3.4.30 \(\int \frac {(1-c^2 x^2)^{3/2}}{(a+b \cosh ^{-1}(c x))^2} \, dx\) [330]

Optimal. Leaf size=246 \[ -\frac {\sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )^{3/2}}{b c \left (a+b \cosh ^{-1}(c x)\right )}-\frac {\sqrt {1-c x} \text {Chi}\left (\frac {2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right ) \sinh \left (\frac {2 a}{b}\right )}{b^2 c \sqrt {-1+c x}}+\frac {\sqrt {1-c x} \text {Chi}\left (\frac {4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right ) \sinh \left (\frac {4 a}{b}\right )}{2 b^2 c \sqrt {-1+c x}}+\frac {\sqrt {1-c x} \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{b^2 c \sqrt {-1+c x}}-\frac {\sqrt {1-c x} \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{2 b^2 c \sqrt {-1+c x}} \]

[Out]

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)-1/2*cosh(4*a/b)*Shi(4*(a+b*arccosh(
c*x))/b)*(-c*x+1)^(1/2)/b^2/c/(c*x-1)^(1/2)-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)+1/2*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)-(-c^2*x^2+1)^(3/2)*(c*
x-1)^(1/2)*(c*x+1)^(1/2)/b/c/(a+b*arccosh(c*x))

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Rubi [A]
time = 0.24, antiderivative size = 246, normalized size of antiderivative = 1.00, number of steps used = 11, 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 {\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 )}{b^2 c \sqrt {c x-1}}+\frac {\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 )}{2 b^2 c \sqrt {c x-1}}+\frac {\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 )}{b^2 c \sqrt {c x-1}}-\frac {\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 )}{2 b^2 c \sqrt {c x-1}}-\frac {\sqrt {c x-1} \sqrt {c x+1} \left (1-c^2 x^2\right )^{3/2}}{b c \left (a+b \cosh ^{-1}(c x)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - c^2*x^2)^(3/2)/(a + b*ArcCosh[c*x])^2,x]

[Out]

-((Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(1 - c^2*x^2)^(3/2))/(b*c*(a + b*ArcCosh[c*x]))) - (Sqrt[1 - c*x]*CoshIntegral
[(2*(a + b*ArcCosh[c*x]))/b]*Sinh[(2*a)/b])/(b^2*c*Sqrt[-1 + c*x]) + (Sqrt[1 - c*x]*CoshIntegral[(4*(a + b*Arc
Cosh[c*x]))/b]*Sinh[(4*a)/b])/(2*b^2*c*Sqrt[-1 + c*x]) + (Sqrt[1 - c*x]*Cosh[(2*a)/b]*SinhIntegral[(2*(a + b*A
rcCosh[c*x]))/b])/(b^2*c*Sqrt[-1 + c*x]) - (Sqrt[1 - c*x]*Cosh[(4*a)/b]*SinhIntegral[(4*(a + b*ArcCosh[c*x]))/
b])/(2*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 )^{3/2}}{\left (a+b \cosh ^{-1}(c x)\right )^2} \, dx &=-\frac {\sqrt {1-c^2 x^2} \int \frac {(-1+c x)^{3/2} (1+c x)^{3/2}}{\left (a+b \cosh ^{-1}(c x)\right )^2} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {(1-c x)^2 (1+c x)^{3/2} \sqrt {1-c^2 x^2}}{b c \sqrt {-1+c x} \left (a+b \cosh ^{-1}(c x)\right )}-\frac {\left (4 c \sqrt {1-c^2 x^2}\right ) \int \frac {x \left (-1+c^2 x^2\right )}{a+b \cosh ^{-1}(c x)} \, dx}{b \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {(1-c x)^2 (1+c x)^{3/2} \sqrt {1-c^2 x^2}}{b c \sqrt {-1+c x} \left (a+b \cosh ^{-1}(c x)\right )}-\frac {\left (4 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\cosh (x) \sinh ^3(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)^2 (1+c x)^{3/2} \sqrt {1-c^2 x^2}}{b c \sqrt {-1+c x} \left (a+b \cosh ^{-1}(c x)\right )}-\frac {\left (4 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \left (-\frac {\sinh (2 x)}{4 (a+b x)}+\frac {\sinh (4 x)}{8 (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)^2 (1+c x)^{3/2} \sqrt {1-c^2 x^2}}{b c \sqrt {-1+c x} \left (a+b \cosh ^{-1}(c x)\right )}-\frac {\sqrt {1-c^2 x^2} \text {Subst}\left (\int \frac {\sinh (4 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{2 b c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\sqrt {1-c^2 x^2} \text {Subst}\left (\int \frac {\sinh (2 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)^2 (1+c x)^{3/2} \sqrt {1-c^2 x^2}}{b c \sqrt {-1+c x} \left (a+b \cosh ^{-1}(c x)\right )}+\frac {\left (\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 )}{b c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (\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 )}{2 b c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (\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 )}{b c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (\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 )}{2 b c \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {(1-c x)^2 (1+c x)^{3/2} \sqrt {1-c^2 x^2}}{b c \sqrt {-1+c x} \left (a+b \cosh ^{-1}(c x)\right )}-\frac {\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 )}{b^2 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\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 )}{2 b^2 c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\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 )}{b^2 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\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 )}{2 b^2 c \sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}

