\(\int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{x^3} \, dx\) [270]

Optimal result

Integrand size = 28, antiderivative size = 358 \[ \int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{x^3} \, dx=-\frac {b c \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{x \sqrt {1+c^2 x^2}}-\frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 x^2}-\frac {c^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2 \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}}-\frac {b^2 c^2 \sqrt {d+c^2 d x^2} \text {arctanh}\left (\sqrt {1+c^2 x^2}\right )}{\sqrt {1+c^2 x^2}}-\frac {b c^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}}+\frac {b c^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}}+\frac {b^2 c^2 \sqrt {d+c^2 d x^2} \operatorname {PolyLog}\left (3,-e^{\text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}}-\frac {b^2 c^2 \sqrt {d+c^2 d x^2} \operatorname {PolyLog}\left (3,e^{\text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}} \] Output:

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

Mathematica [A] (verified)

Time = 3.63 (sec) , antiderivative size = 446, normalized size of antiderivative = 1.25 \[ \int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{x^3} \, dx=\frac {1}{8} \left (-\frac {4 a^2 \sqrt {d+c^2 d x^2}}{x^2}+4 a^2 c^2 \sqrt {d} \log (x)-4 a^2 c^2 \sqrt {d} \log \left (d+\sqrt {d} \sqrt {d+c^2 d x^2}\right )+\frac {2 a b c^2 \sqrt {d+c^2 d x^2} \left (-2 \coth \left (\frac {1}{2} \text {arcsinh}(c x)\right )-\text {arcsinh}(c x) \text {csch}^2\left (\frac {1}{2} \text {arcsinh}(c x)\right )+4 \text {arcsinh}(c x) \log \left (1-e^{-\text {arcsinh}(c x)}\right )-4 \text {arcsinh}(c x) \log \left (1+e^{-\text {arcsinh}(c x)}\right )+4 \operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(c x)}\right )-4 \operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(c x)}\right )-\text {arcsinh}(c x) \text {sech}^2\left (\frac {1}{2} \text {arcsinh}(c x)\right )+2 \tanh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )}{\sqrt {1+c^2 x^2}}+\frac {b^2 c^2 \sqrt {d+c^2 d x^2} \left (-4 \text {arcsinh}(c x) \coth \left (\frac {1}{2} \text {arcsinh}(c x)\right )-\text {arcsinh}(c x)^2 \text {csch}^2\left (\frac {1}{2} \text {arcsinh}(c x)\right )+4 \text {arcsinh}(c x)^2 \log \left (1-e^{-\text {arcsinh}(c x)}\right )-4 \text {arcsinh}(c x)^2 \log \left (1+e^{-\text {arcsinh}(c x)}\right )+8 \log \left (\tanh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )+8 \text {arcsinh}(c x) \operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(c x)}\right )-8 \text {arcsinh}(c x) \operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(c x)}\right )+8 \operatorname {PolyLog}\left (3,-e^{-\text {arcsinh}(c x)}\right )-8 \operatorname {PolyLog}\left (3,e^{-\text {arcsinh}(c x)}\right )-\text {arcsinh}(c x)^2 \text {sech}^2\left (\frac {1}{2} \text {arcsinh}(c x)\right )+4 \text {arcsinh}(c x) \tanh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )}{\sqrt {1+c^2 x^2}}\right ) \] Input:

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

Output:

((-4*a^2*Sqrt[d + c^2*d*x^2])/x^2 + 4*a^2*c^2*Sqrt[d]*Log[x] - 4*a^2*c^2*S 
qrt[d]*Log[d + Sqrt[d]*Sqrt[d + c^2*d*x^2]] + (2*a*b*c^2*Sqrt[d + c^2*d*x^ 
2]*(-2*Coth[ArcSinh[c*x]/2] - ArcSinh[c*x]*Csch[ArcSinh[c*x]/2]^2 + 4*ArcS 
inh[c*x]*Log[1 - E^(-ArcSinh[c*x])] - 4*ArcSinh[c*x]*Log[1 + E^(-ArcSinh[c 
*x])] + 4*PolyLog[2, -E^(-ArcSinh[c*x])] - 4*PolyLog[2, E^(-ArcSinh[c*x])] 
 - ArcSinh[c*x]*Sech[ArcSinh[c*x]/2]^2 + 2*Tanh[ArcSinh[c*x]/2]))/Sqrt[1 + 
 c^2*x^2] + (b^2*c^2*Sqrt[d + c^2*d*x^2]*(-4*ArcSinh[c*x]*Coth[ArcSinh[c*x 
]/2] - ArcSinh[c*x]^2*Csch[ArcSinh[c*x]/2]^2 + 4*ArcSinh[c*x]^2*Log[1 - E^ 
(-ArcSinh[c*x])] - 4*ArcSinh[c*x]^2*Log[1 + E^(-ArcSinh[c*x])] + 8*Log[Tan 
h[ArcSinh[c*x]/2]] + 8*ArcSinh[c*x]*PolyLog[2, -E^(-ArcSinh[c*x])] - 8*Arc 
Sinh[c*x]*PolyLog[2, E^(-ArcSinh[c*x])] + 8*PolyLog[3, -E^(-ArcSinh[c*x])] 
 - 8*PolyLog[3, E^(-ArcSinh[c*x])] - ArcSinh[c*x]^2*Sech[ArcSinh[c*x]/2]^2 
 + 4*ArcSinh[c*x]*Tanh[ArcSinh[c*x]/2]))/Sqrt[1 + c^2*x^2])/8
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 2.07 (sec) , antiderivative size = 224, normalized size of antiderivative = 0.63, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {6220, 6191, 243, 73, 221, 6231, 3042, 26, 4670, 3011, 2720, 7143}

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.

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

\(\Big \downarrow \) 6220

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

\(\Big \downarrow \) 6191

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 73

\(\displaystyle \frac {c^2 \sqrt {c^2 d x^2+d} \int \frac {(a+b \text {arcsinh}(c x))^2}{x \sqrt {c^2 x^2+1}}dx}{2 \sqrt {c^2 x^2+1}}+\frac {b c \sqrt {c^2 d x^2+d} \left (\frac {b \int \frac {1}{\frac {x^4}{c^2}-\frac {1}{c^2}}d\sqrt {c^2 x^2+1}}{c}-\frac {a+b \text {arcsinh}(c x)}{x}\right )}{\sqrt {c^2 x^2+1}}-\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{2 x^2}\)

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 6231

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4670

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

Input:

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

Output:

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

Defintions of rubi rules used

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 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

rule 2720

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] 
   Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct 
ionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ 
[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) 
*(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
 

rule 3011

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) 
*(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + 
b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F]))   Int[(f + g*x)^( 
m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e 
, f, g, n}, x] && GtQ[m, 0]
 

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 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]
 

rule 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d 
, e, n, p}, x] && EqQ[b*d, a*e]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(754\) vs. \(2(371)=742\).

Time = 1.18 (sec) , antiderivative size = 755, normalized size of antiderivative = 2.11

