3.2.51 \(\int \frac {(a+b \text {arcsinh}(c+d x))^4}{c e+d e x} \, dx\) [151]

3.2.51.1 Optimal result

Integrand size = 23, antiderivative size = 186 \[ \int \frac {(a+b \text {arcsinh}(c+d x))^4}{c e+d e x} \, dx=\frac {(a+b \text {arcsinh}(c+d x))^5}{5 b d e}+\frac {(a+b \text {arcsinh}(c+d x))^4 \log \left (1-e^{-2 \text {arcsinh}(c+d x)}\right )}{d e}-\frac {2 b (a+b \text {arcsinh}(c+d x))^3 \operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}(c+d x)}\right )}{d e}-\frac {3 b^2 (a+b \text {arcsinh}(c+d x))^2 \operatorname {PolyLog}\left (3,e^{-2 \text {arcsinh}(c+d x)}\right )}{d e}-\frac {3 b^3 (a+b \text {arcsinh}(c+d x)) \operatorname {PolyLog}\left (4,e^{-2 \text {arcsinh}(c+d x)}\right )}{d e}-\frac {3 b^4 \operatorname {PolyLog}\left (5,e^{-2 \text {arcsinh}(c+d x)}\right )}{2 d e} \]

output

1/5*(a+b*arcsinh(d*x+c))^5/b/d/e+(a+b*arcsinh(d*x+c))^4*ln(1-1/(d*x+c+(1+( 
d*x+c)^2)^(1/2))^2)/d/e-2*b*(a+b*arcsinh(d*x+c))^3*polylog(2,1/(d*x+c+(1+( 
d*x+c)^2)^(1/2))^2)/d/e-3*b^2*(a+b*arcsinh(d*x+c))^2*polylog(3,1/(d*x+c+(1 
+(d*x+c)^2)^(1/2))^2)/d/e-3*b^3*(a+b*arcsinh(d*x+c))*polylog(4,1/(d*x+c+(1 
+(d*x+c)^2)^(1/2))^2)/d/e-3/2*b^4*polylog(5,1/(d*x+c+(1+(d*x+c)^2)^(1/2))^ 
2)/d/e
 

3.2.51.2 Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.84 \[ \int \frac {(a+b \text {arcsinh}(c+d x))^4}{c e+d e x} \, dx=\frac {-\frac {(a+b \text {arcsinh}(c+d x))^5}{5 b}+(a+b \text {arcsinh}(c+d x))^4 \log \left (1-e^{2 \text {arcsinh}(c+d x)}\right )+2 b (a+b \text {arcsinh}(c+d x))^3 \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c+d x)}\right )-3 b^2 (a+b \text {arcsinh}(c+d x))^2 \operatorname {PolyLog}\left (3,e^{2 \text {arcsinh}(c+d x)}\right )+3 b^3 (a+b \text {arcsinh}(c+d x)) \operatorname {PolyLog}\left (4,e^{2 \text {arcsinh}(c+d x)}\right )-\frac {3}{2} b^4 \operatorname {PolyLog}\left (5,e^{2 \text {arcsinh}(c+d x)}\right )}{d e} \]

input

Integrate[(a + b*ArcSinh[c + d*x])^4/(c*e + d*e*x),x]
 

output

(-1/5*(a + b*ArcSinh[c + d*x])^5/b + (a + b*ArcSinh[c + d*x])^4*Log[1 - E^ 
(2*ArcSinh[c + d*x])] + 2*b*(a + b*ArcSinh[c + d*x])^3*PolyLog[2, E^(2*Arc 
Sinh[c + d*x])] - 3*b^2*(a + b*ArcSinh[c + d*x])^2*PolyLog[3, E^(2*ArcSinh 
[c + d*x])] + 3*b^3*(a + b*ArcSinh[c + d*x])*PolyLog[4, E^(2*ArcSinh[c + d 
*x])] - (3*b^4*PolyLog[5, E^(2*ArcSinh[c + d*x])])/2)/(d*e)
 

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

Result contains complex when optimal does not.

