Integrand size = 18, antiderivative size = 120 \[ \int (e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right ) \, dx=\frac {b f (d e-c f) x}{d^2}+\frac {b f^2 (c+d x)^2}{6 d^3}+\frac {(e+f x)^3 \left (a+b \coth ^{-1}(c+d x)\right )}{3 f}+\frac {b (d e+f-c f)^3 \log (1-c-d x)}{6 d^3 f}-\frac {b (d e-(1+c) f)^3 \log (1+c+d x)}{6 d^3 f} \]
[Out]
Time = 0.16 (sec) , antiderivative size = 120, 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} \[ \int (e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right ) \, dx=\frac {(e+f x)^3 \left (a+b \coth ^{-1}(c+d x)\right )}{3 f}+\frac {b (-c f+d e+f)^3 \log (-c-d x+1)}{6 d^3 f}-\frac {b (d e-(c+1) f)^3 \log (c+d x+1)}{6 d^3 f}+\frac {b f^2 (c+d x)^2}{6 d^3}+\frac {b f x (d e-c f)}{d^2} \]
[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 )^2 \left (a+b \coth ^{-1}(x)\right ) \, dx,x,c+d x\right )}{d} \\ & = \frac {(e+f x)^3 \left (a+b \coth ^{-1}(c+d x)\right )}{3 f}-\frac {b \text {Subst}\left (\int \frac {\left (\frac {d e-c f}{d}+\frac {f x}{d}\right )^3}{1-x^2} \, dx,x,c+d x\right )}{3 f} \\ & = \frac {(e+f x)^3 \left (a+b \coth ^{-1}(c+d x)\right )}{3 f}-\frac {b \text {Subst}\left (\int \left (-\frac {3 f^2 (d e-c f)}{d^3}-\frac {f^3 x}{d^3}+\frac {(d e-c f) \left (d^2 e^2-2 c d e f+3 f^2+c^2 f^2\right )+f \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) x}{d^3 \left (1-x^2\right )}\right ) \, dx,x,c+d x\right )}{3 f} \\ & = \frac {b f (d e-c f) x}{d^2}+\frac {b f^2 (c+d x)^2}{6 d^3}+\frac {(e+f x)^3 \left (a+b \coth ^{-1}(c+d x)\right )}{3 f}-\frac {b \text {Subst}\left (\int \frac {(d e-c f) \left (d^2 e^2-2 c d e f+3 f^2+c^2 f^2\right )+f \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) x}{1-x^2} \, dx,x,c+d x\right )}{3 d^3 f} \\ & = \frac {b f (d e-c f) x}{d^2}+\frac {b f^2 (c+d x)^2}{6 d^3}+\frac {(e+f x)^3 \left (a+b \coth ^{-1}(c+d x)\right )}{3 f}-\frac {\left (b (d e+f-c f)^3\right ) \text {Subst}\left (\int \frac {1}{1-x} \, dx,x,c+d x\right )}{6 d^3 f}+\frac {\left (b (d e-(1+c) f)^3\right ) \text {Subst}\left (\int \frac {1}{-1-x} \, dx,x,c+d x\right )}{6 d^3 f} \\ & = \frac {b f (d e-c f) x}{d^2}+\frac {b f^2 (c+d x)^2}{6 d^3}+\frac {(e+f x)^3 \left (a+b \coth ^{-1}(c+d x)\right )}{3 f}+\frac {b (d e+f-c f)^3 \log (1-c-d x)}{6 d^3 f}-\frac {b (d e-(1+c) f)^3 \log (1+c+d x)}{6 d^3 f} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.45 \[ \int (e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right ) \, dx=\frac {2 d \left (3 a d^2 e^2+b f (3 d e-2 c f)\right ) x+d^2 f (6 a d e+b f) x^2+2 a d^3 f^2 x^3+2 b d^3 x \left (3 e^2+3 e f x+f^2 x^2\right ) \coth ^{-1}(c+d x)-b (-1+c) \left (3 d^2 e^2-3 (-1+c) d e f+(-1+c)^2 f^2\right ) \log (1-c-d x)+b (1+c) \left (3 d^2 e^2-3 (1+c) d e f+(1+c)^2 f^2\right ) \log (1+c+d x)}{6 d^3} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(350\) vs. \(2(112)=224\).
