3.4.37 \(\int \frac {(1-c^2 x^2)^{5/2}}{(a+b \cosh ^{-1}(c x))^2} \, dx\) [337]

Optimal. Leaf size=351 \[ -\frac {\sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )^{5/2}}{b c \left (a+b \cosh ^{-1}(c x)\right )}-\frac {15 \sqrt {1-c x} \text {Chi}\left (\frac {2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right ) \sinh \left (\frac {2 a}{b}\right )}{16 b^2 c \sqrt {-1+c x}}+\frac {3 \sqrt {1-c x} \text {Chi}\left (\frac {4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right ) \sinh \left (\frac {4 a}{b}\right )}{4 b^2 c \sqrt {-1+c x}}-\frac {3 \sqrt {1-c x} \text {Chi}\left (\frac {6 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right ) \sinh \left (\frac {6 a}{b}\right )}{16 b^2 c \sqrt {-1+c x}}+\frac {15 \sqrt {1-c x} \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{16 b^2 c \sqrt {-1+c x}}-\frac {3 \sqrt {1-c x} \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{4 b^2 c \sqrt {-1+c x}}+\frac {3 \sqrt {1-c x} \cosh \left (\frac {6 a}{b}\right ) \text {Shi}\left (\frac {6 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{16 b^2 c \sqrt {-1+c x}} \]

[Out]

15/16*cosh(2*a/b)*Shi(2*(a+b*arccosh(c*x))/b)*(-c*x+1)^(1/2)/b^2/c/(c*x-1)^(1/2)-3/4*cosh(4*a/b)*Shi(4*(a+b*ar
ccosh(c*x))/b)*(-c*x+1)^(1/2)/b^2/c/(c*x-1)^(1/2)+3/16*cosh(6*a/b)*Shi(6*(a+b*arccosh(c*x))/b)*(-c*x+1)^(1/2)/
b^2/c/(c*x-1)^(1/2)-15/16*Chi(2*(a+b*arccosh(c*x))/b)*sinh(2*a/b)*(-c*x+1)^(1/2)/b^2/c/(c*x-1)^(1/2)+3/4*Chi(4
*(a+b*arccosh(c*x))/b)*sinh(4*a/b)*(-c*x+1)^(1/2)/b^2/c/(c*x-1)^(1/2)-3/16*Chi(6*(a+b*arccosh(c*x))/b)*sinh(6*
a/b)*(-c*x+1)^(1/2)/b^2/c/(c*x-1)^(1/2)-(-c^2*x^2+1)^(5/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/b/c/(a+b*arccosh(c*x))

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Rubi [A]
time = 0.35, antiderivative size = 351, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 7, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {5904, 5912, 5952, 5556, 3384, 3379, 3382} \begin {gather*} -\frac {15 \sqrt {1-c x} \sinh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{16 b^2 c \sqrt {c x-1}}+\frac {3 \sqrt {1-c x} \sinh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{4 b^2 c \sqrt {c x-1}}-\frac {3 \sqrt {1-c x} \sinh \left (\frac {6 a}{b}\right ) \text {Chi}\left (\frac {6 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{16 b^2 c \sqrt {c x-1}}+\frac {15 \sqrt {1-c x} \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{16 b^2 c \sqrt {c x-1}}-\frac {3 \sqrt {1-c x} \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{4 b^2 c \sqrt {c x-1}}+\frac {3 \sqrt {1-c x} \cosh \left (\frac {6 a}{b}\right ) \text {Shi}\left (\frac {6 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{16 b^2 c \sqrt {c x-1}}-\frac {\sqrt {c x-1} \sqrt {c x+1} \left (1-c^2 x^2\right )^{5/2}}{b c \left (a+b \cosh ^{-1}(c x)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - c^2*x^2)^(5/2)/(a + b*ArcCosh[c*x])^2,x]

[Out]

