Integrand size = 18, antiderivative size = 168 \[ \int (e+f x)^3 \left (a+b \coth ^{-1}(c+d x)\right ) \, dx=\frac {b f \left (6 d^2 e^2-12 c d e f+\left (1+6 c^2\right ) f^2\right ) x}{4 d^3}+\frac {b f^2 (d e-c f) (c+d x)^2}{2 d^4}+\frac {b f^3 (c+d x)^3}{12 d^4}+\frac {(e+f x)^4 \left (a+b \coth ^{-1}(c+d x)\right )}{4 f}+\frac {b (d e+f-c f)^4 \log (1-c-d x)}{8 d^4 f}-\frac {b (d e-f-c f)^4 \log (1+c+d x)}{8 d^4 f} \]
[Out]
Time = 0.25 (sec) , antiderivative size = 168, 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 = {6247, 6064, 716, 647, 31} \[ \int (e+f x)^3 \left (a+b \coth ^{-1}(c+d x)\right ) \, dx=\frac {(e+f x)^4 \left (a+b \coth ^{-1}(c+d x)\right )}{4 f}+\frac {b f x \left (\left (6 c^2+1\right ) f^2-12 c d e f+6 d^2 e^2\right )}{4 d^3}+\frac {b f^2 (c+d x)^2 (d e-c f)}{2 d^4}-\frac {b (-c f+d e-f)^4 \log (c+d x+1)}{8 d^4 f}+\frac {b (-c f+d e+f)^4 \log (-c-d x+1)}{8 d^4 f}+\frac {b f^3 (c+d x)^3}{12 d^4} \]
[In]
[Out]
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 )^3 \left (a+b \coth ^{-1}(x)\right ) \, dx,x,c+d x\right )}{d} \\ & = \frac {(e+f x)^4 \left (a+b \coth ^{-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 \coth ^{-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) \left (d^2 e^2-2 c d e f+f^2+c^2 f^2\right ) 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 \coth ^{-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) \left (d^2 e^2-2 c d e f+f^2+c^2 f^2\right ) 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 \coth ^{-1}(c+d x)\right )}{4 f}+\frac {\left (b (d e-f-c f)^4\right ) \text {Subst}\left (\int \frac {1}{-1-x} \, dx,x,c+d x\right )}{8 d^4 f}-\frac {\left (b (d e+f-c f)^4\right ) \text {Subst}\left (\int \frac {1}{1-x} \, dx,x,c+d x\right )}{8 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 \coth ^{-1}(c+d x)\right )}{4 f}+\frac {b (d e+f-c f)^4 \log (1-c-d x)}{8 d^4 f}-\frac {b (d e-f-c f)^4 \log (1+c+d x)}{8 d^4 f} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.61 \[ \int (e+f x)^3 \left (a+b \coth ^{-1}(c+d x)\right ) \, dx=\frac {6 d \left (4 a d^3 e^3+b f \left (6 d^2 e^2-8 c d e f+\left (1+3 c^2\right ) f^2\right )\right ) x+6 d^2 f \left (6 a d^2 e^2+b f (2 d e-c f)\right ) x^2+2 d^3 f^2 (12 a d e+b f) x^3+6 a d^4 f^3 x^4+6 b d^4 x \left (4 e^3+6 e^2 f x+4 e f^2 x^2+f^3 x^3\right ) \coth ^{-1}(c+d x)-3 b (-1+c) \left (4 d^3 e^3-6 (-1+c) d^2 e^2 f+4 (-1+c)^2 d e f^2-(-1+c)^3 f^3\right ) \log (1-c-d x)-3 b (1+c) \left (-4 d^3 e^3+6 (1+c) d^2 e^2 f-4 (1+c)^2 d e f^2+(1+c)^3 f^3\right ) \log (1+c+d x)}{24 d^4} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(598\) vs. \(2(156)=312\).
