Integrand size = 22, antiderivative size = 66 \[ \int e^{4 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^5 \, dx=\frac {c^5 (1+a x)^8}{a}-\frac {4 c^5 (1+a x)^9}{3 a}+\frac {3 c^5 (1+a x)^{10}}{5 a}-\frac {c^5 (1+a x)^{11}}{11 a} \]
[Out]
Time = 0.07 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6302, 6275, 45} \[ \int e^{4 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^5 \, dx=-\frac {c^5 (a x+1)^{11}}{11 a}+\frac {3 c^5 (a x+1)^{10}}{5 a}-\frac {4 c^5 (a x+1)^9}{3 a}+\frac {c^5 (a x+1)^8}{a} \]
[In]
[Out]
Rule 45
Rule 6275
Rule 6302
Rubi steps \begin{align*} \text {integral}& = \int e^{4 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^5 \, dx \\ & = c^5 \int (1-a x)^3 (1+a x)^7 \, dx \\ & = c^5 \int \left (8 (1+a x)^7-12 (1+a x)^8+6 (1+a x)^9-(1+a x)^{10}\right ) \, dx \\ & = \frac {c^5 (1+a x)^8}{a}-\frac {4 c^5 (1+a x)^9}{3 a}+\frac {3 c^5 (1+a x)^{10}}{5 a}-\frac {c^5 (1+a x)^{11}}{11 a} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.59 \[ \int e^{4 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^5 \, dx=-\frac {c^5 (1+a x)^8 \left (-29+67 a x-54 a^2 x^2+15 a^3 x^3\right )}{165 a} \]
[In]
[Out]
Time = 0.69 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.14
method | result | size |
default | \(c^{5} \left (-\frac {1}{11} a^{10} x^{11}-\frac {2}{5} a^{9} x^{10}-\frac {1}{3} a^{8} x^{9}+a^{7} x^{8}+2 a^{6} x^{7}-\frac {14}{5} a^{4} x^{5}-2 a^{3} x^{4}+a^{2} x^{3}+2 a \,x^{2}+x \right )\) | \(75\) |
gosper | \(-\frac {c^{5} x \left (15 a^{10} x^{10}+66 a^{9} x^{9}+55 a^{8} x^{8}-165 a^{7} x^{7}-330 a^{6} x^{6}+462 a^{4} x^{4}+330 a^{3} x^{3}-165 a^{2} x^{2}-330 a x -165\right )}{165}\) | \(77\) |
risch | \(-\frac {1}{11} a^{10} c^{5} x^{11}-\frac {2}{5} a^{9} c^{5} x^{10}-\frac {1}{3} a^{8} c^{5} x^{9}+a^{7} c^{5} x^{8}+2 a^{6} c^{5} x^{7}-\frac {14}{5} a^{4} c^{5} x^{5}-2 a^{3} c^{5} x^{4}+c^{5} a^{2} x^{3}+2 a \,c^{5} x^{2}+c^{5} x\) | \(102\) |
parallelrisch | \(-\frac {1}{11} a^{10} c^{5} x^{11}-\frac {2}{5} a^{9} c^{5} x^{10}-\frac {1}{3} a^{8} c^{5} x^{9}+a^{7} c^{5} x^{8}+2 a^{6} c^{5} x^{7}-\frac {14}{5} a^{4} c^{5} x^{5}-2 a^{3} c^{5} x^{4}+c^{5} a^{2} x^{3}+2 a \,c^{5} x^{2}+c^{5} x\) | \(102\) |
norman | \(\frac {-c^{5} x +a^{7} c^{5} x^{8}+c^{5} a^{2} x^{3}-a \,c^{5} x^{2}+3 a^{3} c^{5} x^{4}+\frac {4}{5} a^{4} c^{5} x^{5}-\frac {14}{5} a^{5} c^{5} x^{6}-2 a^{6} c^{5} x^{7}+\frac {4}{3} a^{8} c^{5} x^{9}+\frac {1}{15} a^{9} c^{5} x^{10}-\frac {17}{55} a^{10} c^{5} x^{11}-\frac {1}{11} a^{11} c^{5} x^{12}}{a x -1}\) | \(133\) |
meijerg | \(\frac {c^{5} \left (-\frac {x a \left (-2730 x^{11} a^{11}-3276 a^{10} x^{10}-4004 a^{9} x^{9}-5005 a^{8} x^{8}-6435 a^{7} x^{7}-8580 a^{6} x^{6}-12012 a^{5} x^{5}-18018 a^{4} x^{4}-30030 a^{3} x^{3}-60060 a^{2} x^{2}-180180 a x +360360\right )}{30030 \left (-a x +1\right )}-12 \ln \left (-a x +1\right )\right )}{a}-\frac {4 c^{5} \left (-\frac {x a \left (-308 a^{9} x^{9}-385 a^{8} x^{8}-495 a^{7} x^{7}-660 a^{6} x^{6}-924 a^{5} x^{5}-1386 a^{4} x^{4}-2310 a^{3} x^{3}-4620 a^{2} x^{2}-13860 a x +27720\right )}{2772 \left (-a x +1\right )}-10 \ln \left (-a x +1\right )\right )}{a}+\frac {5 c^{5} \left (-\frac {a x \left (-45 a^{7} x^{7}-60 a^{6} x^{6}-84 a^{5} x^{5}-126 a^{4} x^{4}-210 a^{3} x^{3}-420 a^{2} x^{2}-1260 a x +2520\right )}{315 \left (-a x +1\right )}-8 \ln \left (-a x +1\right )\right )}{a}-\frac {5 c^{5} \left (-\frac {a x \left (-5 a^{3} x^{3}-10 a^{2} x^{2}-30 a x +60\right )}{15 \left (-a x +1\right )}-4 \ln \left (-a x +1\right )\right )}{a}+\frac {4 c^{5} \left (-\frac {a x \left (-3 a x +6\right )}{3 \left (-a x +1\right )}-2 \ln \left (-a x +1\right )\right )}{a}-\frac {2 c^{5} \left (\frac {x a \left (-252 a^{10} x^{10}-308 a^{9} x^{9}-385 a^{8} x^{8}-495 a^{7} x^{7}-660 a^{6} x^{6}-924 a^{5} x^{5}-1386 a^{4} x^{4}-2310 a^{3} x^{3}-4620 a^{2} x^{2}-13860 a x +27720\right )}{-2520 a x +2520}+11 \ln \left (-a x +1\right )\right )}{a}+\frac {10 c^{5} \left (\frac {a x \left (-35 a^{8} x^{8}-45 a^{7} x^{7}-60 a^{6} x^{6}-84 a^{5} x^{5}-126 a^{4} x^{4}-210 a^{3} x^{3}-420 a^{2} x^{2}-1260 a x +2520\right )}{-280 a x +280}+9 \ln \left (-a x +1\right )\right )}{a}-\frac {20 c^{5} \left (\frac {a x \left (-20 a^{6} x^{6}-28 a^{5} x^{5}-42 a^{4} x^{4}-70 a^{3} x^{3}-140 a^{2} x^{2}-420 a x +840\right )}{-120 a x +120}+7 \ln \left (-a x +1\right )\right )}{a}+\frac {20 c^{5} \left (\frac {a x \left (-3 a^{4} x^{4}-5 a^{3} x^{3}-10 a^{2} x^{2}-30 a x +60\right )}{-12 a x +12}+5 \ln \left (-a x +1\right )\right )}{a}-\frac {10 c^{5} \left (\frac {a x \left (-2 a^{2} x^{2}-6 a x +12\right )}{-4 a x +4}+3 \ln \left (-a x +1\right )\right )}{a}+\frac {2 c^{5} \left (\frac {a x}{-a x +1}+\ln \left (-a x +1\right )\right )}{a}+\frac {c^{5} x}{-a x +1}\) | \(809\) |
[In]
[Out]
none
Time = 0.23 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.53 \[ \int e^{4 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^5 \, dx=-\frac {1}{11} \, a^{10} c^{5} x^{11} - \frac {2}{5} \, a^{9} c^{5} x^{10} - \frac {1}{3} \, a^{8} c^{5} x^{9} + a^{7} c^{5} x^{8} + 2 \, a^{6} c^{5} x^{7} - \frac {14}{5} \, a^{4} c^{5} x^{5} - 2 \, a^{3} c^{5} x^{4} + a^{2} c^{5} x^{3} + 2 \, a c^{5} x^{2} + c^{5} x \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 109 vs. \(2 (54) = 108\).
Time = 0.05 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.65 \[ \int e^{4 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^5 \, dx=- \frac {a^{10} c^{5} x^{11}}{11} - \frac {2 a^{9} c^{5} x^{10}}{5} - \frac {a^{8} c^{5} x^{9}}{3} + a^{7} c^{5} x^{8} + 2 a^{6} c^{5} x^{7} - \frac {14 a^{4} c^{5} x^{5}}{5} - 2 a^{3} c^{5} x^{4} + a^{2} c^{5} x^{3} + 2 a c^{5} x^{2} + c^{5} x \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.53 \[ \int e^{4 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^5 \, dx=-\frac {1}{11} \, a^{10} c^{5} x^{11} - \frac {2}{5} \, a^{9} c^{5} x^{10} - \frac {1}{3} \, a^{8} c^{5} x^{9} + a^{7} c^{5} x^{8} + 2 \, a^{6} c^{5} x^{7} - \frac {14}{5} \, a^{4} c^{5} x^{5} - 2 \, a^{3} c^{5} x^{4} + a^{2} c^{5} x^{3} + 2 \, a c^{5} x^{2} + c^{5} x \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.55 \[ \int e^{4 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^5 \, dx=-\frac {{\left (15 \, c^{5} + \frac {231 \, c^{5}}{a x - 1} + \frac {1540 \, c^{5}}{{\left (a x - 1\right )}^{2}} + \frac {5775 \, c^{5}}{{\left (a x - 1\right )}^{3}} + \frac {13200 \, c^{5}}{{\left (a x - 1\right )}^{4}} + \frac {18480 \, c^{5}}{{\left (a x - 1\right )}^{5}} + \frac {14784 \, c^{5}}{{\left (a x - 1\right )}^{6}} + \frac {5280 \, c^{5}}{{\left (a x - 1\right )}^{7}}\right )} {\left (a x - 1\right )}^{11}}{165 \, a} \]
[In]
[Out]
Time = 3.97 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.53 \[ \int e^{4 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^5 \, dx=-\frac {a^{10}\,c^5\,x^{11}}{11}-\frac {2\,a^9\,c^5\,x^{10}}{5}-\frac {a^8\,c^5\,x^9}{3}+a^7\,c^5\,x^8+2\,a^6\,c^5\,x^7-\frac {14\,a^4\,c^5\,x^5}{5}-2\,a^3\,c^5\,x^4+a^2\,c^5\,x^3+2\,a\,c^5\,x^2+c^5\,x \]
[In]
[Out]