3.3.85 \(\int \frac {x^2 (1-c^2 x^2)^{5/2}}{a+b \cosh ^{-1}(c x)} \, dx\) [285]

Optimal. Leaf size=439 \[ \frac {\sqrt {1-c x} \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{32 b c^3 \sqrt {-1+c x}}+\frac {\sqrt {1-c x} \cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{32 b c^3 \sqrt {-1+c x}}-\frac {\sqrt {1-c x} \cosh \left (\frac {6 a}{b}\right ) \text {Chi}\left (\frac {6 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{32 b c^3 \sqrt {-1+c x}}+\frac {\sqrt {1-c x} \cosh \left (\frac {8 a}{b}\right ) \text {Chi}\left (\frac {8 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{128 b c^3 \sqrt {-1+c x}}-\frac {5 \sqrt {1-c x} \log \left (a+b \cosh ^{-1}(c x)\right )}{128 b c^3 \sqrt {-1+c x}}-\frac {\sqrt {1-c x} \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{32 b c^3 \sqrt {-1+c x}}-\frac {\sqrt {1-c x} \sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{32 b c^3 \sqrt {-1+c x}}+\frac {\sqrt {1-c x} \sinh \left (\frac {6 a}{b}\right ) \text {Shi}\left (\frac {6 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{32 b c^3 \sqrt {-1+c x}}-\frac {\sqrt {1-c x} \sinh \left (\frac {8 a}{b}\right ) \text {Shi}\left (\frac {8 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{128 b c^3 \sqrt {-1+c x}} \]

[Out]

1/32*Chi(2*(a+b*arccosh(c*x))/b)*cosh(2*a/b)*(-c*x+1)^(1/2)/b/c^3/(c*x-1)^(1/2)+1/32*Chi(4*(a+b*arccosh(c*x))/
b)*cosh(4*a/b)*(-c*x+1)^(1/2)/b/c^3/(c*x-1)^(1/2)-1/32*Chi(6*(a+b*arccosh(c*x))/b)*cosh(6*a/b)*(-c*x+1)^(1/2)/
b/c^3/(c*x-1)^(1/2)+1/128*Chi(8*(a+b*arccosh(c*x))/b)*cosh(8*a/b)*(-c*x+1)^(1/2)/b/c^3/(c*x-1)^(1/2)-5/128*ln(
a+b*arccosh(c*x))*(-c*x+1)^(1/2)/b/c^3/(c*x-1)^(1/2)-1/32*Shi(2*(a+b*arccosh(c*x))/b)*sinh(2*a/b)*(-c*x+1)^(1/
2)/b/c^3/(c*x-1)^(1/2)-1/32*Shi(4*(a+b*arccosh(c*x))/b)*sinh(4*a/b)*(-c*x+1)^(1/2)/b/c^3/(c*x-1)^(1/2)+1/32*Sh
i(6*(a+b*arccosh(c*x))/b)*sinh(6*a/b)*(-c*x+1)^(1/2)/b/c^3/(c*x-1)^(1/2)-1/128*Shi(8*(a+b*arccosh(c*x))/b)*sin
h(8*a/b)*(-c*x+1)^(1/2)/b/c^3/(c*x-1)^(1/2)

