Integrand size = 18, antiderivative size = 233 \[ \int (e+f x)^3 \left (a+b \cot ^{-1}(c+d x)\right ) \, dx=\frac {b f \left (6 d^2 e^2-12 c d e f-\left (1-6 c^2\right ) f^2\right ) x}{4 d^3}+\frac {b f^2 (d e-c f) (c+d x)^2}{2 d^4}+\frac {b f^3 (c+d x)^3}{12 d^4}+\frac {(e+f x)^4 \left (a+b \cot ^{-1}(c+d x)\right )}{4 f}+\frac {b \left (d^4 e^4-4 c d^3 e^3 f-6 \left (1-c^2\right ) d^2 e^2 f^2+4 c \left (3-c^2\right ) d e f^3+\left (1-6 c^2+c^4\right ) f^4\right ) \arctan (c+d x)}{4 d^4 f}+\frac {b (d e-c f) (d e+f-c f) (d e-(1+c) f) \log \left (1+(c+d x)^2\right )}{2 d^4} \]
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Time = 0.27 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5156, 4973, 716, 649, 209, 266} \[ \int (e+f x)^3 \left (a+b \cot ^{-1}(c+d x)\right ) \, dx=\frac {(e+f x)^4 \left (a+b \cot ^{-1}(c+d x)\right )}{4 f}+\frac {b \arctan (c+d x) \left (-6 \left (1-c^2\right ) d^2 e^2 f^2+4 c \left (3-c^2\right ) d e f^3+\left (c^4-6 c^2+1\right ) f^4-4 c d^3 e^3 f+d^4 e^4\right )}{4 d^4 f}+\frac {b f x \left (-\left (1-6 c^2\right ) f^2-12 c d e f+6 d^2 e^2\right )}{4 d^3}+\frac {b f^2 (c+d x)^2 (d e-c f)}{2 d^4}+\frac {b (d e-c f) (-c f+d e+f) (d e-(c+1) f) \log \left ((c+d x)^2+1\right )}{2 d^4}+\frac {b f^3 (c+d x)^3}{12 d^4} \]
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Rule 209
Rule 266
Rule 649
Rule 716
Rule 4973
Rule 5156
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \left (\frac {d e-c f}{d}+\frac {f x}{d}\right )^3 \left (a+b \cot ^{-1}(x)\right ) \, dx,x,c+d x\right )}{d} \\ & = \frac {(e+f x)^4 \left (a+b \cot ^{-1}(c+d x)\right )}{4 f}+\frac {b \text {Subst}\left (\int \frac {\left (\frac {d e-c f}{d}+\frac {f x}{d}\right )^4}{1+x^2} \, dx,x,c+d x\right )}{4 f} \\ & = \frac {(e+f x)^4 \left (a+b \cot ^{-1}(c+d x)\right )}{4 f}+\frac {b \text {Subst}\left (\int \left (\frac {f^2 \left (6 d^2 e^2-12 c d e f-\left (1-6 c^2\right ) f^2\right )}{d^4}+\frac {4 f^3 (d e-c f) x}{d^4}+\frac {f^4 x^2}{d^4}+\frac {d^4 e^4-4 c d^3 e^3 f-6 \left (1-c^2\right ) d^2 e^2 f^2+4 c \left (3-c^2\right ) d e f^3+\left (1-6 c^2+c^4\right ) f^4+4 f (d e-c f) (d e-f-c f) (d e+f-c f) x}{d^4 \left (1+x^2\right )}\right ) \, dx,x,c+d x\right )}{4 f} \\ & = \frac {b f \left (6 d^2 e^2-12 c d e f-\left (1-6 c^2\right ) f^2\right ) x}{4 d^3}+\frac {b f^2 (d e-c f) (c+d x)^2}{2 d^4}+\frac {b f^3 (c+d x)^3}{12 d^4}+\frac {(e+f x)^4 \left (a+b \cot ^{-1}(c+d x)\right )}{4 f}+\frac {b \text {Subst}\left (\int \frac {d^4 e^4-4 c d^3 e^3 f-6 \left (1-c^2\right ) d^2 e^2 f^2+4 c \left (3-c^2\right ) d e f^3+\left (1-6 c^2+c^4\right ) f^4+4 f (d e-c f) (d e-f-c f) (d e+f-c f) x}{1+x^2} \, dx,x,c+d x\right )}{4 d^4 f} \\ & = \frac {b f \left (6 d^2 e^2-12 c d e f-\left (1-6 c^2\right ) f^2\right ) x}{4 d^3}+\frac {b f^2 (d e-c f) (c+d