\(\int \frac {a+b \text {arccosh}(c x)}{x^3 (d-c^2 d x^2)^{5/2}} \, dx\) [132]

Optimal result

Integrand size = 27, antiderivative size = 479 \[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {3 b c \sqrt {-1+c x} \sqrt {1+c x}}{4 d^2 x \sqrt {d-c^2 d x^2}}-\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{4 d^2 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {5 b c^3 x \sqrt {-1+c x} \sqrt {1+c x}}{12 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {5 c^2 (a+b \text {arccosh}(c x))}{6 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {a+b \text {arccosh}(c x)}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}+\frac {5 c^2 (a+b \text {arccosh}(c x))}{2 d^2 \sqrt {d-c^2 d x^2}}+\frac {5 c^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \arctan \left (e^{\text {arccosh}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {13 b c^2 \sqrt {-1+c x} \sqrt {1+c x} \text {arctanh}(c x)}{6 d^2 \sqrt {d-c^2 d x^2}}-\frac {5 i b c^2 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right )}{2 d^2 \sqrt {d-c^2 d x^2}}+\frac {5 i b c^2 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right )}{2 d^2 \sqrt {d-c^2 d x^2}} \]

[Out]

5/6*c^2*(a+b*arccosh(c*x))/d/(-c^2*d*x^2+d)^(3/2)+1/2*(-a-b*arccosh(c*x))/d/x^2/(-c^2*d*x^2+d)^(3/2)+5/2*c^2*(
a+b*arccosh(c*x))/d^2/(-c^2*d*x^2+d)^(1/2)+3/4*b*c*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^2/x/(-c^2*d*x^2+d)^(1/2)-1/4*
b*c*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^2/x/(-c^2*x^2+1)/(-c^2*d*x^2+d)^(1/2)+5/12*b*c^3*x*(c*x-1)^(1/2)*(c*x+1)^(1/
2)/d^2/(-c^2*x^2+1)/(-c^2*d*x^2+d)^(1/2)+5*c^2*(a+b*arccosh(c*x))*arctan(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*(c*x
-1)^(1/2)*(c*x+1)^(1/2)/d^2/(-c^2*d*x^2+d)^(1/2)+13/6*b*c^2*arctanh(c*x)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^2/(-c^2
*d*x^2+d)^(1/2)-5/2*I*b*c^2*polylog(2,-I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^2/(-
c^2*d*x^2+d)^(1/2)+5/2*I*b*c^2*polylog(2,I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^2/
(-c^2*d*x^2+d)^(1/2)

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 479, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.481, Rules used = {5932, 5936, 5946, 4265, 2317, 2438, 35, 213, 41, 205, 74, 296, 331} \[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {5 c^2 \sqrt {c x-1} \sqrt {c x+1} \arctan \left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))}{d^2 \sqrt {d-c^2 d x^2}}+\frac {5 c^2 (a+b \text {arccosh}(c x))}{2 d^2 \sqrt {d-c^2 d x^2}}+\frac {5 c^2 (a+b \text {arccosh}(c x))}{6 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {a+b \text {arccosh}(c x)}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}-\frac {5 i b c^2 \sqrt {c x-1} \sqrt {c x+1} \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right )}{2 d^2 \sqrt {d-c^2 d x^2}}+\frac {5 i b c^2 \sqrt {c x-1} \sqrt {c x+1} \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right )}{2 d^2 \sqrt {d-c^2 d x^2}}+\frac {13 b c^2 \sqrt {c x-1} \sqrt {c x+1} \text {arctanh}(c x)}{6 d^2 \sqrt {d-c^2 d x^2}}+\frac {3 b c \sqrt {c x-1} \sqrt {c x+1}}{4 d^2 x \sqrt {d-c^2 d x^2}}-\frac {b c \sqrt {c x-1} \sqrt {c x+1}}{4 d^2 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {5 b c^3 x \sqrt {c x-1} \sqrt {c x+1}}{12 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}} \]

[In]

Int[(a + b*ArcCosh[c*x])/(x^3*(d - c^2*d*x^2)^(5/2)),x]

[Out]

