3.1.92 \(\int \frac {(d-c^2 d x^2)^{5/2} (a+b \cosh ^{-1}(c x))}{x^6} \, dx\) [92]

Optimal. Leaf size=293 \[ -\frac {b c d^2 \sqrt {d-c^2 d x^2}}{20 x^4 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {11 b c^3 d^2 \sqrt {d-c^2 d x^2}}{30 x^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {c^4 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{x}+\frac {c^2 d \left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}-\frac {\left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )}{5 x^5}+\frac {c^5 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b \sqrt {-1+c x} \sqrt {1+c x}}+\frac {23 b c^5 d^2 \sqrt {d-c^2 d x^2} \log (x)}{15 \sqrt {-1+c x} \sqrt {1+c x}} \]

[Out]

1/3*c^2*d*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))/x^3-1/5*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))/x^5-c^4*d^2*
(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/x-1/20*b*c*d^2*(-c^2*d*x^2+d)^(1/2)/x^4/(c*x-1)^(1/2)/(c*x+1)^(1/2)+11
/30*b*c^3*d^2*(-c^2*d*x^2+d)^(1/2)/x^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/2*c^5*d^2*(a+b*arccosh(c*x))^2*(-c^2*d*x^
2+d)^(1/2)/b/(c*x-1)^(1/2)/(c*x+1)^(1/2)+23/15*b*c^5*d^2*ln(x)*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2
)

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Rubi [A]
time = 0.31, antiderivative size = 293, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {5928, 5924, 29, 5893, 74, 14, 272, 45} \begin {gather*} -\frac {\left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )}{5 x^5}+\frac {c^2 d \left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}+\frac {c^5 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b \sqrt {c x-1} \sqrt {c x+1}}-\frac {c^4 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{x}-\frac {b c d^2 \sqrt {d-c^2 d x^2}}{20 x^4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {23 b c^5 d^2 \log (x) \sqrt {d-c^2 d x^2}}{15 \sqrt {c x-1} \sqrt {c x+1}}+\frac {11 b c^3 d^2 \sqrt {d-c^2 d x^2}}{30 x^2 \sqrt {c x-1} \sqrt {c x+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/x^6,x]

[Out]

-1/20*(b*c*d^2*Sqrt[d - c^2*d*x^2])/(x^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (11*b*c^3*d^2*Sqrt[d - c^2*d*x^2])/(3
0*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (c^4*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/x + (c^2*d*(d - c^2*d
*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/(3*x^3) - ((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/(5*x^5) + (c^5*d^2*Sq
rt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(2*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (23*b*c^5*d^2*Sqrt[d - c^2*d*x^
2]*Log[x])/(15*Sqrt[-1 + c*x]*Sqrt[1 + c*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 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 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 272

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

Rule 5893

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

Rule 5924

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

Rule 5928

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

Rubi steps

\begin {align*} \int \frac {\left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )}{x^6} \, dx &=\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {(-1+c x)^{5/2} (1+c x)^{5/2} \left (a+b \cosh ^{-1}(c x)\right )}{x^6} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {d^2 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{5 x^5}+\frac {\left (b c d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (-1+c^2 x^2\right )^2}{x^5} \, dx}{5 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {(-1+c x)^{3/2} (1+c x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right )}{x^4} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {c^2 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}-\frac {d^2 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{5 x^5}+\frac {\left (b c d^2 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {\left (-1+c^2 x\right )^2}{x^3} \, dx,x,x^2\right )}{10 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b c^3 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {-1+c^2 x^2}{x^3} \, dx}{3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (c^4 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {\sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{x^2} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {c^4 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{x}+\frac {c^2 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}-\frac {d^2 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{5 x^5}+\frac {\left (b c d^2 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \left (\frac {1}{x^3}-\frac {2 c^2}{x^2}+\frac {c^4}{x}\right ) \, dx,x,x^2\right )}{10 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b c^3 d^2 \sqrt {d-c^2 d x^2}\right ) \int \left (-\frac {1}{x^3}+\frac {c^2}{x}\right ) \, dx}{3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b c^5 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{x} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (c^6 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {a+b \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {b c d^2 \sqrt {d-c^2 d x^2}}{20 x^4 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {11 b c^3 d^2 \sqrt {d-c^2 d x^2}}{30 x^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {c^4 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{x}+\frac {c^2 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 x^3}-\frac {d^2 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{5 x^5}+\frac {c^5 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b \sqrt {-1+c x} \sqrt {1+c x}}+\frac {23 b c^5 d^2 \sqrt {d-c^2 d x^2} \log (x)}{15 \sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}