x)^2}{2 d^4}+\frac {b f^3 (c+d x)^3}{12 d^4}+\frac {(e+f x)^4 \left (a+b \cot ^{-1}(c+d x)\right )}{4 f}+\frac {(b (d e-c f) (d e+f-c f) (d e-(1+c) f)) \text {Subst}\left (\int \frac {x}{1+x^2} \, dx,x,c+d x\right )}{d^4}+\frac {\left (b \left (d^4 e^4-4 c d^3 e^3 f-6 \left (1-c^2\right ) d^2 e^2 f^2+4 c \left (3-c^2\right ) d e f^3+\left (1-6 c^2+c^4\right ) f^4\right )\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,c+d x\right )}{4 d^4 f} \\ & = \frac {b f \left (6 d^2 e^2-12 c d e f-\left (1-6 c^2\right ) f^2\right ) x}{4 d^3}+\frac {b f^2 (d e-c f) (c+d x)^2}{2 d^4}+\frac {b f^3 (c+d x)^3}{12 d^4}+\frac {(e+f x)^4 \left (a+b \cot ^{-1}(c+d x)\right )}{4 f}+\frac {b \left (d^4 e^4-4 c d^3 e^3 f-6 \left (1-c^2\right ) d^2 e^2 f^2+4 c \left (3-c^2\right ) d e f^3+\left (1-6 c^2+c^4\right ) f^4\right ) \arctan (c+d x)}{4 d^4 f}+\frac {b (d e-c f) (d e+f-c f) (d e-(1+c) f) \log \left (1+(c+d x)^2\right )}{2 d^4} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.20 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.67 \[ \int (e+f x)^3 \left (a+b \cot ^{-1}(c+d x)\right ) \, dx=\frac {(e+f x)^4 \left (a+b \cot ^{-1}(c+d x)\right )+\frac {b \left (6 d f^2 \left (6 d^2 e^2-12 c d e f+\left (-1+6 c^2\right ) f^2\right ) x+12 f^3 (d e-c f) (c+d x)^2+2 f^4 (c+d x)^3-3 i (d e-(-i+c) f)^4 \log (i-c-d x)+3 i (d e-(i+c) f)^4 \log (i+c+d x)\right )}{6 d^4}}{4 f} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(495\) vs. \(2(221)=442\).
Time = 0.88 (sec) , antiderivative size = 496, normalized size of antiderivative = 2.13
method | result | size |
parts | \(\frac {a \left (f x +e \right )^{4}}{4 f}-\frac {f^{3} b c \,x^{2}}{4 d^{2}}+\frac {f^{2} b e \,x^{2}}{2 d}+\frac {3 f^{3} b \,c^{2} x}{4 d^{3}}+\frac {3 f b \,e^{2} x}{2 d}+\frac {b \,f^{3} \operatorname {arccot}\left (d x +c \right ) x^{4}}{4}+b \,\operatorname {arccot}\left (d x +c \right ) x \,e^{3}+\frac {b \,\operatorname {arccot}\left (d x +c \right ) e^{4}}{4 f}-\frac {3 f b \,e^{2} \arctan \left (d x +c \right )}{2 d^{2}}+\frac {f^{3} b \,c^{4} \arctan \left (d x +c \right )}{4 d^{4}}-\frac {3 f^{3} b \,c^{2} \arctan \left (d x +c \right )}{2 d^{4}}-\frac {b c \,e^{3} \arctan \left (d x +c \right )}{d}-\frac {b \,f^{3} c}{4 d^{4}}+\frac {13 b \,f^{3} c^{3}}{12 d^{4}}+\frac {b \ln \left (1+\left (d x +c \right )^{2}\right ) e^{3}}{2 d}+\frac {f^{3} b \arctan \left (d x +c \right )}{4 d^{4}}+\frac {f^{3} b \,x^{3}}{12 d}-\frac {f^{3} b x}{4 d^{3}}+\frac {b \,e^{4} \arctan \left (d x +c \right )}{4 f}+\frac {3 b \,f^{2} \ln \left (1+\left (d x +c \right )^{2}\right ) c^{2} e}{2 d^{3}}-\frac {b \,f^{3} \ln \left (1+\left (d x +c \right )^{2}\right ) c^{3}}{2 d^{4}}+\frac {3 b f \,\operatorname {arccot}\left (d x +c \right ) e^{2} x^{2}}{2}+b \,f^{2} \operatorname {arccot}\left (d x +c \right ) e \,x^{3}+\frac {b \,f^{3} \ln \left (1+\left (d x +c \right )^{2}\right ) c}{2 d^{4}}-\frac {b \,f^{2} \ln \left (1+\left (d x +c \right )^{2}\right ) e}{2 d^{3}}-\frac {2 f^{2} b c e x}{d^{2}}-\frac {f^{2} b \,c^{3} e \arctan \left (d x +c \right )}{d^{3}}+\frac {3 f b \,c^{2} e^{2} \arctan \left (d x +c \right )}{2 d^{2}}+\frac {3 f^{2} b c e \arctan \left (d x +c \right )}{d^{3}}-\frac {5 b \,f^{2} c^{2} e}{2 d^{3}}+\frac {3 b f c \,e^{2}}{2 d^{2}}-\frac {3 b f \ln \left (1+\left (d x +c \right )^{2}\right ) c \,e^{2}}{2 d^{2}}\) | \(496\) |
derivativedivides | \(\frac {\frac {a \left (c f -d e -f \left (d x +c \right )\right )^{4}}{4 d^{3} f}-\frac {b \left (-\frac {f^{3} \operatorname {arccot}\left (d x +c \right ) c^{4}}{4}+f^{2} \operatorname {arccot}\left (d x +c \right ) c^{3} d e +f^{3} \operatorname {arccot}\left (d x +c \right ) c^{3} \left (d x +c \right )-\frac {3 f \,\operatorname {arccot}\left (d x +c \right ) c^{2} d^{2} e^{2}}{2}-3 f^{2} \operatorname {arccot}\left (d x +c \right ) c^{2} d e \left (d x +c \right )-\frac {3 f^{3} \operatorname {arccot}\left (d x +c \right ) c^{2} \left (d x +c \right )^{2}}{2}+\operatorname {arccot}\left (d x +c \right ) c \,d^{3} e^{3}+3 f \,\operatorname {arccot}\left (d x +c \right ) c \,d^{2} e^{2} \left (d x +c \right )+3 f^{2} \operatorname {arccot}\left (d x +c \right ) c d e \left (d x +c \right )^{2}+f^{3} \operatorname {arccot}\left (d x +c \right ) c \left (d x +c \right )^{3}-\frac {\operatorname {arccot}\left (d x +c \right ) d^{4} e^{4}}{4 f}-\operatorname {arccot}\left (d x +c \right ) d^{3} e^{3} \left (d x +c \right )-\frac {3 f \,\operatorname {arccot}\left (d x +c \right ) d^{2} e^{2} \left (d x +c \right )^{2}}{2}-f^{2} \operatorname {arccot}\left (d x +c \right ) d e \left (d x +c \right )^{3}-\frac {f^{3} \operatorname {arccot}\left (d x +c \right ) \left (d x +c \right )^{4}}{4}-\frac {6 c^{2} f^{4} \left (d x +c \right )-12 c d e \,f^{3} \left (d x +c \right )-2 c \,f^{4} \left (d x +c \right )^{2}+6 d^{2} e^{2} f^{2} \left (d x +c \right )+2 d e \,f^{3} \left (d x +c \right )^{2}+\frac {f^{4} \left (d x +c \right )^{3}}{3}-f^{4} \left (d x +c \right )+\frac {\left (-4 c^{3} f^{4}+12 c^{2} d e \,f^{3}-12 c \,d^{2} e^{2} f^{2}+4 f \,e^{3} d^{3}+4 c \,f^{4}-4 e \,f^{3} d \right ) \ln \left (1+\left (d x +c \right )^{2}\right )}{2}+\left (c^{4} f^{4}-4 c^{3} d e \,f^{3}+6 c^{2} d^{2} e^{2} f^{2}-4 c \,d^{3} e^{3} f +d^{4} e^{4}-6 c^{2} f^{4}+12 c d e \,f^{3}-6 d^{2} e^{2} f^{2}+f^{4}\right ) \arctan \left (d x +c \right )}{4 f}\right )}{d^{3}}}{d}\) | \(565\) |
