Integrand size = 18, antiderivative size = 227 \[ \int \frac {a+b \arctan (c+d x)}{(e+f x)^3} \, dx=-\frac {b d}{2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right ) (e+f x)}+\frac {b d^2 (d e+f-c f) (d e-(1+c) f) \arctan (c+d x)}{2 f \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )^2}-\frac {a+b \arctan (c+d x)}{2 f (e+f x)^2}+\frac {b d^2 (d e-c f) \log (e+f x)}{\left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )^2}-\frac {b d^2 (d e-c f) \log \left (1+c^2+2 c d x+d^2 x^2\right )}{2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )^2} \]
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Time = 0.21 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {5153, 2007, 723, 814, 648, 632, 210, 642} \[ \int \frac {a+b \arctan (c+d x)}{(e+f x)^3} \, dx=-\frac {a+b \arctan (c+d x)}{2 f (e+f x)^2}+\frac {b d^2 \arctan (c+d x) (-c f+d e+f) (d e-(c+1) f)}{2 f \left (\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2\right )^2}-\frac {b d^2 (d e-c f) \log \left (c^2+2 c d x+d^2 x^2+1\right )}{2 \left (\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2\right )^2}-\frac {b d}{2 (e+f x) \left (\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2\right )}+\frac {b d^2 (d e-c f) \log (e+f x)}{\left (\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2\right )^2} \]
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Rule 210
Rule 632
Rule 642
Rule 648
Rule 723
Rule 814
Rule 2007
Rule 5153
Rubi steps \begin{align*} \text {integral}& = -\frac {a+b \arctan (c+d x)}{2 f (e+f x)^2}+\frac {(b d) \int \frac {1}{(e+f x)^2 \left (1+(c+d x)^2\right )} \, dx}{2 f} \\ & = -\frac {a+b \arctan (c+d x)}{2 f (e+f x)^2}+\frac {(b d) \int \frac {1}{(e+f x)^2 \left (1+c^2+2 c d x+d^2 x^2\right )} \, dx}{2 f} \\ & = -\frac {b d}{2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right ) (e+f x)}-\frac {a+b \arctan (c+d x)}{2 f (e+f x)^2}+\frac {(b d) \int \frac {d (d e-2 c f)-d^2 f x}{(e+f x) \left (1+c^2+2 c d x+d^2 x^2\right )} \, dx}{2 f \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )} \\ & = -\frac {b d}{2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right ) (e+f x)}-\frac {a+b \arctan (c+d x)}{2 f (e+f x)^2}+\frac {(b d) \int \left (\frac {2 d f^2 (d e-c f)}{\left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right ) (e+f x)}+\frac {d^2 \left (d^2 e^2-4 c d e f-\left (1-3 c^2\right ) f^2-2 d f (d e-c f) x\right )}{\left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right ) \left (1+c^2+2 c d x+d^2 x^2\right )}\right ) \, dx}{2 f \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )} \\ & = -\frac {b d}{2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right ) (e+f x)}-\frac {a+b \arctan (c+d x)}{2 f (e+f x)^2}+\frac {b d^2 (d e-c f) \log (e+f x)}{\left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )^2}+\frac {\left (b d^3\right ) \int \frac {d^2 e^2-4 c d e f-\left (1-3 c^2\right ) f^2-2 d f (d e-c f) x}{1+c^2+2 c d x+d^2 x^2} \, dx}{2 f \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )^2} \\ & = -\frac {b d}{2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right ) (e+f x)}-\frac {a+b \arctan (c+d x)}{2 f (e+f x)^2}+\frac {b d^2 (d e-c f) \log (e+f x)}{\left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )^2}-\frac {\left (b d^2 (d e-c f)\right ) \int \frac {2 c d+2 d^2 x}{1+c^2+2 c d x+d^2 x^2} \, dx}{2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )^2}+\frac {\left (b d \left (4 c d^2 f (d e-c f)+2 d^2 \left (d^2 e^2-4 c d