Integrand size = 23, antiderivative size = 234 \[ \int \frac {(a+b \text {arcsinh}(c+d x))^4}{(c e+d e x)^2} \, dx=-\frac {(a+b \text {arcsinh}(c+d x))^4}{d e^2 (c+d x)}-\frac {8 b (a+b \text {arcsinh}(c+d x))^3 \text {arctanh}\left (e^{\text {arcsinh}(c+d x)}\right )}{d e^2}-\frac {12 b^2 (a+b \text {arcsinh}(c+d x))^2 \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c+d x)}\right )}{d e^2}+\frac {12 b^2 (a+b \text {arcsinh}(c+d x))^2 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c+d x)}\right )}{d e^2}+\frac {24 b^3 (a+b \text {arcsinh}(c+d x)) \operatorname {PolyLog}\left (3,-e^{\text {arcsinh}(c+d x)}\right )}{d e^2}-\frac {24 b^3 (a+b \text {arcsinh}(c+d x)) \operatorname {PolyLog}\left (3,e^{\text {arcsinh}(c+d x)}\right )}{d e^2}-\frac {24 b^4 \operatorname {PolyLog}\left (4,-e^{\text {arcsinh}(c+d x)}\right )}{d e^2}+\frac {24 b^4 \operatorname {PolyLog}\left (4,e^{\text {arcsinh}(c+d x)}\right )}{d e^2} \]
-(a+b*arcsinh(d*x+c))^4/d/e^2/(d*x+c)-8*b*(a+b*arcsinh(d*x+c))^3*arctanh(d *x+c+(1+(d*x+c)^2)^(1/2))/d/e^2-12*b^2*(a+b*arcsinh(d*x+c))^2*polylog(2,-d *x-c-(1+(d*x+c)^2)^(1/2))/d/e^2+12*b^2*(a+b*arcsinh(d*x+c))^2*polylog(2,d* x+c+(1+(d*x+c)^2)^(1/2))/d/e^2+24*b^3*(a+b*arcsinh(d*x+c))*polylog(3,-d*x- c-(1+(d*x+c)^2)^(1/2))/d/e^2-24*b^3*(a+b*arcsinh(d*x+c))*polylog(3,d*x+c+( 1+(d*x+c)^2)^(1/2))/d/e^2-24*b^4*polylog(4,-d*x-c-(1+(d*x+c)^2)^(1/2))/d/e ^2+24*b^4*polylog(4,d*x+c+(1+(d*x+c)^2)^(1/2))/d/e^2
Leaf count is larger than twice the leaf count of optimal. \(510\) vs. \(2(234)=468\).
Time = 1.32 (sec) , antiderivative size = 510, normalized size of antiderivative = 2.18 \[ \int \frac {(a+b \text {arcsinh}(c+d x))^4}{(c e+d e x)^2} \, dx=\frac {-\frac {2 a^4}{c+d x}-8 a^3 b \left (\frac {\text {arcsinh}(c+d x)}{c+d x}+\log \left (\frac {1}{2} (c+d x) \text {csch}\left (\frac {1}{2} \text {arcsinh}(c+d x)\right )\right )-\log \left (\sinh \left (\frac {1}{2} \text {arcsinh}(c+d x)\right )\right )\right )+12 a^2 b^2 \left (\text {arcsinh}(c+d x) \left (-\frac {\text {arcsinh}(c+d x)}{c+d x}+2 \log \left (1-e^{-\text {arcsinh}(c+d x)}\right )-2 \log \left (1+e^{-\text {arcsinh}(c+d x)}\right )\right )+2 \operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(c+d x)}\right )-2 \operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(c+d x)}\right )\right )+8 a b^3 \left (-\frac {\text {arcsinh}(c+d x)^3}{c+d x}+3 \text {arcsinh}(c+d x)^2 \log \left (1-e^{-\text {arcsinh}(c+d x)}\right )-3 \text {arcsinh}(c+d x)^2 \log \left (1+e^{-\text {arcsinh}(c+d x)}\right )+6 \text {arcsinh}(c+d x) \operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(c+d x)}\right )-6 \text {arcsinh}(c+d x) \operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(c+d x)}\right )+6 \operatorname {PolyLog}\left (3,-e^{-\text {arcsinh}(c+d x)}\right )-6 \operatorname {PolyLog}\left (3,e^{-\text {arcsinh}(c+d x)}\right )\right )+b^4 \left (\pi ^4-2 \text {arcsinh}(c+d x)^4-\frac {2 \text {arcsinh}(c+d x)^4}{c+d x}-8 \text {arcsinh}(c+d x)^3 \log \left (1+e^{-\text {arcsinh}(c+d x)}\right )+8 \text {arcsinh}(c+d x)^3 \log \left (1-e^{\text {arcsinh}(c+d x)}\right )+24 \text {arcsinh}(c+d x)^2 \operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(c+d x)}\right )+24 \text {arcsinh}(c+d x)^2 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c+d x)}\right )+48 \text {arcsinh}(c+d x) \operatorname {PolyLog}\left (3,-e^{-\text {arcsinh}(c+d x)}\right )-48 \text {arcsinh}(c+d x) \operatorname {PolyLog}\left (3,e^{\text {arcsinh}(c+d x)}\right )+48 \operatorname {PolyLog}\left (4,-e^{-\text {arcsinh}(c+d x)}\right )+48 \operatorname {PolyLog}\left (4,e^{\text {arcsinh}(c+d x)}\right )\right )}{2 d e^2} \]
((-2*a^4)/(c + d*x) - 8*a^3*b*(ArcSinh[c + d*x]/(c + d*x) + Log[((c + d*x) *Csch[ArcSinh[c + d*x]/2])/2] - Log[Sinh[ArcSinh[c + d*x]/2]]) + 12*a^2*b^ 2*(ArcSinh[c + d*x]*(-(ArcSinh[c + d*x]/(c + d*x)) + 2*Log[1 - E^(-ArcSinh [c + d*x])] - 2*Log[1 + E^(-ArcSinh[c + d*x])]) + 2*PolyLog[2, -E^(-ArcSin h[c + d*x])] - 2*PolyLog[2, E^(-ArcSinh[c + d*x])]) + 8*a*b^3*(-(ArcSinh[c + d*x]^3/(c + d*x)) + 3*ArcSinh[c + d*x]^2*Log[1 - E^(-ArcSinh[c + d*x])] - 3*ArcSinh[c + d*x]^2*Log[1 + E^(-ArcSinh[c + d*x])] + 6*ArcSinh[c + d*x ]*PolyLog[2, -E^(-ArcSinh[c + d*x])] - 6*ArcSinh[c + d*x]*PolyLog[2, E^(-A rcSinh[c + d*x])] + 6*PolyLog[3, -E^(-ArcSinh[c + d*x])] - 6*PolyLog[3, E^ (-ArcSinh[c + d*x])]) + b^4*(Pi^4 - 2*ArcSinh[c + d*x]^4 - (2*ArcSinh[c + d*x]^4)/(c + d*x) - 8*ArcSinh[c + d*x]^3*Log[1 + E^(-ArcSinh[c + d*x])] + 8*ArcSinh[c + d*x]^3*Log[1 - E^ArcSinh[c + d*x]] + 24*ArcSinh[c + d*x]^2*P olyLog[2, -E^(-ArcSinh[c + d*x])] + 24*ArcSinh[c + d*x]^2*PolyLog[2, E^Arc Sinh[c + d*x]] + 48*ArcSinh[c + d*x]*PolyLog[3, -E^(-ArcSinh[c + d*x])] - 48*ArcSinh[c + d*x]*PolyLog[3, E^ArcSinh[c + d*x]] + 48*PolyLog[4, -E^(-Ar cSinh[c + d*x])] + 48*PolyLog[4, E^ArcSinh[c + d*x]]))/(2*d*e^2)
Result contains complex when optimal does not.
