3.1.89 \(\int (d-c^2 d x^2)^{5/2} (a+b \cosh ^{-1}(c x)) \, dx\) [89]

Optimal. Leaf size=293 \[ -\frac {25 b c d^2 x^2 \sqrt {d-c^2 d x^2}}{96 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 b c^3 d^2 x^4 \sqrt {d-c^2 d x^2}}{96 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d^2 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{36 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5}{16} d^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {5}{24} d x \left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{6} x \left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {5 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{32 b c \sqrt {-1+c x} \sqrt {1+c x}} \]

[Out]

5/24*d*x*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))+1/6*x*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))+5/16*d^2*x*(a+b
*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)-25/96*b*c*d^2*x^2*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+5/96*b*
c^3*d^2*x^4*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/36*b*d^2*(-c^2*x^2+1)^3*(-c^2*d*x^2+d)^(1/2)/c/
(c*x-1)^(1/2)/(c*x+1)^(1/2)-5/32*d^2*(a+b*arccosh(c*x))^2*(-c^2*d*x^2+d)^(1/2)/b/c/(c*x-1)^(1/2)/(c*x+1)^(1/2)

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Rubi [A]
time = 0.16, antiderivative size = 293, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {5897, 5895, 5893, 30, 74, 14, 267} \begin {gather*} \frac {5}{16} d^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {5 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{32 b c \sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{6} x \left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {5}{24} d x \left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {25 b c d^2 x^2 \sqrt {d-c^2 d x^2}}{96 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b d^2 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{36 c \sqrt {c x-1} \sqrt {c x+1}}+\frac {5 b c^3 d^2 x^4 \sqrt {d-c^2 d x^2}}{96 \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]

[Out]

(-25*b*c*d^2*x^2*Sqrt[d - c^2*d*x^2])/(96*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (5*b*c^3*d^2*x^4*Sqrt[d - c^2*d*x^2]
)/(96*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*d^2*(1 - c^2*x^2)^3*Sqrt[d - c^2*d*x^2])/(36*c*Sqrt[-1 + c*x]*Sqrt[1
+ c*x]) + (5*d^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/16 + (5*d*x*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[
c*x]))/24 + (x*(d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/6 - (5*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])
^2)/(32*b*c*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 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

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 267

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

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 5895

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[x*Sqrt[d + e*x^2]*(
(a + b*ArcCosh[c*x])^n/2), x] + (-Dist[(1/2)*Simp[Sqrt[d + e*x^2]/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])], Int[(a + b*
ArcCosh[c*x])^n/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x] - Dist[b*c*(n/2)*Simp[Sqrt[d + e*x^2]/(Sqrt[1 + c*x]*Sq
rt[-1 + c*x])], Int[x*(a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0]
&& GtQ[n, 0]

Rule 5897

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[x*(d + e*x^2)^p*(
(a + b*ArcCosh[c*x])^n/(2*p + 1)), x] + (Dist[2*d*(p/(2*p + 1)), Int[(d + e*x^2)^(p - 1)*(a + b*ArcCosh[c*x])^
n, x], x] - Dist[b*c*(n/(2*p + 1))*Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)], Int[x*(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}, x] && EqQ[c^2*d + e, 0] &&
GtQ[n, 0] && GtQ[p, 0]

Rubi steps

\begin {align*} \int \left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \int (-1+c x)^{5/2} (1+c x)^{5/2} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {1}{6} d^2 x (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (5 d^2 \sqrt {d-c^2 d x^2}\right ) \int (-1+c x)^{3/2} (1+c x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{6 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (b c d^2 \sqrt {d-c^2 d x^2}\right ) \int x \left (-1+c^2 x^2\right )^2 \, dx}{6 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {b d^2 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{36 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5}{24} d^2 x (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{6} d^2 x (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {\left (5 d^2 \sqrt {d-c^2 d x^2}\right ) \int \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{8 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (5 b c d^2 \sqrt {d-c^2 d x^2}\right ) \int x \left (-1+c^2 x^2\right ) \, dx}{24 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {b d^2 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{36 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5}{16} d^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {5}{24} d^2 x (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{6} d^2 x (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (5 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}{16 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (5 b c d^2 \sqrt {d-c^2 d x^2}\right ) \int \left (-x+c^2 x^3\right ) \, dx}{24 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (5 b c d^2 \sqrt {d-c^2 d x^2}\right ) \int x \, dx}{16 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {25 b c d^2 x^2 \sqrt {d-c^2 d x^2}}{96 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 b c^3 d^2 x^4 \sqrt {d-c^2 d x^2}}{96 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d^2 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{36 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5}{16} d^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {5}{24} d^2 x (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{6} d^2 x (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {5 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{32 b c \sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}