e f-\left (1-3 c^2\right ) f^2\right )\right )\right ) \int \frac {1}{1+c^2+2 c d x+d^2 x^2} \, dx}{4 f \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )^2} \\ & = -\frac {b d}{2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right ) (e+f x)}-\frac {a+b \arctan (c+d x)}{2 f (e+f x)^2}+\frac {b d^2 (d e-c f) \log (e+f x)}{\left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )^2}-\frac {b d^2 (d e-c f) \log \left (1+c^2+2 c d x+d^2 x^2\right )}{2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )^2}-\frac {\left (b d \left (4 c d^2 f (d e-c f)+2 d^2 \left (d^2 e^2-4 c d e f-\left (1-3 c^2\right ) f^2\right )\right )\right ) \text {Subst}\left (\int \frac {1}{-4 d^2-x^2} \, dx,x,2 c d+2 d^2 x\right )}{2 f \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )^2} \\ & = -\frac {b d}{2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right ) (e+f x)}+\frac {b d^2 (d e-f-c f) (d e+f-c f) \arctan (c+d x)}{2 f \left (d^2 e^2-2 c d e f+f^2+c^2 f^2\right )^2}-\frac {a+b \arctan (c+d x)}{2 f (e+f x)^2}+\frac {b d^2 (d e-c f) \log (e+f x)}{\left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )^2}-\frac {b d^2 (d e-c f) \log \left (1+c^2+2 c d x+d^2 x^2\right )}{2 \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )^2} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.68 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.77 \[ \int \frac {a+b \arctan (c+d x)}{(e+f x)^3} \, dx=\frac {-\frac {a+b \arctan (c+d x)}{(e+f x)^2}+\frac {1}{2} b d^2 \left (-\frac {2 f}{d \left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right ) (e+f x)}-\frac {i \log (i-c-d x)}{(d e-(-i+c) f)^2}+\frac {i \log (i+c+d x)}{(d e-(i+c) f)^2}-\frac {4 f (-d e+c f) \log (d (e+f x))}{\left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right )^2}\right )}{2 f} \]
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Time = 0.60 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.08
method | result | size |
parts | \(-\frac {a}{2 \left (f x +e \right )^{2} f}+\frac {b \left (-\frac {d^{3} \arctan \left (d x +c \right )}{2 \left (f \left (d x +c \right )-c f +d e \right )^{2} f}+\frac {d^{3} \left (-\frac {f}{\left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}\right ) \left (f \left (d x +c \right )-c f +d e \right )}-\frac {2 \left (c f -d e \right ) f \ln \left (f \left (d x +c \right )-c f +d e \right )}{\left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}\right )^{2}}+\frac {\frac {\left (2 c \,f^{2}-2 d e f \right ) \ln \left (1+\left (d x +c \right )^{2}\right )}{2}+\left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}-f^{2}\right ) \arctan \left (d x +c \right )}{\left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}\right )^{2}}\right )}{2 f}\right )}{d}\) | \(245\) |
derivativedivides | \(\frac {-\frac {a \,d^{3}}{2 \left (c f -d e -f \left (d x +c \right )\right )^{2} f}-b \,d^{3} \left (\frac {\arctan \left (d x +c \right )}{2 \left (c f -d e -f \left (d x +c \right )\right )^{2} f}-\frac {\frac {f}{\left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}\right ) \left (c f -d e -f \left (d x +c \right )\right )}-\frac {2 f \left (c f -d e \right ) \ln \left (c f -d e -f \left (d x +c \right )\right )}{\left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}\right )^{2}}+\frac {\frac {\left (2 c \,f^{2}-2 d e f \right ) \ln \left (1+\left (d x +c \right )^{2}\right )}{2}+\left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}-f^{2}\right ) \arctan \left (d x +c \right )}{\left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}\right )^{2}}}{2 f}\right )}{d}\) | \(260\) |
