\(\int \frac {x (a+b \text {arcsinh}(c x))}{(d+c^2 d x^2)^2} \, dx\) [40]

Optimal result

Integrand size = 22, antiderivative size = 55 \[ \int \frac {x (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^2} \, dx=\frac {b x}{2 c d^2 \sqrt {1+c^2 x^2}}-\frac {a+b \text {arcsinh}(c x)}{2 c^2 d^2 \left (1+c^2 x^2\right )} \] Output:

1/2*b*x/c/d^2/(c^2*x^2+1)^(1/2)-1/2*(a+b*arcsinh(c*x))/c^2/d^2/(c^2*x^2+1)
 

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.35 \[ \int \frac {x (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^2} \, dx=-\frac {a}{2 c^2 d^2 \left (1+c^2 x^2\right )}+\frac {b x}{2 c d^2 \sqrt {1+c^2 x^2}}-\frac {b \text {arcsinh}(c x)}{2 c^2 d^2 \left (1+c^2 x^2\right )} \] Input:

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

Output:

-1/2*a/(c^2*d^2*(1 + c^2*x^2)) + (b*x)/(2*c*d^2*Sqrt[1 + c^2*x^2]) - (b*Ar 
cSinh[c*x])/(2*c^2*d^2*(1 + c^2*x^2))
 

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {6213, 208}

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.

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

\(\Big \downarrow \) 6213

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

\(\Big \downarrow \) 208

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 208

Int[((a_) + (b_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[x/(a*Sqrt[a + b*x^2]), 
x] /; FreeQ[{a, b}, x]
 

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 = 0.34 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.11

method	result	size
derivativedivides	\(\frac {-\frac {a}{2 d^{2} \left (c^{2} x^{2}+1\right )}+\frac {b \left (-\frac {\operatorname {arcsinh}\left (x c \right )}{2 \left (c^{2} x^{2}+1\right )}+\frac {x c}{2 \sqrt {c^{2} x^{2}+1}}\right )}{d^{2}}}{c^{2}}\)	\(61\)
default	\(\frac {-\frac {a}{2 d^{2} \left (c^{2} x^{2}+1\right )}+\frac {b \left (-\frac {\operatorname {arcsinh}\left (x c \right )}{2 \left (c^{2} x^{2}+1\right )}+\frac {x c}{2 \sqrt {c^{2} x^{2}+1}}\right )}{d^{2}}}{c^{2}}\)	\(61\)
parts	\(-\frac {a}{2 d^{2} c^{2} \left (c^{2} x^{2}+1\right )}+\frac {b \left (-\frac {\operatorname {arcsinh}\left (x c \right )}{2 \left (c^{2} x^{2}+1\right )}+\frac {x c}{2 \sqrt {c^{2} x^{2}+1}}\right )}{d^{2} c^{2}}\)	\(63\)
orering	\(\frac {\left (c^{2} x^{2}+1\right ) \left (3 c^{2} x^{2}-2\right ) \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )}{2 c^{2} \left (c^{2} d \,x^{2}+d \right )^{2}}+\frac {\left (c^{2} x^{2}+1\right )^{2} \left (\frac {a +b \,\operatorname {arcsinh}\left (x c \right )}{\left (c^{2} d \,x^{2}+d \right )^{2}}+\frac {x b c}{\sqrt {c^{2} x^{2}+1}\, \left (c^{2} d \,x^{2}+d \right )^{2}}-\frac {4 x^{2} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) c^{2} d}{\left (c^{2} d \,x^{2}+d \right )^{3}}\right )}{2 c^{2}}\)	\(140\)

Input:

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

Output:

1/c^2*(-1/2*a/d^2/(c^2*x^2+1)+b/d^2*(-1/2/(c^2*x^2+1)*arcsinh(x*c)+1/2*x*c 
/(c^2*x^2+1)^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.18 \[ \int \frac {x (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^2} \, dx=\frac {a c^{2} x^{2} + \sqrt {c^{2} x^{2} + 1} b c x - b \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )}{2 \, {\left (c^{4} d^{2} x^{2} + c^{2} d^{2}\right )}} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

-1/4*b*((2*log(c*x + sqrt(c^2*x^2 + 1)) + 1)/(c^4*d^2*x^2 + c^2*d^2) - 4*i 
ntegrate(1/2/(c^6*d^2*x^5 + 2*c^4*d^2*x^3 + c^2*d^2*x + (c^5*d^2*x^4 + 2*c 
^3*d^2*x^2 + c*d^2)*sqrt(c^2*x^2 + 1)), x)) - 1/2*a/(c^4*d^2*x^2 + c^2*d^2 
)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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