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Mathematica [A]
time = 1.62, size = 347, normalized size = 1.18 \begin {gather*} \frac {48 a c d^2 x \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \sqrt {d-c^2 d x^2} \left (33-26 c^2 x^2+8 c^4 x^4\right )-720 a d^{5/2} \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {ArcTan}\left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )-288 b d^2 \sqrt {d-c^2 d x^2} \left (\cosh \left (2 \cosh ^{-1}(c x)\right )+2 \cosh ^{-1}(c x) \left (\cosh ^{-1}(c x)-\sinh \left (2 \cosh ^{-1}(c x)\right )\right )\right )+36 b d^2 \sqrt {d-c^2 d x^2} \left (8 \cosh ^{-1}(c x)^2+\cosh \left (4 \cosh ^{-1}(c x)\right )-4 \cosh ^{-1}(c x) \sinh \left (4 \cosh ^{-1}(c x)\right )\right )+b d^2 \sqrt {d-c^2 d x^2} \left (-72 \cosh ^{-1}(c x)^2+18 \cosh \left (2 \cosh ^{-1}(c x)\right )-9 \cosh \left (4 \cosh ^{-1}(c x)\right )-2 \cosh \left (6 \cosh ^{-1}(c x)\right )+12 \cosh ^{-1}(c x) \left (-3 \sinh \left (2 \cosh ^{-1}(c x)\right )+3 \sinh \left (4 \cosh ^{-1}(c x)\right )+\sinh \left (6 \cosh ^{-1}(c x)\right )\right )\right )}{2304 c \sqrt {\frac {-1+c x}{1+c x}} (1+c x)} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(48*a*c*d^2*x*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*Sqrt[d - c^2*d*x^2]*(33 - 26*c^2*x^2 + 8*c^4*x^4) - 720*a*d
^(5/2)*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] - 288*b
*d^2*Sqrt[d - c^2*d*x^2]*(Cosh[2*ArcCosh[c*x]] + 2*ArcCosh[c*x]*(ArcCosh[c*x] - Sinh[2*ArcCosh[c*x]])) + 36*b*
d^2*Sqrt[d - c^2*d*x^2]*(8*ArcCosh[c*x]^2 + Cosh[4*ArcCosh[c*x]] - 4*ArcCosh[c*x]*Sinh[4*ArcCosh[c*x]]) + b*d^
2*Sqrt[d - c^2*d*x^2]*(-72*ArcCosh[c*x]^2 + 18*Cosh[2*ArcCosh[c*x]] - 9*Cosh[4*ArcCosh[c*x]] - 2*Cosh[6*ArcCos
h[c*x]] + 12*ArcCosh[c*x]*(-3*Sinh[2*ArcCosh[c*x]] + 3*Sinh[4*ArcCosh[c*x]] + Sinh[6*ArcCosh[c*x]])))/(2304*c*
Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(884\) vs. \(2(249)=498\).
time = 2.53, size = 885, normalized size = 3.02

method	result	size
default	\(\frac {a x \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{6}+\frac {5 a d x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{24}+\frac {5 a \,d^{2} x \sqrt {-c^{2} d \,x^{2}+d}}{16}+\frac {5 a \,d^{3} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{16 \sqrt {c^{2} d}}+b \left (-\frac {5 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right )^{2} d^{2}}{32 \sqrt {c x -1}\, \sqrt {c x +1}\, c}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (32 c^{7} x^{7}-64 c^{5} x^{5}+32 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{6} c^{6}+38 c^{3} x^{3}-48 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{4} c^{4}-6 c x +18 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{2} c^{2}-\sqrt {c x -1}\, \sqrt {c x +1}\right ) \left (-1+6 \,\mathrm {arccosh}\left (c x \right )\right ) d^{2}}{2304 \left (c x +1\right ) \left (c x -1\right ) c}-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (8 c^{5} x^{5}-12 c^{3} x^{3}+8 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{4} c^{4}+4 c x -8 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{2} c^{2}+\sqrt {c x -1}\, \sqrt {c x +1}\right ) \left (-1+4 \,\mathrm {arccosh}\left (c x \right )\right ) d^{2}}{512 \left (c x +1\right ) \left (c x -1\right ) c}+\frac {15 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 c^{3} x^{3}-2 c x +2 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{2} c^{2}-\sqrt {c x -1}\, \sqrt {c x +1}\right ) \left (-1+2 \,\mathrm {arccosh}\left (c x \right )\right ) d^{2}}{256 \left (c x +1\right ) \left (c x -1\right ) c}+\frac {15 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-2 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{2} c^{2}+2 c^{3} x^{3}+\sqrt {c x -1}\, \sqrt {c x +1}-2 c x \right ) \left (1+2 \,\mathrm {arccosh}\left (c x \right )\right ) d^{2}}{256 \left (c x +1\right ) \left (c x -1\right ) c}-\frac {3 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-8 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{4} c^{4}+8 c^{5} x^{5}+8 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{2} c^{2}-12 c^{3} x^{3}-\sqrt {c x -1}\, \sqrt {c x +1}+4 c x \right ) \left (1+4 \,\mathrm {arccosh}\left (c x \right )\right ) d^{2}}{512 \left (c x +1\right ) \left (c x -1\right ) c}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-32 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{6} c^{6}+32 c^{7} x^{7}+48 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{4} c^{4}-64 c^{5} x^{5}-18 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{2} c^{2}+38 c^{3} x^{3}+\sqrt {c x -1}\, \sqrt {c x +1}-6 c x \right ) \left (1+6 \,\mathrm {arccosh}\left (c x \right )\right ) d^{2}}{2304 \left (c x +1\right ) \left (c x -1\right ) c}\right )\)	\(885\)