default | \(\frac {-\frac {a \,d^{3}}{2 \left (c f -d e -f \left (d x +c \right )\right )^{2} f}-b \,d^{3} \left (\frac {\arctan \left (d x +c \right )}{2 \left (c f -d e -f \left (d x +c \right )\right )^{2} f}-\frac {\frac {f}{\left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}\right ) \left (c f -d e -f \left (d x +c \right )\right )}-\frac {2 f \left (c f -d e \right ) \ln \left (c f -d e -f \left (d x +c \right )\right )}{\left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}\right )^{2}}+\frac {\frac {\left (2 c \,f^{2}-2 d e f \right ) \ln \left (1+\left (d x +c \right )^{2}\right )}{2}+\left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}-f^{2}\right ) \arctan \left (d x +c \right )}{\left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}+f^{2}\right )^{2}}}{2 f}\right )}{d}\) | \(260\) |
parallelrisch | \(-\frac {2 x^{2} \arctan \left (d x +c \right ) b c \,d^{5} e \,f^{4}-2 x \arctan \left (d x +c \right ) b \,c^{2} d^{4} e \,f^{4}+4 x \arctan \left (d x +c \right ) b c \,d^{5} e^{2} f^{3}+4 \ln \left (f x +e \right ) x b c \,d^{4} e \,f^{4}-2 \ln \left (d^{2} x^{2}+2 c d x +c^{2}+1\right ) x b c \,d^{4} e \,f^{4}+x b \,d^{3} f^{5}+\arctan \left (d x +c \right ) b \,d^{2} f^{5}+x^{2} \arctan \left (d x +c \right ) b \,d^{4} f^{5}+\arctan \left (d x +c \right ) b \,c^{4} d^{2} f^{5}+x b \,c^{2} d^{3} f^{5}+x b \,d^{5} e^{2} f^{3}+2 \arctan \left (d x +c \right ) b \,c^{2} d^{2} f^{5}+3 \arctan \left (d x +c \right ) b \,d^{4} e^{2} f^{3}-2 \ln \left (f x +e \right ) b \,d^{5} e^{3} f^{2}+\ln \left (d^{2} x^{2}+2 c d x +c^{2}+1\right ) b \,d^{5} e^{3} f^{2}+2 a \,d^{4} e^{2} f^{3}+b \,d^{3} e \,f^{4}+a \,c^{4} d^{2} f^{5}+2 a \,c^{2} d^{2} f^{5}+a \,d^{2} f^{5}-4 \arctan \left (d x +c \right ) b \,c^{3} d^{3} e \,f^{4}+5 \arctan \left (d x +c \right ) b \,c^{2} d^{4} e^{2} f^{3}-2 \arctan \left (d x +c \right ) b c \,d^{5} e^{3} f^{2}+2 x \arctan \left (d x +c \right ) b \,d^{4} e \,f^{4}-4 \arctan \left (d x +c \right ) b c \,d^{3} e \,f^{4}-2 x b c \,d^{4} e \,f^{4}+2 \ln \left (f x +e \right ) x^{2} b c \,d^{4} f^{5}-2 \ln \left (f x +e \right ) x^{2} b \,d^{5} e \,f^{4}-\ln \left (d^{2} x^{2}+2 c d x +c^{2}+1\right ) x^{2} b c \,d^{4} f^{5}+\ln \left (d^{2} x^{2}+2 c d x +c^{2}+1\right ) x^{2} b \,d^{5} e \,f^{4}-4 \ln \left (f x +e \right ) x b \,d^{5} e^{2} f^{3}+2 \ln \left (d^{2} x^{2}+2 c d x +c^{2}+1\right ) x b \,d^{5} e^{2} f^{3}+2 \ln \left (f x +e \right ) b c \,d^{4} e^{2} f^{3}-\ln \left (d^{2} x^{2}+2 c d x +c^{2}+1\right ) b c \,d^{4} e^{2} f^{3}-x^{2} \arctan \left (d x +c \right ) b \,c^{2} d^{4} f^{5}-x^{2} \arctan \left (d x +c \right ) b \,d^{6} e^{2} f^{3}-2 x \arctan \left (d x +c \right ) b \,d^{6} e^{3} f^{2}-4 a \,c^{3} d^{3} e \,f^{4}+6 a \,c^{2} d^{4} e^{2} f^{3}+b \,c^{2} d^{3} e \,f^{4}-4 a c \,d^{3} e \,f^{4}-4 a c \,d^{5} e^{3} f^{2}-2 b c \,d^{4} e^{2} f^{3}+a \,e^{4} f \,d^{6}+b \,e^{3} f^{2} d^{5}}{2 \left (f x +e \right )^{2} \left (c^{4} f^{4}-4 c^{3} d e \,f^{3}+6 c^{2} d^{2} e^{2} f^{2}-4 c \,d^{3} e^{3} f +d^{4} e^{4}+2 c^{2} f^{4}-4 c d e \,f^{3}+2 e^{2} f^{2} d^{2}+f^{4}\right ) f^{2} d^{2}}\) | \(884\) |
risch | \(\text {Expression too large to display}\) | \(13218\) |
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Leaf count of result is larger than twice the leaf count of optimal. 682 vs. \(2 (219) = 438\).
