\(\int \frac {x (a+b \text {arcsinh}(c x))^2}{d+c^2 d x^2} \, dx\) [240]

Optimal result

Integrand size = 24, antiderivative size = 105 \[ \int \frac {x (a+b \text {arcsinh}(c x))^2}{d+c^2 d x^2} \, dx=-\frac {(a+b \text {arcsinh}(c x))^3}{3 b c^2 d}+\frac {(a+b \text {arcsinh}(c x))^2 \log \left (1+e^{2 \text {arcsinh}(c x)}\right )}{c^2 d}+\frac {b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{c^2 d}-\frac {b^2 \operatorname {PolyLog}\left (3,-e^{2 \text {arcsinh}(c x)}\right )}{2 c^2 d} \] Output:

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(281\) vs. \(2(105)=210\).

Time = 0.19 (sec) , antiderivative size = 281, normalized size of antiderivative = 2.68 \[ \int \frac {x (a+b \text {arcsinh}(c x))^2}{d+c^2 d x^2} \, dx=\frac {-6 a b \text {arcsinh}(c x)^2-2 b^2 \text {arcsinh}(c x)^3+12 a b \text {arcsinh}(c x) \log \left (1+\frac {c e^{\text {arcsinh}(c x)}}{\sqrt {-c^2}}\right )+6 b^2 \text {arcsinh}(c x)^2 \log \left (1+\frac {c e^{\text {arcsinh}(c x)}}{\sqrt {-c^2}}\right )+12 a b \text {arcsinh}(c x) \log \left (1+\frac {\sqrt {-c^2} e^{\text {arcsinh}(c x)}}{c}\right )+6 b^2 \text {arcsinh}(c x)^2 \log \left (1+\frac {\sqrt {-c^2} e^{\text {arcsinh}(c x)}}{c}\right )+3 a^2 \log \left (1+c^2 x^2\right )+12 b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,\frac {c e^{\text {arcsinh}(c x)}}{\sqrt {-c^2}}\right )+12 b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,\frac {\sqrt {-c^2} e^{\text {arcsinh}(c x)}}{c}\right )-12 b^2 \operatorname {PolyLog}\left (3,\frac {c e^{\text {arcsinh}(c x)}}{\sqrt {-c^2}}\right )-12 b^2 \operatorname {PolyLog}\left (3,\frac {\sqrt {-c^2} e^{\text {arcsinh}(c x)}}{c}\right )}{6 c^2 d} \] Input:

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

Output:

(-6*a*b*ArcSinh[c*x]^2 - 2*b^2*ArcSinh[c*x]^3 + 12*a*b*ArcSinh[c*x]*Log[1 
+ (c*E^ArcSinh[c*x])/Sqrt[-c^2]] + 6*b^2*ArcSinh[c*x]^2*Log[1 + (c*E^ArcSi 
nh[c*x])/Sqrt[-c^2]] + 12*a*b*ArcSinh[c*x]*Log[1 + (Sqrt[-c^2]*E^ArcSinh[c 
*x])/c] + 6*b^2*ArcSinh[c*x]^2*Log[1 + (Sqrt[-c^2]*E^ArcSinh[c*x])/c] + 3* 
a^2*Log[1 + c^2*x^2] + 12*b*(a + b*ArcSinh[c*x])*PolyLog[2, (c*E^ArcSinh[c 
*x])/Sqrt[-c^2]] + 12*b*(a + b*ArcSinh[c*x])*PolyLog[2, (Sqrt[-c^2]*E^ArcS 
inh[c*x])/c] - 12*b^2*PolyLog[3, (c*E^ArcSinh[c*x])/Sqrt[-c^2]] - 12*b^2*P 
olyLog[3, (Sqrt[-c^2]*E^ArcSinh[c*x])/c])/(6*c^2*d)
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.59 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6212, 3042, 26, 4201, 2620, 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.

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

\(\Big \downarrow \) 6212

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

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int -i (a+b \text {arcsinh}(c x))^2 \tan (i \text {arcsinh}(c x))d\text {arcsinh}(c x)}{c^2 d}\)

\(\Big \downarrow \) 26

\(\displaystyle -\frac {i \int (a+b \text {arcsinh}(c x))^2 \tan (i \text {arcsinh}(c x))d\text {arcsinh}(c x)}{c^2 d}\)

\(\Big \downarrow \) 4201

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

Input:

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

Output:

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

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 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 6212

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

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 [A] (verified)

Time = 0.99 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.71

method	result	size
derivativedivides	\(\frac {\frac {a^{2} \ln \left (c^{2} x^{2}+1\right )}{2 d}+\frac {b^{2} \left (-\frac {\operatorname {arcsinh}\left (x c \right )^{3}}{3}+\operatorname {arcsinh}\left (x c \right )^{2} \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+\operatorname {arcsinh}\left (x c \right ) \operatorname {polylog}\left (2, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )-\frac {\operatorname {polylog}\left (3, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2}\right )}{d}+\frac {2 a b \left (-\frac {\operatorname {arcsinh}\left (x c \right )^{2}}{2}+\operatorname {arcsinh}\left (x c \right ) \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (2, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2}\right )}{d}}{c^{2}}\)	\(180\)
default	\(\frac {\frac {a^{2} \ln \left (c^{2} x^{2}+1\right )}{2 d}+\frac {b^{2} \left (-\frac {\operatorname {arcsinh}\left (x c \right )^{3}}{3}+\operatorname {arcsinh}\left (x c \right )^{2} \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+\operatorname {arcsinh}\left (x c \right ) \operatorname {polylog}\left (2, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )-\frac {\operatorname {polylog}\left (3, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2}\right )}{d}+\frac {2 a b \left (-\frac {\operatorname {arcsinh}\left (x c \right )^{2}}{2}+\operatorname {arcsinh}\left (x c \right ) \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (2, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2}\right )}{d}}{c^{2}}\)	\(180\)
parts	\(\frac {a^{2} \ln \left (c^{2} x^{2}+1\right )}{2 d \,c^{2}}+\frac {b^{2} \left (-\frac {\operatorname {arcsinh}\left (x c \right )^{3}}{3}+\operatorname {arcsinh}\left (x c \right )^{2} \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+\operatorname {arcsinh}\left (x c \right ) \operatorname {polylog}\left (2, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )-\frac {\operatorname {polylog}\left (3, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2}\right )}{d \,c^{2}}+\frac {2 a b \left (-\frac {\operatorname {arcsinh}\left (x c \right )^{2}}{2}+\operatorname {arcsinh}\left (x c \right ) \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (2, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2}\right )}{d \,c^{2}}\)	\(185\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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