Integrand size = 18, antiderivative size = 115 \[ \int \frac {a+b \coth ^{-1}(c+d x)}{(e+f x)^2} \, dx=-\frac {a+b \coth ^{-1}(c+d x)}{f (e+f x)}-\frac {b d \log (1-c-d x)}{2 f (d e+f-c f)}+\frac {b d \log (1+c+d x)}{2 f (d e-f-c f)}-\frac {b d \log (e+f x)}{(d e+f-c f) (d e-(1+c) f)} \]
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Time = 0.14 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {6245, 2007, 719, 31, 646} \[ \int \frac {a+b \coth ^{-1}(c+d x)}{(e+f x)^2} \, dx=-\frac {a+b \coth ^{-1}(c+d x)}{f (e+f x)}-\frac {b d \log (-c-d x+1)}{2 f (-c f+d e+f)}+\frac {b d \log (c+d x+1)}{2 f (-c f+d e-f)}-\frac {b d \log (e+f x)}{(-c f+d e+f) (d e-(c+1) f)} \]
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Rule 31
Rule 646
Rule 719
Rule 2007
Rule 6245
Rubi steps \begin{align*} \text {integral}& = -\frac {a+b \coth ^{-1}(c+d x)}{f (e+f x)}+\frac {(b d) \int \frac {1}{(e+f x) \left (1-(c+d x)^2\right )} \, dx}{f} \\ & = -\frac {a+b \coth ^{-1}(c+d x)}{f (e+f x)}+\frac {(b d) \int \frac {1}{(e+f x) \left (1-c^2-2 c d x-d^2 x^2\right )} \, dx}{f} \\ & = -\frac {a+b \coth ^{-1}(c+d x)}{f (e+f x)}+\frac {(b d) \int \frac {-d^2 e+2 c d f+d^2 f x}{1-c^2-2 c d x-d^2 x^2} \, dx}{f \left (-d^2 e^2+2 c d e f+\left (1-c^2\right ) f^2\right )}+\frac {(b d f) \int \frac {1}{e+f x} \, dx}{-d^2 e^2+2 c d e f+\left (1-c^2\right ) f^2} \\ & = -\frac {a+b \coth ^{-1}(c+d x)}{f (e+f x)}-\frac {b d \log (e+f x)}{(d e-f-c f) (d e+f-c f)}-\frac {\left (b d^3\right ) \int \frac {1}{-d-c d-d^2 x} \, dx}{2 f (d e-f-c f)}+\frac {\left (b d^3\right ) \int \frac {1}{d-c d-d^2 x} \, dx}{2 f (d e+f-c f)} \\ & = -\frac {a+b \coth ^{-1}(c+d x)}{f (e+f x)}-\frac {b d \log (1-c-d x)}{2 f (d e+f-c f)}+\frac {b d \log (1+c+d x)}{2 f (d e-f-c f)}-\frac {b d \log (e+f x)}{(d e-f-c f) (d e+f-c f)} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.09 \[ \int \frac {a+b \coth ^{-1}(c+d x)}{(e+f x)^2} \, dx=\frac {1}{2} \left (-\frac {2 a}{f (e+f x)}-\frac {2 b \coth ^{-1}(c+d x)}{f (e+f x)}+\frac {b d \log (1-c-d x)}{f (-d e+(-1+c) f)}-\frac {b d \log (1+c+d x)}{f (-d e+f+c f)}-\frac {2 b d \log (e+f x)}{d^2 e^2-2 c d e f+\left (-1+c^2\right ) f^2}\right ) \]
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Time = 0.63 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.19
method | result | size |
parts | \(-\frac {a}{\left (f x +e \right ) f}-\frac {b d \,\operatorname {arccoth}\left (d x +c \right )}{\left (d f x +d e \right ) f}-\frac {b d \ln \left (f \left (d x +c \right )-c f +d e \right )}{\left (c f -d e -f \right ) \left (c f -d e +f \right )}+\frac {b d \ln \left (d x +c -1\right )}{f \left (2 c f -2 d e -2 f \right )}-\frac {b d \ln \left (d x +c +1\right )}{f \left (2 c f -2 d e +2 f \right )}\) | \(137\) |
derivativedivides | \(\frac {\frac {a \,d^{2}}{\left (c f -d e -f \left (d x +c \right )\right ) f}+b \,d^{2} \left (\frac {\operatorname {arccoth}\left (d x +c \right )}{\left (c f -d e -f \left (d x +c \right )\right ) f}+\frac {\frac {\ln \left (d x +c -1\right )}{2 c f -2 d e -2 f}-\frac {f \ln \left (c f -d e -f \left (d x +c \right )\right )}{\left (c f -d e -f \right ) \left (c f -d e +f \right )}-\frac {\ln \left (d x +c +1\right )}{2 c f -2 d e +2 f}}{f}\right )}{d}\) | \(160\) |
