\(\int \frac {(c+a^2 c x^2)^2}{\text {arcsinh}(a x)} \, dx\) [18]

Optimal result

Integrand size = 19, antiderivative size = 50 \[ \int \frac {\left (c+a^2 c x^2\right )^2}{\text {arcsinh}(a x)} \, dx=\frac {5 c^2 \text {Chi}(\text {arcsinh}(a x))}{8 a}+\frac {5 c^2 \text {Chi}(3 \text {arcsinh}(a x))}{16 a}+\frac {c^2 \text {Chi}(5 \text {arcsinh}(a x))}{16 a} \] Output:

5/8*c^2*Chi(arcsinh(a*x))/a+5/16*c^2*Chi(3*arcsinh(a*x))/a+1/16*c^2*Chi(5* 
arcsinh(a*x))/a
 

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.68 \[ \int \frac {\left (c+a^2 c x^2\right )^2}{\text {arcsinh}(a x)} \, dx=\frac {c^2 (10 \text {Chi}(\text {arcsinh}(a x))+5 \text {Chi}(3 \text {arcsinh}(a x))+\text {Chi}(5 \text {arcsinh}(a x)))}{16 a} \] Input:

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

Output:

(c^2*(10*CoshIntegral[ArcSinh[a*x]] + 5*CoshIntegral[3*ArcSinh[a*x]] + Cos 
hIntegral[5*ArcSinh[a*x]]))/(16*a)
 

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.78, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {6206, 3042, 3793, 2009}

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.

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

\(\Big \downarrow \) 6206

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

\(\Big \downarrow \) 3042

\(\displaystyle \frac {c^2 \int \frac {\sin \left (i \text {arcsinh}(a x)+\frac {\pi }{2}\right )^5}{\text {arcsinh}(a x)}d\text {arcsinh}(a x)}{a}\)

\(\Big \downarrow \) 3793

\(\displaystyle \frac {c^2 \int \left (\frac {5 \cosh (3 \text {arcsinh}(a x))}{16 \text {arcsinh}(a x)}+\frac {\cosh (5 \text {arcsinh}(a x))}{16 \text {arcsinh}(a x)}+\frac {5 \sqrt {a^2 x^2+1}}{8 \text {arcsinh}(a x)}\right )d\text {arcsinh}(a x)}{a}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {c^2 \left (\frac {5}{8} \text {Chi}(\text {arcsinh}(a x))+\frac {5}{16} \text {Chi}(3 \text {arcsinh}(a x))+\frac {1}{16} \text {Chi}(5 \text {arcsinh}(a x))\right )}{a}\)

Input:

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

Output:

(c^2*((5*CoshIntegral[ArcSinh[a*x]])/8 + (5*CoshIntegral[3*ArcSinh[a*x]])/ 
16 + CoshIntegral[5*ArcSinh[a*x]]/16))/a
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3793

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In 
t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f 
, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
 

rule 6206

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

Maple [A] (verified)

Time = 0.59 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.66

method	result	size
derivativedivides	\(\frac {c^{2} \left (10 \,\operatorname {Chi}\left (\operatorname {arcsinh}\left (x a \right )\right )+5 \,\operatorname {Chi}\left (3 \,\operatorname {arcsinh}\left (x a \right )\right )+\operatorname {Chi}\left (5 \,\operatorname {arcsinh}\left (x a \right )\right )\right )}{16 a}\)	\(33\)
default	\(\frac {c^{2} \left (10 \,\operatorname {Chi}\left (\operatorname {arcsinh}\left (x a \right )\right )+5 \,\operatorname {Chi}\left (3 \,\operatorname {arcsinh}\left (x a \right )\right )+\operatorname {Chi}\left (5 \,\operatorname {arcsinh}\left (x a \right )\right )\right )}{16 a}\)	\(33\)

Input:

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

Output:

1/16/a*c^2*(10*Chi(arcsinh(x*a))+5*Chi(3*arcsinh(x*a))+Chi(5*arcsinh(x*a)) 
)
 

Fricas [F]

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

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

Output:

integral((a^4*c^2*x^4 + 2*a^2*c^2*x^2 + c^2)/arcsinh(a*x), x)
 

Sympy [F]

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

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

Output:

c**2*(Integral(2*a**2*x**2/asinh(a*x), x) + Integral(a**4*x**4/asinh(a*x), 
 x) + Integral(1/asinh(a*x), x))
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

c**2*(int(x**4/asinh(a*x),x)*a**4 + 2*int(x**2/asinh(a*x),x)*a**2 + int(1/ 
asinh(a*x),x))