default | \(\frac {\frac {a \left (c f -d e -f \left (d x +c \right )\right )^{4}}{4 d^{3} f}-\frac {b \left (-\frac {f^{3} \operatorname {arccot}\left (d x +c \right ) c^{4}}{4}+f^{2} \operatorname {arccot}\left (d x +c \right ) c^{3} d e +f^{3} \operatorname {arccot}\left (d x +c \right ) c^{3} \left (d x +c \right )-\frac {3 f \,\operatorname {arccot}\left (d x +c \right ) c^{2} d^{2} e^{2}}{2}-3 f^{2} \operatorname {arccot}\left (d x +c \right ) c^{2} d e \left (d x +c \right )-\frac {3 f^{3} \operatorname {arccot}\left (d x +c \right ) c^{2} \left (d x +c \right )^{2}}{2}+\operatorname {arccot}\left (d x +c \right ) c \,d^{3} e^{3}+3 f \,\operatorname {arccot}\left (d x +c \right ) c \,d^{2} e^{2} \left (d x +c \right )+3 f^{2} \operatorname {arccot}\left (d x +c \right ) c d e \left (d x +c \right )^{2}+f^{3} \operatorname {arccot}\left (d x +c \right ) c \left (d x +c \right )^{3}-\frac {\operatorname {arccot}\left (d x +c \right ) d^{4} e^{4}}{4 f}-\operatorname {arccot}\left (d x +c \right ) d^{3} e^{3} \left (d x +c \right )-\frac {3 f \,\operatorname {arccot}\left (d x +c \right ) d^{2} e^{2} \left (d x +c \right )^{2}}{2}-f^{2} \operatorname {arccot}\left (d x +c \right ) d e \left (d x +c \right )^{3}-\frac {f^{3} \operatorname {arccot}\left (d x +c \right ) \left (d x +c \right )^{4}}{4}-\frac {6 c^{2} f^{4} \left (d x +c \right )-12 c d e \,f^{3} \left (d x +c \right )-2 c \,f^{4} \left (d x +c \right )^{2}+6 d^{2} e^{2} f^{2} \left (d x +c \right )+2 d e \,f^{3} \left (d x +c \right )^{2}+\frac {f^{4} \left (d x +c \right )^{3}}{3}-f^{4} \left (d x +c \right )+\frac {\left (-4 c^{3} f^{4}+12 c^{2} d e \,f^{3}-12 c \,d^{2} e^{2} f^{2}+4 f \,e^{3} d^{3}+4 c \,f^{4}-4 e \,f^{3} d \right ) \ln \left (1+\left (d x +c \right )^{2}\right )}{2}+\left (c^{4} f^{4}-4 c^{3} d e \,f^{3}+6 c^{2} d^{2} e^{2} f^{2}-4 c \,d^{3} e^{3} f +d^{4} e^{4}-6 c^{2} f^{4}+12 c d e \,f^{3}-6 d^{2} e^{2} f^{2}+f^{4}\right ) \arctan \left (d x +c \right )}{4 f}\right )}{d^{3}}}{d}\) | \(565\) |
parallelrisch | \(-\frac {-42 b \,c^{2} d e \,f^{2}+24 a c \,d^{3} e^{3}+6 \ln \left (d^{2} x^{2}+2 c d x +c^{2}+1\right ) b \,c^{3} f^{3}-6 \ln \left (d^{2} x^{2}+2 c d x +c^{2}+1\right ) b \,d^{3} e^{3}-6 \ln \left (d^{2} x^{2}+2 c d x +c^{2}+1\right ) b c \,f^{3}-3 x^{4} a \,d^{4} f^{3}-12 x a \,d^{4} e^{3}+3 x b d \,f^{3}+3 \,\operatorname {arccot}\left (d x +c \right ) b \,c^{4} f^{3}-18 \,\operatorname {arccot}\left (d x +c \right ) b \,c^{2} f^{3}+15 b \,c^{3} f^{3}-6 b \,d^{3} e \,f^{2} x^{2}-18 b \,d^{3} e^{2} f x +6 f^{2} e b d -9 b c \,f^{3}+18 a \,d^{2} e^{2} f +3 \,\operatorname {arccot}\left (d x +c \right ) b \,f^{3}-b \,d^{3} f^{3} x^{3}+36 b c \,d^{2} e^{2} f -12 x^{3} \operatorname {arccot}\left (d x +c \right ) b \,d^{4} e \,f^{2}-18 x^{2} \operatorname {arccot}\left (d x +c \right ) b \,d^{4} e^{2} f +24 x b c \,d^{2} e \,f^{2}-12 \,\operatorname {arccot}\left (d x +c \right ) b \,c^{3} d e \,f^{2}+36 \,\operatorname {arccot}\left (d x +c \right ) b c d e \,f^{2}+18 \,\operatorname {arccot}\left (d x +c \right ) b \,c^{2} d^{2} e^{2} f -18 \ln \left (d^{2} x^{2}+2 c d x +c^{2}+1\right ) b \,c^{2} d e \,f^{2}+18 \ln \left (d^{2} x^{2}+2 c d x +c^{2}+1\right ) b c \,d^{2} e^{2} f -18 x^{2} a \,d^{4} e^{2} f +6 \ln \left (d^{2} x^{2}+2 c d x +c^{2}+1\right ) b d e \,f^{2}-3 x^{4} \operatorname {arccot}\left (d x +c \right ) b \,d^{4} f^{3}-12 x \,\operatorname {arccot}\left (d x +c \right ) b \,d^{4} e^{3}-9 x b \,c^{2} d \,f^{3}-12 x^{3} a \,d^{4} e \,f^{2}-12 \,\operatorname {arccot}\left (d x +c \right ) b c \,d^{3} e^{3}-18 \,\operatorname {arccot}\left (d x +c \right ) b \,d^{2} e^{2} f +3 x^{2} b c \,d^{2} f^{3}+18 a \,c^{2} d^{2} e^{2} f}{12 d^{4}}\) | \(574\) |
risch | \(\frac {x^{4} f^{3} a}{4}+x \,e^{3} a -\frac {f^{3} b c \,x^{2}}{4 d^{2}}+\frac {f^{2} b e \,x^{2}}{2 d}+\frac {3 f^{3} b \,c^{2} x}{4 d^{3}}+\frac {3 f b \,e^{2} x}{2 d}-\frac {3 f b \,e^{2} \arctan \left (d x +c \right )}{2 d^{2}}-\frac {f^{2} b e \ln \left (d^{2} x^{2}+2 c d x +c^{2}+1\right )}{2 d^{3}}+\frac {f^{3} b c \ln \left (d^{2} x^{2}+2 c d x +c^{2}+1\right )}{2 d^{4}}-\frac {f^{3} b \,c^{3} \ln \left (d^{2} x^{2}+2 c d x +c^{2}+1\right )}{2 d^{4}}+\frac {f^{3} b \,c^{4} \arctan \left (d x +c \right )}{4 d^{4}}-\frac {3 f^{3} b \,c^{2} \arctan \left (d x +c \right )}{2 d^{4}}+\frac {x^{3} f^{2} e b \pi }{2}+\frac {3 x^{2} f \,e^{2} b \pi }{4}-\frac {b c \,e^{3} \arctan \left (d x +c \right )}{d}-\frac {i b \,e^{3} x \ln \left (1-i \left (d x +c \right )\right )}{2}-\frac {i f^{3} b \,x^{4} \ln \left (1-i \left (d x +c \right )\right )}{8}-\frac {i b \,e^{4} \ln \left (d^{2} x^{2}+2 c d x +c^{2}+1\right )}{16 f}+\frac {i \left (f x +e \right )^{4} b \ln \left (1+i \left (d x +c \right )\right )}{8 f}-\frac {3 i f b \,e^{2} x^{2} \ln \left (1-i \left (d x +c \right )\right )}{4}-\frac {i f^{2} b e \,x^{3} \ln \left (1-i \left (d x +c \right )\right )}{2}+x^{3} f^{2} e a +\frac {3 x^{2} f \,e^{2} a}{2}+\frac {f^{3} b \arctan \left (d x +c \right )}{4 d^{4}}+\frac {x^{4} f^{3} b \pi }{8}+\frac {x \,e^{3} b \pi }{2}+\frac {f^{3} b \,x^{3}}{12 d}-\frac {f^{3} b x}{4 d^{3}}+\frac {b \,e^{3} \ln \left (d^{2} x^{2}+2 c d x +c^{2}+1\right )}{2 d}+\frac {b \,e^{4} \arctan \left (d x +c \right )}{8 f}-\frac {2 f^{2} b c e x}{d^{2}}-\frac {f^{2} b \,c^{3} e \arctan \left (d x +c \right )}{d^{3}}+\frac {3 f b \,c^{2} e^{2} \arctan \left (d x +c \right )}{2 d^{2}}+\frac {3 f^{2} b c e \arctan \left (d x +c \right )}{d^{3}}-\frac {3 f b c \,e^{2} \ln \left (d^{2} x^{2}+2 c d x +c^{2}+1\right )}{2 d^{2}}+\frac {3 f^{2} b \,c^{2} e \ln \left (d^{2} x^{2}+2 c d x +c^{2}+1\right )}{2 d^{3}}\) | \(623\) |