Time = 1.10 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.42, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.565, Rules used = {6274, 27, 6190, 25, 3042, 26, 4201, 2620, 3011, 7163, 7163, 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.

\(\displaystyle \int \frac {(a+b \text {arcsinh}(c+d x))^4}{c e+d e x} \, dx\)

\(\Big \downarrow \) 6274

\(\displaystyle \frac {\int \frac {(a+b \text {arcsinh}(c+d x))^4}{e (c+d x)}d(c+d x)}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {(a+b \text {arcsinh}(c+d x))^4}{c+d x}d(c+d x)}{d e}\)

\(\Big \downarrow \) 6190

\(\displaystyle \frac {\int -(a+b \text {arcsinh}(c+d x))^4 \coth \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )d(a+b \text {arcsinh}(c+d x))}{b d e}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\int (a+b \text {arcsinh}(c+d x))^4 \coth \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )d(a+b \text {arcsinh}(c+d x))}{b d e}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\int -i (a+b \text {arcsinh}(c+d x))^4 \tan \left (\frac {i a}{b}-\frac {i (a+b \text {arcsinh}(c+d x))}{b}+\frac {\pi }{2}\right )d(a+b \text {arcsinh}(c+d x))}{b d e}\)

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4201

\(\displaystyle \frac {i \left (2 i \int \frac {e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}-i \pi } (a+b \text {arcsinh}(c+d x))^4}{1+e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}-i \pi }}d(a+b \text {arcsinh}(c+d x))-\frac {1}{5} i (a+b \text {arcsinh}(c+d x))^5\right )}{b d e}\)

\(\Big \downarrow \) 2620

\(\displaystyle \frac {i \left (2 i \left (2 b \int (a+b \text {arcsinh}(c+d x))^3 \log \left (1+e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}-i \pi }\right )d(a+b \text {arcsinh}(c+d x))-\frac {1}{2} b (a+b \text {arcsinh}(c+d x))^4 \log \left (1+\exp \left (-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}+\frac {2 a}{b}-i \pi \right )\right )\right )-\frac {1}{5} i (a+b \text {arcsinh}(c+d x))^5\right )}{b d e}\)

\(\Big \downarrow \) 3011

\(\displaystyle \frac {i \left (2 i \left (2 b \left (\frac {1}{2} b (a+b \text {arcsinh}(c+d x))^3 \operatorname {PolyLog}\left (2,-e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}-i \pi }\right )-\frac {3}{2} b \int (a+b \text {arcsinh}(c+d x))^2 \operatorname {PolyLog}\left (2,-e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}-i \pi }\right )d(a+b \text {arcsinh}(c+d x))\right )-\frac {1}{2} b (a+b \text {arcsinh}(c+d x))^4 \log \left (1+\exp \left (-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}+\frac {2 a}{b}-i \pi \right )\right )\right )-\frac {1}{5} i (a+b \text {arcsinh}(c+d x))^5\right )}{b d e}\)

\(\Big \downarrow \) 7163

\(\displaystyle \frac {i \left (2 i \left (2 b \left (\frac {1}{2} b (a+b \text {arcsinh}(c+d x))^3 \operatorname {PolyLog}\left (2,-e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}-i \pi }\right )-\frac {3}{2} b \left (b \int (a+b \text {arcsinh}(c+d x)) \operatorname {PolyLog}\left (3,-e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}-i \pi }\right )d(a+b \text {arcsinh}(c+d x))-\frac {1}{2} b (a+b \text {arcsinh}(c+d x))^2 \operatorname {PolyLog}\left (3,-e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}-i \pi }\right )\right )\right )-\frac {1}{2} b (a+b \text {arcsinh}(c+d x))^4 \log \left (1+\exp \left (-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}+\frac {2 a}{b}-i \pi \right )\right )\right )-\frac {1}{5} i (a+b \text {arcsinh}(c+d x))^5\right )}{b d e}\)