Time = 0.48 (sec) , antiderivative size = 351, normalized size of antiderivative = 2.92
method | result | size |
parallelrisch | \(\frac {6 x^{2} a \,d^{3} e f +6 x \,\operatorname {arccoth}\left (d x +c \right ) b \,d^{3} e^{2}+2 x^{3} \operatorname {arccoth}\left (d x +c \right ) b \,d^{3} f^{2}+6 \,\operatorname {arccoth}\left (d x +c \right ) b c \,d^{2} e^{2}-6 \,\operatorname {arccoth}\left (d x +c \right ) b d e f -4 x b c d \,f^{2}+6 x b \,d^{2} e f -6 a \,c^{2} e f d +x^{2} b \,d^{2} f^{2}+2 \,\operatorname {arccoth}\left (d x +c \right ) b \,c^{3} f^{2}+6 \,\operatorname {arccoth}\left (d x +c \right ) b \,c^{2} f^{2}+6 \,\operatorname {arccoth}\left (d x +c \right ) b \,d^{2} e^{2}+6 \,\operatorname {arccoth}\left (d x +c \right ) b c \,f^{2}+6 x a \,d^{3} e^{2}+2 x^{3} a \,d^{3} f^{2}+6 \ln \left (d x +c -1\right ) b \,c^{2} f^{2}+6 \ln \left (d x +c -1\right ) b \,d^{2} e^{2}-12 a c \,e^{2} d^{2}-12 \ln \left (d x +c -1\right ) b c d e f +6 x^{2} \operatorname {arccoth}\left (d x +c \right ) b \,d^{3} e f -12 \,\operatorname {arccoth}\left (d x +c \right ) b c d e f -6 \,\operatorname {arccoth}\left (d x +c \right ) b \,c^{2} d e f +b \,f^{2}+7 b \,c^{2} f^{2}+6 a e f d -12 b c d e f +2 \ln \left (d x +c -1\right ) b \,f^{2}+2 \,\operatorname {arccoth}\left (d x +c \right ) b \,f^{2}}{6 d^{3}}\) | \(351\) |
parts | \(\frac {a \left (f x +e \right )^{3}}{3 f}+\frac {b \left (\frac {f^{2} \operatorname {arccoth}\left (d x +c \right ) \left (d x +c \right )^{3}}{3 d^{2}}-\frac {f^{2} \operatorname {arccoth}\left (d x +c \right ) \left (d x +c \right )^{2} c}{d^{2}}+\frac {f \,\operatorname {arccoth}\left (d x +c \right ) \left (d x +c \right )^{2} e}{d}+\frac {f^{2} \operatorname {arccoth}\left (d x +c \right ) \left (d x +c \right ) c^{2}}{d^{2}}-\frac {2 f \,\operatorname {arccoth}\left (d x +c \right ) \left (d x +c \right ) c e}{d}+\operatorname {arccoth}\left (d x +c \right ) \left (d x +c \right ) e^{2}-\frac {f^{2} \operatorname {arccoth}\left (d x +c \right ) c^{3}}{3 d^{2}}+\frac {f \,\operatorname {arccoth}\left (d x +c \right ) c^{2} e}{d}-\operatorname {arccoth}\left (d x +c \right ) c \,e^{2}+\frac {d \,\operatorname {arccoth}\left (d x +c \right ) e^{3}}{3 f}+\frac {\frac {f^{3} \left (d x +c \right )^{2}}{2}-3 c \,f^{3} \left (d x +c \right )+3 d e \,f^{2} \left (d x +c \right )+\frac {\left (-c^{3} f^{3}+3 c^{2} d e \,f^{2}-3 c \,d^{2} e^{2} f +d^{3} e^{3}+3 c^{2} f^{3}-6 c d e \,f^{2}+3 d^{2} e^{2} f -3 c \,f^{3}+3 d e \,f^{2}+f^{3}\right ) \ln \left (d x +c -1\right )}{2}-\frac {\left (-c^{3} f^{3}+3 c^{2} d e \,f^{2}-3 c \,d^{2} e^{2} f +d^{3} e^{3}-3 c^{2} f^{3}+6 c d e \,f^{2}-3 d^{2} e^{2} f -3 c \,f^{3}+3 d e \,f^{2}-f^{3}\right ) \ln \left (d x +c +1\right )}{2}}{3 d^{2} f}\right )}{d}\) | \(415\) |
derivativedivides | \(\frac {-\frac {a \left (c f -d e -f \left (d x +c \right )\right )^{3}}{3 d^{2} f}+\frac {b \left (-\frac {f^{2} \operatorname {arccoth}\left (d x +c \right ) c^{3}}{3}+f \,\operatorname {arccoth}\left (d x +c \right ) c^{2} d e +f^{2} \operatorname {arccoth}\left (d x +c \right ) c^{2} \left (d x +c \right )-\operatorname {arccoth}\left (d x +c \right ) c \,d^{2} e^{2}-2 f \,\operatorname {arccoth}\left (d x +c \right ) c d e \left (d x +c \right )-f^{2} \operatorname {arccoth}\left (d x +c \right ) c \left (d x +c \right )^{2}+\frac {\operatorname {arccoth}\left (d x +c \right ) d^{3} e^{3}}{3 f}+\operatorname {arccoth}\left (d x +c \right ) d^{2} e^{2} \left (d x +c \right )+f \,\operatorname {arccoth}\left (d x +c \right ) d e \left (d x +c \right )^{2}+\frac {f^{2} \operatorname {arccoth}\left (d x +c \right ) \left (d x +c \right )^{3}}{3}-\frac {3 c \,f^{3} \left (d x +c \right )-3 d e \,f^{2} \left (d x +c \right )-\frac {f^{3} \left (d x +c \right )^{2}}{2}+\frac {\left (c^{3} f^{3}-3 c^{2} d e \,f^{2}+3 c \,d^{2} e^{2} f -d^{3} e^{3}-3 c^{2} f^{3}+6 c d e \,f^{2}-3 d^{2} e^{2} f +3 c \,f^{3}-3 d e \,f^{2}-f^{3}\right ) \ln \left (d x +c -1\right )}{2}-\frac {\left (c^{3} f^{3}-3 c^{2} d e \,f^{2}+3 c \,d^{2} e^{2} f -d^{3} e^{3}+3 c^{2} f^{3}-6 c d e \,f^{2}+3 d^{2} e^{2} f +3 c \,f^{3}-3 d e \,f^{2}+f^{3}\right ) \ln \left (d x +c +1\right )}{2}}{3 f}\right )}{d^{2}}}{d}\) | \(420\) |
default | \(\frac {-\frac {a \left (c f -d e -f \left (d x +c \right )\right )^{3}}{3 d^{2} f}+\frac {b \left (-\frac {f^{2} \operatorname {arccoth}\left (d x +c \right ) c^{3}}{3}+f \,\operatorname {arccoth}\left (d x +c \right ) c^{2} d e +f^{2} \operatorname {arccoth}\left (d x +c \right ) c^{2} \left (d x +c \right )-\operatorname {arccoth}\left (d x +c \right ) c \,d^{2} e^{2}-2 f \,\operatorname {arccoth}\left (d x +c \right ) c d e \left (d x +c \right )-f^{2} \operatorname {arccoth}\left (d x +c \right ) c \left (d x +c \right )^{2}+\frac {\operatorname {arccoth}\left (d x +c \right ) d^{3} e^{3}}{3 f}+\operatorname {arccoth}\left (d x +c \right ) d^{2} e^{2} \left (d x +c \right )+f \,\operatorname {arccoth}\left (d x +c \right ) d e \left (d x +c \right )^{2}+\frac {f^{2} \operatorname {arccoth}\left (d x +c \right ) \left (d x +c \right )^{3}}{3}-\frac {3 c \,f^{3} \left (d x +c \right )-3 d e \,f^{2} \left (d x +c \right )-\frac {f^{3} \left (d x +c \right )^{2}}{2}+\frac {\left (c^{3} f^{3}-3 c^{2} d e \,f^{2}+3 c \,d^{2} e^{2} f -d^{3} e^{3}-3 c^{2} f^{3}+6 c d e \,f^{2}-3 d^{2} e^{2} f +3 c \,f^{3}-3 d e \,f^{2}-f^{3}\right ) \ln \left (d x +c -1\right )}{2}-\frac {\left (c^{3} f^{3}-3 c^{2} d e \,f^{2}+3 c \,d^{2} e^{2} f -d^{3} e^{3}+3 c^{2} f^{3}-6 c d e \,f^{2}+3 d^{2} e^{2} f +3 c \,f^{3}-3 d e \,f^{2}+f^{3}\right ) \ln \left (d x +c +1\right )}{2}}{3 f}\right )}{d^{2}}}{d}\) | \(420\) |
risch | \(-\frac {\ln \left (-d x -c -1\right ) b \,e^{3}}{6 f}-\frac {b \,e^{2} x \ln \left (d x +c -1\right )}{2}+\frac {f^{2} a \,x^{3}}{3}+\frac {f^{2} b \,x^{2}}{6 d}+\frac {f^{2} \ln \left (-d x -c -1\right ) b}{6 d^{3}}+\frac {f^{2} \ln \left (d x +c -1\right ) b}{6 d^{3}}+\frac {\ln \left (-d x -c -1\right ) b \,e^{2}}{2 d}+\frac {\ln \left (d x +c -1\right ) b \,e^{2}}{2 d}-\frac {f^{2} b \,x^{3} \ln \left (d x +c -1\right )}{6}+a \,e^{2} x -\frac {2 f^{2} b c x}{3 d^{2}}+\frac {f b e x}{d}-\frac {f^{2} \ln \left (d x +c -1\right ) b \,c^{3}}{6 d^{3}}+\frac {f^{2} \ln \left (-d x -c -1\right ) b \,c^{2}}{2 d^{3}}+\frac {f^{2} \ln \left (d x +c -1\right ) b \,c^{2}}{2 d^{3}}+\frac {f^{2} \ln \left (-d x -c -1\right ) b c}{2 d^{3}}-\frac {f \ln \left (-d x -c -1\right ) b e}{2 d^{2}}-\frac {f^{2} \ln \left (d x +c -1\right ) b c}{2 d^{3}}+\frac {f \ln \left (d x +c -1\right ) b e}{2 d^{2}}+\frac {f^{2} \ln \left (-d x -c -1\right ) b \,c^{3}}{6 d^{3}}-\frac {\ln \left (d x +c -1\right ) b c \,e^{2}}{2 d}-\frac {f b e \,x^{2} \ln \left (d x +c -1\right )}{2}+\frac {\ln \left (-d x -c -1\right ) b c \,e^{2}}{2 d}+f a e \,x^{2}+\frac {\left (f x +e \right )^{3} b \ln \left (d x +c +1\right )}{6 f}+\frac {f \ln \left (d x +c -1\right ) b \,c^{2} e}{2 d^{2}}-\frac {f \ln \left (-d x -c -1\right ) b c e}{d^{2}}-\frac {f \ln \left (d x +c -1\right ) b c e}{d^{2}}-\frac {f \ln \left (-d x -c -1\right ) b \,c^{2} e}{2 d^{2}}\) | \(472\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 241 vs. \(2 (112) = 224\).
