Integrand size = 18, antiderivative size = 176 \[ \int e^{\text {arctanh}(a x)} (c-a c x)^{9/2} \, dx=\frac {4096 c^6 \left (1-a^2 x^2\right )^{3/2}}{3465 a (c-a c x)^{3/2}}+\frac {1024 c^5 \left (1-a^2 x^2\right )^{3/2}}{1155 a \sqrt {c-a c x}}+\frac {128 c^4 \sqrt {c-a c x} \left (1-a^2 x^2\right )^{3/2}}{231 a}+\frac {32 c^3 (c-a c x)^{3/2} \left (1-a^2 x^2\right )^{3/2}}{99 a}+\frac {2 c^2 (c-a c x)^{5/2} \left (1-a^2 x^2\right )^{3/2}}{11 a} \]
4096/3465*c^6*(-a^2*x^2+1)^(3/2)/a/(-a*c*x+c)^(3/2)+32/99*c^3*(-a*c*x+c)^( 3/2)*(-a^2*x^2+1)^(3/2)/a+2/11*c^2*(-a*c*x+c)^(5/2)*(-a^2*x^2+1)^(3/2)/a+1 024/1155*c^5*(-a^2*x^2+1)^(3/2)/a/(-a*c*x+c)^(1/2)+128/231*c^4*(-a^2*x^2+1 )^(3/2)*(-a*c*x+c)^(1/2)/a
Time = 0.04 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.40 \[ \int e^{\text {arctanh}(a x)} (c-a c x)^{9/2} \, dx=\frac {2 c^4 (1+a x)^{3/2} \sqrt {c-a c x} \left (5419-6396 a x+4530 a^2 x^2-1820 a^3 x^3+315 a^4 x^4\right )}{3465 a \sqrt {1-a x}} \]
(2*c^4*(1 + a*x)^(3/2)*Sqrt[c - a*c*x]*(5419 - 6396*a*x + 4530*a^2*x^2 - 1 820*a^3*x^3 + 315*a^4*x^4))/(3465*a*Sqrt[1 - a*x])
Time = 0.36 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.07, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6677, 459, 459, 459, 459, 458}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int e^{\text {arctanh}(a x)} (c-a c x)^{9/2} \, dx\) |
\(\Big \downarrow \) 6677 |
\(\displaystyle c \int (c-a c x)^{7/2} \sqrt {1-a^2 x^2}dx\) |
\(\Big \downarrow \) 459 |
\(\displaystyle c \left (\frac {16}{11} c \int (c-a c x)^{5/2} \sqrt {1-a^2 x^2}dx+\frac {2 c \left (1-a^2 x^2\right )^{3/2} (c-a c x)^{5/2}}{11 a}\right )\) |
\(\Big \downarrow \) 459 |
\(\displaystyle c \left (\frac {16}{11} c \left (\frac {4}{3} c \int (c-a c x)^{3/2} \sqrt {1-a^2 x^2}dx+\frac {2 c \left (1-a^2 x^2\right )^{3/2} (c-a c x)^{3/2}}{9 a}\right )+\frac {2 c \left (1-a^2 x^2\right )^{3/2} (c-a c x)^{5/2}}{11 a}\right )\) |
\(\Big \downarrow \) 459 |
\(\displaystyle c \left (\frac {16}{11} c \left (\frac {4}{3} c \left (\frac {8}{7} c \int \sqrt {c-a c x} \sqrt {1-a^2 x^2}dx+\frac {2 c \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a c x}}{7 a}\right )+\frac {2 c \left (1-a^2 x^2\right )^{3/2} (c-a c x)^{3/2}}{9 a}\right )+\frac {2 c \left (1-a^2 x^2\right )^{3/2} (c-a c x)^{5/2}}{11 a}\right )\) |
\(\Big \downarrow \) 459 |
\(\displaystyle c \left (\frac {16}{11} c \left (\frac {4}{3} c \left (\frac {8}{7} c \left (\frac {4}{5} c \int \frac {\sqrt {1-a^2 x^2}}{\sqrt {c-a c x}}dx+\frac {2 c \left (1-a^2 x^2\right )^{3/2}}{5 a \sqrt {c-a c x}}\right )+\frac {2 c \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a c x}}{7 a}\right )+\frac {2 c \left (1-a^2 x^2\right )^{3/2} (c-a c x)^{3/2}}{9 a}\right )+\frac {2 c \left (1-a^2 x^2\right )^{3/2} (c-a c x)^{5/2}}{11 a}\right )\) |