(3*b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(4*d^2*x*Sqrt[d - c^2*d*x^2]) - (b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(4*d^2
*x*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]) + (5*b*c^3*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(12*d^2*(1 - c^2*x^2)*Sqrt[d
- c^2*d*x^2]) + (5*c^2*(a + b*ArcCosh[c*x]))/(6*d*(d - c^2*d*x^2)^(3/2)) - (a + b*ArcCosh[c*x])/(2*d*x^2*(d -
c^2*d*x^2)^(3/2)) + (5*c^2*(a + b*ArcCosh[c*x]))/(2*d^2*Sqrt[d - c^2*d*x^2]) + (5*c^2*Sqrt[-1 + c*x]*Sqrt[1 +
c*x]*(a + b*ArcCosh[c*x])*ArcTan[E^ArcCosh[c*x]])/(d^2*Sqrt[d - c^2*d*x^2]) + (13*b*c^2*Sqrt[-1 + c*x]*Sqrt[1
+ c*x]*ArcTanh[c*x])/(6*d^2*Sqrt[d - c^2*d*x^2]) - (((5*I)/2)*b*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, (-
I)*E^ArcCosh[c*x]])/(d^2*Sqrt[d - c^2*d*x^2]) + (((5*I)/2)*b*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, I*E^A
rcCosh[c*x]])/(d^2*Sqrt[d - c^2*d*x^2])

Rule 35

Int[1/(((a_) + (b_.)*(x_))*((c_) + (d_.)*(x_))), x_Symbol] :> Int[1/(a*c + b*d*x^2), x] /; FreeQ[{a, b, c, d},
x] && EqQ[b*c + a*d, 0]

Rule 41

Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[(a*c + b*d*x^2)^m, x] /; FreeQ[{a, b
, c, d, m}, x] && EqQ[b*c + a*d, 0] && (IntegerQ[m] || (GtQ[a, 0] && GtQ[c, 0]))

Rule 74

Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[(a*c + b*
d*x^2)^m*(e + f*x)^p, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[n, m] && Integer
Q[m] && (NeQ[m, -1] || (EqQ[e, 0] && (EqQ[p, 1] ||  !IntegerQ[p])))

Rule 205

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Dist[(n*(p
+ 1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (
IntegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[
p])

Rule 213

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(-1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])]
, x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rule 296

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-(c*x)^(m + 1))*((a + b*x^n)^(p + 1)/
(a*c*n*(p + 1))), x] + Dist[(m + n*(p + 1) + 1)/(a*n*(p + 1)), Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; Free
Q[{a, b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 331

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c
*(m + 1))), x] - Dist[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

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 4265

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] + (-Dist[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] + Dist[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]

Rule 5932

Int[((a_.) + ArcCosh[(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*ArcCosh[c*x])^n/(d*f*(m + 1))), x] + (Dist[c^2*((m + 2*p + 3)/(f^2*(
m + 1))), Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcCosh[c*x])^n, x], x] + Dist[b*c*(n/(f*(m + 1)))*Simp[(d +
e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)], Int[(f*x)^(m + 1)*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCos
h[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && ILtQ[m, -1]

Rule 5936

Int[((a_.) + ArcCosh[(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*ArcCosh[c*x])^n/(2*d*f*(p + 1))), x] + (Dist[(m + 2*p + 3)/(2*d*(
p + 1)), Int[(f*x)^m*(d + e*x^2)^(p + 1)*(a + b*ArcCosh[c*x])^n, x], x] - Dist[b*c*(n/(2*f*(p + 1)))*Simp[(d +
e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)], Int[(f*x)^(m + 1)*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCo
sh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] &
&  !GtQ[m, 1] && (IntegerQ[m] || IntegerQ[p] || EqQ[n, 1])

Rule 5946

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Dist[(1/c^(m
+ 1))*Simp[Sqrt[1 + c*x]*(Sqrt[-1 + c*x]/Sqrt[d + e*x^2])], Subst[Int[(a + b*x)^n*Cosh[x]^m, x], x, ArcCosh[c
*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0] && IntegerQ[m]

