\(\int \frac {(d+c^2 d x^2) (a+b \text {arcsinh}(c x))^2}{x^4} \, dx\) [218]

Optimal result

Integrand size = 24, antiderivative size = 158 \[ \int \frac {\left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2}{x^4} \, dx=-\frac {b^2 c^2 d}{3 x}-\frac {b c d \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{3 x^2}-\frac {2 c^2 d (a+b \text {arcsinh}(c x))^2}{3 x}-\frac {d \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{3 x^3}-\frac {10}{3} b c^3 d (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )-\frac {5}{3} b^2 c^3 d \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )+\frac {5}{3} b^2 c^3 d \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right ) \] Output:

-1/3*b^2*c^2*d/x-1/3*b*c*d*(c^2*x^2+1)^(1/2)*(a+b*arcsinh(c*x))/x^2-2/3*c^ 
2*d*(a+b*arcsinh(c*x))^2/x-1/3*d*(c^2*x^2+1)*(a+b*arcsinh(c*x))^2/x^3-10/3 
*b*c^3*d*(a+b*arcsinh(c*x))*arctanh(c*x+(c^2*x^2+1)^(1/2))-5/3*b^2*c^3*d*p 
olylog(2,-c*x-(c^2*x^2+1)^(1/2))+5/3*b^2*c^3*d*polylog(2,c*x+(c^2*x^2+1)^( 
1/2))
 

Mathematica [A] (verified)

Time = 0.56 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.55 \[ \int \frac {\left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2}{x^4} \, dx=-\frac {d \left (a^2+3 a^2 c^2 x^2+b^2 c^2 x^2+a b c x \sqrt {1+c^2 x^2}+2 a b \text {arcsinh}(c x)+6 a b c^2 x^2 \text {arcsinh}(c x)+b^2 c x \sqrt {1+c^2 x^2} \text {arcsinh}(c x)+b^2 \text {arcsinh}(c x)^2+3 b^2 c^2 x^2 \text {arcsinh}(c x)^2+5 a b c^3 x^3 \text {arctanh}\left (\sqrt {1+c^2 x^2}\right )-5 b^2 c^3 x^3 \text {arcsinh}(c x) \log \left (1-e^{-\text {arcsinh}(c x)}\right )+5 b^2 c^3 x^3 \text {arcsinh}(c x) \log \left (1+e^{-\text {arcsinh}(c x)}\right )-5 b^2 c^3 x^3 \operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(c x)}\right )+5 b^2 c^3 x^3 \operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(c x)}\right )\right )}{3 x^3} \] Input:

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

Output:

-1/3*(d*(a^2 + 3*a^2*c^2*x^2 + b^2*c^2*x^2 + a*b*c*x*Sqrt[1 + c^2*x^2] + 2 
*a*b*ArcSinh[c*x] + 6*a*b*c^2*x^2*ArcSinh[c*x] + b^2*c*x*Sqrt[1 + c^2*x^2] 
*ArcSinh[c*x] + b^2*ArcSinh[c*x]^2 + 3*b^2*c^2*x^2*ArcSinh[c*x]^2 + 5*a*b* 
c^3*x^3*ArcTanh[Sqrt[1 + c^2*x^2]] - 5*b^2*c^3*x^3*ArcSinh[c*x]*Log[1 - E^ 
(-ArcSinh[c*x])] + 5*b^2*c^3*x^3*ArcSinh[c*x]*Log[1 + E^(-ArcSinh[c*x])] - 
 5*b^2*c^3*x^3*PolyLog[2, -E^(-ArcSinh[c*x])] + 5*b^2*c^3*x^3*PolyLog[2, E 
^(-ArcSinh[c*x])]))/x^3
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 1.45 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.32, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6222, 6191, 6220, 15, 6231, 3042, 26, 4670, 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.

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

\(\Big \downarrow \) 6222

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

\(\Big \downarrow \) 6191

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

\(\Big \downarrow \) 6220

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

\(\Big \downarrow \) 15

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

\(\Big \downarrow \) 6231

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4670

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

-1/3*(d*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/x^3 + (2*c^2*d*(-((a + b*Arc 
Sinh[c*x])^2/x) + (2*I)*b*c*((2*I)*(a + b*ArcSinh[c*x])*ArcTanh[E^ArcSinh[ 
c*x]] + I*b*PolyLog[2, -E^ArcSinh[c*x]] - I*b*PolyLog[2, E^ArcSinh[c*x]])) 
)/3 + (2*b*c*d*(-1/2*(b*c)/x - (Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(2 
*x^2) + (I/2)*c^2*((2*I)*(a + b*ArcSinh[c*x])*ArcTanh[E^ArcSinh[c*x]] + I* 
b*PolyLog[2, -E^ArcSinh[c*x]] - I*b*PolyLog[2, E^ArcSinh[c*x]])))/3
 

Defintions of rubi rules used

rule 15

Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ 
{a, m}, x] && NeQ[m, -1]
 

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 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 4670

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

rule 6191

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] 
 :> Simp[(d*x)^(m + 1)*((a + b*ArcSinh[c*x])^n/(d*(m + 1))), x] - Simp[b*c* 
(n/(d*(m + 1)))   Int[(d*x)^(m + 1)*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[1 + 
c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]
 

rule 6220

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

rule 6222

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*((a + b*Arc 
Sinh[c*x])^n/(f*(m + 1))), 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])^n, x], 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}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1]
 

rule 6231

Int[(((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/Sqrt[(d_) + (e_.) 
*(x_)^2], x_Symbol] :> Simp[(1/c^(m + 1))*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e 
*x^2]]   Subst[Int[(a + b*x)^n*Sinh[x]^m, x], x, ArcSinh[c*x]], x] /; FreeQ 
[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[n, 0] && IntegerQ[m]
 

