3.64 \(\int \frac{\sqrt{d-c^2 d x^2} (a+b \cosh ^{-1}(c x))}{x^8} \, dx\)
Optimal. Leaf size=279 \[ -\frac{8 c^4 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )}{105 d x^3}-\frac{4 c^2 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )}{35 d x^5}-\frac{\left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )}{7 d x^7}+\frac{2 b c^5 \sqrt{d-c^2 d x^2}}{105 x^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b c^3 \sqrt{d-c^2 d x^2}}{140 x^4 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c \sqrt{d-c^2 d x^2}}{42 x^6 \sqrt{c x-1} \sqrt{c x+1}}-\frac{8 b c^7 \log (x) \sqrt{d-c^2 d x^2}}{105 \sqrt{c x-1} \sqrt{c x+1}} \]
[Out]
-(b*c*Sqrt[d - c^2*d*x^2])/(42*x^6*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c^3*Sqrt[d - c^2*d*x^2])/(140*x^4*Sqrt[-
1 + c*x]*Sqrt[1 + c*x]) + (2*b*c^5*Sqrt[d - c^2*d*x^2])/(105*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - ((d - c^2*d*x
^2)^(3/2)*(a + b*ArcCosh[c*x]))/(7*d*x^7) - (4*c^2*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/(35*d*x^5) - (8
*c^4*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/(105*d*x^3) - (8*b*c^7*Sqrt[d - c^2*d*x^2]*Log[x])/(105*Sqrt[
-1 + c*x]*Sqrt[1 + c*x])
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Rubi [A] time = 0.377553, antiderivative size = 303, normalized size of antiderivative =
1.09, number of steps used = 5, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259, Rules
used = {5798, 97, 12, 103, 95, 5733, 14} \[ \frac{8 c^6 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{105 x}+\frac{4 c^4 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{105 x^3}+\frac{c^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{35 x^5}-\frac{\sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{7 x^7}+\frac{2 b c^5 \sqrt{d-c^2 d x^2}}{105 x^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{b c^3 \sqrt{d-c^2 d x^2}}{140 x^4 \sqrt{c x-1} \sqrt{c x+1}}-\frac{b c \sqrt{d-c^2 d x^2}}{42 x^6 \sqrt{c x-1} \sqrt{c x+1}}-\frac{8 b c^7 \log (x) \sqrt{d-c^2 d x^2}}{105 \sqrt{c x-1} \sqrt{c x+1}} \]
Antiderivative was successfully verified.
[In]
Int[(Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/x^8,x]
[Out]
-(b*c*Sqrt[d - c^2*d*x^2])/(42*x^6*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c^3*Sqrt[d - c^2*d*x^2])/(140*x^4*Sqrt[-
1 + c*x]*Sqrt[1 + c*x]) + (2*b*c^5*Sqrt[d - c^2*d*x^2])/(105*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (Sqrt[d - c^2
*d*x^2]*(a + b*ArcCosh[c*x]))/(7*x^7) + (c^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(35*x^5) + (4*c^4*Sqrt[
d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(105*x^3) + (8*c^6*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(105*x) - (8
*b*c^7*Sqrt[d - c^2*d*x^2]*Log[x])/(105*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 97
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((a + b
*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p)/(b*(m + 1)), x] - Dist[1/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n
- 1)*(e + f*x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && LtQ[m
, -1] && GtQ[n, 0] && GtQ[p, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
Rule 12
Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]
Rule 103
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[1/((m + 1)*(b*
c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] &&
IntegerQ[m] && (IntegerQ[n] || IntegersQ[2*n, 2*p])
Rule 95