-((Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(1 - c^2*x^2)^(5/2))/(b*c*(a + b*ArcCosh[c*x]))) - (15*Sqrt[1 - c*x]*CoshInteg
ral[(2*(a + b*ArcCosh[c*x]))/b]*Sinh[(2*a)/b])/(16*b^2*c*Sqrt[-1 + c*x]) + (3*Sqrt[1 - c*x]*CoshIntegral[(4*(a
+ b*ArcCosh[c*x]))/b]*Sinh[(4*a)/b])/(4*b^2*c*Sqrt[-1 + c*x]) - (3*Sqrt[1 - c*x]*CoshIntegral[(6*(a + b*ArcCo
sh[c*x]))/b]*Sinh[(6*a)/b])/(16*b^2*c*Sqrt[-1 + c*x]) + (15*Sqrt[1 - c*x]*Cosh[(2*a)/b]*SinhIntegral[(2*(a + b
*ArcCosh[c*x]))/b])/(16*b^2*c*Sqrt[-1 + c*x]) - (3*Sqrt[1 - c*x]*Cosh[(4*a)/b]*SinhIntegral[(4*(a + b*ArcCosh[
c*x]))/b])/(4*b^2*c*Sqrt[-1 + c*x]) + (3*Sqrt[1 - c*x]*Cosh[(6*a)/b]*SinhIntegral[(6*(a + b*ArcCosh[c*x]))/b])
/(16*b^2*c*Sqrt[-1 + c*x])

Rule 3379

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[I*(SinhIntegral[c*f*(fz/
d) + f*fz*x]/d), x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*e - c*f*fz*I, 0]

Rule 3382

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CoshIntegral[c*f*(fz/d)
+ f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]

Rule 3384

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[c*(f/d) + f*x]
/(c + d*x), x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[c*(f/d) + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f},
x] && NeQ[d*e - c*f, 0]

Rule 5556

Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int
[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n,
0] && IGtQ[p, 0]

Rule 5904

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

Rule 5912

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_.)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(
x_))^(p_.), x_Symbol] :> Int[(f*x)^m*(d1*d2 + e1*e2*x^2)^p*(a + b*ArcCosh[c*x])^n, x] /; FreeQ[{a, b, c, d1, e
1, d2, e2, f, m, n}, x] && EqQ[d2*e1 + d1*e2, 0] && IntegerQ[p]

