\(\int \frac {e^{3 \text {arctanh}(a x)}}{(c-\frac {c}{a^2 x^2})^3} \, dx\) [692]

Optimal result

Integrand size = 22, antiderivative size = 154 \[ \int \frac {e^{3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^3} \, dx=-\frac {(1+a x)^3}{7 a c^3 \left (1-a^2 x^2\right )^{7/2}}+\frac {76 (1+a x)}{35 a c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac {175+137 a x}{35 a c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {245+181 a x}{35 a c^3 \sqrt {1-a^2 x^2}}+\frac {\sqrt {1-a^2 x^2}}{a c^3}-\frac {3 \arcsin (a x)}{a c^3} \] Output:

-1/7*(a*x+1)^3/a/c^3/(-a^2*x^2+1)^(7/2)+76/35*(a*x+1)/a/c^3/(-a^2*x^2+1)^( 
5/2)-1/35*(137*a*x+175)/a/c^3/(-a^2*x^2+1)^(3/2)+1/35*(181*a*x+245)/a/c^3/ 
(-a^2*x^2+1)^(1/2)+(-a^2*x^2+1)^(1/2)/a/c^3-3*arcsin(a*x)/a/c^3
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.62 \[ \int \frac {e^{3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^3} \, dx=\frac {-176+423 a x-125 a^2 x^2-368 a^3 x^3+286 a^4 x^4-35 a^5 x^5-105 (-1+a x)^3 \sqrt {1-a^2 x^2} \arcsin (a x)}{35 a c^3 (-1+a x)^3 \sqrt {1-a^2 x^2}} \] Input:

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

Output:

(-176 + 423*a*x - 125*a^2*x^2 - 368*a^3*x^3 + 286*a^4*x^4 - 35*a^5*x^5 - 1 
05*(-1 + a*x)^3*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(35*a*c^3*(-1 + a*x)^3*Sqrt 
[1 - a^2*x^2])
 

Rubi [A] (verified)

Time = 1.33 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.03, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {6707, 6698, 529, 2166, 2166, 27, 2345, 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.

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

\(\Big \downarrow \) 6707

\(\displaystyle -\frac {a^6 \int \frac {e^{3 \text {arctanh}(a x)} x^6}{\left (1-a^2 x^2\right )^3}dx}{c^3}\)

\(\Big \downarrow \) 6698

\(\displaystyle -\frac {a^6 \int \frac {x^6 (a x+1)^3}{\left (1-a^2 x^2\right )^{9/2}}dx}{c^3}\)

\(\Big \downarrow \) 529

\(\displaystyle -\frac {a^6 \left (\frac {(a x+1)^3}{7 a^7 \left (1-a^2 x^2\right )^{7/2}}-\frac {1}{7} \int \frac {(a x+1)^2 \left (\frac {7 x^5}{a}+\frac {7 x^4}{a^2}+\frac {7 x^3}{a^3}+\frac {7 x^2}{a^4}+\frac {7 x}{a^5}+\frac {3}{a^6}\right )}{\left (1-a^2 x^2\right )^{7/2}}dx\right )}{c^3}\)

\(\Big \downarrow \) 2166

\(\displaystyle -\frac {a^6 \left (\frac {1}{7} \left (\frac {1}{5} \int \frac {(a x+1) \left (\frac {35 x^4}{a^2}+\frac {70 x^3}{a^3}+\frac {105 x^2}{a^4}+\frac {140 x}{a^5}+\frac {61}{a^6}\right )}{\left (1-a^2 x^2\right )^{5/2}}dx-\frac {38 (a x+1)^2}{5 a^7 \left (1-a^2 x^2\right )^{5/2}}\right )+\frac {(a x+1)^3}{7 a^7 \left (1-a^2 x^2\right )^{7/2}}\right )}{c^3}\)

