3.5.96 \(\int \frac {a+b \cosh ^{-1}(c x)}{x^3 (d+e x^2)} \, dx\) [496]

Optimal. Leaf size=550 \[ \frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{2 d x}-\frac {a+b \cosh ^{-1}(c x)}{2 d x^2}-\frac {e \left (a+b \cosh ^{-1}(c x)\right )^2}{b d^2}-\frac {e \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{-2 \cosh ^{-1}(c x)}\right )}{d^2}+\frac {e \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 d^2}+\frac {e \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 d^2}+\frac {e \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 d^2}+\frac {e \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 d^2}+\frac {b e \text {PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c x)}\right )}{2 d^2}+\frac {b e \text {PolyLog}\left (2,-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 d^2}+\frac {b e \text {PolyLog}\left (2,\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 d^2}+\frac {b e \text {PolyLog}\left (2,-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 d^2}+\frac {b e \text {PolyLog}\left (2,\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 d^2} \]

[Out]

1/2*(-a-b*arccosh(c*x))/d/x^2-e*(a+b*arccosh(c*x))^2/b/d^2-e*(a+b*arccosh(c*x))*ln(1+1/(c*x+(c*x-1)^(1/2)*(c*x
+1)^(1/2))^2)/d^2+1/2*e*(a+b*arccosh(c*x))*ln(1-(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*e^(1/2)/(c*(-d)^(1/2)-(-c^2*
d-e)^(1/2)))/d^2+1/2*e*(a+b*arccosh(c*x))*ln(1+(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*e^(1/2)/(c*(-d)^(1/2)-(-c^2*d
-e)^(1/2)))/d^2+1/2*e*(a+b*arccosh(c*x))*ln(1-(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*e^(1/2)/(c*(-d)^(1/2)+(-c^2*d-
e)^(1/2)))/d^2+1/2*e*(a+b*arccosh(c*x))*ln(1+(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*e^(1/2)/(c*(-d)^(1/2)+(-c^2*d-e
)^(1/2)))/d^2+1/2*b*e*polylog(2,-1/(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)/d^2+1/2*b*e*polylog(2,-(c*x+(c*x-1)^(1
/2)*(c*x+1)^(1/2))*e^(1/2)/(c*(-d)^(1/2)-(-c^2*d-e)^(1/2)))/d^2+1/2*b*e*polylog(2,(c*x+(c*x-1)^(1/2)*(c*x+1)^(
1/2))*e^(1/2)/(c*(-d)^(1/2)-(-c^2*d-e)^(1/2)))/d^2+1/2*b*e*polylog(2,-(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*e^(1/2
)/(c*(-d)^(1/2)+(-c^2*d-e)^(1/2)))/d^2+1/2*b*e*polylog(2,(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*e^(1/2)/(c*(-d)^(1/
2)+(-c^2*d-e)^(1/2)))/d^2+1/2*b*c*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d/x

________________________________________________________________________________________

Rubi [A]
time = 0.71, antiderivative size = 550, normalized size of antiderivative = 1.00, number of steps used = 27, number of rules used = 10, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {5959, 5883, 97, 5882, 3799, 2221, 2317, 2438, 5962, 5681} \begin {gather*} \frac {e \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {c^2 (-d)-e}}\right )}{2 d^2}+\frac {e \left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {c^2 (-d)-e}}+1\right )}{2 d^2}+\frac {e \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {c^2 (-d)-e}+c \sqrt {-d}}\right )}{2 d^2}+\frac {e \left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {c^2 (-d)-e}+c \sqrt {-d}}+1\right )}{2 d^2}-\frac {e \left (a+b \cosh ^{-1}(c x)\right )^2}{b d^2}-\frac {e \log \left (e^{-2 \cosh ^{-1}(c x)}+1\right ) \left (a+b \cosh ^{-1}(c x)\right )}{d^2}-\frac {a+b \cosh ^{-1}(c x)}{2 d x^2}+\frac {b e \text {Li}_2\left (-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right )}{2 d^2}+\frac {b e \text {Li}_2\left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right )}{2 d^2}+\frac {b e \text {Li}_2\left (-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )}{2 d^2}+\frac {b e \text {Li}_2\left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )}{2 d^2}+\frac {b e \text {Li}_2\left (-e^{-2 \cosh ^{-1}(c x)}\right )}{2 d^2}+\frac {b c \sqrt {c x-1} \sqrt {c x+1}}{2 d x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcCosh[c*x])/(x^3*(d + e*x^2)),x]

