\(\int \frac {x (a+b \text {arcsinh}(c x))^2}{(d+c^2 d x^2)^{3/2}} \, dx\) [311]

Optimal result

Integrand size = 26, antiderivative size = 188 \[ \int \frac {x (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=-\frac {(a+b \text {arcsinh}(c x))^2}{c^2 d \sqrt {d+c^2 d x^2}}+\frac {4 b \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{c^2 d \sqrt {d+c^2 d x^2}}-\frac {2 i b^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{c^2 d \sqrt {d+c^2 d x^2}}+\frac {2 i b^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{c^2 d \sqrt {d+c^2 d x^2}} \] Output:

-(a+b*arcsinh(c*x))^2/c^2/d/(c^2*d*x^2+d)^(1/2)+4*b*(c^2*x^2+1)^(1/2)*(a+b 
*arcsinh(c*x))*arctan(c*x+(c^2*x^2+1)^(1/2))/c^2/d/(c^2*d*x^2+d)^(1/2)-2*I 
*b^2*(c^2*x^2+1)^(1/2)*polylog(2,-I*(c*x+(c^2*x^2+1)^(1/2)))/c^2/d/(c^2*d* 
x^2+d)^(1/2)+2*I*b^2*(c^2*x^2+1)^(1/2)*polylog(2,I*(c*x+(c^2*x^2+1)^(1/2)) 
)/c^2/d/(c^2*d*x^2+d)^(1/2)
 

Mathematica [A] (verified)

Time = 0.80 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.15 \[ \int \frac {x (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=-\frac {a^2+2 a b \text {arcsinh}(c x)+b^2 \text {arcsinh}(c x)^2-4 a b \sqrt {1+c^2 x^2} \arctan \left (\tanh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )+2 i b^2 \sqrt {1+c^2 x^2} \text {arcsinh}(c x) \log \left (1-i e^{-\text {arcsinh}(c x)}\right )-2 i b^2 \sqrt {1+c^2 x^2} \text {arcsinh}(c x) \log \left (1+i e^{-\text {arcsinh}(c x)}\right )+2 i b^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-i e^{-\text {arcsinh}(c x)}\right )-2 i b^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,i e^{-\text {arcsinh}(c x)}\right )}{c^2 d \sqrt {d+c^2 d x^2}} \] Input:

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

Output:

-((a^2 + 2*a*b*ArcSinh[c*x] + b^2*ArcSinh[c*x]^2 - 4*a*b*Sqrt[1 + c^2*x^2] 
*ArcTan[Tanh[ArcSinh[c*x]/2]] + (2*I)*b^2*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*L 
og[1 - I/E^ArcSinh[c*x]] - (2*I)*b^2*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*Log[1 
+ I/E^ArcSinh[c*x]] + (2*I)*b^2*Sqrt[1 + c^2*x^2]*PolyLog[2, (-I)/E^ArcSin 
h[c*x]] - (2*I)*b^2*Sqrt[1 + c^2*x^2]*PolyLog[2, I/E^ArcSinh[c*x]])/(c^2*d 
*Sqrt[d + c^2*d*x^2]))
 

Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.64, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {6213, 6204, 3042, 4668, 2715, 2838}

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.

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

\(\Big \downarrow \) 6213

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

\(\Big \downarrow \) 6204

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4668

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

\(\displaystyle -\frac {(a+b \text {arcsinh}(c x))^2}{c^2 d \sqrt {c^2 d x^2+d}}+\frac {2 b \sqrt {c^2 x^2+1} \left (2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))-i b \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )+i b \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )\right )}{c^2 d \sqrt {c^2 d x^2+d}}\)

Input:

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

Output:

-((a + b*ArcSinh[c*x])^2/(c^2*d*Sqrt[d + c^2*d*x^2])) + (2*b*Sqrt[1 + c^2* 
x^2]*(2*(a + b*ArcSinh[c*x])*ArcTan[E^ArcSinh[c*x]] - I*b*PolyLog[2, (-I)* 
E^ArcSinh[c*x]] + I*b*PolyLog[2, I*E^ArcSinh[c*x]]))/(c^2*d*Sqrt[d + c^2*d 
*x^2])
 

