)\right ) c^2-\frac{12 \coth ^{-1}(c+d x)^2 c}{(c+d x) \sqrt{1-\frac{1}{(c+d x)^2}}}-6 \coth ^{-1}(c+d x) \cosh \left (3 \coth ^{-1}(c+d x)\right ) c-\frac{18 \log \left (\frac{1}{(c+d x) \sqrt{1-\frac{1}{(c+d x)^2}}}\right ) c}{(c+d x) \sqrt{1-\frac{1}{(c+d x)^2}}}+6 \log \left (\frac{1}{(c+d x) \sqrt{1-\frac{1}{(c+d x)^2}}}\right ) \sinh \left (3 \coth ^{-1}(c+d x)\right ) c+\frac{3 \coth ^{-1}(c+d x)^2}{(c+d x) \sqrt{1-\frac{1}{(c+d x)^2}}}+\frac{4 \coth ^{-1}(c+d x)}{(c+d x) \sqrt{1-\frac{1}{(c+d x)^2}}}+\frac{-3 c^2 \coth ^{-1}(c+d x)^2+3 \coth ^{-1}(c+d x)^2+6 c \coth ^{-1}(c+d x)-1}{\sqrt{1-\frac{1}{(c+d x)^2}}}+\coth ^{-1}(c+d x)^2 \cosh \left (3 \coth ^{-1}(c+d x)\right )+\cosh \left (3 \coth ^{-1}(c+d x)\right )+\frac{6 \coth ^{-1}(c+d x) \log \left (1-e^{-2 \coth ^{-1}(c+d x)}\right )}{(c+d x) \sqrt{1-\frac{1}{(c+d x)^2}}}+\frac{4 \left (3 c^2+1\right ) \text{PolyLog}\left (2,e^{-2 \coth ^{-1}(c+d x)}\right )}{(c+d x)^3 \left (1-\frac{1}{(c+d x)^2}\right )^{3/2}}-\coth ^{-1}(c+d x)^2 \sinh \left (3 \coth ^{-1}(c+d x)\right )-2 \coth ^{-1}(c+d x) \log \left (1-e^{-2 \coth ^{-1}(c+d x)}\right ) \sinh \left (3 \coth ^{-1}(c+d x)\right )\right )}{12 d^3} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.069, size = 2694, normalized size = 7.2 \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 = 1.97234, size = 1068, normalized size = 2.86 \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} f^{2} x^{2} + 2 \, a^{2} e f x + a^{2} e^{2} +{\left (b^{2} f^{2} x^{2} + 2 \, b^{2} e f x + b^{2} e^{2}\right )} \operatorname{arcoth}\left (d x + c\right )^{2} + 2 \,{\left (a b f^{2} x^{2} + 2 \, a b e f x + a b e^{2}\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 )}^{2}{\left (b \operatorname{arcoth}\left (d x + c\right ) + a\right )}^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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