3.93 \(\int \frac{(d-c^2 d x^2)^{5/2} (a+b \cosh ^{-1}(c x))}{x^8} \, dx\)
Optimal. Leaf size=219 \[ -\frac{\left (d-c^2 d x^2\right )^{7/2} \left (a+b \cosh ^{-1}(c x)\right )}{7 d x^7}-\frac{3 b c^5 d^2 \sqrt{d-c^2 d x^2}}{14 x^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{3 b c^3 d^2 \sqrt{d-c^2 d x^2}}{28 x^4 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c d^2 \sqrt{d-c^2 d x^2}}{42 x^6 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c^7 d^2 \log (x) \sqrt{d-c^2 d x^2}}{7 \sqrt{c x-1} \sqrt{c x+1}} \]
[Out]
-(b*c*d^2*Sqrt[d - c^2*d*x^2])/(42*x^6*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (3*b*c^3*d^2*Sqrt[d - c^2*d*x^2])/(28*x
^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (3*b*c^5*d^2*Sqrt[d - c^2*d*x^2])/(14*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (
(d - c^2*d*x^2)^(7/2)*(a + b*ArcCosh[c*x]))/(7*d*x^7) - (b*c^7*d^2*Sqrt[d - c^2*d*x^2]*Log[x])/(7*Sqrt[-1 + c*
x]*Sqrt[1 + c*x])
________________________________________________________________________________________
Rubi [A] time = 0.378647, antiderivative size = 234, normalized size of antiderivative =
1.07, number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules
used = {5798, 5724, 266, 43} \[ -\frac{d^2 (1-c x)^3 (c x+1)^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{7 x^7}-\frac{3 b c^5 d^2 \sqrt{d-c^2 d x^2}}{14 x^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{3 b c^3 d^2 \sqrt{d-c^2 d x^2}}{28 x^4 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c d^2 \sqrt{d-c^2 d x^2}}{42 x^6 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c^7 d^2 \log (x) \sqrt{d-c^2 d x^2}}{7 \sqrt{c x-1} \sqrt{c x+1}} \]
Antiderivative was successfully verified.
[In]
Int[((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/x^8,x]
[Out]
-(b*c*d^2*Sqrt[d - c^2*d*x^2])/(42*x^6*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (3*b*c^3*d^2*Sqrt[d - c^2*d*x^2])/(28*x
^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (3*b*c^5*d^2*Sqrt[d - c^2*d*x^2])/(14*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (
d^2*(1 - c*x)^3*(1 + c*x)^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(7*x^7) - (b*c^7*d^2*Sqrt[d - c^2*d*x^2]
*Log[x])/(7*Sqrt[-1 + c*x]*Sqrt[1 + c*x])
Rule 5798
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Dist
[((-d)^IntPart[p]*(d + e*x^2)^FracPart[p])/((1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p]), Int[(f*x)^m*(1 + c*
x)^p*(-1 + c*x)^p*(a + b*ArcCosh[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[c^2*d + e, 0]
&& !IntegerQ[p]
Rule 5724
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x
_))^(p_.), x_Symbol] :> Simp[((f*x)^(m + 1)*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n)/(d
1*d2*f*(m + 1)), x] + Dist[(b*c*n*(-(d1*d2))^IntPart[p]*(d1 + e1*x)^FracPart[p]*(d2 + e2*x)^FracPart[p])/(f*(m
+ 1)*(1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p]), Int[(f*x)^(m + 1)*(-1 + c^2*x^2)^(p + 1/2)*(a + b*ArcCosh
[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, m, p}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2,
0] && GtQ[n, 0] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1] && IntegerQ[p + 1/2]
Rule 266
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 43
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])
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^8} \, 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^8} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{d^2 (1-c x)^3 (1+c x)^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{7 x^7}-\frac{\left (b c d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{\left (-1+c^2 x^2\right )^3}{x^7} \, dx}{7 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{d^2 (1-c x)^3 (1+c x)^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{7 x^7}-\frac{\left (b c d^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{\left (-1+c^2 x\right )^3}{x^4} \, dx,x,x^2\right )}{14 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{d^2 (1-c x)^3 (1+c x)^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{7 x^7}-\frac{\left (b c d^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \left (-\frac{1}{x^4}+\frac{3 c^2}{x^3}-\frac{3 c^4}{x^2}+\frac{c^6}{x}\right ) \, dx,x,x^2\right )}{14 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{b c d^2 \sqrt{d-c^2 d x^2}}{42 x^6 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{3 b c^3 d^2 \sqrt{d-c^2 d x^2}}{28 x^4 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{3 b c^5 d^2 \sqrt{d-c^2 d x^2}}{14 x^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{d^2 (1-c x)^3 (1+c x)^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{7 x^7}-\frac{b c^7 d^2 \sqrt{d-c^2 d x^2} \log (x)}{7 \sqrt{-1+c x} \sqrt{1+c x}}\\ \end{align*}
Mathematica [A] time = 0.0949244, size = 105, normalized size = 0.48 \[ \frac{d^2 \sqrt{d-c^2 d x^2} \left (12 (c x-1)^{7/2} (c x+1)^{7/2} \left (a+b \cosh ^{-1}(c x)\right )-b c x \left (18 c^4 x^4-9 c^2 x^2+12 c^6 x^6 \log (x)+2\right )\right )}{84 x^7 \sqrt{c x-1} \sqrt{c x+1}} \]
Antiderivative was successfully verified.
