\(\int e^{-2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^3 \, dx\) [827]

Optimal result

Integrand size = 27, antiderivative size = 130 \[ \int e^{-2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^3 \, dx=\frac {4 \sqrt {c-\frac {c}{a^2 x^2}} x}{3 a^3}-\frac {7 \sqrt {c-\frac {c}{a^2 x^2}} x^2}{8 a^2}+\frac {2 \sqrt {c-\frac {c}{a^2 x^2}} x^3}{3 a}-\frac {1}{4} \sqrt {c-\frac {c}{a^2 x^2}} x^4-\frac {7 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {c}}\right )}{8 a^4} \] Output:

4/3*(c-c/a^2/x^2)^(1/2)*x/a^3-7/8*(c-c/a^2/x^2)^(1/2)*x^2/a^2+2/3*(c-c/a^2 
/x^2)^(1/2)*x^3/a-1/4*(c-c/a^2/x^2)^(1/2)*x^4-7/8*c^(1/2)*arctanh((c-c/a^2 
/x^2)^(1/2)/c^(1/2))/a^4
 

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.72 \[ \int e^{-2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^3 \, dx=-\frac {\sqrt {c-\frac {c}{a^2 x^2}} x \left (\sqrt {-1+a^2 x^2} \left (-32+21 a x-16 a^2 x^2+6 a^3 x^3\right )+21 \log \left (a x+\sqrt {-1+a^2 x^2}\right )\right )}{24 a^3 \sqrt {-1+a^2 x^2}} \] Input:

Integrate[(Sqrt[c - c/(a^2*x^2)]*x^3)/E^(2*ArcTanh[a*x]),x]
 

Output:

-1/24*(Sqrt[c - c/(a^2*x^2)]*x*(Sqrt[-1 + a^2*x^2]*(-32 + 21*a*x - 16*a^2* 
x^2 + 6*a^3*x^3) + 21*Log[a*x + Sqrt[-1 + a^2*x^2]]))/(a^3*Sqrt[-1 + a^2*x 
^2])
 

Rubi [A] (verified)

Time = 0.79 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.14, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.481, Rules used = {6709, 570, 541, 25, 27, 533, 25, 27, 533, 25, 27, 455, 223}

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.

\(\displaystyle \int x^3 e^{-2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} \, dx\)

\(\Big \downarrow \) 6709

\(\displaystyle \frac {x \sqrt {c-\frac {c}{a^2 x^2}} \int \frac {x^2 \left (1-a^2 x^2\right )^{3/2}}{(a x+1)^2}dx}{\sqrt {1-a^2 x^2}}\)

\(\Big \downarrow \) 570

\(\displaystyle \frac {x \sqrt {c-\frac {c}{a^2 x^2}} \int \frac {x^2 (1-a x)^2}{\sqrt {1-a^2 x^2}}dx}{\sqrt {1-a^2 x^2}}\)

\(\Big \downarrow \) 541

\(\displaystyle \frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (-\frac {\int -\frac {a^2 x^2 (7-8 a x)}{\sqrt {1-a^2 x^2}}dx}{4 a^2}-\frac {1}{4} x^3 \sqrt {1-a^2 x^2}\right )}{\sqrt {1-a^2 x^2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (\frac {\int \frac {a^2 x^2 (7-8 a x)}{\sqrt {1-a^2 x^2}}dx}{4 a^2}-\frac {1}{4} x^3 \sqrt {1-a^2 x^2}\right )}{\sqrt {1-a^2 x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (\frac {1}{4} \int \frac {x^2 (7-8 a x)}{\sqrt {1-a^2 x^2}}dx-\frac {1}{4} x^3 \sqrt {1-a^2 x^2}\right )}{\sqrt {1-a^2 x^2}}\)

