Integrand size = 23, antiderivative size = 169 \[ \int \frac {(a+b \text {arcsinh}(c+d x))^2}{(c e+d e x)^4} \, dx=-\frac {b^2}{3 d e^4 (c+d x)}-\frac {b \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))}{3 d e^4 (c+d x)^2}-\frac {(a+b \text {arcsinh}(c+d x))^2}{3 d e^4 (c+d x)^3}+\frac {2 b (a+b \text {arcsinh}(c+d x)) \text {arctanh}\left (e^{\text {arcsinh}(c+d x)}\right )}{3 d e^4}+\frac {b^2 \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c+d x)}\right )}{3 d e^4}-\frac {b^2 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c+d x)}\right )}{3 d e^4} \]
-1/3*b^2/d/e^4/(d*x+c)-1/3*(a+b*arcsinh(d*x+c))^2/d/e^4/(d*x+c)^3+2/3*b*(a +b*arcsinh(d*x+c))*arctanh(d*x+c+(1+(d*x+c)^2)^(1/2))/d/e^4+1/3*b^2*polylo g(2,-d*x-c-(1+(d*x+c)^2)^(1/2))/d/e^4-1/3*b^2*polylog(2,d*x+c+(1+(d*x+c)^2 )^(1/2))/d/e^4-1/3*b*(a+b*arcsinh(d*x+c))*(1+(d*x+c)^2)^(1/2)/d/e^4/(d*x+c )^2
Time = 1.31 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.36 \[ \int \frac {(a+b \text {arcsinh}(c+d x))^2}{(c e+d e x)^4} \, dx=-\frac {4 a^2+a b \left (8 \text {arcsinh}(c+d x)+2 \sinh (2 \text {arcsinh}(c+d x))+\left (\log \left (\cosh \left (\frac {1}{2} \text {arcsinh}(c+d x)\right )\right )-\log \left (\sinh \left (\frac {1}{2} \text {arcsinh}(c+d x)\right )\right )\right ) (3 c+3 d x-\sinh (3 \text {arcsinh}(c+d x)))\right )+b^2 \left (4 (c+d x)^2+4 \text {arcsinh}(c+d x)^2+4 (c+d x)^3 \operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(c+d x)}\right )-4 (c+d x)^3 \operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(c+d x)}\right )+\text {arcsinh}(c+d x) \left (2 \sinh (2 \text {arcsinh}(c+d x))+\left (\log \left (1-e^{-\text {arcsinh}(c+d x)}\right )-\log \left (1+e^{-\text {arcsinh}(c+d x)}\right )\right ) (-3 (c+d x)+\sinh (3 \text {arcsinh}(c+d x)))\right )\right )}{12 d e^4 (c+d x)^3} \]
-1/12*(4*a^2 + a*b*(8*ArcSinh[c + d*x] + 2*Sinh[2*ArcSinh[c + d*x]] + (Log [Cosh[ArcSinh[c + d*x]/2]] - Log[Sinh[ArcSinh[c + d*x]/2]])*(3*c + 3*d*x - Sinh[3*ArcSinh[c + d*x]])) + b^2*(4*(c + d*x)^2 + 4*ArcSinh[c + d*x]^2 + 4*(c + d*x)^3*PolyLog[2, -E^(-ArcSinh[c + d*x])] - 4*(c + d*x)^3*PolyLog[2 , E^(-ArcSinh[c + d*x])] + ArcSinh[c + d*x]*(2*Sinh[2*ArcSinh[c + d*x]] + (Log[1 - E^(-ArcSinh[c + d*x])] - Log[1 + E^(-ArcSinh[c + d*x])])*(-3*(c + d*x) + Sinh[3*ArcSinh[c + d*x]]))))/(d*e^4*(c + d*x)^3)
Result contains complex when optimal does not.
Time = 0.73 (sec) , antiderivative size = 143, 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, 6224, 15, 6231, 3042, 26, 4670, 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.