Defintions of rubi rules used

rule 2715

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] 
:> Simp[1/(d*e*n*Log[F])   Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) 
))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
 

rule 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

rule 3042

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

rule 4668

Int[csc[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_ 
))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)/E^( 
I*k*Pi)]/(f*fz*I)), x] + (-Simp[d*(m/(f*fz*I))   Int[(c + d*x)^(m - 1)*Log[ 
1 - E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x] + Simp[d*(m/(f*fz*I))   Int[(c 
+ d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x]) /; FreeQ[{c 
, d, e, f, fz}, x] && IntegerQ[2*k] && IGtQ[m, 0]
 

rule 6204

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2), x_Symb 
ol] :> Simp[1/(c*d)   Subst[Int[(a + b*x)^n*Sech[x], x], x, ArcSinh[c*x]], 
x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[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 = 1.14 (sec) , antiderivative size = 318, normalized size of antiderivative = 1.69

method	result	size
default	\(-\frac {a^{2}}{c^{2} d \sqrt {c^{2} d \,x^{2}+d}}-\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 i \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (x c \right ) \ln \left (1+i \left (x c +\sqrt {c^{2} x^{2}+1}\right )\right )-2 i \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (x c \right ) \ln \left (1-i \left (x c +\sqrt {c^{2} x^{2}+1}\right )\right )+2 i \sqrt {c^{2} x^{2}+1}\, \operatorname {dilog}\left (1+i \left (x c +\sqrt {c^{2} x^{2}+1}\right )\right )-2 i \sqrt {c^{2} x^{2}+1}\, \operatorname {dilog}\left (1-i \left (x c +\sqrt {c^{2} x^{2}+1}\right )\right )+\operatorname {arcsinh}\left (x c \right )^{2}\right )}{d^{2} c^{2} \left (c^{2} x^{2}+1\right )}-\frac {2 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (-i \sqrt {c^{2} x^{2}+1}\, \ln \left (x c +\sqrt {c^{2} x^{2}+1}+i\right )+i \sqrt {c^{2} x^{2}+1}\, \ln \left (x c +\sqrt {c^{2} x^{2}+1}-i\right )+\operatorname {arcsinh}\left (x c \right )\right )}{d^{2} c^{2} \left (c^{2} x^{2}+1\right )}\)	\(318\)
parts	\(-\frac {a^{2}}{c^{2} d \sqrt {c^{2} d \,x^{2}+d}}-\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 i \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (x c \right ) \ln \left (1+i \left (x c +\sqrt {c^{2} x^{2}+1}\right )\right )-2 i \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (x c \right ) \ln \left (1-i \left (x c +\sqrt {c^{2} x^{2}+1}\right )\right )+2 i \sqrt {c^{2} x^{2}+1}\, \operatorname {dilog}\left (1+i \left (x c +\sqrt {c^{2} x^{2}+1}\right )\right )-2 i \sqrt {c^{2} x^{2}+1}\, \operatorname {dilog}\left (1-i \left (x c +\sqrt {c^{2} x^{2}+1}\right )\right )+\operatorname {arcsinh}\left (x c \right )^{2}\right )}{d^{2} c^{2} \left (c^{2} x^{2}+1\right )}-\frac {2 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (-i \sqrt {c^{2} x^{2}+1}\, \ln \left (x c +\sqrt {c^{2} x^{2}+1}+i\right )+i \sqrt {c^{2} x^{2}+1}\, \ln \left (x c +\sqrt {c^{2} x^{2}+1}-i\right )+\operatorname {arcsinh}\left (x c \right )\right )}{d^{2} c^{2} \left (c^{2} x^{2}+1\right )}\)	\(318\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

Integral(x*(a + b*asinh(c*x))**2/(d*(c**2*x**2 + 1))**(3/2), x)
 

Maxima [F]

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

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

Output:

-a^2/(sqrt(c^2*d*x^2 + d)*c^2*d) + integrate(b^2*x*log(c*x + sqrt(c^2*x^2 
+ 1))^2/(c^2*d*x^2 + d)^(3/2) + 2*a*b*x*log(c*x + sqrt(c^2*x^2 + 1))/(c^2* 
d*x^2 + d)^(3/2), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - sqrt(c**2*x**2 + 1)*a**2 + 2*int((asinh(c*x)*x)/(sqrt(c**2*x**2 + 1)*c 
**2*x**2 + sqrt(c**2*x**2 + 1)),x)*a*b*c**4*x**2 + 2*int((asinh(c*x)*x)/(s 
qrt(c**2*x**2 + 1)*c**2*x**2 + sqrt(c**2*x**2 + 1)),x)*a*b*c**2 + int((asi 
nh(c*x)**2*x)/(sqrt(c**2*x**2 + 1)*c**2*x**2 + sqrt(c**2*x**2 + 1)),x)*b** 
2*c**4*x**2 + int((asinh(c*x)**2*x)/(sqrt(c**2*x**2 + 1)*c**2*x**2 + sqrt( 
c**2*x**2 + 1)),x)*b**2*c**2)/(sqrt(d)*c**2*d*(c**2*x**2 + 1))