\(\int \frac {x (a+b \text {arcsinh}(c x))^2}{(d+c^2 d x^2)^3} \, dx\) [258]

Optimal result

Integrand size = 24, antiderivative size = 145 \[ \int \frac {x (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^3} \, dx=\frac {b^2}{12 c^2 d^3 \left (1+c^2 x^2\right )}+\frac {b x (a+b \text {arcsinh}(c x))}{6 c d^3 \left (1+c^2 x^2\right )^{3/2}}+\frac {b x (a+b \text {arcsinh}(c x))}{3 c d^3 \sqrt {1+c^2 x^2}}-\frac {(a+b \text {arcsinh}(c x))^2}{4 c^2 d^3 \left (1+c^2 x^2\right )^2}-\frac {b^2 \log \left (1+c^2 x^2\right )}{6 c^2 d^3} \] Output:

1/12*b^2/c^2/d^3/(c^2*x^2+1)+1/6*b*x*(a+b*arcsinh(c*x))/c/d^3/(c^2*x^2+1)^ 
(3/2)+1/3*b*x*(a+b*arcsinh(c*x))/c/d^3/(c^2*x^2+1)^(1/2)-1/4*(a+b*arcsinh( 
c*x))^2/c^2/d^3/(c^2*x^2+1)^2-1/6*b^2*ln(c^2*x^2+1)/c^2/d^3
 

Mathematica [A] (verified)

Time = 0.49 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.05 \[ \int \frac {x (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^3} \, dx=\frac {-3 a^2+b^2+b^2 c^2 x^2+6 a b c x \sqrt {1+c^2 x^2}+4 a b c^3 x^3 \sqrt {1+c^2 x^2}+2 b \left (-3 a+b c x \sqrt {1+c^2 x^2} \left (3+2 c^2 x^2\right )\right ) \text {arcsinh}(c x)-3 b^2 \text {arcsinh}(c x)^2-2 \left (b+b c^2 x^2\right )^2 \log \left (1+c^2 x^2\right )}{12 d^3 \left (c+c^3 x^2\right )^2} \] Input:

Integrate[(x*(a + b*ArcSinh[c*x])^2)/(d + c^2*d*x^2)^3,x]
 

Output:

(-3*a^2 + b^2 + b^2*c^2*x^2 + 6*a*b*c*x*Sqrt[1 + c^2*x^2] + 4*a*b*c^3*x^3* 
Sqrt[1 + c^2*x^2] + 2*b*(-3*a + b*c*x*Sqrt[1 + c^2*x^2]*(3 + 2*c^2*x^2))*A 
rcSinh[c*x] - 3*b^2*ArcSinh[c*x]^2 - 2*(b + b*c^2*x^2)^2*Log[1 + c^2*x^2]) 
/(12*d^3*(c + c^3*x^2)^2)
 

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.93, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {6213, 6203, 241, 6202, 240}

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.

\(\displaystyle \int \frac {x (a+b \text {arcsinh}(c x))^2}{\left (c^2 d x^2+d\right )^3} \, dx\)

\(\Big \downarrow \) 6213

\(\displaystyle \frac {b \int \frac {a+b \text {arcsinh}(c x)}{\left (c^2 x^2+1\right )^{5/2}}dx}{2 c d^3}-\frac {(a+b \text {arcsinh}(c x))^2}{4 c^2 d^3 \left (c^2 x^2+1\right )^2}\)

\(\Big \downarrow \) 6203

\(\displaystyle \frac {b \left (\frac {2}{3} \int \frac {a+b \text {arcsinh}(c x)}{\left (c^2 x^2+1\right )^{3/2}}dx-\frac {1}{3} b c \int \frac {x}{\left (c^2 x^2+1\right )^2}dx+\frac {x (a+b \text {arcsinh}(c x))}{3 \left (c^2 x^2+1\right )^{3/2}}\right )}{2 c d^3}-\frac {(a+b \text {arcsinh}(c x))^2}{4 c^2 d^3 \left (c^2 x^2+1\right )^2}\)

