\(\int e^{3 \text {arctanh}(a x)} (c-a^2 c x^2)^2 \, dx\) [1169]

Optimal result

Integrand size = 22, antiderivative size = 108 \[ \int e^{3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^2 \, dx=\frac {7}{8} c^2 x \sqrt {1-a^2 x^2}-\frac {17 c^2 \left (1-a^2 x^2\right )^{3/2}}{15 a}-\frac {3}{4} c^2 x \left (1-a^2 x^2\right )^{3/2}-\frac {1}{5} a c^2 x^2 \left (1-a^2 x^2\right )^{3/2}+\frac {7 c^2 \arcsin (a x)}{8 a} \] Output:

7/8*c^2*x*(-a^2*x^2+1)^(1/2)-17/15*c^2*(-a^2*x^2+1)^(3/2)/a-3/4*c^2*x*(-a^ 
2*x^2+1)^(3/2)-1/5*a*c^2*x^2*(-a^2*x^2+1)^(3/2)+7/8*c^2*arcsin(a*x)/a
 

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.69 \[ \int e^{3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^2 \, dx=\frac {c^2 \left (\sqrt {1-a^2 x^2} \left (-136+15 a x+112 a^2 x^2+90 a^3 x^3+24 a^4 x^4\right )-210 \arcsin \left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{120 a} \] Input:

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

Output:

(c^2*(Sqrt[1 - a^2*x^2]*(-136 + 15*a*x + 112*a^2*x^2 + 90*a^3*x^3 + 24*a^4 
*x^4) - 210*ArcSin[Sqrt[1 - a*x]/Sqrt[2]]))/(120*a)
 

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.11, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {6688, 469, 469, 455, 211, 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.

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

\(\Big \downarrow \) 6688

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

\(\Big \downarrow \) 469

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

\(\Big \downarrow \) 469

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

\(\Big \downarrow \) 455

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

\(\Big \downarrow \) 211

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

\(\Big \downarrow \) 223

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

Input:

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

Output:

c^2*(-1/5*((1 + a*x)^2*(1 - a^2*x^2)^(3/2))/a + (7*(-1/4*((1 + a*x)*(1 - a 
^2*x^2)^(3/2))/a + (5*((x*Sqrt[1 - a^2*x^2])/2 - (1 - a^2*x^2)^(3/2)/(3*a) 
 + ArcSin[a*x]/(2*a)))/4))/5)
 

Defintions of rubi rules used

rule 211

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 
)), x] + Simp[2*a*(p/(2*p + 1))   Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ 
{a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
 

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 469

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* 
((n + p)/(n + 2*p + 1))   Int[(c + d*x)^(n - 1)*(a + b*x^2)^p, x], x] /; Fr 
eeQ[{a, b, c, d, p}, x] && EqQ[b*c^2 + a*d^2, 0] && GtQ[n, 0] && NeQ[n + 2* 
p + 1, 0] && IntegerQ[2*p]
 

rule 6688

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

Maple [A] (verified)

