\(\int \frac {x^3 (a+b \text {arccosh}(c x))}{(d+e x^2)^2} \, dx\) [498]

Optimal result

Integrand size = 21, antiderivative size = 562 \[ \int \frac {x^3 (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^2} \, dx=\frac {d (a+b \text {arccosh}(c x))}{2 e^2 \left (d+e x^2\right )}-\frac {(a+b \text {arccosh}(c x))^2}{2 b e^2}-\frac {b c \sqrt {d} \sqrt {-1+c^2 x^2} \text {arctanh}\left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{2 e^2 \sqrt {c^2 d+e} \sqrt {-1+c x} \sqrt {1+c x}}+\frac {(a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e^2}+\frac {(a+b \text {arccosh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e^2}+\frac {(a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e^2}+\frac {(a+b \text {arccosh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e^2}+\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e^2}+\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e^2}+\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e^2}+\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.70 (sec) , antiderivative size = 562, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {5959, 5957, 533, 385, 214, 5962, 5681, 2221, 2317, 2438} \[ \int \frac {x^3 (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^2} \, dx=\frac {(a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {c^2 (-d)-e}}\right )}{2 e^2}+\frac {(a+b \text {arccosh}(c x)) \log \left (\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {c^2 (-d)-e}}+1\right )}{2 e^2}+\frac {(a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {c^2 (-d)-e}+c \sqrt {-d}}\right )}{2 e^2}+\frac {(a+b \text {arccosh}(c x)) \log \left (\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {c^2 (-d)-e}+c \sqrt {-d}}+1\right )}{2 e^2}+\frac {d (a+b \text {arccosh}(c x))}{2 e^2 \left (d+e x^2\right )}-\frac {(a+b \text {arccosh}(c x))^2}{2 b e^2}+\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right )}{2 e^2}+\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right )}{2 e^2}+\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )}{2 e^2}+\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )}{2 e^2}-\frac {b c \sqrt {d} \sqrt {c^2 x^2-1} \text {arctanh}\left (\frac {x \sqrt {c^2 d+e}}{\sqrt {d} \sqrt {c^2 x^2-1}}\right )}{2 e^2 \sqrt {c x-1} \sqrt {c x+1} \sqrt {c^2 d+e}} \]

[In]

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

[Out]

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

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 385

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 533

Int[(u_.)*((c_) + (d_.)*(x_)^(n_.))^(q_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p_)*((a2_) + (b2_.)*(x_)^(non2_.))^(
p_), x_Symbol] :> Dist[(a1 + b1*x^(n/2))^FracPart[p]*((a2 + b2*x^(n/2))^FracPart[p]/(a1*a2 + b1*b2*x^n)^FracPa
rt[p]), Int[u*(a1*a2 + b1*b2*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a1, b1, a2, b2, c, d, n, p, q}, x] && EqQ[
non2, n/2] && EqQ[a2*b1 + a1*b2, 0] &&  !(EqQ[n, 2] && IGtQ[q, 0])

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 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 5957

