Integrand size = 26, antiderivative size = 204 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^2 \left (d+c^2 d x^2\right )} \, dx=-\frac {(a+b \text {arcsinh}(c x))^2}{d x}-\frac {2 c (a+b \text {arcsinh}(c x))^2 \arctan \left (e^{\text {arcsinh}(c x)}\right )}{d}-\frac {4 b c (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{d}-\frac {2 b^2 c \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )}{d}+\frac {2 i b c (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{d}-\frac {2 i b c (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{d}+\frac {2 b^2 c \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )}{d}-\frac {2 i b^2 c \operatorname {PolyLog}\left (3,-i e^{\text {arcsinh}(c x)}\right )}{d}+\frac {2 i b^2 c \operatorname {PolyLog}\left (3,i e^{\text {arcsinh}(c x)}\right )}{d} \]
-(a+b*arcsinh(c*x))^2/d/x-2*c*(a+b*arcsinh(c*x))^2*arctan(c*x+(c^2*x^2+1)^ (1/2))/d-4*b*c*(a+b*arcsinh(c*x))*arctanh(c*x+(c^2*x^2+1)^(1/2))/d-2*b^2*c *polylog(2,-c*x-(c^2*x^2+1)^(1/2))/d+2*I*b*c*(a+b*arcsinh(c*x))*polylog(2, -I*(c*x+(c^2*x^2+1)^(1/2)))/d-2*I*b*c*(a+b*arcsinh(c*x))*polylog(2,I*(c*x+ (c^2*x^2+1)^(1/2)))/d+2*b^2*c*polylog(2,c*x+(c^2*x^2+1)^(1/2))/d-2*I*b^2*c *polylog(3,-I*(c*x+(c^2*x^2+1)^(1/2)))/d+2*I*b^2*c*polylog(3,I*(c*x+(c^2*x ^2+1)^(1/2)))/d
Time = 0.80 (sec) , antiderivative size = 363, normalized size of antiderivative = 1.78 \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^2 \left (d+c^2 d x^2\right )} \, dx=-\frac {\frac {a^2}{x}+\frac {2 a b \text {arcsinh}(c x)}{x}+a^2 c \arctan (c x)+2 a b c \text {arctanh}\left (\sqrt {1+c^2 x^2}\right )+\frac {1}{2} i a b c \left (\text {arcsinh}(c x) \left (\text {arcsinh}(c x)-4 \log \left (1+i e^{\text {arcsinh}(c x)}\right )\right )-4 \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )\right )-\frac {1}{2} i a b c \left (\text {arcsinh}(c x) \left (\text {arcsinh}(c x)-4 \log \left (1-i e^{\text {arcsinh}(c x)}\right )\right )-4 \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )\right )-b^2 c \left (-\frac {\text {arcsinh}(c x)^2}{c x}+2 \text {arcsinh}(c x) \log \left (1-e^{-\text {arcsinh}(c x)}\right )+i \text {arcsinh}(c x)^2 \log \left (1-i e^{-\text {arcsinh}(c x)}\right )-i \text {arcsinh}(c x)^2 \log \left (1+i e^{-\text {arcsinh}(c x)}\right )-2 \text {arcsinh}(c x) \log \left (1+e^{-\text {arcsinh}(c x)}\right )+2 \operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(c x)}\right )+2 i \text {arcsinh}(c x) \operatorname {PolyLog}\left (2,-i e^{-\text {arcsinh}(c x)}\right )-2 i \text {arcsinh}(c x) \operatorname {PolyLog}\left (2,i e^{-\text {arcsinh}(c x)}\right )-2 \operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(c x)}\right )+2 i \operatorname {PolyLog}\left (3,-i e^{-\text {arcsinh}(c x)}\right )-2 i \operatorname {PolyLog}\left (3,i e^{-\text {arcsinh}(c x)}\right )\right )}{d} \]
-((a^2/x + (2*a*b*ArcSinh[c*x])/x + a^2*c*ArcTan[c*x] + 2*a*b*c*ArcTanh[Sq rt[1 + c^2*x^2]] + (I/2)*a*b*c*(ArcSinh[c*x]*(ArcSinh[c*x] - 4*Log[1 + I*E ^ArcSinh[c*x]]) - 4*PolyLog[2, (-I)*E^ArcSinh[c*x]]) - (I/2)*a*b*c*(ArcSin h[c*x]*(ArcSinh[c*x] - 4*Log[1 - I*E^ArcSinh[c*x]]) - 4*PolyLog[2, I*E^Arc Sinh[c*x]]) - b^2*c*(-(ArcSinh[c*x]^2/(c*x)) + 2*ArcSinh[c*x]*Log[1 - E^(- ArcSinh[c*x])] + I*ArcSinh[c*x]^2*Log[1 - I/E^ArcSinh[c*x]] - I*ArcSinh[c* x]^2*Log[1 + I/E^ArcSinh[c*x]] - 2*ArcSinh[c*x]*Log[1 + E^(-ArcSinh[c*x])] + 2*PolyLog[2, -E^(-ArcSinh[c*x])] + (2*I)*ArcSinh[c*x]*PolyLog[2, (-I)/E ^ArcSinh[c*x]] - (2*I)*ArcSinh[c*x]*PolyLog[2, I/E^ArcSinh[c*x]] - 2*PolyL og[2, E^(-ArcSinh[c*x])] + (2*I)*PolyLog[3, (-I)/E^ArcSinh[c*x]] - (2*I)*P olyLog[3, I/E^ArcSinh[c*x]]))/d)
Time = 1.49 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.91, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {6224, 27, 6204, 3042, 4668, 3011, 2720, 6231, 3042, 26, 4670, 2715, 2838, 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.
