\(\int \frac {(d+c^2 d x^2) (a+b \text {arcsinh}(c x))^2}{x} \, dx\) [215]

Optimal result

Integrand size = 24, antiderivative size = 165 \[ \int \frac {\left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2}{x} \, dx=\frac {1}{4} b^2 c^2 d x^2-\frac {1}{2} b c d x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))-\frac {1}{4} d (a+b \text {arcsinh}(c x))^2+\frac {1}{2} d \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2-\frac {d (a+b \text {arcsinh}(c x))^3}{3 b}+d (a+b \text {arcsinh}(c x))^2 \log \left (1-e^{2 \text {arcsinh}(c x)}\right )+b d (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )-\frac {1}{2} b^2 d \operatorname {PolyLog}\left (3,e^{2 \text {arcsinh}(c x)}\right ) \] Output:

1/4*b^2*c^2*d*x^2-1/2*b*c*d*x*(c^2*x^2+1)^(1/2)*(a+b*arcsinh(c*x))-1/4*d*( 
a+b*arcsinh(c*x))^2+1/2*d*(c^2*x^2+1)*(a+b*arcsinh(c*x))^2-1/3*d*(a+b*arcs 
inh(c*x))^3/b+d*(a+b*arcsinh(c*x))^2*ln(1-(c*x+(c^2*x^2+1)^(1/2))^2)+b*d*( 
a+b*arcsinh(c*x))*polylog(2,(c*x+(c^2*x^2+1)^(1/2))^2)-1/2*b^2*d*polylog(3 
,(c*x+(c^2*x^2+1)^(1/2))^2)
 

Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.37 \[ \int \frac {\left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2}{x} \, dx=\frac {1}{2} d \left (a^2 c^2 x^2+2 a b c^2 x^2 \text {arcsinh}(c x)+\frac {1}{4} b^2 \left (1+2 \text {arcsinh}(c x)^2\right ) \cosh (2 \text {arcsinh}(c x))-2 a b \text {arcsinh}(c x) \left (\text {arcsinh}(c x)-2 \log \left (1-e^{2 \text {arcsinh}(c x)}\right )\right )+2 a^2 \log (x)-a b \left (c x \sqrt {1+c^2 x^2}+\log \left (-c x+\sqrt {1+c^2 x^2}\right )\right )+2 a b \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )+2 b^2 \left (-\frac {1}{3} \text {arcsinh}(c x)^3+\text {arcsinh}(c x)^2 \log \left (1-e^{2 \text {arcsinh}(c x)}\right )+\text {arcsinh}(c x) \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )-\frac {1}{2} \operatorname {PolyLog}\left (3,e^{2 \text {arcsinh}(c x)}\right )\right )-\frac {1}{2} b^2 \text {arcsinh}(c x) \sinh (2 \text {arcsinh}(c x))\right ) \] Input:

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

Output:

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

Rubi [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 1.32 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.38, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.542, Rules used = {6223, 6190, 25, 3042, 26, 4201, 2620, 3011, 2720, 6200, 15, 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.

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

\(\Big \downarrow \) 6223

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

\(\Big \downarrow \) 6190

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4201

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 6200

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

\(\Big \downarrow \) 15

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

\(\Big \downarrow \) 6198

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

\(\Big \downarrow \) 7143

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

Input:

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

Output:

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

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 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 6200

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_ 
Symbol] :> Simp[x*Sqrt[d + e*x^2]*((a + b*ArcSinh[c*x])^n/2), x] + (Simp[(1 
/2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]]   Int[(a + b*ArcSinh[c*x])^n/Sq 
rt[1 + c^2*x^2], x], x] - Simp[b*c*(n/2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2* 
x^2]]   Int[x*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e 
}, x] && EqQ[e, c^2*d] && GtQ[n, 0]
 

rule 6223

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 + 2*p + 1))), x] + (Simp[2*d*(p/(m + 2*p + 1))   Int[(f* 
x)^m*(d + e*x^2)^(p - 1)*(a + b*ArcSinh[c*x])^n, x], x] - Simp[b*c*(n/(f*(m 
 + 2*p + 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}, 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. \(414\) vs. \(2(178)=356\).