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*(x_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x^2)^(p + 1
)*((a + b*ArcCosh[c*x])/(2*e*(p + 1))), x] - Dist[b*(c/(2*e*(p + 1))), Int[(d + e*x^2)^(p + 1)/(Sqrt[1 + c*x]*
Sqrt[-1 + c*x]), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[c^2*d + e, 0] && NeQ[p, -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*} \text {integral}& = \int \left (-\frac {d x (a+b \text {arccosh}(c x))}{e \left (d+e x^2\right )^2}+\frac {x (a+b \text {arccosh}(c x))}{e \left (d+e x^2\right )}\right ) \, dx \\ & = \frac {\int \frac {x (a+b \text {arccosh}(c x))}{d+e x^2} \, dx}{e}-\frac {d \int \frac {x (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^2} \, dx}{e} \\ & = \frac {d (a+b \text {arccosh}(c x))}{2 e^2 \left (d+e x^2\right )}-\frac {(b c d) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x} \left (d+e x^2\right )} \, dx}{2 e^2}+\frac {\int \left (-\frac {a+b \text {arccosh}(c x)}{2 \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \text {arccosh}(c x)}{2 \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}\right ) \, dx}{e} \\ & = \frac {d (a+b \text {arccosh}(c x))}{2 e^2 \left (d+e x^2\right )}-\frac {\int \frac {a+b \text {arccosh}(c x)}{\sqrt {-d}-\sqrt {e} x} \, dx}{2 e^{3/2}}+\frac {\int \frac {a+b \text {arccosh}(c x)}{\sqrt {-d}+\sqrt {e} x} \, dx}{2 e^{3/2}}-\frac {\left (b c d \sqrt {-1+c^2 x^2}\right ) \int \frac {1}{\sqrt {-1+c^2 x^2} \left (d+e x^2\right )} \, dx}{2 e^2 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {d (a+b \text {arccosh}(c x))}{2 e^2 \left (d+e x^2\right )}-\frac {\text {Subst}\left (\int \frac {(a+b x) \sinh (x)}{c \sqrt {-d}-\sqrt {e} \cosh (x)} \, dx,x,\text {arccosh}(c x)\right )}{2 e^{3/2}}+\frac {\text {Subst}\left (\int \frac {(a+b x) \sinh (x)}{c \sqrt {-d}+\sqrt {e} \cosh (x)} \, dx,x,\text {arccosh}(c x)\right )}{2 e^{3/2}}-\frac {\left (b c d \sqrt {-1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{d-\left (c^2 d+e\right ) x^2} \, dx,x,\frac {x}{\sqrt {-1+c^2 x^2}}\right )}{2 e^2 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {d (a+b \text {arccosh}(c x))}{2 e^2 \left (d+e x^2\right )}-\frac {(a+b \text {arccosh}(c x))^2}{2 b e^2}-\frac {b c \sqrt {d} \sqrt {-1+c^2 x^2} \text {arctanh}\left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{2 e^2 \sqrt {c^2 d+e} \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\text {Subst}\left (\int \frac {e^x (a+b x)}{c \sqrt {-d}-\sqrt {-c^2 d-e}-\sqrt {e} e^x} \, dx,x,\text {arccosh}(c x)\right )}{2 e^{3/2}}-\frac {\text {Subst}\left (\int \frac {e^x (a+b x)}{c \sqrt {-d}+\sqrt {-c^2 d-e}-\sqrt {e} e^x} \, dx,x,\text {arccosh}(c x)\right )}{2 e^{3/2}}+\frac {\text {Subst}\left (\int \frac {e^x (a+b x)}{c \sqrt {-d}-\sqrt {-c^2 d-e}+\sqrt {e} e^x} \, dx,x,\text {arccosh}(c x)\right )}{2 e^{3/2}}+\frac {\text {Subst}\left (\int \frac {e^x (a+b x)}{c \sqrt {-d}+\sqrt {-c^2 d-e}+\sqrt {e} e^x} \, dx,x,\text {arccosh}(c x)\right )}{2 e^{3/2}} \\ & = \frac {d (a+b \text {arccosh}(c x))}{2 e^2 \left (d+e x^2\right )}-\frac {(a+b \text {arccosh}(c x))^2}{2 b e^2}-\frac {b c \sqrt {d} \sqrt {-1+c^2 x^2} \text {arctanh}\left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{2 e^2 \sqrt {c^2 d+e} \sqrt {-1+c x} \sqrt {1+c x}}+\frac {(a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e^2}+\frac {(a+b \text {arccosh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e^2}+\frac {(a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e^2}+\frac {(a+b \text {arccosh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e^2}-\frac {b \text {Subst}\left (\int \log \left (1-\frac {\sqrt {e} e^x}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right ) \, dx,x,\text {arccosh}(c x)\right )}{2 e^2}-\frac {b \text {Subst}\left (\int \log \left (1+\frac {\sqrt {e} e^x}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right ) \, dx,x,\text {arccosh}(c x)\right )}{2 e^2}-\frac {b \text {Subst}\left (\int \log \left (1-\frac {\sqrt {e} e^x}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right ) \, dx,x,\text {arccosh}(c x)\right )}{2 e^2}-\frac {b \text {Subst}\left (\int \log \left (1+\frac {\sqrt {e} e^x}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right ) \, dx,x,\text {arccosh}(c x)\right )}{2 e^2} \\ & = \frac {d (a+b \text {arccosh}(c x))}{2 e^2 \left (d+e x^2\right )}-\frac {(a+b \text {arccosh}(c x))^2}{2 b e^2}-\frac {b c \sqrt {d} \sqrt {-1+c^2 x^2} \text {arctanh}\left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{2 e^2 \sqrt {c^2 d+e} \sqrt {-1+c x} \sqrt {1+c x}}+\frac {(a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e^2}+\frac {(a+b \text {arccosh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e^2}+\frac {(a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e^2}+\frac {(a+b \text {arccosh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e^2}-\frac {b \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^{\text {arccosh}(c x)}\right )}{2 e^2}-\frac {b \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^{\text {arccosh}(c x)}\right )}{2 e^2}-\frac {b \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^{\text {arccosh}(c x)}\right )}{2 e^2}-\frac {b \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^{\text {arccosh}(c x)}\right )}{2 e^2} \\ & = \frac {d (a+b \text {arccosh}(c x))}{2 e^2 \left (d+e x^2\right )}-\frac {(a+b \text {arccosh}(c x))^2}{2 b e^2}-\frac {b c \sqrt {d} \sqrt {-1+c^2 x^2} \text {arctanh}\left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {-1+c^2 x^2}}\right )}{2 e^2 \sqrt {c^2 d+e} \sqrt {-1+c x} \sqrt {1+c x}}+\frac {(a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e^2}+\frac {(a+b \text {arccosh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e^2}+\frac {(a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e^2}+\frac {(a+b \text {arccosh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e^2}+\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e^2}+\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e^2}+\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e^2}+\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e^2} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.38 (sec) , antiderivative size = 694, normalized size of antiderivative = 1.23 \[ \int \frac {x^3 (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^2} \, dx=\frac {\frac {2 a d}{d+e x^2}+2 a \log \left (d+e x^2\right )+b \left (-2 \text {arccosh}(c x)^2+2 \text {arccosh}(c x) \left (\log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{-i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )+\log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )\right )+2 \text {arccosh}(c x) \left (\log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{i c \sqrt {d}-\sqrt {-c^2 d-e}}\right )+\log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )\right )-i \sqrt {d} \left (\frac {\text {arccosh}(c x)}{-i \sqrt {d}+\sqrt {e} x}+\frac {c \log \left (\frac {2 e \left (i \sqrt {e}+c^2 \sqrt {d} x-i \sqrt {-c^2 d-e} \sqrt {-1+c x} \sqrt {1+c x}\right )}{c \sqrt {-c^2 d-e} \left (\sqrt {d}+i \sqrt {e} x\right )}\right )}{\sqrt {-c^2 d-e}}\right )-i \sqrt {d} \left (-\frac {\text {arccosh}(c x)}{i \sqrt {d}+\sqrt {e} x}-\frac {c \log \left (\frac {2 e \left (-\sqrt {e}-i c^2 \sqrt {d} x+\sqrt {-c^2 d-e} \sqrt {-1+c x} \sqrt {1+c x}\right )}{c \sqrt {-c^2 d-e} \left (i \sqrt {d}+\sqrt {e} x\right )}\right )}{\sqrt {-c^2 d-e}}\right )+2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{i c \sqrt {d}-\sqrt {-c^2 d-e}}\right )+2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{-i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )+2 \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )+2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )\right )}{4 e^2} \]