Time = 0.95 (sec) , antiderivative size = 200, normalized size of antiderivative = 0.85, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {6274, 27, 6191, 6231, 3042, 26, 4670, 3011, 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.
| \(\displaystyle \int \frac {(a+b \text {arcsinh}(c+d x))^4}{(c e+d e x)^2} \, dx\) |
| \(\Big \downarrow \) 6274 |
| \(\displaystyle \frac {\int \frac {(a+b \text {arcsinh}(c+d x))^4}{e^2 (c+d x)^2}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(a+b \text {arcsinh}(c+d x))^4}{(c+d x)^2}d(c+d x)}{d e^2}\) |
| \(\Big \downarrow \) 6191 |
| \(\displaystyle \frac {4 b \int \frac {(a+b \text {arcsinh}(c+d x))^3}{(c+d x) \sqrt {(c+d x)^2+1}}d(c+d x)-\frac {(a+b \text {arcsinh}(c+d x))^4}{c+d x}}{d e^2}\) |
| \(\Big \downarrow \) 6231 |
| \(\displaystyle \frac {4 b \int \frac {(a+b \text {arcsinh}(c+d x))^3}{c+d x}d\text {arcsinh}(c+d x)-\frac {(a+b \text {arcsinh}(c+d x))^4}{c+d x}}{d e^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {(a+b \text {arcsinh}(c+d x))^4}{c+d x}+4 b \int i (a+b \text {arcsinh}(c+d x))^3 \csc (i \text {arcsinh}(c+d x))d\text {arcsinh}(c+d x)}{d e^2}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {-\frac {(a+b \text {arcsinh}(c+d x))^4}{c+d x}+4 i b \int (a+b \text {arcsinh}(c+d x))^3 \csc (i \text {arcsinh}(c+d x))d\text {arcsinh}(c+d x)}{d e^2}\) |
| \(\Big \downarrow \) 4670 |
| \(\displaystyle \frac {-\frac {(a+b \text {arcsinh}(c+d x))^4}{c+d x}+4 i b \left (3 i b \int (a+b \text {arcsinh}(c+d x))^2 \log \left (1-e^{\text {arcsinh}(c+d x)}\right )d\text {arcsinh}(c+d x)-3 i b \int (a+b \text {arcsinh}(c+d x))^2 \log \left (1+e^{\text {arcsinh}(c+d x)}\right )d\text {arcsinh}(c+d x)+2 i \text {arctanh}\left (e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))^3\right )}{d e^2}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle \frac {-\frac {(a+b \text {arcsinh}(c+d x))^4}{c+d x}+4 i b \left (-3 i b \left (2 b \int (a+b \text {arcsinh}(c+d x)) \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c+d x)}\right )d\text {arcsinh}(c+d x)-\operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))^2\right )+3 i b \left (2 b \int (a+b \text {arcsinh}(c+d x)) \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c+d x)}\right )d\text {arcsinh}(c+d x)-\operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))^2\right )+2 i \text {arctanh}\left (e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))^3\right )}{d e^2}\) |
| \(\Big \downarrow \) 7163 |
| \(\displaystyle \frac {-\frac {(a+b \text {arcsinh}(c+d x))^4}{c+d x}+4 i b \left (-3 i b \left (2 b \left (\operatorname {PolyLog}\left (3,-e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))-b \int \operatorname {PolyLog}\left (3,-e^{\text {arcsinh}(c+d x)}\right )d\text {arcsinh}(c+d x)\right )-\operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))^2\right )+3 i b \left (2 b \left (\operatorname {PolyLog}\left (3,e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))-b \int \operatorname {PolyLog}\left (3,e^{\text {arcsinh}(c+d x)}\right )d\text {arcsinh}(c+d x)\right )-\operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))^2\right )+2 i \text {arctanh}\left (e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))^3\right )}{d e^2}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle \frac {-\frac {(a+b \text {arcsinh}(c+d x))^4}{c+d x}+4 i b \left (3 i b \left (2 b \left (\operatorname {PolyLog}\left (3,e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))-b \int e^{-\text {arcsinh}(c+d x)} \operatorname {PolyLog}\left (3,e^{\text {arcsinh}(c+d x)}\right )de^{\text {arcsinh}(c+d x)}\right )-\operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))^2\right )-3 i b \left (2 b \left (\operatorname {PolyLog}\left (3,-e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))-b \int e^{-\text {arcsinh}(c+d x)} \operatorname {PolyLog}(3,-c-d x)de^{\text {arcsinh}(c+d x)}\right )-\operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))^2\right )+2 i \text {arctanh}\left (e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))^3\right )}{d e^2}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle \frac {-\frac {(a+b \text {arcsinh}(c+d x))^4}{c+d x}+4 i b \left (2 i \text {arctanh}\left (e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))^3+3 i b \left (2 b \left (\operatorname {PolyLog}\left (3,e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))-b \operatorname {PolyLog}\left (4,e^{\text {arcsinh}(c+d x)}\right )\right )-\operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))^2\right )-3 i b \left (2 b \left (\operatorname {PolyLog}\left (3,-e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))-b \operatorname {PolyLog}(4,-c-d x)\right )-\operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))^2\right )\right )}{d e^2}\) |