Time = 0.55 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.84

method	result	size
risch	\(-\frac {\left (24 a^{4} x^{4}+90 a^{3} x^{3}+112 a^{2} x^{2}+15 a x -136\right ) \left (a^{2} x^{2}-1\right ) c^{2}}{120 a \sqrt {-a^{2} x^{2}+1}}+\frac {7 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right ) c^{2}}{8 \sqrt {a^{2}}}\)	\(91\)
meijerg	\(-\frac {5 c^{2} \left (\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {5}{2}} \left (-5 a^{2} x^{2}+15\right )}{10 a^{4} \sqrt {-a^{2} x^{2}+1}}-\frac {3 \sqrt {\pi }\, \left (-a^{2}\right )^{\frac {5}{2}} \arcsin \left (a x \right )}{2 a^{5}}\right )}{\sqrt {\pi }\, \sqrt {-a^{2}}}-\frac {c^{2} \left (\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {3}{2}}}{a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {\sqrt {\pi }\, \left (-a^{2}\right )^{\frac {3}{2}} \arcsin \left (a x \right )}{a^{3}}\right )}{\sqrt {\pi }\, \sqrt {-a^{2}}}+\frac {c^{2} x}{\sqrt {-a^{2} x^{2}+1}}+\frac {c^{2} \left (-\frac {16 \sqrt {\pi }}{5}+\frac {\sqrt {\pi }\, \left (-8 x^{6} a^{6}-16 a^{4} x^{4}-64 a^{2} x^{2}+128\right )}{40 \sqrt {-a^{2} x^{2}+1}}\right )}{a \sqrt {\pi }}-\frac {c^{2} \left (\frac {8 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (-2 a^{4} x^{4}-8 a^{2} x^{2}+16\right )}{6 \sqrt {-a^{2} x^{2}+1}}\right )}{a \sqrt {\pi }}-\frac {5 c^{2} \left (-2 \sqrt {\pi }+\frac {\sqrt {\pi }\, \left (-4 a^{2} x^{2}+8\right )}{4 \sqrt {-a^{2} x^{2}+1}}\right )}{a \sqrt {\pi }}-\frac {3 c^{2} \left (\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {7}{2}} \left (-14 a^{4} x^{4}-35 a^{2} x^{2}+105\right )}{56 a^{6} \sqrt {-a^{2} x^{2}+1}}-\frac {15 \sqrt {\pi }\, \left (-a^{2}\right )^{\frac {7}{2}} \arcsin \left (a x \right )}{8 a^{7}}\right )}{\sqrt {\pi }\, \sqrt {-a^{2}}}-\frac {3 c^{2} \left (\sqrt {\pi }-\frac {\sqrt {\pi }}{\sqrt {-a^{2} x^{2}+1}}\right )}{a \sqrt {\pi }}\)	\(422\)
default	\(c^{2} \left (\frac {x}{\sqrt {-a^{2} x^{2}+1}}+a^{2} \left (\frac {x}{a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{2} \sqrt {a^{2}}}\right )+a^{5} \left (-\frac {x^{4}}{3 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {-\frac {4 x^{2}}{3 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {8}{3 a^{4} \sqrt {-a^{2} x^{2}+1}}}{a^{2}}\right )+a^{7} \left (-\frac {x^{6}}{5 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {-\frac {2 x^{4}}{5 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {6 \left (-\frac {4 x^{2}}{3 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {8}{3 a^{4} \sqrt {-a^{2} x^{2}+1}}\right )}{5 a^{2}}}{a^{2}}\right )+\frac {3}{a \sqrt {-a^{2} x^{2}+1}}-5 a^{3} \left (-\frac {x^{2}}{a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {2}{a^{4} \sqrt {-a^{2} x^{2}+1}}\right )-5 a^{4} \left (-\frac {x^{3}}{2 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {\frac {3 x}{2 a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a^{2} \sqrt {a^{2}}}}{a^{2}}\right )+3 a^{6} \left (-\frac {x^{5}}{4 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {-\frac {5 x^{3}}{8 a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {5 \left (\frac {3 x}{2 a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a^{2} \sqrt {a^{2}}}\right )}{4 a^{2}}}{a^{2}}\right )\right )\)	\(478\)

Input:

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

Output:

-1/120*(24*a^4*x^4+90*a^3*x^3+112*a^2*x^2+15*a*x-136)*(a^2*x^2-1)/a/(-a^2* 
x^2+1)^(1/2)*c^2+7/8/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))* 
c^2
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.86 \[ \int e^{3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^2 \, dx=-\frac {210 \, c^{2} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - {\left (24 \, a^{4} c^{2} x^{4} + 90 \, a^{3} c^{2} x^{3} + 112 \, a^{2} c^{2} x^{2} + 15 \, a c^{2} x - 136 \, c^{2}\right )} \sqrt {-a^{2} x^{2} + 1}}{120 \, a} \] Input:

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

Output:

-1/120*(210*c^2*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - (24*a^4*c^2*x^4 + 
 90*a^3*c^2*x^3 + 112*a^2*c^2*x^2 + 15*a*c^2*x - 136*c^2)*sqrt(-a^2*x^2 + 
1))/a
                                                                                    
                                                                                    
 

Sympy [A] (verification not implemented)

Time = 7.98 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.65 \[ \int e^{3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^2 \, dx=\begin {cases} \frac {7 c^{2} \log {\left (- 2 a^{2} x + 2 \sqrt {- a^{2}} \sqrt {- a^{2} x^{2} + 1} \right )}}{8 \sqrt {- a^{2}}} + \sqrt {- a^{2} x^{2} + 1} \left (\frac {a^{3} c^{2} x^{4}}{5} + \frac {3 a^{2} c^{2} x^{3}}{4} + \frac {14 a c^{2} x^{2}}{15} + \frac {c^{2} x}{8} - \frac {17 c^{2}}{15 a}\right ) & \text {for}\: a^{2} \neq 0 \\- \frac {a^{5} c^{2} x^{6}}{6} - \frac {3 a^{4} c^{2} x^{5}}{5} - \frac {a^{3} c^{2} x^{4}}{2} + \frac {2 a^{2} c^{2} x^{3}}{3} + \frac {3 a c^{2} x^{2}}{2} + c^{2} x & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((7*c**2*log(-2*a**2*x + 2*sqrt(-a**2)*sqrt(-a**2*x**2 + 1))/(8*s 
qrt(-a**2)) + sqrt(-a**2*x**2 + 1)*(a**3*c**2*x**4/5 + 3*a**2*c**2*x**3/4 
+ 14*a*c**2*x**2/15 + c**2*x/8 - 17*c**2/(15*a)), Ne(a**2, 0)), (-a**5*c** 
2*x**6/6 - 3*a**4*c**2*x**5/5 - a**3*c**2*x**4/2 + 2*a**2*c**2*x**3/3 + 3* 
a*c**2*x**2/2 + c**2*x, True))
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.52 \[ \int e^{3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^2 \, dx=-\frac {a^{5} c^{2} x^{6}}{5 \, \sqrt {-a^{2} x^{2} + 1}} - \frac {3 \, a^{4} c^{2} x^{5}}{4 \, \sqrt {-a^{2} x^{2} + 1}} - \frac {11 \, a^{3} c^{2} x^{4}}{15 \, \sqrt {-a^{2} x^{2} + 1}} + \frac {5 \, a^{2} c^{2} x^{3}}{8 \, \sqrt {-a^{2} x^{2} + 1}} + \frac {31 \, a c^{2} x^{2}}{15 \, \sqrt {-a^{2} x^{2} + 1}} + \frac {c^{2} x}{8 \, \sqrt {-a^{2} x^{2} + 1}} + \frac {7 \, c^{2} \arcsin \left (a x\right )}{8 \, a} - \frac {17 \, c^{2}}{15 \, \sqrt {-a^{2} x^{2} + 1} a} \] Input:

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

Output:

-1/5*a^5*c^2*x^6/sqrt(-a^2*x^2 + 1) - 3/4*a^4*c^2*x^5/sqrt(-a^2*x^2 + 1) - 
 11/15*a^3*c^2*x^4/sqrt(-a^2*x^2 + 1) + 5/8*a^2*c^2*x^3/sqrt(-a^2*x^2 + 1) 
 + 31/15*a*c^2*x^2/sqrt(-a^2*x^2 + 1) + 1/8*c^2*x/sqrt(-a^2*x^2 + 1) + 7/8 
*c^2*arcsin(a*x)/a - 17/15*c^2/(sqrt(-a^2*x^2 + 1)*a)
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.72 \[ \int e^{3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^2 \, dx=\frac {7 \, c^{2} \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{8 \, {\left | a \right |}} + \frac {1}{120} \, \sqrt {-a^{2} x^{2} + 1} {\left ({\left (15 \, c^{2} + 2 \, {\left (56 \, a c^{2} + 3 \, {\left (4 \, a^{3} c^{2} x + 15 \, a^{2} c^{2}\right )} x\right )} x\right )} x - \frac {136 \, c^{2}}{a}\right )} \] Input:

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

Output:

7/8*c^2*arcsin(a*x)*sgn(a)/abs(a) + 1/120*sqrt(-a^2*x^2 + 1)*((15*c^2 + 2* 
(56*a*c^2 + 3*(4*a^3*c^2*x + 15*a^2*c^2)*x)*x)*x - 136*c^2/a)
 

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.19 \[ \int e^{3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^2 \, dx=\frac {c^2\,x\,\sqrt {1-a^2\,x^2}}{8}+\frac {7\,c^2\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{8\,\sqrt {-a^2}}-\frac {17\,c^2\,\sqrt {1-a^2\,x^2}}{15\,a}+\frac {14\,a\,c^2\,x^2\,\sqrt {1-a^2\,x^2}}{15}+\frac {3\,a^2\,c^2\,x^3\,\sqrt {1-a^2\,x^2}}{4}+\frac {a^3\,c^2\,x^4\,\sqrt {1-a^2\,x^2}}{5} \] Input:

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

Output:

(c^2*x*(1 - a^2*x^2)^(1/2))/8 + (7*c^2*asinh(x*(-a^2)^(1/2)))/(8*(-a^2)^(1 
/2)) - (17*c^2*(1 - a^2*x^2)^(1/2))/(15*a) + (14*a*c^2*x^2*(1 - a^2*x^2)^( 
1/2))/15 + (3*a^2*c^2*x^3*(1 - a^2*x^2)^(1/2))/4 + (a^3*c^2*x^4*(1 - a^2*x 
^2)^(1/2))/5
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.94 \[ \int e^{3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^2 \, dx=\frac {c^{2} \left (105 \mathit {asin} \left (a x \right )+24 \sqrt {-a^{2} x^{2}+1}\, a^{4} x^{4}+90 \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}+112 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}+15 \sqrt {-a^{2} x^{2}+1}\, a x -136 \sqrt {-a^{2} x^{2}+1}+136\right )}{120 a} \] Input:

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

Output:

(c**2*(105*asin(a*x) + 24*sqrt( - a**2*x**2 + 1)*a**4*x**4 + 90*sqrt( - a* 
*2*x**2 + 1)*a**3*x**3 + 112*sqrt( - a**2*x**2 + 1)*a**2*x**2 + 15*sqrt( - 
 a**2*x**2 + 1)*a*x - 136*sqrt( - a**2*x**2 + 1) + 136))/(120*a)