Integrand size = 18, antiderivative size = 221 \[ \int (e+f x) \left (a+b \coth ^{-1}(c+d x)\right )^2 \, dx=\frac {a b f x}{d}+\frac {b^2 f (c+d x) \coth ^{-1}(c+d x)}{d^2}+\frac {(d e-c f) \left (a+b \coth ^{-1}(c+d x)\right )^2}{d^2}-\frac {\left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 d^2 f}+\frac {(e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 f}-\frac {2 b (d e-c f) \left (a+b \coth ^{-1}(c+d x)\right ) \log \left (\frac {2}{1-c-d x}\right )}{d^2}+\frac {b^2 f \log \left (1-(c+d x)^2\right )}{2 d^2}-\frac {b^2 (d e-c f) \operatorname {PolyLog}\left (2,-\frac {1+c+d x}{1-c-d x}\right )}{d^2} \]
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Time = 0.35 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {6247, 6066, 6022, 266, 6196, 6096, 6132, 6056, 2449, 2352} \[ \int (e+f x) \left (a+b \coth ^{-1}(c+d x)\right )^2 \, dx=\frac {\left (-\frac {\left (c^2+1\right ) f}{d}+2 c e-\frac {d e^2}{f}\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 d}+\frac {(d e-c f) \left (a+b \coth ^{-1}(c+d x)\right )^2}{d^2}-\frac {2 b (d e-c f) \log \left (\frac {2}{-c-d x+1}\right ) \left (a+b \coth ^{-1}(c+d x)\right )}{d^2}+\frac {(e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 f}+\frac {a b f x}{d}-\frac {b^2 (d e-c f) \operatorname {PolyLog}\left (2,-\frac {c+d x+1}{-c-d x+1}\right )}{d^2}+\frac {b^2 f \log \left (1-(c+d x)^2\right )}{2 d^2}+\frac {b^2 f (c+d x) \coth ^{-1}(c+d x)}{d^2} \]
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Rule 266
Rule 2352
Rule 2449
Rule 6022
Rule 6056
Rule 6066
Rule 6096
Rule 6132
Rule 6196
Rule 6247
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \left (\frac {d e-c f}{d}+\frac {f x}{d}\right ) \left (a+b \coth ^{-1}(x)\right )^2 \, dx,x,c+d x\right )}{d} \\ & = \frac {(e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 f}-\frac {b \text {Subst}\left (\int \left (-\frac {f^2 \left (a+b \coth ^{-1}(x)\right )}{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 )}{d^2 \left (1-x^2\right )}\right ) \, dx,x,c+d x\right )}{f} \\ & = \frac {(e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 f}-\frac {b \text {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 )}{1-x^2} \, dx,x,c+d x\right )}{d^2 f}+\frac {(b f) \text {Subst}\left (\int \left (a+b \coth ^{-1}(x)\right ) \, dx,x,c+d x\right )}{d^2} \\ & = \frac {a b f x}{d}+\frac {(e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 f}-\frac {b \text {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 )}{1-x^2}-\frac {2 f (-d e+c f) x \left (a+b \coth ^{-1}(x)\right )}{1-x^2}\right ) \, dx,x,c+d x\right )}{d^2 f}+\frac {\left (b^2 f\right ) \text {Subst}\left (\int \coth ^{-1}(x) \, dx,x,c+d x\right )}{d^2} \\ & = \frac {a b f x}{d}+\frac {b^2 f (c+d x) \coth ^{-1}(c+d x)}{d^2}+\frac {(e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 f}-\frac {\left (b^2 f\right ) \text {Subst}\left (\int \frac {x}{1-x^2} \, dx,x,c+d x\right )}{d^2}-\frac {(2 b (d e-c f)) \text {Subst}\left (\int \frac {x \left (a+b \coth ^{-1}(x)\right )}{1-x^2} \, dx,x,c+d x\right )}{d^2}-\frac {\left (b \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )\right ) \text {Subst}\left (\int \frac {a+b \coth ^{-1}(x)}{1-x^2} \, dx,x,c+d x\right )}{d^2 f} \\ & = \frac {a b f x}{d}+\frac {b^2 f (c+d x) \coth ^{-1}(c+d x)}{d^2}+\frac {(d e-c f) \left (a+b \coth ^{-1}(c+d x)\right )^2}{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 )^2}{2 d^2 f}+\frac {(e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 f}+\frac {b^2 f \log \left (1-(c+d x)^2\right )}{2 d^2}-\frac {(2 b (d e-c f)) \text {Subst}\left (\int \frac {a+b \coth ^{-1}(x)}{1-x} \, dx,x,c+d x\right )}{d^2} \\ & = \frac {a b f x}{d}+\frac {b^2 f (c+d x) \coth ^{-1}(c+d x)}{d^2}+\frac {(d e-c f) \left (a+b \coth ^{-1}(c+d x)\right )^2}{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 )^2}{2 d^2 f}+\frac {(e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 f}-\frac {2 b (d e-c f) \left (a+b \coth ^{-1}(c+d x)\right ) \log \left (\frac {2}{1-c-d x}\right )}{d^2}+\frac {b^2 f \log \left (1-(c+d x)^2\right )}{2 d^2}+\frac {\left (2 b^2 (d e-c f)\right ) \text {Subst}\left (\int \frac {\log \left (\frac {2}{1-x}\right )}{1-x^2} \, dx,x,c+d x\right )}{d^2} \\ & = \frac {a b f x}{d}+\frac {b^2 f (c+d x) \coth ^{-1}(c+d x)}{d^2}+\frac {(d e-c f) \left (a+b \coth ^{-1}(c+d x)\right )^2}{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 )^2}{2 d^2 f}+\frac {(e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 f}-\frac {2 b (d e-c f) \left (a+b \coth ^{-1}(c+d x)\right ) \log \left (\frac {2}{1-c-d x}\right )}{d^2}+\frac {b^2 f \log \left (1-(c+d x)^2\right )}{2 d^2}-\frac {\left (2 b^2 (d e-c f)\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-c-d x}\right )}{d^2} \\ & = \frac {a b f x}{d}+\frac {b^2 f (c+d x) \coth ^{-1}(c+d x)}{d^2}+\frac {(d e-c f) \left (a+b \coth ^{-1}(c+d x)\right )^2}{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 )^2}{2 d^2 f}+\frac {(e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 f}-\frac {2 b (d e-c f) \left (a+b \coth ^{-1}(c+d x)\right ) \log \left (\frac {2}{1-c-d x}\right )}{d^2}+\frac {b^2 f \log \left (1-(c+d x)^2\right )}{2 d^2}-\frac {b^2 (d e-c f) \operatorname {PolyLog}\left (2,1-\frac {2}{1-c-d x}\right )}{d^2} \\ \end{align*}
Time = 0.92 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.33 \[ \int (e+f x) \left (a+b \coth ^{-1}(c+d x)\right )^2 \, dx=\frac {2 a^2 c d e+2 a b c f-a^2 c^2 f+2 a^2 d^2 e x+2 a b d f x+a^2 d^2 f x^2+b^2 (-1+c+d x) (2 d e+f-c f+d f x) \coth ^{-1}(c+d x)^2+2 b \coth ^{-1}(c+d x) \left (-((c+d x) (-b f+a c f-a d (2 e+f x)))-2 b (d e-c f) \log \left (1-e^{-2 \coth ^{-1}(c+d x)}\right )\right )+a b f \log (1-c-d x)-a b f \log (1+c+d x)-4 a b d e \log \left (\frac {1}{(c+d x) \sqrt {1-\frac {1}{(c+d x)^2}}}\right )-2 b^2 f \log \left (\frac {1}{(c+d x) \sqrt {1-\frac {1}{(c+d x)^2}}}\right )+4 a b c f \log \left (\frac {1}{(c+d x) \sqrt {1-\frac {1}{(c+d x)^2}}}\right )+2 b^2 (d e-c f) \operatorname {PolyLog}\left (2,e^{-2 \coth ^{-1}(c+d x)}\right )}{2 d^2} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(449\) vs. \(2(217)=434\).