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] /; FreeQ[{a, b, c, d,
e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && EqQ[a*d*f*(m + 1) + b*c*f*(n + 1) + b*d*e*(p + 1), 0
] && NeQ[m, -1]
Rule 5733
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*(x_)^(m_)*((d1_) + (e1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_))^(p_), x_Sym
bol] :> With[{u = IntHide[x^m*(1 + c*x)^p*(-1 + c*x)^p, x]}, Dist[(-(d1*d2))^p*(a + b*ArcCosh[c*x]), u, x] - D
ist[b*c*(-(d1*d2))^p, Int[SimplifyIntegrand[u/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x], x]] /; FreeQ[{a, b, c, d
1, e1, d2, e2}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IntegerQ[p - 1/2] && (IGtQ[(m + 1)/2, 0] || IL
tQ[(m + 2*p + 3)/2, 0]) && NeQ[p, -2^(-1)] && GtQ[d1, 0] && LtQ[d2, 0]
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]]
Rubi steps
\begin{align*} \int \frac{\sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{x^8} \, dx &=\frac{\sqrt{d-c^2 d x^2} \int \frac{\sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{x^8} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{\sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{7 x^7}+\frac{c^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{35 x^5}+\frac{4 c^4 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{105 x^3}+\frac{8 c^6 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{105 x}-\frac{\left (b c \sqrt{d-c^2 d x^2}\right ) \int \frac{-15+3 c^2 x^2+4 c^4 x^4+8 c^6 x^6}{105 x^7} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{\sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{7 x^7}+\frac{c^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{35 x^5}+\frac{4 c^4 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{105 x^3}+\frac{8 c^6 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{105 x}-\frac{\left (b c \sqrt{d-c^2 d x^2}\right ) \int \frac{-15+3 c^2 x^2+4 c^4 x^4+8 c^6 x^6}{x^7} \, dx}{105 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{\sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{7 x^7}+\frac{c^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{35 x^5}+\frac{4 c^4 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{105 x^3}+\frac{8 c^6 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{105 x}-\frac{\left (b c \sqrt{d-c^2 d x^2}\right ) \int \left (-\frac{15}{x^7}+\frac{3 c^2}{x^5}+\frac{4 c^4}{x^3}+\frac{8 c^6}{x}\right ) \, dx}{105 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{b c \sqrt{d-c^2 d x^2}}{42 x^6 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c^3 \sqrt{d-c^2 d x^2}}{140 x^4 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{2 b c^5 \sqrt{d-c^2 d x^2}}{105 x^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{7 x^7}+\frac{c^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{35 x^5}+\frac{4 c^4 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{105 x^3}+\frac{8 c^6 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{105 x}-\frac{8 b c^7 \sqrt{d-c^2 d x^2} \log (x)}{105 \sqrt{-1+c x} \sqrt{1+c x}}\\ \end{align*}
Mathematica [A] time = 0.257163, size = 146, normalized size = 0.52 \[ \frac{\sqrt{d-c^2 d x^2} \left (16 c^2 x^2 (c x-1)^{3/2} (c x+1)^{3/2} \left (2 c^2 x^2+3\right ) \left (a+b \cosh ^{-1}(c x)\right )+60 (c x-1)^{3/2} (c x+1)^{3/2} \left (a+b \cosh ^{-1}(c x)\right )-b c x \left (-8 c^4 x^4-3 c^2 x^2+32 c^6 x^6 \log (x)+10\right )\right )}{420 x^7 \sqrt{c x-1} \sqrt{c x+1}} \]
Antiderivative was successfully verified.