Rule 5952

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

Rubi steps

\begin {align*} \int \frac {\left (1-c^2 x^2\right )^{5/2}}{\left (a+b \cosh ^{-1}(c x)\right )^2} \, dx &=\frac {\sqrt {1-c^2 x^2} \int \frac {(-1+c x)^{5/2} (1+c x)^{5/2}}{\left (a+b \cosh ^{-1}(c x)\right )^2} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {(1-c x)^3 (1+c x)^{5/2} \sqrt {1-c^2 x^2}}{b c \sqrt {-1+c x} \left (a+b \cosh ^{-1}(c x)\right )}+\frac {\left (6 c \sqrt {1-c^2 x^2}\right ) \int \frac {x \left (-1+c^2 x^2\right )^2}{a+b \cosh ^{-1}(c x)} \, dx}{b \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {(1-c x)^3 (1+c x)^{5/2} \sqrt {1-c^2 x^2}}{b c \sqrt {-1+c x} \left (a+b \cosh ^{-1}(c x)\right )}+\frac {\left (6 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\cosh (x) \sinh ^5(x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{b c \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {(1-c x)^3 (1+c x)^{5/2} \sqrt {1-c^2 x^2}}{b c \sqrt {-1+c x} \left (a+b \cosh ^{-1}(c x)\right )}+\frac {\left (6 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \left (\frac {5 \sinh (2 x)}{32 (a+b x)}-\frac {\sinh (4 x)}{8 (a+b x)}+\frac {\sinh (6 x)}{32 (a+b x)}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{b c \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {(1-c x)^3 (1+c x)^{5/2} \sqrt {1-c^2 x^2}}{b c \sqrt {-1+c x} \left (a+b \cosh ^{-1}(c x)\right )}+\frac {\left (3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\sinh (6 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{16 b c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\sinh (4 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{4 b c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (15 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\sinh (2 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{16 b c \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {(1-c x)^3 (1+c x)^{5/2} \sqrt {1-c^2 x^2}}{b c \sqrt {-1+c x} \left (a+b \cosh ^{-1}(c x)\right )}+\frac {\left (15 \sqrt {1-c^2 x^2} \cosh \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {2 a}{b}+2 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{16 b c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (3 \sqrt {1-c^2 x^2} \cosh \left (\frac {4 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {4 a}{b}+4 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{4 b c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (3 \sqrt {1-c^2 x^2} \cosh \left (\frac {6 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {6 a}{b}+6 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{16 b c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (15 \sqrt {1-c^2 x^2} \sinh \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {2 a}{b}+2 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{16 b c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (3 \sqrt {1-c^2 x^2} \sinh \left (\frac {4 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {4 a}{b}+4 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{4 b c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (3 \sqrt {1-c^2 x^2} \sinh \left (\frac {6 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {6 a}{b}+6 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{16 b c \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {(1-c x)^3 (1+c x)^{5/2} \sqrt {1-c^2 x^2}}{b c \sqrt {-1+c x} \left (a+b \cosh ^{-1}(c x)\right )}-\frac {15 \sqrt {1-c^2 x^2} \text {Chi}\left (\frac {2 a}{b}+2 \cosh ^{-1}(c x)\right ) \sinh \left (\frac {2 a}{b}\right )}{16 b^2 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 \sqrt {1-c^2 x^2} \text {Chi}\left (\frac {4 a}{b}+4 \cosh ^{-1}(c x)\right ) \sinh \left (\frac {4 a}{b}\right )}{4 b^2 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 \sqrt {1-c^2 x^2} \text {Chi}\left (\frac {6 a}{b}+6 \cosh ^{-1}(c x)\right ) \sinh \left (\frac {6 a}{b}\right )}{16 b^2 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {15 \sqrt {1-c^2 x^2} \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \cosh ^{-1}(c x)\right )}{16 b^2 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 \sqrt {1-c^2 x^2} \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 a}{b}+4 \cosh ^{-1}(c x)\right )}{4 b^2 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 \sqrt {1-c^2 x^2} \cosh \left (\frac {6 a}{b}\right ) \text {Shi}\left (\frac {6 a}{b}+6 \cosh ^{-1}(c x)\right )}{16 b^2 c \sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}

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Mathematica [A]
time = 0.72, size = 343, normalized size = 0.98 \begin {gather*} \frac {\sqrt {-1+c x} \sqrt {1+c x} \left (-16 b+48 b c^2 x^2-48 b c^4 x^4+16 b c^6 x^6+15 \left (a+b \cosh ^{-1}(c x)\right ) \text {Chi}\left (2 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right ) \sinh \left (\frac {2 a}{b}\right )-12 \left (a+b \cosh ^{-1}(c x)\right ) \text {Chi}\left (4 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right ) \sinh \left (\frac {4 a}{b}\right )+3 a \text {Chi}\left (6 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right ) \sinh \left (\frac {6 a}{b}\right )+3 b \cosh ^{-1}(c x) \text {Chi}\left (6 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right ) \sinh \left (\frac {6 a}{b}\right )-15 a \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )-15 b \cosh ^{-1}(c x) \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )+12 a \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (4 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )+12 b \cosh ^{-1}(c x) \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (4 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )-3 a \cosh \left (\frac {6 a}{b}\right ) \text {Shi}\left (6 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )-3 b \cosh ^{-1}(c x) \cosh \left (\frac {6 a}{b}\right ) \text {Shi}\left (6 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )\right )}{16 b^2 c \sqrt {1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 - c^2*x^2)^(5/2)/(a + b*ArcCosh[c*x])^2,x]

[Out]

(Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-16*b + 48*b*c^2*x^2 - 48*b*c^4*x^4 + 16*b*c^6*x^6 + 15*(a + b*ArcCosh[c*x])*Co
shIntegral[2*(a/b + ArcCosh[c*x])]*Sinh[(2*a)/b] - 12*(a + b*ArcCosh[c*x])*CoshIntegral[4*(a/b + ArcCosh[c*x])
]*Sinh[(4*a)/b] + 3*a*CoshIntegral[6*(a/b + ArcCosh[c*x])]*Sinh[(6*a)/b] + 3*b*ArcCosh[c*x]*CoshIntegral[6*(a/
b + ArcCosh[c*x])]*Sinh[(6*a)/b] - 15*a*Cosh[(2*a)/b]*SinhIntegral[2*(a/b + ArcCosh[c*x])] - 15*b*ArcCosh[c*x]
*Cosh[(2*a)/b]*SinhIntegral[2*(a/b + ArcCosh[c*x])] + 12*a*Cosh[(4*a)/b]*SinhIntegral[4*(a/b + ArcCosh[c*x])]
+ 12*b*ArcCosh[c*x]*Cosh[(4*a)/b]*SinhIntegral[4*(a/b + ArcCosh[c*x])] - 3*a*Cosh[(6*a)/b]*SinhIntegral[6*(a/b
+ ArcCosh[c*x])] - 3*b*ArcCosh[c*x]*Cosh[(6*a)/b]*SinhIntegral[6*(a/b + ArcCosh[c*x])]))/(16*b^2*c*Sqrt[1 - c
^2*x^2]*(a + b*ArcCosh[c*x]))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1175\) vs. \(2(309)=618\).
time = 4.03, size = 1176, normalized size = 3.35

method	result	size
default	\(\text {Expression too large to display}\)	\(1176\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-c^2*x^2+1)^(5/2)/(a+b*arccosh(c*x))^2,x,method=_RETURNVERBOSE)

[Out]

1/64*(-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)/(c*x+1)/(
c*x-1)/c/b/(a+b*arccosh(c*x))-3/32*(-c^2*x^2+1)^(1/2)*(-(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c+c^2*x^2-1)*Ei(1,6*arcc
osh(c*x)+6*a/b)*exp((b*arccosh(c*x)+6*a)/b)/(c*x+1)/(c*x-1)/c/b^2-1/64/(c*x+1)^(1/2)/(c*x-1)^(1/2)*(-c^2*x^2+1
)^(1/2)*(32*(c*x-1)^(1/2)*(c*x+1)^(1/2)*b*c^5*x^5+32*b*c^6*x^6-32*(c*x+1)^(1/2)*(c*x-1)^(1/2)*b*c^3*x^3-48*b*c
^4*x^4+6*(c*x+1)^(1/2)*(c*x-1)^(1/2)*b*c*x+18*b*c^2*x^2+6*arccosh(c*x)*Ei(1,-6*arccosh(c*x)-6*a/b)*exp(-6*a/b)
*b+6*Ei(1,-6*arccosh(c*x)-6*a/b)*exp(-6*a/b)*a-b)/c/b^2/(a+b*arccosh(c*x))+5/16*(-c^2*x^2+1)^(1/2)/(c*x-1)^(1/
2)/(c*x+1)^(1/2)/c/(a+b*arccosh(c*x))/b-3/32*(-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)/(c*x+1)/(c*x-1)/c/b/(a
+b*arccosh(c*x))+3/8*(-c^2*x^2+1)^(1/2)*(-(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c+c^2*x^2-1)*Ei(1,4*arccosh(c*x)+4*a/b
)*exp((b*arccosh(c*x)+4*a)/b)/(c*x+1)/(c*x-1)/c/b^2+15/64*(-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)/(c*x+1)/(c*x-1)/c/(a+b*arccosh(c*x))/b-15/32*(-c^2*x^2+1)^
(1/2)*(-(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c+c^2*x^2-1)*Ei(1,2*arccosh(c*x)+2*a/b)*exp((b*arccosh(c*x)+2*a)/b)/(c*x
+1)/(c*x-1)/c/b^2-15/64*(-c^2*x^2+1)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)*(2*(c*x+1)^(1/2)*(c*x-1)^(1/2)*b*c*x+2*
b*c^2*x^2+2*arccosh(c*x)*Ei(1,-2*arccosh(c*x)-2*a/b)*exp(-2*a/b)*b+2*Ei(1,-2*arccosh(c*x)-2*a/b)*exp(-2*a/b)*a
-b)/c/b^2/(a+b*arccosh(c*x))+3/32/(c*x+1)^(1/2)/(c*x-1)^(1/2)*(-c^2*x^2+1)^(1/2)*(8*(c*x+1)^(1/2)*(c*x-1)^(1/2
)*b*c^3*x^3+8*b*c^4*x^4-4*(c*x+1)^(1/2)*(c*x-1)^(1/2)*b*c*x-8*b*c^2*x^2+4*arccosh(c*x)*exp(-4*a/b)*Ei(1,-4*arc
cosh(c*x)-4*a/b)*b+4*exp(-4*a/b)*Ei(1,-4*arccosh(c*x)-4*a/b)*a+b)/c/b^2/(a+b*arccosh(c*x))