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Mathematica [A]
time = 2.20, size = 400, normalized size = 1.37 \begin {gather*} \frac {d^2 \left (8 a d \sqrt {\frac {-1+c x}{1+c x}} \left (-1+c^2 x^2\right ) \left (3-11 c^2 x^2+23 c^4 x^4\right )+120 a c^5 \sqrt {d} x^5 \sqrt {\frac {-1+c x}{1+c x}} \sqrt {d-c^2 d x^2} \text {ArcTan}\left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )+40 b c^2 d x^2 (1-c x) \left (c x-2 \left (\frac {-1+c x}{1+c x}\right )^{3/2} (1+c x)^3 \cosh ^{-1}(c x)+2 c^3 x^3 \log (c x)\right )-60 b c^4 d x^4 (1-c x) \left (2 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \cosh ^{-1}(c x)-c x \left (\cosh ^{-1}(c x)^2+2 \log (c x)\right )\right )-b d (1-c x) \left (20 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \cosh ^{-1}(c x)+\cosh \left (5 \cosh ^{-1}(c x)\right ) \log (c x)+\cosh \left (3 \cosh ^{-1}(c x)\right ) (-1+5 \log (c x))+c x (3+10 \log (c x))-5 \cosh ^{-1}(c x) \sinh \left (3 \cosh ^{-1}(c x)\right )-\cosh ^{-1}(c x) \sinh \left (5 \cosh ^{-1}(c x)\right )\right )\right )}{120 x^5 \sqrt {\frac {-1+c x}{1+c x}} \sqrt {d-c^2 d x^2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/x^6,x]

[Out]

(d^2*(8*a*d*Sqrt[(-1 + c*x)/(1 + c*x)]*(-1 + c^2*x^2)*(3 - 11*c^2*x^2 + 23*c^4*x^4) + 120*a*c^5*Sqrt[d]*x^5*Sq
rt[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + 40*b
*c^2*d*x^2*(1 - c*x)*(c*x - 2*((-1 + c*x)/(1 + c*x))^(3/2)*(1 + c*x)^3*ArcCosh[c*x] + 2*c^3*x^3*Log[c*x]) - 60
*b*c^4*d*x^4*(1 - c*x)*(2*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x] - c*x*(ArcCosh[c*x]^2 + 2*Log[c*x]
)) - b*d*(1 - c*x)*(20*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x] + Cosh[5*ArcCosh[c*x]]*Log[c*x] + Cos
h[3*ArcCosh[c*x]]*(-1 + 5*Log[c*x]) + c*x*(3 + 10*Log[c*x]) - 5*ArcCosh[c*x]*Sinh[3*ArcCosh[c*x]] - ArcCosh[c*
x]*Sinh[5*ArcCosh[c*x]])))/(120*x^5*Sqrt[(-1 + c*x)/(1 + c*x)]*Sqrt[d - c^2*d*x^2])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2428\) vs. \(2(251)=502\).
time = 8.07, size = 2429, normalized size = 8.29

method	result	size
default	\(\text {Expression too large to display}\)	\(2429\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))/x^6,x,method=_RETURNVERBOSE)

[Out]