\(\Big \downarrow \) 7163

\(\displaystyle \frac {i \left (2 i \left (2 b \left (\frac {1}{2} b (a+b \text {arcsinh}(c+d x))^3 \operatorname {PolyLog}\left (2,-e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}-i \pi }\right )-\frac {3}{2} b \left (b \left (\frac {1}{2} b \int \operatorname {PolyLog}\left (4,-e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}-i \pi }\right )d(a+b \text {arcsinh}(c+d x))-\frac {1}{2} b (a+b \text {arcsinh}(c+d x)) \operatorname {PolyLog}\left (4,-e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}-i \pi }\right )\right )-\frac {1}{2} b (a+b \text {arcsinh}(c+d x))^2 \operatorname {PolyLog}\left (3,-e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}-i \pi }\right )\right )\right )-\frac {1}{2} b (a+b \text {arcsinh}(c+d x))^4 \log \left (1+\exp \left (-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}+\frac {2 a}{b}-i \pi \right )\right )\right )-\frac {1}{5} i (a+b \text {arcsinh}(c+d x))^5\right )}{b d e}\)

\(\Big \downarrow \) 2720

\(\displaystyle \frac {i \left (2 i \left (2 b \left (\frac {1}{2} b (a+b \text {arcsinh}(c+d x))^3 \operatorname {PolyLog}\left (2,-e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}-i \pi }\right )-\frac {3}{2} b \left (b \left (-\frac {1}{4} b^2 \int \exp \left (-\frac {2 a}{b}+\frac {2 (a+b \text {arcsinh}(c+d x))}{b}+i \pi \right ) \operatorname {PolyLog}(4,-c-d x)de^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}-i \pi }-\frac {1}{2} b (a+b \text {arcsinh}(c+d x)) \operatorname {PolyLog}\left (4,-e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}-i \pi }\right )\right )-\frac {1}{2} b (a+b \text {arcsinh}(c+d x))^2 \operatorname {PolyLog}\left (3,-e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}-i \pi }\right )\right )\right )-\frac {1}{2} b (a+b \text {arcsinh}(c+d x))^4 \log \left (1+\exp \left (-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}+\frac {2 a}{b}-i \pi \right )\right )\right )-\frac {1}{5} i (a+b \text {arcsinh}(c+d x))^5\right )}{b d e}\)

\(\Big \downarrow \) 7143

\(\displaystyle \frac {i \left (2 i \left (2 b \left (\frac {1}{2} b (a+b \text {arcsinh}(c+d x))^3 \operatorname {PolyLog}\left (2,-e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}-i \pi }\right )-\frac {3}{2} b \left (b \left (-\frac {1}{4} b^2 \operatorname {PolyLog}(5,-c-d x)-\frac {1}{2} b (a+b \text {arcsinh}(c+d x)) \operatorname {PolyLog}\left (4,-e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}-i \pi }\right )\right )-\frac {1}{2} b (a+b \text {arcsinh}(c+d x))^2 \operatorname {PolyLog}\left (3,-e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}-i \pi }\right )\right )\right )-\frac {1}{2} b (a+b \text {arcsinh}(c+d x))^4 \log \left (1+\exp \left (-\frac {2 (a+b \text {arcsinh}(c+d x))}{b}+\frac {2 a}{b}-i \pi \right )\right )\right )-\frac {1}{5} i (a+b \text {arcsinh}(c+d x))^5\right )}{b d e}\)

input

Int[(a + b*ArcSinh[c + d*x])^4/(c*e + d*e*x),x]
 

output

(I*((-1/5*I)*(a + b*ArcSinh[c + d*x])^5 + (2*I)*(-1/2*(b*(a + b*ArcSinh[c 
+ d*x])^4*Log[1 + E^((2*a)/b - I*Pi - (2*(a + b*ArcSinh[c + d*x]))/b)]) + 
2*b*((b*(a + b*ArcSinh[c + d*x])^3*PolyLog[2, -E^((2*a)/b - I*Pi - (2*(a + 
 b*ArcSinh[c + d*x]))/b)])/2 - (3*b*(-1/2*(b*(a + b*ArcSinh[c + d*x])^2*Po 
lyLog[3, -E^((2*a)/b - I*Pi - (2*(a + b*ArcSinh[c + d*x]))/b)]) + b*(-1/2* 
(b*(a + b*ArcSinh[c + d*x])*PolyLog[4, -E^((2*a)/b - I*Pi - (2*(a + b*ArcS 
inh[c + d*x]))/b)]) - (b^2*PolyLog[5, -c - d*x])/4)))/2))))/(b*d*e)
 

