Integrand size = 26, antiderivative size = 194 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^3 \left (d+c^2 d x^2\right )} \, dx=-\frac {b c \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{d x}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d x^2}+\frac {2 c^2 (a+b \text {arcsinh}(c x))^2 \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right )}{d}+\frac {b^2 c^2 \log (x)}{d}+\frac {b c^2 (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{d}-\frac {b c^2 (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )}{d}-\frac {b^2 c^2 \operatorname {PolyLog}\left (3,-e^{2 \text {arcsinh}(c x)}\right )}{2 d}+\frac {b^2 c^2 \operatorname {PolyLog}\left (3,e^{2 \text {arcsinh}(c x)}\right )}{2 d} \]
-1/2*(a+b*arcsinh(c*x))^2/d/x^2+2*c^2*(a+b*arcsinh(c*x))^2*arctanh((c*x+(c ^2*x^2+1)^(1/2))^2)/d+b^2*c^2*ln(x)/d+b*c^2*(a+b*arcsinh(c*x))*polylog(2,- (c*x+(c^2*x^2+1)^(1/2))^2)/d-b*c^2*(a+b*arcsinh(c*x))*polylog(2,(c*x+(c^2* x^2+1)^(1/2))^2)/d-1/2*b^2*c^2*polylog(3,-(c*x+(c^2*x^2+1)^(1/2))^2)/d+1/2 *b^2*c^2*polylog(3,(c*x+(c^2*x^2+1)^(1/2))^2)/d-b*c*(a+b*arcsinh(c*x))*(c^ 2*x^2+1)^(1/2)/d/x
Result contains complex when optimal does not.
Time = 1.04 (sec) , antiderivative size = 411, normalized size of antiderivative = 2.12 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^3 \left (d+c^2 d x^2\right )} \, dx=-\frac {\frac {a^2}{x^2}+\frac {2 a b \left (c x \sqrt {1+c^2 x^2}+\text {arcsinh}(c x)\right )}{x^2}+2 a^2 c^2 \log (x)-a^2 c^2 \log \left (1+c^2 x^2\right )+a b c^2 \left (\text {arcsinh}(c x) \left (\text {arcsinh}(c x)-4 \log \left (1+i e^{\text {arcsinh}(c x)}\right )\right )-4 \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )\right )+a b c^2 \left (\text {arcsinh}(c x) \left (\text {arcsinh}(c x)-4 \log \left (1-i e^{\text {arcsinh}(c x)}\right )\right )-4 \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )\right )-2 a b c^2 \left (\text {arcsinh}(c x) \left (\text {arcsinh}(c x)-2 \log \left (1-e^{2 \text {arcsinh}(c x)}\right )\right )-\operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )\right )-2 b^2 c^2 \left (-\frac {i \pi ^3}{24}-\frac {\sqrt {1+c^2 x^2} \text {arcsinh}(c x)}{c x}-\frac {\text {arcsinh}(c x)^2}{2 c^2 x^2}+\frac {2}{3} \text {arcsinh}(c x)^3+\text {arcsinh}(c x)^2 \log \left (1+e^{-2 \text {arcsinh}(c x)}\right )-\text {arcsinh}(c x)^2 \log \left (1-e^{2 \text {arcsinh}(c x)}\right )+\log \left (\frac {c x}{\sqrt {1+c^2 x^2}}\right )+\frac {1}{2} \log \left (1+c^2 x^2\right )-\text {arcsinh}(c x) \operatorname {PolyLog}\left (2,-e^{-2 \text {arcsinh}(c x)}\right )-\text {arcsinh}(c x) \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )-\frac {1}{2} \operatorname {PolyLog}\left (3,-e^{-2 \text {arcsinh}(c x)}\right )+\frac {1}{2} \operatorname {PolyLog}\left (3,e^{2 \text {arcsinh}(c x)}\right )\right )}{2 d} \]
-1/2*(a^2/x^2 + (2*a*b*(c*x*Sqrt[1 + c^2*x^2] + ArcSinh[c*x]))/x^2 + 2*a^2 *c^2*Log[x] - a^2*c^2*Log[1 + c^2*x^2] + a*b*c^2*(ArcSinh[c*x]*(ArcSinh[c* x] - 4*Log[1 + I*E^ArcSinh[c*x]]) - 4*PolyLog[2, (-I)*E^ArcSinh[c*x]]) + a *b*c^2*(ArcSinh[c*x]*(ArcSinh[c*x] - 4*Log[1 - I*E^ArcSinh[c*x]]) - 4*Poly Log[2, I*E^ArcSinh[c*x]]) - 2*a*b*c^2*(ArcSinh[c*x]*(ArcSinh[c*x] - 2*Log[ 1 - E^(2*ArcSinh[c*x])]) - PolyLog[2, E^(2*ArcSinh[c*x])]) - 2*b^2*c^2*((- 1/24*I)*Pi^3 - (Sqrt[1 + c^2*x^2]*ArcSinh[c*x])/(c*x) - ArcSinh[c*x]^2/(2* c^2*x^2) + (2*ArcSinh[c*x]^3)/3 + ArcSinh[c*x]^2*Log[1 + E^(-2*ArcSinh[c*x ])] - ArcSinh[c*x]^2*Log[1 - E^(2*ArcSinh[c*x])] + Log[(c*x)/Sqrt[1 + c^2* x^2]] + Log[1 + c^2*x^2]/2 - ArcSinh[c*x]*PolyLog[2, -E^(-2*ArcSinh[c*x])] - ArcSinh[c*x]*PolyLog[2, E^(2*ArcSinh[c*x])] - PolyLog[3, -E^(-2*ArcSinh [c*x])]/2 + PolyLog[3, E^(2*ArcSinh[c*x])]/2))/d