\(\Big \downarrow \) 458 |
\(\displaystyle c \left (\frac {16}{11} c \left (\frac {4}{3} c \left (\frac {8}{7} c \left (\frac {8 c^2 \left (1-a^2 x^2\right )^{3/2}}{15 a (c-a c x)^{3/2}}+\frac {2 c \left (1-a^2 x^2\right )^{3/2}}{5 a \sqrt {c-a c x}}\right )+\frac {2 c \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a c x}}{7 a}\right )+\frac {2 c \left (1-a^2 x^2\right )^{3/2} (c-a c x)^{3/2}}{9 a}\right )+\frac {2 c \left (1-a^2 x^2\right )^{3/2} (c-a c x)^{5/2}}{11 a}\right )\) |
c*((2*c*(c - a*c*x)^(5/2)*(1 - a^2*x^2)^(3/2))/(11*a) + (16*c*((2*c*(c - a *c*x)^(3/2)*(1 - a^2*x^2)^(3/2))/(9*a) + (4*c*((2*c*Sqrt[c - a*c*x]*(1 - a ^2*x^2)^(3/2))/(7*a) + (8*c*((8*c^2*(1 - a^2*x^2)^(3/2))/(15*a*(c - a*c*x) ^(3/2)) + (2*c*(1 - a^2*x^2)^(3/2))/(5*a*Sqrt[c - a*c*x])))/7))/3))/11)
3.3.26.3.1 Defintions of rubi rules used
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ d*(c + d*x)^(n - 1)*((a + b*x^2)^(p + 1)/(b*(p + 1))), x] /; FreeQ[{a, b, c , d, n, p}, x] && EqQ[b*c^2 + a*d^2, 0] && EqQ[n + p, 0]
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ d*(c + d*x)^(n - 1)*((a + b*x^2)^(p + 1)/(b*(n + 2*p + 1))), x] + Simp[2*c* (Simplify[n + p]/(n + 2*p + 1)) Int[(c + d*x)^(n - 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, n, p}, x] && EqQ[b*c^2 + a*d^2, 0] && IGtQ[Simplif y[n + p], 0]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_))^(p_.), x_Symbol] :> S imp[c^n Int[(c + d*x)^(p - n)*(1 - a^2*x^2)^(n/2), x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[a*c + d, 0] && IntegerQ[(n - 1)/2] && IntegerQ[2*p]
Time = 0.08 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.40
method | result | size |
gosper | \(\frac {2 \left (a x +1\right )^{2} \left (315 a^{4} x^{4}-1820 a^{3} x^{3}+4530 a^{2} x^{2}-6396 a x +5419\right ) \left (-a c x +c \right )^{\frac {9}{2}}}{3465 a \left (a x -1\right )^{4} \sqrt {-a^{2} x^{2}+1}}\) | \(71\) |
default | \(-\frac {2 \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}\, c^{4} \left (a x +1\right ) \left (315 a^{4} x^{4}-1820 a^{3} x^{3}+4530 a^{2} x^{2}-6396 a x +5419\right )}{3465 \left (a x -1\right ) a}\) | \(73\) |
risch | \(-\frac {2 \sqrt {-\frac {\left (-a^{2} x^{2}+1\right ) c}{a x -1}}\, \left (a x -1\right ) c^{5} \left (315 a^{5} x^{5}-1505 a^{4} x^{4}+2710 a^{3} x^{3}-1866 a^{2} x^{2}-977 a x +5419\right ) \left (a x +1\right )}{3465 \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}\, a \sqrt {c \left (a x +1\right )}}\) | \(110\) |
2/3465*(a*x+1)^2*(315*a^4*x^4-1820*a^3*x^3+4530*a^2*x^2-6396*a*x+5419)*(-a *c*x+c)^(9/2)/a/(a*x-1)^4/(-a^2*x^2+1)^(1/2)