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Time = 0.28 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.39 \[ \int (e+f x)^3 \left (a+b \cot ^{-1}(c+d x)\right ) \, dx=\frac {3 \, a d^{4} f^{3} x^{4} + {\left (12 \, a d^{4} e f^{2} + b d^{3} f^{3}\right )} x^{3} + 3 \, {\left (6 \, a d^{4} e^{2} f + 2 \, b d^{3} e f^{2} - b c d^{2} f^{3}\right )} x^{2} + 3 \, {\left (4 \, a d^{4} e^{3} + 6 \, b d^{3} e^{2} f - 8 \, b c d^{2} e f^{2} + {\left (3 \, b c^{2} - b\right )} d f^{3}\right )} x + 3 \, {\left (b d^{4} f^{3} x^{4} + 4 \, b d^{4} e f^{2} x^{3} + 6 \, b d^{4} e^{2} f x^{2} + 4 \, b d^{4} e^{3} x\right )} \operatorname {arccot}\left (d x + c\right ) - 3 \, {\left (4 \, b c d^{3} e^{3} - 6 \, {\left (b c^{2} - b\right )} d^{2} e^{2} f + 4 \, {\left (b c^{3} - 3 \, b c\right )} d e f^{2} - {\left (b c^{4} - 6 \, b c^{2} + b\right )} f^{3}\right )} \arctan \left (d x + c\right ) + 6 \, {\left (b d^{3} e^{3} - 3 \, b c d^{2} e^{2} f + {\left (3 \, b c^{2} - b\right )} d e f^{2} - {\left (b c^{3} - b c\right )} f^{3}\right )} \log \left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )}{12 \, d^{4}} \]
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Timed out. \[ \int (e+f x)^3 \left (a+b \cot ^{-1}(c+d x)\right ) \, dx=\text {Timed out} \]
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Time = 0.28 (sec) , antiderivative size = 341, normalized size of antiderivative = 1.46 \[ \int (e+f x)^3 \left (a+b \cot ^{-1}(c+d x)\right ) \, dx=\frac {1}{4} \, a f^{3} x^{4} + a e f^{2} x^{3} + \frac {3}{2} \, a e^{2} f x^{2} + \frac {3}{2} \, {\left (x^{2} \operatorname {arccot}\left (d x + c\right ) + d {\left (\frac {x}{d^{2}} + \frac {{\left (c^{2} - 1\right )} \arctan \left (\frac {d^{2} x + c d}{d}\right )}{d^{3}} - \frac {c \log \left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )}{d^{3}}\right )}\right )} b e^{2} f + \frac {1}{2} \, {\left (2 \, x^{3} \operatorname {arccot}\left (d x + c\right ) + d {\left (\frac {d x^{2} - 4 \, c x}{d^{3}} - \frac {2 \, {\left (c^{3} - 3 \, c\right )} \arctan \left (\frac {d^{2} x + c d}{d}\right )}{d^{4}} + \frac {{\left (3 \, c^{2} - 1\right )} \log \left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )}{d^{4}}\right )}\right )} b e f^{2} + \frac {1}{12} \, {\left (3 \, x^{4} \operatorname {arccot}\left (d x + c\right ) + d {\left (\frac {d^{2} x^{3} - 3 \, c d x^{2} + 3 \, {\left (3 \, c^{2} - 1\right )} x}{d^{4}} + \frac {3 \, {\left (c^{4} - 6 \, c^{2} + 1\right )} \arctan \left (\frac {d^{2} x + c d}{d}\right )}{d^{5}} - \frac {6 \, {\left (c^{3} - c\right )} \log \left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )}{d^{5}}\right )}\right )} b f^{3} + a e^{3} x + \frac {{\left (2 \, {\left (d x + c\right )} \operatorname {arccot}\left (d x + c\right ) + \log \left ({\left (d x + c\right )}^{2} + 1\right )\right )} b e^{3}}{2 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 2265 vs. \(2 (216) = 432\).