[Out]

(b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(2*d*x) - (a + b*ArcCosh[c*x])/(2*d*x^2) - (e*(a + b*ArcCosh[c*x])^2)/(b*d^
2) - (e*(a + b*ArcCosh[c*x])*Log[1 + E^(-2*ArcCosh[c*x])])/d^2 + (e*(a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^Ar
cCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(2*d^2) + (e*(a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c
*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(2*d^2) + (e*(a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c
*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/(2*d^2) + (e*(a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-
d] + Sqrt[-(c^2*d) - e])])/(2*d^2) + (b*e*PolyLog[2, -E^(-2*ArcCosh[c*x])])/(2*d^2) + (b*e*PolyLog[2, -((Sqrt[
e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e]))])/(2*d^2) + (b*e*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*
Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(2*d^2) + (b*e*PolyLog[2, -((Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2
*d) - e]))])/(2*d^2) + (b*e*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/(2*d^2)

Rule 97

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] /; FreeQ[{a, b, c, d,
e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && EqQ[a*d*f*(m + 1) + b*c*f*(n + 1) + b*d*e*(p + 1), 0
] && NeQ[m, -1]

Rule 2221

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]
- 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 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[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]

Rule 2438

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]

Rule 3799

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (Complex[0, fz_])*(f_.)*(x_)], x_Symbol] :> Simp[(-I)*((c + d*x)^(m
+ 1)/(d*(m + 1))), x] + Dist[2*I, Int[(c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*fz*x)))), x]
, x] /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]

Rule 5681

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 5882

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Dist[1/b, Subst[Int[x^n*Tanh[-a/b + x/b], x],
x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0]

Rule 5883

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcC
osh[c*x])^n/(d*(m + 1))), x] - Dist[b*c*(n/(d*(m + 1))), Int[(d*x)^(m + 1)*((a + b*ArcCosh[c*x])^(n - 1)/(Sqrt
[1 + c*x]*Sqrt[-1 + c*x])), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 5959

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Int
[ExpandIntegrand[(a + b*ArcCosh[c*x])^n, (f*x)^m*(d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[
c^2*d + e, 0] && IGtQ[n, 0] && IntegerQ[p] && IntegerQ[m]

Rule 5962

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]