Maple [A] (verified)

Time = 1.06 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.52

method	result	size
parts	\(a^{2} d \left (-\frac {1}{3 x^{3}}-\frac {c^{2}}{x}\right )+b^{2} d \,c^{3} \left (-\frac {3 \operatorname {arcsinh}\left (x c \right )^{2} x^{2} c^{2}+\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x c +\operatorname {arcsinh}\left (x c \right )^{2}+c^{2} x^{2}}{3 x^{3} c^{3}}-\frac {5 \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )}{3}-\frac {5 \operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right )}{3}+\frac {5 \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right )}{3}+\frac {5 \operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right )}{3}\right )+2 a b d \,c^{3} \left (-\frac {\operatorname {arcsinh}\left (x c \right )}{3 x^{3} c^{3}}-\frac {\operatorname {arcsinh}\left (x c \right )}{x c}-\frac {\sqrt {c^{2} x^{2}+1}}{6 x^{2} c^{2}}-\frac {5 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )}{6}\right )\)	\(240\)
derivativedivides	\(c^{3} \left (a^{2} d \left (-\frac {1}{3 x^{3} c^{3}}-\frac {1}{x c}\right )+b^{2} d \left (-\frac {3 \operatorname {arcsinh}\left (x c \right )^{2} x^{2} c^{2}+\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x c +\operatorname {arcsinh}\left (x c \right )^{2}+c^{2} x^{2}}{3 x^{3} c^{3}}-\frac {5 \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )}{3}-\frac {5 \operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right )}{3}+\frac {5 \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right )}{3}+\frac {5 \operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right )}{3}\right )+2 a b d \left (-\frac {\operatorname {arcsinh}\left (x c \right )}{3 x^{3} c^{3}}-\frac {\operatorname {arcsinh}\left (x c \right )}{x c}-\frac {\sqrt {c^{2} x^{2}+1}}{6 x^{2} c^{2}}-\frac {5 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )}{6}\right )\right )\)	\(241\)
default	\(c^{3} \left (a^{2} d \left (-\frac {1}{3 x^{3} c^{3}}-\frac {1}{x c}\right )+b^{2} d \left (-\frac {3 \operatorname {arcsinh}\left (x c \right )^{2} x^{2} c^{2}+\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x c +\operatorname {arcsinh}\left (x c \right )^{2}+c^{2} x^{2}}{3 x^{3} c^{3}}-\frac {5 \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )}{3}-\frac {5 \operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right )}{3}+\frac {5 \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right )}{3}+\frac {5 \operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right )}{3}\right )+2 a b d \left (-\frac {\operatorname {arcsinh}\left (x c \right )}{3 x^{3} c^{3}}-\frac {\operatorname {arcsinh}\left (x c \right )}{x c}-\frac {\sqrt {c^{2} x^{2}+1}}{6 x^{2} c^{2}}-\frac {5 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )}{6}\right )\right )\)	\(241\)

Input:

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

Output:

a^2*d*(-1/3/x^3-c^2/x)+b^2*d*c^3*(-1/3*(3*arcsinh(x*c)^2*x^2*c^2+arcsinh(x 
*c)*(c^2*x^2+1)^(1/2)*x*c+arcsinh(x*c)^2+c^2*x^2)/x^3/c^3-5/3*arcsinh(x*c) 
*ln(1+x*c+(c^2*x^2+1)^(1/2))-5/3*polylog(2,-x*c-(c^2*x^2+1)^(1/2))+5/3*arc 
sinh(x*c)*ln(1-x*c-(c^2*x^2+1)^(1/2))+5/3*polylog(2,x*c+(c^2*x^2+1)^(1/2)) 
)+2*a*b*d*c^3*(-1/3*arcsinh(x*c)/x^3/c^3-arcsinh(x*c)/x/c-1/6/x^2/c^2*(c^2 
*x^2+1)^(1/2)-5/6*arctanh(1/(c^2*x^2+1)^(1/2)))
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

-2*(c*arcsinh(1/(c*abs(x))) + arcsinh(c*x)/x)*a*b*c^2*d + 1/3*((c^2*arcsin 
h(1/(c*abs(x))) - sqrt(c^2*x^2 + 1)/x^2)*c - 2*arcsinh(c*x)/x^3)*a*b*d - a 
^2*c^2*d/x - 1/3*a^2*d/x^3 - 1/3*(3*b^2*c^2*d*x^2 + b^2*d)*log(c*x + sqrt( 
c^2*x^2 + 1))^2/x^3 + integrate(2/3*(3*b^2*c^5*d*x^4 + 4*b^2*c^3*d*x^2 + b 
^2*c*d + (3*b^2*c^4*d*x^3 + b^2*c^2*d*x)*sqrt(c^2*x^2 + 1))*log(c*x + sqrt 
(c^2*x^2 + 1))/(c^3*x^6 + c*x^4 + (c^2*x^5 + x^3)*sqrt(c^2*x^2 + 1)), x)
 

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2}{x^4} \, dx=\frac {d \left (-6 \mathit {asinh} \left (c x \right ) a b \,c^{2} x^{2}-2 \mathit {asinh} \left (c x \right ) a b -\sqrt {c^{2} x^{2}+1}\, a b c x +3 \left (\int \frac {\mathit {asinh} \left (c x \right )^{2}}{x^{4}}d x \right ) b^{2} x^{3}+3 \left (\int \frac {\mathit {asinh} \left (c x \right )^{2}}{x^{2}}d x \right ) b^{2} c^{2} x^{3}+5 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x -1\right ) a b \,c^{3} x^{3}-5 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x +1\right ) a b \,c^{3} x^{3}-3 a^{2} c^{2} x^{2}-a^{2}\right )}{3 x^{3}} \] Input:

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

Output:

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