(-((a + b*ArcSinh[c + d*x])^4/(c + d*x)) + (4*I)*b*((2*I)*(a + b*ArcSinh[c + d*x])^3*ArcTanh[E^ArcSinh[c + d*x]] + (3*I)*b*(-((a + b*ArcSinh[c + d*x ])^2*PolyLog[2, E^ArcSinh[c + d*x]]) + 2*b*((a + b*ArcSinh[c + d*x])*PolyL og[3, E^ArcSinh[c + d*x]] - b*PolyLog[4, E^ArcSinh[c + d*x]])) - (3*I)*b*( -((a + b*ArcSinh[c + d*x])^2*PolyLog[2, -E^ArcSinh[c + d*x]]) + 2*b*((a + b*ArcSinh[c + d*x])*PolyLog[3, -E^ArcSinh[c + d*x]] - b*PolyLog[4, -c - d* x]))))/(d*e^2)
3.2.52.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]]]
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]
Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x _Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] + (-Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*fz*x )], x], x] + Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
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]
Int[(((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/Sqrt[(d_) + (e_.) *(x_)^2], x_Symbol] :> Simp[(1/c^(m + 1))*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e *x^2]] Subst[Int[(a + b*x)^n*Sinh[x]^m, x], x, ArcSinh[c*x]], x] /; FreeQ [{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[n, 0] && IntegerQ[m]
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]
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]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(629\) vs. \(2(299)=598\).
Time = 0.47 (sec) , antiderivative size = 630, normalized size of antiderivative = 2.69
| method | result | size |
| derivativedivides | \(\frac {-\frac {a^{4}}{e^{2} \left (d x +c \right )}+\frac {b^{4} \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )^{4}}{d x +c}-4 \operatorname {arcsinh}\left (d x +c \right )^{3} \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )-12 \operatorname {arcsinh}\left (d x +c \right )^{2} \operatorname {polylog}\left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+24 \,\operatorname {arcsinh}\left (d x +c \right ) \operatorname {polylog}\left (3, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )-24 \operatorname {polylog}\left (4, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+4 \operatorname {arcsinh}\left (d x +c \right )^{3} \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+12 \operatorname {arcsinh}\left (d x +c \right )^{2} \operatorname {polylog}\left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )-24 \,\operatorname {arcsinh}\left (d x +c \right ) \operatorname {polylog}\left (3, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )+24 \operatorname {polylog}\left (4, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )\right )}{e^{2}}+\frac {4 a \,b^{3} \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )^{3}}{d x +c}-3 \operatorname {arcsinh}\left (d x +c \right )^{2} \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )-6 \,\operatorname {arcsinh}\left (d x +c \right ) \operatorname {polylog}\left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+6 \operatorname {polylog}\left (3, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+3 \operatorname {arcsinh}\left (d x +c \right )^{2} \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+6 \,\operatorname {arcsinh}\left (d x +c \right ) \operatorname {polylog}\left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )-6 \operatorname {polylog}\left (3, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )\right )}{e^{2}}+\frac {6 a^{2} b^{2} \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )^{2}}{d x +c}-2 \,\operatorname {arcsinh}\left (d x +c \right ) \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )-2 \operatorname {polylog}\left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+2 \,\operatorname {arcsinh}\left (d x +c \right ) \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+2 \operatorname {polylog}\left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )\right )}{e^{2}}+\frac {4 b \,a^{3} \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )}{d x +c}-\operatorname {arctanh}\left (\frac {1}{\sqrt {1+\left (d x +c \right )^{2}}}\right )\right )}{e^{2}}}{d}\) | \(630\) |