Time = 1.22 (sec) , antiderivative size = 682, normalized size of antiderivative = 3.00 \[ \int \frac {a+b \arctan (c+d x)}{(e+f x)^3} \, dx=-\frac {a d^{4} e^{4} - {\left (4 \, a c - b\right )} d^{3} e^{3} f + 2 \, {\left (3 \, a c^{2} - b c + a\right )} d^{2} e^{2} f^{2} - {\left (4 \, a c^{3} - b c^{2} + 4 \, a c - b\right )} d e f^{3} + {\left (a c^{4} + 2 \, a c^{2} + a\right )} f^{4} + {\left (b d^{3} e^{2} f^{2} - 2 \, b c d^{2} e f^{3} + {\left (b c^{2} + b\right )} d f^{4}\right )} x - {\left (2 \, b c d^{3} e^{3} f - {\left (5 \, b c^{2} + 3 \, b\right )} d^{2} e^{2} f^{2} + 4 \, {\left (b c^{3} + b c\right )} d e f^{3} - {\left (b c^{4} + 2 \, b c^{2} + b\right )} f^{4} + {\left (b d^{4} e^{2} f^{2} - 2 \, b c d^{3} e f^{3} + {\left (b c^{2} - b\right )} d^{2} f^{4}\right )} x^{2} + 2 \, {\left (b d^{4} e^{3} f - 2 \, b c d^{3} e^{2} f^{2} + {\left (b c^{2} - b\right )} d^{2} e f^{3}\right )} x\right )} \arctan \left (d x + c\right ) + {\left (b d^{3} e^{3} f - b c d^{2} e^{2} f^{2} + {\left (b d^{3} e f^{3} - b c d^{2} f^{4}\right )} x^{2} + 2 \, {\left (b d^{3} e^{2} f^{2} - b c d^{2} e f^{3}\right )} x\right )} \log \left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right ) - 2 \, {\left (b d^{3} e^{3} f - b c d^{2} e^{2} f^{2} + {\left (b d^{3} e f^{3} - b c d^{2} f^{4}\right )} x^{2} + 2 \, {\left (b d^{3} e^{2} f^{2} - b c d^{2} e f^{3}\right )} x\right )} \log \left (f x + e\right )}{2 \, {\left (d^{4} e^{6} f - 4 \, c d^{3} e^{5} f^{2} + 2 \, {\left (3 \, c^{2} + 1\right )} d^{2} e^{4} f^{3} - 4 \, {\left (c^{3} + c\right )} d e^{3} f^{4} + {\left (c^{4} + 2 \, c^{2} + 1\right )} e^{2} f^{5} + {\left (d^{4} e^{4} f^{3} - 4 \, c d^{3} e^{3} f^{4} + 2 \, {\left (3 \, c^{2} + 1\right )} d^{2} e^{2} f^{5} - 4 \, {\left (c^{3} + c\right )} d e f^{6} + {\left (c^{4} + 2 \, c^{2} + 1\right )} f^{7}\right )} x^{2} + 2 \, {\left (d^{4} e^{5} f^{2} - 4 \, c d^{3} e^{4} f^{3} + 2 \, {\left (3 \, c^{2} + 1\right )} d^{2} e^{3} f^{4} - 4 \, {\left (c^{3} + c\right )} d e^{2} f^{5} + {\left (c^{4} + 2 \, c^{2} + 1\right )} e f^{6}\right )} x\right )}} \]
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Timed out. \[ \int \frac {a+b \arctan (c+d x)}{(e+f x)^3} \, dx=\text {Timed out} \]
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Time = 0.27 (sec) , antiderivative size = 409, normalized size of antiderivative = 1.80 \[ \int \frac {a+b \arctan (c+d x)}{(e+f x)^3} \, dx=-\frac {1}{2} \, {\left (d {\left (\frac {{\left (d^{2} e - c d f\right )} \log \left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )}{d^{4} e^{4} - 4 \, c d^{3} e^{3} f + 2 \, {\left (3 \, c^{2} + 1\right )} d^{2} e^{2} f^{2} - 