3.78 \(\int \frac {(d-c^2 d x^2)^{3/2} (a+b \cosh ^{-1}(c x))}{x^{10}} \, dx\)

Optimal. Leaf size=328 \[ -\frac {\left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )}{9 d x^9}-\frac {4 c^2 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )}{63 d x^7}-\frac {8 c^4 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )}{315 d x^5}-\frac {b c d \sqrt {d-c^2 d x^2}}{72 x^8 \sqrt {c x-1} \sqrt {c x+1}}+\frac {8 b c^9 d \log (x) \sqrt {d-c^2 d x^2}}{315 \sqrt {c x-1} \sqrt {c x+1}}-\frac {2 b c^7 d \sqrt {d-c^2 d x^2}}{315 x^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c^5 d \sqrt {d-c^2 d x^2}}{420 x^4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {5 b c^3 d \sqrt {d-c^2 d x^2}}{189 x^6 \sqrt {c x-1} \sqrt {c x+1}} \]

[Out]

-1/9*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))/d/x^9-4/63*c^2*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))/d/x^7-8/31
5*c^4*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))/d/x^5-1/72*b*c*d*(-c^2*d*x^2+d)^(1/2)/x^8/(c*x-1)^(1/2)/(c*x+1)^
(1/2)+5/189*b*c^3*d*(-c^2*d*x^2+d)^(1/2)/x^6/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/420*b*c^5*d*(-c^2*d*x^2+d)^(1/2)/x^
4/(c*x-1)^(1/2)/(c*x+1)^(1/2)-2/315*b*c^7*d*(-c^2*d*x^2+d)^(1/2)/x^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)+8/315*b*c^9*d
*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 = 0.51, antiderivative size = 401, normalized size of antiderivative = 1.22, number of steps used = 6, number of rules used = 8, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {5798, 97, 12, 103, 95, 5733, 1251, 893} \[ -\frac {8 c^8 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{315 x}-\frac {4 c^6 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{315 x^3}-\frac {c^4 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{105 x^5}+\frac {c^2 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{21 x^7}-\frac {d (1-c x) (c x+1) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 x^9}-\frac {2 b c^7 d \sqrt {d-c^2 d x^2}}{315 x^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c^5 d \sqrt {d-c^2 d x^2}}{420 x^4 \sqrt {c x-1} \sqrt {c x+1}}+\frac {5 b c^3 d \sqrt {d-c^2 d x^2}}{189 x^6 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c d \sqrt {d-c^2 d x^2}}{72 x^8 \sqrt {c x-1} \sqrt {c x+1}}+\frac {8 b c^9 d \log (x) \sqrt {d-c^2 d x^2}}{315 \sqrt {c x-1} \sqrt {c x+1}} \]

Antiderivative was successfully verified.

[In]

Int[((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/x^10,x]

[Out]

-(b*c*d*Sqrt[d - c^2*d*x^2])/(72*x^8*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (5*b*c^3*d*Sqrt[d - c^2*d*x^2])/(189*x^6*
Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c^5*d*Sqrt[d - c^2*d*x^2])/(420*x^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (2*b*c^
7*d*Sqrt[d - c^2*d*x^2])/(315*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (c^2*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*
x]))/(21*x^7) - (c^4*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(105*x^5) - (4*c^6*d*Sqrt[d - c^2*d*x^2]*(a +
b*ArcCosh[c*x]))/(315*x^3) - (8*c^8*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(315*x) - (d*(1 - c*x)*(1 + c
*x)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(9*x^9) + (8*b*c^9*d*Sqrt[d - c^2*d*x^2]*Log[x])/(315*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 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 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 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 893

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &
& NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p] && ((EqQ[p, 1] && I
ntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))

Rule 1251

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2,
Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] &&
IntegerQ[(m - 1)/2]

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 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]

