Integrand size = 20, antiderivative size = 171 \[ \int e^{-\text {arctanh}(a x)} (c-a c x)^{7/2} \, dx=\frac {4096 c^4 \sqrt {1-a^2 x^2}}{315 a \sqrt {c-a c x}}+\frac {1024 c^3 \sqrt {c-a c x} \sqrt {1-a^2 x^2}}{315 a}+\frac {128 c^2 (c-a c x)^{3/2} \sqrt {1-a^2 x^2}}{105 a}+\frac {32 c (c-a c x)^{5/2} \sqrt {1-a^2 x^2}}{63 a}+\frac {2 (c-a c x)^{7/2} \sqrt {1-a^2 x^2}}{9 a} \] Output:
4096/315*c^4*(-a^2*x^2+1)^(1/2)/a/(-a*c*x+c)^(1/2)+1024/315*c^3*(-a*c*x+c) ^(1/2)*(-a^2*x^2+1)^(1/2)/a+128/105*c^2*(-a*c*x+c)^(3/2)*(-a^2*x^2+1)^(1/2 )/a+32/63*c*(-a*c*x+c)^(5/2)*(-a^2*x^2+1)^(1/2)/a+2/9*(-a*c*x+c)^(7/2)*(-a ^2*x^2+1)^(1/2)/a
Time = 0.05 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.38 \[ \int e^{-\text {arctanh}(a x)} (c-a c x)^{7/2} \, dx=\frac {2 c^4 \sqrt {1-a^2 x^2} \left (2867-1276 a x+642 a^2 x^2-220 a^3 x^3+35 a^4 x^4\right )}{315 a \sqrt {c-a c x}} \] Input:
Integrate[(c - a*c*x)^(7/2)/E^ArcTanh[a*x],x]
Output:
(2*c^4*Sqrt[1 - a^2*x^2]*(2867 - 1276*a*x + 642*a^2*x^2 - 220*a^3*x^3 + 35 *a^4*x^4))/(315*a*Sqrt[c - a*c*x])
Time = 0.58 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.11, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, 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.
\(\displaystyle \int e^{-\text {arctanh}(a x)} (c-a c x)^{7/2} \, dx\) |
\(\Big \downarrow \) 6677 |
\(\displaystyle \frac {\int \frac {(c-a c x)^{9/2}}{\sqrt {1-a^2 x^2}}dx}{c}\) |
\(\Big \downarrow \) 459 |
\(\displaystyle \frac {\frac {16}{9} c \int \frac {(c-a c x)^{7/2}}{\sqrt {1-a^2 x^2}}dx+\frac {2 c \sqrt {1-a^2 x^2} (c-a c x)^{7/2}}{9 a}}{c}\) |
\(\Big \downarrow \) 459 |
\(\displaystyle \frac {\frac {16}{9} c \left (\frac {12}{7} c \int \frac {(c-a c x)^{5/2}}{\sqrt {1-a^2 x^2}}dx+\frac {2 c \sqrt {1-a^2 x^2} (c-a c x)^{5/2}}{7 a}\right )+\frac {2 c \sqrt {1-a^2 x^2} (c-a c x)^{7/2}}{9 a}}{c}\) |
\(\Big \downarrow \) 459 |
\(\displaystyle \frac {\frac {16}{9} c \left (\frac {12}{7} c \left (\frac {8}{5} c \int \frac {(c-a c x)^{3/2}}{\sqrt {1-a^2 x^2}}dx+\frac {2 c \sqrt {1-a^2 x^2} (c-a c x)^{3/2}}{5 a}\right )+\frac {2 c \sqrt {1-a^2 x^2} (c-a c x)^{5/2}}{7 a}\right )+\frac {2 c \sqrt {1-a^2 x^2} (c-a c x)^{7/2}}{9 a}}{c}\) |
\(\Big \downarrow \) 459 |
\(\displaystyle \frac {\frac {16}{9} c \left (\frac {12}{7} c \left (\frac {8}{5} c \left (\frac {4}{3} c \int \frac {\sqrt {c-a c x}}{\sqrt {1-a^2 x^2}}dx+\frac {2 c \sqrt {1-a^2 x^2} \sqrt {c-a c x}}{3 a}\right )+\frac {2 c \sqrt {1-a^2 x^2} (c-a c x)^{3/2}}{5 a}\right )+\frac {2 c \sqrt {1-a^2 x^2} (c-a c x)^{5/2}}{7 a}\right )+\frac {2 c \sqrt {1-a^2 x^2} (c-a c x)^{7/2}}{9 a}}{c}\) |