Time = 0.25 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.52 \[ \int e^{\text {arctanh}(a x)} (c-a c x)^{9/2} \, dx=-\frac {2 \, {\left (315 \, a^{5} c^{4} x^{5} - 1505 \, a^{4} c^{4} x^{4} + 2710 \, a^{3} c^{4} x^{3} - 1866 \, a^{2} c^{4} x^{2} - 977 \, a c^{4} x + 5419 \, c^{4}\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c}}{3465 \, {\left (a^{2} x - a\right )}} \]
-2/3465*(315*a^5*c^4*x^5 - 1505*a^4*c^4*x^4 + 2710*a^3*c^4*x^3 - 1866*a^2* c^4*x^2 - 977*a*c^4*x + 5419*c^4)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)/(a^2 *x - a)
\[ \int e^{\text {arctanh}(a x)} (c-a c x)^{9/2} \, dx=\int \frac {\left (- c \left (a x - 1\right )\right )^{\frac {9}{2}} \left (a x + 1\right )}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]
Time = 0.20 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.85 \[ \int e^{\text {arctanh}(a x)} (c-a c x)^{9/2} \, dx=\frac {2 \, {\left (35 \, a^{6} c^{\frac {9}{2}} x^{6} - 175 \, a^{5} c^{\frac {9}{2}} x^{5} + 360 \, a^{4} c^{\frac {9}{2}} x^{4} - 422 \, a^{3} c^{\frac {9}{2}} x^{3} + 459 \, a^{2} c^{\frac {9}{2}} x^{2} - 1451 \, a c^{\frac {9}{2}} x - 2902 \, c^{\frac {9}{2}}\right )}}{385 \, \sqrt {a x + 1} a} + \frac {2 \, {\left (35 \, a^{5} c^{\frac {9}{2}} x^{5} - 185 \, a^{4} c^{\frac {9}{2}} x^{4} + 422 \, a^{3} c^{\frac {9}{2}} x^{3} - 634 \, a^{2} c^{\frac {9}{2}} x^{2} + 1591 \, a c^{\frac {9}{2}} x + 2867 \, c^{\frac {9}{2}}\right )}}{315 \, \sqrt {a x + 1} a} \]
2/385*(35*a^6*c^(9/2)*x^6 - 175*a^5*c^(9/2)*x^5 + 360*a^4*c^(9/2)*x^4 - 42 2*a^3*c^(9/2)*x^3 + 459*a^2*c^(9/2)*x^2 - 1451*a*c^(9/2)*x - 2902*c^(9/2)) /(sqrt(a*x + 1)*a) + 2/315*(35*a^5*c^(9/2)*x^5 - 185*a^4*c^(9/2)*x^4 + 422 *a^3*c^(9/2)*x^3 - 634*a^2*c^(9/2)*x^2 + 1591*a*c^(9/2)*x + 2867*c^(9/2))/ (sqrt(a*x + 1)*a)
Time = 0.30 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.49 \[ \int e^{\text {arctanh}(a x)} (c-a c x)^{9/2} \, dx=-\frac {2 \, {\left (4096 \, \sqrt {2} c^{\frac {7}{2}} - \frac {315 \, {\left (a c x + c\right )}^{\frac {11}{2}} - 3080 \, {\left (a c x + c\right )}^{\frac {9}{2}} c + 11880 \, {\left (a c x + c\right )}^{\frac {7}{2}} c^{2} - 22176 \, {\left (a c x + c\right )}^{\frac {5}{2}} c^{3} + 18480 \, {\left (a c x + c\right )}^{\frac {3}{2}} c^{4}}{c^{2}}\right )} c^{2}}{3465 \, a {\left | c \right |}} \]
-2/3465*(4096*sqrt(2)*c^(7/2) - (315*(a*c*x + c)^(11/2) - 3080*(a*c*x + c) ^(9/2)*c + 11880*(a*c*x + c)^(7/2)*c^2 - 22176*(a*c*x + c)^(5/2)*c^3 + 184 80*(a*c*x + c)^(3/2)*c^4)/c^2)*c^2/(a*abs(c))
Time = 3.88 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.51 \[ \int e^{\text {arctanh}(a x)} (c-a c x)^{9/2} \, dx=\frac {\sqrt {c-a\,c\,x}\,\left (\frac {8884\,c^4\,x}{3465}+\frac {10838\,c^4}{3465\,a}-\frac {5686\,a\,c^4\,x^2}{3465}+\frac {1688\,a^2\,c^4\,x^3}{3465}+\frac {482\,a^3\,c^4\,x^4}{693}-\frac {68\,a^4\,c^4\,x^5}{99}+\frac {2\,a^5\,c^4\,x^6}{11}\right )}{\sqrt {1-a^2\,x^2}} \]