Rubi steps

\begin {align*} \int \frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )}{x^{10}} \, dx &=-\frac {\left (d \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^{10}} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {c^2 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{21 x^7}-\frac {c^4 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{105 x^5}-\frac {4 c^6 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{315 x^3}-\frac {8 c^8 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{315 x}-\frac {d (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 x^9}+\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (1-c^2 x^2\right )^2 \left (35+20 c^2 x^2+8 c^4 x^4\right )}{315 x^9} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {c^2 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{21 x^7}-\frac {c^4 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{105 x^5}-\frac {4 c^6 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{315 x^3}-\frac {8 c^8 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{315 x}-\frac {d (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 x^9}+\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (1-c^2 x^2\right )^2 \left (35+20 c^2 x^2+8 c^4 x^4\right )}{x^9} \, dx}{315 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {c^2 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{21 x^7}-\frac {c^4 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{105 x^5}-\frac {4 c^6 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{315 x^3}-\frac {8 c^8 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{315 x}-\frac {d (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 x^9}+\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {\left (1-c^2 x\right )^2 \left (35+20 c^2 x+8 c^4 x^2\right )}{x^5} \, dx,x,x^2\right )}{630 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {c^2 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{21 x^7}-\frac {c^4 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{105 x^5}-\frac {4 c^6 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{315 x^3}-\frac {8 c^8 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{315 x}-\frac {d (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 x^9}+\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {35}{x^5}-\frac {50 c^2}{x^4}+\frac {3 c^4}{x^3}+\frac {4 c^6}{x^2}+\frac {8 c^8}{x}\right ) \, dx,x,x^2\right )}{630 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {b c d \sqrt {d-c^2 d x^2}}{72 x^8 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 b c^3 d \sqrt {d-c^2 d x^2}}{189 x^6 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d \sqrt {d-c^2 d x^2}}{420 x^4 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c^7 d \sqrt {d-c^2 d x^2}}{315 x^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {c^2 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{21 x^7}-\frac {c^4 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{105 x^5}-\frac {4 c^6 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{315 x^3}-\frac {8 c^8 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{315 x}-\frac {d (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 x^9}+\frac {8 b c^9 d \sqrt {d-c^2 d x^2} \log (x)}{315 \sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}

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Mathematica [A]  time = 0.33, size = 154, normalized size = 0.47 \[ -\frac {d \sqrt {d-c^2 d x^2} \left (96 c^2 x^2 (c x-1)^{5/2} \left (2 c^2 x^2+5\right ) (c x+1)^{5/2} \left (a+b \cosh ^{-1}(c x)\right )+840 (c x-1)^{5/2} (c x+1)^{5/2} \left (a+b \cosh ^{-1}(c x)\right )+b c x \left (-192 c^8 x^8 \log (x)+48 c^6 x^6+18 c^4 x^4-200 c^2 x^2+105\right )\right )}{7560 x^9 \sqrt {c x-1} \sqrt {c x+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/x^10,x]

[Out]

-1/7560*(d*Sqrt[d - c^2*d*x^2]*(840*(-1 + c*x)^(5/2)*(1 + c*x)^(5/2)*(a + b*ArcCosh[c*x]) + 96*c^2*x^2*(-1 + c
*x)^(5/2)*(1 + c*x)^(5/2)*(5 + 2*c^2*x^2)*(a + b*ArcCosh[c*x]) + b*c*x*(105 - 200*c^2*x^2 + 18*c^4*x^4 + 48*c^
6*x^6 - 192*c^8*x^8*Log[x])))/(x^9*Sqrt[-1 + c*x]*Sqrt[1 + c*x])