Result contains complex when optimal does not.
Time = 1.44 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.94, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {6224, 27, 6214, 5984, 3042, 26, 4670, 3011, 2720, 6215, 14, 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.
| \(\displaystyle \int \frac {(a+b \text {arcsinh}(c x))^2}{x^3 \left (c^2 d x^2+d\right )} \, dx\) |
| \(\Big \downarrow \) 6224 |
| \(\displaystyle c^2 \left (-\int \frac {(a+b \text {arcsinh}(c x))^2}{d x \left (c^2 x^2+1\right )}dx\right )+\frac {b c \int \frac {a+b \text {arcsinh}(c x)}{x^2 \sqrt {c^2 x^2+1}}dx}{d}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d x^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {c^2 \int \frac {(a+b \text {arcsinh}(c x))^2}{x \left (c^2 x^2+1\right )}dx}{d}+\frac {b c \int \frac {a+b \text {arcsinh}(c x)}{x^2 \sqrt {c^2 x^2+1}}dx}{d}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d x^2}\) |
| \(\Big \downarrow \) 6214 |
| \(\displaystyle -\frac {c^2 \int \frac {(a+b \text {arcsinh}(c x))^2}{c x \sqrt {c^2 x^2+1}}d\text {arcsinh}(c x)}{d}+\frac {b c \int \frac {a+b \text {arcsinh}(c x)}{x^2 \sqrt {c^2 x^2+1}}dx}{d}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d x^2}\) |
| \(\Big \downarrow \) 5984 |
| \(\displaystyle \frac {b c \int \frac {a+b \text {arcsinh}(c x)}{x^2 \sqrt {c^2 x^2+1}}dx}{d}-\frac {2 c^2 \int (a+b \text {arcsinh}(c x))^2 \text {csch}(2 \text {arcsinh}(c x))d\text {arcsinh}(c x)}{d}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d x^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {b c \int \frac {a+b \text {arcsinh}(c x)}{x^2 \sqrt {c^2 x^2+1}}dx}{d}-\frac {2 c^2 \int i (a+b \text {arcsinh}(c x))^2 \csc (2 i \text {arcsinh}(c x))d\text {arcsinh}(c x)}{d}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d x^2}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {b c \int \frac {a+b \text {arcsinh}(c x)}{x^2 \sqrt {c^2 x^2+1}}dx}{d}-\frac {2 i c^2 \int (a+b \text {arcsinh}(c x))^2 \csc (2 i \text {arcsinh}(c x))d\text {arcsinh}(c x)}{d}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d x^2}\) |
| \(\Big \downarrow \) 4670 |
| \(\displaystyle -\frac {2 i c^2 \left (i b \int (a+b \text {arcsinh}(c x)) \log \left (1-e^{2 \text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)-i b \int (a+b \text {arcsinh}(c x)) \log \left (1+e^{2 \text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)+i \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2\right )}{d}+\frac {b c \int \frac {a+b \text {arcsinh}(c x)}{x^2 \sqrt {c^2 x^2+1}}dx}{d}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d x^2}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle -\frac {2 i c^2 \left (-i b \left (\frac {1}{2} b \int \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)-\frac {1}{2} \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+i b \left (\frac {1}{2} b \int \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)-\frac {1}{2} \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+i \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2\right )}{d}+\frac {b c \int \frac {a+b \text {arcsinh}(c x)}{x^2 \sqrt {c^2 x^2+1}}dx}{d}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d x^2}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle -\frac {2 i c^2 \left (-i b \left (\frac {1}{4} b \int e^{-2 \text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )de^{2 \text {arcsinh}(c x)}-\frac {1}{2} \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+i b \left (\frac {1}{4} b \int e^{-2 \text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )de^{2 \text {arcsinh}(c x)}-\frac {1}{2} \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+i \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2\right )}{d}+\frac {b c \int \frac {a+b \text {arcsinh}(c x)}{x^2 \sqrt {c^2 x^2+1}}dx}{d}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d x^2}\) |