| \(\displaystyle \int \frac {(a+b \text {arcsinh}(c+d x))^2}{(c e+d e x)^4} \, dx\) |
| \(\Big \downarrow \) 6274 |
| \(\displaystyle \frac {\int \frac {(a+b \text {arcsinh}(c+d x))^2}{e^4 (c+d x)^4}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(a+b \text {arcsinh}(c+d x))^2}{(c+d x)^4}d(c+d x)}{d e^4}\) |
| \(\Big \downarrow \) 6191 |
| \(\displaystyle \frac {\frac {2}{3} b \int \frac {a+b \text {arcsinh}(c+d x)}{(c+d x)^3 \sqrt {(c+d x)^2+1}}d(c+d x)-\frac {(a+b \text {arcsinh}(c+d x))^2}{3 (c+d x)^3}}{d e^4}\) |
| \(\Big \downarrow \) 6224 |
| \(\displaystyle \frac {\frac {2}{3} b \left (-\frac {1}{2} \int \frac {a+b \text {arcsinh}(c+d x)}{(c+d x) \sqrt {(c+d x)^2+1}}d(c+d x)+\frac {1}{2} b \int \frac {1}{(c+d x)^2}d(c+d x)-\frac {\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))}{2 (c+d x)^2}\right )-\frac {(a+b \text {arcsinh}(c+d x))^2}{3 (c+d x)^3}}{d e^4}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {\frac {2}{3} b \left (-\frac {1}{2} \int \frac {a+b \text {arcsinh}(c+d x)}{(c+d x) \sqrt {(c+d x)^2+1}}d(c+d x)-\frac {\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))}{2 (c+d x)^2}-\frac {b}{2 (c+d x)}\right )-\frac {(a+b \text {arcsinh}(c+d x))^2}{3 (c+d x)^3}}{d e^4}\) |
| \(\Big \downarrow \) 6231 |
| \(\displaystyle \frac {\frac {2}{3} b \left (-\frac {1}{2} \int \frac {a+b \text {arcsinh}(c+d x)}{c+d x}d\text {arcsinh}(c+d x)-\frac {\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))}{2 (c+d x)^2}-\frac {b}{2 (c+d x)}\right )-\frac {(a+b \text {arcsinh}(c+d x))^2}{3 (c+d x)^3}}{d e^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {(a+b \text {arcsinh}(c+d x))^2}{3 (c+d x)^3}+\frac {2}{3} b \left (-\frac {1}{2} \int i (a+b \text {arcsinh}(c+d x)) \csc (i \text {arcsinh}(c+d x))d\text {arcsinh}(c+d x)-\frac {\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))}{2 (c+d x)^2}-\frac {b}{2 (c+d x)}\right )}{d e^4}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {-\frac {(a+b \text {arcsinh}(c+d x))^2}{3 (c+d x)^3}+\frac {2}{3} b \left (-\frac {1}{2} i \int (a+b \text {arcsinh}(c+d x)) \csc (i \text {arcsinh}(c+d x))d\text {arcsinh}(c+d x)-\frac {\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))}{2 (c+d x)^2}-\frac {b}{2 (c+d x)}\right )}{d e^4}\) |
| \(\Big \downarrow \) 4670 |
| \(\displaystyle \frac {-\frac {(a+b \text {arcsinh}(c+d x))^2}{3 (c+d x)^3}+\frac {2}{3} b \left (-\frac {1}{2} i \left (i b \int \log \left (1-e^{\text {arcsinh}(c+d x)}\right )d\text {arcsinh}(c+d x)-i b \int \log \left (1+e^{\text {arcsinh}(c+d x)}\right )d\text {arcsinh}(c+d x)+2 i \text {arctanh}\left (e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))\right )-\frac {\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))}{2 (c+d x)^2}-\frac {b}{2 (c+d x)}\right )}{d e^4}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {-\frac {(a+b \text {arcsinh}(c+d x))^2}{3 (c+d x)^3}+\frac {2}{3} b \left (-\frac {1}{2} i \left (-i b \int e^{-\text {arcsinh}(c+d x)} \log \left (1+e^{\text {arcsinh}(c+d x)}\right )de^{\text {arcsinh}(c+d x)}+i b \int e^{-\text {arcsinh}(c+d x)} \log (-c-d x+1)de^{\text {arcsinh}(c+d x)}+2 i \text {arctanh}\left (e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))\right )-\frac {\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))}{2 (c+d x)^2}-\frac {b}{2 (c+d x)}\right )}{d e^4}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {-\frac {(a+b \text {arcsinh}(c+d x))^2}{3 (c+d x)^3}+\frac {2}{3} b \left (-\frac {1}{2} i \left (2 i \text {arctanh}\left (e^{\text {arcsinh}(c+d x)}\right ) (a+b \text {arcsinh}(c+d x))-i b \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c+d x)}\right )+i b \operatorname {PolyLog}(2,-c-d x)\right )-\frac {\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))}{2 (c+d x)^2}-\frac {b}{2 (c+d x)}\right )}{d e^4}\) |