[In]
Integrate[(Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/x^8,x]
[Out]
(Sqrt[d - c^2*d*x^2]*(60*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2)*(a + b*ArcCosh[c*x]) + 16*c^2*x^2*(-1 + c*x)^(3/2)*(
1 + c*x)^(3/2)*(3 + 2*c^2*x^2)*(a + b*ArcCosh[c*x]) - b*c*x*(10 - 3*c^2*x^2 - 8*c^4*x^4 + 32*c^6*x^6*Log[x])))
/(420*x^7*Sqrt[-1 + c*x]*Sqrt[1 + c*x])
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Maple [B] time = 0.363, size = 2534, normalized size = 9.1 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/x^8,x)
[Out]
-8/105*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^7-73/2
0*b*(-d*(c^2*x^2-1))^(1/2)/(280*c^8*x^8-105*c^6*x^6-21*c^4*x^4-315*c^2*x^2+225)/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c^
7+16/105*b*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)*arccosh(c*x)*c^7+128/105*b*(-d*(c^2*x^2-1))^(1/2
)/(280*c^8*x^8-105*c^6*x^6-21*c^4*x^4-315*c^2*x^2+225)*x^11*c^18-64/3*b*(-d*(c^2*x^2-1))^(1/2)/(280*c^8*x^8-10
5*c^6*x^6-21*c^4*x^4-315*c^2*x^2+225)*x^8/(c*x+1)^(1/2)/(c*x-1)^(1/2)*arccosh(c*x)*c^15+8*b*(-d*(c^2*x^2-1))^(
1/2)/(280*c^8*x^8-105*c^6*x^6-21*c^4*x^4-315*c^2*x^2+225)*x^6/(c*x+1)^(1/2)/(c*x-1)^(1/2)*arccosh(c*x)*c^13+8/
5*b*(-d*(c^2*x^2-1))^(1/2)/(280*c^8*x^8-105*c^6*x^6-21*c^4*x^4-315*c^2*x^2+225)*x^4/(c*x+1)^(1/2)/(c*x-1)^(1/2
)*arccosh(c*x)*c^11+24*b*(-d*(c^2*x^2-1))^(1/2)/(280*c^8*x^8-105*c^6*x^6-21*c^4*x^4-315*c^2*x^2+225)*x^2/(c*x+
1)^(1/2)/(c*x-1)^(1/2)*arccosh(c*x)*c^9+3057/35*b*(-d*(c^2*x^2-1))^(1/2)/(280*c^8*x^8-105*c^6*x^6-21*c^4*x^4-3
15*c^2*x^2+225)*x/(c*x+1)/(c*x-1)*arccosh(c*x)*c^8-594/35*b*(-d*(c^2*x^2-1))^(1/2)/(280*c^8*x^8-105*c^6*x^6-21
*c^4*x^4-315*c^2*x^2+225)/x/(c*x+1)/(c*x-1)*arccosh(c*x)*c^6+342/7*b*(-d*(c^2*x^2-1))^(1/2)/(280*c^8*x^8-105*c
^6*x^6-21*c^4*x^4-315*c^2*x^2+225)/x^3/(c*x+1)/(c*x-1)*arccosh(c*x)*c^4-585/7*b*(-d*(c^2*x^2-1))^(1/2)/(280*c^
8*x^8-105*c^6*x^6-21*c^4*x^4-315*c^2*x^2+225)/x^5/(c*x+1)/(c*x-1)*arccosh(c*x)*c^2+64/3*b*(-d*(c^2*x^2-1))^(1/
2)/(280*c^8*x^8-105*c^6*x^6-21*c^4*x^4-315*c^2*x^2+225)*x^9/(c*x+1)/(c*x-1)*arccosh(c*x)*c^16-56/3*b*(-d*(c^2*
x^2-1))^(1/2)/(280*c^8*x^8-105*c^6*x^6-21*c^4*x^4-315*c^2*x^2+225)*x^7/(c*x+1)/(c*x-1)*arccosh(c*x)*c^14-4/15*
b*(-d*(c^2*x^2-1))^(1/2)/(280*c^8*x^8-105*c^6*x^6-21*c^4*x^4-315*c^2*x^2+225)*x^5/(c*x+1)/(c*x-1)*arccosh(c*x)
*c^12-351/5*b*(-d*(c^2*x^2-1))^(1/2)/(280*c^8*x^8-105*c^6*x^6-21*c^4*x^4-315*c^2*x^2+225)*x^3/(c*x+1)/(c*x-1)*
arccosh(c*x)*c^10+71/28*b*(-d*(c^2*x^2-1))^(1/2)/(280*c^8*x^8-105*c^6*x^6-21*c^4*x^4-315*c^2*x^2+225)/x^2/(c*x
+1)^(1/2)/(c*x-1)^(1/2)*c^5+255/28*b*(-d*(c^2*x^2-1))^(1/2)/(280*c^8*x^8-105*c^6*x^6-21*c^4*x^4-315*c^2*x^2+22