default | \(\frac {\frac {a \,d^{2}}{\left (c f -d e -f \left (d x +c \right )\right ) f}+b \,d^{2} \left (\frac {\operatorname {arccoth}\left (d x +c \right )}{\left (c f -d e -f \left (d x +c \right )\right ) f}+\frac {\frac {\ln \left (d x +c -1\right )}{2 c f -2 d e -2 f}-\frac {f \ln \left (c f -d e -f \left (d x +c \right )\right )}{\left (c f -d e -f \right ) \left (c f -d e +f \right )}-\frac {\ln \left (d x +c +1\right )}{2 c f -2 d e +2 f}}{f}\right )}{d}\) | \(160\) |
parallelrisch | \(-\frac {a \,d^{4} e^{2}-a \,d^{2} f^{2}-2 a c \,d^{3} e f +a \,c^{2} d^{2} f^{2}-x \,\operatorname {arccoth}\left (d x +c \right ) b \,d^{3} f^{2}+\operatorname {arccoth}\left (d x +c \right ) b \,c^{2} d^{2} f^{2}+x \,\operatorname {arccoth}\left (d x +c \right ) b c \,d^{3} f^{2}-x \,\operatorname {arccoth}\left (d x +c \right ) b \,d^{4} e f -\operatorname {arccoth}\left (d x +c \right ) b c \,d^{3} e f -\operatorname {arccoth}\left (d x +c \right ) b \,d^{2} f^{2}-\operatorname {arccoth}\left (d x +c \right ) b \,d^{3} e f +\ln \left (f x +e \right ) x b \,d^{3} f^{2}-\ln \left (d x +c -1\right ) x b \,d^{3} f^{2}+\ln \left (f x +e \right ) b \,d^{3} e f -\ln \left (d x +c -1\right ) b \,d^{3} e f}{\left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}-f^{2}\right ) \left (f x +e \right ) d^{2} f}\) | \(250\) |
risch | \(-\frac {b \ln \left (d x +c +1\right )}{2 f \left (f x +e \right )}-\frac {\ln \left (d x +c +1\right ) b c d \,f^{2} x -\ln \left (d x +c +1\right ) b \,d^{2} e f x -\ln \left (-d x -c +1\right ) b c d \,f^{2} x +\ln \left (-d x -c +1\right ) b \,d^{2} e f x +2 \ln \left (-f x -e \right ) b d \,f^{2} x +\ln \left (d x +c +1\right ) b c d e f -\ln \left (d x +c +1\right ) b \,d^{2} e^{2}-\ln \left (d x +c +1\right ) b d \,f^{2} x -\ln \left (-d x -c +1\right ) b c d e f +\ln \left (-d x -c +1\right ) b \,d^{2} e^{2}-\ln \left (-d x -c +1\right ) b d \,f^{2} x -\ln \left (d x +c -1\right ) b \,c^{2} f^{2}+2 \ln \left (d x +c -1\right ) b c d e f -\ln \left (d x +c -1\right ) b \,d^{2} e^{2}+2 \ln \left (-f x -e \right ) b d e f -\ln \left (d x +c +1\right ) b d e f -\ln \left (-d x -c +1\right ) b d e f +2 a \,c^{2} f^{2}-4 a c d e f +2 e^{2} a \,d^{2}+\ln \left (d x +c -1\right ) b \,f^{2}-2 f^{2} a}{2 \left (c f -d e -f \right ) \left (c f -d e +f \right ) \left (f x +e \right ) f}\) | \(374\) |
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Leaf count of result is larger than twice the leaf count of optimal. 262 vs. \(2 (112) = 224\).
Time = 0.34 (sec) , antiderivative size = 262, normalized size of antiderivative = 2.28 \[ \int \frac {a+b \coth ^{-1}(c+d x)}{(e+f x)^2} \, dx=-\frac {2 \, a d^{2} e^{2} - 4 \, a c d e f + 2 \, {\left (a c^{2} - a\right )} f^{2} - {\left (b d^{2} e^{2} - {\left (b c - b\right )} d e f + {\left (b d^{2} e f - {\left (b c - b\right )} d f^{2}\right )} x\right )} \log \left (d x + c + 1\right ) + {\left (b d^{2} e^{2} - {\left (b c + b\right )} d e f + {\left (b d^{2} e f - {\left (b c + b\right )} d f^{2}\right )} x\right )} \log \left (d x + c - 1\right ) + 2 \, {\left (b d f^{2} x + b d e f\right )} \log \left (f x + e\right ) + {\left (b d^{2} e^{2} - 2 \, b c d e f + {\left (b c^{2} - b\right )} f^{2}\right )} \log \left (\frac {d x + c + 1}{d x + c - 1}\right )}{2 \, {\left (d^{2} e^{3} f - 2 \, c d e^{2} f^{2} + {\left (c^{2} - 1\right )} e f^{3} + {\left (d^{2} e^{2} f^{2} - 2 \, c d e f^{3} + {\left (c^{2} - 1\right )} f^{4}\right )} x\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1605 vs. \(2 (92) = 184\).