\(\Big \downarrow \) 2166

\(\displaystyle -\frac {a^6 \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {137 (a x+1)}{a^7 \left (1-a^2 x^2\right )^{3/2}}-\frac {1}{3} \int \frac {3 \left (\frac {35 x^3}{a^3}+\frac {105 x^2}{a^4}+\frac {210 x}{a^5}+\frac {76}{a^6}\right )}{\left (1-a^2 x^2\right )^{3/2}}dx\right )-\frac {38 (a x+1)^2}{5 a^7 \left (1-a^2 x^2\right )^{5/2}}\right )+\frac {(a x+1)^3}{7 a^7 \left (1-a^2 x^2\right )^{7/2}}\right )}{c^3}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {a^6 \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {137 (a x+1)}{a^7 \left (1-a^2 x^2\right )^{3/2}}-\int \frac {\frac {35 x^3}{a^3}+\frac {105 x^2}{a^4}+\frac {210 x}{a^5}+\frac {76}{a^6}}{\left (1-a^2 x^2\right )^{3/2}}dx\right )-\frac {38 (a x+1)^2}{5 a^7 \left (1-a^2 x^2\right )^{5/2}}\right )+\frac {(a x+1)^3}{7 a^7 \left (1-a^2 x^2\right )^{7/2}}\right )}{c^3}\)

\(\Big \downarrow \) 2345

\(\displaystyle -\frac {a^6 \left (\frac {1}{7} \left (\frac {1}{5} \left (\int \frac {35 (a x+3)}{a^6 \sqrt {1-a^2 x^2}}dx+\frac {137 (a x+1)}{a^7 \left (1-a^2 x^2\right )^{3/2}}-\frac {181 a x+245}{a^7 \sqrt {1-a^2 x^2}}\right )-\frac {38 (a x+1)^2}{5 a^7 \left (1-a^2 x^2\right )^{5/2}}\right )+\frac {(a x+1)^3}{7 a^7 \left (1-a^2 x^2\right )^{7/2}}\right )}{c^3}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {a^6 \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {35 \int \frac {a x+3}{\sqrt {1-a^2 x^2}}dx}{a^6}+\frac {137 (a x+1)}{a^7 \left (1-a^2 x^2\right )^{3/2}}-\frac {181 a x+245}{a^7 \sqrt {1-a^2 x^2}}\right )-\frac {38 (a x+1)^2}{5 a^7 \left (1-a^2 x^2\right )^{5/2}}\right )+\frac {(a x+1)^3}{7 a^7 \left (1-a^2 x^2\right )^{7/2}}\right )}{c^3}\)

\(\Big \downarrow \) 455

\(\displaystyle -\frac {a^6 \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {35 \left (3 \int \frac {1}{\sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2}}{a}\right )}{a^6}+\frac {137 (a x+1)}{a^7 \left (1-a^2 x^2\right )^{3/2}}-\frac {181 a x+245}{a^7 \sqrt {1-a^2 x^2}}\right )-\frac {38 (a x+1)^2}{5 a^7 \left (1-a^2 x^2\right )^{5/2}}\right )+\frac {(a x+1)^3}{7 a^7 \left (1-a^2 x^2\right )^{7/2}}\right )}{c^3}\)

\(\Big \downarrow \) 223

\(\displaystyle -\frac {a^6 \left (\frac {(a x+1)^3}{7 a^7 \left (1-a^2 x^2\right )^{7/2}}+\frac {1}{7} \left (\frac {1}{5} \left (\frac {137 (a x+1)}{a^7 \left (1-a^2 x^2\right )^{3/2}}-\frac {181 a x+245}{a^7 \sqrt {1-a^2 x^2}}+\frac {35 \left (\frac {3 \arcsin (a x)}{a}-\frac {\sqrt {1-a^2 x^2}}{a}\right )}{a^6}\right )-\frac {38 (a x+1)^2}{5 a^7 \left (1-a^2 x^2\right )^{5/2}}\right )\right )}{c^3}\)

Input:

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

Output:

-((a^6*((1 + a*x)^3/(7*a^7*(1 - a^2*x^2)^(7/2)) + ((-38*(1 + a*x)^2)/(5*a^ 
7*(1 - a^2*x^2)^(5/2)) + ((137*(1 + a*x))/(a^7*(1 - a^2*x^2)^(3/2)) - (245 
 + 181*a*x)/(a^7*Sqrt[1 - a^2*x^2]) + (35*(-(Sqrt[1 - a^2*x^2]/a) + (3*Arc 
Sin[a*x])/a))/a^6)/5)/7))/c^3)
 