4 \, {\left (c^{3} + c\right )} d e f^{3} + {\left (c^{4} + 2 \, c^{2} + 1\right )} f^{4}} - \frac {2 \, {\left (d^{2} e - c d f\right )} \log \left (f x + e\right )}{d^{4} e^{4} - 4 \, c d^{3} e^{3} f + 2 \, {\left (3 \, c^{2} + 1\right )} d^{2} e^{2} f^{2} - 4 \, {\left (c^{3} + c\right )} d e f^{3} + {\left (c^{4} + 2 \, c^{2} + 1\right )} f^{4}} - \frac {{\left (d^{4} e^{2} - 2 \, c d^{3} e f + {\left (c^{2} - 1\right )} d^{2} f^{2}\right )} \arctan \left (\frac {d^{2} x + c d}{d}\right )}{{\left (d^{4} e^{4} f - 4 \, c d^{3} e^{3} f^{2} + 2 \, {\left (3 \, c^{2} + 1\right )} d^{2} e^{2} f^{3} - 4 \, {\left (c^{3} + c\right )} d e f^{4} + {\left (c^{4} + 2 \, c^{2} + 1\right )} f^{5}\right )} d} + \frac {1}{d^{2} e^{3} - 2 \, c d e^{2} f + {\left (c^{2} + 1\right )} e f^{2} + {\left (d^{2} e^{2} f - 2 \, c d e f^{2} + {\left (c^{2} + 1\right )} f^{3}\right )} x}\right )} + \frac {\arctan \left (d x + c\right )}{f^{3} x^{2} + 2 \, e f^{2} x + e^{2} f}\right )} b - \frac {a}{2 \, {\left (f^{3} x^{2} + 2 \, e f^{2} x + e^{2} f\right )}} \]
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\[ \int \frac {a+b \arctan (c+d x)}{(e+f x)^3} \, dx=\int { \frac {b \arctan \left (d x + c\right ) + a}{{\left (f x + e\right )}^{3}} \,d x } \]
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Time = 8.02 (sec) , antiderivative size = 399, normalized size of antiderivative = 1.76 \[ \int \frac {a+b \arctan (c+d x)}{(e+f x)^3} \, dx=\frac {b\,d^3\,e\,\ln \left (e+f\,x\right )}{{\left (c^2\,f^2-2\,c\,d\,e\,f+d^2\,e^2+f^2\right )}^2}-\frac {a\,f}{2\,{\left (e+f\,x\right )}^2\,\left (c^2\,f^2-2\,c\,d\,e\,f+d^2\,e^2+f^2\right )}-\frac {b\,d\,e}{2\,{\left (e+f\,x\right )}^2\,\left (c^2\,f^2-2\,c\,d\,e\,f+d^2\,e^2+f^2\right )}-\frac {a\,c^2\,f}{2\,{\left (e+f\,x\right )}^2\,\left (c^2\,f^2-2\,c\,d\,e\,f+d^2\,e^2+f^2\right )}-\frac {b\,\mathrm {atan}\left (c+d\,x\right )}{2\,f\,{\left (e+f\,x\right )}^2}-\frac {b\,c\,d^2\,f\,\ln \left (e+f\,x\right )}{{\left (c^2\,f^2-2\,c\,d\,e\,f+d^2\,e^2+f^2\right )}^2}+\frac {a\,c\,d\,e}{{\left (e+f\,x\right )}^2\,\left (c^2\,f^2-2\,c\,d\,e\,f+d^2\,e^2+f^2\right )}-\frac {b\,d\,f\,x}{2\,{\left (e+f\,x\right )}^2\,\left (c^2\,f^2-2\,c\,d\,e\,f+d^2\,e^2+f^2\right )}-\frac {a\,d^2\,e^2}{2\,f\,{\left (e+f\,x\right )}^2\,\left (c^2\,f^2-2\,c\,d\,e\,f+d^2\,e^2+f^2\right )}-\frac {b\,d^2\,\ln \left (c+d\,x-\mathrm {i}\right )\,1{}\mathrm {i}}{4\,f\,{\left (d\,e-c\,f+f\,1{}\mathrm {i}\right )}^2}+\frac {b\,d^2\,\ln \left (c+d\,x+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{4\,f\,{\left (c\,f-d\,e+f\,1{}\mathrm {i}\right )}^2} \]
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