\(\Big \downarrow \) 458 |
\(\displaystyle \frac {\frac {16}{9} c \left (\frac {12}{7} c \left (\frac {8}{5} c \left (\frac {8 c^2 \sqrt {1-a^2 x^2}}{3 a \sqrt {c-a c x}}+\frac {2 c \sqrt {1-a^2 x^2} \sqrt {c-a c x}}{3 a}\right )+\frac {2 c \sqrt {1-a^2 x^2} (c-a c x)^{3/2}}{5 a}\right )+\frac {2 c \sqrt {1-a^2 x^2} (c-a c x)^{5/2}}{7 a}\right )+\frac {2 c \sqrt {1-a^2 x^2} (c-a c x)^{7/2}}{9 a}}{c}\) |
Input:
Int[(c - a*c*x)^(7/2)/E^ArcTanh[a*x],x]
Output:
((2*c*(c - a*c*x)^(7/2)*Sqrt[1 - a^2*x^2])/(9*a) + (16*c*((2*c*(c - a*c*x) ^(5/2)*Sqrt[1 - a^2*x^2])/(7*a) + (12*c*((2*c*(c - a*c*x)^(3/2)*Sqrt[1 - a ^2*x^2])/(5*a) + (8*c*((8*c^2*Sqrt[1 - a^2*x^2])/(3*a*Sqrt[c - a*c*x]) + ( 2*c*Sqrt[c - a*c*x]*Sqrt[1 - a^2*x^2])/(3*a)))/5))/7))/9)/c
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 = 0.10 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.37
method | result | size |
gosper | \(\frac {2 \sqrt {-a^{2} x^{2}+1}\, \left (-a c x +c \right )^{\frac {7}{2}} \left (35 a^{4} x^{4}-220 a^{3} x^{3}+642 a^{2} x^{2}-1276 a x +2867\right )}{315 \left (a x -1\right )^{4} a}\) | \(64\) |
orering | \(\frac {2 \sqrt {-a^{2} x^{2}+1}\, \left (-a c x +c \right )^{\frac {7}{2}} \left (35 a^{4} x^{4}-220 a^{3} x^{3}+642 a^{2} x^{2}-1276 a x +2867\right )}{315 \left (a x -1\right )^{4} a}\) | \(64\) |
default | \(-\frac {2 \sqrt {-c \left (a x -1\right )}\, \sqrt {-a^{2} x^{2}+1}\, c^{3} \left (35 a^{4} x^{4}-220 a^{3} x^{3}+642 a^{2} x^{2}-1276 a x +2867\right )}{315 \left (a x -1\right ) a}\) | \(68\) |
risch | \(-\frac {2 \sqrt {-\frac {\left (-a^{2} x^{2}+1\right ) c}{a x -1}}\, \left (a x -1\right ) c^{4} \left (35 a^{4} x^{4}-220 a^{3} x^{3}+642 a^{2} x^{2}-1276 a x +2867\right ) \left (a x +1\right )}{315 \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a x -1\right )}\, a \sqrt {c \left (a x +1\right )}}\) | \(102\) |
Input:
int((-a*c*x+c)^(7/2)/(a*x+1)*(-a^2*x^2+1)^(1/2),x,method=_RETURNVERBOSE)
Output:
2/315*(-a^2*x^2+1)^(1/2)*(-a*c*x+c)^(7/2)*(35*a^4*x^4-220*a^3*x^3+642*a^2* x^2-1276*a*x+2867)/(a*x-1)^4/a
Time = 0.07 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.47 \[ \int e^{-\text {arctanh}(a x)} (c-a c x)^{7/2} \, dx=-\frac {2 \, {\left (35 \, a^{4} c^{3} x^{4} - 220 \, a^{3} c^{3} x^{3} + 642 \, a^{2} c^{3} x^{2} - 1276 \, a c^{3} x + 2867 \, c^{3}\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {-a c x + c}}{315 \, {\left (a^{2} x - a\right )}} \] Input:
integrate((-a*c*x+c)^(7/2)/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm="fricas ")
Output:
-2/315*(35*a^4*c^3*x^4 - 220*a^3*c^3*x^3 + 642*a^2*c^3*x^2 - 1276*a*c^3*x + 2867*c^3)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)/(a^2*x - a)
\[ \int e^{-\text {arctanh}(a x)} (c-a c x)^{7/2} \, dx=\int \frac {\left (- c \left (a x - 1\right )\right )^{\frac {7}{2}} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}{a x + 1}\, dx \] Input:
integrate((-a*c*x+c)**(7/2)/(a*x+1)*(-a**2*x**2+1)**(1/2),x)
Output:
Integral((-c*(a*x - 1))**(7/2)*sqrt(-(a*x - 1)*(a*x + 1))/(a*x + 1), x)
Time = 0.04 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.42 \[ \int e^{-\text {arctanh}(a x)} (c-a c x)^{7/2} \, dx=\frac {2 \, {\left (35 \, a^{4} c^{\frac {7}{2}} x^{4} - 220 \, a^{3} c^{\frac {7}{2}} x^{3} + 642 \, a^{2} c^{\frac {7}{2}} x^{2} - 1276 \, a c^{\frac {7}{2}} x + 2867 \, c^{\frac {7}{2}}\right )} \sqrt {a x + 1} {\left (a x - 1\right )}}{315 \, {\left (a^{2} x - a\right )}} \] Input:
integrate((-a*c*x+c)^(7/2)/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm="maxima ")
Output:
2/315*(35*a^4*c^(7/2)*x^4 - 220*a^3*c^(7/2)*x^3 + 642*a^2*c^(7/2)*x^2 - 12 76*a*c^(7/2)*x + 2867*c^(7/2))*sqrt(a*x + 1)*(a*x - 1)/(a^2*x - a)
Exception generated. \[ \int e^{-\text {arctanh}(a x)} (c-a c x)^{7/2} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((-a*c*x+c)^(7/2)/(a*x+1)*(-a^2*x^2+1)^(1/2),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Time = 22.84 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.51 \[ \int e^{-\text {arctanh}(a x)} (c-a c x)^{7/2} \, dx=-\frac {4096\,c^3\,\sqrt {1-a^2\,x^2}\,\sqrt {c-a\,c\,x}}{315\,a\,\left (a\,x-1\right )}-\frac {2\,c^3\,\sqrt {1-a^2\,x^2}\,\sqrt {c-a\,c\,x}\,\left (35\,a^3\,x^3-185\,a^2\,x^2+457\,a\,x-819\right )}{315\,a} \] Input:
int(((1 - a^2*x^2)^(1/2)*(c - a*c*x)^(7/2))/(a*x + 1),x)
Output:
- (4096*c^3*(1 - a^2*x^2)^(1/2)*(c - a*c*x)^(1/2))/(315*a*(a*x - 1)) - (2* c^3*(1 - a^2*x^2)^(1/2)*(c - a*c*x)^(1/2)*(457*a*x - 185*a^2*x^2 + 35*a^3* x^3 - 819))/(315*a)
Time = 0.17 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.27 \[ \int e^{-\text {arctanh}(a x)} (c-a c x)^{7/2} \, dx=\frac {2 \sqrt {c}\, \sqrt {a x +1}\, c^{3} \left (35 a^{4} x^{4}-220 a^{3} x^{3}+642 a^{2} x^{2}-1276 a x +2867\right )}{315 a} \] Input:
int((-a*c*x+c)^(7/2)/(a*x+1)*(-a^2*x^2+1)^(1/2),x)
Output:
(2*sqrt(c)*sqrt(a*x + 1)*c**3*(35*a**4*x**4 - 220*a**3*x**3 + 642*a**2*x** 2 - 1276*a*x + 2867))/(315*a)