Defintions of rubi rules used

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 529

Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo 
l] :> With[{Qx = PolynomialQuotient[x^m, a*d + b*c*x, x], R = PolynomialRem 
ainder[x^m, a*d + b*c*x, x]}, Simp[(-c)*R*(c + d*x)^n*((a + b*x^2)^(p + 1)/ 
(2*a*d*(p + 1))), x] + Simp[c/(2*a*(p + 1))   Int[(c + d*x)^(n - 1)*(a + b* 
x^2)^(p + 1)*ExpandToSum[2*a*d*(p + 1)*Qx + R*(n + 2*p + 2), x], x], x]] /; 
 FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && IGtQ[m, 1] && LtQ[p, -1] && EqQ[b* 
c^2 + a*d^2, 0]
 

rule 2166

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : 
> With[{Qx = PolynomialQuotient[Pq, a*e + b*d*x, x], R = PolynomialRemainde 
r[Pq, a*e + b*d*x, x]}, Simp[(-d)*R*(d + e*x)^m*((a + b*x^2)^(p + 1)/(2*a*e 
*(p + 1))), x] + Simp[d/(2*a*(p + 1))   Int[(d + e*x)^(m - 1)*(a + b*x^2)^( 
p + 1)*ExpandToSum[2*a*e*(p + 1)*Qx + R*(m + 2*p + 2), x], x], x]] /; FreeQ 
[{a, b, d, e}, x] && PolyQ[Pq, x] && EqQ[b*d^2 + a*e^2, 0] && ILtQ[p + 1/2, 
 0] && GtQ[m, 0]
 

rule 2345

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuot 
ient[Pq, a + b*x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 
 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[(a*g - b 
*f*x)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1))   In 
t[(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] 
/; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]
 

rule 6698

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

rule 6707

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

Maple [A] (verified)

Time = 0.36 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.74

method	result	size
risch	\(-\frac {a^{2} x^{2}-1}{a \sqrt {-a^{2} x^{2}+1}\, c^{3}}+\frac {\left (-\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{6} \sqrt {a^{2}}}-\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{14 a^{11} \left (x -\frac {1}{a}\right )^{4}}-\frac {71 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{140 a^{10} \left (x -\frac {1}{a}\right )^{3}}-\frac {477 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{280 a^{9} \left (x -\frac {1}{a}\right )^{2}}-\frac {2931 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{560 a^{8} \left (x -\frac {1}{a}\right )}+\frac {\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{16 a^{8} \left (x +\frac {1}{a}\right )}\right ) a^{6}}{c^{3}}\)	\(268\)
default	\(\frac {a^{6} \left (\frac {-\frac {x^{2}}{a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {2}{a^{4} \sqrt {-a^{2} x^{2}+1}}}{a^{3}}+\frac {\frac {1}{7 a \left (x -\frac {1}{a}\right )^{3} \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}-\frac {4 a \left (\frac {1}{5 a \left (x -\frac {1}{a}\right )^{2} \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}-\frac {3 a \left (\frac {1}{3 a \left (x -\frac {1}{a}\right ) \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}+\frac {-2 \left (x -\frac {1}{a}\right ) a^{2}-2 a}{3 a \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}\right )}{5}\right )}{7}}{a^{9}}+\frac {10 x}{a^{6} \sqrt {-a^{2} x^{2}+1}}+\frac {\frac {3 x}{a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{2} \sqrt {a^{2}}}}{a^{4}}+\frac {6}{a^{7} \sqrt {-a^{2} x^{2}+1}}+\frac {\frac {6}{5 a \left (x -\frac {1}{a}\right )^{2} \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}-\frac {18 a \left (\frac {1}{3 a \left (x -\frac {1}{a}\right ) \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}+\frac {-2 \left (x -\frac {1}{a}\right ) a^{2}-2 a}{3 a \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}\right )}{5}}{a^{8}}+\frac {\frac {5}{a \left (x -\frac {1}{a}\right ) \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}+\frac {5 \left (-2 \left (x -\frac {1}{a}\right ) a^{2}-2 a \right )}{a \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}}{a^{7}}\right )}{c^{3}}\)	\(558\)

Input:

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

Output:

-1/a*(a^2*x^2-1)/(-a^2*x^2+1)^(1/2)/c^3+(-3/a^6/(a^2)^(1/2)*arctan((a^2)^( 
1/2)*x/(-a^2*x^2+1)^(1/2))-1/14/a^11/(x-1/a)^4*(-(x-1/a)^2*a^2-2*a*(x-1/a) 
)^(1/2)-71/140/a^10/(x-1/a)^3*(-(x-1/a)^2*a^2-2*a*(x-1/a))^(1/2)-477/280/a 
^9/(x-1/a)^2*(-(x-1/a)^2*a^2-2*a*(x-1/a))^(1/2)-2931/560/a^8/(x-1/a)*(-(x- 
1/a)^2*a^2-2*a*(x-1/a))^(1/2)+1/16/a^8/(x+1/a)*(-a^2*(x+1/a)^2+2*a*(x+1/a) 
)^(1/2))*a^6/c^3
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.38 \[ \int \frac {e^{3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^3} \, dx=\frac {176 \, a^{5} x^{5} - 528 \, a^{4} x^{4} + 352 \, a^{3} x^{3} + 352 \, a^{2} x^{2} - 528 \, a x + 210 \, {\left (a^{5} x^{5} - 3 \, a^{4} x^{4} + 2 \, a^{3} x^{3} + 2 \, a^{2} x^{2} - 3 \, a x + 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (35 \, a^{5} x^{5} - 286 \, a^{4} x^{4} + 368 \, a^{3} x^{3} + 125 \, a^{2} x^{2} - 423 \, a x + 176\right )} \sqrt {-a^{2} x^{2} + 1} + 176}{35 \, {\left (a^{6} c^{3} x^{5} - 3 \, a^{5} c^{3} x^{4} + 2 \, a^{4} c^{3} x^{3} + 2 \, a^{3} c^{3} x^{2} - 3 \, a^{2} c^{3} x + a c^{3}\right )}} \] Input:

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

Output:

1/35*(176*a^5*x^5 - 528*a^4*x^4 + 352*a^3*x^3 + 352*a^2*x^2 - 528*a*x + 21 
0*(a^5*x^5 - 3*a^4*x^4 + 2*a^3*x^3 + 2*a^2*x^2 - 3*a*x + 1)*arctan((sqrt(- 
a^2*x^2 + 1) - 1)/(a*x)) + (35*a^5*x^5 - 286*a^4*x^4 + 368*a^3*x^3 + 125*a 
^2*x^2 - 423*a*x + 176)*sqrt(-a^2*x^2 + 1) + 176)/(a^6*c^3*x^5 - 3*a^5*c^3 
*x^4 + 2*a^4*c^3*x^3 + 2*a^3*c^3*x^2 - 3*a^2*c^3*x + a*c^3)
 

Sympy [F]

\[ \int \frac {e^{3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^3} \, dx=\frac {a^{6} \int \frac {x^{6}}{- a^{5} x^{5} \sqrt {- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 2 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + 3 a x \sqrt {- a^{2} x^{2} + 1} - \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{3}} \] Input:

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

Output:

a**6*Integral(x**6/(-a**5*x**5*sqrt(-a**2*x**2 + 1) + 3*a**4*x**4*sqrt(-a* 
*2*x**2 + 1) - 2*a**3*x**3*sqrt(-a**2*x**2 + 1) - 2*a**2*x**2*sqrt(-a**2*x 
**2 + 1) + 3*a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x)/c**3
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [B] (verification not implemented)

Time = 24.51 (sec) , antiderivative size = 1548, normalized size of antiderivative = 10.05 \[ \int \frac {e^{3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