Time = 0.56 (sec) , antiderivative size = 599, normalized size of antiderivative = 3.57
method | result | size |
parallelrisch | \(-\frac {12 \ln \left (d x +c -1\right ) b \,c^{3} f^{3}-12 \ln \left (d x +c -1\right ) b \,d^{3} e^{3}+12 \ln \left (d x +c -1\right ) b c \,f^{3}+24 x b c \,d^{2} e \,f^{2}-42 b \,c^{2} d e \,f^{2}+24 a c \,d^{3} e^{3}+36 b c \,d^{2} e^{2} f +3 \,\operatorname {arccoth}\left (d x +c \right ) b \,f^{3}+15 b \,c^{3} f^{3}-6 b e \,f^{2} d +9 b c \,f^{3}+36 \,\operatorname {arccoth}\left (d x +c \right ) b c \,d^{2} e^{2} f -18 a \,e^{2} f \,d^{2}-12 \ln \left (d x +c -1\right ) b d e \,f^{2}-36 \ln \left (d x +c -1\right ) b \,c^{2} d e \,f^{2}+36 \ln \left (d x +c -1\right ) b c \,d^{2} e^{2} f -3 x^{4} a \,d^{4} f^{3}-12 x a \,d^{4} e^{3}-3 x b d \,f^{3}-x^{3} b \,d^{3} f^{3}+3 \,\operatorname {arccoth}\left (d x +c \right ) b \,c^{4} f^{3}-12 \,\operatorname {arccoth}\left (d x +c \right ) b \,d^{3} e^{3}+18 \,\operatorname {arccoth}\left (d x +c \right ) b \,c^{2} f^{3}+12 \,\operatorname {arccoth}\left (d x +c \right ) b c \,f^{3}+12 \,\operatorname {arccoth}\left (d x +c \right ) b \,c^{3} f^{3}-12 x^{3} a \,d^{4} e \,f^{2}-12 \,\operatorname {arccoth}\left (d x +c \right ) b c \,d^{3} e^{3}+18 \,\operatorname {arccoth}\left (d x +c \right ) b \,d^{2} e^{2} f -12 \,\operatorname {arccoth}\left (d x +c \right ) b d e \,f^{2}-12 x \,\operatorname {arccoth}\left (d x +c \right ) b \,d^{4} e^{3}-18 x^{2} a \,d^{4} e^{2} f +3 x^{2} b c \,d^{2} f^{3}-6 x^{2} b \,d^{3} e \,f^{2}-3 x^{4} \operatorname {arccoth}\left (d x +c \right ) b \,d^{4} f^{3}-9 x b \,c^{2} d \,f^{3}-18 x b \,d^{3} e^{2} f -12 x^{3} \operatorname {arccoth}\left (d x +c \right ) b \,d^{4} e \,f^{2}+18 a \,c^{2} d^{2} e^{2} f -18 x^{2} \operatorname {arccoth}\left (d x +c \right ) b \,d^{4} e^{2} f +18 \,\operatorname {arccoth}\left (d x +c \right ) b \,c^{2} d^{2} e^{2} f -36 \,\operatorname {arccoth}\left (d x +c \right ) b \,c^{2} d e \,f^{2}-36 \,\operatorname {arccoth}\left (d x +c \right ) b c d e \,f^{2}-12 \,\operatorname {arccoth}\left (d x +c \right ) b \,c^{3} d e \,f^{2}}{12 d^{4}}\) | \(599\) |
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 {arccoth}\left (d x +c \right ) c^{4}}{4}+f^{2} \operatorname {arccoth}\left (d x +c \right ) c^{3} d e +f^{3} \operatorname {arccoth}\left (d x +c \right ) c^{3} \left (d x +c \right )-\frac {3 f \,\operatorname {arccoth}\left (d x +c \right ) c^{2} d^{2} e^{2}}{2}-3 f^{2} \operatorname {arccoth}\left (d x +c \right ) c^{2} d e \left (d x +c \right )-\frac {3 f^{3} \operatorname {arccoth}\left (d x +c \right ) c^{2} \left (d x +c \right )^{2}}{2}+\operatorname {arccoth}\left (d x +c \right ) c \,d^{3} e^{3}+3 f \,\operatorname {arccoth}\left (d x +c \right ) c \,d^{2} e^{2} \left (d x +c \right )+3 f^{2} \operatorname {arccoth}\left (d x +c \right ) c d e \left (d x +c \right )^{2}+f^{3} \operatorname {arccoth}\left (d x +c \right ) c \left (d x +c \right )^{3}-\frac {\operatorname {arccoth}\left (d x +c \right ) d^{4} e^{4}}{4 f}-\operatorname {arccoth}\left (d x +c \right ) d^{3} e^{3} \left (d x +c \right )-\frac {3 f \,\operatorname {arccoth}\left (d x +c \right ) d^{2} e^{2} \left (d x +c \right )^{2}}{2}-f^{2} \operatorname {arccoth}\left (d x +c \right ) d e \left (d x +c \right )^{3}-\frac {f^{3} \operatorname {arccoth}\left (d x +c \right ) \left (d x +c \right )^{4}}{4}-\frac {-12 c d e \,f^{3} \left (d x +c \right )+2 d e \,f^{3} \left (d x +c \right )^{2}+6 d^{2} e^{2} f^{2} \left (d x +c \right )-2 c \,f^{4} \left (d x +c \right )^{2}+6 c^{2} f^{4} \left (d x +c \right )+\frac {f^{4} \left (d x +c \right )^{3}}{3}+f^{4} \left (d x +c \right )+\frac {\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}-4 c^{3} f^{4}+12 c^{2} d e \,f^{3}-12 c \,d^{2} e^{2} f^{2}+4 d^{3} e^{3} f +6 c^{2} f^{4}-12 c d e \,f^{3}+6 d^{2} e^{2} f^{2}-4 c \,f^{4}+4 d e \,f^{3}+f^{4}\right ) \ln \left (d x +c -1\right )}{2}-\frac {\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}+4 c^{3} f^{4}-12 c^{2} d e \,f^{3}+12 c \,d^{2} e^{2} f^{2}-4 d^{3} e^{3} f +6 c^{2} f^{4}-12 c d e \,f^{3}+6 d^{2} e^{2} f^{2}+4 c \,f^{4}-4 d e \,f^{3}+f^{4}\right ) \ln \left (d x +c +1\right )}{2}}{4 f}\right )}{d^{3}}}{d}\) | \(693\) |
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 {arccoth}\left (d x +c \right ) c^{4}}{4}+f^{2} \operatorname {arccoth}\left (d x +c \right ) c^{3} d e +f^{3} \operatorname {arccoth}\left (d x +c \right ) c^{3} \left (d x +c \right )-\frac {3 f \,\operatorname {arccoth}\left (d x +c \right ) c^{2} d^{2} e^{2}}{2}-3 f^{2} \operatorname {arccoth}\left (d x +c \right ) c^{2} d e \left (d x +c \right )-\frac {3 f^{3} \operatorname {arccoth}\left (d x +c \right ) c^{2} \left (d x +c \right )^{2}}{2}+\operatorname {arccoth}\left (d x +c \right ) c \,d^{3} e^{3}+3 f \,\operatorname {arccoth}\left (d x +c \right ) c \,d^{2} e^{2} \left (d x +c \right )+3 f^{2} \operatorname {arccoth}\left (d x +c \right ) c d e \left (d x +c \right )^{2}+f^{3} \operatorname {arccoth}\left (d x +c \right ) c \left (d x +c \right )^{3}-\frac {\operatorname {arccoth}\left (d x +c \right ) d^{4} e^{4}}{4 f}-\operatorname {arccoth}\left (d x +c \right ) d^{3} e^{3} \left (d x +c \right )-\frac {3 f \,\operatorname {arccoth}\left (d x +c \right ) d^{2} e^{2} \left (d x +c \right )^{2}}{2}-f^{2} \operatorname {arccoth}\left (d x +c \right ) d e \left (d x +c \right )^{3}-\frac {f^{3} \operatorname {arccoth}\left (d x +c \right ) \left (d x +c \right )^{4}}{4}-\frac {-12 c d e \,f^{3} \left (d x +c \right )+2 d e \,f^{3} \left (d x +c \right )^{2}+6 d^{2} e^{2} f^{2} \left (d x +c \right )-2 c \,f^{4} \left (d x +c \right )^{2}+6 c^{2} f^{4} \left (d x +c \right )+\frac {f^{4} \left (d x +c \right )^{3}}{3}+f^{4} \left (d x +c \right )+\frac {\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}-4 c^{3} f^{4}+12 c^{2} d e \,f^{3}-12 c \,d^{2} e^{2} f^{2}+4 d^{3} e^{3} f +6 c^{2} f^{4}-12 c d e \,f^{3}+6 d^{2} e^{2} f^{2}-4 c \,f^{4}+4 d e \,f^{3}+f^{4}\right ) \ln \left (d x +c -1\right )}{2}-\frac {\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}+4 c^{3} f^{4}-12 c^{2} d e \,f^{3}+12 c \,d^{2} e^{2} f^{2}-4 d^{3} e^{3} f +6 c^{2} f^{4}-12 c d e \,f^{3}+6 d^{2} e^{2} f^{2}+4 c \,f^{4}-4 d e \,f^{3}+f^{4}\right ) \ln \left (d x +c +1\right )}{2}}{4 f}\right )}{d^{3}}}{d}\) | \(693\) |
parts | \(\frac {a \left (f x +e \right )^{4}}{4 f}+\frac {b \left (\frac {f^{3} \operatorname {arccoth}\left (d x +c \right ) \left (d x +c \right )^{4}}{4 d^{3}}-\frac {f^{3} \operatorname {arccoth}\left (d x +c \right ) \left (d x +c \right )^{3} c}{d^{3}}+\frac {f^{2} \operatorname {arccoth}\left (d x +c \right ) \left (d x +c \right )^{3} e}{d^{2}}+\frac {3 f^{3} \operatorname {arccoth}\left (d x +c \right ) \left (d x +c \right )^{2} c^{2}}{2 d^{3}}-\frac {3 f^{2} \operatorname {arccoth}\left (d x +c \right ) \left (d x +c \right )^{2} c e}{d^{2}}+\frac {3 f \,\operatorname {arccoth}\left (d x +c \right ) \left (d x +c \right )^{2} e^{2}}{2 d}-\frac {f^{3} \operatorname {arccoth}\left (d x +c \right ) \left (d x +c \right ) c^{3}}{d^{3}}+\frac {3 f^{2} \operatorname {arccoth}\left (d x +c \right ) \left (d x +c \right ) c^{2} e}{d^{2}}-\frac {3 f \,\operatorname {arccoth}\left (d x +c \right ) \left (d x +c \right ) c \,e^{2}}{d}+\operatorname {arccoth}\left (d x +c \right ) \left (d x +c \right ) e^{3}+\frac {f^{3} \operatorname {arccoth}\left (d x +c \right ) c^{4}}{4 d^{3}}-\frac {f^{2} \operatorname {arccoth}\left (d x +c \right ) c^{3} e}{d^{2}}+\frac {3 f \,\operatorname {arccoth}\left (d x +c \right ) c^{2} e^{2}}{2 d}-\operatorname {arccoth}\left (d x +c \right ) c \,e^{3}+\frac {d \,\operatorname {arccoth}\left (d x +c \right ) e^{4}}{4 f}+\frac {-12 c d e \,f^{3} \left (d x +c \right )+2 d e \,f^{3} \left (d x +c \right )^{2}+6 d^{2} e^{2} f^{2} \left (d x +c \right )-2 c \,f^{4} \left (d x +c \right )^{2}+6 c^{2} f^{4} \left (d x +c \right )+\frac {f^{4} \left (d x +c \right )^{3}}{3}+f^{4} \left (d x +c \right )+\frac {\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}-4 c^{3} f^{4}+12 c^{2} d e \,f^{3}-12 c \,d^{2} e^{2} f^{2}+4 d^{3} e^{3} f +6 c^{2} f^{4}-12 c d e \,f^{3}+6 d^{2} e^{2} f^{2}-4 c \,f^{4}+4 d e \,f^{3}+f^{4}\right ) \ln \left (d x +c -1\right )}{2}-\frac {\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}+4 c^{3} f^{4}-12 c^{2} d e \,f^{3}+12 c \,d^{2} e^{2} f^{2}-4 d^{3} e^{3} f +6 c^{2} f^{4}-12 c d e \,f^{3}+6 d^{2} e^{2} f^{2}+4 c \,f^{4}-4 d e \,f^{3}+f^{4}\right ) \ln \left (d x +c +1\right )}{2}}{4 d^{3} f}\right )}{d}\) | \(694\) |