| \(\Big \downarrow \) 6215 |
| \(\displaystyle -\frac {2 i c^2 \left (-i b \left (\frac {1}{4} b \int e^{-2 \text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )de^{2 \text {arcsinh}(c x)}-\frac {1}{2} \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+i b \left (\frac {1}{4} b \int e^{-2 \text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )de^{2 \text {arcsinh}(c x)}-\frac {1}{2} \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+i \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2\right )}{d}+\frac {b c \left (b c \int \frac {1}{x}dx-\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{x}\right )}{d}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d x^2}\) |
| \(\Big \downarrow \) 14 |
| \(\displaystyle -\frac {2 i c^2 \left (-i b \left (\frac {1}{4} b \int e^{-2 \text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )de^{2 \text {arcsinh}(c x)}-\frac {1}{2} \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+i b \left (\frac {1}{4} b \int e^{-2 \text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )de^{2 \text {arcsinh}(c x)}-\frac {1}{2} \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+i \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2\right )}{d}+\frac {b c \left (b c \log (x)-\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{x}\right )}{d}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d x^2}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle -\frac {2 i c^2 \left (i \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2-i b \left (\frac {1}{4} b \operatorname {PolyLog}\left (3,-e^{2 \text {arcsinh}(c x)}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+i b \left (\frac {1}{4} b \operatorname {PolyLog}\left (3,e^{2 \text {arcsinh}(c x)}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )\right )}{d}+\frac {b c \left (b c \log (x)-\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{x}\right )}{d}-\frac {(a+b \text {arcsinh}(c x))^2}{2 d x^2}\) |
-1/2*(a + b*ArcSinh[c*x])^2/(d*x^2) + (b*c*(-((Sqrt[1 + c^2*x^2]*(a + b*Ar cSinh[c*x]))/x) + b*c*Log[x]))/d - ((2*I)*c^2*(I*(a + b*ArcSinh[c*x])^2*Ar cTanh[E^(2*ArcSinh[c*x])] - I*b*(-1/2*((a + b*ArcSinh[c*x])*PolyLog[2, -E^ (2*ArcSinh[c*x])]) + (b*PolyLog[3, -E^(2*ArcSinh[c*x])])/4) + I*b*(-1/2*(( a + b*ArcSinh[c*x])*PolyLog[2, E^(2*ArcSinh[c*x])]) + (b*PolyLog[3, E^(2*A rcSinh[c*x])])/4)))/d
3.3.32.3.1 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[Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Simp[2^n Int[(c + d*x)^m*Csch[2*a + 2*b*x ]^n, x], x] /; FreeQ[{a, b, c, d}, x] && RationalQ[m] && IntegerQ[n]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/((x_)*((d_) + (e_.)*(x_)^2)), x_Symbol] :> Simp[1/d Subst[Int[(a + b*x)^n/(Cosh[x]*Sinh[x]), x], x, Ar cSinh[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[n, 0]
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]
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[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]
Leaf count of result is larger than twice the leaf count of optimal. \(539\) vs. \(2(231)=462\).