| default | \(\frac {-\frac {a^{4}}{e^{2} \left (d x +c \right )}+\frac {b^{4} \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )^{4}}{d x +c}-4 \operatorname {arcsinh}\left (d x +c \right )^{3} \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )-12 \operatorname {arcsinh}\left (d x +c \right )^{2} \operatorname {polylog}\left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+24 \,\operatorname {arcsinh}\left (d x +c \right ) \operatorname {polylog}\left (3, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )-24 \operatorname {polylog}\left (4, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+4 \operatorname {arcsinh}\left (d x +c \right )^{3} \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+12 \operatorname {arcsinh}\left (d x +c \right )^{2} \operatorname {polylog}\left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )-24 \,\operatorname {arcsinh}\left (d x +c \right ) \operatorname {polylog}\left (3, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )+24 \operatorname {polylog}\left (4, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )\right )}{e^{2}}+\frac {4 a \,b^{3} \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )^{3}}{d x +c}-3 \operatorname {arcsinh}\left (d x +c \right )^{2} \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )-6 \,\operatorname {arcsinh}\left (d x +c \right ) \operatorname {polylog}\left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+6 \operatorname {polylog}\left (3, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+3 \operatorname {arcsinh}\left (d x +c \right )^{2} \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+6 \,\operatorname {arcsinh}\left (d x +c \right ) \operatorname {polylog}\left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )-6 \operatorname {polylog}\left (3, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )\right )}{e^{2}}+\frac {6 a^{2} b^{2} \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )^{2}}{d x +c}-2 \,\operatorname {arcsinh}\left (d x +c \right ) \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )-2 \operatorname {polylog}\left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+2 \,\operatorname {arcsinh}\left (d x +c \right ) \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+2 \operatorname {polylog}\left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )\right )}{e^{2}}+\frac {4 b \,a^{3} \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )}{d x +c}-\operatorname {arctanh}\left (\frac {1}{\sqrt {1+\left (d x +c \right )^{2}}}\right )\right )}{e^{2}}}{d}\) | \(630\) |
| parts | \(-\frac {a^{4}}{e^{2} \left (d x +c \right ) d}+\frac {b^{4} \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )^{4}}{d x +c}-4 \operatorname {arcsinh}\left (d x +c \right )^{3} \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )-12 \operatorname {arcsinh}\left (d x +c \right )^{2} \operatorname {polylog}\left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+24 \,\operatorname {arcsinh}\left (d x +c \right ) \operatorname {polylog}\left (3, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )-24 \operatorname {polylog}\left (4, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+4 \operatorname {arcsinh}\left (d x +c \right )^{3} \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+12 \operatorname {arcsinh}\left (d x +c \right )^{2} \operatorname {polylog}\left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )-24 \,\operatorname {arcsinh}\left (d x +c \right ) \operatorname {polylog}\left (3, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )+24 \operatorname {polylog}\left (4, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )\right )}{e^{2} d}+\frac {4 a \,b^{3} \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )^{3}}{d x +c}-3 \operatorname {arcsinh}\left (d x +c \right )^{2} \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )-6 \,\operatorname {arcsinh}\left (d x +c \right ) \operatorname {polylog}\left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+6 \operatorname {polylog}\left (3, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+3 \operatorname {arcsinh}\left (d x +c \right )^{2} \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+6 \,\operatorname {arcsinh}\left (d x +c \right ) \operatorname {polylog}\left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )-6 \operatorname {polylog}\left (3, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )\right )}{e^{2} d}+\frac {6 a^{2} b^{2} \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )^{2}}{d x +c}-2 \,\operatorname {arcsinh}\left (d x +c \right ) \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )-2 \operatorname {polylog}\left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+2 \,\operatorname {arcsinh}\left (d x +c \right ) \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )+2 \operatorname {polylog}\left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )\right )}{e^{2} d}+\frac {4 b \,a^{3} \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )}{d x +c}-\operatorname {arctanh}\left (\frac {1}{\sqrt {1+\left (d x +c \right )^{2}}}\right )\right )}{e^{2} d}\) | \(641\) |
1/d*(-a^4/e^2/(d*x+c)+b^4/e^2*(-1/(d*x+c)*arcsinh(d*x+c)^4-4*arcsinh(d*x+c )^3*ln(1+d*x+c+(1+(d*x+c)^2)^(1/2))-12*arcsinh(d*x+c)^2*polylog(2,-d*x-c-( 1+(d*x+c)^2)^(1/2))+24*arcsinh(d*x+c)*polylog(3,-d*x-c-(1+(d*x+c)^2)^(1/2) )-24*polylog(4,-d*x-c-(1+(d*x+c)^2)^(1/2))+4*arcsinh(d*x+c)^3*ln(1-d*x-c-( 1+(d*x+c)^2)^(1/2))+12*arcsinh(d*x+c)^2*polylog(2,d*x+c+(1+(d*x+c)^2)^(1/2 ))-24*arcsinh(d*x+c)*polylog(3,d*x+c+(1+(d*x+c)^2)^(1/2))+24*polylog(4,d*x +c+(1+(d*x+c)^2)^(1/2)))+4*a*b^3/e^2*(-1/(d*x+c)*arcsinh(d*x+c)^3-3*arcsin h(d*x+c)^2*ln(1+d*x+c+(1+(d*x+c)^2)^(1/2))-6*arcsinh(d*x+c)*polylog(2,-d*x -c-(1+(d*x+c)^2)^(1/2))+6*polylog(3,-d*x-c-(1+(d*x+c)^2)^(1/2))+3*arcsinh( d*x+c)^2*ln(1-d*x-c-(1+(d*x+c)^2)^(1/2))+6*arcsinh(d*x+c)*polylog(2,d*x+c+ (1+(d*x+c)^2)^(1/2))-6*polylog(3,d*x+c+(1+(d*x+c)^2)^(1/2)))+6*a^2*b^2/e^2 *(-1/(d*x+c)*arcsinh(d*x+c)^2-2*arcsinh(d*x+c)*ln(1+d*x+c+(1+(d*x+c)^2)^(1 /2))-2*polylog(2,-d*x-c-(1+(d*x+c)^2)^(1/2))+2*arcsinh(d*x+c)*ln(1-d*x-c-( 1+(d*x+c)^2)^(1/2))+2*polylog(2,d*x+c+(1+(d*x+c)^2)^(1/2)))+4*b*a^3/e^2*(- 1/(d*x+c)*arcsinh(d*x+c)-arctanh(1/(1+(d*x+c)^2)^(1/2))))
\[ \int \frac {(a+b \text {arcsinh}(c+d x))^4}{(c e+d e x)^2} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{4}}{{\left (d e x + c e\right )}^{2}} \,d x } \]
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^2*e^2*x^2 + 2*c*d* e^2*x + c^2*e^2), x)
\[ \int \frac {(a+b \text {arcsinh}(c+d x))^4}{(c e+d e x)^2} \, dx=\frac {\int \frac {a^{4}}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx + \int \frac {b^{4} \operatorname {asinh}^{4}{\left (c + d x \right )}}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx + \int \frac {4 a b^{3} \operatorname {asinh}^{3}{\left (c + d x \right )}}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx + \int \frac {6 a^{2} b^{2} \operatorname {asinh}^{2}{\left (c + d x \right )}}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx + \int \frac {4 a^{3} b \operatorname {asinh}{\left (c + d x \right )}}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx}{e^{2}} \]
(Integral(a**4/(c**2 + 2*c*d*x + d**2*x**2), x) + Integral(b**4*asinh(c + d*x)**4/(c**2 + 2*c*d*x + d**2*x**2), x) + Integral(4*a*b**3*asinh(c + d*x )**3/(c**2 + 2*c*d*x + d**2*x**2), x) + Integral(6*a**2*b**2*asinh(c + d*x )**2/(c**2 + 2*c*d*x + d**2*x**2), x) + Integral(4*a**3*b*asinh(c + d*x)/( c**2 + 2*c*d*x + d**2*x**2), x))/e**2
Exception generated. \[ \int \frac {(a+b \text {arcsinh}(c+d x))^4}{(c e+d e x)^2} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
\[ \int \frac {(a+b \text {arcsinh}(c+d x))^4}{(c e+d e x)^2} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{4}}{{\left (d e x + c e\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {(a+b \text {arcsinh}(c+d x))^4}{(c e+d e x)^2} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^4}{{\left (c\,e+d\,e\,x\right )}^2} \,d x \]