(-1/3*(a + b*ArcSinh[c + d*x])^2/(c + d*x)^3 + (2*b*(-1/2*b/(c + d*x) - (S qrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x]))/(2*(c + d*x)^2) - (I/2)*((2 *I)*(a + b*ArcSinh[c + d*x])*ArcTanh[E^ArcSinh[c + d*x]] - I*b*PolyLog[2, E^ArcSinh[c + d*x]] + I*b*PolyLog[2, -c - d*x])))/3)/(d*e^4)
3.2.35.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
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[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]
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]
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_.)*((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[c^2*((m + 2*p + 3)/(f^2*(m + 1))) Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n, x], x] - Sim p[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, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && ILtQ[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]
Time = 0.53 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.40
| method | result | size |
| derivativedivides | \(\frac {-\frac {a^{2}}{3 e^{4} \left (d x +c \right )^{3}}+\frac {b^{2} \left (-\frac {\sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )+\operatorname {arcsinh}\left (d x +c \right )^{2}+\left (d x +c \right )^{2}}{3 \left (d x +c \right )^{3}}+\frac {\operatorname {arcsinh}\left (d x +c \right ) \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{3}+\frac {\operatorname {polylog}\left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )}{3}-\frac {\operatorname {arcsinh}\left (d x +c \right ) \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )}{3}-\frac {\operatorname {polylog}\left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{3}\right )}{e^{4}}+\frac {2 a b \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )}{3 \left (d x +c \right )^{3}}-\frac {\sqrt {1+\left (d x +c \right )^{2}}}{6 \left (d x +c \right )^{2}}+\frac {\operatorname {arctanh}\left (\frac {1}{\sqrt {1+\left (d x +c \right )^{2}}}\right )}{6}\right )}{e^{4}}}{d}\) | \(236\) |
| default | \(\frac {-\frac {a^{2}}{3 e^{4} \left (d x +c \right )^{3}}+\frac {b^{2} \left (-\frac {\sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )+\operatorname {arcsinh}\left (d x +c \right )^{2}+\left (d x +c \right )^{2}}{3 \left (d x +c \right )^{3}}+\frac {\operatorname {arcsinh}\left (d x +c \right ) \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{3}+\frac {\operatorname {polylog}\left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )}{3}-\frac {\operatorname {arcsinh}\left (d x +c \right ) \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )}{3}-\frac {\operatorname {polylog}\left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{3}\right )}{e^{4}}+\frac {2 a b \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )}{3 \left (d x +c \right )^{3}}-\frac {\sqrt {1+\left (d x +c \right )^{2}}}{6 \left (d x +c \right )^{2}}+\frac {\operatorname {arctanh}\left (\frac {1}{\sqrt {1+\left (d x +c \right )^{2}}}\right )}{6}\right )}{e^{4}}}{d}\) | \(236\) |
