3.80 \(\int \frac {(a+b \cosh ^{-1}(c x)) \log (h (f+g x)^m)}{\sqrt {1-c^2 x^2}} \, dx\)

Optimal. Leaf size=600 \[ \frac {m \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )^3}{6 b^2 c \sqrt {1-c^2 x^2}}-\frac {m \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (-\frac {e^{\cosh ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{c \sqrt {1-c^2 x^2}}-\frac {m \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (-\frac {e^{\cosh ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{c \sqrt {1-c^2 x^2}}-\frac {m \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (\frac {g e^{\cosh ^{-1}(c x)}}{c f-\sqrt {c^2 f^2-g^2}}+1\right )}{2 b c \sqrt {1-c^2 x^2}}-\frac {m \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (\frac {g e^{\cosh ^{-1}(c x)}}{\sqrt {c^2 f^2-g^2}+c f}+1\right )}{2 b c \sqrt {1-c^2 x^2}}+\frac {\sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (h (f+g x)^m\right )}{2 b c \sqrt {1-c^2 x^2}}+\frac {b m \sqrt {c x-1} \sqrt {c x+1} \text {Li}_3\left (-\frac {e^{\cosh ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{c \sqrt {1-c^2 x^2}}+\frac {b m \sqrt {c x-1} \sqrt {c x+1} \text {Li}_3\left (-\frac {e^{\cosh ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{c \sqrt {1-c^2 x^2}} \]

[Out]

1/6*m*(a+b*arccosh(c*x))^3*(c*x-1)^(1/2)*(c*x+1)^(1/2)/b^2/c/(-c^2*x^2+1)^(1/2)+1/2*(a+b*arccosh(c*x))^2*ln(h*
(g*x+f)^m)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/b/c/(-c^2*x^2+1)^(1/2)-1/2*m*(a+b*arccosh(c*x))^2*ln(1+(c*x+(c*x-1)^(1/
2)*(c*x+1)^(1/2))*g/(c*f-(c^2*f^2-g^2)^(1/2)))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/b/c/(-c^2*x^2+1)^(1/2)-1/2*m*(a+b*a
rccosh(c*x))^2*ln(1+(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*g/(c*f+(c^2*f^2-g^2)^(1/2)))*(c*x-1)^(1/2)*(c*x+1)^(1/2)
/b/c/(-c^2*x^2+1)^(1/2)-m*(a+b*arccosh(c*x))*polylog(2,-(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*g/(c*f-(c^2*f^2-g^2)
^(1/2)))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c/(-c^2*x^2+1)^(1/2)-m*(a+b*arccosh(c*x))*polylog(2,-(c*x+(c*x-1)^(1/2)*(
c*x+1)^(1/2))*g/(c*f+(c^2*f^2-g^2)^(1/2)))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c/(-c^2*x^2+1)^(1/2)+b*m*polylog(3,-(c*
x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*g/(c*f-(c^2*f^2-g^2)^(1/2)))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c/(-c^2*x^2+1)^(1/2)+b
*m*polylog(3,-(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*g/(c*f+(c^2*f^2-g^2)^(1/2)))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c/(-c
^2*x^2+1)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 1.13, antiderivative size = 600, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 10, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.303, Rules used = {5713, 5676, 5841, 5839, 5800, 5562, 2190, 2531, 2282, 6589} \[ -\frac {m \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right ) \text {PolyLog}\left (2,-\frac {g e^{\cosh ^{-1}(c x)}}{c f-\sqrt {c^2 f^2-g^2}}\right )}{c \sqrt {1-c^2 x^2}}-\frac {m \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right ) \text {PolyLog}\left (2,-\frac {g e^{\cosh ^{-1}(c x)}}{\sqrt {c^2 f^2-g^2}+c f}\right )}{c \sqrt {1-c^2 x^2}}+\frac {b m \sqrt {c x-1} \sqrt {c x+1} \text {PolyLog}\left (3,-\frac {g e^{\cosh ^{-1}(c x)}}{c f-\sqrt {c^2 f^2-g^2}}\right )}{c \sqrt {1-c^2 x^2}}+\frac {b m \sqrt {c x-1} \sqrt {c x+1} \text {PolyLog}\left (3,-\frac {g e^{\cosh ^{-1}(c x)}}{\sqrt {c^2 f^2-g^2}+c f}\right )}{c \sqrt {1-c^2 x^2}}+\frac {m \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )^3}{6 b^2 c \sqrt {1-c^2 x^2}}-\frac {m \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (\frac {g e^{\cosh ^{-1}(c x)}}{c f-\sqrt {c^2 f^2-g^2}}+1\right )}{2 b c \sqrt {1-c^2 x^2}}-\frac {m \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (\frac {g e^{\cosh ^{-1}(c x)}}{\sqrt {c^2 f^2-g^2}+c f}+1\right )}{2 b c \sqrt {1-c^2 x^2}}+\frac {\sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (h (f+g x)^m\right )}{2 b c \sqrt {1-c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*ArcCosh[c*x])*Log[h*(f + g*x)^m])/Sqrt[1 - c^2*x^2],x]