Rubi steps \begin{align*} \text {integral}& = -\frac {a+b \text {arccosh}(c x)}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}+\frac {1}{2} \left (5 c^2\right ) \int \frac {a+b \text {arccosh}(c x)}{x \left (d-c^2 d x^2\right )^{5/2}} \, dx-\frac {\left (b c \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {1}{x^2 (-1+c x)^2 (1+c x)^2} \, dx}{2 d^2 \sqrt {d-c^2 d x^2}} \\ & = \frac {5 c^2 (a+b \text {arccosh}(c x))}{6 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {a+b \text {arccosh}(c x)}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}+\frac {\left (5 c^2\right ) \int \frac {a+b \text {arccosh}(c x)}{x \left (d-c^2 d x^2\right )^{3/2}} \, dx}{2 d}-\frac {\left (b c \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {1}{x^2 \left (-1+c^2 x^2\right )^2} \, dx}{2 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (5 b c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {1}{(-1+c x)^2 (1+c x)^2} \, dx}{6 d^2 \sqrt {d-c^2 d x^2}} \\ & = -\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{4 d^2 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {5 c^2 (a+b \text {arccosh}(c x))}{6 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {a+b \text {arccosh}(c x)}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}+\frac {5 c^2 (a+b \text {arccosh}(c x))}{2 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (5 c^2\right ) \int \frac {a+b \text {arccosh}(c x)}{x \sqrt {d-c^2 d x^2}} \, dx}{2 d^2}+\frac {\left (3 b c \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {1}{x^2 \left (-1+c^2 x^2\right )} \, dx}{4 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (5 b c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {1}{\left (-1+c^2 x^2\right )^2} \, dx}{6 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (5 b c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {1}{(-1+c x) (1+c x)} \, dx}{2 d^2 \sqrt {d-c^2 d x^2}} \\ & = \frac {3 b c \sqrt {-1+c x} \sqrt {1+c x}}{4 d^2 x \sqrt {d-c^2 d x^2}}-\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{4 d^2 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {5 b c^3 x \sqrt {-1+c x} \sqrt {1+c x}}{12 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {5 c^2 (a+b \text {arccosh}(c x))}{6 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {a+b \text {arccosh}(c x)}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}+\frac {5 c^2 (a+b \text {arccosh}(c x))}{2 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (5 c^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}(\int (a+b x) \text {sech}(x) \, dx,x,\text {arccosh}(c x))}{2 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (5 b c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {1}{-1+c^2 x^2} \, dx}{12 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (3 b c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {1}{-1+c^2 x^2} \, dx}{4 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (5 b c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {1}{-1+c^2 x^2} \, dx}{2 d^2 \sqrt {d-c^2 d x^2}} \\ & = \frac {3 b c \sqrt {-1+c x} \sqrt {1+c x}}{4 d^2 x \sqrt {d-c^2 d x^2}}-\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{4 d^2 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {5 b c^3 x \sqrt {-1+c x} \sqrt {1+c x}}{12 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {5 c^2 (a+b \text {arccosh}(c x))}{6 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {a+b \text {arccosh}(c x)}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}+\frac {5 c^2 (a+b \text {arccosh}(c x))}{2 d^2 \sqrt {d-c^2 d x^2}}+\frac {5 c^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \arctan \left (e^{\text {arccosh}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {13 b c^2 \sqrt {-1+c x} \sqrt {1+c x} \text {arctanh}(c x)}{6 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (5 i b c^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\text {arccosh}(c x)\right )}{2 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (5 i b c^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\text {arccosh}(c x)\right )}{2 d^2 \sqrt {d-c^2 d x^2}} \\ & = \frac {3 b c \sqrt {-1+c x} \sqrt {1+c x}}{4 d^2 x \sqrt {d-c^2 d x^2}}-\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{4 d^2 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {5 b c^3 x \sqrt {-1+c x} \sqrt {1+c x}}{12 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {5 c^2 (a+b \text {arccosh}(c x))}{6 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {a+b \text {arccosh}(c x)}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}+\frac {5 c^2 (a+b \text {arccosh}(c x))}{2 d^2 \sqrt {d-c^2 d x^2}}+\frac {5 c^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \arctan \left (e^{\text {arccosh}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {13 b c^2 \sqrt {-1+c x} \sqrt {1+c x} \text {arctanh}(c x)}{6 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (5 i b c^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{\text {arccosh}(c x)}\right )}{2 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (5 i b c^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{\text {arccosh}(c x)}\right )}{2 d^2 \sqrt {d-c^2 d x^2}} \\ & = \frac {3 b c \sqrt {-1+c x} \sqrt {1+c x}}{4 d^2 x \sqrt {d-c^2 d x^2}}-\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{4 d^2 x \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {5 b c^3 x \sqrt {-1+c x} \sqrt {1+c x}}{12 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {5 c^2 (a+b \text {arccosh}(c x))}{6 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {a+b \text {arccosh}(c x)}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}}+\frac {5 c^2 (a+b \text {arccosh}(c x))}{2 d^2 \sqrt {d-c^2 d x^2}}+\frac {5 c^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \arctan \left (e^{\text {arccosh}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {13 b c^2 \sqrt {-1+c x} \sqrt {1+c x} \text {arctanh}(c x)}{6 d^2 \sqrt {d-c^2 d x^2}}-\frac {5 i b c^2 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right )}{2 d^2 \sqrt {d-c^2 d x^2}}+\frac {5 i b c^2 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right )}{2 d^2 \sqrt {d-c^2 d x^2}} \\ \end{align*}