Time = 0.27 (sec) , antiderivative size = 540, normalized size of antiderivative = 2.78
| method | result | size |
| derivativedivides | \(c^{2} \left (\frac {a^{2} \left (-\frac {1}{2 c^{2} x^{2}}-\ln \left (c x \right )+\frac {\ln \left (c^{2} x^{2}+1\right )}{2}\right )}{d}+\frac {b^{2} \left (-\frac {\operatorname {arcsinh}\left (c x \right ) \left (-2 c^{2} x^{2}+2 c x \sqrt {c^{2} x^{2}+1}+\operatorname {arcsinh}\left (c x \right )\right )}{2 c^{2} x^{2}}+\ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )-2 \ln \left (c x +\sqrt {c^{2} x^{2}+1}\right )+\ln \left (c x +\sqrt {c^{2} x^{2}+1}-1\right )-\operatorname {arcsinh}\left (c x \right )^{2} \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )-2 \,\operatorname {arcsinh}\left (c x \right ) \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (3, -c x -\sqrt {c^{2} x^{2}+1}\right )+\operatorname {arcsinh}\left (c x \right )^{2} \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+\operatorname {arcsinh}\left (c x \right ) \operatorname {polylog}\left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )-\frac {\operatorname {polylog}\left (3, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2}-\operatorname {arcsinh}\left (c x \right )^{2} \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )-2 \,\operatorname {arcsinh}\left (c x \right ) \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (3, c x +\sqrt {c^{2} x^{2}+1}\right )\right )}{d}+\frac {2 a b \left (-\frac {c x \sqrt {c^{2} x^{2}+1}-c^{2} x^{2}+\operatorname {arcsinh}\left (c x \right )}{2 c^{2} x^{2}}-\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )-\operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+\operatorname {arcsinh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2}-\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )-\operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )\right )}{d}\right )\) | \(540\) |
| default | \(c^{2} \left (\frac {a^{2} \left (-\frac {1}{2 c^{2} x^{2}}-\ln \left (c x \right )+\frac {\ln \left (c^{2} x^{2}+1\right )}{2}\right )}{d}+\frac {b^{2} \left (-\frac {\operatorname {arcsinh}\left (c x \right ) \left (-2 c^{2} x^{2}+2 c x \sqrt {c^{2} x^{2}+1}+\operatorname {arcsinh}\left (c x \right )\right )}{2 c^{2} x^{2}}+\ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )-2 \ln \left (c x +\sqrt {c^{2} x^{2}+1}\right )+\ln \left (c x +\sqrt {c^{2} x^{2}+1}-1\right )-\operatorname {arcsinh}\left (c x \right )^{2} \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )-2 \,\operatorname {arcsinh}\left (c x \right ) \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (3, -c x -\sqrt {c^{2} x^{2}+1}\right )+\operatorname {arcsinh}\left (c x \right )^{2} \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+\operatorname {arcsinh}\left (c x \right ) \operatorname {polylog}\left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )-\frac {\operatorname {polylog}\left (3, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2}-\operatorname {arcsinh}\left (c x \right )^{2} \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )-2 \,\operatorname {arcsinh}\left (c x \right ) \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (3, c x +\sqrt {c^{2} x^{2}+1}\right )\right )}{d}+\frac {2 a b \left (-\frac {c x \sqrt {c^{2} x^{2}+1}-c^{2} x^{2}+\operatorname {arcsinh}\left (c x \right )}{2 c^{2} x^{2}}-\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )-\operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+\operatorname {arcsinh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2}-\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )-\operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )\right )}{d}\right )\) | \(540\) |