[In]

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

[Out]

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

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 3.01 (sec) , antiderivative size = 2132, normalized size of antiderivative = 3.79

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(2132\)
default	\(\text {Expression too large to display}\)	\(2132\)
parts	\(\text {Expression too large to display}\)	\(2144\)

[In]

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

[Out]

1/c^4*(1/2*a*c^6/e^2*d/(c^2*e*x^2+c^2*d)+1/2*a*c^4/e^2*ln(c^2*e*x^2+c^2*d)+b*c^4*(-1/8*(-2*(d*c^2*(c^2*d+e))^(
1/2)*c^2*d+2*c^4*d^2+2*c^2*d*e-(d*c^2*(c^2*d+e))^(1/2)*e)/c^2/d/e^2/(c^2*d+e)*polylog(2,e*(c*x+(c*x-1)^(1/2)*(
c*x+1)^(1/2))^2/(-2*c^2*d-2*(d*c^2*(c^2*d+e))^(1/2)-e))+(-2*(d*c^2*(c^2*d+e))^(1/2)*c^2*d+2*c^4*d^2+2*c^2*d*e-
(d*c^2*(c^2*d+e))^(1/2)*e)*c^2*d/e^4/(c^2*d+e)*arccosh(c*x)^2-1/2*(-2*(d*c^2*(c^2*d+e))^(1/2)*c^2*d+2*c^4*d^2+
2*c^2*d*e-(d*c^2*(c^2*d+e))^(1/2)*e)*c^2*d/e^4/(c^2*d+e)*polylog(2,e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2/(-2*c
^2*d-2*(d*c^2*(c^2*d+e))^(1/2)-e))+1/4*(d*c^2*(c^2*d+e))^(1/2)/e/d/c^2/(c^2*d+e)*arccosh(c*x)^2+1/4*(-2*(d*c^2
*(c^2*d+e))^(1/2)*c^2*d+2*c^4*d^2+2*c^2*d*e-(d*c^2*(c^2*d+e))^(1/2)*e)/c^2/d/e^2/(c^2*d+e)*arccosh(c*x)^2+(2*c
^2*d-2*(d*c^2*(c^2*d+e))^(1/2)+e)/e^4*ln(1-e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2/(-2*c^2*d-2*(d*c^2*(c^2*d+e))
^(1/2)-e))*c^2*d*arccosh(c*x)+1/2*(d*c^2*(c^2*d+e))^(1/2)/e^2/(c^2*d+e)*arctanh(1/4*(4*c^2*d+2*e*(c*x+(c*x-1)^
(1/2)*(c*x+1)^(1/2))^2+2*e)/(c^4*d^2+c^2*d*e)^(1/2))-1/4*(d*c^2*(c^2*d+e))^(1/2)/e^2/(c^2*d+e)*polylog(2,e*(c*
x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2/(-2*c^2*d+2*(d*c^2*(c^2*d+e))^(1/2)-e))-1/8*(d*c^2*(c^2*d+e))^(1/2)/e/d/c^2/(
c^2*d+e)*polylog(2,e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2/(-2*c^2*d+2*(d*c^2*(c^2*d+e))^(1/2)-e))+(-2*(d*c^2*(c
^2*d+e))^(1/2)*c^2*d+2*c^4*d^2+2*c^2*d*e-(d*c^2*(c^2*d+e))^(1/2)*e)/e^3/(c^2*d+e)*arccosh(c*x)^2+1/2*(2*c^2*d-
2*(d*c^2*(c^2*d+e))^(1/2)+e)/e^3*ln(1-e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2/(-2*c^2*d-2*(d*c^2*(c^2*d+e))^(1/2
)-e))*arccosh(c*x)+1/2*(d*c^2*(c^2*d+e))^(1/2)/e^2/(c^2*d+e)*arccosh(c*x)^2-1/2*(-2*(d*c^2*(c^2*d+e))^(1/2)*c^
2*d+2*c^4*d^2+2*c^2*d*e-(d*c^2*(c^2*d+e))^(1/2)*e)/e^3/(c^2*d+e)*polylog(2,e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))
^2/(-2*c^2*d-2*(d*c^2*(c^2*d+e))^(1/2)-e))-(-2*(d*c^2*(c^2*d+e))^(1/2)*c^2*d+2*c^4*d^2+2*c^2*d*e-(d*c^2*(c^2*d
+e))^(1/2)*e)*c^2*d/e^4/(c^2*d+e)*ln(1-e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2/(-2*c^2*d-2*(d*c^2*(c^2*d+e))^(1/
2)-e))*arccosh(c*x)-1/4*(-2*(d*c^2*(c^2*d+e))^(1/2)*c^2*d+2*c^4*d^2+2*c^2*d*e-(d*c^2*(c^2*d+e))^(1/2)*e)/c^2/d