-a*c^6*d^3/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))-a*c^6*d^2*x*(-c^2*d*x^2+d)^(1/2)+2/15*a*
c^2/d/x^3*(-c^2*d*x^2+d)^(7/2)-46/15*b*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)*arccosh(c*x)*d^2*c^5
+1/2*b*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)*arccosh(c*x)^2*d^2*c^5+23/15*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)*d^2*c^5-175/4*b*(-d*(c^2*x^2-1))^(1/2)
*d^2/(1035*c^8*x^8-765*c^6*x^6+325*c^4*x^4-75*c^2*x^2+9)/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c^5-8/15*a*c^4/d/x*(-c^2*
d*x^2+d)^(7/2)-2/3*a*c^6*d*x*(-c^2*d*x^2+d)^(3/2)+9/5*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(1035*c^8*x^8-765*c^6*x^6+3
25*c^4*x^4-75*c^2*x^2+9)/x^5/(c*x+1)/(c*x-1)*arccosh(c*x)-69/20*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(1035*c^8*x^8-765
*c^6*x^6+325*c^4*x^4-75*c^2*x^2+9)*x/(c*x+1)/(c*x-1)*c^6-5819/30*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(1035*c^8*x^8-76
5*c^6*x^6+325*c^4*x^4-75*c^2*x^2+9)*x^9/(c*x+1)/(c*x-1)*c^14+69/5*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(1035*c^8*x^8-7
65*c^6*x^6+325*c^4*x^4-75*c^2*x^2+9)/(c*x+1)^(1/2)/(c*x-1)^(1/2)*arccosh(c*x)*c^5-1329/4*b*(-d*(c^2*x^2-1))^(1
/2)*d^2/(1035*c^8*x^8-765*c^6*x^6+325*c^4*x^4-75*c^2*x^2+9)*x^4/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c^9+759/2*b*(-d*(c
^2*x^2-1))^(1/2)*d^2/(1035*c^8*x^8-765*c^6*x^6+325*c^4*x^4-75*c^2*x^2+9)*x^6/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c^11+
1889/12*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(1035*c^8*x^8-765*c^6*x^6+325*c^4*x^4-75*c^2*x^2+9)*x^2/(c*x+1)^(1/2)/(c*
x-1)^(1/2)*c^7+141/20*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(1035*c^8*x^8-765*c^6*x^6+325*c^4*x^4-75*c^2*x^2+9)/x^2/(c*
x+1)^(1/2)/(c*x-1)^(1/2)*c^3-9/20*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(1035*c^8*x^8-765*c^6*x^6+325*c^4*x^4-75*c^2*x^
2+9)/x^4/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c+18791/60*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(1035*c^8*x^8-765*c^6*x^6+325*c^4
*x^4-75*c^2*x^2+9)*x^7/(c*x+1)/(c*x-1)*c^12-943/6*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(1035*c^8*x^8-765*c^6*x^6+325*c
^4*x^4-75*c^2*x^2+9)*x^5/(c*x+1)/(c*x-1)*c^10+207/5*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(1035*c^8*x^8-765*c^6*x^6+325
*c^4*x^4-75*c^2*x^2+9)*x^3/(c*x+1)/(c*x-1)*c^8-1/5*a/d/x^5*(-c^2*d*x^2+d)^(7/2)-8/15*a*c^6*x*(-c^2*d*x^2+d)^(5
/2)-115*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(1035*c^8*x^8-765*c^6*x^6+325*c^4*x^4-75*c^2*x^2+9)*x^2/(c*x+1)^(1/2)/(c*
x-1)^(1/2)*arccosh(c*x)*c^7+777/5*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(1035*c^8*x^8-765*c^6*x^6+325*c^4*x^4-75*c^2*x^
2+9)/x/(c*x+1)/(c*x-1)*arccosh(c*x)*c^4+5318/3*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(1035*c^8*x^8-765*c^6*x^6+325*c^4*
x^4-75*c^2*x^2+9)*x^3/(c*x+1)/(c*x-1)*arccosh(c*x)*c^8-1587*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(1035*c^8*x^8-765*c^6
*x^6+325*c^4*x^4-75*c^2*x^2+9)*x^9/(c*x+1)/(c*x-1)*arccosh(c*x)*c^14+3519*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(1035*c
^8*x^8-765*c^6*x^6+325*c^4*x^4-75*c^2*x^2+9)*x^7/(c*x+1)/(c*x-1)*arccosh(c*x)*c^12-117/5*b*(-d*(c^2*x^2-1))^(1
/2)*d^2/(1035*c^8*x^8-765*c^6*x^6+325*c^4*x^4-75*c^2*x^2+9)/x^3/(c*x+1)/(c*x-1)*arccosh(c*x)*c^2-9595/3*b*(-d*
(c^2*x^2-1))^(1/2)*d^2/(1035*c^8*x^8-765*c^6*x^6+325*c^4*x^4-75*c^2*x^2+9)*x^5/(c*x+1)/(c*x-1)*arccosh(c*x)*c^
10-9602/15*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(1035*c^8*x^8-765*c^6*x^6+325*c^4*x^4-75*c^2*x^2+9)*x/(c*x+1)/(c*x-1)*
arccosh(c*x)*c^6+1587*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(1035*c^8*x^8-765*c^6*x^6+325*c^4*x^4-75*c^2*x^2+9)*x^8/(c*
x+1)^(1/2)/(c*x-1)^(1/2)*arccosh(c*x)*c^13-1173*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(1035*c^8*x^8-765*c^6*x^6+325*c^4
*x^4-75*c^2*x^2+9)*x^6/(c*x+1)^(1/2)/(c*x-1)^(1/2)*arccosh(c*x)*c^11+1495/3*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(1035
*c^8*x^8-765*c^6*x^6+325*c^4*x^4-75*c^2*x^2+9)*x^4/(c*x+1)^(1/2)/(c*x-1)^(1/2)*arccosh(c*x)*c^9-69/20*b*(-d*(c
^2*x^2-1))^(1/2)*d^2/(1035*c^8*x^8-765*c^6*x^6+325*c^4*x^4-75*c^2*x^2+9)*x*c^6+5819/30*b*(-d*(c^2*x^2-1))^(1/2
)*d^2/(1035*c^8*x^8-765*c^6*x^6+325*c^4*x^4-75*c^2*x^2+9)*x^7*c^12-7153/60*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(1035*
c^8*x^8-765*c^6*x^6+325*c^4*x^4-75*c^2*x^2+9)*x^5*c^10+759/20*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(1035*c^8*x^8-765*c
^6*x^6+325*c^4*x^4-75*c^2*x^2+9)*x^3*c^8