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,method=_RETURNVERBOSE)

[Out]

1/6*a*x*(-c^2*d*x^2+d)^(5/2)+5/24*a*d*x*(-c^2*d*x^2+d)^(3/2)+5/16*a*d^2*x*(-c^2*d*x^2+d)^(1/2)+5/16*a*d^3/(c^2
*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))+b*(-5/32*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(
1/2)/c*arccosh(c*x)^2*d^2+1/2304*(-d*(c^2*x^2-1))^(1/2)*(32*c^7*x^7-64*c^5*x^5+32*(c*x+1)^(1/2)*(c*x-1)^(1/2)*
x^6*c^6+38*c^3*x^3-48*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^4*c^4-6*c*x+18*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^2*c^2-(c*x-1)
^(1/2)*(c*x+1)^(1/2))*(-1+6*arccosh(c*x))*d^2/(c*x+1)/(c*x-1)/c-3/512*(-d*(c^2*x^2-1))^(1/2)*(8*c^5*x^5-12*c^3
*x^3+8*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^4*c^4+4*c*x-8*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^2*c^2+(c*x-1)^(1/2)*(c*x+1)^(
1/2))*(-1+4*arccosh(c*x))*d^2/(c*x+1)/(c*x-1)/c+15/256*(-d*(c^2*x^2-1))^(1/2)*(2*c^3*x^3-2*c*x+2*(c*x+1)^(1/2)
*(c*x-1)^(1/2)*x^2*c^2-(c*x-1)^(1/2)*(c*x+1)^(1/2))*(-1+2*arccosh(c*x))*d^2/(c*x+1)/(c*x-1)/c+15/256*(-d*(c^2*
x^2-1))^(1/2)*(-2*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^2*c^2+2*c^3*x^3+(c*x-1)^(1/2)*(c*x+1)^(1/2)-2*c*x)*(1+2*arccos
h(c*x))*d^2/(c*x+1)/(c*x-1)/c-3/512*(-d*(c^2*x^2-1))^(1/2)*(-8*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^4*c^4+8*c^5*x^5+8
*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^2*c^2-12*c^3*x^3-(c*x-1)^(1/2)*(c*x+1)^(1/2)+4*c*x)*(1+4*arccosh(c*x))*d^2/(c*x
+1)/(c*x-1)/c+1/2304*(-d*(c^2*x^2-1))^(1/2)*(-32*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^6*c^6+32*c^7*x^7+48*(c*x+1)^(1/
2)*(c*x-1)^(1/2)*x^4*c^4-64*c^5*x^5-18*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^2*c^2+38*c^3*x^3+(c*x-1)^(1/2)*(c*x+1)^(1
/2)-6*c*x)*(1+6*arccosh(c*x))*d^2/(c*x+1)/(c*x-1)/c)

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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, algorithm="maxima")

[Out]

1/48*(8*(-c^2*d*x^2 + d)^(5/2)*x + 10*(-c^2*d*x^2 + d)^(3/2)*d*x + 15*sqrt(-c^2*d*x^2 + d)*d^2*x + 15*d^(5/2)*
arcsin(c*x)/c)*a + b*integrate((-c^2*d*x^2 + d)^(5/2)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1)), 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, 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)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \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 )\, 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)

[Out]

Integral((-d*(c*x - 1)*(c*x + 1))**(5/2)*(a + b*acosh(c*x)), 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, 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 \left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{5/2} \,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)

[Out]

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

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