/e^2/(c^2*d+e)*ln(1-e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2/(-2*c^2*d-2*(d*c^2*(c^2*d+e))^(1/2)-e))*arccosh(c*x)
-1/4*(d*c^2*(c^2*d+e))^(1/2)/e/d/c^2/(c^2*d+e)*arccosh(c*x)*ln(1-e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2/(-2*c^2
*d+2*(d*c^2*(c^2*d+e))^(1/2)-e))+1/4*(2*c^2*d-2*(d*c^2*(c^2*d+e))^(1/2)+e)/e^3*polylog(2,e*(c*x+(c*x-1)^(1/2)*
(c*x+1)^(1/2))^2/(-2*c^2*d-2*(d*c^2*(c^2*d+e))^(1/2)-e))-1/2*(2*c^2*d-2*(d*c^2*(c^2*d+e))^(1/2)+e)/e^3*arccosh
(c*x)^2+1/2*arccosh(c*x)/e^2*d*c^2/(c^2*e*x^2+c^2*d)-(-2*(d*c^2*(c^2*d+e))^(1/2)*c^2*d+2*c^4*d^2+2*c^2*d*e-(d*
c^2*(c^2*d+e))^(1/2)*e)/e^3/(c^2*d+e)*ln(1-e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2/(-2*c^2*d-2*(d*c^2*(c^2*d+e))
^(1/2)-e))*arccosh(c*x)-1/2*(d*c^2*(c^2*d+e))^(1/2)/e^2/(c^2*d+e)*arccosh(c*x)*ln(1-e*(c*x+(c*x-1)^(1/2)*(c*x+
1)^(1/2))^2/(-2*c^2*d+2*(d*c^2*(c^2*d+e))^(1/2)-e))-(2*c^2*d-2*(d*c^2*(c^2*d+e))^(1/2)+e)/e^4*d*c^2*arccosh(c*
x)^2+1/2*(2*c^2*d-2*(d*c^2*(c^2*d+e))^(1/2)+e)/e^4*d*c^2*polylog(2,e*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2/(-2*c
^2*d-2*(d*c^2*(c^2*d+e))^(1/2)-e))-1/2*arccosh(c*x)^2/e^2+1/2/e^2*sum((_R1^2*e+4*c^2*d+2*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))))

Fricas [F]

\[ \int \frac {x^3 (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^2} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{3}}{{\left (e x^{2} + d\right )}^{2}} \,d x } \]

[In]

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

[Out]

integral((b*x^3*arccosh(c*x) + a*x^3)/(e^2*x^4 + 2*d*e*x^2 + d^2), x)

Sympy [F]

\[ \int \frac {x^3 (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^2} \, dx=\int \frac {x^{3} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )}{\left (d + e x^{2}\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {x^3 (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^2} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{3}}{{\left (e x^{2} + d\right )}^{2}} \,d x } \]

[In]

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

[Out]

1/2*a*(d/(e^3*x^2 + d*e^2) + log(e*x^2 + d)/e^2) + b*integrate(x^3*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/(e^2
*x^4 + 2*d*e*x^2 + d^2), x)

Giac [F(-2)]

Exception generated. \[ \int \frac {x^3 (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^2} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

Timed out. \[ \int \frac {x^3 (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^2} \, dx=\int \frac {x^3\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}{{\left (e\,x^2+d\right )}^2} \,d x \]

[In]

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

[Out]

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