Integrand size = 16, antiderivative size = 97 \[ \int (e+f x) \left (a+b \coth ^{-1}(c+d x)\right ) \, dx=\frac {b f x}{2 d}+\frac {(e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right )}{2 f}+\frac {b (d e+f-c f)^2 \log (1-c-d x)}{4 d^2 f}-\frac {b (d e-(1+c) f)^2 \log (1+c+d x)}{4 d^2 f} \]
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Time = 0.14 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {6247, 6064, 716, 647, 31} \[ \int (e+f x) \left (a+b \coth ^{-1}(c+d x)\right ) \, dx=\frac {(e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right )}{2 f}+\frac {b (-c f+d e+f)^2 \log (-c-d x+1)}{4 d^2 f}-\frac {b (d e-(c+1) f)^2 \log (c+d x+1)}{4 d^2 f}+\frac {b f x}{2 d} \]
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Rule 31
Rule 647
Rule 716
Rule 6064
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 ) \, dx,x,c+d x\right )}{d} \\ & = \frac {(e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right )}{2 f}-\frac {b \text {Subst}\left (\int \frac {\left (\frac {d e-c f}{d}+\frac {f x}{d}\right )^2}{1-x^2} \, dx,x,c+d x\right )}{2 f} \\ & = \frac {(e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right )}{2 f}-\frac {b \text {Subst}\left (\int \left (-\frac {f^2}{d^2}+\frac {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2+2 f (d e-c f) x}{d^2 \left (1-x^2\right )}\right ) \, dx,x,c+d x\right )}{2 f} \\ & = \frac {b f x}{2 d}+\frac {(e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right )}{2 f}-\frac {b \text {Subst}\left (\int \frac {d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2+2 f (d e-c f) x}{1-x^2} \, dx,x,c+d x\right )}{2 d^2 f} \\ & = \frac {b f x}{2 d}+\frac {(e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right )}{2 f}-\frac {\left (b (d e+f-c f)^2\right ) \text {Subst}\left (\int \frac {1}{1-x} \, dx,x,c+d x\right )}{4 d^2 f}+\frac {\left (b (d e-(1+c) f)^2\right ) \text {Subst}\left (\int \frac {1}{-1-x} \, dx,x,c+d x\right )}{4 d^2 f} \\ & = \frac {b f x}{2 d}+\frac {(e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right )}{2 f}+\frac {b (d e+f-c f)^2 \log (1-c-d x)}{4 d^2 f}-\frac {b (d e-(1+c) f)^2 \log (1+c+d x)}{4 d^2 f} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.42 \[ \int (e+f x) \left (a+b \coth ^{-1}(c+d x)\right ) \, dx=a e x+\frac {b f x}{2 d}+\frac {1}{2} a f x^2+b e x \coth ^{-1}(c+d x)+\frac {1}{2} b f x^2 \coth ^{-1}(c+d x)+\frac {b \left (1-2 c+c^2\right ) f \log (1-c-d x)}{4 d^2}+\frac {b \left (-1-2 c-c^2\right ) f \log (1+c+d x)}{4 d^2}+\frac {b e (-((-1+c) \log (1-c-d x))+(1+c) \log (1+c+d x))}{2 d} \]
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Time = 0.18 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.25
method | result | size |
parts | \(a \left (\frac {1}{2} f \,x^{2}+e x \right )+\frac {b \left (\frac {\operatorname {arccoth}\left (d x +c \right ) \left (d x +c \right )^{2} f}{2 d}-\frac {\operatorname {arccoth}\left (d x +c \right ) c f \left (d x +c \right )}{d}+\operatorname {arccoth}\left (d x +c \right ) e \left (d x +c \right )+\frac {f \left (d x +c \right )+\frac {\left (-2 c f +2 d e +f \right ) \ln \left (d x +c -1\right )}{2}-\frac {\left (2 c f -2 d e +f \right ) \ln \left (d x +c +1\right )}{2}}{2 d}\right )}{d}\) | \(121\) |
