\(\int \frac {c+a^2 c x^2}{\text {arcsinh}(a x)^2} \, dx\) [24]

Optimal result

Integrand size = 17, antiderivative size = 54 \[ \int \frac {c+a^2 c x^2}{\text {arcsinh}(a x)^2} \, dx=-\frac {c \left (1+a^2 x^2\right )^{3/2}}{a \text {arcsinh}(a x)}+\frac {3 c \text {Shi}(\text {arcsinh}(a x))}{4 a}+\frac {3 c \text {Shi}(3 \text {arcsinh}(a x))}{4 a} \] Output:

-c*(a^2*x^2+1)^(3/2)/a/arcsinh(a*x)+3/4*c*Shi(arcsinh(a*x))/a+3/4*c*Shi(3* 
arcsinh(a*x))/a
 

Mathematica [A] (verified)

Time = 0.33 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00 \[ \int \frac {c+a^2 c x^2}{\text {arcsinh}(a x)^2} \, dx=\frac {c \left (-4 \left (1+a^2 x^2\right )^{3/2}+3 \text {arcsinh}(a x) \text {Shi}(\text {arcsinh}(a x))+3 \text {arcsinh}(a x) \text {Shi}(3 \text {arcsinh}(a x))\right )}{4 a \text {arcsinh}(a x)} \] Input:

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

Output:

(c*(-4*(1 + a^2*x^2)^(3/2) + 3*ArcSinh[a*x]*SinhIntegral[ArcSinh[a*x]] + 3 
*ArcSinh[a*x]*SinhIntegral[3*ArcSinh[a*x]]))/(4*a*ArcSinh[a*x])
 

Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.98, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {6205, 6234, 5971, 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.

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

\(\Big \downarrow \) 6205

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

\(\Big \downarrow \) 6234

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

\(\Big \downarrow \) 5971

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-((c*(1 + a^2*x^2)^(3/2))/(a*ArcSinh[a*x])) + (3*c*(SinhIntegral[ArcSinh[a 
*x]]/4 + SinhIntegral[3*ArcSinh[a*x]]/4))/a
 

Defintions of rubi rules used

rule 2009

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

rule 5971

Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + 
(b_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sinh[a + 
b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & 
& IGtQ[p, 0]
 

rule 6205

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

rule 6234

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

Maple [A] (verified)

Time = 0.83 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.11

method	result	size
derivativedivides	\(\frac {c \left (3 \,\operatorname {Shi}\left (\operatorname {arcsinh}\left (x a \right )\right ) \operatorname {arcsinh}\left (x a \right )+3 \,\operatorname {Shi}\left (3 \,\operatorname {arcsinh}\left (x a \right )\right ) \operatorname {arcsinh}\left (x a \right )-\cosh \left (3 \,\operatorname {arcsinh}\left (x a \right )\right )-3 \sqrt {a^{2} x^{2}+1}\right )}{4 a \,\operatorname {arcsinh}\left (x a \right )}\)	\(60\)
default	\(\frac {c \left (3 \,\operatorname {Shi}\left (\operatorname {arcsinh}\left (x a \right )\right ) \operatorname {arcsinh}\left (x a \right )+3 \,\operatorname {Shi}\left (3 \,\operatorname {arcsinh}\left (x a \right )\right ) \operatorname {arcsinh}\left (x a \right )-\cosh \left (3 \,\operatorname {arcsinh}\left (x a \right )\right )-3 \sqrt {a^{2} x^{2}+1}\right )}{4 a \,\operatorname {arcsinh}\left (x a \right )}\)	\(60\)

Input:

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

Output:

1/4/a*c*(3*Shi(arcsinh(x*a))*arcsinh(x*a)+3*Shi(3*arcsinh(x*a))*arcsinh(x* 
a)-cosh(3*arcsinh(x*a))-3*(a^2*x^2+1)^(1/2))/arcsinh(x*a)
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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