\(\int \frac {(d+c^2 d x^2) (a+b \text {arcsinh}(c x))^2}{x^3} \, dx\) [217]

Optimal result

Integrand size = 24, antiderivative size = 179 \[ \int \frac {\left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2}{x^3} \, dx=-\frac {b c d \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{x}+\frac {1}{2} c^2 d (a+b \text {arcsinh}(c x))^2-\frac {d \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2}{2 x^2}-\frac {c^2 d (a+b \text {arcsinh}(c x))^3}{3 b}+c^2 d (a+b \text {arcsinh}(c x))^2 \log \left (1-e^{2 \text {arcsinh}(c x)}\right )+b^2 c^2 d \log (x)+b c^2 d (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )-\frac {1}{2} b^2 c^2 d \operatorname {PolyLog}\left (3,e^{2 \text {arcsinh}(c x)}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.35 \[ \int \frac {\left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2}{x^3} \, dx=-\frac {a^2 d}{2 x^2}+2 a b c^2 d \left (-\frac {\sqrt {1+c^2 x^2}}{2 c x}-\frac {\text {arcsinh}(c x)}{2 c^2 x^2}\right )+a^2 c^2 d \log (x)+b^2 c^2 d \left (-\frac {\sqrt {1+c^2 x^2} \text {arcsinh}(c x)}{c x}-\frac {\text {arcsinh}(c x)^2}{2 c^2 x^2}+\log (c x)\right )+2 a b c^2 d \left (-\frac {1}{2} \text {arcsinh}(c x)^2+\text {arcsinh}(c x) \log \left (1-e^{2 \text {arcsinh}(c x)}\right )+\frac {1}{2} \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )\right )+b^2 c^2 d \left (-\frac {1}{3} \text {arcsinh}(c x)^3+\text {arcsinh}(c x)^2 \log \left (1-e^{2 \text {arcsinh}(c x)}\right )+\text {arcsinh}(c x) \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )-\frac {1}{2} \operatorname {PolyLog}\left (3,e^{2 \text {arcsinh}(c x)}\right )\right ) \] Input:

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

Output:

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

Rubi [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 1.64 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.27, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.542, Rules used = {6222, 6190, 25, 3042, 26, 4201, 2620, 3011, 2720, 6220, 14, 6198, 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.

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

\(\Big \downarrow \) 6222

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

\(\Big \downarrow \) 6190

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4201

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 6220

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

\(\Big \downarrow \) 14

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

\(\Big \downarrow \) 6198

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

\(\Big \downarrow \) 7143

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 14

Int[(a_.)/(x_), x_Symbol] :> Simp[a*Log[x], x] /; FreeQ[a, x]
 

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 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 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 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 6198

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_ 
Symbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]]*( 
a + b*ArcSinh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c 
^2*d] && NeQ[n, -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 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. \(403\) vs. \(2(196)=392\).

Time = 1.14 (sec) , antiderivative size = 404, normalized size of antiderivative = 2.26