derivativedivides | \(\frac {-\frac {a \left (f c \left (d x +c \right )-e d \left (d x +c \right )-\frac {f \left (d x +c \right )^{2}}{2}\right )}{d}-\frac {b \left (\operatorname {arccoth}\left (d x +c \right ) f c \left (d x +c \right )-\operatorname {arccoth}\left (d x +c \right ) e d \left (d x +c \right )-\frac {\operatorname {arccoth}\left (d x +c \right ) f \left (d x +c \right )^{2}}{2}-\frac {f \left (d x +c \right )}{2}+\frac {\left (2 c f -2 d e -f \right ) \ln \left (d x +c -1\right )}{4}-\frac {\left (-2 c f +2 d e -f \right ) \ln \left (d x +c +1\right )}{4}\right )}{d}}{d}\) | \(142\) |
default | \(\frac {-\frac {a \left (f c \left (d x +c \right )-e d \left (d x +c \right )-\frac {f \left (d x +c \right )^{2}}{2}\right )}{d}-\frac {b \left (\operatorname {arccoth}\left (d x +c \right ) f c \left (d x +c \right )-\operatorname {arccoth}\left (d x +c \right ) e d \left (d x +c \right )-\frac {\operatorname {arccoth}\left (d x +c \right ) f \left (d x +c \right )^{2}}{2}-\frac {f \left (d x +c \right )}{2}+\frac {\left (2 c f -2 d e -f \right ) \ln \left (d x +c -1\right )}{4}-\frac {\left (-2 c f +2 d e -f \right ) \ln \left (d x +c +1\right )}{4}\right )}{d}}{d}\) | \(142\) |
parallelrisch | \(-\frac {-\operatorname {arccoth}\left (d x +c \right ) b \,d^{2} f \,x^{2}-a \,d^{2} f \,x^{2}-2 x \,\operatorname {arccoth}\left (d x +c \right ) b \,d^{2} e -2 a \,d^{2} e x +\operatorname {arccoth}\left (d x +c \right ) b \,c^{2} f -2 \,\operatorname {arccoth}\left (d x +c \right ) b c d e +2 \ln \left (d x +c -1\right ) b c f -2 \ln \left (d x +c -1\right ) b d e -b d f x +2 \,\operatorname {arccoth}\left (d x +c \right ) b c f -2 \,\operatorname {arccoth}\left (d x +c \right ) b d e +a \,c^{2} f +4 a c d e +\operatorname {arccoth}\left (d x +c \right ) b f +2 b c f -a f}{2 d^{2}}\) | \(161\) |
risch | \(\frac {b x \left (f x +2 e \right ) \ln \left (d x +c +1\right )}{4}-\frac {b f \,x^{2} \ln \left (d x +c -1\right )}{4}-\frac {b e x \ln \left (d x +c -1\right )}{2}+\frac {a f \,x^{2}}{2}+\frac {\ln \left (-d x -c +1\right ) b \,c^{2} f}{4 d^{2}}-\frac {\ln \left (-d x -c +1\right ) b c e}{2 d}-\frac {\ln \left (d x +c +1\right ) b \,c^{2} f}{4 d^{2}}+\frac {\ln \left (d x +c +1\right ) b c e}{2 d}+a e x -\frac {\ln \left (-d x -c +1\right ) b c f}{2 d^{2}}+\frac {\ln \left (-d x -c +1\right ) b e}{2 d}-\frac {\ln \left (d x +c +1\right ) b c f}{2 d^{2}}+\frac {\ln \left (d x +c +1\right ) b e}{2 d}+\frac {b f x}{2 d}+\frac {\ln \left (-d x -c +1\right ) b f}{4 d^{2}}-\frac {\ln \left (d x +c +1\right ) b f}{4 d^{2}}\) | \(230\) |
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Time = 0.26 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.37 \[ \int (e+f x) \left (a+b \coth ^{-1}(c+d x)\right ) \, dx=\frac {2 \, a d^{2} f x^{2} + 2 \, {\left (2 \, a d^{2} e + b d f\right )} x + {\left (2 \, {\left (b c + b\right )} d e - {\left (b c^{2} + 2 \, b c + b\right )} f\right )} \log \left (d x + c + 1\right ) - {\left (2 \, {\left (b c - b\right )} d e - {\left (b c^{2} - 2 \, b c + b\right )} f\right )} \log \left (d x + c - 1\right ) + {\left (b d^{2} f x^{2} + 2 \, b d^{2} e x\right )} \log \left (\frac {d x + c + 1}{d x + c - 1}\right )}{4 \, d^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 173 vs. \(2 (82) = 164\).