\(\Big \downarrow \) 533

\(\displaystyle \frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (\frac {1}{4} \left (\frac {\int -\frac {a x (16-21 a x)}{\sqrt {1-a^2 x^2}}dx}{3 a^2}+\frac {8 x^2 \sqrt {1-a^2 x^2}}{3 a}\right )-\frac {1}{4} x^3 \sqrt {1-a^2 x^2}\right )}{\sqrt {1-a^2 x^2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (\frac {1}{4} \left (\frac {8 x^2 \sqrt {1-a^2 x^2}}{3 a}-\frac {\int \frac {a x (16-21 a x)}{\sqrt {1-a^2 x^2}}dx}{3 a^2}\right )-\frac {1}{4} x^3 \sqrt {1-a^2 x^2}\right )}{\sqrt {1-a^2 x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (\frac {1}{4} \left (\frac {8 x^2 \sqrt {1-a^2 x^2}}{3 a}-\frac {\int \frac {x (16-21 a x)}{\sqrt {1-a^2 x^2}}dx}{3 a}\right )-\frac {1}{4} x^3 \sqrt {1-a^2 x^2}\right )}{\sqrt {1-a^2 x^2}}\)

\(\Big \downarrow \) 533

\(\displaystyle \frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (\frac {1}{4} \left (\frac {8 x^2 \sqrt {1-a^2 x^2}}{3 a}-\frac {\frac {\int -\frac {a (21-32 a x)}{\sqrt {1-a^2 x^2}}dx}{2 a^2}+\frac {21 x \sqrt {1-a^2 x^2}}{2 a}}{3 a}\right )-\frac {1}{4} x^3 \sqrt {1-a^2 x^2}\right )}{\sqrt {1-a^2 x^2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (\frac {1}{4} \left (\frac {8 x^2 \sqrt {1-a^2 x^2}}{3 a}-\frac {\frac {21 x \sqrt {1-a^2 x^2}}{2 a}-\frac {\int \frac {a (21-32 a x)}{\sqrt {1-a^2 x^2}}dx}{2 a^2}}{3 a}\right )-\frac {1}{4} x^3 \sqrt {1-a^2 x^2}\right )}{\sqrt {1-a^2 x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (\frac {1}{4} \left (\frac {8 x^2 \sqrt {1-a^2 x^2}}{3 a}-\frac {\frac {21 x \sqrt {1-a^2 x^2}}{2 a}-\frac {\int \frac {21-32 a x}{\sqrt {1-a^2 x^2}}dx}{2 a}}{3 a}\right )-\frac {1}{4} x^3 \sqrt {1-a^2 x^2}\right )}{\sqrt {1-a^2 x^2}}\)

\(\Big \downarrow \) 455

\(\displaystyle \frac {x \sqrt {c-\frac {c}{a^2 x^2}} \left (\frac {1}{4} \left (\frac {8 x^2 \sqrt {1-a^2 x^2}}{3 a}-\frac {\frac {21 x \sqrt {1-a^2 x^2}}{2 a}-\frac {21 \int \frac {1}{\sqrt {1-a^2 x^2}}dx+\frac {32 \sqrt {1-a^2 x^2}}{a}}{2 a}}{3 a}\right )-\frac {1}{4} x^3 \sqrt {1-a^2 x^2}\right )}{\sqrt {1-a^2 x^2}}\)

\(\Big \downarrow \) 223

\(\displaystyle \frac {x \left (\frac {1}{4} \left (\frac {8 x^2 \sqrt {1-a^2 x^2}}{3 a}-\frac {\frac {21 x \sqrt {1-a^2 x^2}}{2 a}-\frac {\frac {32 \sqrt {1-a^2 x^2}}{a}+\frac {21 \arcsin (a x)}{a}}{2 a}}{3 a}\right )-\frac {1}{4} x^3 \sqrt {1-a^2 x^2}\right ) \sqrt {c-\frac {c}{a^2 x^2}}}{\sqrt {1-a^2 x^2}}\)

Input:

Int[(Sqrt[c - c/(a^2*x^2)]*x^3)/E^(2*ArcTanh[a*x]),x]
 

Output:

(Sqrt[c - c/(a^2*x^2)]*x*(-1/4*(x^3*Sqrt[1 - a^2*x^2]) + ((8*x^2*Sqrt[1 - 
a^2*x^2])/(3*a) - ((21*x*Sqrt[1 - a^2*x^2])/(2*a) - ((32*Sqrt[1 - a^2*x^2] 
)/a + (21*ArcSin[a*x])/a)/(2*a))/(3*a))/4))/Sqrt[1 - a^2*x^2]
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt 
[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
 

rule 455

Int[((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[d*(( 
a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] + Simp[c   Int[(a + b*x^2)^p, x], x] 
/; FreeQ[{a, b, c, d, p}, x] &&  !LeQ[p, -1]
 

rule 533

Int[(x_)^(m_.)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> 
 Simp[d*x^m*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 2))), x] - Simp[1/(b*(m + 2* 
p + 2))   Int[x^(m - 1)*(a + b*x^2)^p*Simp[a*d*m - b*c*(m + 2*p + 2)*x, x], 
 x], x] /; FreeQ[{a, b, c, d, p}, x] && IGtQ[m, 0] && GtQ[p, -1] && Integer 
Q[2*p]
 

rule 541

Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo 
l] :> Simp[d^n*x^(m + n - 1)*((a + b*x^2)^(p + 1)/(b*(m + n + 2*p + 1))), x 
] + Simp[1/(b*(m + n + 2*p + 1))   Int[x^m*(a + b*x^2)^p*ExpandToSum[b*(m + 
 n + 2*p + 1)*(c + d*x)^n - b*d^n*(m + n + 2*p + 1)*x^n - a*d^n*(m + n - 1) 
*x^(n - 2), x], x], x] /; FreeQ[{a, b, c, d, m, p}, x] && IGtQ[n, 1] && IGt 
Q[m, -2] && GtQ[p, -1] && IntegerQ[2*p]
 

rule 570

Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), 
x_Symbol] :> Simp[c^(2*n)/a^n   Int[(e*x)^m*((a + b*x^2)^(n + p)/(c - d*x)^ 
n), x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[b*c^2 + a*d^2, 0] && I 
LtQ[n, -1] &&  !(IGtQ[m, 0] && ILtQ[m + n, 0] &&  !GtQ[p, 1])
 

rule 6709

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_)^2)^(p_), x_Symbo 
l] :> Simp[x^(2*p)*((c + d/x^2)^p/(1 - a^2*x^2)^p)   Int[u*((1 + a*x)^n/(x^ 
(2*p)*(1 - a^2*x^2)^(n/2 - p))), x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[c 
+ a^2*d, 0] &&  !IntegerQ[p] && IntegerQ[n/2] &&  !GtQ[c, 0]
 

Maple [A] (verified)

Time = 0.16 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.03

method	result	size
risch	\(-\frac {\left (6 a^{3} x^{3}-16 a^{2} x^{2}+21 a x -32\right ) x \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}}{24 a^{3}}-\frac {7 \ln \left (\frac {a^{2} c x}{\sqrt {a^{2} c}}+\sqrt {a^{2} c \,x^{2}-c}\right ) x \sqrt {c \left (a^{2} x^{2}-1\right )}\, \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}}{8 a^{2} \sqrt {a^{2} c}\, \left (a^{2} x^{2}-1\right )}\)	\(134\)
default	\(-\frac {\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}}\, x \left (6 x {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {3}{2}} a^{4}-16 {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}\right )}^{\frac {3}{2}} a^{3}+27 \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{2} c x -27 c^{\frac {3}{2}} \ln \left (x \sqrt {c}+\sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\right )+48 c^{\frac {3}{2}} \ln \left (\frac {\sqrt {c}\, \sqrt {\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}}+x c}{\sqrt {c}}\right )-48 \sqrt {\frac {c \left (a x -1\right ) \left (a x +1\right )}{a^{2}}}\, a c \right )}{24 \sqrt {\frac {c \left (a^{2} x^{2}-1\right )}{a^{2}}}\, a^{4} c}\)	\(196\)