Time = 1.32 (sec) , antiderivative size = 415, normalized size of antiderivative = 2.52

method	result	size
derivativedivides	\(a^{2} d \left (\frac {c^{2} x^{2}}{2}+\ln \left (x c \right )\right )+b^{2} d \left (-\frac {\operatorname {arcsinh}\left (x c \right )^{3}}{3}+\frac {\left (2 \operatorname {arcsinh}\left (x c \right )^{2}-2 \,\operatorname {arcsinh}\left (x c \right )+1\right ) \left (2 c^{2} x^{2}+2 \sqrt {c^{2} x^{2}+1}\, x c +1\right )}{16}+\frac {\left (-2 \sqrt {c^{2} x^{2}+1}\, x c +2 c^{2} x^{2}+1\right ) \left (2 \operatorname {arcsinh}\left (x c \right )^{2}+2 \,\operatorname {arcsinh}\left (x c \right )+1\right )}{16}+\operatorname {arcsinh}\left (x c \right )^{2} \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right )+2 \,\operatorname {arcsinh}\left (x c \right ) \operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, x c +\sqrt {c^{2} x^{2}+1}\right )+\operatorname {arcsinh}\left (x c \right )^{2} \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )+2 \,\operatorname {arcsinh}\left (x c \right ) \operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, -x c -\sqrt {c^{2} x^{2}+1}\right )\right )-a b d \operatorname {arcsinh}\left (x c \right )^{2}+a b d \,\operatorname {arcsinh}\left (x c \right ) x^{2} c^{2}-\frac {a b d x c \sqrt {c^{2} x^{2}+1}}{2}+\frac {a b d \,\operatorname {arcsinh}\left (x c \right )}{2}+2 a b d \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right )+2 a b d \operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right )+2 a b d \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )+2 a b d \operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right )\)	\(415\)
default	\(a^{2} d \left (\frac {c^{2} x^{2}}{2}+\ln \left (x c \right )\right )+b^{2} d \left (-\frac {\operatorname {arcsinh}\left (x c \right )^{3}}{3}+\frac {\left (2 \operatorname {arcsinh}\left (x c \right )^{2}-2 \,\operatorname {arcsinh}\left (x c \right )+1\right ) \left (2 c^{2} x^{2}+2 \sqrt {c^{2} x^{2}+1}\, x c +1\right )}{16}+\frac {\left (-2 \sqrt {c^{2} x^{2}+1}\, x c +2 c^{2} x^{2}+1\right ) \left (2 \operatorname {arcsinh}\left (x c \right )^{2}+2 \,\operatorname {arcsinh}\left (x c \right )+1\right )}{16}+\operatorname {arcsinh}\left (x c \right )^{2} \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right )+2 \,\operatorname {arcsinh}\left (x c \right ) \operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, x c +\sqrt {c^{2} x^{2}+1}\right )+\operatorname {arcsinh}\left (x c \right )^{2} \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )+2 \,\operatorname {arcsinh}\left (x c \right ) \operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right )-2 \operatorname {polylog}\left (3, -x c -\sqrt {c^{2} x^{2}+1}\right )\right )-a b d \operatorname {arcsinh}\left (x c \right )^{2}+a b d \,\operatorname {arcsinh}\left (x c \right ) x^{2} c^{2}-\frac {a b d x c \sqrt {c^{2} x^{2}+1}}{2}+\frac {a b d \,\operatorname {arcsinh}\left (x c \right )}{2}+2 a b d \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right )+2 a b d \operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right )+2 a b d \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )+2 a b d \operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right )\)	\(415\)
parts	\(a^{2} d \left (\frac {c^{2} x^{2}}{2}+\ln \left (x \right )\right )-\frac {b^{2} d \operatorname {arcsinh}\left (x c \right )^{3}}{3}+\frac {b^{2} d \operatorname {arcsinh}\left (x c \right )^{2} x^{2} c^{2}}{2}-\frac {b^{2} d \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x c}{2}+\frac {d \,b^{2} c^{2} x^{2}}{4}+\frac {b^{2} d \operatorname {arcsinh}\left (x c \right )^{2}}{4}+\frac {b^{2} d}{8}+b^{2} d \operatorname {arcsinh}\left (x c \right )^{2} \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right )+2 b^{2} d \,\operatorname {arcsinh}\left (x c \right ) \operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right )-2 b^{2} d \operatorname {polylog}\left (3, x c +\sqrt {c^{2} x^{2}+1}\right )+b^{2} d \operatorname {arcsinh}\left (x c \right )^{2} \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )+2 b^{2} d \,\operatorname {arcsinh}\left (x c \right ) \operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right )-2 b^{2} d \operatorname {polylog}\left (3, -x c -\sqrt {c^{2} x^{2}+1}\right )-a b d \operatorname {arcsinh}\left (x c \right )^{2}+a b d \,\operatorname {arcsinh}\left (x c \right ) x^{2} c^{2}-\frac {a b d x c \sqrt {c^{2} x^{2}+1}}{2}+\frac {a b d \,\operatorname {arcsinh}\left (x c \right )}{2}+2 a b d \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right )+2 a b d \operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right )+2 a b d \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )+2 a b d \operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right )\)	\(420\)