risch | \(-\frac {b \,e^{4} \ln \left (d x +c -1\right )}{8 f}+\frac {f^{3} a \,x^{4}}{4}+\frac {f^{3} b \,x^{3}}{12 d}+\frac {f^{3} b x}{4 d^{3}}+\frac {f^{3} \ln \left (-d x -c +1\right ) b}{8 d^{4}}-\frac {f^{3} \ln \left (d x +c +1\right ) b}{8 d^{4}}-\frac {b \,e^{3} x \ln \left (d x +c -1\right )}{2}+\frac {\ln \left (-d x -c +1\right ) b \,e^{3}}{2 d}+\frac {\ln \left (d x +c +1\right ) b \,e^{3}}{2 d}-\frac {f^{3} b \,x^{4} \ln \left (d x +c -1\right )}{8}+\frac {\ln \left (-d x -c +1\right ) b \,e^{4}}{8 f}-\frac {\ln \left (d x +c +1\right ) b \,e^{4}}{8 f}-\frac {2 f^{2} b c e x}{d^{2}}-\frac {3 f \ln \left (d x +c +1\right ) b \,c^{2} e^{2}}{4 d^{2}}+\frac {3 f^{2} \ln \left (-d x -c +1\right ) b \,c^{2} e}{2 d^{3}}-\frac {3 f \ln \left (-d x -c +1\right ) b c \,e^{2}}{2 d^{2}}+\frac {3 f^{2} \ln \left (d x +c +1\right ) b \,c^{2} e}{2 d^{3}}-\frac {3 f \ln \left (d x +c +1\right ) b c \,e^{2}}{2 d^{2}}-\frac {3 f^{2} \ln \left (-d x -c +1\right ) b c e}{2 d^{3}}+\frac {3 f^{2} \ln \left (d x +c +1\right ) b c e}{2 d^{3}}-\frac {f^{2} \ln \left (-d x -c +1\right ) b \,c^{3} e}{2 d^{3}}+\frac {3 f \ln \left (-d x -c +1\right ) b \,c^{2} e^{2}}{4 d^{2}}+\frac {f^{2} \ln \left (d x +c +1\right ) b \,c^{3} e}{2 d^{3}}+\frac {\left (f x +e \right )^{4} b \ln \left (d x +c +1\right )}{8 f}+a \,e^{3} x -\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}+f^{2} a e \,x^{3}+\frac {3 f a \,e^{2} x^{2}}{2}+\frac {\ln \left (d x +c +1\right ) b c \,e^{3}}{2 d}-\frac {f^{2} b e \,x^{3} \ln \left (d x +c -1\right )}{2}-\frac {3 f b \,e^{2} x^{2} \ln \left (d x +c -1\right )}{4}-\frac {\ln \left (-d x -c +1\right ) b c \,e^{3}}{2 d}+\frac {3 f^{3} \ln \left (-d x -c +1\right ) b \,c^{2}}{4 d^{4}}+\frac {3 f \ln \left (-d x -c +1\right ) b \,e^{2}}{4 d^{2}}-\frac {3 f^{3} \ln \left (d x +c +1\right ) b \,c^{2}}{4 d^{4}}-\frac {3 f \ln \left (d x +c +1\right ) b \,e^{2}}{4 d^{2}}-\frac {f^{3} \ln \left (-d x -c +1\right ) b \,c^{3}}{2 d^{4}}-\frac {f^{3} \ln \left (d x +c +1\right ) b \,c^{3}}{2 d^{4}}+\frac {f^{3} \ln \left (-d x -c +1\right ) b \,c^{4}}{8 d^{4}}-\frac {f^{3} \ln \left (d x +c +1\right ) b \,c^{4}}{8 d^{4}}-\frac {f^{3} \ln \left (-d x -c +1\right ) b c}{2 d^{4}}+\frac {f^{2} \ln \left (-d x -c +1\right ) b e}{2 d^{3}}-\frac {f^{3} \ln \left (d x +c +1\right ) b c}{2 d^{4}}+\frac {f^{2} \ln \left (d x +c +1\right ) b e}{2 d^{3}}\) | \(803\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 385 vs. \(2 (156) = 312\).