[In]
Integrate[((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/x^8,x]
[Out]
(d^2*Sqrt[d - c^2*d*x^2]*(12*(-1 + c*x)^(7/2)*(1 + c*x)^(7/2)*(a + b*ArcCosh[c*x]) - b*c*x*(2 - 9*c^2*x^2 + 18
*c^4*x^4 + 12*c^6*x^6*Log[x])))/(84*x^7*Sqrt[-1 + c*x]*Sqrt[1 + c*x])
________________________________________________________________________________________
Maple [B] time = 0.301, size = 3775, normalized size = 17.2 \begin{align*} \text{output too large to display} \end{align*}
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^8,x)
[Out]
-3/14*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(7*c^12*x^12-21*c^10*x^10+35*c^8*x^8-35*c^6*x^6+21*c^4*x^4-7*c^2*x^2+1)*x^1
1*c^18+1/7*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(7*c^12*x^12-21*c^10*x^10+35*c^8*x^8-35*c^6*x^6+21*c^4*x^4-7*c^2*x^2+1
)/x^7/(c*x+1)/(c*x-1)*arccosh(c*x)+3/14*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(7*c^12*x^12-21*c^10*x^10+35*c^8*x^8-35*c
^6*x^6+21*c^4*x^4-7*c^2*x^2+1)*x^13/(c*x+1)/(c*x-1)*c^20-27/28*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(7*c^12*x^12-21*c^
10*x^10+35*c^8*x^8-35*c^6*x^6+21*c^4*x^4-7*c^2*x^2+1)*x^11/(c*x+1)/(c*x-1)*c^18-41/28*b*(-d*(c^2*x^2-1))^(1/2)
*d^2/(7*c^12*x^12-21*c^10*x^10+35*c^8*x^8-35*c^6*x^6+21*c^4*x^4-7*c^2*x^2+1)/x^2/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c
^5+23/84*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(7*c^12*x^12-21*c^10*x^10+35*c^8*x^8-35*c^6*x^6+21*c^4*x^4-7*c^2*x^2+1)/
x^4/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c^3-1/42*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(7*c^12*x^12-21*c^10*x^10+35*c^8*x^8-35*
c^6*x^6+21*c^4*x^4-7*c^2*x^2+1)/x^6/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c-1/7*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(7*c^12*x^1
2-21*c^10*x^10+35*c^8*x^8-35*c^6*x^6+21*c^4*x^4-7*c^2*x^2+1)/(c*x+1)^(1/2)/(c*x-1)^(1/2)*arccosh(c*x)*c^7-3/2*
b*(-d*(c^2*x^2-1))^(1/2)*d^2/(7*c^12*x^12-21*c^10*x^10+35*c^8*x^8-35*c^6*x^6+21*c^4*x^4-7*c^2*x^2+1)*x^10/(c*x
+1)^(1/2)/(c*x-1)^(1/2)*c^17+21/4*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(7*c^12*x^12-21*c^10*x^10+35*c^8*x^8-35*c^6*x^6
+21*c^4*x^4-7*c^2*x^2+1)*x^8/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c^15-119/12*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(7*c^12*x^12
-21*c^10*x^10+35*c^8*x^8-35*c^6*x^6+21*c^4*x^4-7*c^2*x^2+1)*x^6/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c^13+47/4*b*(-d*(c
^2*x^2-1))^(1/2)*d^2/(7*c^12*x^12-21*c^10*x^10+35*c^8*x^8-35*c^6*x^6+21*c^4*x^4-7*c^2*x^2+1)*x^4/(c*x+1)^(1/2)
/(c*x-1)^(1/2)*c^11-109/12*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(7*c^12*x^12-21*c^10*x^10+35*c^8*x^8-35*c^6*x^6+21*c^4