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fricas [A]  time = 0.67, size = 720, normalized size = 2.20 \[ \left [-\frac {24 \, {\left (8 \, b c^{10} d x^{10} - 4 \, b c^{8} d x^{8} - b c^{6} d x^{6} - 53 \, b c^{4} d x^{4} + 85 \, b c^{2} d x^{2} - 35 \, b d\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - 96 \, {\left (b c^{11} d x^{11} - b c^{9} d x^{9}\right )} \sqrt {-d} \log \left (\frac {c^{2} d x^{6} + c^{2} d x^{2} - d x^{4} - \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} {\left (x^{4} - 1\right )} \sqrt {-d} - d}{c^{2} x^{4} - x^{2}}\right ) + {\left (48 \, b c^{7} d x^{7} + 18 \, b c^{5} d x^{5} - {\left (48 \, b c^{7} + 18 \, b c^{5} - 200 \, b c^{3} + 105 \, b c\right )} d x^{9} - 200 \, b c^{3} d x^{3} + 105 \, b c d x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} + 24 \, {\left (8 \, a c^{10} d x^{10} - 4 \, a c^{8} d x^{8} - a c^{6} d x^{6} - 53 \, a c^{4} d x^{4} + 85 \, a c^{2} d x^{2} - 35 \, a d\right )} \sqrt {-c^{2} d x^{2} + d}}{7560 \, {\left (c^{2} x^{11} - x^{9}\right )}}, \frac {192 \, {\left (b c^{11} d x^{11} - b c^{9} d x^{9}\right )} \sqrt {d} \arctan \left (\frac {\sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} {\left (x^{2} + 1\right )} \sqrt {d}}{c^{2} d x^{4} - {\left (c^{2} + 1\right )} d x^{2} + d}\right ) - 24 \, {\left (8 \, b c^{10} d x^{10} - 4 \, b c^{8} d x^{8} - b c^{6} d x^{6} - 53 \, b c^{4} d x^{4} + 85 \, b c^{2} d x^{2} - 35 \, b d\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (48 \, b c^{7} d x^{7} + 18 \, b c^{5} d x^{5} - {\left (48 \, b c^{7} + 18 \, b c^{5} - 200 \, b c^{3} + 105 \, b c\right )} d x^{9} - 200 \, b c^{3} d x^{3} + 105 \, b c d x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} - 24 \, {\left (8 \, a c^{10} d x^{10} - 4 \, a c^{8} d x^{8} - a c^{6} d x^{6} - 53 \, a c^{4} d x^{4} + 85 \, a c^{2} d x^{2} - 35 \, a d\right )} \sqrt {-c^{2} d x^{2} + d}}{7560 \, {\left (c^{2} x^{11} - x^{9}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))/x^10,x, algorithm="fricas")

[Out]

[-1/7560*(24*(8*b*c^10*d*x^10 - 4*b*c^8*d*x^8 - b*c^6*d*x^6 - 53*b*c^4*d*x^4 + 85*b*c^2*d*x^2 - 35*b*d)*sqrt(-
c^2*d*x^2 + d)*log(c*x + sqrt(c^2*x^2 - 1)) - 96*(b*c^11*d*x^11 - b*c^9*d*x^9)*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)) + (48*b*c^7*d*x
^7 + 18*b*c^5*d*x^5 - (48*b*c^7 + 18*b*c^5 - 200*b*c^3 + 105*b*c)*d*x^9 - 200*b*c^3*d*x^3 + 105*b*c*d*x)*sqrt(
-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1) + 24*(8*a*c^10*d*x^10 - 4*a*c^8*d*x^8 - a*c^6*d*x^6 - 53*a*c^4*d*x^4 + 85*a*
c^2*d*x^2 - 35*a*d)*sqrt(-c^2*d*x^2 + d))/(c^2*x^11 - x^9), 1/7560*(192*(b*c^11*d*x^11 - b*c^9*d*x^9)*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)) - 24*(8*b*c
^10*d*x^10 - 4*b*c^8*d*x^8 - b*c^6*d*x^6 - 53*b*c^4*d*x^4 + 85*b*c^2*d*x^2 - 35*b*d)*sqrt(-c^2*d*x^2 + d)*log(
c*x + sqrt(c^2*x^2 - 1)) - (48*b*c^7*d*x^7 + 18*b*c^5*d*x^5 - (48*b*c^7 + 18*b*c^5 - 200*b*c^3 + 105*b*c)*d*x^
9 - 200*b*c^3*d*x^3 + 105*b*c*d*x)*sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1) - 24*(8*a*c^10*d*x^10 - 4*a*c^8*d*x^
8 - a*c^6*d*x^6 - 53*a*c^4*d*x^4 + 85*a*c^2*d*x^2 - 35*a*d)*sqrt(-c^2*d*x^2 + d))/(c^2*x^11 - x^9)]

