Integrand size = 34, antiderivative size = 438 \[ \int \frac {(a+b \text {arcsinh}(c x))^2 \log \left (h (f+g x)^m\right )}{\sqrt {1+c^2 x^2}} \, dx=\frac {m (a+b \text {arcsinh}(c x))^4}{12 b^2 c}-\frac {m (a+b \text {arcsinh}(c x))^3 \log \left (1+\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{3 b c}-\frac {m (a+b \text {arcsinh}(c x))^3 \log \left (1+\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{3 b c}+\frac {(a+b \text {arcsinh}(c x))^3 \log \left (h (f+g x)^m\right )}{3 b c}-\frac {m (a+b \text {arcsinh}(c x))^2 \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{c}-\frac {m (a+b \text {arcsinh}(c x))^2 \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{c}+\frac {2 b m (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (3,-\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{c}+\frac {2 b m (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (3,-\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{c}-\frac {2 b^2 m \operatorname {PolyLog}\left (4,-\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{c}-\frac {2 b^2 m \operatorname {PolyLog}\left (4,-\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{c} \]
1/12*m*(a+b*arcsinh(c*x))^4/b^2/c+1/3*(a+b*arcsinh(c*x))^3*ln(h*(g*x+f)^m) /b/c-1/3*m*(a+b*arcsinh(c*x))^3*ln(1+(c*x+(c^2*x^2+1)^(1/2))*g/(c*f-(c^2*f ^2+g^2)^(1/2)))/b/c-1/3*m*(a+b*arcsinh(c*x))^3*ln(1+(c*x+(c^2*x^2+1)^(1/2) )*g/(c*f+(c^2*f^2+g^2)^(1/2)))/b/c-m*(a+b*arcsinh(c*x))^2*polylog(2,-(c*x+ (c^2*x^2+1)^(1/2))*g/(c*f-(c^2*f^2+g^2)^(1/2)))/c-m*(a+b*arcsinh(c*x))^2*p olylog(2,-(c*x+(c^2*x^2+1)^(1/2))*g/(c*f+(c^2*f^2+g^2)^(1/2)))/c+2*b*m*(a+ b*arcsinh(c*x))*polylog(3,-(c*x+(c^2*x^2+1)^(1/2))*g/(c*f-(c^2*f^2+g^2)^(1 /2)))/c+2*b*m*(a+b*arcsinh(c*x))*polylog(3,-(c*x+(c^2*x^2+1)^(1/2))*g/(c*f +(c^2*f^2+g^2)^(1/2)))/c-2*b^2*m*polylog(4,-(c*x+(c^2*x^2+1)^(1/2))*g/(c*f -(c^2*f^2+g^2)^(1/2)))/c-2*b^2*m*polylog(4,-(c*x+(c^2*x^2+1)^(1/2))*g/(c*f +(c^2*f^2+g^2)^(1/2)))/c
Time = 0.14 (sec) , antiderivative size = 397, normalized size of antiderivative = 0.91 \[ \int \frac {(a+b \text {arcsinh}(c x))^2 \log \left (h (f+g x)^m\right )}{\sqrt {1+c^2 x^2}} \, dx=-\frac {-\frac {m (a+b \text {arcsinh}(c x))^4}{4 b}+m (a+b \text {arcsinh}(c x))^3 \log \left (1+\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )+m (a+b \text {arcsinh}(c x))^3 \log \left (1+\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )-(a+b \text {arcsinh}(c x))^3 \log \left (h (f+g x)^m\right )+3 b m \left ((a+b \text {arcsinh}(c x))^2 \operatorname {PolyLog}\left (2,\frac {e^{\text {arcsinh}(c x)} g}{-c f+\sqrt {c^2 f^2+g^2}}\right )-2 b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (3,\frac {e^{\text {arcsinh}(c x)} g}{-c f+\sqrt {c^2 f^2+g^2}}\right )+2 b^2 \operatorname {PolyLog}\left (4,\frac {e^{\text {arcsinh}(c x)} g}{-c f+\sqrt {c^2 f^2+g^2}}\right )\right )+3 b m \left ((a+b \text {arcsinh}(c x))^2 \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )-2 b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (3,-\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )+2 b^2 \operatorname {PolyLog}\left (4,-\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )\right )}{3 b c} \]