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Mathematica [A]
time = 0.43, size = 232, normalized size = 0.94 \begin {gather*} \frac {\sqrt {-1+c x} \sqrt {1+c x} \left (-2 b+4 b c^2 x^2-2 b c^4 x^4+2 \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 )-\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 )-2 a \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )-2 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 )+a \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (4 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )+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 )\right )}{2 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)^(3/2)/(a + b*ArcCosh[c*x])^2,x]

[Out]

(Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-2*b + 4*b*c^2*x^2 - 2*b*c^4*x^4 + 2*(a + b*ArcCosh[c*x])*CoshIntegral[2*(a/b +
ArcCosh[c*x])]*Sinh[(2*a)/b] - (a + b*ArcCosh[c*x])*CoshIntegral[4*(a/b + ArcCosh[c*x])]*Sinh[(4*a)/b] - 2*a*
Cosh[(2*a)/b]*SinhIntegral[2*(a/b + ArcCosh[c*x])] - 2*b*ArcCosh[c*x]*Cosh[(2*a)/b]*SinhIntegral[2*(a/b + ArcC
osh[c*x])] + a*Cosh[(4*a)/b]*SinhIntegral[4*(a/b + ArcCosh[c*x])] + b*ArcCosh[c*x]*Cosh[(4*a)/b]*SinhIntegral[
4*(a/b + ArcCosh[c*x])]))/(2*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. \(736\) vs. \(2(220)=440\).
time = 4.68, size = 737, normalized size = 3.00