| \(\displaystyle \int \frac {(a+b \text {arcsinh}(c x))^2}{x^2 \left (c^2 d x^2+d\right )} \, dx\) |
| \(\Big \downarrow \) 6224 |
| \(\displaystyle c^2 \left (-\int \frac {(a+b \text {arcsinh}(c x))^2}{d \left (c^2 x^2+1\right )}dx\right )+\frac {2 b c \int \frac {a+b \text {arcsinh}(c x)}{x \sqrt {c^2 x^2+1}}dx}{d}-\frac {(a+b \text {arcsinh}(c x))^2}{d x}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {c^2 \int \frac {(a+b \text {arcsinh}(c x))^2}{c^2 x^2+1}dx}{d}+\frac {2 b c \int \frac {a+b \text {arcsinh}(c x)}{x \sqrt {c^2 x^2+1}}dx}{d}-\frac {(a+b \text {arcsinh}(c x))^2}{d x}\) |
| \(\Big \downarrow \) 6204 |
| \(\displaystyle \frac {2 b c \int \frac {a+b \text {arcsinh}(c x)}{x \sqrt {c^2 x^2+1}}dx}{d}-\frac {c \int \frac {(a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}d\text {arcsinh}(c x)}{d}-\frac {(a+b \text {arcsinh}(c x))^2}{d x}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 b c \int \frac {a+b \text {arcsinh}(c x)}{x \sqrt {c^2 x^2+1}}dx}{d}-\frac {c \int (a+b \text {arcsinh}(c x))^2 \csc \left (i \text {arcsinh}(c x)+\frac {\pi }{2}\right )d\text {arcsinh}(c x)}{d}-\frac {(a+b \text {arcsinh}(c x))^2}{d x}\) |
| \(\Big \downarrow \) 4668 |
| \(\displaystyle -\frac {c \left (-2 i b \int (a+b \text {arcsinh}(c x)) \log \left (1-i e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)+2 i b \int (a+b \text {arcsinh}(c x)) \log \left (1+i e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2\right )}{d}+\frac {2 b c \int \frac {a+b \text {arcsinh}(c x)}{x \sqrt {c^2 x^2+1}}dx}{d}-\frac {(a+b \text {arcsinh}(c x))^2}{d x}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle -\frac {c \left (2 i b \left (b \int \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)-\operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-2 i b \left (b \int \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)-\operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2\right )}{d}+\frac {2 b c \int \frac {a+b \text {arcsinh}(c x)}{x \sqrt {c^2 x^2+1}}dx}{d}-\frac {(a+b \text {arcsinh}(c x))^2}{d x}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle -\frac {c \left (2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2\right )}{d}+\frac {2 b c \int \frac {a+b \text {arcsinh}(c x)}{x \sqrt {c^2 x^2+1}}dx}{d}-\frac {(a+b \text {arcsinh}(c x))^2}{d x}\) |
| \(\Big \downarrow \) 6231 |
| \(\displaystyle -\frac {c \left (2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2\right )}{d}+\frac {2 b c \int \frac {a+b \text {arcsinh}(c x)}{c x}d\text {arcsinh}(c x)}{d}-\frac {(a+b \text {arcsinh}(c x))^2}{d x}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {c \left (2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2\right )}{d}+\frac {2 b c \int i (a+b \text {arcsinh}(c x)) \csc (i \text {arcsinh}(c x))d\text {arcsinh}(c x)}{d}-\frac {(a+b \text {arcsinh}(c x))^2}{d x}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {c \left (2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2\right )}{d}+\frac {2 i b c \int (a+b \text {arcsinh}(c x)) \csc (i \text {arcsinh}(c x))d\text {arcsinh}(c x)}{d}-\frac {(a+b \text {arcsinh}(c x))^2}{d x}\) |
| \(\Big \downarrow \) 4670 |
| \(\displaystyle -\frac {c \left (2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2\right )}{d}+\frac {2 i b c \left (i b \int \log \left (1-e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)-i b \int \log \left (1+e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)+2 i \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )}{d}-\frac {(a+b \text {arcsinh}(c x))^2}{d x}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle -\frac {c \left (2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2\right )}{d}+\frac {2 i b c \left (i b \int e^{-\text {arcsinh}(c x)} \log \left (1-e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-i b \int e^{-\text {arcsinh}(c x)} \log \left (1+e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}+2 i \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )}{d}-\frac {(a+b \text {arcsinh}(c x))^2}{d x}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -\frac {c \left (2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2\right )}{d}+\frac {2 i b c \left (2 i \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )\right )}{d}-\frac {(a+b \text {arcsinh}(c x))^2}{d x}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle -\frac {c \left (2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2+2 i b \left (b \operatorname {PolyLog}\left (3,-i e^{\text {arcsinh}(c x)}\right )-\operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )-2 i b \left (b \operatorname {PolyLog}\left (3,i e^{\text {arcsinh}(c x)}\right )-\operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )\right )}{d}+\frac {2 i b c \left (2 i \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))+i b \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )\right )}{d}-\frac {(a+b \text {arcsinh}(c x))^2}{d x}\) |
-((a + b*ArcSinh[c*x])^2/(d*x)) + ((2*I)*b*c*((2*I)*(a + b*ArcSinh[c*x])*A rcTanh[E^ArcSinh[c*x]] + I*b*PolyLog[2, -E^ArcSinh[c*x]] - I*b*PolyLog[2, E^ArcSinh[c*x]]))/d - (c*(2*(a + b*ArcSinh[c*x])^2*ArcTan[E^ArcSinh[c*x]] + (2*I)*b*(-((a + b*ArcSinh[c*x])*PolyLog[2, (-I)*E^ArcSinh[c*x]]) + b*Pol yLog[3, (-I)*E^ArcSinh[c*x]]) - (2*I)*b*(-((a + b*ArcSinh[c*x])*PolyLog[2, I*E^ArcSinh[c*x]]) + b*PolyLog[3, I*E^ArcSinh[c*x]])))/d
3.3.31.3.1 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[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[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[(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[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_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_ ))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)/E^( I*k*Pi)]/(f*fz*I)), x] + (-Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[ 1 - E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x] + Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x]) /; FreeQ[{c , d, e, f, fz}, x] && IntegerQ[2*k] && IGtQ[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_) + (e_.)*(x_)^2), x_Symb ol] :> Simp[1/(c*d) Subst[Int[(a + b*x)^n*Sech[x], x], x, ArcSinh[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[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[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]
\[\int \frac {\left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )^{2}}{x^{2} \left (c^{2} d \,x^{2}+d \right )}d x\]
\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^2 \left (d+c^2 d x^2\right )} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} + d\right )} x^{2}} \,d x } \]
\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^2 \left (d+c^2 d x^2\right )} \, dx=\frac {\int \frac {a^{2}}{c^{2} x^{4} + x^{2}}\, dx + \int \frac {b^{2} \operatorname {asinh}^{2}{\left (c x \right )}}{c^{2} x^{4} + x^{2}}\, dx + \int \frac {2 a b \operatorname {asinh}{\left (c x \right )}}{c^{2} x^{4} + x^{2}}\, dx}{d} \]
(Integral(a**2/(c**2*x**4 + x**2), x) + Integral(b**2*asinh(c*x)**2/(c**2* x**4 + x**2), x) + Integral(2*a*b*asinh(c*x)/(c**2*x**4 + x**2), x))/d
\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^2 \left (d+c^2 d x^2\right )} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} + d\right )} x^{2}} \,d x } \]
-a^2*(c*arctan(c*x)/d + 1/(d*x)) + integrate(b^2*log(c*x + sqrt(c^2*x^2 + 1))^2/(c^2*d*x^4 + d*x^2) + 2*a*b*log(c*x + sqrt(c^2*x^2 + 1))/(c^2*d*x^4 + d*x^2), x)
\[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^2 \left (d+c^2 d x^2\right )} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} + d\right )} x^{2}} \,d x } \]
Timed out. \[ \int \frac {(a+b \text {arcsinh}(c x))^2}{x^2 \left (d+c^2 d x^2\right )} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{x^2\,\left (d\,c^2\,x^2+d\right )} \,d x \]