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Rubi [A]
time = 0.35, antiderivative size = 439, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {5952, 5556, 3384, 3379, 3382} \begin {gather*} \frac {\sqrt {1-c x} \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{32 b c^3 \sqrt {c x-1}}+\frac {\sqrt {1-c x} \cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{32 b c^3 \sqrt {c x-1}}-\frac {\sqrt {1-c x} \cosh \left (\frac {6 a}{b}\right ) \text {Chi}\left (\frac {6 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{32 b c^3 \sqrt {c x-1}}+\frac {\sqrt {1-c x} \cosh \left (\frac {8 a}{b}\right ) \text {Chi}\left (\frac {8 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{128 b c^3 \sqrt {c x-1}}-\frac {\sqrt {1-c x} \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{32 b c^3 \sqrt {c x-1}}-\frac {\sqrt {1-c x} \sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{32 b c^3 \sqrt {c x-1}}+\frac {\sqrt {1-c x} \sinh \left (\frac {6 a}{b}\right ) \text {Shi}\left (\frac {6 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{32 b c^3 \sqrt {c x-1}}-\frac {\sqrt {1-c x} \sinh \left (\frac {8 a}{b}\right ) \text {Shi}\left (\frac {8 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{128 b c^3 \sqrt {c x-1}}-\frac {5 \sqrt {1-c x} \log \left (a+b \cosh ^{-1}(c x)\right )}{128 b c^3 \sqrt {c x-1}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 - c*x]*Cosh[(2*a)/b]*CoshIntegral[(2*(a + b*ArcCosh[c*x]))/b])/(32*b*c^3*Sqrt[-1 + c*x]) + (Sqrt[1 - c
*x]*Cosh[(4*a)/b]*CoshIntegral[(4*(a + b*ArcCosh[c*x]))/b])/(32*b*c^3*Sqrt[-1 + c*x]) - (Sqrt[1 - c*x]*Cosh[(6
*a)/b]*CoshIntegral[(6*(a + b*ArcCosh[c*x]))/b])/(32*b*c^3*Sqrt[-1 + c*x]) + (Sqrt[1 - c*x]*Cosh[(8*a)/b]*Cosh
Integral[(8*(a + b*ArcCosh[c*x]))/b])/(128*b*c^3*Sqrt[-1 + c*x]) - (5*Sqrt[1 - c*x]*Log[a + b*ArcCosh[c*x]])/(
128*b*c^3*Sqrt[-1 + c*x]) - (Sqrt[1 - c*x]*Sinh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcCosh[c*x]))/b])/(32*b*c^3*S
qrt[-1 + c*x]) - (Sqrt[1 - c*x]*Sinh[(4*a)/b]*SinhIntegral[(4*(a + b*ArcCosh[c*x]))/b])/(32*b*c^3*Sqrt[-1 + c*
x]) + (Sqrt[1 - c*x]*Sinh[(6*a)/b]*SinhIntegral[(6*(a + b*ArcCosh[c*x]))/b])/(32*b*c^3*Sqrt[-1 + c*x]) - (Sqrt
[1 - c*x]*Sinh[(8*a)/b]*SinhIntegral[(8*(a + b*ArcCosh[c*x]))/b])/(128*b*c^3*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 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 {x^2 \left (1-c^2 x^2\right )^{5/2}}{a+b \cosh ^{-1}(c x)} \, dx &=\frac {\sqrt {1-c^2 x^2} \int \frac {x^2 (-1+c x)^{5/2} (1+c x)^{5/2}}{a+b \cosh ^{-1}(c x)} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {\sqrt {1-c^2 x^2} \text {Subst}\left (\int \frac {\cosh ^2(x) \sinh ^6(x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{c^3 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {\sqrt {1-c^2 x^2} \text {Subst}\left (\int \left (-\frac {5}{128 (a+b x)}+\frac {\cosh (2 x)}{32 (a+b x)}+\frac {\cosh (4 x)}{32 (a+b x)}-\frac {\cosh (6 x)}{32 (a+b x)}+\frac {\cosh (8 x)}{128 (a+b x)}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{c^3 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {5 \sqrt {1-c^2 x^2} \log \left (a+b \cosh ^{-1}(c x)\right )}{128 b c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\sqrt {1-c^2 x^2} \text {Subst}\left (\int \frac {\cosh (8 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{128 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\sqrt {1-c^2 x^2} \text {Subst}\left (\int \frac {\cosh (2 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{32 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\sqrt {1-c^2 x^2} \text {Subst}\left (\int \frac {\cosh (4 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{32 c^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\sqrt {1-c^2 x^2} \text {Subst}\left (\int \frac {\cosh (6 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{32 c^3 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {5 \sqrt {1-c^2 x^2} \log \left (a+b \cosh ^{-1}(c x)\right )}{128 b c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (\sqrt {1-c^2 x^2} \cosh \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 )}{32 c^3 \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 {\cosh \left (\frac {4 a}{b}+4 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{32 c^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (\sqrt {1-c^2 x^2} \cosh \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 )}{32 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (\sqrt {1-c^2 x^2} \cosh \left (\frac {8 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {8 a}{b}+8 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{128 c^3 \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 {\sinh \left (\frac {2 a}{b}+2 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{32 c^3 \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 {\sinh \left (\frac {4 a}{b}+4 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{32 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (\sqrt {1-c^2 x^2} \sinh \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 )}{32 c^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (\sqrt {1-c^2 x^2} \sinh \left (\frac {8 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {8 a}{b}+8 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{128 c^3 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {\sqrt {1-c^2 x^2} \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 a}{b}+2 \cosh ^{-1}(c x)\right )}{32 b c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\sqrt {1-c^2 x^2} \cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 a}{b}+4 \cosh ^{-1}(c x)\right )}{32 b c^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\sqrt {1-c^2 x^2} \cosh \left (\frac {6 a}{b}\right ) \text {Chi}\left (\frac {6 a}{b}+6 \cosh ^{-1}(c x)\right )}{32 b c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\sqrt {1-c^2 x^2} \cosh \left (\frac {8 a}{b}\right ) \text {Chi}\left (\frac {8 a}{b}+8 \cosh ^{-1}(c x)\right )}{128 b c^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {5 \sqrt {1-c^2 x^2} \log \left (a+b \cosh ^{-1}(c x)\right )}{128 b c^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\sqrt {1-c^2 x^2} \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \cosh ^{-1}(c x)\right )}{32 b c^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\sqrt {1-c^2 x^2} \sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 a}{b}+4 \cosh ^{-1}(c x)\right )}{32 b c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\sqrt {1-c^2 x^2} \sinh \left (\frac {6 a}{b}\right ) \text {Shi}\left (\frac {6 a}{b}+6 \cosh ^{-1}(c x)\right )}{32 b c^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\sqrt {1-c^2 x^2} \sinh \left (\frac {8 a}{b}\right ) \text {Shi}\left (\frac {8 a}{b}+8 \cosh ^{-1}(c x)\right )}{128 b c^3 \sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}