\(\Big \downarrow \) 241

\(\displaystyle \frac {b \left (\frac {2}{3} \int \frac {a+b \text {arcsinh}(c x)}{\left (c^2 x^2+1\right )^{3/2}}dx+\frac {x (a+b \text {arcsinh}(c x))}{3 \left (c^2 x^2+1\right )^{3/2}}+\frac {b}{6 c \left (c^2 x^2+1\right )}\right )}{2 c d^3}-\frac {(a+b \text {arcsinh}(c x))^2}{4 c^2 d^3 \left (c^2 x^2+1\right )^2}\)

\(\Big \downarrow \) 6202

\(\displaystyle \frac {b \left (\frac {2}{3} \left (\frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}-b c \int \frac {x}{c^2 x^2+1}dx\right )+\frac {x (a+b \text {arcsinh}(c x))}{3 \left (c^2 x^2+1\right )^{3/2}}+\frac {b}{6 c \left (c^2 x^2+1\right )}\right )}{2 c d^3}-\frac {(a+b \text {arcsinh}(c x))^2}{4 c^2 d^3 \left (c^2 x^2+1\right )^2}\)

\(\Big \downarrow \) 240

\(\displaystyle \frac {b \left (\frac {x (a+b \text {arcsinh}(c x))}{3 \left (c^2 x^2+1\right )^{3/2}}+\frac {2}{3} \left (\frac {x (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}-\frac {b \log \left (c^2 x^2+1\right )}{2 c}\right )+\frac {b}{6 c \left (c^2 x^2+1\right )}\right )}{2 c d^3}-\frac {(a+b \text {arcsinh}(c x))^2}{4 c^2 d^3 \left (c^2 x^2+1\right )^2}\)

Input:

Int[(x*(a + b*ArcSinh[c*x])^2)/(d + c^2*d*x^2)^3,x]
 

Output:

-1/4*(a + b*ArcSinh[c*x])^2/(c^2*d^3*(1 + c^2*x^2)^2) + (b*(b/(6*c*(1 + c^ 
2*x^2)) + (x*(a + b*ArcSinh[c*x]))/(3*(1 + c^2*x^2)^(3/2)) + (2*((x*(a + b 
*ArcSinh[c*x]))/Sqrt[1 + c^2*x^2] - (b*Log[1 + c^2*x^2])/(2*c)))/3))/(2*c* 
d^3)
 

Defintions of rubi rules used

rule 240

Int[(x_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[Log[RemoveContent[a + b*x 
^2, x]]/(2*b), x] /; FreeQ[{a, b}, x]
 

rule 241

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

rule 6202

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2)^(3/2), 
x_Symbol] :> Simp[x*((a + b*ArcSinh[c*x])^n/(d*Sqrt[d + e*x^2])), x] - Simp 
[b*c*(n/d)*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]]   Int[x*((a + b*ArcSinh[ 
c*x])^(n - 1)/(1 + c^2*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, 
 c^2*d] && GtQ[n, 0]
 

rule 6203

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_), x 
_Symbol] :> Simp[(-x)*(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(2*d*(p + 
 1))), x] + (Simp[(2*p + 3)/(2*d*(p + 1))   Int[(d + e*x^2)^(p + 1)*(a + b* 
ArcSinh[c*x])^n, x], x] + Simp[b*c*(n/(2*(p + 1)))*Simp[(d + e*x^2)^p/(1 + 
c^2*x^2)^p]   Int[x*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x 
], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && LtQ[p, 
 -1] && NeQ[p, -3/2]
 

rule 6213

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(2*e*(p 
+ 1))), x] - Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p] 
 Int[(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x] /; FreeQ[ 
{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && NeQ[p, -1]
 

Maple [A] (verified)