Rubi steps

\begin {align*} \int \frac {a+b \cosh ^{-1}(c x)}{x^3 \left (d+e x^2\right )} \, dx &=\int \left (\frac {a+b \cosh ^{-1}(c x)}{d x^3}-\frac {e \left (a+b \cosh ^{-1}(c x)\right )}{d^2 x}+\frac {e^2 x \left (a+b \cosh ^{-1}(c x)\right )}{d^2 \left (d+e x^2\right )}\right ) \, dx\\ &=\frac {\int \frac {a+b \cosh ^{-1}(c x)}{x^3} \, dx}{d}-\frac {e \int \frac {a+b \cosh ^{-1}(c x)}{x} \, dx}{d^2}+\frac {e^2 \int \frac {x \left (a+b \cosh ^{-1}(c x)\right )}{d+e x^2} \, dx}{d^2}\\ &=-\frac {a+b \cosh ^{-1}(c x)}{2 d x^2}+\frac {(b c) \int \frac {1}{x^2 \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{2 d}-\frac {e \text {Subst}\left (\int (a+b x) \tanh (x) \, dx,x,\cosh ^{-1}(c x)\right )}{d^2}+\frac {e^2 \int \left (-\frac {a+b \cosh ^{-1}(c x)}{2 \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \cosh ^{-1}(c x)}{2 \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}\right ) \, dx}{d^2}\\ &=\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{2 d x}-\frac {a+b \cosh ^{-1}(c x)}{2 d x^2}+\frac {e \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b d^2}-\frac {(2 e) \text {Subst}\left (\int \frac {e^{2 x} (a+b x)}{1+e^{2 x}} \, dx,x,\cosh ^{-1}(c x)\right )}{d^2}-\frac {e^{3/2} \int \frac {a+b \cosh ^{-1}(c x)}{\sqrt {-d}-\sqrt {e} x} \, dx}{2 d^2}+\frac {e^{3/2} \int \frac {a+b \cosh ^{-1}(c x)}{\sqrt {-d}+\sqrt {e} x} \, dx}{2 d^2}\\ &=\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{2 d x}-\frac {a+b \cosh ^{-1}(c x)}{2 d x^2}+\frac {e \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b d^2}-\frac {e \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c x)}\right )}{d^2}+\frac {(b e) \text {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{d^2}-\frac {e^{3/2} \text {Subst}\left (\int \frac {(a+b x) \sinh (x)}{c \sqrt {-d}-\sqrt {e} \cosh (x)} \, dx,x,\cosh ^{-1}(c x)\right )}{2 d^2}+\frac {e^{3/2} \text {Subst}\left (\int \frac {(a+b x) \sinh (x)}{c \sqrt {-d}+\sqrt {e} \cosh (x)} \, dx,x,\cosh ^{-1}(c x)\right )}{2 d^2}\\ &=\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{2 d x}-\frac {a+b \cosh ^{-1}(c x)}{2 d x^2}-\frac {e \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c x)}\right )}{d^2}+\frac {(b e) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \cosh ^{-1}(c x)}\right )}{2 d^2}-\frac {e^{3/2} \text {Subst}\left (\int \frac {e^x (a+b x)}{c \sqrt {-d}-\sqrt {-c^2 d-e}-\sqrt {e} e^x} \, dx,x,\cosh ^{-1}(c x)\right )}{2 d^2}-\frac {e^{3/2} \text {Subst}\left (\int \frac {e^x (a+b x)}{c \sqrt {-d}+\sqrt {-c^2 d-e}-\sqrt {e} e^x} \, dx,x,\cosh ^{-1}(c x)\right )}{2 d^2}+\frac {e^{3/2} \text {Subst}\left (\int \frac {e^x (a+b x)}{c \sqrt {-d}-\sqrt {-c^2 d-e}+\sqrt {e} e^x} \, dx,x,\cosh ^{-1}(c x)\right )}{2 d^2}+\frac {e^{3/2} \text {Subst}\left (\int \frac {e^x (a+b x)}{c \sqrt {-d}+\sqrt {-c^2 d-e}+\sqrt {e} e^x} \, dx,x,\cosh ^{-1}(c x)\right )}{2 d^2}\\ &=\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{2 d x}-\frac {a+b \cosh ^{-1}(c x)}{2 d x^2}+\frac {e \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 d^2}+\frac {e \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 d^2}+\frac {e \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 d^2}+\frac {e \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 d^2}-\frac {e \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c x)}\right )}{d^2}-\frac {b e \text {Li}_2\left (-e^{2 \cosh ^{-1}(c x)}\right )}{2 d^2}-\frac {(b e) \text {Subst}\left (\int \log \left (1-\frac {\sqrt {e} e^x}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{2 d^2}-\frac {(b e) \text {Subst}\left (\int \log \left (1+\frac {\sqrt {e} e^x}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{2 d^2}-\frac {(b e) \text {Subst}\left (\int \log \left (1-\frac {\sqrt {e} e^x}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{2 d^2}-\frac {(b e) \text {Subst}\left (\int \log \left (1+\frac {\sqrt {e} e^x}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{2 d^2}\\ &=\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{2 d x}-\frac {a+b \cosh ^{-1}(c x)}{2 d x^2}+\frac {e \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 d^2}+\frac {e \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 d^2}+\frac {e \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 d^2}+\frac {e \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 d^2}-\frac {e \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c x)}\right )}{d^2}-\frac {b e \text {Li}_2\left (-e^{2 \cosh ^{-1}(c x)}\right )}{2 d^2}-\frac {(b e) \text {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {e} x}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{2 d^2}-\frac {(b e) \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {e} x}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{2 d^2}-\frac {(b e) \text {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {e} x}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{2 d^2}-\frac {(b e) \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {e} x}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{2 d^2}\\ &=\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{2 d x}-\frac {a+b \cosh ^{-1}(c x)}{2 d x^2}+\frac {e \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 d^2}+\frac {e \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 d^2}+\frac {e \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 d^2}+\frac {e \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 d^2}-\frac {e \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c x)}\right )}{d^2}+\frac {b e \text {Li}_2\left (-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 d^2}+\frac {b e \text {Li}_2\left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 d^2}+\frac {b e \text {Li}_2\left (-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 d^2}+\frac {b e \text {Li}_2\left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 d^2}-\frac {b e \text {Li}_2\left (-e^{2 \cosh ^{-1}(c x)}\right )}{2 d^2}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.96, size = 479, normalized size = 0.87 \begin {gather*} \frac {-\frac {2 a d}{x^2}-4 a e \log (x)+2 a e \log \left (d+e x^2\right )+b \left (\frac {2 c d \sqrt {\frac {-1+c x}{1+c x}} (1+c x)}{x}-\frac {2 d \cosh ^{-1}(c x)}{x^2}+4 i e \text {ArcSin}\left (\sqrt {1+\frac {c^2 d}{e}}\right ) \tanh ^{-1}\left (\frac {c d \sqrt {\frac {-1+c x}{1+c x}} (1+c x)}{\sqrt {c^2 d \left (c^2 d+e\right )} x}\right )-4 e \cosh ^{-1}(c x) \log \left (1+e^{-2 \cosh ^{-1}(c x)}\right )+2 e \cosh ^{-1}(c x) \log \left (1+\frac {\left (2 c^2 d+e-2 \sqrt {c^2 d \left (c^2 d+e\right )}\right ) e^{-2 \cosh ^{-1}(c x)}}{e}\right )-2 i e \text {ArcSin}\left (\sqrt {1+\frac {c^2 d}{e}}\right ) \log \left (1+\frac {\left (2 c^2 d+e-2 \sqrt {c^2 d \left (c^2 d+e\right )}\right ) e^{-2 \cosh ^{-1}(c x)}}{e}\right )+2 e \cosh ^{-1}(c x) \log \left (1+\frac {\left (2 c^2 d+e+2 \sqrt {c^2 d \left (c^2 d+e\right )}\right ) e^{-2 \cosh ^{-1}(c x)}}{e}\right )+2 i e \text {ArcSin}\left (\sqrt {1+\frac {c^2 d}{e}}\right ) \log \left (1+\frac {\left (2 c^2 d+e+2 \sqrt {c^2 d \left (c^2 d+e\right )}\right ) e^{-2 \cosh ^{-1}(c x)}}{e}\right )+2 e \text {PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c x)}\right )-e \text {PolyLog}\left (2,-\frac {\left (2 c^2 d+e-2 \sqrt {c^2 d \left (c^2 d+e\right )}\right ) e^{-2 \cosh ^{-1}(c x)}}{e}\right )-e \text {PolyLog}\left (2,-\frac {\left (2 c^2 d+e+2 \sqrt {c^2 d \left (c^2 d+e\right )}\right ) e^{-2 \cosh ^{-1}(c x)}}{e}\right )\right )}{4 d^2} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*ArcCosh[c*x])/(x^3*(d + e*x^2)),x]