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))/x^10,x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:sym2
poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [B]  time = 1.04, size = 4259, normalized size = 12.98 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))/x^10,x)

[Out]

-35/9*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x
^2+1225)*x*c^10-128/315*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*
c^4*x^4-4725*c^2*x^2+1225)*x^15*c^24-16/45*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8
-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^13*c^22+1384/945*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945
*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^11*c^20+2306/945*b*(-d*(c^2*x^2-1))^(1/2
)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^9*c^18-40/63*b*(-d
*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^
7*c^16-2189/189*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-
4725*c^2*x^2+1225)*x^5*c^14+350/27*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^
6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^3*c^12-16/315*b*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)*arc
cosh(c*x)*d*c^9+8/315*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*c^9-30055/504*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c
^4*x^4-4725*c^2*x^2+1225)/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c^9-43264/63*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-9
45*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x/(c*x+1)/(c*x-1)*arccosh(c*x)*c^10+1135
94/63*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x
^2+1225)/x/(c*x+1)/(c*x-1)*arccosh(c*x)*c^8-174520/63*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+
189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)/x^3/(c*x+1)/(c*x-1)*arccosh(c*x)*c^6+19540/9*b*(-d*(c
^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)/x^5/(
c*x+1)/(c*x-1)*arccosh(c*x)*c^4-7700/9*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-273
0*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)/x^7/(c*x+1)/(c*x-1)*arccosh(c*x)*c^2-4/63*a*c^2/d/x^7*(-c^2*d*x^2+d)
^(5/2)-829/56*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-47
25*c^2*x^2+1225)*x^2/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c^11+25915/126*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*
c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)/x^2/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c^7+1225/9*
b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+122
5)/x^9/(c*x+1)/(c*x-1)*arccosh(c*x)-1225/72*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^
8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)/x^8/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c-16/3*b*(-d*(c^2*x^2-1))^(1/2)
*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^10/(c*x+1)^(1/2)/(c
*x-1)^(1/2)*c^19+4*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x
^4-4725*c^2*x^2+1225)*x^8/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c^17+4189/180*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-
945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^6/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c^15-11
87/60*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x
^2+1225)*x^4/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c^13-1285/6*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+1
89*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)/x^4/(c*x+1)^(1/2)/(c*x-1)^(1/2)*c^5+21175/216*b*(-d*(c
^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)/x^6/(
c*x+1)^(1/2)/(c*x-1)^(1/2)*c^3+280/9*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*
c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)/(c*x+1)^(1/2)/(c*x-1)^(1/2)*arccosh(c*x)*c^9+128/315*b*(-d*(c^2*x^2-1)
)^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^17/(c*x+1)/(
c*x-1)*c^26-16/315*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x
^4-4725*c^2*x^2+1225)*x^15/(c*x+1)/(c*x-1)*c^24-344/189*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^1
0+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^13/(c*x+1)/(c*x-1)*c^22-922/945*b*(-d*(c^2*x^2-1)
)^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^11/(c*x+1)/(
c*x-1)*c^20+2906/945*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4
*x^4-4725*c^2*x^2+1225)*x^9/(c*x+1)/(c*x-1)*c^18+2069/189*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x
^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^7/(c*x+1)/(c*x-1)*c^16-4639/189*b*(-d*(c^2*x^2-
1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^5/(c*x+1)/
(c*x-1)*c^14+455/27*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*
x^4-4725*c^2*x^2+1225)*x^3/(c*x+1)/(c*x-1)*c^12-35/9*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+1
89*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x/(c*x+1)/(c*x-1)*c^10+64/3*b*(-d*(c^2*x^2-1))^(1/2)*d
/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^12/(c*x+1)^(1/2)/(c*x
-1)^(1/2)*arccosh(c*x)*c^21-24*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^
6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^10/(c*x+1)^(1/2)/(c*x-1)^(1/2)*arccosh(c*x)*c^19+24/5*b*(-d*(c^2*x^2-1))^(
1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^8/(c*x+1)^(1/2)
/(c*x-1)^(1/2)*arccosh(c*x)*c^17-208/3*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-273
0*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^6/(c*x+1)^(1/2)/(c*x-1)^(1/2)*arccosh(c*x)*c^15+1104/7*b*(-d*(c^2*
x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^4/(c*x
+1)^(1/2)/(c*x-1)^(1/2)*arccosh(c*x)*c^13-120*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*
x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^2/(c*x+1)^(1/2)/(c*x-1)^(1/2)*arccosh(c*x)*c^11-64/3*b*(-d*
(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^1
3/(c*x+1)/(c*x-1)*arccosh(c*x)*c^22+104/3*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-
2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^11/(c*x+1)/(c*x-1)*arccosh(c*x)*c^20-212/15*b*(-d*(c^2*x^2-1))^
(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^9/(c*x+1)/(c*x
-1)*arccosh(c*x)*c^18+3151/15*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6
+6210*c^4*x^4-4725*c^2*x^2+1225)*x^7/(c*x+1)/(c*x-1)*arccosh(c*x)*c^16-60632/105*b*(-d*(c^2*x^2-1))^(1/2)*d/(8
40*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^4*x^4-4725*c^2*x^2+1225)*x^5/(c*x+1)/(c*x-1)*arccos
h(c*x)*c^14+59884/105*b*(-d*(c^2*x^2-1))^(1/2)*d/(840*c^12*x^12-945*c^10*x^10+189*c^8*x^8-2730*c^6*x^6+6210*c^
4*x^4-4725*c^2*x^2+1225)*x^3/(c*x+1)/(c*x-1)*arccosh(c*x)*c^12-8/315*a*c^4/d/x^5*(-c^2*d*x^2+d)^(5/2)-1/9*a/d/
x^9*(-c^2*d*x^2+d)^(5/2)