| parts | \(-\frac {a^{2}}{3 e^{4} \left (d x +c \right )^{3} d}+\frac {b^{2} \left (-\frac {\sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )+\operatorname {arcsinh}\left (d x +c \right )^{2}+\left (d x +c \right )^{2}}{3 \left (d x +c \right )^{3}}+\frac {\operatorname {arcsinh}\left (d x +c \right ) \ln \left (1+d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{3}+\frac {\operatorname {polylog}\left (2, -d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )}{3}-\frac {\operatorname {arcsinh}\left (d x +c \right ) \ln \left (1-d x -c -\sqrt {1+\left (d x +c \right )^{2}}\right )}{3}-\frac {\operatorname {polylog}\left (2, d x +c +\sqrt {1+\left (d x +c \right )^{2}}\right )}{3}\right )}{e^{4} d}+\frac {2 a b \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )}{3 \left (d x +c \right )^{3}}-\frac {\sqrt {1+\left (d x +c \right )^{2}}}{6 \left (d x +c \right )^{2}}+\frac {\operatorname {arctanh}\left (\frac {1}{\sqrt {1+\left (d x +c \right )^{2}}}\right )}{6}\right )}{e^{4} d}\) | \(241\) |
1/d*(-1/3*a^2/e^4/(d*x+c)^3+b^2/e^4*(-1/3*((1+(d*x+c)^2)^(1/2)*(d*x+c)*arc sinh(d*x+c)+arcsinh(d*x+c)^2+(d*x+c)^2)/(d*x+c)^3+1/3*arcsinh(d*x+c)*ln(1+ d*x+c+(1+(d*x+c)^2)^(1/2))+1/3*polylog(2,-d*x-c-(1+(d*x+c)^2)^(1/2))-1/3*a rcsinh(d*x+c)*ln(1-d*x-c-(1+(d*x+c)^2)^(1/2))-1/3*polylog(2,d*x+c+(1+(d*x+ c)^2)^(1/2)))+2*a*b/e^4*(-1/3/(d*x+c)^3*arcsinh(d*x+c)-1/6/(d*x+c)^2*(1+(d *x+c)^2)^(1/2)+1/6*arctanh(1/(1+(d*x+c)^2)^(1/2))))
\[ \int \frac {(a+b \text {arcsinh}(c+d x))^2}{(c e+d e x)^4} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{2}}{{\left (d e x + c e\right )}^{4}} \,d x } \]
integral((b^2*arcsinh(d*x + c)^2 + 2*a*b*arcsinh(d*x + c) + a^2)/(d^4*e^4* x^4 + 4*c*d^3*e^4*x^3 + 6*c^2*d^2*e^4*x^2 + 4*c^3*d*e^4*x + c^4*e^4), x)
\[ \int \frac {(a+b \text {arcsinh}(c+d x))^2}{(c e+d e x)^4} \, dx=\frac {\int \frac {a^{2}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx + \int \frac {b^{2} \operatorname {asinh}^{2}{\left (c + d x \right )}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx + \int \frac {2 a b \operatorname {asinh}{\left (c + d x \right )}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx}{e^{4}} \]
(Integral(a**2/(c**4 + 4*c**3*d*x + 6*c**2*d**2*x**2 + 4*c*d**3*x**3 + d** 4*x**4), x) + Integral(b**2*asinh(c + d*x)**2/(c**4 + 4*c**3*d*x + 6*c**2* d**2*x**2 + 4*c*d**3*x**3 + d**4*x**4), x) + Integral(2*a*b*asinh(c + d*x) /(c**4 + 4*c**3*d*x + 6*c**2*d**2*x**2 + 4*c*d**3*x**3 + d**4*x**4), x))/e **4
\[ \int \frac {(a+b \text {arcsinh}(c+d x))^2}{(c e+d e x)^4} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{2}}{{\left (d e x + c e\right )}^{4}} \,d x } \]
-1/3*b^2*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))^2/(d^4*e^4*x^3 + 3*c*d^3*e^4*x^2 + 3*c^2*d^2*e^4*x + c^3*d*e^4) - 1/3*a^2/(d^4*e^4*x^3 + 3 *c*d^3*e^4*x^2 + 3*c^2*d^2*e^4*x + c^3*d*e^4) + integrate(2/3*((3*a*b*d^3 + b^2*d^3)*x^3 + 3*(c^3 + c)*a*b + (c^3 + c)*b^2 + 3*(3*a*b*c*d^2 + b^2*c* d^2)*x^2 + (3*(3*c^2*d + d)*a*b + (3*c^2*d + d)*b^2)*x + (b^2*c^2 + 3*(c^2 + 1)*a*b + (3*a*b*d^2 + b^2*d^2)*x^2 + 2*(3*a*b*c*d + b^2*c*d)*x)*sqrt(d^ 2*x^2 + 2*c*d*x + c^2 + 1))*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1 ))/(d^7*e^4*x^7 + 7*c*d^6*e^4*x^6 + c^7*e^4 + c^5*e^4 + (21*c^2*d^5*e^4 + d^5*e^4)*x^5 + 5*(7*c^3*d^4*e^4 + c*d^4*e^4)*x^4 + 5*(7*c^4*d^3*e^4 + 2*c^ 2*d^3*e^4)*x^3 + (21*c^5*d^2*e^4 + 10*c^3*d^2*e^4)*x^2 + (7*c^6*d*e^4 + 5* c^4*d*e^4)*x + (d^6*e^4*x^6 + 6*c*d^5*e^4*x^5 + c^6*e^4 + c^4*e^4 + (15*c^ 2*d^4*e^4 + d^4*e^4)*x^4 + 4*(5*c^3*d^3*e^4 + c*d^3*e^4)*x^3 + 3*(5*c^4*d^ 2*e^4 + 2*c^2*d^2*e^4)*x^2 + 2*(3*c^5*d*e^4 + 2*c^3*d*e^4)*x)*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)), x)
\[ \int \frac {(a+b \text {arcsinh}(c+d x))^2}{(c e+d e x)^4} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{2}}{{\left (d e x + c e\right )}^{4}} \,d x } \]
Timed out. \[ \int \frac {(a+b \text {arcsinh}(c+d x))^2}{(c e+d e x)^4} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^2}{{\left (c\,e+d\,e\,x\right )}^4} \,d x \]