(71*(1 - a^2*x^2)^(1/2))/(280*(-a^2)^(1/2)*(c^3*1i + 3*c^3*x*(-a^2)^(1/2) 
+ a^2*c^3*x^2*3i + a^2*c^3*x^3*(-a^2)^(1/2))) - (3*a^7*(1 - a^2*x^2)^(1/2) 
)/(112*(a^8*c^3 + c^3*x*(-a^2)^(9/2)*4i + 6*a^10*c^3*x^2 + a^12*c^3*x^4 + 
a^2*c^3*x^3*(-a^2)^(9/2)*4i)) - (a^9*(1 - a^2*x^2)^(1/2))/(112*(a^10*c^3 - 
 c^3*x*(-a^2)^(11/2)*4i + 6*a^12*c^3*x^2 + a^14*c^3*x^4 - a^2*c^3*x^3*(-a^ 
2)^(11/2)*4i)) - (a^9*(1 - a^2*x^2)^(1/2))/(112*(a^10*c^3 + c^3*x*(-a^2)^( 
11/2)*4i + 6*a^12*c^3*x^2 + a^14*c^3*x^4 + a^2*c^3*x^3*(-a^2)^(11/2)*4i)) 
- (537*a^7*(1 - a^2*x^2)^(1/2))/(1120*(a^8*c^3 - c^3*x*(-a^2)^(9/2)*2i + a 
^10*c^3*x^2)) - (537*a^7*(1 - a^2*x^2)^(1/2))/(1120*(a^8*c^3 + c^3*x*(-a^2 
)^(9/2)*2i + a^10*c^3*x^2)) - (417*a^9*(1 - a^2*x^2)^(1/2))/(1120*(a^10*c^ 
3 - c^3*x*(-a^2)^(11/2)*2i + a^12*c^3*x^2)) - (417*a^9*(1 - a^2*x^2)^(1/2) 
)/(1120*(a^10*c^3 + c^3*x*(-a^2)^(11/2)*2i + a^12*c^3*x^2)) - (3*asinh(x*( 
-a^2)^(1/2)))/(c^3*(-a^2)^(1/2)) - (3*a^7*(1 - a^2*x^2)^(1/2))/(112*(a^8*c 
^3 - c^3*x*(-a^2)^(9/2)*4i + 6*a^10*c^3*x^2 + a^12*c^3*x^4 - a^2*c^3*x^3*( 
-a^2)^(9/2)*4i)) - (71*(1 - a^2*x^2)^(1/2))/(280*(-a^2)^(1/2)*(c^3*1i - 3* 
c^3*x*(-a^2)^(1/2) + a^2*c^3*x^2*3i - a^2*c^3*x^3*(-a^2)^(1/2))) + (a^7*(1 
 - a^2*x^2)^(1/2)*6i)/(35*(a^8*c^3*1i + 3*c^3*x*(-a^2)^(9/2) + a^10*c^3*x^ 
2*3i + a^2*c^3*x^3*(-a^2)^(9/2))) + (a^7*(1 - a^2*x^2)^(1/2)*6i)/(35*(a^8* 
c^3*1i - 3*c^3*x*(-a^2)^(9/2) + a^10*c^3*x^2*3i - a^2*c^3*x^3*(-a^2)^(9/2) 
)) + (a^9*(1 - a^2*x^2)^(1/2)*23i)/(280*(a^10*c^3*1i + 3*c^3*x*(-a^2)^(...
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.49 \[ \int \frac {e^{3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^3} \, dx=\frac {-105 \sqrt {-a^{2} x^{2}+1}\, \mathit {asin} \left (a x \right ) a^{3} x^{3}+315 \sqrt {-a^{2} x^{2}+1}\, \mathit {asin} \left (a x \right ) a^{2} x^{2}-315 \sqrt {-a^{2} x^{2}+1}\, \mathit {asin} \left (a x \right ) a x +105 \sqrt {-a^{2} x^{2}+1}\, \mathit {asin} \left (a x \right )+141 \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}-423 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}+423 \sqrt {-a^{2} x^{2}+1}\, a x -141 \sqrt {-a^{2} x^{2}+1}-35 a^{5} x^{5}+286 a^{4} x^{4}-368 a^{3} x^{3}-125 a^{2} x^{2}+423 a x -176}{35 \sqrt {-a^{2} x^{2}+1}\, a \,c^{3} \left (a^{3} x^{3}-3 a^{2} x^{2}+3 a x -1\right )} \] Input:

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

Output:

( - 105*sqrt( - a**2*x**2 + 1)*asin(a*x)*a**3*x**3 + 315*sqrt( - a**2*x**2 
 + 1)*asin(a*x)*a**2*x**2 - 315*sqrt( - a**2*x**2 + 1)*asin(a*x)*a*x + 105 
*sqrt( - a**2*x**2 + 1)*asin(a*x) + 141*sqrt( - a**2*x**2 + 1)*a**3*x**3 - 
 423*sqrt( - a**2*x**2 + 1)*a**2*x**2 + 423*sqrt( - a**2*x**2 + 1)*a*x - 1 
41*sqrt( - a**2*x**2 + 1) - 35*a**5*x**5 + 286*a**4*x**4 - 368*a**3*x**3 - 
 125*a**2*x**2 + 423*a*x - 176)/(35*sqrt( - a**2*x**2 + 1)*a*c**3*(a**3*x* 
*3 - 3*a**2*x**2 + 3*a*x - 1))