Time = 0.27 (sec) , antiderivative size = 241, normalized size of antiderivative = 2.01 \[ \int (e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right ) \, dx=\frac {2 \, a d^{3} f^{2} x^{3} + {\left (6 \, a d^{3} e f + b d^{2} f^{2}\right )} x^{2} + 2 \, {\left (3 \, a d^{3} e^{2} + 3 \, b d^{2} e f - 2 \, b c d f^{2}\right )} x + {\left (3 \, {\left (b c + b\right )} d^{2} e^{2} - 3 \, {\left (b c^{2} + 2 \, b c + b\right )} d e f + {\left (b c^{3} + 3 \, b c^{2} + 3 \, b c + b\right )} f^{2}\right )} \log \left (d x + c + 1\right ) - {\left (3 \, {\left (b c - b\right )} d^{2} e^{2} - 3 \, {\left (b c^{2} - 2 \, b c + b\right )} d e f + {\left (b c^{3} - 3 \, b c^{2} + 3 \, b c - b\right )} f^{2}\right )} \log \left (d x + c - 1\right ) + {\left (b d^{3} f^{2} x^{3} + 3 \, b d^{3} e f x^{2} + 3 \, b d^{3} e^{2} x\right )} \log \left (\frac {d x + c + 1}{d x + c - 1}\right )}{6 \, d^{3}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 369 vs. \(2 (105) = 210\).
Time = 0.52 (sec) , antiderivative size = 369, normalized size of antiderivative = 3.08 \[ \int (e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right ) \, dx=\begin {cases} a e^{2} x + a e f x^{2} + \frac {a f^{2} x^{3}}{3} + \frac {b c^{3} f^{2} \operatorname {acoth}{\left (c + d x \right )}}{3 d^{3}} - \frac {b c^{2} e f \operatorname {acoth}{\left (c + d x \right )}}{d^{2}} + \frac {b c^{2} f^{2} \log {\left (\frac {c}{d} + x + \frac {1}{d} \right )}}{d^{3}} - \frac {b c^{2} f^{2} \operatorname {acoth}{\left (c + d x \right )}}{d^{3}} + \frac {b c e^{2} \operatorname {acoth}{\left (c + d x \right )}}{d} - \frac {2 b c e f \log {\left (\frac {c}{d} + x + \frac {1}{d} \right )}}{d^{2}} + \frac {2 b c e f \operatorname {acoth}{\left (c + d x \right )}}{d^{2}} - \frac {2 b c f^{2} x}{3 d^{2}} + \frac {b c f^{2} \operatorname {acoth}{\left (c + d x \right )}}{d^{3}} + b e^{2} x \operatorname {acoth}{\left (c + d x \right )} + b e f x^{2} \operatorname {acoth}{\left (c + d x \right )} + \frac {b f^{2} x^{3} \operatorname {acoth}{\left (c + d x \right )}}{3} + \frac {b e^{2} \log {\left (\frac {c}{d} + x + \frac {1}{d} \right )}}{d} - \frac {b e^{2} \operatorname {acoth}{\left (c + d x \right )}}{d} + \frac {b e f x}{d} + \frac {b f^{2} x^{2}}{6 d} - \frac {b e f \operatorname {acoth}{\left (c + d x \right )}}{d^{2}} + \frac {b f^{2} \log {\left (\frac {c}{d} + x + \frac {1}{d} \right )}}{3 d^{3}} - \frac {b f^{2} \operatorname {acoth}{\left (c + d x \right )}}{3 d^{3}} & \text {for}\: d \neq 0 \\\left (a + b \operatorname {acoth}{\left (c \right )}\right ) \left (e^{2} x + e f x^{2} + \frac {f^{2} x^{3}}{3}\right ) & \text {otherwise} \end {cases} \]
[In]
[Out]
none