method	result	size
derivativedivides	\(c^{2} \left (a^{2} d \left (\ln \left (x c \right )-\frac {1}{2 c^{2} x^{2}}\right )+b^{2} d \left (-\frac {\operatorname {arcsinh}\left (x c \right )^{3}}{3}-\frac {\operatorname {arcsinh}\left (x c \right ) \left (-2 c^{2} x^{2}+2 \sqrt {c^{2} x^{2}+1}\, x c +\operatorname {arcsinh}\left (x c \right )\right )}{2 x^{2} c^{2}}-2 \ln \left (x c +\sqrt {c^{2} x^{2}+1}\right )+\ln \left (x c +\sqrt {c^{2} x^{2}+1}-1\right )+\ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )+\operatorname {arcsinh}\left (x c \right )^{2} \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right )+2 \,\operatorname {arcsinh}\left (x c \right ) \operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, x c +\sqrt {c^{2} x^{2}+1}\right )+\operatorname {arcsinh}\left (x c \right )^{2} \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )+2 \,\operatorname {arcsinh}\left (x c \right ) \operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, -x c -\sqrt {c^{2} x^{2}+1}\right )\right )+2 a b d \left (-\frac {\operatorname {arcsinh}\left (x c \right )^{2}}{2}-\frac {\sqrt {c^{2} x^{2}+1}\, x c -c^{2} x^{2}+\operatorname {arcsinh}\left (x c \right )}{2 x^{2} c^{2}}+\operatorname {arcsinh}\left (x c \right ) \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right )+\operatorname {arcsinh}\left (x c \right ) \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right )\right )\right )\)	\(404\)
default	\(c^{2} \left (a^{2} d \left (\ln \left (x c \right )-\frac {1}{2 c^{2} x^{2}}\right )+b^{2} d \left (-\frac {\operatorname {arcsinh}\left (x c \right )^{3}}{3}-\frac {\operatorname {arcsinh}\left (x c \right ) \left (-2 c^{2} x^{2}+2 \sqrt {c^{2} x^{2}+1}\, x c +\operatorname {arcsinh}\left (x c \right )\right )}{2 x^{2} c^{2}}-2 \ln \left (x c +\sqrt {c^{2} x^{2}+1}\right )+\ln \left (x c +\sqrt {c^{2} x^{2}+1}-1\right )+\ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )+\operatorname {arcsinh}\left (x c \right )^{2} \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right )+2 \,\operatorname {arcsinh}\left (x c \right ) \operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, x c +\sqrt {c^{2} x^{2}+1}\right )+\operatorname {arcsinh}\left (x c \right )^{2} \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )+2 \,\operatorname {arcsinh}\left (x c \right ) \operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, -x c -\sqrt {c^{2} x^{2}+1}\right )\right )+2 a b d \left (-\frac {\operatorname {arcsinh}\left (x c \right )^{2}}{2}-\frac {\sqrt {c^{2} x^{2}+1}\, x c -c^{2} x^{2}+\operatorname {arcsinh}\left (x c \right )}{2 x^{2} c^{2}}+\operatorname {arcsinh}\left (x c \right ) \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right )+\operatorname {arcsinh}\left (x c \right ) \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right )\right )\right )\)	\(404\)
parts	\(a^{2} d \left (c^{2} \ln \left (x \right )-\frac {1}{2 x^{2}}\right )+b^{2} d \,c^{2} \left (-\frac {\operatorname {arcsinh}\left (x c \right )^{3}}{3}-\frac {\operatorname {arcsinh}\left (x c \right ) \left (-2 c^{2} x^{2}+2 \sqrt {c^{2} x^{2}+1}\, x c +\operatorname {arcsinh}\left (x c \right )\right )}{2 x^{2} c^{2}}-2 \ln \left (x c +\sqrt {c^{2} x^{2}+1}\right )+\ln \left (x c +\sqrt {c^{2} x^{2}+1}-1\right )+\ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )+\operatorname {arcsinh}\left (x c \right )^{2} \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right )+2 \,\operatorname {arcsinh}\left (x c \right ) \operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, x c +\sqrt {c^{2} x^{2}+1}\right )+\operatorname {arcsinh}\left (x c \right )^{2} \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )+2 \,\operatorname {arcsinh}\left (x c \right ) \operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, -x c -\sqrt {c^{2} x^{2}+1}\right )\right )+2 a b d \,c^{2} \left (-\frac {\operatorname {arcsinh}\left (x c \right )^{2}}{2}-\frac {\sqrt {c^{2} x^{2}+1}\, x c -c^{2} x^{2}+\operatorname {arcsinh}\left (x c \right )}{2 x^{2} c^{2}}+\operatorname {arcsinh}\left (x c \right ) \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right )+\operatorname {arcsinh}\left (x c \right ) \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right )\right )\)	\(405\)

Input:

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

Output:

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

Fricas [F]

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

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

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

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

Output:

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