method	result	size
default	\(a^{2} \left (-\frac {\left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{2 d \,x^{2}}+\frac {c^{2} \left (\sqrt {c^{2} d \,x^{2}+d}-\sqrt {d}\, \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {c^{2} d \,x^{2}+d}}{x}\right )\right )}{2}\right )+b^{2} \left (-\frac {\left (\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}+2 \sqrt {c^{2} x^{2}+1}\, x c +\operatorname {arcsinh}\left (x c \right )\right ) \operatorname {arcsinh}\left (x c \right ) \sqrt {d \left (c^{2} x^{2}+1\right )}}{2 x^{2} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right )^{2} \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right ) c^{2}}{2 \sqrt {c^{2} x^{2}+1}}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right ) \operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right ) c^{2}}{\sqrt {c^{2} x^{2}+1}}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {polylog}\left (3, x c +\sqrt {c^{2} x^{2}+1}\right ) c^{2}}{\sqrt {c^{2} x^{2}+1}}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right )^{2} \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right ) c^{2}}{2 \sqrt {c^{2} x^{2}+1}}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right ) \operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right ) c^{2}}{\sqrt {c^{2} x^{2}+1}}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {polylog}\left (3, -x c -\sqrt {c^{2} x^{2}+1}\right ) c^{2}}{\sqrt {c^{2} x^{2}+1}}-\frac {2 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arctanh}\left (x c +\sqrt {c^{2} x^{2}+1}\right ) c^{2}}{\sqrt {c^{2} x^{2}+1}}\right )+2 a b \left (-\frac {\left (\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}\, x c +\operatorname {arcsinh}\left (x c \right )\right ) \sqrt {d \left (c^{2} x^{2}+1\right )}}{2 x^{2} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right ) \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right ) c^{2}}{2 \sqrt {c^{2} x^{2}+1}}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right ) c^{2}}{2 \sqrt {c^{2} x^{2}+1}}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right ) \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right ) c^{2}}{2 \sqrt {c^{2} x^{2}+1}}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right ) c^{2}}{2 \sqrt {c^{2} x^{2}+1}}\right )\)	\(755\)
parts	\(a^{2} \left (-\frac {\left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{2 d \,x^{2}}+\frac {c^{2} \left (\sqrt {c^{2} d \,x^{2}+d}-\sqrt {d}\, \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {c^{2} d \,x^{2}+d}}{x}\right )\right )}{2}\right )+b^{2} \left (-\frac {\left (\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}+2 \sqrt {c^{2} x^{2}+1}\, x c +\operatorname {arcsinh}\left (x c \right )\right ) \operatorname {arcsinh}\left (x c \right ) \sqrt {d \left (c^{2} x^{2}+1\right )}}{2 x^{2} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right )^{2} \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right ) c^{2}}{2 \sqrt {c^{2} x^{2}+1}}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right ) \operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right ) c^{2}}{\sqrt {c^{2} x^{2}+1}}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {polylog}\left (3, x c +\sqrt {c^{2} x^{2}+1}\right ) c^{2}}{\sqrt {c^{2} x^{2}+1}}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right )^{2} \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right ) c^{2}}{2 \sqrt {c^{2} x^{2}+1}}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right ) \operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right ) c^{2}}{\sqrt {c^{2} x^{2}+1}}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {polylog}\left (3, -x c -\sqrt {c^{2} x^{2}+1}\right ) c^{2}}{\sqrt {c^{2} x^{2}+1}}-\frac {2 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arctanh}\left (x c +\sqrt {c^{2} x^{2}+1}\right ) c^{2}}{\sqrt {c^{2} x^{2}+1}}\right )+2 a b \left (-\frac {\left (\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}\, x c +\operatorname {arcsinh}\left (x c \right )\right ) \sqrt {d \left (c^{2} x^{2}+1\right )}}{2 x^{2} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right ) \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right ) c^{2}}{2 \sqrt {c^{2} x^{2}+1}}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right ) c^{2}}{2 \sqrt {c^{2} x^{2}+1}}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right ) \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right ) c^{2}}{2 \sqrt {c^{2} x^{2}+1}}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right ) c^{2}}{2 \sqrt {c^{2} x^{2}+1}}\right )\)	\(755\)

Input:

int((c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(x*c))^2/x^3,x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [F]

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

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate((c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))^2/x^3,x, algorithm="maxim 
a")
 

Output:

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

Giac [F(-2)]

Exception generated. \[ \int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{x^3} \, dx=\text {Exception raised: TypeError} \] Input:

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

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

Output:

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

Reduce [F]

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

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

Output:

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