Time = 0.36 (sec) , antiderivative size = 450, normalized size of antiderivative = 2.04
method | result | size |
parts | \(a^{2} \left (\frac {1}{2} f \,x^{2}+e x \right )+\frac {b^{2} \left (\frac {\operatorname {arccoth}\left (d x +c \right )^{2} \left (d x +c \right )^{2} f}{2 d}-\frac {\operatorname {arccoth}\left (d x +c \right )^{2} c f \left (d x +c \right )}{d}+\operatorname {arccoth}\left (d x +c \right )^{2} e \left (d x +c \right )+\frac {\operatorname {arccoth}\left (d x +c \right ) \left (d x +c \right ) f -\operatorname {arccoth}\left (d x +c \right ) \ln \left (d x +c -1\right ) c f +\operatorname {arccoth}\left (d x +c \right ) \ln \left (d x +c -1\right ) d e +\frac {\operatorname {arccoth}\left (d x +c \right ) \ln \left (d x +c -1\right ) f}{2}-\operatorname {arccoth}\left (d x +c \right ) \ln \left (d x +c +1\right ) c f +\operatorname {arccoth}\left (d x +c \right ) \ln \left (d x +c +1\right ) d e -\frac {\operatorname {arccoth}\left (d x +c \right ) \ln \left (d x +c +1\right ) f}{2}+\frac {\ln \left (d x +c -1\right ) f}{2}+\frac {\ln \left (d x +c +1\right ) f}{2}+\frac {\left (-2 c f +2 d e +f \right ) \left (\frac {\ln \left (d x +c -1\right )^{2}}{4}-\frac {\operatorname {dilog}\left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (d x +c -1\right ) \ln \left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )}{2}\right )}{2}+\frac {\left (-2 c f +2 d e -f \right ) \left (-\frac {\ln \left (d x +c +1\right )^{2}}{4}+\frac {\left (\ln \left (d x +c +1\right )-\ln \left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {d x}{2}-\frac {c}{2}+\frac {1}{2}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )}{2}\right )}{2}}{d}\right )}{d}+\frac {2 a b \left (\frac {\operatorname {arccoth}\left (d x +c \right ) \left (d x +c \right )^{2} f}{2 d}-\frac {\operatorname {arccoth}\left (d x +c \right ) c f \left (d x +c \right )}{d}+\operatorname {arccoth}\left (d x +c \right ) e \left (d x +c \right )+\frac {f \left (d x +c \right )+\frac {\left (-2 c f +2 d e +f \right ) \ln \left (d x +c -1\right )}{2}-\frac {\left (2 c f -2 d e +f \right ) \ln \left (d x +c +1\right )}{2}}{2 d}\right )}{d}\) | \(450\) |
derivativedivides | \(\frac {-\frac {a^{2} \left (f c \left (d x +c \right )-e d \left (d x +c \right )-\frac {f \left (d x +c \right )^{2}}{2}\right )}{d}-\frac {b^{2} \left (\operatorname {arccoth}\left (d x +c \right )^{2} f c \left (d x +c \right )-\operatorname {arccoth}\left (d x +c \right )^{2} e d \left (d x +c \right )-\frac {\operatorname {arccoth}\left (d x +c \right )^{2} f \left (d x +c \right )^{2}}{2}-\operatorname {arccoth}\left (d x +c \right ) \left (d x +c \right ) f +\operatorname {arccoth}\left (d x +c \right ) \ln \left (d x +c -1\right ) c f -\operatorname {arccoth}\left (d x +c \right ) \ln \left (d x +c -1\right ) d e -\frac {\operatorname {arccoth}\left (d x +c \right ) \ln \left (d x +c -1\right ) f}{2}+\operatorname {arccoth}\left (d x +c \right ) \ln \left (d x +c +1\right ) c f -\operatorname {arccoth}\left (d x +c \right ) \ln \left (d x +c +1\right ) d e +\frac {\operatorname {arccoth}\left (d x +c \right ) \ln \left (d x +c +1\right ) f}{2}+\frac {\left (2 c f -2 d e -f \right ) \left (\frac {\ln \left (d x +c -1\right )^{2}}{4}-\frac {\operatorname {dilog}\left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (d x +c -1\right ) \ln \left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )}{2}\right )}{2}+\frac {\left (2 c f -2 d e +f \right ) \left (-\frac {\ln \left (d x +c +1\right )^{2}}{4}+\frac {\left (\ln \left (d x +c +1\right )-\ln \left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {d x}{2}-\frac {c}{2}+\frac {1}{2}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )}{2}\right )}{2}-\frac {\ln \left (d x +c -1\right ) f}{2}-\frac {\ln \left (d x +c +1\right ) f}{2}\right )}{d}-\frac {2 a b \left (\operatorname {arccoth}\left (d x +c \right ) f c \left (d x +c \right )-\operatorname {arccoth}\left (d x +c \right ) e d \left (d x +c \right )-\frac {\operatorname {arccoth}\left (d x +c \right ) f \left (d x +c \right )^{2}}{2}-\frac {f \left (d x +c \right )}{2}+\frac {\left (2 c f -2 d e -f \right ) \ln \left (d x +c -1\right )}{4}-\frac {\left (-2 c f +2 d e -f \right ) \ln \left (d x +c +1\right )}{4}\right )}{d}}{d}\) | \(462\) |
default | \(\frac {-\frac {a^{2} \left (f c \left (d x +c \right )-e d \left (d x +c \right )-\frac {f \left (d x +c \right )^{2}}{2}\right )}{d}-\frac {b^{2} \left (\operatorname {arccoth}\left (d x +c \right )^{2} f c \left (d x +c \right )-\operatorname {arccoth}\left (d x +c \right )^{2} e d \left (d x +c \right )-\frac {\operatorname {arccoth}\left (d x +c \right )^{2} f \left (d x +c \right )^{2}}{2}-\operatorname {arccoth}\left (d x +c \right ) \left (d x +c \right ) f +\operatorname {arccoth}\left (d x +c \right ) \ln \left (d x +c -1\right ) c f -\operatorname {arccoth}\left (d x +c \right ) \ln \left (d x +c -1\right ) d e -\frac {\operatorname {arccoth}\left (d x +c \right ) \ln \left (d x +c -1\right ) f}{2}+\operatorname {arccoth}\left (d x +c \right ) \ln \left (d x +c +1\right ) c f -\operatorname {arccoth}\left (d x +c \right ) \ln \left (d x +c +1\right ) d e +\frac {\operatorname {arccoth}\left (d x +c \right ) \ln \left (d x +c +1\right ) f}{2}+\frac {\left (2 c f -2 d e -f \right ) \left (\frac {\ln \left (d x +c -1\right )^{2}}{4}-\frac {\operatorname {dilog}\left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (d x +c -1\right ) \ln \left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )}{2}\right )}{2}+\frac {\left (2 c f -2 d e +f \right ) \left (-\frac {\ln \left (d x +c +1\right )^{2}}{4}+\frac {\left (\ln \left (d x +c +1\right )-\ln \left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {d x}{2}-\frac {c}{2}+\frac {1}{2}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )}{2}\right )}{2}-\frac {\ln \left (d x +c -1\right ) f}{2}-\frac {\ln \left (d x +c +1\right ) f}{2}\right )}{d}-\frac {2 a b \left (\operatorname {arccoth}\left (d x +c \right ) f c \left (d x +c \right )-\operatorname {arccoth}\left (d x +c \right ) e d \left (d x +c \right )-\frac {\operatorname {arccoth}\left (d x +c \right ) f \left (d x +c \right )^{2}}{2}-\frac {f \left (d x +c \right )}{2}+\frac {\left (2 c f -2 d e -f \right ) \ln \left (d x +c -1\right )}{4}-\frac {\left (-2 c f +2 d e -f \right ) \ln \left (d x +c +1\right )}{4}\right )}{d}}{d}\) | \(462\) |
risch | \(\frac {a^{2} c e}{d}-\frac {b a f}{d^{2}}+\frac {a^{2} c f}{d^{2}}-\frac {a^{2} f \,c^{2}}{2 d^{2}}+\frac {b^{2} \operatorname {dilog}\left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right ) c f}{d^{2}}-\frac {b^{2} \ln \left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right ) \ln \left (d x +c -1\right ) e}{d}+\frac {b \ln \left (d x +c +1\right ) a e}{d}+\frac {b^{2} \ln \left (d x +c +1\right ) c f}{2 d^{2}}-\frac {b \ln \left (d x +c +1\right ) a f}{2 d^{2}}-\frac {b^{2} f \ln \left (d x +c -1\right )^{2} c^{2}}{8 d^{2}}+\frac {b^{2} f \ln \left (d x +c -1\right )^{2} c}{4 d^{2}}-\frac {b^{2} f \ln \left (d x +c -1\right ) x}{2 d}-\frac {b^{2} f \ln \left (d x +c -1\right ) c}{2 d^{2}}+\frac {\ln \left (d x +c -1\right ) a b e}{d}+\frac {\ln \left (d x +c -1\right )^{2} b^{2} c e}{4 d}+\frac {b a f \ln \left (d x +c -1\right )}{2 d^{2}}-\ln \left (d x +c -1\right ) a b e x -\frac {b a f \ln \left (d x +c -1\right ) x^{2}}{2}-\frac {b^{2} \left (-d^{2} f \,x^{2}-2 d^{2} e x +c^{2} f -2 c d e +2 c f -2 d e +f \right ) \ln \left (d x +c +1\right )^{2}}{8 d^{2}}-\frac {a^{2} e}{d}-\frac {a^{2} f}{2 d^{2}}-\frac {b a f \ln \left (d x +c -1\right ) c}{d^{2}}-\frac {\ln \left (d x +c -1\right ) a b c e}{d}+\frac {b a f \ln \left (d x +c -1\right ) c^{2}}{2 d^{2}}-\frac {b \ln \left (d x +c +1\right ) a \,c^{2} f}{2 d^{2}}+\frac {b \ln \left (d x +c +1\right ) a c e}{d}-\frac {b \ln \left (d x +c +1\right ) a c f}{d^{2}}+\frac {b a c f}{d^{2}}+\frac {b^{2} \ln \left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right ) \ln \left (d x +c -1\right ) c f}{d^{2}}+\left (-\frac {b^{2} x \left (f x +2 e \right ) \ln \left (d x +c -1\right )}{4}-\frac {b \left (-2 a \,d^{2} f \,x^{2}-4 a \,d^{2} e x -\ln \left (d x +c -1\right ) b \,c^{2} f +2 \ln \left (d x +c -1\right ) b c d e +2 \ln \left (d x +c -1\right ) b c f -2 \ln \left (d x +c -1\right ) b d e -2 b d f x -\ln \left (d x +c -1\right ) b f \right )}{4 d^{2}}\right ) \ln \left (d x +c +1\right )-\frac {b^{2} \operatorname {dilog}\left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right ) e}{d}+\frac {b^{2} \ln \left (d x +c +1\right ) f}{2 d^{2}}+\frac {a^{2} f \,x^{2}}{2}+a^{2} e x +\frac {b^{2} f \ln \left (d x +c -1\right )^{2} x^{2}}{8}+\frac {\ln \left (d x +c -1\right )^{2} b^{2} e x}{4}-\frac {\ln \left (d x +c -1\right )^{2} b^{2} e}{4 d}-\frac {b^{2} f \ln \left (d x +c -1\right )^{2}}{8 d^{2}}+\frac {b^{2} f \ln \left (d x +c -1\right )}{2 d^{2}}+\frac {a b f x}{d}\) | \(751\) |