Time = 1.12 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.57

method	result	size
derivativedivides	\(\frac {-\frac {a^{2}}{4 d^{3} \left (c^{2} x^{2}+1\right )^{2}}+\frac {b^{2} \left (\frac {2 \,\operatorname {arcsinh}\left (x c \right )}{3}-\frac {-4 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}+4 \,\operatorname {arcsinh}\left (x c \right ) c^{4} x^{4}-6 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x c +8 \,\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}+3 \operatorname {arcsinh}\left (x c \right )^{2}-c^{2} x^{2}+4 \,\operatorname {arcsinh}\left (x c \right )-1}{12 \left (c^{4} x^{4}+2 c^{2} x^{2}+1\right )}-\frac {\ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{3}\right )}{d^{3}}+\frac {2 a b \left (-\frac {\operatorname {arcsinh}\left (x c \right )}{4 \left (c^{2} x^{2}+1\right )^{2}}+\frac {x c}{12 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {x c}{6 \sqrt {c^{2} x^{2}+1}}\right )}{d^{3}}}{c^{2}}\)	\(227\)
default	\(\frac {-\frac {a^{2}}{4 d^{3} \left (c^{2} x^{2}+1\right )^{2}}+\frac {b^{2} \left (\frac {2 \,\operatorname {arcsinh}\left (x c \right )}{3}-\frac {-4 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}+4 \,\operatorname {arcsinh}\left (x c \right ) c^{4} x^{4}-6 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x c +8 \,\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}+3 \operatorname {arcsinh}\left (x c \right )^{2}-c^{2} x^{2}+4 \,\operatorname {arcsinh}\left (x c \right )-1}{12 \left (c^{4} x^{4}+2 c^{2} x^{2}+1\right )}-\frac {\ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{3}\right )}{d^{3}}+\frac {2 a b \left (-\frac {\operatorname {arcsinh}\left (x c \right )}{4 \left (c^{2} x^{2}+1\right )^{2}}+\frac {x c}{12 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {x c}{6 \sqrt {c^{2} x^{2}+1}}\right )}{d^{3}}}{c^{2}}\)	\(227\)
parts	\(-\frac {a^{2}}{4 d^{3} c^{2} \left (c^{2} x^{2}+1\right )^{2}}+\frac {b^{2} \left (\frac {2 \,\operatorname {arcsinh}\left (x c \right )}{3}-\frac {-4 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}+4 \,\operatorname {arcsinh}\left (x c \right ) c^{4} x^{4}-6 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x c +8 \,\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}+3 \operatorname {arcsinh}\left (x c \right )^{2}-c^{2} x^{2}+4 \,\operatorname {arcsinh}\left (x c \right )-1}{12 \left (c^{4} x^{4}+2 c^{2} x^{2}+1\right )}-\frac {\ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{3}\right )}{d^{3} c^{2}}+\frac {2 a b \left (-\frac {\operatorname {arcsinh}\left (x c \right )}{4 \left (c^{2} x^{2}+1\right )^{2}}+\frac {x c}{12 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {x c}{6 \sqrt {c^{2} x^{2}+1}}\right )}{d^{3} c^{2}}\)	\(232\)

Input:

int(x*(a+b*arcsinh(x*c))^2/(c^2*d*x^2+d)^3,x,method=_RETURNVERBOSE)
 

Output:

1/c^2*(-1/4*a^2/d^3/(c^2*x^2+1)^2+b^2/d^3*(2/3*arcsinh(x*c)-1/12*(-4*arcsi 
nh(x*c)*(c^2*x^2+1)^(1/2)*x^3*c^3+4*arcsinh(x*c)*c^4*x^4-6*arcsinh(x*c)*(c 
^2*x^2+1)^(1/2)*x*c+8*arcsinh(x*c)*c^2*x^2+3*arcsinh(x*c)^2-c^2*x^2+4*arcs 
inh(x*c)-1)/(c^4*x^4+2*c^2*x^2+1)-1/3*ln(1+(x*c+(c^2*x^2+1)^(1/2))^2))+2*a 
*b/d^3*(-1/4*arcsinh(x*c)/(c^2*x^2+1)^2+1/12/(c^2*x^2+1)^(3/2)*x*c+1/6*x*c 
/(c^2*x^2+1)^(1/2)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 273 vs. \(2 (131) = 262\).

