\(\int \frac {x (a+b \text {arcsinh}(c x))^2}{(d+c^2 d x^2)^2} \, dx\) [249]

Optimal result

Integrand size = 24, antiderivative size = 85 \[ \int \frac {x (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^2} \, dx=\frac {b x (a+b \text {arcsinh}(c x))}{c d^2 \sqrt {1+c^2 x^2}}-\frac {(a+b \text {arcsinh}(c x))^2}{2 c^2 d^2 \left (1+c^2 x^2\right )}-\frac {b^2 \log \left (1+c^2 x^2\right )}{2 c^2 d^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.49 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.71 \[ \int \frac {x (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^2} \, dx=-\frac {a^2}{2 c^2 d^2 \left (1+c^2 x^2\right )}+\frac {a b x}{c d^2 \sqrt {1+c^2 x^2}}+\frac {b \left (-a+b c x \sqrt {1+c^2 x^2}\right ) \text {arcsinh}(c x)}{c^2 d^2 \left (1+c^2 x^2\right )}-\frac {b^2 \text {arcsinh}(c x)^2}{2 c^2 d^2 \left (1+c^2 x^2\right )}-\frac {b^2 \log \left (1+c^2 x^2\right )}{2 c^2 d^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.96, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6213, 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.

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

\(\Big \downarrow \) 6213

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

\(\Big \downarrow \) 6202

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

\(\Big \downarrow \) 240

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

Input:

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

Output:

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

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 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 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 = 1.18 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.73

method	result	size
derivativedivides	\(\frac {-\frac {a^{2}}{2 d^{2} \left (c^{2} x^{2}+1\right )}+\frac {b^{2} \left (2 \,\operatorname {arcsinh}\left (x c \right )-\frac {\left (2 c^{2} x^{2}-2 \sqrt {c^{2} x^{2}+1}\, x c +\operatorname {arcsinh}\left (x c \right )+2\right ) \operatorname {arcsinh}\left (x c \right )}{2 \left (c^{2} x^{2}+1\right )}-\ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )\right )}{d^{2}}+\frac {2 a b \left (-\frac {\operatorname {arcsinh}\left (x c \right )}{2 \left (c^{2} x^{2}+1\right )}+\frac {x c}{2 \sqrt {c^{2} x^{2}+1}}\right )}{d^{2}}}{c^{2}}\)	\(147\)
default	\(\frac {-\frac {a^{2}}{2 d^{2} \left (c^{2} x^{2}+1\right )}+\frac {b^{2} \left (2 \,\operatorname {arcsinh}\left (x c \right )-\frac {\left (2 c^{2} x^{2}-2 \sqrt {c^{2} x^{2}+1}\, x c +\operatorname {arcsinh}\left (x c \right )+2\right ) \operatorname {arcsinh}\left (x c \right )}{2 \left (c^{2} x^{2}+1\right )}-\ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )\right )}{d^{2}}+\frac {2 a b \left (-\frac {\operatorname {arcsinh}\left (x c \right )}{2 \left (c^{2} x^{2}+1\right )}+\frac {x c}{2 \sqrt {c^{2} x^{2}+1}}\right )}{d^{2}}}{c^{2}}\)	\(147\)
parts	\(-\frac {a^{2}}{2 d^{2} c^{2} \left (c^{2} x^{2}+1\right )}+\frac {b^{2} \left (2 \,\operatorname {arcsinh}\left (x c \right )-\frac {\left (2 c^{2} x^{2}-2 \sqrt {c^{2} x^{2}+1}\, x c +\operatorname {arcsinh}\left (x c \right )+2\right ) \operatorname {arcsinh}\left (x c \right )}{2 \left (c^{2} x^{2}+1\right )}-\ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )\right )}{d^{2} c^{2}}+\frac {2 a b \left (-\frac {\operatorname {arcsinh}\left (x c \right )}{2 \left (c^{2} x^{2}+1\right )}+\frac {x c}{2 \sqrt {c^{2} x^{2}+1}}\right )}{d^{2} c^{2}}\)	\(152\)

Input:

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

Output:

1/c^2*(-1/2*a^2/d^2/(c^2*x^2+1)+b^2/d^2*(2*arcsinh(x*c)-1/2*(2*c^2*x^2-2*( 
c^2*x^2+1)^(1/2)*x*c+arcsinh(x*c)+2)*arcsinh(x*c)/(c^2*x^2+1)-ln(1+(x*c+(c 
^2*x^2+1)^(1/2))^2))+2*a*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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 185 vs. \(2 (79) = 158\).

Time = 0.11 (sec) , antiderivative size = 185, normalized size of antiderivative = 2.18 \[ \int \frac {x (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^2} \, dx=\frac {2 \, a b c^{2} x^{2} + 2 \, \sqrt {c^{2} x^{2} + 1} a b c x - b^{2} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} - a^{2} + 2 \, a b - {\left (b^{2} c^{2} x^{2} + b^{2}\right )} \log \left (c^{2} x^{2} + 1\right ) + 2 \, {\left (a b c^{2} x^{2} + \sqrt {c^{2} x^{2} + 1} b^{2} c x\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + 2 \, {\left (a b c^{2} x^{2} + a b\right )} \log \left (-c x + \sqrt {c^{2} x^{2} + 1}\right )}{2 \, {\left (c^{4} d^{2} x^{2} + c^{2} d^{2}\right )}} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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