\(\int \frac {(d-c^2 d x^2)^{3/2} (a+b \text {arccosh}(c x))}{x} \, dx\) [84]

Optimal result

Integrand size = 27, antiderivative size = 292 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x} \, dx=-\frac {4 b c d x \sqrt {d-c^2 d x^2}}{3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d x^3 \sqrt {d-c^2 d x^2}}{9 \sqrt {-1+c x} \sqrt {1+c x}}+d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))-\frac {2 d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \arctan \left (e^{\text {arccosh}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {i b d \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {i b d \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \]

[Out]

1/3*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))+d*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)-4/3*b*c*d*x*(-c^2*d*x^2+
d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/9*b*c^3*d*x^3*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-2*d*(a+b
*arccosh(c*x))*arctan(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+I*b*d*
polylog(2,-I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-I*b*d*polylog
(2,I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {5930, 5926, 5947, 4265, 2317, 2438, 8, 41} \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x} \, dx=-\frac {2 d \sqrt {d-c^2 d x^2} \arctan \left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))+d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))+\frac {i b d \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {i b d \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {4 b c d x \sqrt {d-c^2 d x^2}}{3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b c^3 d x^3 \sqrt {d-c^2 d x^2}}{9 \sqrt {c x-1} \sqrt {c x+1}} \]

[In]

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

[Out]

(-4*b*c*d*x*Sqrt[d - c^2*d*x^2])/(3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c^3*d*x^3*Sqrt[d - c^2*d*x^2])/(9*Sqrt[
-1 + c*x]*Sqrt[1 + c*x]) + d*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]) + ((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[
c*x]))/3 - (2*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])*ArcTan[E^ArcCosh[c*x]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]
) + (I*b*d*Sqrt[d - c^2*d*x^2]*PolyLog[2, (-I)*E^ArcCosh[c*x]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (I*b*d*Sqrt[d
- c^2*d*x^2]*PolyLog[2, I*E^ArcCosh[c*x]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

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 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 5926

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

Rule 5930

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

Rule 5947

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]
), x_Symbol] :> Dist[(1/c^(m + 1))*Simp[Sqrt[1 + c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]], S
ubst[Int[(a + b*x)^n*Cosh[x]^m, x], x, ArcCosh[c*x]], x] /; FreeQ[{a, b, c, d1, e1, d2, e2}, x] && EqQ[e1, c*d
1] && EqQ[e2, (-c)*d2] && IGtQ[n, 0] && IntegerQ[m]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))+d \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{x} \, dx+\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \int (-1+c x) (1+c x) \, dx}{3 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))-\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \int \frac {a+b \text {arccosh}(c x)}{x \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \int \left (-1+c^2 x^2\right ) \, dx}{3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \int 1 \, dx}{\sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {4 b c d x \sqrt {d-c^2 d x^2}}{3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d x^3 \sqrt {d-c^2 d x^2}}{9 \sqrt {-1+c x} \sqrt {1+c x}}+d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))-\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \text {Subst}(\int (a+b x) \text {sech}(x) \, dx,x,\text {arccosh}(c x))}{\sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {4 b c d x \sqrt {d-c^2 d x^2}}{3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d x^3 \sqrt {d-c^2 d x^2}}{9 \sqrt {-1+c x} \sqrt {1+c x}}+d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))-\frac {2 d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \arctan \left (e^{\text {arccosh}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (i b d \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\text {arccosh}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (i b d \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\text {arccosh}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {4 b c d x \sqrt {d-c^2 d x^2}}{3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d x^3 \sqrt {d-c^2 d x^2}}{9 \sqrt {-1+c x} \sqrt {1+c x}}+d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))-\frac {2 d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \arctan \left (e^{\text {arccosh}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (i b d \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{\text {arccosh}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (i b d \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{\text {arccosh}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {4 b c d x \sqrt {d-c^2 d x^2}}{3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d x^3 \sqrt {d-c^2 d x^2}}{9 \sqrt {-1+c x} \sqrt {1+c x}}+d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))+\frac {1}{3} \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))-\frac {2 d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \arctan \left (e^{\text {arccosh}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {i b d \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {i b d \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \\ \end{align*}

Mathematica [A] (warning: unable to verify)

Time = 0.96 (sec) , antiderivative size = 336, normalized size of antiderivative = 1.15 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x} \, dx=-\frac {1}{3} a d \left (-4+c^2 x^2\right ) \sqrt {d-c^2 d x^2}-\frac {b d \sqrt {d-c^2 d x^2} \left (9 c x+12 \left (\frac {-1+c x}{1+c x}\right )^{3/2} (1+c x)^3 \text {arccosh}(c x)-\cosh (3 \text {arccosh}(c x))\right )}{36 \sqrt {\frac {-1+c x}{1+c x}} (1+c x)}+a d^{3/2} \log (x)-a d^{3/2} \log \left (d+\sqrt {d} \sqrt {d-c^2 d x^2}\right )+\frac {b d \sqrt {d-c^2 d x^2} \left (-c x+\sqrt {\frac {-1+c x}{1+c x}} \text {arccosh}(c x)+c x \sqrt {\frac {-1+c x}{1+c x}} \text {arccosh}(c x)+i \text {arccosh}(c x) \log \left (1-i e^{-\text {arccosh}(c x)}\right )-i \text {arccosh}(c x) \log \left (1+i e^{-\text {arccosh}(c x)}\right )+i \operatorname {PolyLog}\left (2,-i e^{-\text {arccosh}(c x)}\right )-i \operatorname {PolyLog}\left (2,i e^{-\text {arccosh}(c x)}\right )\right )}{\sqrt {\frac {-1+c x}{1+c x}} (1+c x)} \]

[In]

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

[Out]

