\(\int \frac {(a+b \text {arcsinh}(c x))^2}{d+e x} \, dx\) [16]

Optimal result

Integrand size = 18, antiderivative size = 291 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{d+e x} \, dx=-\frac {(a+b \text {arcsinh}(c x))^3}{3 b e}+\frac {(a+b \text {arcsinh}(c x))^2 \log \left (1+\frac {e e^{\text {arcsinh}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}\right )}{e}+\frac {(a+b \text {arcsinh}(c x))^2 \log \left (1+\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )}{e}+\frac {2 b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arcsinh}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}\right )}{e}+\frac {2 b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )}{e}-\frac {2 b^2 \operatorname {PolyLog}\left (3,-\frac {e e^{\text {arcsinh}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}\right )}{e}-\frac {2 b^2 \operatorname {PolyLog}\left (3,-\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )}{e} \] Output:

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

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 273, normalized size of antiderivative = 0.94 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{d+e x} \, dx=\frac {-\frac {(a+b \text {arcsinh}(c x))^3}{b}+3 (a+b \text {arcsinh}(c x))^2 \log \left (1+\frac {e e^{\text {arcsinh}(c x)}}{c d-\sqrt {c^2 d^2+e^2}}\right )+3 (a+b \text {arcsinh}(c x))^2 \log \left (1+\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )+6 b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,\frac {e e^{\text {arcsinh}(c x)}}{-c d+\sqrt {c^2 d^2+e^2}}\right )+6 b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )-6 b^2 \operatorname {PolyLog}\left (3,\frac {e e^{\text {arcsinh}(c x)}}{-c d+\sqrt {c^2 d^2+e^2}}\right )-6 b^2 \operatorname {PolyLog}\left (3,-\frac {e e^{\text {arcsinh}(c x)}}{c d+\sqrt {c^2 d^2+e^2}}\right )}{3 e} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.71 (sec) , antiderivative size = 285, normalized size of antiderivative = 0.98, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6242, 6095, 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.

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

\(\Big \downarrow \) 6242

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

\(\Big \downarrow \) 6095

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

Input:

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

Output:

-1/3*(a + b*ArcSinh[c*x])^3/(b*e) + ((a + b*ArcSinh[c*x])^2*Log[1 + (e*E^A 
rcSinh[c*x])/(c*d - Sqrt[c^2*d^2 + e^2])])/e + ((a + b*ArcSinh[c*x])^2*Log 
[1 + (e*E^ArcSinh[c*x])/(c*d + Sqrt[c^2*d^2 + e^2])])/e - (2*b*(-((a + b*A 
rcSinh[c*x])*PolyLog[2, -((e*E^ArcSinh[c*x])/(c*d - Sqrt[c^2*d^2 + e^2]))] 
) + b*PolyLog[3, -((e*E^ArcSinh[c*x])/(c*d - Sqrt[c^2*d^2 + e^2]))]))/e - 
(2*b*(-((a + b*ArcSinh[c*x])*PolyLog[2, -((e*E^ArcSinh[c*x])/(c*d + Sqrt[c 
^2*d^2 + e^2]))]) + b*PolyLog[3, -((e*E^ArcSinh[c*x])/(c*d + Sqrt[c^2*d^2 
+ e^2]))]))/e
 

Defintions of rubi rules used

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 6095

Int[(Cosh[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sin 
h[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[-(e + f*x)^(m + 1)/(b*f*(m + 1)), 
 x] + (Int[(e + f*x)^m*(E^(c + d*x)/(a - Rt[a^2 + b^2, 2] + b*E^(c + d*x))) 
, x] + Int[(e + f*x)^m*(E^(c + d*x)/(a + Rt[a^2 + b^2, 2] + b*E^(c + d*x))) 
, x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[a^2 + b^2, 0]
 

rule 6242

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/((d_.) + (e_.)*(x_)), x_Symbo 
l] :> Subst[Int[(a + b*x)^n*(Cosh[x]/(c*d + e*Sinh[x])), x], x, ArcSinh[c*x 
]] /; FreeQ[{a, b, c, d, e}, x] && 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 [F]

\[\int \frac {\left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2}}{e x +d}d x\]

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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