[Out]

((-2*a*d)/x^2 - 4*a*e*Log[x] + 2*a*e*Log[d + e*x^2] + b*((2*c*d*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))/x - (2*d
*ArcCosh[c*x])/x^2 + (4*I)*e*ArcSin[Sqrt[1 + (c^2*d)/e]]*ArcTanh[(c*d*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))/(S
qrt[c^2*d*(c^2*d + e)]*x)] - 4*e*ArcCosh[c*x]*Log[1 + E^(-2*ArcCosh[c*x])] + 2*e*ArcCosh[c*x]*Log[1 + (2*c^2*d
+ e - 2*Sqrt[c^2*d*(c^2*d + e)])/(e*E^(2*ArcCosh[c*x]))] - (2*I)*e*ArcSin[Sqrt[1 + (c^2*d)/e]]*Log[1 + (2*c^2
*d + e - 2*Sqrt[c^2*d*(c^2*d + e)])/(e*E^(2*ArcCosh[c*x]))] + 2*e*ArcCosh[c*x]*Log[1 + (2*c^2*d + e + 2*Sqrt[c
^2*d*(c^2*d + e)])/(e*E^(2*ArcCosh[c*x]))] + (2*I)*e*ArcSin[Sqrt[1 + (c^2*d)/e]]*Log[1 + (2*c^2*d + e + 2*Sqrt
[c^2*d*(c^2*d + e)])/(e*E^(2*ArcCosh[c*x]))] + 2*e*PolyLog[2, -E^(-2*ArcCosh[c*x])] - e*PolyLog[2, -((2*c^2*d
+ e - 2*Sqrt[c^2*d*(c^2*d + e)])/(e*E^(2*ArcCosh[c*x])))] - e*PolyLog[2, -((2*c^2*d + e + 2*Sqrt[c^2*d*(c^2*d
+ e)])/(e*E^(2*ArcCosh[c*x])))]))/(4*d^2)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 6.61, size = 495, normalized size = 0.90

method	result	size
derivativedivides	\(c^{2} \left (\frac {a e \ln \left (c^{2} e \,x^{2}+c^{2} d \right )}{2 c^{2} d^{2}}-\frac {a}{2 d \,c^{2} x^{2}}-\frac {a e \ln \left (c x \right )}{c^{2} d^{2}}+\frac {b \sqrt {c x +1}\, \sqrt {c x -1}}{2 d c x}-\frac {b}{2 d}-\frac {b \,\mathrm {arccosh}\left (c x \right )}{2 d \,c^{2} x^{2}}+\frac {b \,e^{2} \left (\munderset {\textit {\_R1} =\RootOf \left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d +2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (\textit {\_R1}^{2}+1\right ) \left (\mathrm {arccosh}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )+\dilog \left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} e +2 c^{2} d +e}\right )}{4 c^{2} d^{2}}-\frac {b e \,\mathrm {arccosh}\left (c x \right ) \ln \left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{c^{2} d^{2}}-\frac {b e \,\mathrm {arccosh}\left (c x \right ) \ln \left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{c^{2} d^{2}}-\frac {b e \dilog \left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{c^{2} d^{2}}-\frac {b e \dilog \left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{c^{2} d^{2}}+\frac {b e \left (\munderset {\textit {\_R1} =\RootOf \left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d +2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (\textit {\_R1}^{2} e +4 c^{2} d +e \right ) \left (\mathrm {arccosh}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )+\dilog \left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} e +2 c^{2} d +e}\right )}{4 c^{2} d^{2}}\right )\)	\(495\)
default	\(c^{2} \left (\frac {a e \ln \left (c^{2} e \,x^{2}+c^{2} d \right )}{2 c^{2} d^{2}}-\frac {a}{2 d \,c^{2} x^{2}}-\frac {a e \ln \left (c x \right )}{c^{2} d^{2}}+\frac {b \sqrt {c x +1}\, \sqrt {c x -1}}{2 d c x}-\frac {b}{2 d}-\frac {b \,\mathrm {arccosh}\left (c x \right )}{2 d \,c^{2} x^{2}}+\frac {b \,e^{2} \left (\munderset {\textit {\_R1} =\RootOf \left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d +2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (\textit {\_R1}^{2}+1\right ) \left (\mathrm {arccosh}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )+\dilog \left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} e +2 c^{2} d +e}\right )}{4 c^{2} d^{2}}-\frac {b e \,\mathrm {arccosh}\left (c x \right ) \ln \left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{c^{2} d^{2}}-\frac {b e \,\mathrm {arccosh}\left (c x \right ) \ln \left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{c^{2} d^{2}}-\frac {b e \dilog \left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{c^{2} d^{2}}-\frac {b e \dilog \left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right )}{c^{2} d^{2}}+\frac {b e \left (\munderset {\textit {\_R1} =\RootOf \left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d +2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (\textit {\_R1}^{2} e +4 c^{2} d +e \right ) \left (\mathrm {arccosh}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )+\dilog \left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} e +2 c^{2} d +e}\right )}{4 c^{2} d^{2}}\right )\)	\(495\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arccosh(c*x))/x^3/(e*x^2+d),x,method=_RETURNVERBOSE)