-1/3*(a*d*(-4 + c^2*x^2)*Sqrt[d - c^2*d*x^2]) - (b*d*Sqrt[d - c^2*d*x^2]*(9*c*x + 12*((-1 + c*x)/(1 + c*x))^(3
/2)*(1 + c*x)^3*ArcCosh[c*x] - Cosh[3*ArcCosh[c*x]]))/(36*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)) + a*d^(3/2)*Lo
g[x] - a*d^(3/2)*Log[d + Sqrt[d]*Sqrt[d - c^2*d*x^2]] + (b*d*Sqrt[d - c^2*d*x^2]*(-(c*x) + Sqrt[(-1 + c*x)/(1
+ c*x)]*ArcCosh[c*x] + c*x*Sqrt[(-1 + c*x)/(1 + c*x)]*ArcCosh[c*x] + I*ArcCosh[c*x]*Log[1 - I/E^ArcCosh[c*x]]
- I*ArcCosh[c*x]*Log[1 + I/E^ArcCosh[c*x]] + I*PolyLog[2, (-I)/E^ArcCosh[c*x]] - I*PolyLog[2, I/E^ArcCosh[c*x]
]))/(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))

Maple [A] (verified)

Time = 2.03 (sec) , antiderivative size = 499, normalized size of antiderivative = 1.71

method	result	size
default	\(\frac {\left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} a}{3}-a \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right ) d^{\frac {3}{2}}+a d \sqrt {-c^{2} d \,x^{2}+d}+\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) d}{\sqrt {c x -1}\, \sqrt {c x +1}}-\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) d}{\sqrt {c x -1}\, \sqrt {c x +1}}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \,c^{3} x^{3}}{9 \sqrt {c x +1}\, \sqrt {c x -1}}-\frac {4 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d c x}{3 \sqrt {c x +1}\, \sqrt {c x -1}}-\frac {4 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \,\operatorname {arccosh}\left (c x \right )}{3 \left (c x +1\right ) \left (c x -1\right )}-\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {dilog}\left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) d}{\sqrt {c x -1}\, \sqrt {c x +1}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \,\operatorname {arccosh}\left (c x \right ) x^{4} c^{4}}{3 \left (c x +1\right ) \left (c x -1\right )}+\frac {5 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \,\operatorname {arccosh}\left (c x \right ) x^{2} c^{2}}{3 \left (c x +1\right ) \left (c x -1\right )}+\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {dilog}\left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) d}{\sqrt {c x -1}\, \sqrt {c x +1}}\)	\(499\)
parts	\(\frac {\left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} a}{3}-a \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right ) d^{\frac {3}{2}}+a d \sqrt {-c^{2} d \,x^{2}+d}+\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) d}{\sqrt {c x -1}\, \sqrt {c x +1}}-\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (c x \right ) \ln \left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) d}{\sqrt {c x -1}\, \sqrt {c x +1}}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \,c^{3} x^{3}}{9 \sqrt {c x +1}\, \sqrt {c x -1}}-\frac {4 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d c x}{3 \sqrt {c x +1}\, \sqrt {c x -1}}-\frac {4 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \,\operatorname {arccosh}\left (c x \right )}{3 \left (c x +1\right ) \left (c x -1\right )}-\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {dilog}\left (1-i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) d}{\sqrt {c x -1}\, \sqrt {c x +1}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \,\operatorname {arccosh}\left (c x \right ) x^{4} c^{4}}{3 \left (c x +1\right ) \left (c x -1\right )}+\frac {5 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \,\operatorname {arccosh}\left (c x \right ) x^{2} c^{2}}{3 \left (c x +1\right ) \left (c x -1\right )}+\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \operatorname {dilog}\left (1+i \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )\right ) d}{\sqrt {c x -1}\, \sqrt {c x +1}}\)	\(499\)

[In]

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

[Out]

1/3*(-c^2*d*x^2+d)^(3/2)*a-a*ln((2*d+2*d^(1/2)*(-c^2*d*x^2+d)^(1/2))/x)*d^(3/2)+a*d*(-c^2*d*x^2+d)^(1/2)+I*b*(
-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)*arccosh(c*x)*ln(1+I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))*d-I*b
*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)*arccosh(c*x)*ln(1-I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))*d+1
/9*b*(-d*(c^2*x^2-1))^(1/2)*d/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c^3*x^3-4/3*b*(-d*(c^2*x^2-1))^(1/2)*d/(c*x+1)^(1/2)
/(c*x-1)^(1/2)*c*x-4/3*b*(-d*(c^2*x^2-1))^(1/2)*d/(c*x+1)/(c*x-1)*arccosh(c*x)-I*b*(-d*(c^2*x^2-1))^(1/2)/(c*x
-1)^(1/2)/(c*x+1)^(1/2)*dilog(1-I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))*d-1/3*b*(-d*(c^2*x^2-1))^(1/2)*d/(c*x+1)/
(c*x-1)*arccosh(c*x)*x^4*c^4+5/3*b*(-d*(c^2*x^2-1))^(1/2)*d/(c*x+1)/(c*x-1)*arccosh(c*x)*x^2*c^2+I*b*(-d*(c^2*
x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)*dilog(1+I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))*d

Fricas [F]

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

[In]

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

[Out]

integral(-(a*c^2*d*x^2 - a*d + (b*c^2*d*x^2 - b*d)*arccosh(c*x))*sqrt(-c^2*d*x^2 + d)/x, x)

Sympy [F]

\[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{x} \, dx=\int \frac {\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )}{x}\, dx \]

[In]

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

[Out]

Integral((-d*(c*x - 1)*(c*x + 1))**(3/2)*(a + b*acosh(c*x))/x, x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-2)]

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

[In]

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

[In]

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

[Out]

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