Time = 2.46 (sec) , antiderivative size = 1605, normalized size of antiderivative = 13.96 \[ \int \frac {a+b \coth ^{-1}(c+d x)}{(e+f x)^2} \, dx=\text {Too large to display} \]
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Time = 0.20 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.05 \[ \int \frac {a+b \coth ^{-1}(c+d x)}{(e+f x)^2} \, dx=\frac {1}{2} \, {\left (d {\left (\frac {\log \left (d x + c + 1\right )}{d e f - {\left (c + 1\right )} f^{2}} - \frac {\log \left (d x + c - 1\right )}{d e f - {\left (c - 1\right )} f^{2}} - \frac {2 \, \log \left (f x + e\right )}{d^{2} e^{2} - 2 \, c d e f + {\left (c^{2} - 1\right )} f^{2}}\right )} - \frac {2 \, \operatorname {arcoth}\left (d x + c\right )}{f^{2} x + e f}\right )} b - \frac {a}{f^{2} x + e f} \]
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Leaf count of result is larger than twice the leaf count of optimal. 472 vs. \(2 (112) = 224\).
Time = 0.28 (sec) , antiderivative size = 472, normalized size of antiderivative = 4.10 \[ \int \frac {a+b \coth ^{-1}(c+d x)}{(e+f x)^2} \, dx=-\frac {1}{2} \, {\left ({\left (c + 1\right )} d - {\left (c - 1\right )} d\right )} {\left (\frac {b \log \left (-\frac {{\left (d x + c + 1\right )} d e}{d x + c - 1} + d e + \frac {{\left (d x + c + 1\right )} c f}{d x + c - 1} - c f - \frac {{\left (d x + c + 1\right )} f}{d x + c - 1} - f\right )}{d^{2} e^{2} - 2 \, c d e f + c^{2} f^{2} - f^{2}} - \frac {b \log \left (\frac {d x + c + 1}{d x + c - 1}\right )}{\frac {{\left (d x + c + 1\right )} d^{2} e^{2}}{d x + c - 1} - d^{2} e^{2} - \frac {2 \, {\left (d x + c + 1\right )} c d e f}{d x + c - 1} + 2 \, c d e f + \frac {{\left (d x + c + 1\right )} c^{2} f^{2}}{d x + c - 1} - c^{2} f^{2} + \frac {2 \, {\left (d x + c + 1\right )} d e f}{d x + c - 1} - \frac {2 \, {\left (d x + c + 1\right )} c f^{2}}{d x + c - 1} + \frac {{\left (d x + c + 1\right )} f^{2}}{d x + c - 1} + f^{2}} - \frac {b \log \left (\frac {d x + c + 1}{d x + c - 1}\right )}{d^{2} e^{2} - 2 \, c d e f + c^{2} f^{2} - f^{2}} - \frac {2 \, a}{\frac {{\left (d x + c + 1\right )} d^{2} e^{2}}{d x + c - 1} - d^{2} e^{2} - \frac {2 \, {\left (d x + c + 1\right )} c d e f}{d x + c - 1} + 2 \, c d e f + \frac {{\left (d x + c + 1\right )} c^{2} f^{2}}{d x + c - 1} - c^{2} f^{2} + \frac {2 \, {\left (d x + c + 1\right )} d e f}{d x + c - 1} - \frac {2 \, {\left (d x + c + 1\right )} c f^{2}}{d x + c - 1} + \frac {{\left (d x + c + 1\right )} f^{2}}{d x + c - 1} + f^{2}}\right )} \]
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Time = 5.35 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.52 \[ \int \frac {a+b \coth ^{-1}(c+d x)}{(e+f x)^2} \, dx=\ln \left (e+f\,x\right )\,\left (\frac {b\,\left (c-1\right )}{2\,e\,\left (d\,e-f\,\left (c-1\right )\right )}-\frac {b\,\left (c+1\right )}{2\,e\,\left (d\,e-f\,\left (c+1\right )\right )}\right )-\frac {a}{x\,f^2+e\,f}-\frac {b\,\ln \left (\frac {1}{c+d\,x}+1\right )}{2\,f\,\left (e+f\,x\right )}-\frac {b\,d\,\ln \left (c+d\,x-1\right )}{2\,f^2-2\,c\,f^2+2\,d\,e\,f}-\frac {b\,d\,\ln \left (c+d\,x+1\right )}{2\,c\,f^2+2\,f^2-2\,d\,e\,f}+\frac {b\,\ln \left (1-\frac {1}{c+d\,x}\right )}{f\,\left (2\,e+2\,f\,x\right )} \]
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