Time = 0.41 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.78 \[ \int (e+f x) \left (a+b \coth ^{-1}(c+d x)\right ) \, dx=\begin {cases} a e x + \frac {a f x^{2}}{2} - \frac {b c^{2} f \operatorname {acoth}{\left (c + d x \right )}}{2 d^{2}} + \frac {b c e \operatorname {acoth}{\left (c + d x \right )}}{d} - \frac {b c f \log {\left (\frac {c}{d} + x + \frac {1}{d} \right )}}{d^{2}} + \frac {b c f \operatorname {acoth}{\left (c + d x \right )}}{d^{2}} + b e x \operatorname {acoth}{\left (c + d x \right )} + \frac {b f x^{2} \operatorname {acoth}{\left (c + d x \right )}}{2} + \frac {b e \log {\left (\frac {c}{d} + x + \frac {1}{d} \right )}}{d} - \frac {b e \operatorname {acoth}{\left (c + d x \right )}}{d} + \frac {b f x}{2 d} - \frac {b f \operatorname {acoth}{\left (c + d x \right )}}{2 d^{2}} & \text {for}\: d \neq 0 \\\left (a + b \operatorname {acoth}{\left (c \right )}\right ) \left (e x + \frac {f x^{2}}{2}\right ) & \text {otherwise} \end {cases} \]
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Time = 0.21 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.12 \[ \int (e+f x) \left (a+b \coth ^{-1}(c+d x)\right ) \, dx=\frac {1}{2} \, a f x^{2} + \frac {1}{4} \, {\left (2 \, x^{2} \operatorname {arcoth}\left (d x + c\right ) + d {\left (\frac {2 \, x}{d^{2}} - \frac {{\left (c^{2} + 2 \, c + 1\right )} \log \left (d x + c + 1\right )}{d^{3}} + \frac {{\left (c^{2} - 2 \, c + 1\right )} \log \left (d x + c - 1\right )}{d^{3}}\right )}\right )} b f + a e x + \frac {{\left (2 \, {\left (d x + c\right )} \operatorname {arcoth}\left (d x + c\right ) + \log \left (-{\left (d x + c\right )}^{2} + 1\right )\right )} b e}{2 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 338 vs. \(2 (89) = 178\).
Time = 0.28 (sec) , antiderivative size = 338, normalized size of antiderivative = 3.48 \[ \int (e+f x) \left (a+b \coth ^{-1}(c+d x)\right ) \, dx=\frac {1}{2} \, {\left ({\left (c + 1\right )} d - {\left (c - 1\right )} d\right )} {\left (\frac {{\left (\frac {{\left (d x + c + 1\right )} b d e}{d x + c - 1} - b d e - \frac {{\left (d x + c + 1\right )} b c f}{d x + c - 1} + b c f + \frac {{\left (d x + c + 1\right )} b f}{d x + c - 1}\right )} \log \left (\frac {d x + c + 1}{d x + c - 1}\right )}{\frac {{\left (d x + c + 1\right )}^{2} d^{3}}{{\left (d x + c - 1\right )}^{2}} - \frac {2 \, {\left (d x + c + 1\right )} d^{3}}{d x + c - 1} + d^{3}} + \frac {\frac {2 \, {\left (d x + c + 1\right )} a d e}{d x + c - 1} - 2 \, a d e - \frac {2 \, {\left (d x + c + 1\right )} a c f}{d x + c - 1} + 2 \, a c f + \frac {2 \, {\left (d x + c + 1\right )} a f}{d x + c - 1} + \frac {{\left (d x + c + 1\right )} b f}{d x + c - 1} - b f}{\frac {{\left (d x + c + 1\right )}^{2} d^{3}}{{\left (d x + c - 1\right )}^{2}} - \frac {2 \, {\left (d x + c + 1\right )} d^{3}}{d x + c - 1} + d^{3}} - \frac {{\left (b d e - b c f\right )} \log \left (\frac {d x + c + 1}{d x + c - 1} - 1\right )}{d^{3}} + \frac {{\left (b d e - b c f\right )} \log \left (\frac {d x + c + 1}{d x + c - 1}\right )}{d^{3}}\right )} \]
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Time = 5.24 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.40 \[ \int (e+f x) \left (a+b \coth ^{-1}(c+d x)\right ) \, dx=a\,e\,x+\frac {a\,f\,x^2}{2}+\frac {b\,e\,\ln \left (c^2+2\,c\,d\,x+d^2\,x^2-1\right )}{2\,d}-\frac {b\,f\,\mathrm {acoth}\left (c+d\,x\right )}{2\,d^2}+\frac {b\,f\,x^2\,\mathrm {acoth}\left (c+d\,x\right )}{2}+\frac {b\,f\,x}{2\,d}+b\,e\,x\,\mathrm {acoth}\left (c+d\,x\right )-\frac {b\,c^2\,f\,\mathrm {acoth}\left (c+d\,x\right )}{2\,d^2}-\frac {b\,c\,f\,\ln \left (c^2+2\,c\,d\,x+d^2\,x^2-1\right )}{2\,d^2}+\frac {b\,c\,e\,\mathrm {acoth}\left (c+d\,x\right )}{d} \]
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