Mathematica [A] (warning: unable to verify)

Time = 7.21 (sec) , antiderivative size = 534, normalized size of antiderivative = 1.11 \[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\sqrt {-d \left (-1+c^2 x^2\right )} \left (-\frac {a}{2 d^3 x^2}+\frac {a c^2}{3 d^3 \left (-1+c^2 x^2\right )^2}-\frac {2 a c^2}{d^3 \left (-1+c^2 x^2\right )}\right )+\frac {5 a c^2 \log (x)}{2 d^{5/2}}-\frac {5 a c^2 \log \left (d+\sqrt {d} \sqrt {-d \left (-1+c^2 x^2\right )}\right )}{2 d^{5/2}}+\frac {b c^2 \left (\frac {6 \sqrt {\frac {-1+c x}{1+c x}} (1+c x)}{c x}+\frac {6 (-1+c x) (1+c x) \text {arccosh}(c x)}{c^2 x^2}+26 \text {arccosh}(c x) \cosh ^2\left (\frac {1}{2} \text {arccosh}(c x)\right )-\coth \left (\frac {1}{2} \text {arccosh}(c x)\right )-\text {arccosh}(c x) \coth ^2\left (\frac {1}{2} \text {arccosh}(c x)\right )-30 i \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {arccosh}(c x) \log \left (1-i e^{-\text {arccosh}(c x)}\right )+30 i \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {arccosh}(c x) \log \left (1+i e^{-\text {arccosh}(c x)}\right )+26 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \log \left (\cosh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )-26 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \log \left (\sinh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )-30 i \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \operatorname {PolyLog}\left (2,-i e^{-\text {arccosh}(c x)}\right )+30 i \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \operatorname {PolyLog}\left (2,i e^{-\text {arccosh}(c x)}\right )-26 \text {arccosh}(c x) \sinh ^2\left (\frac {1}{2} \text {arccosh}(c x)\right )-\tanh \left (\frac {1}{2} \text {arccosh}(c x)\right )-\text {arccosh}(c x) \tanh ^2\left (\frac {1}{2} \text {arccosh}(c x)\right )\right )}{12 d^2 \sqrt {-d (-1+c x) (1+c x)}} \]

[In]

Integrate[(a + b*ArcCosh[c*x])/(x^3*(d - c^2*d*x^2)^(5/2)),x]

[Out]

