Integrand size = 24, antiderivative size = 139 \[ \int e^{3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx=\frac {2 c^3 (1+a x)^6 \sqrt {c-a^2 c x^2}}{3 a \sqrt {1-a^2 x^2}}-\frac {4 c^3 (1+a x)^7 \sqrt {c-a^2 c x^2}}{7 a \sqrt {1-a^2 x^2}}+\frac {c^3 (1+a x)^8 \sqrt {c-a^2 c x^2}}{8 a \sqrt {1-a^2 x^2}} \] Output:
2/3*c^3*(a*x+1)^6*(-a^2*c*x^2+c)^(1/2)/a/(-a^2*x^2+1)^(1/2)-4/7*c^3*(a*x+1 )^7*(-a^2*c*x^2+c)^(1/2)/a/(-a^2*x^2+1)^(1/2)+1/8*c^3*(a*x+1)^8*(-a^2*c*x^ 2+c)^(1/2)/a/(-a^2*x^2+1)^(1/2)
Time = 0.03 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.43 \[ \int e^{3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx=\frac {c^3 (1+a x)^6 \left (37-54 a x+21 a^2 x^2\right ) \sqrt {c-a^2 c x^2}}{168 a \sqrt {1-a^2 x^2}} \] Input:
Integrate[E^(3*ArcTanh[a*x])*(c - a^2*c*x^2)^(7/2),x]
Output:
(c^3*(1 + a*x)^6*(37 - 54*a*x + 21*a^2*x^2)*Sqrt[c - a^2*c*x^2])/(168*a*Sq rt[1 - a^2*x^2])
Time = 0.41 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.55, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6693, 6690, 49, 2009}
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.
\(\displaystyle \int e^{3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx\) |
\(\Big \downarrow \) 6693 |
\(\displaystyle \frac {c^3 \sqrt {c-a^2 c x^2} \int e^{3 \text {arctanh}(a x)} \left (1-a^2 x^2\right )^{7/2}dx}{\sqrt {1-a^2 x^2}}\) |
\(\Big \downarrow \) 6690 |
\(\displaystyle \frac {c^3 \sqrt {c-a^2 c x^2} \int (1-a x)^2 (a x+1)^5dx}{\sqrt {1-a^2 x^2}}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle \frac {c^3 \sqrt {c-a^2 c x^2} \int \left ((a x+1)^7-4 (a x+1)^6+4 (a x+1)^5\right )dx}{\sqrt {1-a^2 x^2}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {c^3 \left (\frac {(a x+1)^8}{8 a}-\frac {4 (a x+1)^7}{7 a}+\frac {2 (a x+1)^6}{3 a}\right ) \sqrt {c-a^2 c x^2}}{\sqrt {1-a^2 x^2}}\) |
Input:
Int[E^(3*ArcTanh[a*x])*(c - a^2*c*x^2)^(7/2),x]
Output:
(c^3*Sqrt[c - a^2*c*x^2]*((2*(1 + a*x)^6)/(3*a) - (4*(1 + a*x)^7)/(7*a) + (1 + a*x)^8/(8*a)))/Sqrt[1 - a^2*x^2]
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[c^p Int[(1 - a*x)^(p - n/2)*(1 + a*x)^(p + n/2), x], x] /; FreeQ[{a , c, d, n, p}, x] && EqQ[a^2*c + d, 0] && (IntegerQ[p] || GtQ[c, 0])
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c^IntPart[p]*((c + d*x^2)^FracPart[p]/(1 - a^2*x^2)^FracPart[p]) Int [(1 - a^2*x^2)^p*E^(n*ArcTanh[a*x]), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c + d, 0] && !(IntegerQ[p] || GtQ[c, 0])
Time = 0.33 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.63
method | result | size |
default | \(\frac {\left (21 a^{7} x^{7}+72 x^{6} a^{6}+28 a^{5} x^{5}-168 a^{4} x^{4}-210 a^{3} x^{3}+56 a^{2} x^{2}+252 a x +168\right ) \sqrt {-c \left (a^{2} x^{2}-1\right )}\, c^{3} x}{168 \sqrt {-a^{2} x^{2}+1}}\) | \(87\) |
