3.2.44 \(\int \frac {(a+b \text {arcsinh}(c+d x))^3}{(c e+d e x)^3} \, dx\) [144]

3.2.44.1 Optimal result

Integrand size = 23, antiderivative size = 157 \[ \int \frac {(a+b \text {arcsinh}(c+d x))^3}{(c e+d e x)^3} \, dx=\frac {3 b (a+b \text {arcsinh}(c+d x))^2}{2 d e^3}-\frac {3 b \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^2}{2 d e^3 (c+d x)}-\frac {(a+b \text {arcsinh}(c+d x))^3}{2 d e^3 (c+d x)^2}+\frac {3 b^2 (a+b \text {arcsinh}(c+d x)) \log \left (1-e^{-2 \text {arcsinh}(c+d x)}\right )}{d e^3}-\frac {3 b^3 \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c+d x)}\right )}{2 d e^3} \]

output

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

3.2.44.2 Mathematica [A] (verified)

Time = 0.72 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.46 \[ \int \frac {(a+b \text {arcsinh}(c+d x))^3}{(c e+d e x)^3} \, dx=-\frac {3 b^2 \left (a+b (c+d x) \left (-c-d x+\sqrt {1+c^2+2 c d x+d^2 x^2}\right )\right ) \text {arcsinh}(c+d x)^2+b^3 \text {arcsinh}(c+d x)^3+3 b \text {arcsinh}(c+d x) \left (a \left (a+2 b (c+d x) \sqrt {1+c^2+2 c d x+d^2 x^2}\right )-2 b^2 (c+d x)^2 \log \left (1-e^{-2 \text {arcsinh}(c+d x)}\right )\right )+a \left (a \left (a+3 b (c+d x) \sqrt {1+c^2+2 c d x+d^2 x^2}\right )-6 b^2 (c+d x)^2 \log (c+d x)\right )+3 b^3 (c+d x)^2 \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c+d x)}\right )}{2 d e^3 (c+d x)^2} \]

input

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

output

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

3.2.44.3 Rubi [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 0.87 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.04, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {6274, 27, 6191, 6215, 6190, 25, 3042, 26, 4201, 2620, 2715, 2838}

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.

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

\(\Big \downarrow \) 6274

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6191

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

\(\Big \downarrow \) 6215

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

\(\Big \downarrow \) 6190

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4201

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

input

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

output

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

3.2.44.3.1 Defintions of rubi rules used

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 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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 2715

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

rule 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

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 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 6215

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 + 1)*((a + 
b*ArcSinh[c*x])^n/(d*f*(m + 1))), 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, m, p}, x] && EqQ 
[e, c^2*d] && GtQ[n, 0] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]
 

rule 6274

Int[((a_.) + ArcSinh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^( 
m_.), x_Symbol] :> Simp[1/d   Subst[Int[((d*e - c*f)/d + f*(x/d))^m*(a + b* 
ArcSinh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
 

3.2.44.4 Maple [A] (verified)

Time = 0.47 (sec) , antiderivative size = 325, normalized size of antiderivative = 2.07