Sqrt[-(d*(-1 + c^2*x^2))]*(-1/2*a/(d^3*x^2) + (a*c^2)/(3*d^3*(-1 + c^2*x^2)^2) - (2*a*c^2)/(d^3*(-1 + c^2*x^2)
)) + (5*a*c^2*Log[x])/(2*d^(5/2)) - (5*a*c^2*Log[d + Sqrt[d]*Sqrt[-(d*(-1 + c^2*x^2))]])/(2*d^(5/2)) + (b*c^2*
((6*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))/(c*x) + (6*(-1 + c*x)*(1 + c*x)*ArcCosh[c*x])/(c^2*x^2) + 26*ArcCosh
[c*x]*Cosh[ArcCosh[c*x]/2]^2 - Coth[ArcCosh[c*x]/2] - ArcCosh[c*x]*Coth[ArcCosh[c*x]/2]^2 - (30*I)*Sqrt[(-1 +
c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]*Log[1 - I/E^ArcCosh[c*x]] + (30*I)*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)
*ArcCosh[c*x]*Log[1 + I/E^ArcCosh[c*x]] + 26*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*Log[Cosh[ArcCosh[c*x]/2]] -
26*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*Log[Sinh[ArcCosh[c*x]/2]] - (30*I)*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x
)*PolyLog[2, (-I)/E^ArcCosh[c*x]] + (30*I)*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*PolyLog[2, I/E^ArcCosh[c*x]] -
26*ArcCosh[c*x]*Sinh[ArcCosh[c*x]/2]^2 - Tanh[ArcCosh[c*x]/2] - ArcCosh[c*x]*Tanh[ArcCosh[c*x]/2]^2))/(12*d^2
*Sqrt[-(d*(-1 + c*x)*(1 + c*x))])

Maple [A] (verified)

Time = 1.22 (sec) , antiderivative size = 655, normalized size of antiderivative = 1.37

method	result	size
default	\(-\frac {a}{2 d \,x^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {5 a \,c^{2}}{6 d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {5 a \,c^{2}}{2 d^{2} \sqrt {-c^{2} d \,x^{2}+d}}-\frac {5 a \,c^{2} \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )}{2 d^{\frac {5}{2}}}+b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (15 c^{4} x^{4} \operatorname {arccosh}\left (c x \right )+2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}-20 c^{2} x^{2} \operatorname {arccosh}\left (c x \right )-3 \sqrt {c x -1}\, \sqrt {c x +1}\, c x +3 \,\operatorname {arccosh}\left (c x \right )\right )}{6 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) x^{2}}-\frac {13 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) c^{2}}{6 d^{3} \left (c^{2} x^{2}-1\right )}+\frac {13 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right ) c^{2}}{6 d^{3} \left (c^{2} x^{2}-1\right )}-\frac {5 i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{2}}{2 d^{3} \left (c^{2} x^{2}-1\right )}+\frac {5 i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {dilog}\left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{2}}{2 d^{3} \left (c^{2} x^{2}-1\right )}-\frac {5 i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {dilog}\left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{2}}{2 d^{3} \left (c^{2} x^{2}-1\right )}+\frac {5 i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{2}}{2 d^{3} \left (c^{2} x^{2}-1\right )}\right )\)	\(655\)
parts	\(-\frac {a}{2 d \,x^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {5 a \,c^{2}}{6 d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {5 a \,c^{2}}{2 d^{2} \sqrt {-c^{2} d \,x^{2}+d}}-\frac {5 a \,c^{2} \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )}{2 d^{\frac {5}{2}}}+b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (15 c^{4} x^{4} \operatorname {arccosh}\left (c x \right )+2 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}-20 c^{2} x^{2} \operatorname {arccosh}\left (c x \right )-3 \sqrt {c x -1}\, \sqrt {c x +1}\, c x +3 \,\operatorname {arccosh}\left (c x \right )\right )}{6 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right ) x^{2}}-\frac {13 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right ) c^{2}}{6 d^{3} \left (c^{2} x^{2}-1\right )}+\frac {13 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \ln \left (\sqrt {c x -1}\, \sqrt {c x +1}+c x -1\right ) c^{2}}{6 d^{3} \left (c^{2} x^{2}-1\right )}-\frac {5 i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{2}}{2 d^{3} \left (c^{2} x^{2}-1\right )}+\frac {5 i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {dilog}\left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{2}}{2 d^{3} \left (c^{2} x^{2}-1\right )}-\frac {5 i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {dilog}\left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{2}}{2 d^{3} \left (c^{2} x^{2}-1\right )}+\frac {5 i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) c^{2}}{2 d^{3} \left (c^{2} x^{2}-1\right )}\right )\)	\(655\)

