\(\int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{x^6} \, dx\) [63]

Optimal result

Integrand size = 27, antiderivative size = 199 \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{x^6} \, dx=-\frac {b c \sqrt {d-c^2 d x^2}}{20 x^4 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 \sqrt {d-c^2 d x^2}}{30 x^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{5 d x^5}-\frac {2 c^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{15 d x^3}-\frac {2 b c^5 \sqrt {d-c^2 d x^2} \log (x)}{15 \sqrt {-1+c x} \sqrt {1+c x}} \]

[Out]

-1/5*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))/d/x^5-2/15*c^2*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))/d/x^3-1/20
*b*c*(-c^2*d*x^2+d)^(1/2)/x^4/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/30*b*c^3*(-c^2*d*x^2+d)^(1/2)/x^2/(c*x-1)^(1/2)/(c
*x+1)^(1/2)-2/15*b*c^5*ln(x)*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {277, 270, 5922, 12, 14} \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{x^6} \, dx=-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{5 d x^5}-\frac {2 c^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{15 d x^3}-\frac {b c \sqrt {d-c^2 d x^2}}{20 x^4 \sqrt {c x-1} \sqrt {c x+1}}-\frac {2 b c^5 \log (x) \sqrt {d-c^2 d x^2}}{15 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b c^3 \sqrt {d-c^2 d x^2}}{30 x^2 \sqrt {c x-1} \sqrt {c x+1}} \]

[In]

Int[(Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/x^6,x]

[Out]

-1/20*(b*c*Sqrt[d - c^2*d*x^2])/(x^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c^3*Sqrt[d - c^2*d*x^2])/(30*x^2*Sqrt[
-1 + c*x]*Sqrt[1 + c*x]) - ((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/(5*d*x^5) - (2*c^2*(d - c^2*d*x^2)^(3/
2)*(a + b*ArcCosh[c*x]))/(15*d*x^3) - (2*b*c^5*Sqrt[d - c^2*d*x^2]*Log[x])/(15*Sqrt[-1 + c*x]*Sqrt[1 + c*x])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 270

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] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rule 277

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x^(m + 1)*((a + b*x^n)^(p + 1)/(a*(m + 1))), x]
- Dist[b*((m + n*(p + 1) + 1)/(a*(m + 1))), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]

Rule 5922

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> With[{u = IntHide[x
^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcCosh[c*x], u, x] - Dist[b*c*Simp[Sqrt[d + e*x^2]/(Sqrt[1 + c*x]*Sqrt[-1 +
c*x])], Int[SimplifyIntegrand[u/Sqrt[d + e*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0
] && IntegerQ[p - 1/2] && NeQ[p, -2^(-1)] && (IGtQ[(m + 1)/2, 0] || ILtQ[(m + 2*p + 3)/2, 0])

Rubi steps \begin{align*} \text {integral}& = -\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{5 d x^5}-\frac {2 c^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{15 d x^3}-\frac {\left (b c \sqrt {d-c^2 d x^2}\right ) \int \frac {-3+c^2 x^2+2 c^4 x^4}{15 x^5} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{5 d x^5}-\frac {2 c^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{15 d x^3}-\frac {\left (b c \sqrt {d-c^2 d x^2}\right ) \int \frac {-3+c^2 x^2+2 c^4 x^4}{x^5} \, dx}{15 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{5 d x^5}-\frac {2 c^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{15 d x^3}-\frac {\left (b c \sqrt {d-c^2 d x^2}\right ) \int \left (-\frac {3}{x^5}+\frac {c^2}{x^3}+\frac {2 c^4}{x}\right ) \, dx}{15 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {b c \sqrt {d-c^2 d x^2}}{20 x^4 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 \sqrt {d-c^2 d x^2}}{30 x^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{5 d x^5}-\frac {2 c^2 \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))}{15 d x^3}-\frac {2 b c^5 \sqrt {d-c^2 d x^2} \log (x)}{15 \sqrt {-1+c x} \sqrt {1+c x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.64 \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{x^6} \, dx=\frac {\sqrt {d-c^2 d x^2} \left (12 (-1+c x)^{3/2} (1+c x)^{3/2} (a+b \text {arccosh}(c x))+8 c^2 x^2 (-1+c x)^{3/2} (1+c x)^{3/2} (a+b \text {arccosh}(c x))-b c x \left (3-2 c^2 x^2+8 c^4 x^4 \log (x)\right )\right )}{60 x^5 \sqrt {-1+c x} \sqrt {1+c x}} \]