[Out]

(m*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^3)/(6*b^2*c*Sqrt[1 - c^2*x^2]) - (m*Sqrt[-1 + c*x]*Sqrt[1
+ c*x]*(a + b*ArcCosh[c*x])^2*Log[1 + (E^ArcCosh[c*x]*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(2*b*c*Sqrt[1 - c^2*x^
2]) - (m*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^2*Log[1 + (E^ArcCosh[c*x]*g)/(c*f + Sqrt[c^2*f^2 -
g^2])])/(2*b*c*Sqrt[1 - c^2*x^2]) + (Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^2*Log[h*(f + g*x)^m])/(
2*b*c*Sqrt[1 - c^2*x^2]) - (m*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*PolyLog[2, -((E^ArcCosh[c*x]*g
)/(c*f - Sqrt[c^2*f^2 - g^2]))])/(c*Sqrt[1 - c^2*x^2]) - (m*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*
PolyLog[2, -((E^ArcCosh[c*x]*g)/(c*f + Sqrt[c^2*f^2 - g^2]))])/(c*Sqrt[1 - c^2*x^2]) + (b*m*Sqrt[-1 + c*x]*Sqr
t[1 + c*x]*PolyLog[3, -((E^ArcCosh[c*x]*g)/(c*f - Sqrt[c^2*f^2 - g^2]))])/(c*Sqrt[1 - c^2*x^2]) + (b*m*Sqrt[-1
+ c*x]*Sqrt[1 + c*x]*PolyLog[3, -((E^ArcCosh[c*x]*g)/(c*f + Sqrt[c^2*f^2 - g^2]))])/(c*Sqrt[1 - c^2*x^2])

Rule 2190

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*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), x]
- Dist[(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]

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2531

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] + Dist[(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]

Rule 5562

Int[(((e_.) + (f_.)*(x_))^(m_.)*Sinh[(c_.) + (d_.)*(x_)])/(Cosh[(c_.) + (d_.)*(x_)]*(b_.) + (a_)), 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]

Rule 5676

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol]
:> Simp[(a + b*ArcCosh[c*x])^(n + 1)/(b*c*Sqrt[-(d1*d2)]*(n + 1)), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n},
x] && EqQ[e1, c*d1] && EqQ[e2, -(c*d2)] && GtQ[d1, 0] && LtQ[d2, 0] && NeQ[n, -1]

Rule 5713

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Dist[((-d)^IntPart[p]*(
d + e*x^2)^FracPart[p])/((1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p]), Int[(1 + c*x)^p*(-1 + c*x)^p*(a + b*Ar
cCosh[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[c^2*d + e, 0] &&  !IntegerQ[p]

Rule 5800

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/((d_.) + (e_.)*(x_)), x_Symbol] :> Subst[Int[((a + b*x)^n*Sinh[x
])/(c*d + e*Cosh[x]), x], x, ArcCosh[c*x]] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[n, 0]

Rule 5839

Int[(Log[(h_.)*((f_.) + (g_.)*(x_))^(m_.)]*((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.))/(Sqrt[(d1_) + (e1_.)*(x_
)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol] :> Simp[(Log[h*(f + g*x)^m]*(a + b*ArcCosh[c*x])^(n + 1))/(b*c*Sqrt[-
(d1*d2)]*(n + 1)), x] - Dist[(g*m)/(b*c*Sqrt[-(d1*d2)]*(n + 1)), Int[(a + b*ArcCosh[c*x])^(n + 1)/(f + g*x), x
], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, g, h, m}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[d1,
0] && LtQ[d2, 0] && IGtQ[n, 0]

Rule 5841

Int[Log[(h_.)*((f_.) + (g_.)*(x_))^(m_.)]*((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_)
, x_Symbol] :> Dist[((-d)^IntPart[p]*(d + e*x^2)^FracPart[p])/((1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p]),
Int[Log[h*(f + g*x)^m]*(1 + c*x)^p*(-1 + c*x)^p*(a + b*ArcCosh[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g,
h, m, n}, x] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2]