Input:

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

Output:

a^2*d*(1/2*c^2*x^2+ln(x*c))+b^2*d*(-1/3*arcsinh(x*c)^3+1/16*(2*arcsinh(x*c 
)^2-2*arcsinh(x*c)+1)*(2*c^2*x^2+2*(c^2*x^2+1)^(1/2)*x*c+1)+1/16*(-2*(c^2* 
x^2+1)^(1/2)*x*c+2*c^2*x^2+1)*(2*arcsinh(x*c)^2+2*arcsinh(x*c)+1)+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*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*polylog(3, 
-x*c-(c^2*x^2+1)^(1/2)))-a*b*d*arcsinh(x*c)^2+a*b*d*arcsinh(x*c)*x^2*c^2-1 
/2*a*b*d*x*c*(c^2*x^2+1)^(1/2)+1/2*a*b*d*arcsinh(x*c)+2*a*b*d*arcsinh(x*c) 
*ln(1-x*c-(c^2*x^2+1)^(1/2))+2*a*b*d*polylog(2,x*c+(c^2*x^2+1)^(1/2))+2*a* 
b*d*arcsinh(x*c)*ln(1+x*c+(c^2*x^2+1)^(1/2))+2*a*b*d*polylog(2,-x*c-(c^2*x 
^2+1)^(1/2))
 

Fricas [F]

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

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2}{x} \, dx=\frac {d \left (2 \mathit {asinh} \left (c x \right )^{2} b^{2} c^{2} x^{2}+\mathit {asinh} \left (c x \right )^{2} b^{2}-2 \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) b^{2} c x +4 \mathit {asinh} \left (c x \right ) a b \,c^{2} x^{2}-2 \sqrt {c^{2} x^{2}+1}\, a b c x +8 \left (\int \frac {\mathit {asinh} \left (c x \right )}{x}d x \right ) a b +4 \left (\int \frac {\mathit {asinh} \left (c x \right )^{2}}{x}d x \right ) b^{2}+2 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x \right ) a b +4 \,\mathrm {log}\left (x \right ) a^{2}+2 a^{2} c^{2} x^{2}+b^{2} c^{2} x^{2}\right )}{4} \] Input:

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

Output:

(d*(2*asinh(c*x)**2*b**2*c**2*x**2 + asinh(c*x)**2*b**2 - 2*sqrt(c**2*x**2 
 + 1)*asinh(c*x)*b**2*c*x + 4*asinh(c*x)*a*b*c**2*x**2 - 2*sqrt(c**2*x**2 
+ 1)*a*b*c*x + 8*int(asinh(c*x)/x,x)*a*b + 4*int(asinh(c*x)**2/x,x)*b**2 + 
 2*log(sqrt(c**2*x**2 + 1) + c*x)*a*b + 4*log(x)*a**2 + 2*a**2*c**2*x**2 + 
 b**2*c**2*x**2))/4