*x^4-7*c^2*x^2+1)*x^2/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c^9+73/42*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(7*c^12*x^12-21*c^10*
x^10+35*c^8*x^8-35*c^6*x^6+21*c^4*x^4-7*c^2*x^2+1)*x^9/(c*x+1)/(c*x-1)*c^16-67/42*b*(-d*(c^2*x^2-1))^(1/2)*d^2
/(7*c^12*x^12-21*c^10*x^10+35*c^8*x^8-35*c^6*x^6+21*c^4*x^4-7*c^2*x^2+1)*x^7/(c*x+1)/(c*x-1)*c^14+11/14*b*(-d*
(c^2*x^2-1))^(1/2)*d^2/(7*c^12*x^12-21*c^10*x^10+35*c^8*x^8-35*c^6*x^6+21*c^4*x^4-7*c^2*x^2+1)*x^5/(c*x+1)/(c*
x-1)*c^12-17/84*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(7*c^12*x^12-21*c^10*x^10+35*c^8*x^8-35*c^6*x^6+21*c^4*x^4-7*c^2*
x^2+1)*x^3/(c*x+1)/(c*x-1)*c^10+1/42*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(7*c^12*x^12-21*c^10*x^10+35*c^8*x^8-35*c^6*
x^6+21*c^4*x^4-7*c^2*x^2+1)*x/(c*x+1)/(c*x-1)*c^8-7*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(7*c^12*x^12-21*c^10*x^10+35*
c^8*x^8-35*c^6*x^6+21*c^4*x^4-7*c^2*x^2+1)*x^11/(c*x+1)/(c*x-1)*arccosh(c*x)*c^18+23*b*(-d*(c^2*x^2-1))^(1/2)*
d^2/(7*c^12*x^12-21*c^10*x^10+35*c^8*x^8-35*c^6*x^6+21*c^4*x^4-7*c^2*x^2+1)*x^9/(c*x+1)/(c*x-1)*arccosh(c*x)*c
^16-47*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(7*c^12*x^12-21*c^10*x^10+35*c^8*x^8-35*c^6*x^6+21*c^4*x^4-7*c^2*x^2+1)*x^
7/(c*x+1)/(c*x-1)*arccosh(c*x)*c^14+66*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(7*c^12*x^12-21*c^10*x^10+35*c^8*x^8-35*c^
6*x^6+21*c^4*x^4-7*c^2*x^2+1)*x^5/(c*x+1)/(c*x-1)*arccosh(c*x)*c^12-66*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(7*c^12*x^
12-21*c^10*x^10+35*c^8*x^8-35*c^6*x^6+21*c^4*x^4-7*c^2*x^2+1)*x^3/(c*x+1)/(c*x-1)*arccosh(c*x)*c^10+330/7*b*(-
d*(c^2*x^2-1))^(1/2)*d^2/(7*c^12*x^12-21*c^10*x^10+35*c^8*x^8-35*c^6*x^6+21*c^4*x^4-7*c^2*x^2+1)*x/(c*x+1)/(c*
x-1)*arccosh(c*x)*c^8-165/7*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(7*c^12*x^12-21*c^10*x^10+35*c^8*x^8-35*c^6*x^6+21*c^
4*x^4-7*c^2*x^2+1)/x/(c*x+1)/(c*x-1)*arccosh(c*x)*c^6+55/7*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(7*c^12*x^12-21*c^10*x
^10+35*c^8*x^8-35*c^6*x^6+21*c^4*x^4-7*c^2*x^2+1)/x^3/(c*x+1)/(c*x-1)*arccosh(c*x)*c^4-11/7*b*(-d*(c^2*x^2-1))
^(1/2)*d^2/(7*c^12*x^12-21*c^10*x^10+35*c^8*x^8-35*c^6*x^6+21*c^4*x^4-7*c^2*x^2+1)/x^5/(c*x+1)/(c*x-1)*arccosh
(c*x)*c^2-3*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(7*c^12*x^12-21*c^10*x^10+35*c^8*x^8-35*c^6*x^6+21*c^4*x^4-7*c^2*x^2+
1)*x^4/(c*x+1)^(1/2)/(c*x-1)^(1/2)*arccosh(c*x)*c^11+b*(-d*(c^2*x^2-1))^(1/2)*d^2/(7*c^12*x^12-21*c^10*x^10+35
*c^8*x^8-35*c^6*x^6+21*c^4*x^4-7*c^2*x^2+1)*x^2/(c*x+1)^(1/2)/(c*x-1)^(1/2)*arccosh(c*x)*c^9-b*(-d*(c^2*x^2-1)
)^(1/2)*d^2/(7*c^12*x^12-21*c^10*x^10+35*c^8*x^8-35*c^6*x^6+21*c^4*x^4-7*c^2*x^2+1)*x^12/(c*x+1)^(1/2)/(c*x-1)