Rule 6589

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> 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]

Rubi steps

\begin {align*} \int \frac {\left (a+b \cosh ^{-1}(c x)\right ) \log \left (h (f+g x)^m\right )}{\sqrt {1-c^2 x^2}} \, dx &=\frac {\left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {\left (a+b \cosh ^{-1}(c x)\right ) \log \left (h (f+g x)^m\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{\sqrt {1-c^2 x^2}}\\ &=\frac {\sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (h (f+g x)^m\right )}{2 b c \sqrt {1-c^2 x^2}}-\frac {\left (g m \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{f+g x} \, dx}{2 b c \sqrt {1-c^2 x^2}}\\ &=\frac {\sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (h (f+g x)^m\right )}{2 b c \sqrt {1-c^2 x^2}}-\frac {\left (g m \sqrt {-1+c x} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \frac {(a+b x)^2 \sinh (x)}{c f+g \cosh (x)} \, dx,x,\cosh ^{-1}(c x)\right )}{2 b c \sqrt {1-c^2 x^2}}\\ &=\frac {m \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^3}{6 b^2 c \sqrt {1-c^2 x^2}}+\frac {\sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (h (f+g x)^m\right )}{2 b c \sqrt {1-c^2 x^2}}-\frac {\left (g m \sqrt {-1+c x} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \frac {e^x (a+b x)^2}{c f+e^x g-\sqrt {c^2 f^2-g^2}} \, dx,x,\cosh ^{-1}(c x)\right )}{2 b c \sqrt {1-c^2 x^2}}-\frac {\left (g m \sqrt {-1+c x} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \frac {e^x (a+b x)^2}{c f+e^x g+\sqrt {c^2 f^2-g^2}} \, dx,x,\cosh ^{-1}(c x)\right )}{2 b c \sqrt {1-c^2 x^2}}\\ &=\frac {m \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^3}{6 b^2 c \sqrt {1-c^2 x^2}}-\frac {m \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (1+\frac {e^{\cosh ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{2 b c \sqrt {1-c^2 x^2}}-\frac {m \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (1+\frac {e^{\cosh ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{2 b c \sqrt {1-c^2 x^2}}+\frac {\sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (h (f+g x)^m\right )}{2 b c \sqrt {1-c^2 x^2}}+\frac {\left (m \sqrt {-1+c x} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int (a+b x) \log \left (1+\frac {e^x g}{c f-\sqrt {c^2 f^2-g^2}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{c \sqrt {1-c^2 x^2}}+\frac {\left (m \sqrt {-1+c x} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int (a+b x) \log \left (1+\frac {e^x g}{c f+\sqrt {c^2 f^2-g^2}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{c \sqrt {1-c^2 x^2}}\\ &=\frac {m \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^3}{6 b^2 c \sqrt {1-c^2 x^2}}-\frac {m \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (1+\frac {e^{\cosh ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{2 b c \sqrt {1-c^2 x^2}}-\frac {m \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (1+\frac {e^{\cosh ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{2 b c \sqrt {1-c^2 x^2}}+\frac {\sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (h (f+g x)^m\right )}{2 b c \sqrt {1-c^2 x^2}}-\frac {m \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (-\frac {e^{\cosh ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{c \sqrt {1-c^2 x^2}}-\frac {m \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (-\frac {e^{\cosh ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{c \sqrt {1-c^2 x^2}}+\frac {\left (b m \sqrt {-1+c x} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (-\frac {e^x g}{c f-\sqrt {c^2 f^2-g^2}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{c \sqrt {1-c^2 x^2}}+\frac {\left (b m \sqrt {-1+c x} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (-\frac {e^x g}{c f+\sqrt {c^2 f^2-g^2}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{c \sqrt {1-c^2 x^2}}\\ &=\frac {m \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^3}{6 b^2 c \sqrt {1-c^2 x^2}}-\frac {m \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (1+\frac {e^{\cosh ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{2 b c \sqrt {1-c^2 x^2}}-\frac {m \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (1+\frac {e^{\cosh ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{2 b c \sqrt {1-c^2 x^2}}+\frac {\sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (h (f+g x)^m\right )}{2 b c \sqrt {1-c^2 x^2}}-\frac {m \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (-\frac {e^{\cosh ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{c \sqrt {1-c^2 x^2}}-\frac {m \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (-\frac {e^{\cosh ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{c \sqrt {1-c^2 x^2}}+\frac {\left (b m \sqrt {-1+c x} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (\frac {g x}{-c f+\sqrt {c^2 f^2-g^2}}\right )}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{c \sqrt {1-c^2 x^2}}+\frac {\left (b m \sqrt {-1+c x} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {g x}{c f+\sqrt {c^2 f^2-g^2}}\right )}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{c \sqrt {1-c^2 x^2}}\\ &=\frac {m \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^3}{6 b^2 c \sqrt {1-c^2 x^2}}-\frac {m \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (1+\frac {e^{\cosh ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{2 b c \sqrt {1-c^2 x^2}}-\frac {m \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (1+\frac {e^{\cosh ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{2 b c \sqrt {1-c^2 x^2}}+\frac {\sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \log \left (h (f+g x)^m\right )}{2 b c \sqrt {1-c^2 x^2}}-\frac {m \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (-\frac {e^{\cosh ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{c \sqrt {1-c^2 x^2}}-\frac {m \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (-\frac {e^{\cosh ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{c \sqrt {1-c^2 x^2}}+\frac {b m \sqrt {-1+c x} \sqrt {1+c x} \text {Li}_3\left (-\frac {e^{\cosh ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{c \sqrt {1-c^2 x^2}}+\frac {b m \sqrt {-1+c x} \sqrt {1+c x} \text {Li}_3\left (-\frac {e^{\cosh ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{c \sqrt {1-c^2 x^2}}\\ \end {align*}