Time = 0.19 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.72 \[ \int (e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right ) \, dx=\frac {1}{3} \, a f^{2} x^{3} + a e f x^{2} + \frac {1}{2} \, {\left (2 \, x^{2} \operatorname {arcoth}\left (d x + c\right ) + d {\left (\frac {2 \, x}{d^{2}} - \frac {{\left (c^{2} + 2 \, c + 1\right )} \log \left (d x + c + 1\right )}{d^{3}} + \frac {{\left (c^{2} - 2 \, c + 1\right )} \log \left (d x + c - 1\right )}{d^{3}}\right )}\right )} b e f + \frac {1}{6} \, {\left (2 \, x^{3} \operatorname {arcoth}\left (d x + c\right ) + d {\left (\frac {d x^{2} - 4 \, c x}{d^{3}} + \frac {{\left (c^{3} + 3 \, c^{2} + 3 \, c + 1\right )} \log \left (d x + c + 1\right )}{d^{4}} - \frac {{\left (c^{3} - 3 \, c^{2} + 3 \, c - 1\right )} \log \left (d x + c - 1\right )}{d^{4}}\right )}\right )} b f^{2} + a e^{2} x + \frac {{\left (2 \, {\left (d x + c\right )} \operatorname {arcoth}\left (d x + c\right ) + \log \left (-{\left (d x + c\right )}^{2} + 1\right )\right )} b e^{2}}{2 \, d} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 973 vs. \(2 (112) = 224\).
Time = 0.30 (sec) , antiderivative size = 973, normalized size of antiderivative = 8.11 \[ \int (e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right ) \, dx=\text {Too large to display} \]
[In]
[Out]
Time = 4.69 (sec) , antiderivative size = 386, normalized size of antiderivative = 3.22 \[ \int (e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right ) \, dx=x^2\,\left (\frac {f\,\left (b\,f+6\,a\,c\,f+6\,a\,d\,e\right )}{6\,d}-\frac {a\,c\,f^2}{d}\right )-\ln \left (1-\frac {1}{c+d\,x}\right )\,\left (\frac {b\,e^2\,x}{2}+\frac {b\,e\,f\,x^2}{2}+\frac {b\,f^2\,x^3}{6}\right )-x\,\left (\frac {2\,c\,\left (\frac {f\,\left (b\,f+6\,a\,c\,f+6\,a\,d\,e\right )}{3\,d}-\frac {2\,a\,c\,f^2}{d}\right )}{d}-\frac {3\,a\,c^2\,f^2+12\,a\,c\,d\,e\,f+3\,a\,d^2\,e^2+3\,b\,d\,e\,f-3\,a\,f^2}{3\,d^2}+\frac {a\,f^2\,\left (3\,c^2-3\right )}{3\,d^2}\right )+\ln \left (\frac {1}{c+d\,x}+1\right )\,\left (\frac {b\,e^2\,x}{2}+\frac {b\,e\,f\,x^2}{2}+\frac {b\,f^2\,x^3}{6}\right )+\frac {a\,f^2\,x^3}{3}+\frac {\ln \left (c+d\,x-1\right )\,\left (\frac {b\,f^2}{6}+d\,\left (\frac {b\,e\,f\,c^2}{2}-b\,e\,f\,c+\frac {b\,e\,f}{2}\right )+d^2\,\left (\frac {b\,e^2}{2}-\frac {b\,c\,e^2}{2}\right )+\frac {b\,c^2\,f^2}{2}-\frac {b\,c^3\,f^2}{6}-\frac {b\,c\,f^2}{2}\right )}{d^3}+\frac {\ln \left (c+d\,x+1\right )\,\left (\frac {b\,f^2}{6}-d\,\left (\frac {b\,e\,f\,c^2}{2}+b\,e\,f\,c+\frac {b\,e\,f}{2}\right )+d^2\,\left (\frac {b\,e^2}{2}+\frac {b\,c\,e^2}{2}\right )+\frac {b\,c^2\,f^2}{2}+\frac {b\,c^3\,f^2}{6}+\frac {b\,c\,f^2}{2}\right )}{d^3} \]
[In]
[Out]