Time = 0.27 (sec) , antiderivative size = 385, normalized size of antiderivative = 2.29 \[ \int (e+f x)^3 \left (a+b \coth ^{-1}(c+d x)\right ) \, dx=\frac {6 \, a d^{4} f^{3} x^{4} + 2 \, {\left (12 \, a d^{4} e f^{2} + b d^{3} f^{3}\right )} x^{3} + 6 \, {\left (6 \, a d^{4} e^{2} f + 2 \, b d^{3} e f^{2} - b c d^{2} f^{3}\right )} x^{2} + 6 \, {\left (4 \, a d^{4} e^{3} + 6 \, b d^{3} e^{2} f - 8 \, b c d^{2} e f^{2} + {\left (3 \, b c^{2} + b\right )} d f^{3}\right )} x + 3 \, {\left (4 \, {\left (b c + b\right )} d^{3} e^{3} - 6 \, {\left (b c^{2} + 2 \, b c + b\right )} d^{2} e^{2} f + 4 \, {\left (b c^{3} + 3 \, b c^{2} + 3 \, b c + b\right )} d e f^{2} - {\left (b c^{4} + 4 \, b c^{3} + 6 \, b c^{2} + 4 \, b c + b\right )} f^{3}\right )} \log \left (d x + c + 1\right ) - 3 \, {\left (4 \, {\left (b c - b\right )} d^{3} e^{3} - 6 \, {\left (b c^{2} - 2 \, b c + b\right )} d^{2} e^{2} f + 4 \, {\left (b c^{3} - 3 \, b c^{2} + 3 \, b c - b\right )} d e f^{2} - {\left (b c^{4} - 4 \, b c^{3} + 6 \, b c^{2} - 4 \, b c + b\right )} f^{3}\right )} \log \left (d x + c - 1\right ) + 3 \, {\left (b d^{4} f^{3} x^{4} + 4 \, b d^{4} e f^{2} x^{3} + 6 \, b d^{4} e^{2} f x^{2} + 4 \, b d^{4} e^{3} x\right )} \log \left (\frac {d x + c + 1}{d x + c - 1}\right )}{24 \, d^{4}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 644 vs. \(2 (151) = 302\).
Time = 0.70 (sec) , antiderivative size = 644, normalized size of antiderivative = 3.83 \[ \int (e+f x)^3 \left (a+b \coth ^{-1}(c+d x)\right ) \, dx=\begin {cases} a e^{3} x + \frac {3 a e^{2} f x^{2}}{2} + a e f^{2} x^{3} + \frac {a f^{3} x^{4}}{4} - \frac {b c^{4} f^{3} \operatorname {acoth}{\left (c + d x \right )}}{4 d^{4}} + \frac {b c^{3} e f^{2} \operatorname {acoth}{\left (c + d x \right )}}{d^{3}} - \frac {b c^{3} f^{3} \log {\left (\frac {c}{d} + x + \frac {1}{d} \right )}}{d^{4}} + \frac {b c^{3} f^{3} \operatorname {acoth}{\left (c + d x \right )}}{d^{4}} - \frac {3 b c^{2} e^{2} f \operatorname {acoth}{\left (c + d x \right )}}{2 d^{2}} + \frac {3 b c^{2} e f^{2} \log {\left (\frac {c}{d} + x + \frac {1}{d} \right )}}{d^{3}} - \frac {3 b c^{2} e f^{2} \operatorname {acoth}{\left (c + d x \right )}}{d^{3}} + \frac {3 b c^{2} f^{3} x}{4 d^{3}} - \frac {3 b c^{2} f^{3} \operatorname {acoth}{\left (c + d x \right )}}{2 d^{4}} + \frac {b c e^{3} \operatorname {acoth}{\left (c + d x \right )}}{d} - \frac {3 b c e^{2} f \log {\left (\frac {c}{d} + x + \frac {1}{d} \right )}}{d^{2}} + \frac {3 b c e^{2} f \operatorname {acoth}{\left (c + d x \right )}}{d^{2}} - \frac {2 b c e f^{2} x}{d^{2}} - \frac {b c f^{3} x^{2}}{4 d^{2}} + \frac {3 b c e f^{2} \operatorname {acoth}{\left (c + d x \right )}}{d^{3}} - \frac {b c f^{3} \log {\left (\frac {c}{d} + x + \frac {1}{d} \right )}}{d^{4}} + \frac {b c