[In]

int((a+b*arccosh(c*x))/x^3/(-c^2*d*x^2+d)^(5/2),x,method=_RETURNVERBOSE)

[Out]

-1/2*a/d/x^2/(-c^2*d*x^2+d)^(3/2)+5/6*a*c^2/d/(-c^2*d*x^2+d)^(3/2)+5/2*a*c^2/d^2/(-c^2*d*x^2+d)^(1/2)-5/2*a*c^
2/d^(5/2)*ln((2*d+2*d^(1/2)*(-c^2*d*x^2+d)^(1/2))/x)+b*(-1/6*(-d*(c^2*x^2-1))^(1/2)*(15*c^4*x^4*arccosh(c*x)+2
*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c^3*x^3-20*c^2*x^2*arccosh(c*x)-3*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x+3*arccosh(c*x))
/d^3/(c^4*x^4-2*c^2*x^2+1)/x^2-13/6*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^3/(c^2*x^2-1)*ln(1+c*
x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*c^2+13/6*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^3/(c^2*x^2-1)*ln(
(c*x-1)^(1/2)*(c*x+1)^(1/2)+c*x-1)*c^2-5/2*I*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^3/(c^2*x^2-1
)*arccosh(c*x)*ln(1-I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))*c^2+5/2*I*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1
)^(1/2)/d^3/(c^2*x^2-1)*dilog(1+I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))*c^2-5/2*I*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^
(1/2)*(c*x+1)^(1/2)/d^3/(c^2*x^2-1)*dilog(1-I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))*c^2+5/2*I*(-d*(c^2*x^2-1))^(1
/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^3/(c^2*x^2-1)*arccosh(c*x)*ln(1+I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))*c^2)

Fricas [F]

\[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{3}} \,d x } \]

[In]

integrate((a+b*arccosh(c*x))/x^3/(-c^2*d*x^2+d)^(5/2),x, algorithm="fricas")

[Out]

integral(-sqrt(-c^2*d*x^2 + d)*(b*arccosh(c*x) + a)/(c^6*d^3*x^9 - 3*c^4*d^3*x^7 + 3*c^2*d^3*x^5 - d^3*x^3), x
)

Sympy [F(-1)]

Timed out. \[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\text {Timed out} \]

[In]

integrate((a+b*acosh(c*x))/x**3/(-c**2*d*x**2+d)**(5/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{3}} \,d x } \]

[In]

integrate((a+b*arccosh(c*x))/x^3/(-c^2*d*x^2+d)^(5/2),x, algorithm="maxima")

[Out]

-1/6*a*(15*c^2*log(2*sqrt(-c^2*d*x^2 + d)*sqrt(d)/abs(x) + 2*d/abs(x))/d^(5/2) - 15*c^2/(sqrt(-c^2*d*x^2 + d)*
d^2) - 5*c^2/((-c^2*d*x^2 + d)^(3/2)*d) + 3/((-c^2*d*x^2 + d)^(3/2)*d*x^2)) + b*integrate(log(c*x + sqrt(c*x +
1)*sqrt(c*x - 1))/((-c^2*d*x^2 + d)^(5/2)*x^3), x)

Giac [F]

\[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{3}} \,d x } \]

[In]

integrate((a+b*arccosh(c*x))/x^3/(-c^2*d*x^2+d)^(5/2),x, algorithm="giac")

[Out]

integrate((b*arccosh(c*x) + a)/((-c^2*d*x^2 + d)^(5/2)*x^3), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \text {arccosh}(c x)}{x^3 \left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{x^3\,{\left (d-c^2\,d\,x^2\right )}^{5/2}} \,d x \]

[In]

int((a + b*acosh(c*x))/(x^3*(d - c^2*d*x^2)^(5/2)),x)

[Out]

int((a + b*acosh(c*x))/(x^3*(d - c^2*d*x^2)^(5/2)), x)