Time = 1.73 (sec) , antiderivative size = 2265, normalized size of antiderivative = 9.72 \[ \int (e+f x)^3 \left (a+b \cot ^{-1}(c+d x)\right ) \, dx=\text {Too large to display} \]
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Time = 1.37 (sec) , antiderivative size = 783, normalized size of antiderivative = 3.36 \[ \int (e+f x)^3 \left (a+b \cot ^{-1}(c+d x)\right ) \, dx=\mathrm {acot}\left (c+d\,x\right )\,\left (b\,e^3\,x+\frac {3\,b\,e^2\,f\,x^2}{2}+b\,e\,f^2\,x^3+\frac {b\,f^3\,x^4}{4}\right )+x\,\left (\frac {e\,\left (6\,a\,c^2\,f^2+12\,a\,c\,d\,e\,f+2\,a\,d^2\,e^2+3\,b\,d\,e\,f+6\,a\,f^2\right )}{2\,d^2}-\frac {\left (4\,c^2+4\right )\,\left (\frac {f^2\,\left (b\,f+8\,a\,c\,f+12\,a\,d\,e\right )}{4\,d}-\frac {2\,a\,c\,f^3}{d}\right )}{4\,d^2}+\frac {2\,c\,\left (\frac {2\,c\,\left (\frac {f^2\,\left (b\,f+8\,a\,c\,f+12\,a\,d\,e\right )}{4\,d}-\frac {2\,a\,c\,f^3}{d}\right )}{d}-\frac {4\,a\,c^2\,f^3+24\,a\,c\,d\,e\,f^2+12\,a\,d^2\,e^2\,f+4\,b\,d\,e\,f^2+4\,a\,f^3}{4\,d^2}+\frac {a\,f^3\,\left (4\,c^2+4\right )}{4\,d^2}\right )}{d}\right )-x^2\,\left (\frac {c\,\left (\frac {f^2\,\left (b\,f+8\,a\,c\,f+12\,a\,d\,e\right )}{4\,d}-\frac {2\,a\,c\,f^3}{d}\right )}{d}-\frac {4\,a\,c^2\,f^3+24\,a\,c\,d\,e\,f^2+12\,a\,d^2\,e^2\,f+4\,b\,d\,e\,f^2+4\,a\,f^3}{8\,d^2}+\frac {a\,f^3\,\left (4\,c^2+4\right )}{8\,d^2}\right )+x^3\,\left (\frac {f^2\,\left (b\,f+8\,a\,c\,f+12\,a\,d\,e\right )}{12\,d}-\frac {2\,a\,c\,f^3}{3\,d}\right )+\frac {a\,f^3\,x^4}{4}+\frac {\ln \left (c^2+2\,c\,d\,x+d^2\,x^2+1\right )\,\left (-64\,b\,c^3\,d^4\,f^3+192\,b\,c^2\,d^5\,e\,f^2-192\,b\,c\,d^6\,e^2\,f+64\,b\,c\,d^4\,f^3+64\,b\,d^7\,e^3-64\,b\,d^5\,e\,f^2\right )}{128\,d^8}+\frac {b\,\mathrm {atan}\left (\frac {4\,d^3\,\left (\frac {c\,\left (c^4\,f^3-4\,c^3\,d\,e\,f^2+6\,c^2\,d^2\,e^2\,f-6\,c^2\,f^3-4\,c\,d^3\,e^3+12\,c\,d\,e\,f^2-6\,d^2\,e^2\,f+f^3\right )}{4\,d^3}+\frac {x\,\left (c^4\,f^3-4\,c^3\,d\,e\,f^2+6\,c^2\,d^2\,e^2\,f-6\,c^2\,f^3-4\,c\,d^3\,e^3+12\,c\,d\,e\,f^2-6\,d^2\,e^2\,f+f^3\right )}{4\,d^2}\right )}{c^4\,f^3-4\,c^3\,d\,e\,f^2+6\,c^2\,d^2\,e^2\,f-6\,c^2\,f^3-4\,c\,d^3\,e^3+12\,c\,d\,e\,f^2-6\,d^2\,e^2\,f+f^3}\right )\,\left (c^4\,f^3-4\,c^3\,d\,e\,f^2+6\,c^2\,d^2\,e^2\,f-6\,c^2\,f^3-4\,c\,d^3\,e^3+12\,c\,d\,e\,f^2-6\,d^2\,e^2\,f+f^3\right )}{4\,d^4} \]
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