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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((-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))/x^6,x, algorithm="maxima")

[Out]

-1/15*(10*(-c^2*d*x^2 + d)^(3/2)*c^6*d*x + 15*sqrt(-c^2*d*x^2 + d)*c^6*d^2*x + 15*c^5*d^(5/2)*arcsin(c*x) + 8*
(-c^2*d*x^2 + d)^(5/2)*c^4/x - 2*(-c^2*d*x^2 + d)^(7/2)*c^2/(d*x^3) + 3*(-c^2*d*x^2 + d)^(7/2)/(d*x^5))*a + b*
integrate((-c^2*d*x^2 + d)^(5/2)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/x^6, 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((-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))/x^6,x, algorithm="fricas")

[Out]

integral((a*c^4*d^2*x^4 - 2*a*c^2*d^2*x^2 + a*d^2 + (b*c^4*d^2*x^4 - 2*b*c^2*d^2*x^2 + b*d^2)*arccosh(c*x))*sq
rt(-c^2*d*x^2 + d)/x^6, x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )}{x^{6}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c**2*d*x**2+d)**(5/2)*(a+b*acosh(c*x))/x**6,x)

[Out]

Integral((-d*(c*x - 1)*(c*x + 1))**(5/2)*(a + b*acosh(c*x))/x**6, x)

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{5/2}}{x^6} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*acosh(c*x))*(d - c^2*d*x^2)^(5/2))/x^6,x)

[Out]

int(((a + b*acosh(c*x))*(d - c^2*d*x^2)^(5/2))/x^6, x)

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