gosper | \(\frac {x \left (21 a^{7} x^{7}+72 x^{6} a^{6}+28 a^{5} x^{5}-168 a^{4} x^{4}-210 a^{3} x^{3}+56 a^{2} x^{2}+252 a x +168\right ) \left (-a^{2} c \,x^{2}+c \right )^{\frac {7}{2}}}{168 \left (a x -1\right )^{2} \left (a x +1\right )^{2} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}\) | \(97\) |
orering | \(\frac {x \left (21 a^{7} x^{7}+72 x^{6} a^{6}+28 a^{5} x^{5}-168 a^{4} x^{4}-210 a^{3} x^{3}+56 a^{2} x^{2}+252 a x +168\right ) \left (-a^{2} c \,x^{2}+c \right )^{\frac {7}{2}}}{168 \left (a x -1\right )^{2} \left (a x +1\right )^{2} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}\) | \(97\) |
Input:
int((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a^2*c*x^2+c)^(7/2),x,method=_RETURNVERB OSE)
Output:
1/168*(21*a^7*x^7+72*a^6*x^6+28*a^5*x^5-168*a^4*x^4-210*a^3*x^3+56*a^2*x^2 +252*a*x+168)/(-a^2*x^2+1)^(1/2)*(-c*(a^2*x^2-1))^(1/2)*c^3*x
Time = 0.07 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.86 \[ \int e^{3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx=-\frac {{\left (21 \, a^{7} c^{3} x^{8} + 72 \, a^{6} c^{3} x^{7} + 28 \, a^{5} c^{3} x^{6} - 168 \, a^{4} c^{3} x^{5} - 210 \, a^{3} c^{3} x^{4} + 56 \, a^{2} c^{3} x^{3} + 252 \, a c^{3} x^{2} + 168 \, c^{3} x\right )} \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1}}{168 \, {\left (a^{2} x^{2} - 1\right )}} \] Input:
integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a^2*c*x^2+c)^(7/2),x, algorithm=" fricas")
Output:
-1/168*(21*a^7*c^3*x^8 + 72*a^6*c^3*x^7 + 28*a^5*c^3*x^6 - 168*a^4*c^3*x^5 - 210*a^3*c^3*x^4 + 56*a^2*c^3*x^3 + 252*a*c^3*x^2 + 168*c^3*x)*sqrt(-a^2 *c*x^2 + c)*sqrt(-a^2*x^2 + 1)/(a^2*x^2 - 1)
\[ \int e^{3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx=\int \frac {\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {7}{2}} \left (a x + 1\right )^{3}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(-a**2*c*x**2+c)**(7/2),x)
Output:
Integral((-c*(a*x - 1)*(a*x + 1))**(7/2)*(a*x + 1)**3/(-(a*x - 1)*(a*x + 1 ))**(3/2), x)
Leaf count of result is larger than twice the leaf count of optimal. 323 vs. \(2 (121) = 242\).
Time = 0.08 (sec) , antiderivative size = 323, normalized size of antiderivative = 2.32 \[ \int e^{3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx=\frac {1}{5} \, a^{4} c^{\frac {7}{2}} x^{5} - \frac {2}{3} \, a^{2} c^{\frac {7}{2}} x^{3} + c^{\frac {7}{2}} x - \frac {1}{24} \, {\left (\frac {3 \, a^{6} c^{4} x^{10}}{\sqrt {a^{4} c x^{4} - 2 \, a^{2} c x^{2} + c}} - \frac {11 \, a^{4} c^{4} x^{8}}{\sqrt {a^{4} c x^{4} - 2 \, a^{2} c x^{2} + c}} + \frac {14 \, a^{2} c^{4} x^{6}}{\sqrt {a^{4} c x^{4} - 2 \, a^{2} c x^{2} + c}} - \frac {6 \, c^{4} x^{4}}{\sqrt {a^{4} c x^{4} - 2 \, a^{2} c x^{2} + c}}\right )} a^{3} + \frac {1}{35} \, {\left (15 \, a^{4} c^{\frac {7}{2}} x^{7} - 42 \, a^{2} c^{\frac {7}{2}} x^{5} + 35 \, c^{\frac {7}{2}} x^{3}\right )} a^{2} - \frac {1}{2} \, {\left (\frac {a^{6} c^{4} x^{8}}{\sqrt {a^{4} c x^{4} - 2 \, a^{2} c x^{2} + c}} - \frac {4 \, a^{4} c^{4} x^{6}}{\sqrt {a^{4} c x^{4} - 2 \, a^{2} c x^{2} + c}} + \frac {6 \, a^{2} c^{4} x^{4}}{\sqrt {a^{4} c x^{4} - 2 \, a^{2} c x^{2} + c}} - \frac {3 \, c^{4}}{\sqrt {a^{4} c x^{4} - 2 \, a^{2} c x^{2} + c} a^{2}}\right )} a \] Input:
integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a^2*c*x^2+c)^(7/2),x, algorithm=" maxima")
Output:
1/5*a^4*c^(7/2)*x^5 - 2/3*a^2*c^(7/2)*x^3 + c^(7/2)*x - 1/24*(3*a^6*c^4*x^ 10/sqrt(a^4*c*x^4 - 2*a^2*c*x^2 + c) - 11*a^4*c^4*x^8/sqrt(a^4*c*x^4 - 2*a ^2*c*x^2 + c) + 14*a^2*c^4*x^6/sqrt(a^4*c*x^4 - 2*a^2*c*x^2 + c) - 6*c^4*x ^4/sqrt(a^4*c*x^4 - 2*a^2*c*x^2 + c))*a^3 + 1/35*(15*a^4*c^(7/2)*x^7 - 42* a^2*c^(7/2)*x^5 + 35*c^(7/2)*x^3)*a^2 - 1/2*(a^6*c^4*x^8/sqrt(a^4*c*x^4 - 2*a^2*c*x^2 + c) - 4*a^4*c^4*x^6/sqrt(a^4*c*x^4 - 2*a^2*c*x^2 + c) + 6*a^2 *c^4*x^4/sqrt(a^4*c*x^4 - 2*a^2*c*x^2 + c) - 3*c^4/(sqrt(a^4*c*x^4 - 2*a^2 *c*x^2 + c)*a^2))*a
\[ \int e^{3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx=\int { \frac {{\left (-a^{2} c x^{2} + c\right )}^{\frac {7}{2}} {\left (a x + 1\right )}^{3}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a^2*c*x^2+c)^(7/2),x, algorithm=" giac")
Output:
integrate((-a^2*c*x^2 + c)^(7/2)*(a*x + 1)^3/(-a^2*x^2 + 1)^(3/2), x)
Time = 28.96 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.77 \[ \int e^{3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx=\frac {\sqrt {c-a^2\,c\,x^2}\,\left (\frac {a^7\,c^3\,x^8}{8}+\frac {3\,a^6\,c^3\,x^7}{7}+\frac {a^5\,c^3\,x^6}{6}-a^4\,c^3\,x^5-\frac {5\,a^3\,c^3\,x^4}{4}+\frac {a^2\,c^3\,x^3}{3}+\frac {3\,a\,c^3\,x^2}{2}+c^3\,x\right )}{\sqrt {1-a^2\,x^2}} \] Input:
int(((c - a^2*c*x^2)^(7/2)*(a*x + 1)^3)/(1 - a^2*x^2)^(3/2),x)
Output:
((c - a^2*c*x^2)^(1/2)*(c^3*x + (3*a*c^3*x^2)/2 + (a^2*c^3*x^3)/3 - (5*a^3 *c^3*x^4)/4 - a^4*c^3*x^5 + (a^5*c^3*x^6)/6 + (3*a^6*c^3*x^7)/7 + (a^7*c^3 *x^8)/8))/(1 - a^2*x^2)^(1/2)
Time = 0.14 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.45 \[ \int e^{3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx=\frac {\sqrt {c}\, c^{3} x \left (21 a^{7} x^{7}+72 a^{6} x^{6}+28 a^{5} x^{5}-168 a^{4} x^{4}-210 a^{3} x^{3}+56 a^{2} x^{2}+252 a x +168\right )}{168} \] Input:
int((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a^2*c*x^2+c)^(7/2),x)
Output:
(sqrt(c)*c**3*x*(21*a**7*x**7 + 72*a**6*x**6 + 28*a**5*x**5 - 168*a**4*x** 4 - 210*a**3*x**3 + 56*a**2*x**2 + 252*a*x + 168))/168