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Mathematica [A]
time = 0.88, size = 233, normalized size = 0.53 \begin {gather*} \frac {\sqrt {1-c^2 x^2} \left (4 \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (2 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )+4 \cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (4 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )-4 \cosh \left (\frac {6 a}{b}\right ) \text {Chi}\left (6 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )+\cosh \left (\frac {8 a}{b}\right ) \text {Chi}\left (8 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )-5 \log \left (a+b \cosh ^{-1}(c x)\right )-4 \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )-4 \sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (4 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )+4 \sinh \left (\frac {6 a}{b}\right ) \text {Shi}\left (6 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )-\sinh \left (\frac {8 a}{b}\right ) \text {Shi}\left (8 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )\right )}{128 c^3 \sqrt {\frac {-1+c x}{1+c x}} (b+b c x)} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Sqrt[1 - c^2*x^2]*(4*Cosh[(2*a)/b]*CoshIntegral[2*(a/b + ArcCosh[c*x])] + 4*Cosh[(4*a)/b]*CoshIntegral[4*(a/b
+ ArcCosh[c*x])] - 4*Cosh[(6*a)/b]*CoshIntegral[6*(a/b + ArcCosh[c*x])] + Cosh[(8*a)/b]*CoshIntegral[8*(a/b +
ArcCosh[c*x])] - 5*Log[a + b*ArcCosh[c*x]] - 4*Sinh[(2*a)/b]*SinhIntegral[2*(a/b + ArcCosh[c*x])] - 4*Sinh[(4
*a)/b]*SinhIntegral[4*(a/b + ArcCosh[c*x])] + 4*Sinh[(6*a)/b]*SinhIntegral[6*(a/b + ArcCosh[c*x])] - Sinh[(8*a
)/b]*SinhIntegral[8*(a/b + ArcCosh[c*x])]))/(128*c^3*Sqrt[(-1 + c*x)/(1 + c*x)]*(b + b*c*x))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(780\) vs. \(2(385)=770\).
time = 3.92, size = 781, normalized size = 1.78

method	result	size
default	\(\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, 8 \,\mathrm {arccosh}\left (c x \right )+\frac {8 a}{b}\right ) {\mathrm e}^{\frac {b \,\mathrm {arccosh}\left (c x \right )+8 a}{b}}}{256 \left (c x +1\right ) c^{3} \left (c x -1\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, -8 \,\mathrm {arccosh}\left (c x \right )-\frac {8 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\mathrm {arccosh}\left (c x \right )+8 a}{b}}}{256 \left (c x +1\right ) c^{3} \left (c x -1\right ) b}-\frac {5 \sqrt {-c^{2} x^{2}+1}\, \ln \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )}{128 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} 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, 6 \,\mathrm {arccosh}\left (c x \right )+\frac {6 a}{b}\right ) {\mathrm e}^{\frac {b \,\mathrm {arccosh}\left (c x \right )+6 a}{b}}}{64 \left (c x +1\right ) c^{3} \left (c x -1\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, 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}}}{64 \left (c x +1\right ) c^{3} \left (c x -1\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}}}{64 \left (c x +1\right ) c^{3} \left (c x -1\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}}}{64 \left (c x +1\right ) c^{3} \left (c x -1\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, -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}}}{64 \left (c x +1\right ) c^{3} \left (c x -1\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, -6 \,\mathrm {arccosh}\left (c x \right )-\frac {6 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\mathrm {arccosh}\left (c x \right )+6 a}{b}}}{64 \left (c x +1\right ) c^{3} \left (c x -1\right ) b}\)	\(781\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/256*(-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(1,8*arccosh(c*x)+8*a/b)*exp((b*arccos
h(c*x)+8*a)/b)/(c*x+1)/c^3/(c*x-1)/b+1/256*(-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,-8*arccosh(c*x)-8*a/b)*exp(-(-b*arccosh(c*x)+8*a)/b)/(c*x+1)/c^3/(c*x-1)/b-5/128*(-c^2*x^2+1)^(1/2)/(c*x-1)^
(1/2)/(c*x+1)^(1/2)/c^3*ln(a+b*arccosh(c*x))/b-1/64*(-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*arccosh(c*x)+6*a/b)*exp((b*arccosh(c*x)+6*a)/b)/(c*x+1)/c^3/(c*x-1)/b+1/64*(-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^3/(
c*x-1)/b+1/64*(-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^3/(c*x-1)/b+1/64*(-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^3/(c*x-1)/b+1/64*(-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
^3/(c*x-1)/b-1/64*(-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*arccosh(c*x)-6*a/b)*
exp(-(-b*arccosh(c*x)+6*a)/b)/(c*x+1)/c^3/(c*x-1)/b

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

[Out]

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

[Out]

integral((c^4*x^6 - 2*c^2*x^4 + x^2)*sqrt(-c^2*x^2 + 1)/(b*arccosh(c*x) + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**2*(-(c*x - 1)*(c*x + 1))**(5/2)/(a + b*acosh(c*x)), 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(x^2*(-c^2*x^2+1)^(5/2)/(a+b*arccosh(c*x)),x, algorithm="giac")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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