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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*x^2+1)^(5/2)/(a+b*arccosh(c*x))^2,x, algorithm="maxima")

[Out]

-((c^6*x^6 - 3*c^4*x^4 + 3*c^2*x^2 - 1)*(c*x + 1)*sqrt(c*x - 1) + (c^7*x^7 - 3*c^5*x^5 + 3*c^3*x^3 - c*x)*sqrt
(c*x + 1))*sqrt(-c*x + 1)/(a*b*c^3*x^2 + sqrt(c*x + 1)*sqrt(c*x - 1)*a*b*c^2*x - a*b*c + (b^2*c^3*x^2 + sqrt(c
*x + 1)*sqrt(c*x - 1)*b^2*c^2*x - b^2*c)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))) + integrate(((6*c^6*x^6 - 11*
c^4*x^4 + 4*c^2*x^2 + 1)*(c*x + 1)^(3/2)*(c*x - 1) + 6*(2*c^7*x^7 - 5*c^5*x^5 + 4*c^3*x^3 - c*x)*(c*x + 1)*sqr
t(c*x - 1) + (6*c^8*x^8 - 19*c^6*x^6 + 21*c^4*x^4 - 9*c^2*x^2 + 1)*sqrt(c*x + 1))*sqrt(-c*x + 1)/(a*b*c^4*x^4
+ (c*x + 1)*(c*x - 1)*a*b*c^2*x^2 - 2*a*b*c^2*x^2 + 2*(a*b*c^3*x^3 - a*b*c*x)*sqrt(c*x + 1)*sqrt(c*x - 1) + a*
b + (b^2*c^4*x^4 + (c*x + 1)*(c*x - 1)*b^2*c^2*x^2 - 2*b^2*c^2*x^2 + 2*(b^2*c^3*x^3 - b^2*c*x)*sqrt(c*x + 1)*s
qrt(c*x - 1) + b^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*x^2+1)^(5/2)/(a+b*arccosh(c*x))^2,x, algorithm="fricas")

[Out]

integral((c^4*x^4 - 2*c^2*x^2 + 1)*sqrt(-c^2*x^2 + 1)/(b^2*arccosh(c*x)^2 + 2*a*b*arccosh(c*x) + a^2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c**2*x**2+1)**(5/2)/(a+b*acosh(c*x))**2,x)

[Out]

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

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Giac [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*x^2+1)^(5/2)/(a+b*arccosh(c*x))^2,x, algorithm="giac")

[Out]

integrate((-c^2*x^2 + 1)^(5/2)/(b*arccosh(c*x) + a)^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - c^2*x^2)^(5/2)/(a + b*acosh(c*x))^2,x)

[Out]

int((1 - c^2*x^2)^(5/2)/(a + b*acosh(c*x))^2, x)

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