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Mathematica [F]  time = 1.94, size = 0, normalized size = 0.00 \[ \int \frac {\left (a+b \cosh ^{-1}(c x)\right ) \log \left (h (f+g x)^m\right )}{\sqrt {1-c^2 x^2}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[((a + b*ArcCosh[c*x])*Log[h*(f + g*x)^m])/Sqrt[1 - c^2*x^2],x]

[Out]

Integrate[((a + b*ArcCosh[c*x])*Log[h*(f + g*x)^m])/Sqrt[1 - c^2*x^2], x]

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fricas [F]  time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-c^{2} x^{2} + 1} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} \log \left ({\left (g x + f\right )}^{m} h\right )}{c^{2} x^{2} - 1}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(c*x))*log(h*(g*x+f)^m)/(-c^2*x^2+1)^(1/2),x, algorithm="fricas")

[Out]

integral(-sqrt(-c^2*x^2 + 1)*(b*arccosh(c*x) + a)*log((g*x + f)^m*h)/(c^2*x^2 - 1), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} \log \left ({\left (g x + f\right )}^{m} h\right )}{\sqrt {-c^{2} x^{2} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(c*x))*log(h*(g*x+f)^m)/(-c^2*x^2+1)^(1/2),x, algorithm="giac")

[Out]

integrate((b*arccosh(c*x) + a)*log((g*x + f)^m*h)/sqrt(-c^2*x^2 + 1), x)

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maple [F]  time = 4.15, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \,\mathrm {arccosh}\left (c x \right )\right ) \ln \left (h \left (g x +f \right )^{m}\right )}{\sqrt {-c^{2} x^{2}+1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arccosh(c*x))*ln(h*(g*x+f)^m)/(-c^2*x^2+1)^(1/2),x)

[Out]

int((a+b*arccosh(c*x))*ln(h*(g*x+f)^m)/(-c^2*x^2+1)^(1/2),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} \log \left ({\left (g x + f\right )}^{m} h\right )}{\sqrt {-c^{2} x^{2} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(c*x))*log(h*(g*x+f)^m)/(-c^2*x^2+1)^(1/2),x, algorithm="maxima")

[Out]

integrate((b*arccosh(c*x) + a)*log((g*x + f)^m*h)/sqrt(-c^2*x^2 + 1), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\ln \left (h\,{\left (f+g\,x\right )}^m\right )\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}{\sqrt {1-c^2\,x^2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(h*(f + g*x)^m)*(a + b*acosh(c*x)))/(1 - c^2*x^2)^(1/2),x)

[Out]

int((log(h*(f + g*x)^m)*(a + b*acosh(c*x)))/(1 - c^2*x^2)^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \operatorname {acosh}{\left (c x \right )}\right ) \log {\left (h \left (f + g x\right )^{m} \right )}}{\sqrt {- \left (c x - 1\right ) \left (c x + 1\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*acosh(c*x))*ln(h*(g*x+f)**m)/(-c**2*x**2+1)**(1/2),x)

[Out]

Integral((a + b*acosh(c*x))*log(h*(f + g*x)**m)/sqrt(-(c*x - 1)*(c*x + 1)), x)

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