5)/x^4/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c^3-75/14*b*(-d*(c^2*x^2-1))^(1/2)/(280*c^8*x^8-105*c^6*x^6-21*c^4*x^4-315*
c^2*x^2+225)/x^6/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c-469/60*b*(-d*(c^2*x^2-1))^(1/2)/(280*c^8*x^8-105*c^6*x^6-21*c^4
*x^4-315*c^2*x^2+225)*x^2/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c^9-120/7*b*(-d*(c^2*x^2-1))^(1/2)/(280*c^8*x^8-105*c^6*
x^6-21*c^4*x^4-315*c^2*x^2+225)/(c*x+1)^(1/2)/(c*x-1)^(1/2)*arccosh(c*x)*c^7+16/3*b*(-d*(c^2*x^2-1))^(1/2)/(28
0*c^8*x^8-105*c^6*x^6-21*c^4*x^4-315*c^2*x^2+225)*x^6/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c^13+20/7*b*(-d*(c^2*x^2-1))
^(1/2)/(280*c^8*x^8-105*c^6*x^6-21*c^4*x^4-315*c^2*x^2+225)*x/(c*x+1)/(c*x-1)*c^8+225/7*b*(-d*(c^2*x^2-1))^(1/
2)/(280*c^8*x^8-105*c^6*x^6-21*c^4*x^4-315*c^2*x^2+225)/x^7/(c*x+1)/(c*x-1)*arccosh(c*x)-128/105*b*(-d*(c^2*x^
2-1))^(1/2)/(280*c^8*x^8-105*c^6*x^6-21*c^4*x^4-315*c^2*x^2+225)*x^13/(c*x+1)/(c*x-1)*c^20+16/105*b*(-d*(c^2*x
^2-1))^(1/2)/(280*c^8*x^8-105*c^6*x^6-21*c^4*x^4-315*c^2*x^2+225)*x^11/(c*x+1)/(c*x-1)*c^18+40/21*b*(-d*(c^2*x
^2-1))^(1/2)/(280*c^8*x^8-105*c^6*x^6-21*c^4*x^4-315*c^2*x^2+225)*x^9/(c*x+1)/(c*x-1)*c^16+214/105*b*(-d*(c^2*
x^2-1))^(1/2)/(280*c^8*x^8-105*c^6*x^6-21*c^4*x^4-315*c^2*x^2+225)*x^7/(c*x+1)/(c*x-1)*c^14-152/105*b*(-d*(c^2
*x^2-1))^(1/2)/(280*c^8*x^8-105*c^6*x^6-21*c^4*x^4-315*c^2*x^2+225)*x^5/(c*x+1)/(c*x-1)*c^12-30/7*b*(-d*(c^2*x
^2-1))^(1/2)/(280*c^8*x^8-105*c^6*x^6-21*c^4*x^4-315*c^2*x^2+225)*x^3/(c*x+1)/(c*x-1)*c^10-1/7*a/d/x^7*(-c^2*d
*x^2+d)^(3/2)+16/15*b*(-d*(c^2*x^2-1))^(1/2)/(280*c^8*x^8-105*c^6*x^6-21*c^4*x^4-315*c^2*x^2+225)*x^9*c^16+20/
7*b*(-d*(c^2*x^2-1))^(1/2)/(280*c^8*x^8-105*c^6*x^6-21*c^4*x^4-315*c^2*x^2+225)*x*c^8-10/7*b*(-d*(c^2*x^2-1))^
(1/2)/(280*c^8*x^8-105*c^6*x^6-21*c^4*x^4-315*c^2*x^2+225)*x^3*c^10-302/105*b*(-d*(c^2*x^2-1))^(1/2)/(280*c^8*
x^8-105*c^6*x^6-21*c^4*x^4-315*c^2*x^2+225)*x^5*c^12-88/105*b*(-d*(c^2*x^2-1))^(1/2)/(280*c^8*x^8-105*c^6*x^6-
21*c^4*x^4-315*c^2*x^2+225)*x^7*c^14-4/35*a*c^2/d/x^5*(-c^2*d*x^2+d)^(3/2)-8/105*a*c^4/d/x^3*(-c^2*d*x^2+d)^(3
/2)
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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((a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/x^8,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 2.26926, size = 1320, normalized size = 4.73 \begin{align*} \left [\frac{4 \,{\left (8 \, b c^{8} x^{8} - 4 \, b c^{6} x^{6} - b c^{4} x^{4} - 18 \, b c^{2} x^{2} + 15 \, b\right )} \sqrt{-c^{2} d x^{2} + d} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) + 