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\[ \int (e+f x) \left (a+b \coth ^{-1}(c+d x)\right )^2 \, dx=\int { {\left (f x + e\right )} {\left (b \operatorname {arcoth}\left (d x + c\right ) + a\right )}^{2} \,d x } \]
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\[ \int (e+f x) \left (a+b \coth ^{-1}(c+d x)\right )^2 \, dx=\int \left (a + b \operatorname {acoth}{\left (c + d x \right )}\right )^{2} \left (e + f x\right )\, dx \]
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Time = 0.41 (sec) , antiderivative size = 400, normalized size of antiderivative = 1.81 \[ \int (e+f x) \left (a+b \coth ^{-1}(c+d x)\right )^2 \, dx=\frac {1}{2} \, a^{2} f x^{2} + \frac {1}{2} \, {\left (2 \, x^{2} \operatorname {arcoth}\left (d x + c\right ) + d {\left (\frac {2 \, x}{d^{2}} - \frac {{\left (c^{2} + 2 \, c + 1\right )} \log \left (d x + c + 1\right )}{d^{3}} + \frac {{\left (c^{2} - 2 \, c + 1\right )} \log \left (d x + c - 1\right )}{d^{3}}\right )}\right )} a b f + a^{2} e x + \frac {{\left (2 \, {\left (d x + c\right )} \operatorname {arcoth}\left (d x + c\right ) + \log \left (-{\left (d x + c\right )}^{2} + 1\right )\right )} a b e}{d} - \frac {{\left (d e - c f\right )} {\left (\log \left (d x + c - 1\right ) \log \left (\frac {1}{2} \, d x + \frac {1}{2} \, c + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, d x - \frac {1}{2} \, c + \frac {1}{2}\right )\right )} b^{2}}{d^{2}} + \frac {{\left (c f + f\right )} b^{2} \log \left (d x + c + 1\right )}{2 \, d^{2}} + \frac {{\left (b^{2} d^{2} f x^{2} + 2 \, b^{2} d^{2} e x - {\left (c^{2} f - 2 \, {\left (d e - f\right )} c - 2 \, d e + f\right )} b^{2}\right )} \log \left (d x + c + 1\right )^{2} + {\left (b^{2} d^{2} f x^{2} + 2 \, b^{2} d^{2} e x - {\left (c^{2} f - 2 \, {\left (d e + f\right )} c + 2 \, d e + f\right )} b^{2}\right )} \log \left (d x + c - 1\right )^{2} + 2 \, {\left (2 \, b^{2} d f x - {\left (b^{2} d^{2} f x^{2} + 2 \, b^{2} d^{2} e x - {\left (c^{2} f - 2 \, {\left (d e + f\right )} c + 2 \, d e + f\right )} b^{2}\right )} \log \left (d x + c - 1\right )\right )} \log \left (d x + c + 1\right ) - 4 \, {\left (b^{2} d f x + {\left (c f - f\right )} b^{2}\right )} \log \left (d x + c - 1\right )}{8 \, d^{2}} \]
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\[ \int (e+f x) \left (a+b \coth ^{-1}(c+d x)\right )^2 \, dx=\int { {\left (f x + e\right )} {\left (b \operatorname {arcoth}\left (d x + c\right ) + a\right )}^{2} \,d x } \]
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Timed out. \[ \int (e+f x) \left (a+b \coth ^{-1}(c+d x)\right )^2 \, dx=\int \left (e+f\,x\right )\,{\left (a+b\,\mathrm {acoth}\left (c+d\,x\right )\right )}^2 \,d x \]
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