method	result	size
derivativedivides	\(\frac {-\frac {a^{3}}{2 e^{3} \left (d x +c \right )^{2}}+\frac {b^{3} \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )^{2} \left (3 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}-3 \left (d x +c \right )^{2}+\operatorname {arcsinh}\left (d x +c \right )\right )}{2 \left (d x +c \right )^{2}}-3 \operatorname {arcsinh}\left (d x +c \right )^{2}+3 \,\operatorname {arcsinh}\left (d x +c \right ) \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )+3 \operatorname {polylog}\left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+3 \,\operatorname {arcsinh}\left (d x +c \right ) \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+3 \operatorname {polylog}\left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )\right )}{e^{3}}+\frac {3 a \,b^{2} \left (-2 \,\operatorname {arcsinh}\left (d x +c \right )-\frac {\operatorname {arcsinh}\left (d x +c \right ) \left (-2 \left (d x +c \right )^{2}+2 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}+\operatorname {arcsinh}\left (d x +c \right )\right )}{2 \left (d x +c \right )^{2}}+\ln \left (\left (d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )^{2}-1\right )\right )}{e^{3}}+\frac {3 a^{2} b \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )}{2 \left (d x +c \right )^{2}}-\frac {\sqrt {1+\left (d x +c \right )^{2}}}{2 \left (d x +c \right )}\right )}{e^{3}}}{d}\)	\(325\)
default	\(\frac {-\frac {a^{3}}{2 e^{3} \left (d x +c \right )^{2}}+\frac {b^{3} \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )^{2} \left (3 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}-3 \left (d x +c \right )^{2}+\operatorname {arcsinh}\left (d x +c \right )\right )}{2 \left (d x +c \right )^{2}}-3 \operatorname {arcsinh}\left (d x +c \right )^{2}+3 \,\operatorname {arcsinh}\left (d x +c \right ) \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )+3 \operatorname {polylog}\left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+3 \,\operatorname {arcsinh}\left (d x +c \right ) \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+3 \operatorname {polylog}\left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )\right )}{e^{3}}+\frac {3 a \,b^{2} \left (-2 \,\operatorname {arcsinh}\left (d x +c \right )-\frac {\operatorname {arcsinh}\left (d x +c \right ) \left (-2 \left (d x +c \right )^{2}+2 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}+\operatorname {arcsinh}\left (d x +c \right )\right )}{2 \left (d x +c \right )^{2}}+\ln \left (\left (d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )^{2}-1\right )\right )}{e^{3}}+\frac {3 a^{2} b \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )}{2 \left (d x +c \right )^{2}}-\frac {\sqrt {1+\left (d x +c \right )^{2}}}{2 \left (d x +c \right )}\right )}{e^{3}}}{d}\)	\(325\)
parts	\(-\frac {a^{3}}{2 e^{3} \left (d x +c \right )^{2} d}+\frac {b^{3} \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )^{2} \left (3 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}-3 \left (d x +c \right )^{2}+\operatorname {arcsinh}\left (d x +c \right )\right )}{2 \left (d x +c \right )^{2}}-3 \operatorname {arcsinh}\left (d x +c \right )^{2}+3 \,\operatorname {arcsinh}\left (d x +c \right ) \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )+3 \operatorname {polylog}\left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+3 \,\operatorname {arcsinh}\left (d x +c \right ) \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+3 \operatorname {polylog}\left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )\right )}{e^{3} d}+\frac {3 a \,b^{2} \left (-2 \,\operatorname {arcsinh}\left (d x +c \right )-\frac {\operatorname {arcsinh}\left (d x +c \right ) \left (-2 \left (d x +c \right )^{2}+2 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}+\operatorname {arcsinh}\left (d x +c \right )\right )}{2 \left (d x +c \right )^{2}}+\ln \left (\left (d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )^{2}-1\right )\right )}{e^{3} d}+\frac {3 a^{2} b \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )}{2 \left (d x +c \right )^{2}}-\frac {\sqrt {1+\left (d x +c \right )^{2}}}{2 \left (d x +c \right )}\right )}{e^{3} d}\)	\(333\)

input

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

output

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

3.2.44.5 Fricas [F]

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

input

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

output

integral((b^3*arcsinh(d*x + c)^3 + 3*a*b^2*arcsinh(d*x + c)^2 + 3*a^2*b*ar 
csinh(d*x + c) + a^3)/(d^3*e^3*x^3 + 3*c*d^2*e^3*x^2 + 3*c^2*d*e^3*x + c^3 
*e^3), x)
 

3.2.44.6 Sympy [F]

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

input

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

output

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

3.2.44.7 Maxima [F]

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

input

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

output

-3*(sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*d*arcsinh(d*x + c)/(d^3*e^3*x + c*d^ 
2*e^3) - log(d*x + c)/(d*e^3))*a*b^2 - 1/2*(log(d*x + c + sqrt(d^2*x^2 + 2 
*c*d*x + c^2 + 1))^3/(d^3*e^3*x^2 + 2*c*d^2*e^3*x + c^2*d*e^3) - 2*integra 
te(3/2*(d^2*x^2 + 2*c*d*x + c^2 + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*(d*x + 
 c) + 1)*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))^2/(d^5*e^3*x^5 + 
 5*c*d^4*e^3*x^4 + c^5*e^3 + c^3*e^3 + (10*c^2*d^3*e^3 + d^3*e^3)*x^3 + (1 
0*c^3*d^2*e^3 + 3*c*d^2*e^3)*x^2 + (5*c^4*d*e^3 + 3*c^2*d*e^3)*x + (d^4*e^ 
3*x^4 + 4*c*d^3*e^3*x^3 + c^4*e^3 + c^2*e^3 + (6*c^2*d^2*e^3 + d^2*e^3)*x^ 
2 + 2*(2*c^3*d*e^3 + c*d*e^3)*x)*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)), x))*b 
^3 - 3/2*a^2*b*(sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*d/(d^3*e^3*x + c*d^2*e^3 
) + arcsinh(d*x + c)/(d^3*e^3*x^2 + 2*c*d^2*e^3*x + c^2*d*e^3)) - 3/2*a*b^ 
2*arcsinh(d*x + c)^2/(d^3*e^3*x^2 + 2*c*d^2*e^3*x + c^2*d*e^3) - 1/2*a^3/( 
d^3*e^3*x^2 + 2*c*d^2*e^3*x + c^2*d*e^3)
 

3.2.44.8 Giac [F]

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

input

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

output

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

3.2.44.9 Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \text {arcsinh}(c+d x))^3}{(c e+d e x)^3} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^3}{{\left (c\,e+d\,e\,x\right )}^3} \,d x \]

input

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

output

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