-1/3*(-1/4*(m*(a + b*ArcSinh[c*x])^4)/b + m*(a + b*ArcSinh[c*x])^3*Log[1 + (E^ArcSinh[c*x]*g)/(c*f - Sqrt[c^2*f^2 + g^2])] + m*(a + b*ArcSinh[c*x])^ 3*Log[1 + (E^ArcSinh[c*x]*g)/(c*f + Sqrt[c^2*f^2 + g^2])] - (a + b*ArcSinh [c*x])^3*Log[h*(f + g*x)^m] + 3*b*m*((a + b*ArcSinh[c*x])^2*PolyLog[2, (E^ ArcSinh[c*x]*g)/(-(c*f) + Sqrt[c^2*f^2 + g^2])] - 2*b*(a + b*ArcSinh[c*x]) *PolyLog[3, (E^ArcSinh[c*x]*g)/(-(c*f) + Sqrt[c^2*f^2 + g^2])] + 2*b^2*Pol yLog[4, (E^ArcSinh[c*x]*g)/(-(c*f) + Sqrt[c^2*f^2 + g^2])]) + 3*b*m*((a + b*ArcSinh[c*x])^2*PolyLog[2, -((E^ArcSinh[c*x]*g)/(c*f + Sqrt[c^2*f^2 + g^ 2]))] - 2*b*(a + b*ArcSinh[c*x])*PolyLog[3, -((E^ArcSinh[c*x]*g)/(c*f + Sq rt[c^2*f^2 + g^2]))] + 2*b^2*PolyLog[4, -((E^ArcSinh[c*x]*g)/(c*f + Sqrt[c ^2*f^2 + g^2]))]))/(b*c)
Time = 1.71 (sec) , antiderivative size = 426, normalized size of antiderivative = 0.97, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {6261, 6242, 6095, 2620, 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.
\(\displaystyle \int \frac {(a+b \text {arcsinh}(c x))^2 \log \left (h (f+g x)^m\right )}{\sqrt {c^2 x^2+1}} \, dx\) |
\(\Big \downarrow \) 6261 |
\(\displaystyle \frac {(a+b \text {arcsinh}(c x))^3 \log \left (h (f+g x)^m\right )}{3 b c}-\frac {g m \int \frac {(a+b \text {arcsinh}(c x))^3}{f+g x}dx}{3 b c}\) |
\(\Big \downarrow \) 6242 |
\(\displaystyle \frac {(a+b \text {arcsinh}(c x))^3 \log \left (h (f+g x)^m\right )}{3 b c}-\frac {g m \int \frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^3}{c f+c g x}d\text {arcsinh}(c x)}{3 b c}\) |
\(\Big \downarrow \) 6095 |
\(\displaystyle \frac {(a+b \text {arcsinh}(c x))^3 \log \left (h (f+g x)^m\right )}{3 b c}-\frac {g m \left (\int \frac {e^{\text {arcsinh}(c x)} (a+b \text {arcsinh}(c x))^3}{c f+e^{\text {arcsinh}(c x)} g-\sqrt {c^2 f^2+g^2}}d\text {arcsinh}(c x)+\int \frac {e^{\text {arcsinh}(c x)} (a+b \text {arcsinh}(c x))^3}{c f+e^{\text {arcsinh}(c x)} g+\sqrt {c^2 f^2+g^2}}d\text {arcsinh}(c x)-\frac {(a+b \text {arcsinh}(c x))^4}{4 b g}\right )}{3 b c}\) |
\(\Big \downarrow \) 2620 |
\(\displaystyle \frac {(a+b \text {arcsinh}(c x))^3 \log \left (h (f+g x)^m\right )}{3 b c}-\frac {g m \left (-\frac {3 b \int (a+b \text {arcsinh}(c x))^2 \log \left (\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}+1\right )d\text {arcsinh}(c x)}{g}-\frac {3 b \int (a+b \text {arcsinh}(c x))^2 \log \left (\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}+1\right )d\text {arcsinh}(c x)}{g}+\frac {(a+b \text {arcsinh}(c x))^3 \log \left (\frac {g e^{\text {arcsinh}(c x)}}{c f-\sqrt {c^2 f^2+g^2}}+1\right )}{g}+\frac {(a+b \text {arcsinh}(c x))^3 \log \left (\frac {g e^{\text {arcsinh}(c x)}}{\sqrt {c^2 f^2+g^2}+c f}+1\right )}{g}-\frac {(a+b \text {arcsinh}(c x))^4}{4 b g}\right )}{3 b c}\) |
\(\Big \downarrow \) 3011 |
\(\displaystyle \frac {(a+b \text {arcsinh}(c x))^3 \log \left (h (f+g x)^m\right )}{3 b c}-\frac {g m \left (-\frac {3 b \left (2 b \int (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )d\text {arcsinh}(c x)-(a+b \text {arcsinh}(c x))^2 \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )\right )}{g}-\frac {3 b \left (2 b \int (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )d\text {arcsinh}(c x)-(a+b \text {arcsinh}(c x))^2 \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )\right )}{g}+\frac {(a+b \text {arcsinh}(c x))^3 \log \left (\frac {g e^{\text {arcsinh}(c x)}}{c f-\sqrt {c^2 f^2+g^2}}+1\right )}{g}+\frac {(a+b \text {arcsinh}(c x))^3 \log \left (\frac {g e^{\text {arcsinh}(c x)}}{\sqrt {c^2 f^2+g^2}+c f}+1\right )}{g}-\frac {(a+b \text {arcsinh}(c x))^4}{4 b g}\right )}{3 b c}\) |
\(\Big \downarrow \) 7163 |