f^{3} \operatorname {acoth}{\left (c + d x \right )}}{d^{4}} + b e^{3} x \operatorname {acoth}{\left (c + d x \right )} + \frac {3 b e^{2} f x^{2} \operatorname {acoth}{\left (c + d x \right )}}{2} + b e f^{2} x^{3} \operatorname {acoth}{\left (c + d x \right )} + \frac {b f^{3} x^{4} \operatorname {acoth}{\left (c + d x \right )}}{4} + \frac {b e^{3} \log {\left (\frac {c}{d} + x + \frac {1}{d} \right )}}{d} - \frac {b e^{3} \operatorname {acoth}{\left (c + d x \right )}}{d} + \frac {3 b e^{2} f x}{2 d} + \frac {b e f^{2} x^{2}}{2 d} + \frac {b f^{3} x^{3}}{12 d} - \frac {3 b e^{2} f \operatorname {acoth}{\left (c + d x \right )}}{2 d^{2}} + \frac {b e f^{2} \log {\left (\frac {c}{d} + x + \frac {1}{d} \right )}}{d^{3}} - \frac {b e f^{2} \operatorname {acoth}{\left (c + d x \right )}}{d^{3}} + \frac {b f^{3} x}{4 d^{3}} - \frac {b f^{3} \operatorname {acoth}{\left (c + d x \right )}}{4 d^{4}} & \text {for}\: d \neq 0 \\\left (a + b \operatorname {acoth}{\left (c \right )}\right ) \left (e^{3} x + \frac {3 e^{2} f x^{2}}{2} + e f^{2} x^{3} + \frac {f^{3} x^{4}}{4}\right ) & \text {otherwise} \end {cases} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 333 vs. \(2 (156) = 312\).
Time = 0.21 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.98 \[ \int (e+f x)^3 \left (a+b \coth ^{-1}(c+d x)\right ) \, dx=\frac {1}{4} \, a f^{3} x^{4} + a e f^{2} x^{3} + \frac {3}{2} \, a e^{2} f x^{2} + \frac {3}{4} \, {\left (2 \, x^{2} \operatorname {arcoth}\left (d x + c\right ) + d {\left (\frac {2 \, x}{d^{2}} - \frac {{\left (c^{2} + 2 \, c + 1\right )} \log \left (d x + c + 1\right )}{d^{3}} + \frac {{\left (c^{2} - 2 \, c + 1\right )} \log \left (d x + c - 1\right )}{d^{3}}\right )}\right )} b e^{2} f + \frac {1}{2} \, {\left (2 \, x^{3} \operatorname {arcoth}\left (d x + c\right ) + d {\left (\frac {d x^{2} - 4 \, c x}{d^{3}} + \frac {{\left (c^{3} + 3 \, c^{2} + 3 \, c + 1\right )} \log \left (d x + c + 1\right )}{d^{4}} - \frac {{\left (c^{3} - 3 \, c^{2} + 3 \, c - 1\right )} \log \left (d x + c - 1\right )}{d^{4}}\right )}\right )} b e f^{2} + \frac {1}{24} \, {\left (6 \, x^{4} \operatorname {arcoth}\left (d x + c\right ) + d {\left (\frac {2 \, {\left (d^{2} x^{3} - 3 \, c d x^{2} + 3 \, {\left (3 \, c^{2} + 1\right )} x\right )}}{d^{4}} - \frac {3 \, {\left (c^{4} + 4 \, c^{3} + 6 \, c^{2} + 4 \, c + 1\right )} \log \left (d x + c + 1\right )}{d^{5}} + \frac {3 \, {\left (c^{4} - 4 \, c^{3} + 6 \, c^{2} - 4 \, c + 1\right )} \log \left (d x + c - 1\right )}{d^{5}}\right )}\right )} b f^{3} + a e^{3} x + \frac {{\left (2 \, {\left (d x + c\right )} \operatorname {arcoth}\left (d x + c\right ) + \log \left (-{\left (d x + c\right )}^{2} + 1\right )\right )} b e^{3}}{2 \, d} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 2333 vs. \(2 (156) = 312\).