Time = 0.11 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.88 \[ \int \frac {x (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^3} \, dx=\frac {4 \, a b c^{4} x^{4} + {\left (8 \, a b + b^{2}\right )} c^{2} x^{2} - 3 \, b^{2} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} - 3 \, a^{2} + 4 \, a b + b^{2} - 2 \, {\left (b^{2} c^{4} x^{4} + 2 \, b^{2} c^{2} x^{2} + b^{2}\right )} \log \left (c^{2} x^{2} + 1\right ) + 2 \, {\left (3 \, a b c^{4} x^{4} + 6 \, a b c^{2} x^{2} + {\left (2 \, b^{2} c^{3} x^{3} + 3 \, b^{2} c x\right )} \sqrt {c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + 6 \, {\left (a b c^{4} x^{4} + 2 \, a b c^{2} x^{2} + a b\right )} \log \left (-c x + \sqrt {c^{2} x^{2} + 1}\right ) + 2 \, {\left (2 \, a b c^{3} x^{3} + 3 \, a b c x\right )} \sqrt {c^{2} x^{2} + 1}}{12 \, {\left (c^{6} d^{3} x^{4} + 2 \, c^{4} d^{3} x^{2} + c^{2} d^{3}\right )}} \] Input:

integrate(x*(a+b*arcsinh(c*x))^2/(c^2*d*x^2+d)^3,x, algorithm="fricas")
 

Output:

1/12*(4*a*b*c^4*x^4 + (8*a*b + b^2)*c^2*x^2 - 3*b^2*log(c*x + sqrt(c^2*x^2 
 + 1))^2 - 3*a^2 + 4*a*b + b^2 - 2*(b^2*c^4*x^4 + 2*b^2*c^2*x^2 + b^2)*log 
(c^2*x^2 + 1) + 2*(3*a*b*c^4*x^4 + 6*a*b*c^2*x^2 + (2*b^2*c^3*x^3 + 3*b^2* 
c*x)*sqrt(c^2*x^2 + 1))*log(c*x + sqrt(c^2*x^2 + 1)) + 6*(a*b*c^4*x^4 + 2* 
a*b*c^2*x^2 + a*b)*log(-c*x + sqrt(c^2*x^2 + 1)) + 2*(2*a*b*c^3*x^3 + 3*a* 
b*c*x)*sqrt(c^2*x^2 + 1))/(c^6*d^3*x^4 + 2*c^4*d^3*x^2 + c^2*d^3)
 

Sympy [F]

\[ \int \frac {x (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^3} \, dx=\frac {\int \frac {a^{2} x}{c^{6} x^{6} + 3 c^{4} x^{4} + 3 c^{2} x^{2} + 1}\, dx + \int \frac {b^{2} x \operatorname {asinh}^{2}{\left (c x \right )}}{c^{6} x^{6} + 3 c^{4} x^{4} + 3 c^{2} x^{2} + 1}\, dx + \int \frac {2 a b x \operatorname {asinh}{\left (c x \right )}}{c^{6} x^{6} + 3 c^{4} x^{4} + 3 c^{2} x^{2} + 1}\, dx}{d^{3}} \] Input:

integrate(x*(a+b*asinh(c*x))**2/(c**2*d*x**2+d)**3,x)
 

Output:

(Integral(a**2*x/(c**6*x**6 + 3*c**4*x**4 + 3*c**2*x**2 + 1), x) + Integra 
l(b**2*x*asinh(c*x)**2/(c**6*x**6 + 3*c**4*x**4 + 3*c**2*x**2 + 1), x) + I 
ntegral(2*a*b*x*asinh(c*x)/(c**6*x**6 + 3*c**4*x**4 + 3*c**2*x**2 + 1), x) 
)/d**3
 

Maxima [F]

\[ \int \frac {x (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^3} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x}{{\left (c^{2} d x^{2} + d\right )}^{3}} \,d x } \] Input:

integrate(x*(a+b*arcsinh(c*x))^2/(c^2*d*x^2+d)^3,x, algorithm="maxima")
 

Output:

-1/4*b^2*log(c*x + sqrt(c^2*x^2 + 1))^2/(c^6*d^3*x^4 + 2*c^4*d^3*x^2 + c^2 
*d^3) - 1/4*a^2/(c^6*d^3*x^4 + 2*c^4*d^3*x^2 + c^2*d^3) + integrate(1/2*(( 
4*a*b*c^2 + b^2*c^2)*x^2 + sqrt(c^2*x^2 + 1)*(4*a*b*c + b^2*c)*x + b^2)*lo 
g(c*x + sqrt(c^2*x^2 + 1))/(c^8*d^3*x^7 + 3*c^6*d^3*x^5 + 3*c^4*d^3*x^3 + 
c^2*d^3*x + (c^7*d^3*x^6 + 3*c^5*d^3*x^4 + 3*c^3*d^3*x^2 + c*d^3)*sqrt(c^2 
*x^2 + 1)), x)
 

Giac [F]

\[ \int \frac {x (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^3} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x}{{\left (c^{2} d x^{2} + d\right )}^{3}} \,d x } \] Input:

integrate(x*(a+b*arcsinh(c*x))^2/(c^2*d*x^2+d)^3,x, algorithm="giac")
 

Output:

integrate((b*arcsinh(c*x) + a)^2*x/(c^2*d*x^2 + d)^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^3} \, dx=\int \frac {x\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{{\left (d\,c^2\,x^2+d\right )}^3} \,d x \] Input:

int((x*(a + b*asinh(c*x))^2)/(d + c^2*d*x^2)^3,x)
 

Output:

int((x*(a + b*asinh(c*x))^2)/(d + c^2*d*x^2)^3, x)
 

Reduce [F]

\[ \int \frac {x (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^3} \, dx=\frac {8 \left (\int \frac {\mathit {asinh} \left (c x \right ) x}{c^{6} x^{6}+3 c^{4} x^{4}+3 c^{2} x^{2}+1}d x \right ) a b \,c^{6} x^{4}+16 \left (\int \frac {\mathit {asinh} \left (c x \right ) x}{c^{6} x^{6}+3 c^{4} x^{4}+3 c^{2} x^{2}+1}d x \right ) a b \,c^{4} x^{2}+8 \left (\int \frac {\mathit {asinh} \left (c x \right ) x}{c^{6} x^{6}+3 c^{4} x^{4}+3 c^{2} x^{2}+1}d x \right ) a b \,c^{2}+4 \left (\int \frac {\mathit {asinh} \left (c x \right )^{2} x}{c^{6} x^{6}+3 c^{4} x^{4}+3 c^{2} x^{2}+1}d x \right ) b^{2} c^{6} x^{4}+8 \left (\int \frac {\mathit {asinh} \left (c x \right )^{2} x}{c^{6} x^{6}+3 c^{4} x^{4}+3 c^{2} x^{2}+1}d x \right ) b^{2} c^{4} x^{2}+4 \left (\int \frac {\mathit {asinh} \left (c x \right )^{2} x}{c^{6} x^{6}+3 c^{4} x^{4}+3 c^{2} x^{2}+1}d x \right ) b^{2} c^{2}-a^{2}}{4 c^{2} d^{3} \left (c^{4} x^{4}+2 c^{2} x^{2}+1\right )} \] Input:

int(x*(a+b*asinh(c*x))^2/(c^2*d*x^2+d)^3,x)
 

Output:

(8*int((asinh(c*x)*x)/(c**6*x**6 + 3*c**4*x**4 + 3*c**2*x**2 + 1),x)*a*b*c 
**6*x**4 + 16*int((asinh(c*x)*x)/(c**6*x**6 + 3*c**4*x**4 + 3*c**2*x**2 + 
1),x)*a*b*c**4*x**2 + 8*int((asinh(c*x)*x)/(c**6*x**6 + 3*c**4*x**4 + 3*c* 
*2*x**2 + 1),x)*a*b*c**2 + 4*int((asinh(c*x)**2*x)/(c**6*x**6 + 3*c**4*x** 
4 + 3*c**2*x**2 + 1),x)*b**2*c**6*x**4 + 8*int((asinh(c*x)**2*x)/(c**6*x** 
6 + 3*c**4*x**4 + 3*c**2*x**2 + 1),x)*b**2*c**4*x**2 + 4*int((asinh(c*x)** 
2*x)/(c**6*x**6 + 3*c**4*x**4 + 3*c**2*x**2 + 1),x)*b**2*c**2 - a**2)/(4*c 
**2*d**3*(c**4*x**4 + 2*c**2*x**2 + 1))