Input:

int((c-c/a^2/x^2)^(1/2)*x^3/(a*x+1)^2*(-a^2*x^2+1),x,method=_RETURNVERBOSE 
)
 

Output:

-1/24*(6*a^3*x^3-16*a^2*x^2+21*a*x-32)/a^3*x*(c*(a^2*x^2-1)/a^2/x^2)^(1/2) 
-7/8/a^2*ln(a^2*c*x/(a^2*c)^(1/2)+(a^2*c*x^2-c)^(1/2))/(a^2*c)^(1/2)*x*(c* 
(a^2*x^2-1))^(1/2)*(c*(a^2*x^2-1)/a^2/x^2)^(1/2)/(a^2*x^2-1)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.71 \[ \int e^{-2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^3 \, dx=\left [-\frac {2 \, {\left (6 \, a^{4} x^{4} - 16 \, a^{3} x^{3} + 21 \, a^{2} x^{2} - 32 \, a x\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - 21 \, \sqrt {c} \log \left (2 \, a^{2} c x^{2} - 2 \, a^{2} \sqrt {c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - c\right )}{48 \, a^{4}}, -\frac {{\left (6 \, a^{4} x^{4} - 16 \, a^{3} x^{3} + 21 \, a^{2} x^{2} - 32 \, a x\right )} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}} - 21 \, \sqrt {-c} \arctan \left (\frac {a^{2} \sqrt {-c} x^{2} \sqrt {\frac {a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{2} - c}\right )}{24 \, a^{4}}\right ] \] Input:

integrate((c-c/a^2/x^2)^(1/2)*x^3/(a*x+1)^2*(-a^2*x^2+1),x, algorithm="fri 
cas")
 

Output:

[-1/48*(2*(6*a^4*x^4 - 16*a^3*x^3 + 21*a^2*x^2 - 32*a*x)*sqrt((a^2*c*x^2 - 
 c)/(a^2*x^2)) - 21*sqrt(c)*log(2*a^2*c*x^2 - 2*a^2*sqrt(c)*x^2*sqrt((a^2* 
c*x^2 - c)/(a^2*x^2)) - c))/a^4, -1/24*((6*a^4*x^4 - 16*a^3*x^3 + 21*a^2*x 
^2 - 32*a*x)*sqrt((a^2*c*x^2 - c)/(a^2*x^2)) - 21*sqrt(-c)*arctan(a^2*sqrt 
(-c)*x^2*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^2*c*x^2 - c)))/a^4]
 

Sympy [F]

\[ \int e^{-2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^3 \, dx=- \int \left (- \frac {x^{3} \sqrt {c - \frac {c}{a^{2} x^{2}}}}{a x + 1}\right )\, dx - \int \frac {a x^{4} \sqrt {c - \frac {c}{a^{2} x^{2}}}}{a x + 1}\, dx \] Input:

integrate((c-c/a**2/x**2)**(1/2)*x**3/(a*x+1)**2*(-a**2*x**2+1),x)
 

Output:

-Integral(-x**3*sqrt(c - c/(a**2*x**2))/(a*x + 1), x) - Integral(a*x**4*sq 
rt(c - c/(a**2*x**2))/(a*x + 1), x)
 

Maxima [F]

\[ \int e^{-2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^3 \, dx=\int { -\frac {{\left (a^{2} x^{2} - 1\right )} \sqrt {c - \frac {c}{a^{2} x^{2}}} x^{3}}{{\left (a x + 1\right )}^{2}} \,d x } \] Input:

integrate((c-c/a^2/x^2)^(1/2)*x^3/(a*x+1)^2*(-a^2*x^2+1),x, algorithm="max 
ima")
 