3.2.51.3.1 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 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 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]
 

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]
 

rule 7163

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

3.2.51.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(860\) vs. \(2(222)=444\).

Time = 0.54 (sec) , antiderivative size = 861, normalized size of antiderivative = 4.63

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(861\)
default	\(\text {Expression too large to display}\)	\(861\)
parts	\(\text {Expression too large to display}\)	\(872\)

input

int((a+b*arcsinh(d*x+c))^4/(d*e*x+c*e),x,method=_RETURNVERBOSE)
 

output

1/d*(a^4/e*ln(d*x+c)+b^4/e*(-1/5*arcsinh(d*x+c)^5+arcsinh(d*x+c)^4*ln(1+d* 
x+c+(1+(d*x+c)^2)^(1/2))+4*arcsinh(d*x+c)^3*polylog(2,-d*x-c-(1+(d*x+c)^2) 
^(1/2))-12*arcsinh(d*x+c)^2*polylog(3,-d*x-c-(1+(d*x+c)^2)^(1/2))+24*arcsi 
nh(d*x+c)*polylog(4,-d*x-c-(1+(d*x+c)^2)^(1/2))-24*polylog(5,-d*x-c-(1+(d* 
x+c)^2)^(1/2))+arcsinh(d*x+c)^4*ln(1-d*x-c-(1+(d*x+c)^2)^(1/2))+4*arcsinh( 
d*x+c)^3*polylog(2,d*x+c+(1+(d*x+c)^2)^(1/2))-12*arcsinh(d*x+c)^2*polylog( 
3,d*x+c+(1+(d*x+c)^2)^(1/2))+24*arcsinh(d*x+c)*polylog(4,d*x+c+(1+(d*x+c)^ 
2)^(1/2))-24*polylog(5,d*x+c+(1+(d*x+c)^2)^(1/2)))+4*a*b^3/e*(-1/4*arcsinh 
(d*x+c)^4+arcsinh(d*x+c)^3*ln(1+d*x+c+(1+(d*x+c)^2)^(1/2))+3*arcsinh(d*x+c 
)^2*polylog(2,-d*x-c-(1+(d*x+c)^2)^(1/2))-6*arcsinh(d*x+c)*polylog(3,-d*x- 
c-(1+(d*x+c)^2)^(1/2))+6*polylog(4,-d*x-c-(1+(d*x+c)^2)^(1/2))+arcsinh(d*x 
+c)^3*ln(1-d*x-c-(1+(d*x+c)^2)^(1/2))+3*arcsinh(d*x+c)^2*polylog(2,d*x+c+( 
1+(d*x+c)^2)^(1/2))-6*arcsinh(d*x+c)*polylog(3,d*x+c+(1+(d*x+c)^2)^(1/2))+ 
6*polylog(4,d*x+c+(1+(d*x+c)^2)^(1/2)))+6*a^2*b^2/e*(-1/3*arcsinh(d*x+c)^3 
+arcsinh(d*x+c)^2*ln(1+d*x+c+(1+(d*x+c)^2)^(1/2))+2*arcsinh(d*x+c)*polylog 
(2,-d*x-c-(1+(d*x+c)^2)^(1/2))-2*polylog(3,-d*x-c-(1+(d*x+c)^2)^(1/2))+arc 
sinh(d*x+c)^2*ln(1-d*x-c-(1+(d*x+c)^2)^(1/2))+2*arcsinh(d*x+c)*polylog(2,d 
*x+c+(1+(d*x+c)^2)^(1/2))-2*polylog(3,d*x+c+(1+(d*x+c)^2)^(1/2)))+4*b*a^3/ 
e*(-1/2*arcsinh(d*x+c)^2+arcsinh(d*x+c)*ln(1+d*x+c+(1+(d*x+c)^2)^(1/2))+po 
lylog(2,-d*x-c-(1+(d*x+c)^2)^(1/2))+arcsinh(d*x+c)*ln(1-d*x-c-(1+(d*x+c...
 

