\(\int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{x^4} \, dx\) [271]

Optimal result

Integrand size = 28, antiderivative size = 294 \[ \int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{x^4} \, dx=-\frac {b^2 c^2 \sqrt {d+c^2 d x^2}}{3 x}+\frac {b^2 c^3 \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)}{3 \sqrt {1+c^2 x^2}}-\frac {b c \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{3 x^2}-\frac {c^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{3 \sqrt {1+c^2 x^2}}-\frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{3 d x^3}+\frac {2 b c^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \log \left (1-e^{2 \text {arcsinh}(c x)}\right )}{3 \sqrt {1+c^2 x^2}}+\frac {b^2 c^3 \sqrt {d+c^2 d x^2} \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )}{3 \sqrt {1+c^2 x^2}} \] Output:

-1/3*b^2*c^2*(c^2*d*x^2+d)^(1/2)/x+1/3*b^2*c^3*(c^2*d*x^2+d)^(1/2)*arcsinh 
(c*x)/(c^2*x^2+1)^(1/2)-1/3*b*c*(c^2*x^2+1)^(1/2)*(c^2*d*x^2+d)^(1/2)*(a+b 
*arcsinh(c*x))/x^2-1/3*c^3*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))^2/(c^2*x 
^2+1)^(1/2)-1/3*(c^2*d*x^2+d)^(3/2)*(a+b*arcsinh(c*x))^2/d/x^3+2/3*b*c^3*( 
c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))*ln(1-(c*x+(c^2*x^2+1)^(1/2))^2)/(c^2 
*x^2+1)^(1/2)+1/3*b^2*c^3*(c^2*d*x^2+d)^(1/2)*polylog(2,(c*x+(c^2*x^2+1)^( 
1/2))^2)/(c^2*x^2+1)^(1/2)
 

Mathematica [A] (verified)

Time = 0.67 (sec) , antiderivative size = 240, normalized size of antiderivative = 0.82 \[ \int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{x^4} \, dx=-\frac {\sqrt {d+c^2 d x^2} \left (a b c x+a^2 \sqrt {1+c^2 x^2}+a^2 c^2 x^2 \sqrt {1+c^2 x^2}+b^2 c^2 x^2 \sqrt {1+c^2 x^2}+b^2 \left (-c^3 x^3+\sqrt {1+c^2 x^2}+c^2 x^2 \sqrt {1+c^2 x^2}\right ) \text {arcsinh}(c x)^2-b \text {arcsinh}(c x) \left (-b c x-2 a \left (1+c^2 x^2\right )^{3/2}+2 b c^3 x^3 \log \left (1-e^{-2 \text {arcsinh}(c x)}\right )\right )-2 a b c^3 x^3 \log (c x)+b^2 c^3 x^3 \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c x)}\right )\right )}{3 x^3 \sqrt {1+c^2 x^2}} \] Input:

Integrate[(Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/x^4,x]
 

Output:

-1/3*(Sqrt[d + c^2*d*x^2]*(a*b*c*x + a^2*Sqrt[1 + c^2*x^2] + a^2*c^2*x^2*S 
qrt[1 + c^2*x^2] + b^2*c^2*x^2*Sqrt[1 + c^2*x^2] + b^2*(-(c^3*x^3) + Sqrt[ 
1 + c^2*x^2] + c^2*x^2*Sqrt[1 + c^2*x^2])*ArcSinh[c*x]^2 - b*ArcSinh[c*x]* 
(-(b*c*x) - 2*a*(1 + c^2*x^2)^(3/2) + 2*b*c^3*x^3*Log[1 - E^(-2*ArcSinh[c* 
x])]) - 2*a*b*c^3*x^3*Log[c*x] + b^2*c^3*x^3*PolyLog[2, E^(-2*ArcSinh[c*x] 
)]))/(x^3*Sqrt[1 + c^2*x^2])
 

Rubi [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 1.38 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.74, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {6215, 6217, 247, 222, 6190, 25, 3042, 26, 4201, 2620, 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.