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maxima [A]  time = 0.73, size = 225, normalized size = 0.69 \[ \frac {1}{7560} \, {\left (192 \, c^{8} \sqrt {-d} d \log \relax (x) - \frac {48 \, c^{6} \sqrt {-d} d x^{6} + 18 \, c^{4} \sqrt {-d} d x^{4} - 200 \, c^{2} \sqrt {-d} d x^{2} + 105 \, \sqrt {-d} d}{x^{8}}\right )} b c - \frac {1}{315} \, b {\left (\frac {8 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} c^{4}}{d x^{5}} + \frac {20 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} c^{2}}{d x^{7}} + \frac {35 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}{d x^{9}}\right )} \operatorname {arcosh}\left (c x\right ) - \frac {1}{315} \, a {\left (\frac {8 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} c^{4}}{d x^{5}} + \frac {20 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} c^{2}}{d x^{7}} + \frac {35 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}{d x^{9}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))/x^10,x, algorithm="maxima")

[Out]

1/7560*(192*c^8*sqrt(-d)*d*log(x) - (48*c^6*sqrt(-d)*d*x^6 + 18*c^4*sqrt(-d)*d*x^4 - 200*c^2*sqrt(-d)*d*x^2 +
105*sqrt(-d)*d)/x^8)*b*c - 1/315*b*(8*(-c^2*d*x^2 + d)^(5/2)*c^4/(d*x^5) + 20*(-c^2*d*x^2 + d)^(5/2)*c^2/(d*x^
7) + 35*(-c^2*d*x^2 + d)^(5/2)/(d*x^9))*arccosh(c*x) - 1/315*a*(8*(-c^2*d*x^2 + d)^(5/2)*c^4/(d*x^5) + 20*(-c^
2*d*x^2 + d)^(5/2)*c^2/(d*x^7) + 35*(-c^2*d*x^2 + d)^(5/2)/(d*x^9))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{3/2}}{x^{10}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*acosh(c*x))*(d - c^2*d*x^2)^(3/2))/x^10,x)

[Out]

int(((a + b*acosh(c*x))*(d - c^2*d*x^2)^(3/2))/x^10, x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c**2*d*x**2+d)**(3/2)*(a+b*acosh(c*x))/x**10,x)

[Out]

Timed out

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