[Out]

c^2*(1/2*a/c^2*e/d^2*ln(c^2*e*x^2+c^2*d)-1/2*a/d/c^2/x^2-a/c^2/d^2*e*ln(c*x)+1/2*b/d/c/x*(c*x+1)^(1/2)*(c*x-1)
^(1/2)-1/2*b/d-1/2*b*arccosh(c*x)/d/c^2/x^2+1/4*b/c^2/d^2*e^2*sum((_R1^2+1)/(_R1^2*e+2*c^2*d+e)*(arccosh(c*x)*
ln((_R1-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))/_R1)+dilog((_R1-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))/_R1)),_R1=RootOf(e*_
Z^4+(4*c^2*d+2*e)*_Z^2+e))-b/c^2/d^2*e*arccosh(c*x)*ln(1+I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))-b/c^2/d^2*e*arcc
osh(c*x)*ln(1-I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))-b/c^2/d^2*e*dilog(1+I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))-b/
c^2/d^2*e*dilog(1-I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))+1/4*b/c^2/d^2*e*sum((_R1^2*e+4*c^2*d+e)/(_R1^2*e+2*c^2*
d+e)*(arccosh(c*x)*ln((_R1-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))/_R1)+dilog((_R1-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))/_
R1)),_R1=RootOf(e*_Z^4+(4*c^2*d+2*e)*_Z^2+e)))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(c*x))/x^3/(e*x^2+d),x, algorithm="maxima")

[Out]

1/2*a*(e*log(x^2*e + d)/d^2 - 2*e*log(x)/d^2 - 1/(d*x^2)) + b*integrate(log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))
/(x^5*e + d*x^3), x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(c*x))/x^3/(e*x^2+d),x, algorithm="fricas")

[Out]

integral((b*arccosh(c*x) + a)/(x^5*e + d*x^3), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a + b \operatorname {acosh}{\left (c x \right )}}{x^{3} \left (d + e x^{2}\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*acosh(c*x))/x**3/(e*x**2+d),x)

[Out]

Integral((a + b*acosh(c*x))/(x**3*(d + e*x**2)), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(c*x))/x^3/(e*x^2+d),x, algorithm="giac")

[Out]

integrate((b*arccosh(c*x) + a)/((e*x^2 + d)*x^3), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{x^3\,\left (e\,x^2+d\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*acosh(c*x))/(x^3*(d + e*x^2)),x)

[Out]

int((a + b*acosh(c*x))/(x^3*(d + e*x^2)), x)

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