\(\displaystyle \frac {(a+b \text {arcsinh}(c x))^3 \log \left (h (f+g x)^m\right )}{3 b c}-\frac {g m \left (-\frac {3 b \left (2 b \left ((a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (3,-\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )-b \int \operatorname {PolyLog}\left (3,-\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )d\text {arcsinh}(c x)\right )-(a+b \text {arcsinh}(c x))^2 \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )\right )}{g}-\frac {3 b \left (2 b \left ((a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (3,-\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )-b \int \operatorname {PolyLog}\left (3,-\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )d\text {arcsinh}(c x)\right )-(a+b \text {arcsinh}(c x))^2 \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )\right )}{g}+\frac {(a+b \text {arcsinh}(c x))^3 \log \left (\frac {g e^{\text {arcsinh}(c x)}}{c f-\sqrt {c^2 f^2+g^2}}+1\right )}{g}+\frac {(a+b \text {arcsinh}(c x))^3 \log \left (\frac {g e^{\text {arcsinh}(c x)}}{\sqrt {c^2 f^2+g^2}+c f}+1\right )}{g}-\frac {(a+b \text {arcsinh}(c x))^4}{4 b g}\right )}{3 b c}\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle \frac {(a+b \text {arcsinh}(c x))^3 \log \left (h (f+g x)^m\right )}{3 b c}-\frac {g m \left (-\frac {3 b \left (2 b \left ((a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (3,-\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )-b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (3,-\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )de^{\text {arcsinh}(c x)}\right )-(a+b \text {arcsinh}(c x))^2 \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )\right )}{g}-\frac {3 b \left (2 b \left ((a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (3,-\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )-b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (3,-\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )de^{\text {arcsinh}(c x)}\right )-(a+b \text {arcsinh}(c x))^2 \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )\right )}{g}+\frac {(a+b \text {arcsinh}(c x))^3 \log \left (\frac {g e^{\text {arcsinh}(c x)}}{c f-\sqrt {c^2 f^2+g^2}}+1\right )}{g}+\frac {(a+b \text {arcsinh}(c x))^3 \log \left (\frac {g e^{\text {arcsinh}(c x)}}{\sqrt {c^2 f^2+g^2}+c f}+1\right )}{g}-\frac {(a+b \text {arcsinh}(c x))^4}{4 b g}\right )}{3 b c}\) |
\(\Big \downarrow \) 7143 |
\(\displaystyle \frac {(a+b \text {arcsinh}(c x))^3 \log \left (h (f+g x)^m\right )}{3 b c}-\frac {g m \left (-\frac {3 b \left (2 b \left ((a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (3,-\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )-b \operatorname {PolyLog}\left (4,-\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )\right )-(a+b \text {arcsinh}(c x))^2 \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )\right )}{g}-\frac {3 b \left (2 b \left ((a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (3,-\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )-b \operatorname {PolyLog}\left (4,-\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )\right )-(a+b \text {arcsinh}(c x))^2 \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )\right )}{g}+\frac {(a+b \text {arcsinh}(c x))^3 \log \left (\frac {g e^{\text {arcsinh}(c x)}}{c f-\sqrt {c^2 f^2+g^2}}+1\right )}{g}+\frac {(a+b \text {arcsinh}(c x))^3 \log \left (\frac {g e^{\text {arcsinh}(c x)}}{\sqrt {c^2 f^2+g^2}+c f}+1\right )}{g}-\frac {(a+b \text {arcsinh}(c x))^4}{4 b g}\right )}{3 b c}\) |