\(\displaystyle \int \frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{x^4} \, dx\)

\(\Big \downarrow \) 6215

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

\(\Big \downarrow \) 6217

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

\(\Big \downarrow \) 247

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

\(\Big \downarrow \) 222

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

\(\Big \downarrow \) 6190

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4201

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

\(\displaystyle -\frac {\left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{3 d x^3}+\frac {2 b c \sqrt {c^2 d x^2+d} \left (\frac {i c^2 \left (2 i \left (-\frac {1}{4} b^2 \int e^{-\frac {2 a}{b}+\frac {2 (a+b \text {arcsinh}(c x))}{b}+i \pi } \log \left (1+e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}-i \pi }\right )de^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}-i \pi }-\frac {1}{2} b (a+b \text {arcsinh}(c x)) \log \left (1+e^{-\frac {2 (a+b \text {arcsinh}(c x))}{b}+\frac {2 a}{b}-i \pi }\right )\right )-\frac {1}{2} i (a+b \text {arcsinh}(c x))^2\right )}{b}-\frac {\left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {1}{2} b c \left (c \text {arcsinh}(c x)-\frac {\sqrt {c^2 x^2+1}}{x}\right )\right )}{3 \sqrt {c^2 x^2+1}}\)

\(\Big \downarrow \) 2838

\(\displaystyle -\frac {\left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{3 d x^3}+\frac {2 b c \sqrt {c^2 d x^2+d} \left (\frac {i c^2 \left (2 i \left (\frac {1}{4} b^2 \operatorname {PolyLog}(2,-a-b \text {arcsinh}(c x))-\frac {1}{2} b (a+b \text {arcsinh}(c x)) \log \left (1+e^{-\frac {2 (a+b \text {arcsinh}(c x))}{b}+\frac {2 a}{b}-i \pi }\right )\right )-\frac {1}{2} i (a+b \text {arcsinh}(c x))^2\right )}{b}-\frac {\left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {1}{2} b c \left (c \text {arcsinh}(c x)-\frac {\sqrt {c^2 x^2+1}}{x}\right )\right )}{3 \sqrt {c^2 x^2+1}}\)

Input:

Int[(Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/x^4,x]
 

Output:

-1/3*((d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2)/(d*x^3) + (2*b*c*Sqrt[ 
d + c^2*d*x^2]*(-1/2*((1 + c^2*x^2)*(a + b*ArcSinh[c*x]))/x^2 + (b*c*(-(Sq 
rt[1 + c^2*x^2]/x) + c*ArcSinh[c*x]))/2 + (I*c^2*((-1/2*I)*(a + b*ArcSinh[ 
c*x])^2 + (2*I)*(-1/2*(b*(a + b*ArcSinh[c*x])*Log[1 + E^((2*a)/b - I*Pi - 
(2*(a + b*ArcSinh[c*x]))/b)]) + (b^2*PolyLog[2, -a - b*ArcSinh[c*x]])/4))) 
/b))/(3*Sqrt[1 + c^2*x^2])
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 222

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt 
[a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
 

rule 247

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ 
(m + 1)*((a + b*x^2)^p/(c*(m + 1))), x] - Simp[2*b*(p/(c^2*(m + 1)))   Int[ 
(c*x)^(m + 2)*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && GtQ[p, 
0] && LtQ[m, -1] &&  !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomialQ[a, b, c, 2, 
m, p, x]
 

rule 2620

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ 
((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp 
[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si 
mp[d*(m/(b*f*g*n*Log[F]))   Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x 
)))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
 

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 4201

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

rule 6190

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Simp[1/b 
 Subst[Int[x^n*Coth[-a/b + x/b], x], x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a 
, b, c}, x] && IGtQ[n, 0]
 

rule 6215

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ 
.)*(x_)^2)^(p_), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*((a + 
b*ArcSinh[c*x])^n/(d*f*(m + 1))), x] - Simp[b*c*(n/(f*(m + 1)))*Simp[(d + e 
*x^2)^p/(1 + c^2*x^2)^p]   Int[(f*x)^(m + 1)*(1 + c^2*x^2)^(p + 1/2)*(a + b 
*ArcSinh[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ 
[e, c^2*d] && GtQ[n, 0] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]
 

rule 6217

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1728\) vs. \(2(274)=548\).

Time = 1.24 (sec) , antiderivative size = 1729, normalized size of antiderivative = 5.88

method	result	size
default	\(\text {Expression too large to display}\)	\(1729\)
parts	\(\text {Expression too large to display}\)	\(1729\)

Input:

int((c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(x*c))^2/x^4,x,method=_RETURNVERBOSE)
 

Output:

2/3*b^2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)*polylog(2,-x*c-(c^2*x^2+1) 
^(1/2))*c^3-1/3*b^2*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)*x^3*c^6- 
1/3*a*b*(d*(c^2*x^2+1))^(1/2)*(2*arcsinh(x*c)*x^3*c^3-2*ln((x*c+(c^2*x^2+1 
)^(1/2))^2-1)*x^3*c^3+2*(c^2*x^2+1)^(1/2)*arcsinh(x*c)*x^2*c^2+2*arcsinh(x 
*c)*(c^2*x^2+1)^(1/2)+x*c)/(c^2*x^2+1)^(1/2)/x^3+2/3*b^2*(d*(c^2*x^2+1))^( 
1/2)/(c^2*x^2+1)^(1/2)*arcsinh(x*c)*ln(1+x*c+(c^2*x^2+1)^(1/2))*c^3+1/3*b^ 
2*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)*x*arcsinh(x*c)*c^4+1/3*b^2 
*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)*x^3*arcsinh(x*c)*c^6+2/3*b^ 
2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)*arcsinh(x*c)*ln(1-x*c-(c^2*x^2+1 
)^(1/2))*c^3-2/3*b^2*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)*x^5/(c^ 
2*x^2+1)*c^8-5/3*b^2*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)*x^3/(c^ 
2*x^2+1)*c^6-4/3*b^2*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)*x/(c^2* 
x^2+1)*c^4-1/3*b^2*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)/x/(c^2*x^ 
2+1)*c^2-1/3*b^2*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)/x^3/(c^2*x^ 
2+1)*arcsinh(x*c)^2+1/3*b^2*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)/ 
(c^2*x^2+1)^(1/2)*arcsinh(x*c)^2*c^3-b^2*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+ 
3*c^2*x^2+1)/(c^2*x^2+1)^(1/2)*arcsinh(x*c)*c^3+b^2*(d*(c^2*x^2+1))^(1/2)/ 
(3*c^4*x^4+3*c^2*x^2+1)*x^4/(c^2*x^2+1)^(1/2)*c^7+b^2*(d*(c^2*x^2+1))^(1/2 
)/(3*c^4*x^4+3*c^2*x^2+1)*x^2/(c^2*x^2+1)^(1/2)*c^5-1/3*a^2/d/x^3*(c^2*d*x 
^2+d)^(3/2)+b^2*(d*(c^2*x^2+1))^(1/2)/(3*c^4*x^4+3*c^2*x^2+1)*x^2/(c^2*...
 

Fricas [F]

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

integrate((c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))^2/x^4,x, algorithm="frica 
s")
 

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate((c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))^2/x^4,x, algorithm="maxim 
a")
 

Output:

-1/3*((-1)^(2*c^2*d*x^2 + 2*d)*c^2*d^(3/2)*log(2*c^2*d + 2*d/x^2) - c^2*d^ 
(3/2)*log(x^2 + 1/c^2) + sqrt(c^4*d*x^4 + 2*c^2*d*x^2 + d)*d/x^2)*a*b*c/d 
- 1/3*b^2*((c^2*sqrt(d)*x^2 + sqrt(d))*sqrt(c^2*x^2 + 1)*log(c*x + sqrt(c^ 
2*x^2 + 1))^2/x^3 - 3*integrate(2/3*((c^2*x^2 + 1)*c^2*sqrt(d)*x + (c^3*sq 
rt(d)*x^2 + c*sqrt(d))*sqrt(c^2*x^2 + 1))*log(c*x + sqrt(c^2*x^2 + 1))/(c* 
x^4 + sqrt(c^2*x^2 + 1)*x^3), x)) - 2/3*(c^2*d*x^2 + d)^(3/2)*a*b*arcsinh( 
c*x)/(d*x^3) - 1/3*(c^2*d*x^2 + d)^(3/2)*a^2/(d*x^3)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{x^4} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))^2/x^4,x, algorithm="giac" 
)
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{x^4} \, dx=\frac {\sqrt {d}\, \left (-\sqrt {c^{2} x^{2}+1}\, a^{2} c^{2} x^{2}-\sqrt {c^{2} x^{2}+1}\, a^{2}+6 \left (\int \frac {\sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right )}{x^{4}}d x \right ) a b \,x^{3}+3 \left (\int \frac {\sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right )^{2}}{x^{4}}d x \right ) b^{2} x^{3}-a^{2} c^{3} x^{3}\right )}{3 x^{3}} \] Input:

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

Output:

(sqrt(d)*( - sqrt(c**2*x**2 + 1)*a**2*c**2*x**2 - sqrt(c**2*x**2 + 1)*a**2 
 + 6*int((sqrt(c**2*x**2 + 1)*asinh(c*x))/x**4,x)*a*b*x**3 + 3*int((sqrt(c 
**2*x**2 + 1)*asinh(c*x)**2)/x**4,x)*b**2*x**3 - a**2*c**3*x**3))/(3*x**3)