^(1/2)*arccosh(c*x)*c^19+3*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(7*c^12*x^12-21*c^10*x^10+35*c^8*x^8-35*c^6*x^6+21*c^4
*x^4-7*c^2*x^2+1)*x^10/(c*x+1)^(1/2)/(c*x-1)^(1/2)*arccosh(c*x)*c^17-5*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(7*c^12*x^
12-21*c^10*x^10+35*c^8*x^8-35*c^6*x^6+21*c^4*x^4-7*c^2*x^2+1)*x^8/(c*x+1)^(1/2)/(c*x-1)^(1/2)*arccosh(c*x)*c^1
5+b*(-d*(c^2*x^2-1))^(1/2)*d^2/(7*c^12*x^12-21*c^10*x^10+35*c^8*x^8-35*c^6*x^6+21*c^4*x^4-7*c^2*x^2+1)*x^13/(c
*x+1)/(c*x-1)*arccosh(c*x)*c^20+5*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(7*c^12*x^12-21*c^10*x^10+35*c^8*x^8-35*c^6*x^6
+21*c^4*x^4-7*c^2*x^2+1)*x^6/(c*x+1)^(1/2)/(c*x-1)^(1/2)*arccosh(c*x)*c^13-1/7*a/d/x^7*(-c^2*d*x^2+d)^(7/2)+3/
4*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(7*c^12*x^12-21*c^10*x^10+35*c^8*x^8-35*c^6*x^6+21*c^4*x^4-7*c^2*x^2+1)*x^9*c^1
6-83/84*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(7*c^12*x^12-21*c^10*x^10+35*c^8*x^8-35*c^6*x^6+21*c^4*x^4-7*c^2*x^2+1)*x
^7*c^14+17/28*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(7*c^12*x^12-21*c^10*x^10+35*c^8*x^8-35*c^6*x^6+21*c^4*x^4-7*c^2*x^
2+1)*x^5*c^12-5/28*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(7*c^12*x^12-21*c^10*x^10+35*c^8*x^8-35*c^6*x^6+21*c^4*x^4-7*c
^2*x^2+1)*x^3*c^10+1/42*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(7*c^12*x^12-21*c^10*x^10+35*c^8*x^8-35*c^6*x^6+21*c^4*x^
4-7*c^2*x^2+1)*x*c^8+55/12*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(7*c^12*x^12-21*c^10*x^10+35*c^8*x^8-35*c^6*x^6+21*c^4
*x^4-7*c^2*x^2+1)/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c^7+2/7*b*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)*arc
cosh(c*x)*c^7*d^2-1/7*b*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)*ln((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)
)^2+1)*c^7*d^2
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
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^8,x, algorithm="maxima")
[Out]
Exception raised: ValueError
________________________________________________________________________________________
Fricas [A] time = 2.69859, size = 1473, normalized size = 6.73 \begin{align*} \left [\frac{12 \,{\left (b c^{8} d^{2} x^{8} - 4 \, b c^{6} d^{2} x^{6} + 6 \, b c^{4} d^{2} x^{4} - 4 \, b c^{2} d^{2} x^{2} + b d^{2}\right )} \sqrt{-c^{2} d x^{2} + d} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) + 6 \,{\left (b c^{9} d^{2} x^{9} - b c^{7} d^{2} x^{7}\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 (18 \, b c^{5} d^{2} x^{5} -{\left (18 \, b c^{5} - 9 \, b c^{3} + 2 \, b c\right )} d^{2} x^{7} - 9 \, b c^{3} d^{2} x^{3} + 