| parts | \(\frac {a^{2} \left (\frac {c^{2} \ln \left (c^{2} x^{2}+1\right )}{2}-\frac {1}{2 x^{2}}-c^{2} \ln \left (x \right )\right )}{d}+\frac {b^{2} c^{2} \left (-\frac {\operatorname {arcsinh}\left (c x \right ) \left (-2 c^{2} x^{2}+2 c x \sqrt {c^{2} x^{2}+1}+\operatorname {arcsinh}\left (c x \right )\right )}{2 c^{2} x^{2}}+\ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )-2 \ln \left (c x +\sqrt {c^{2} x^{2}+1}\right )+\ln \left (c x +\sqrt {c^{2} x^{2}+1}-1\right )-\operatorname {arcsinh}\left (c x \right )^{2} \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )-2 \,\operatorname {arcsinh}\left (c x \right ) \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (3, -c x -\sqrt {c^{2} x^{2}+1}\right )+\operatorname {arcsinh}\left (c x \right )^{2} \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+\operatorname {arcsinh}\left (c x \right ) \operatorname {polylog}\left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )-\frac {\operatorname {polylog}\left (3, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2}-\operatorname {arcsinh}\left (c x \right )^{2} \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )-2 \,\operatorname {arcsinh}\left (c x \right ) \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )+2 \operatorname {polylog}\left (3, c x +\sqrt {c^{2} x^{2}+1}\right )\right )}{d}+\frac {2 a b \,c^{2} \left (-\frac {c x \sqrt {c^{2} x^{2}+1}-c^{2} x^{2}+\operatorname {arcsinh}\left (c x \right )}{2 c^{2} x^{2}}-\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )-\operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+\operatorname {arcsinh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2}-\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )-\operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )\right )}{d}\) | \(543\) |
c^2*(a^2/d*(-1/2/c^2/x^2-ln(c*x)+1/2*ln(c^2*x^2+1))+b^2/d*(-1/2*arcsinh(c* x)*(-2*c^2*x^2+2*c*x*(c^2*x^2+1)^(1/2)+arcsinh(c*x))/c^2/x^2+ln(1+c*x+(c^2 *x^2+1)^(1/2))-2*ln(c*x+(c^2*x^2+1)^(1/2))+ln(c*x+(c^2*x^2+1)^(1/2)-1)-arc sinh(c*x)^2*ln(1+c*x+(c^2*x^2+1)^(1/2))-2*arcsinh(c*x)*polylog(2,-c*x-(c^2 *x^2+1)^(1/2))+2*polylog(3,-c*x-(c^2*x^2+1)^(1/2))+arcsinh(c*x)^2*ln(1+(c* x+(c^2*x^2+1)^(1/2))^2)+arcsinh(c*x)*polylog(2,-(c*x+(c^2*x^2+1)^(1/2))^2) -1/2*polylog(3,-(c*x+(c^2*x^2+1)^(1/2))^2)-arcsinh(c*x)^2*ln(1-c*x-(c^2*x^ 2+1)^(1/2))-2*arcsinh(c*x)*polylog(2,c*x+(c^2*x^2+1)^(1/2))+2*polylog(3,c* x+(c^2*x^2+1)^(1/2)))+2*a*b/d*(-1/2*(c*x*(c^2*x^2+1)^(1/2)-c^2*x^2+arcsinh (c*x))/c^2/x^2-arcsinh(c*x)*ln(1+c*x+(c^2*x^2+1)^(1/2))-polylog(2,-c*x-(c^ 2*x^2+1)^(1/2))+arcsinh(c*x)*ln(1+(c*x+(c^2*x^2+1)^(1/2))^2)+1/2*polylog(2 ,-(c*x+(c^2*x^2+1)^(1/2))^2)-arcsinh(c*x)*ln(1-c*x-(c^2*x^2+1)^(1/2))-poly log(2,c*x+(c^2*x^2+1)^(1/2))))
\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^3 \left (d+c^2 d x^2\right )} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} + d\right )} x^{3}} \,d x } \]
\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^3 \left (d+c^2 d x^2\right )} \, dx=\frac {\int \frac {a^{2}}{c^{2} x^{5} + x^{3}}\, dx + \int \frac {b^{2} \operatorname {asinh}^{2}{\left (c x \right )}}{c^{2} x^{5} + x^{3}}\, dx + \int \frac {2 a b \operatorname {asinh}{\left (c x \right )}}{c^{2} x^{5} + x^{3}}\, dx}{d} \]
(Integral(a**2/(c**2*x**5 + x**3), x) + Integral(b**2*asinh(c*x)**2/(c**2* x**5 + x**3), x) + Integral(2*a*b*asinh(c*x)/(c**2*x**5 + x**3), x))/d
\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^3 \left (d+c^2 d x^2\right )} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} + d\right )} x^{3}} \,d x } \]
1/2*(c^2*log(c^2*x^2 + 1)/d - 2*c^2*log(x)/d - 1/(d*x^2))*a^2 + integrate( b^2*log(c*x + sqrt(c^2*x^2 + 1))^2/(c^2*d*x^5 + d*x^3) + 2*a*b*log(c*x + s qrt(c^2*x^2 + 1))/(c^2*d*x^5 + d*x^3), x)
\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^3 \left (d+c^2 d x^2\right )} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} + d\right )} x^{3}} \,d x } \]
Timed out. \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^3 \left (d+c^2 d x^2\right )} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{x^3\,\left (d\,c^2\,x^2+d\right )} \,d x \]