Time = 0.34 (sec) , antiderivative size = 2333, normalized size of antiderivative = 13.89 \[ \int (e+f x)^3 \left (a+b \coth ^{-1}(c+d x)\right ) \, dx=\text {Too large to display} \]
[In]
[Out]
Time = 5.39 (sec) , antiderivative size = 742, normalized size of antiderivative = 4.42 \[ \int (e+f x)^3 \left (a+b \coth ^{-1}(c+d x)\right ) \, dx=x\,\left (\frac {e\,\left (6\,a\,c^2\,f^2+12\,a\,c\,d\,e\,f+2\,a\,d^2\,e^2+3\,b\,d\,e\,f-6\,a\,f^2\right )}{2\,d^2}-\frac {\left (4\,c^2-4\right )\,\left (\frac {f^2\,\left (b\,f+8\,a\,c\,f+12\,a\,d\,e\right )}{4\,d}-\frac {2\,a\,c\,f^3}{d}\right )}{4\,d^2}+\frac {2\,c\,\left (\frac {2\,c\,\left (\frac {f^2\,\left (b\,f+8\,a\,c\,f+12\,a\,d\,e\right )}{4\,d}-\frac {2\,a\,c\,f^3}{d}\right )}{d}-\frac {4\,a\,c^2\,f^3+24\,a\,c\,d\,e\,f^2+12\,a\,d^2\,e^2\,f+4\,b\,d\,e\,f^2-4\,a\,f^3}{4\,d^2}+\frac {a\,f^3\,\left (4\,c^2-4\right )}{4\,d^2}\right )}{d}\right )-\ln \left (1-\frac {1}{c+d\,x}\right )\,\left (\frac {b\,e^3\,x}{2}+\frac {3\,b\,e^2\,f\,x^2}{4}+\frac {b\,e\,f^2\,x^3}{2}+\frac {b\,f^3\,x^4}{8}\right )-x^2\,\left (\frac {c\,\left (\frac {f^2\,\left (b\,f+8\,a\,c\,f+12\,a\,d\,e\right )}{4\,d}-\frac {2\,a\,c\,f^3}{d}\right )}{d}-\frac {4\,a\,c^2\,f^3+24\,a\,c\,d\,e\,f^2+12\,a\,d^2\,e^2\,f+4\,b\,d\,e\,f^2-4\,a\,f^3}{8\,d^2}+\frac {a\,f^3\,\left (4\,c^2-4\right )}{8\,d^2}\right )+x^3\,\left (\frac {f^2\,\left (b\,f+8\,a\,c\,f+12\,a\,d\,e\right )}{12\,d}-\frac {2\,a\,c\,f^3}{3\,d}\right )+\ln \left (\frac {1}{c+d\,x}+1\right )\,\left (\frac {b\,e^3\,x}{2}+\frac {3\,b\,e^2\,f\,x^2}{4}+\frac {b\,e\,f^2\,x^3}{2}+\frac {b\,f^3\,x^4}{8}\right )+\frac {a\,f^3\,x^4}{4}+\frac {\ln \left (c+d\,x-1\right )\,\left (b\,c^4\,f^3-4\,b\,c^3\,d\,e\,f^2-4\,b\,c^3\,f^3+6\,b\,c^2\,d^2\,e^2\,f+12\,b\,c^2\,d\,e\,f^2+6\,b\,c^2\,f^3-4\,b\,c\,d^3\,e^3-12\,b\,c\,d^2\,e^2\,f-12\,b\,c\,d\,e\,f^2-4\,b\,c\,f^3+4\,b\,d^3\,e^3+6\,b\,d^2\,e^2\,f+4\,b\,d\,e\,f^2+b\,f^3\right )}{8\,d^4}-\frac {\ln \left (c+d\,x+1\right )\,\left (b\,c^4\,f^3-4\,b\,c^3\,d\,e\,f^2+4\,b\,c^3\,f^3+6\,b\,c^2\,d^2\,e^2\,f-12\,b\,c^2\,d\,e\,f^2+6\,b\,c^2\,f^3-4\,b\,c\,d^3\,e^3+12\,b\,c\,d^2\,e^2\,f-12\,b\,c\,d\,e\,f^2+4\,b\,c\,f^3-4\,b\,d^3\,e^3+6\,b\,d^2\,e^2\,f-4\,b\,d\,e\,f^2+b\,f^3\right )}{8\,d^4} \]
[In]
[Out]