[In]

Integrate[(Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/x^6,x]

[Out]

(Sqrt[d - c^2*d*x^2]*(12*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2)*(a + b*ArcCosh[c*x]) + 8*c^2*x^2*(-1 + c*x)^(3/2)*(1
+ c*x)^(3/2)*(a + b*ArcCosh[c*x]) - b*c*x*(3 - 2*c^2*x^2 + 8*c^4*x^4*Log[x])))/(60*x^5*Sqrt[-1 + c*x]*Sqrt[1
+ c*x])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1741\) vs. \(2(167)=334\).

Time = 1.14 (sec) , antiderivative size = 1742, normalized size of antiderivative = 8.75

method	result	size
default	\(\text {Expression too large to display}\)	\(1742\)
parts	\(\text {Expression too large to display}\)	\(1742\)

[In]

int((a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/x^6,x,method=_RETURNVERBOSE)

[Out]

-9/20*b*(-d*(c^2*x^2-1))^(1/2)/(15*c^6*x^6-5*c^4*x^4-15*c^2*x^2+9)/x^4/(c*x-1)^(1/2)/(c*x+1)^(1/2)*c+1/2*b*(-d
*(c^2*x^2-1))^(1/2)/(15*c^6*x^6-5*c^4*x^4-15*c^2*x^2+9)*x^4/(c*x-1)^(1/2)/(c*x+1)^(1/2)*c^9-11/12*b*(-d*(c^2*x
^2-1))^(1/2)/(15*c^6*x^6-5*c^4*x^4-15*c^2*x^2+9)*x^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)*c^7+21/20*b*(-d*(c^2*x^2-1))^
(1/2)/(15*c^6*x^6-5*c^4*x^4-15*c^2*x^2+9)/x^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)*c^3+1/6*b*(-d*(c^2*x^2-1))^(1/2)/(15
*c^6*x^6-5*c^4*x^4-15*c^2*x^2+9)*x^5/(c*x-1)/(c*x+1)*c^10-3/5*b*(-d*(c^2*x^2-1))^(1/2)/(15*c^6*x^6-5*c^4*x^4-1
5*c^2*x^2+9)*x^3/(c*x-1)/(c*x+1)*c^8+3/10*b*(-d*(c^2*x^2-1))^(1/2)/(15*c^6*x^6-5*c^4*x^4-15*c^2*x^2+9)*x/(c*x-
1)/(c*x+1)*c^6+9/5*b*(-d*(c^2*x^2-1))^(1/2)/(15*c^6*x^6-5*c^4*x^4-15*c^2*x^2+9)/x^5/(c*x-1)/(c*x+1)*arccosh(c*
x)-2/15*b*(-d*(c^2*x^2-1))^(1/2)/(15*c^6*x^6-5*c^4*x^4-15*c^2*x^2+9)*x^9/(c*x-1)/(c*x+1)*c^14+4/15*b*(-d*(c^2*
x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)*arccosh(c*x)*c^5-2/15*b*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)
^(1/2)*ln(1+(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)*c^5-1/4*b*(-d*(c^2*x^2-1))^(1/2)/(15*c^6*x^6-5*c^4*x^4-15*c^2
*x^2+9)/(c*x-1)^(1/2)/(c*x+1)^(1/2)*c^5+a*(-1/5/d/x^5*(-c^2*d*x^2+d)^(3/2)-2/15*c^2/d/x^3*(-c^2*d*x^2+d)^(3/2)
)+4/15*b*(-d*(c^2*x^2-1))^(1/2)/(15*c^6*x^6-5*c^4*x^4-15*c^2*x^2+9)*x^7/(c*x-1)/(c*x+1)*c^12-6/5*b*(-d*(c^2*x^
2-1))^(1/2)/(15*c^6*x^6-5*c^4*x^4-15*c^2*x^2+9)/(c*x-1)^(1/2)/(c*x+1)^(1/2)*arccosh(c*x)*c^5-3/10*b*(-d*(c^2*x
^2-1))^(1/2)/(15*c^6*x^6-5*c^4*x^4-15*c^2*x^2+9)*x^3*c^8+3/10*b*(-d*(c^2*x^2-1))^(1/2)/(15*c^6*x^6-5*c^4*x^4-1
5*c^2*x^2+9)*x*c^6+2/15*b*(-d*(c^2*x^2-1))^(1/2)/(15*c^6*x^6-5*c^4*x^4-15*c^2*x^2+9)*x^7*c^12-2/15*b*(-d*(c^2*
x^2-1))^(1/2)/(15*c^6*x^6-5*c^4*x^4-15*c^2*x^2+9)*x^5*c^10-27/5*b*(-d*(c^2*x^2-1))^(1/2)/(15*c^6*x^6-5*c^4*x^4
-15*c^2*x^2+9)/x^3/(c*x-1)/(c*x+1)*arccosh(c*x)*c^2+2/3*b*(-d*(c^2*x^2-1))^(1/2)/(15*c^6*x^6-5*c^4*x^4-15*c^2*
x^2+9)*x^4/(c*x-1)^(1/2)/(c*x+1)^(1/2)*arccosh(c*x)*c^9+2*b*(-d*(c^2*x^2-1))^(1/2)/(15*c^6*x^6-5*c^4*x^4-15*c^
2*x^2+9)*x^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)*arccosh(c*x)*c^7-2*b*(-d*(c^2*x^2-1))^(1/2)/(15*c^6*x^6-5*c^4*x^4-15*
c^2*x^2+9)*x^6/(c*x-1)^(1/2)/(c*x+1)^(1/2)*arccosh(c*x)*c^11-5/3*b*(-d*(c^2*x^2-1))^(1/2)/(15*c^6*x^6-5*c^4*x^
4-15*c^2*x^2+9)*x^5/(c*x-1)/(c*x+1)*arccosh(c*x)*c^10-17/3*b*(-d*(c^2*x^2-1))^(1/2)/(15*c^6*x^6-5*c^4*x^4-15*c
^2*x^2+9)*x^3/(c*x-1)/(c*x+1)*arccosh(c*x)*c^8+2*b*(-d*(c^2*x^2-1))^(1/2)/(15*c^6*x^6-5*c^4*x^4-15*c^2*x^2+9)*
x^7/(c*x-1)/(c*x+1)*arccosh(c*x)*c^12+98/15*b*(-d*(c^2*x^2-1))^(1/2)/(15*c^6*x^6-5*c^4*x^4-15*c^2*x^2+9)*x/(c*
x-1)/(c*x+1)*arccosh(c*x)*c^6+12/5*b*(-d*(c^2*x^2-1))^(1/2)/(15*c^6*x^6-5*c^4*x^4-15*c^2*x^2+9)/x/(c*x-1)/(c*x
+1)*arccosh(c*x)*c^4