Output:

-integrate((a^2*x^2 - 1)*sqrt(c - c/(a^2*x^2))*x^3/(a*x + 1)^2, x)
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.98 \[ \int e^{-2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^3 \, dx=-\frac {1}{48} \, {\left (2 \, \sqrt {a^{2} c x^{2} - c} {\left ({\left (2 \, x {\left (\frac {3 \, x \mathrm {sgn}\left (x\right )}{a^{2}} - \frac {8 \, \mathrm {sgn}\left (x\right )}{a^{3}}\right )} + \frac {21 \, \mathrm {sgn}\left (x\right )}{a^{4}}\right )} x - \frac {32 \, \mathrm {sgn}\left (x\right )}{a^{5}}\right )} - \frac {42 \, \sqrt {c} \log \left ({\left | -\sqrt {a^{2} c} x + \sqrt {a^{2} c x^{2} - c} \right |}\right ) \mathrm {sgn}\left (x\right )}{a^{4} {\left | a \right |}} + \frac {{\left (21 \, a \sqrt {c} \log \left ({\left | c \right |}\right ) + 64 \, \sqrt {-c} {\left | a \right |}\right )} \mathrm {sgn}\left (x\right )}{a^{5} {\left | a \right |}}\right )} {\left | a \right |} \] Input:

integrate((c-c/a^2/x^2)^(1/2)*x^3/(a*x+1)^2*(-a^2*x^2+1),x, algorithm="gia 
c")
 

Output:

-1/48*(2*sqrt(a^2*c*x^2 - c)*((2*x*(3*x*sgn(x)/a^2 - 8*sgn(x)/a^3) + 21*sg 
n(x)/a^4)*x - 32*sgn(x)/a^5) - 42*sqrt(c)*log(abs(-sqrt(a^2*c)*x + sqrt(a^ 
2*c*x^2 - c)))*sgn(x)/(a^4*abs(a)) + (21*a*sqrt(c)*log(abs(c)) + 64*sqrt(- 
c)*abs(a))*sgn(x)/(a^5*abs(a)))*abs(a)
 

Mupad [F(-1)]

Timed out. \[ \int e^{-2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^3 \, dx=-\int \frac {x^3\,\sqrt {c-\frac {c}{a^2\,x^2}}\,\left (a^2\,x^2-1\right )}{{\left (a\,x+1\right )}^2} \,d x \] Input:

int(-(x^3*(c - c/(a^2*x^2))^(1/2)*(a^2*x^2 - 1))/(a*x + 1)^2,x)
 

Output:

-int((x^3*(c - c/(a^2*x^2))^(1/2)*(a^2*x^2 - 1))/(a*x + 1)^2, x)
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.67 \[ \int e^{-2 \text {arctanh}(a x)} \sqrt {c-\frac {c}{a^2 x^2}} x^3 \, dx=\frac {\sqrt {c}\, \left (-6 \sqrt {a^{2} x^{2}-1}\, a^{3} x^{3}+16 \sqrt {a^{2} x^{2}-1}\, a^{2} x^{2}-21 \sqrt {a^{2} x^{2}-1}\, a x +32 \sqrt {a^{2} x^{2}-1}-21 \,\mathrm {log}\left (\sqrt {a^{2} x^{2}-1}+a x \right )\right )}{24 a^{4}} \] Input:

int((c-c/a^2/x^2)^(1/2)*x^3/(a*x+1)^2*(-a^2*x^2+1),x)
 

Output:

(sqrt(c)*( - 6*sqrt(a**2*x**2 - 1)*a**3*x**3 + 16*sqrt(a**2*x**2 - 1)*a**2 
*x**2 - 21*sqrt(a**2*x**2 - 1)*a*x + 32*sqrt(a**2*x**2 - 1) - 21*log(sqrt( 
a**2*x**2 - 1) + a*x)))/(24*a**4)