2 \, b c d^{2} x\right )} \sqrt{-c^{2} d x^{2} + d} \sqrt{c^{2} x^{2} - 1} + 12 \,{\left (a c^{8} d^{2} x^{8} - 4 \, a c^{6} d^{2} x^{6} + 6 \, a c^{4} d^{2} x^{4} - 4 \, a c^{2} d^{2} x^{2} + a d^{2}\right )} \sqrt{-c^{2} d x^{2} + d}}{84 \,{\left (c^{2} x^{9} - x^{7}\right )}}, -\frac{12 \,{\left (b c^{9} d^{2} x^{9} - b c^{7} d^{2} x^{7}\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 ) - 12 \,{\left (b c^{8} d^{2} x^{8} - 4 \, b c^{6} d^{2} x^{6} + 6 \, b c^{4} d^{2} x^{4} - 4 \, b c^{2} d^{2} x^{2} + b d^{2}\right )} \sqrt{-c^{2} d x^{2} + d} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) +{\left (18 \, b c^{5} d^{2} x^{5} -{\left (18 \, b c^{5} - 9 \, b c^{3} + 2 \, b c\right )} d^{2} x^{7} - 9 \, b c^{3} d^{2} x^{3} + 2 \, b c d^{2} x\right )} \sqrt{-c^{2} d x^{2} + d} \sqrt{c^{2} x^{2} - 1} - 12 \,{\left (a c^{8} d^{2} x^{8} - 4 \, a c^{6} d^{2} x^{6} + 6 \, a c^{4} d^{2} x^{4} - 4 \, a c^{2} d^{2} x^{2} + a d^{2}\right )} \sqrt{-c^{2} d x^{2} + d}}{84 \,{\left (c^{2} x^{9} - x^{7}\right )}}\right ] \end{align*}
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^8,x, algorithm="fricas")
[Out]
[1/84*(12*(b*c^8*d^2*x^8 - 4*b*c^6*d^2*x^6 + 6*b*c^4*d^2*x^4 - 4*b*c^2*d^2*x^2 + b*d^2)*sqrt(-c^2*d*x^2 + d)*l
og(c*x + sqrt(c^2*x^2 - 1)) + 6*(b*c^9*d^2*x^9 - b*c^7*d^2*x^7)*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)) - (18*b*c^5*d^2*x^5 - (18*b*c^
5 - 9*b*c^3 + 2*b*c)*d^2*x^7 - 9*b*c^3*d^2*x^3 + 2*b*c*d^2*x)*sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1) + 12*(a*c
^8*d^2*x^8 - 4*a*c^6*d^2*x^6 + 6*a*c^4*d^2*x^4 - 4*a*c^2*d^2*x^2 + a*d^2)*sqrt(-c^2*d*x^2 + d))/(c^2*x^9 - x^7
), -1/84*(12*(b*c^9*d^2*x^9 - b*c^7*d^2*x^7)*sqrt(d)*arctan(sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1)*(x^2 + 1)*s
qrt(d)/(c^2*d*x^4 - (c^2 + 1)*d*x^2 + d)) - 12*(b*c^8*d^2*x^8 - 4*b*c^6*d^2*x^6 + 6*b*c^4*d^2*x^4 - 4*b*c^2*d^
2*x^2 + b*d^2)*sqrt(-c^2*d*x^2 + d)*log(c*x + sqrt(c^2*x^2 - 1)) + (18*b*c^5*d^2*x^5 - (18*b*c^5 - 9*b*c^3 + 2
*b*c)*d^2*x^7 - 9*b*c^3*d^2*x^3 + 2*b*c*d^2*x)*sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1) - 12*(a*c^8*d^2*x^8 - 4*
a*c^6*d^2*x^6 + 6*a*c^4*d^2*x^4 - 4*a*c^2*d^2*x^2 + a*d^2)*sqrt(-c^2*d*x^2 + d))/(c^2*x^9 - x^7)]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
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**8,x)
[Out]
Timed out
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-c^{2} d x^{2} + d\right )}^{\frac{5}{2}}{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )}}{x^{8}}\,{d x} \end{align*}
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^8,x, algorithm="giac")
[Out]
integrate((-c^2*d*x^2 + d)^(5/2)*(b*arccosh(c*x) + a)/x^8, x)