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 549, normalized size of antiderivative = 2.76 \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{x^6} \, dx=\left [\frac {4 \, {\left (2 \, b c^{6} x^{6} - b c^{4} x^{4} - 4 \, b c^{2} x^{2} + 3 \, b\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + 4 \, {\left (b c^{7} x^{7} - b c^{5} x^{5}\right )} \sqrt {-d} \log \left (\frac {c^{2} d x^{6} + c^{2} d x^{2} - d x^{4} + \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} {\left (x^{4} - 1\right )} \sqrt {-d} - d}{c^{2} x^{4} - x^{2}}\right ) + {\left (2 \, b c^{3} x^{3} - {\left (2 \, b c^{3} - 3 \, b c\right )} x^{5} - 3 \, b c x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} + 4 \, {\left (2 \, a c^{6} x^{6} - a c^{4} x^{4} - 4 \, a c^{2} x^{2} + 3 \, a\right )} \sqrt {-c^{2} d x^{2} + d}}{60 \, {\left (c^{2} x^{7} - x^{5}\right )}}, -\frac {8 \, {\left (b c^{7} x^{7} - b c^{5} x^{5}\right )} \sqrt {d} \arctan \left (\frac {\sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} {\left (x^{2} + 1\right )} \sqrt {d}}{c^{2} d x^{4} - {\left (c^{2} + 1\right )} d x^{2} + d}\right ) - 4 \, {\left (2 \, b c^{6} x^{6} - b c^{4} x^{4} - 4 \, b c^{2} x^{2} + 3 \, b\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (2 \, b c^{3} x^{3} - {\left (2 \, b c^{3} - 3 \, b c\right )} x^{5} - 3 \, b c x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} - 4 \, {\left (2 \, a c^{6} x^{6} - a c^{4} x^{4} - 4 \, a c^{2} x^{2} + 3 \, a\right )} \sqrt {-c^{2} d x^{2} + d}}{60 \, {\left (c^{2} x^{7} - x^{5}\right )}}\right ] \]