method	result	size
default	\(-\frac {\sqrt {-c^{2} x^{2}+1}\, \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 )}{16 \left (c x +1\right ) \left (c x -1\right ) c b \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )}+\frac {\sqrt {-c^{2} x^{2}+1}\, \left (-\sqrt {c x +1}\, \sqrt {c x -1}\, x c +c^{2} x^{2}-1\right ) \expIntegral \left (1, 4 \,\mathrm {arccosh}\left (c x \right )+\frac {4 a}{b}\right ) {\mathrm e}^{\frac {b \,\mathrm {arccosh}\left (c x \right )+4 a}{b}}}{4 \left (c x +1\right ) \left (c x -1\right ) c \,b^{2}}+\frac {\sqrt {-c^{2} x^{2}+1}\, \left (8 \sqrt {c x +1}\, \sqrt {c x -1}\, b \,c^{3} x^{3}+8 b \,c^{4} x^{4}-4 \sqrt {c x +1}\, \sqrt {c x -1}\, b c x -8 b \,c^{2} x^{2}+4 \,\mathrm {arccosh}\left (c x \right ) {\mathrm e}^{-\frac {4 a}{b}} \expIntegral \left (1, -4 \,\mathrm {arccosh}\left (c x \right )-\frac {4 a}{b}\right ) b +4 \,{\mathrm e}^{-\frac {4 a}{b}} \expIntegral \left (1, -4 \,\mathrm {arccosh}\left (c x \right )-\frac {4 a}{b}\right ) a +b \right )}{16 \sqrt {c x +1}\, \sqrt {c x -1}\, c \,b^{2} \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )}+\frac {3 \sqrt {-c^{2} x^{2}+1}}{8 \sqrt {c x -1}\, \sqrt {c x +1}\, c \left (a +b \,\mathrm {arccosh}\left (c x \right )\right ) b}+\frac {\sqrt {-c^{2} x^{2}+1}\, \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 )}{4 \left (c x +1\right ) \left (c x -1\right ) c \left (a +b \,\mathrm {arccosh}\left (c x \right )\right ) b}-\frac {\sqrt {-c^{2} x^{2}+1}\, \left (-\sqrt {c x +1}\, \sqrt {c x -1}\, x c +c^{2} x^{2}-1\right ) \expIntegral \left (1, 2 \,\mathrm {arccosh}\left (c x \right )+\frac {2 a}{b}\right ) {\mathrm e}^{\frac {b \,\mathrm {arccosh}\left (c x \right )+2 a}{b}}}{2 \left (c x +1\right ) \left (c x -1\right ) c \,b^{2}}-\frac {\sqrt {-c^{2} x^{2}+1}\, \left (2 \sqrt {c x +1}\, \sqrt {c x -1}\, b c x +2 b \,c^{2} x^{2}+2 \,\mathrm {arccosh}\left (c x \right ) \expIntegral \left (1, -2 \,\mathrm {arccosh}\left (c x \right )-\frac {2 a}{b}\right ) {\mathrm e}^{-\frac {2 a}{b}} b +2 \expIntegral \left (1, -2 \,\mathrm {arccosh}\left (c x \right )-\frac {2 a}{b}\right ) {\mathrm e}^{-\frac {2 a}{b}} a -b \right )}{4 \sqrt {c x -1}\, \sqrt {c x +1}\, c \,b^{2} \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )}\)	\(737\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-c^2*x^2+1)^(3/2)/(a+b*arccosh(c*x))^2,x,method=_RETURNVERBOSE)

[Out]

-1/16*(-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))+1/4*(-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+1/16/(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*arccosh(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))+3/8*(-c^2*x^2+1)^(1/2)/(c*x-1)^(1/2
)/(c*x+1)^(1/2)/c/(a+b*arccosh(c*x))/b+1/4*(-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-1/2*(-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
-1/4*(-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*arcc
osh(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)^(3/2)/(a+b*arccosh(c*x))^2,x, algorithm="maxima")

[Out]

((c^4*x^4 - 2*c^2*x^2 + 1)*(c*x + 1)*sqrt(c*x - 1) + (c^5*x^5 - 2*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(((4*c^4*x^4 - 3*c^2*x^2 - 1)*(c*x + 1)^(3/
2)*(c*x - 1) + 4*(2*c^5*x^5 - 3*c^3*x^3 + c*x)*(c*x + 1)*sqrt(c*x - 1) + (4*c^6*x^6 - 9*c^4*x^4 + 6*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)*sqrt(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)^(3/2)/(a+b*arccosh(c*x))^2,x, algorithm="fricas")

[Out]

integral((-c^2*x^2 + 1)^(3/2)/(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 {3}{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)**(3/2)/(a+b*acosh(c*x))**2,x)

[Out]

Integral((-(c*x - 1)*(c*x + 1))**(3/2)/(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)^(3/2)/(a+b*arccosh(c*x))^2,x, algorithm="giac")

[Out]

integrate((-c^2*x^2 + 1)^(3/2)/(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 )}^{3/2}}{{\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)^(3/2)/(a + b*acosh(c*x))^2,x)

[Out]

int((1 - c^2*x^2)^(3/2)/(a + b*acosh(c*x))^2, x)

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