((a + b*ArcSinh[c*x])^3*Log[h*(f + g*x)^m])/(3*b*c) - (g*m*(-1/4*(a + b*Ar cSinh[c*x])^4/(b*g) + ((a + b*ArcSinh[c*x])^3*Log[1 + (E^ArcSinh[c*x]*g)/( c*f - Sqrt[c^2*f^2 + g^2])])/g + ((a + b*ArcSinh[c*x])^3*Log[1 + (E^ArcSin h[c*x]*g)/(c*f + Sqrt[c^2*f^2 + g^2])])/g - (3*b*(-((a + b*ArcSinh[c*x])^2 *PolyLog[2, -((E^ArcSinh[c*x]*g)/(c*f - Sqrt[c^2*f^2 + g^2]))]) + 2*b*((a + b*ArcSinh[c*x])*PolyLog[3, -((E^ArcSinh[c*x]*g)/(c*f - Sqrt[c^2*f^2 + g^ 2]))] - b*PolyLog[4, -((E^ArcSinh[c*x]*g)/(c*f - Sqrt[c^2*f^2 + g^2]))]))) /g - (3*b*(-((a + b*ArcSinh[c*x])^2*PolyLog[2, -((E^ArcSinh[c*x]*g)/(c*f + Sqrt[c^2*f^2 + g^2]))]) + 2*b*((a + b*ArcSinh[c*x])*PolyLog[3, -((E^ArcSi nh[c*x]*g)/(c*f + Sqrt[c^2*f^2 + g^2]))] - b*PolyLog[4, -((E^ArcSinh[c*x]* g)/(c*f + Sqrt[c^2*f^2 + g^2]))])))/g))/(3*b*c)
3.1.54.3.1 Defintions of rubi rules used
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]
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[(Cosh[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sin h[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[-(e + f*x)^(m + 1)/(b*f*(m + 1)), x] + (Int[(e + f*x)^m*(E^(c + d*x)/(a - Rt[a^2 + b^2, 2] + b*E^(c + d*x))) , x] + Int[(e + f*x)^m*(E^(c + d*x)/(a + Rt[a^2 + b^2, 2] + b*E^(c + d*x))) , x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && NeQ[a^2 + b^2, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/((d_.) + (e_.)*(x_)), x_Symbo l] :> Subst[Int[(a + b*x)^n*(Cosh[x]/(c*d + e*Sinh[x])), x], x, ArcSinh[c*x ]] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[n, 0]
Int[(Log[(h_.)*((f_.) + (g_.)*(x_))^(m_.)]*((a_.) + ArcSinh[(c_.)*(x_)]*(b_ .))^(n_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[Log[h*(f + g*x)^m]* ((a + b*ArcSinh[c*x])^(n + 1)/(b*c*Sqrt[d]*(n + 1))), x] - Simp[g*(m/(b*c*S qrt[d]*(n + 1))) Int[(a + b*ArcSinh[c*x])^(n + 1)/(f + g*x), x], x] /; Fr eeQ[{a, b, c, d, e, f, g, h, m}, x] && EqQ[e, c^2*d] && GtQ[d, 0] && IGtQ[n , 0]
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]
\[\int \frac {\left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )^{2} \ln \left (h \left (g x +f \right )^{m}\right )}{\sqrt {c^{2} x^{2}+1}}d x\]
\[ \int \frac {(a+b \text {arcsinh}(c x))^2 \log \left (h (f+g x)^m\right )}{\sqrt {1+c^2 x^2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} \log \left ({\left (g x + f\right )}^{m} h\right )}{\sqrt {c^{2} x^{2} + 1}} \,d x } \]
integral((b^2*arcsinh(c*x)^2 + 2*a*b*arcsinh(c*x) + a^2)*log((g*x + f)^m*h )/sqrt(c^2*x^2 + 1), x)
\[ \int \frac {(a+b \text {arcsinh}(c x))^2 \log \left (h (f+g x)^m\right )}{\sqrt {1+c^2 x^2}} \, dx=\int \frac {\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2} \log {\left (h \left (f + g x\right )^{m} \right )}}{\sqrt {c^{2} x^{2} + 1}}\, dx \]
\[ \int \frac {(a+b \text {arcsinh}(c x))^2 \log \left (h (f+g x)^m\right )}{\sqrt {1+c^2 x^2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} \log \left ({\left (g x + f\right )}^{m} h\right )}{\sqrt {c^{2} x^{2} + 1}} \,d x } \]
Exception generated. \[ \int \frac {(a+b \text {arcsinh}(c x))^2 \log \left (h (f+g x)^m\right )}{\sqrt {1+c^2 x^2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro unding error%%%{720,[0,4,1,1,1,1,4,0]%%%}+%%%{-1260,[0,4,1,1,1,1,3,1]%%%}+ %%%{360,[
Timed out. \[ \int \frac {(a+b \text {arcsinh}(c x))^2 \log \left (h (f+g x)^m\right )}{\sqrt {1+c^2 x^2}} \, dx=\int \frac {\ln \left (h\,{\left (f+g\,x\right )}^m\right )\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{\sqrt {c^2\,x^2+1}} \,d x \]