[In]

integrate((a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/x^6,x, algorithm="fricas")

[Out]

[1/60*(4*(2*b*c^6*x^6 - b*c^4*x^4 - 4*b*c^2*x^2 + 3*b)*sqrt(-c^2*d*x^2 + d)*log(c*x + sqrt(c^2*x^2 - 1)) + 4*(
b*c^7*x^7 - b*c^5*x^5)*sqrt(-d)*log((c^2*d*x^6 + c^2*d*x^2 - d*x^4 + sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1)*(x
^4 - 1)*sqrt(-d) - d)/(c^2*x^4 - x^2)) + (2*b*c^3*x^3 - (2*b*c^3 - 3*b*c)*x^5 - 3*b*c*x)*sqrt(-c^2*d*x^2 + d)*
sqrt(c^2*x^2 - 1) + 4*(2*a*c^6*x^6 - a*c^4*x^4 - 4*a*c^2*x^2 + 3*a)*sqrt(-c^2*d*x^2 + d))/(c^2*x^7 - x^5), -1/
60*(8*(b*c^7*x^7 - b*c^5*x^5)*sqrt(d)*arctan(sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1)*(x^2 + 1)*sqrt(d)/(c^2*d*x
^4 - (c^2 + 1)*d*x^2 + d)) - 4*(2*b*c^6*x^6 - b*c^4*x^4 - 4*b*c^2*x^2 + 3*b)*sqrt(-c^2*d*x^2 + d)*log(c*x + sq
rt(c^2*x^2 - 1)) - (2*b*c^3*x^3 - (2*b*c^3 - 3*b*c)*x^5 - 3*b*c*x)*sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1) - 4*
(2*a*c^6*x^6 - a*c^4*x^4 - 4*a*c^2*x^2 + 3*a)*sqrt(-c^2*d*x^2 + d))/(c^2*x^7 - x^5)]

Sympy [F]

\[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{x^6} \, dx=\int \frac {\sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )}{x^{6}}\, dx \]

[In]

integrate((a+b*acosh(c*x))*(-c**2*d*x**2+d)**(1/2)/x**6,x)

[Out]

Integral(sqrt(-d*(c*x - 1)*(c*x + 1))*(a + b*acosh(c*x))/x**6, x)

Maxima [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.73 \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{x^6} \, dx=-\frac {1}{60} \, {\left (8 \, c^{4} \sqrt {-d} \log \left (x\right ) - \frac {2 \, c^{2} \sqrt {-d} x^{2} - 3 \, \sqrt {-d}}{x^{4}}\right )} b c - \frac {1}{15} \, b {\left (\frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} c^{2}}{d x^{3}} + \frac {3 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}}{d x^{5}}\right )} \operatorname {arcosh}\left (c x\right ) - \frac {1}{15} \, a {\left (\frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} c^{2}}{d x^{3}} + \frac {3 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}}{d x^{5}}\right )} \]

[In]

integrate((a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/x^6,x, algorithm="maxima")

[Out]

-1/60*(8*c^4*sqrt(-d)*log(x) - (2*c^2*sqrt(-d)*x^2 - 3*sqrt(-d))/x^4)*b*c - 1/15*b*(2*(-c^2*d*x^2 + d)^(3/2)*c
^2/(d*x^3) + 3*(-c^2*d*x^2 + d)^(3/2)/(d*x^5))*arccosh(c*x) - 1/15*a*(2*(-c^2*d*x^2 + d)^(3/2)*c^2/(d*x^3) + 3
*(-c^2*d*x^2 + d)^(3/2)/(d*x^5))

Giac [F(-2)]

Exception generated. \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{x^6} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/x^6,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. \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{x^6} \, dx=\int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,\sqrt {d-c^2\,d\,x^2}}{x^6} \,d x \]

[In]

int(((a + b*acosh(c*x))*(d - c^2*d*x^2)^(1/2))/x^6,x)

[Out]

int(((a + b*acosh(c*x))*(d - c^2*d*x^2)^(1/2))/x^6, x)