3.2.51.5 Fricas [F]

\[ \int \frac {(a+b \text {arcsinh}(c+d x))^4}{c e+d e x} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{4}}{d e x + c e} \,d x } \]

input

integrate((a+b*arcsinh(d*x+c))^4/(d*e*x+c*e),x, algorithm="fricas")
 

output

integral((b^4*arcsinh(d*x + c)^4 + 4*a*b^3*arcsinh(d*x + c)^3 + 6*a^2*b^2* 
arcsinh(d*x + c)^2 + 4*a^3*b*arcsinh(d*x + c) + a^4)/(d*e*x + c*e), x)
 

3.2.51.6 Sympy [F]

\[ \int \frac {(a+b \text {arcsinh}(c+d x))^4}{c e+d e x} \, dx=\frac {\int \frac {a^{4}}{c + d x}\, dx + \int \frac {b^{4} \operatorname {asinh}^{4}{\left (c + d x \right )}}{c + d x}\, dx + \int \frac {4 a b^{3} \operatorname {asinh}^{3}{\left (c + d x \right )}}{c + d x}\, dx + \int \frac {6 a^{2} b^{2} \operatorname {asinh}^{2}{\left (c + d x \right )}}{c + d x}\, dx + \int \frac {4 a^{3} b \operatorname {asinh}{\left (c + d x \right )}}{c + d x}\, dx}{e} \]

input

integrate((a+b*asinh(d*x+c))**4/(d*e*x+c*e),x)
 

output

(Integral(a**4/(c + d*x), x) + Integral(b**4*asinh(c + d*x)**4/(c + d*x), 
x) + Integral(4*a*b**3*asinh(c + d*x)**3/(c + d*x), x) + Integral(6*a**2*b 
**2*asinh(c + d*x)**2/(c + d*x), x) + Integral(4*a**3*b*asinh(c + d*x)/(c 
+ d*x), x))/e
 

3.2.51.7 Maxima [F]

\[ \int \frac {(a+b \text {arcsinh}(c+d x))^4}{c e+d e x} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{4}}{d e x + c e} \,d x } \]

input

integrate((a+b*arcsinh(d*x+c))^4/(d*e*x+c*e),x, algorithm="maxima")
 

output

a^4*log(d*e*x + c*e)/(d*e) + integrate(b^4*log(d*x + c + sqrt((d*x + c)^2 
+ 1))^4/(d*e*x + c*e) + 4*a*b^3*log(d*x + c + sqrt((d*x + c)^2 + 1))^3/(d* 
e*x + c*e) + 6*a^2*b^2*log(d*x + c + sqrt((d*x + c)^2 + 1))^2/(d*e*x + c*e 
) + 4*a^3*b*log(d*x + c + sqrt((d*x + c)^2 + 1))/(d*e*x + c*e), x)
 

3.2.51.8 Giac [F]

\[ \int \frac {(a+b \text {arcsinh}(c+d x))^4}{c e+d e x} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{4}}{d e x + c e} \,d x } \]

input

integrate((a+b*arcsinh(d*x+c))^4/(d*e*x+c*e),x, algorithm="giac")
 

output

integrate((b*arcsinh(d*x + c) + a)^4/(d*e*x + c*e), x)
 

3.2.51.9 Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \text {arcsinh}(c+d x))^4}{c e+d e x} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^4}{c\,e+d\,e\,x} \,d x \]

input

int((a + b*asinh(c + d*x))^4/(c*e + d*e*x),x)
 

output

int((a + b*asinh(c + d*x))^4/(c*e + d*e*x), x)