16 \,{\left (b c^{9} x^{9} - b c^{7} 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 (8 \, b c^{5} x^{5} -{\left (8 \, b c^{5} + 3 \, b c^{3} - 10 \, b c\right )} x^{7} + 3 \, b c^{3} x^{3} - 10 \, b c x\right )} \sqrt{-c^{2} d x^{2} + d} \sqrt{c^{2} x^{2} - 1} + 4 \,{\left (8 \, a c^{8} x^{8} - 4 \, a c^{6} x^{6} - a c^{4} x^{4} - 18 \, a c^{2} x^{2} + 15 \, a\right )} \sqrt{-c^{2} d x^{2} + d}}{420 \,{\left (c^{2} x^{9} - x^{7}\right )}}, -\frac{32 \,{\left (b c^{9} x^{9} - b c^{7} 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 ) - 4 \,{\left (8 \, b c^{8} x^{8} - 4 \, b c^{6} x^{6} - b c^{4} x^{4} - 18 \, b c^{2} x^{2} + 15 \, b\right )} \sqrt{-c^{2} d x^{2} + d} \log \left (c x + \sqrt{c^{2} x^{2} - 1}\right ) -{\left (8 \, b c^{5} x^{5} -{\left (8 \, b c^{5} + 3 \, b c^{3} - 10 \, b c\right )} x^{7} + 3 \, b c^{3} x^{3} - 10 \, b c x\right )} \sqrt{-c^{2} d x^{2} + d} \sqrt{c^{2} x^{2} - 1} - 4 \,{\left (8 \, a c^{8} x^{8} - 4 \, a c^{6} x^{6} - a c^{4} x^{4} - 18 \, a c^{2} x^{2} + 15 \, a\right )} \sqrt{-c^{2} d x^{2} + d}}{420 \,{\left (c^{2} x^{9} - x^{7}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/x^8,x, algorithm="fricas")
[Out]
[1/420*(4*(8*b*c^8*x^8 - 4*b*c^6*x^6 - b*c^4*x^4 - 18*b*c^2*x^2 + 15*b)*sqrt(-c^2*d*x^2 + d)*log(c*x + sqrt(c^
2*x^2 - 1)) + 16*(b*c^9*x^9 - b*c^7*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)*sq
rt(c^2*x^2 - 1)*(x^4 - 1)*sqrt(-d) - d)/(c^2*x^4 - x^2)) + (8*b*c^5*x^5 - (8*b*c^5 + 3*b*c^3 - 10*b*c)*x^7 + 3
*b*c^3*x^3 - 10*b*c*x)*sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1) + 4*(8*a*c^8*x^8 - 4*a*c^6*x^6 - a*c^4*x^4 - 18*
a*c^2*x^2 + 15*a)*sqrt(-c^2*d*x^2 + d))/(c^2*x^9 - x^7), -1/420*(32*(b*c^9*x^9 - b*c^7*x^7)*sqrt(d)*arctan(sqr
t(-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*(8*b*c^8*x^8 - 4*
b*c^6*x^6 - b*c^4*x^4 - 18*b*c^2*x^2 + 15*b)*sqrt(-c^2*d*x^2 + d)*log(c*x + sqrt(c^2*x^2 - 1)) - (8*b*c^5*x^5
- (8*b*c^5 + 3*b*c^3 - 10*b*c)*x^7 + 3*b*c^3*x^3 - 10*b*c*x)*sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1) - 4*(8*a*c
^8*x^8 - 4*a*c^6*x^6 - a*c^4*x^4 - 18*a*c^2*x^2 + 15*a)*sqrt(-c^2*d*x^2 + d))/(c^2*x^9 - x^7)]
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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((a+b*acosh(c*x))*(-c**2*d*x**2+d)**(1/2)/x**8,x)
[Out]
Timed out
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-c^{2} d x^{2} + d}{\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((a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/x^8,x, algorithm="giac")
[Out]
integrate(sqrt(-c^2*d*x^2 + d)*(b*arccosh(c*x) + a)/x^8, x)